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Download Hi-Res ImageDownload to MS-PowerPointCite This:J. Chem. Theory Comput. 2024, XXXX, XXX, XXX-XXX
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Strategies to Obtain Reliable Energy Landscapes from Embedded Multireference Correlated Wavefunction Methods for Surface Reactions

Cite this: J. Chem. Theory Comput. 2024, XXXX, XXX, XXX-XXX
Publication Date (Web):July 14, 2024
https://doi.org/10.1021/acs.jctc.4c00558
© 2024 American Chemical Society
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Abstract

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Embedded correlated wavefunction (ECW) theory is a powerful tool for studying ground- and excited-state reaction mechanisms and associated energetics in heterogeneous catalysis. Several factors are important to obtaining reliable ECW energies, critically the construction of consistent active spaces (ASs) along reaction pathways when using a multireference correlated wavefunction (CW) method that relies on a subset of orbital spaces in the configuration interaction expansion to account for static electron correlation, e.g., complete AS self-consistent field theory, in addition to the adequate partitioning of the system into a cluster and environment, as well as the choice of a suitable basis set and number of states included in excited-state simulations. Here, we conducted a series of systematic studies to develop best-practice guidelines for ground- and excited-state ECW theory simulations, utilizing the decomposition of NH3 on Pd(111) as an example. We determine that ECW theory results are relatively insensitive to cluster size, the aug-cc-pVDZ basis set provides an adequate compromise between computational complexity and accuracy, and that a fixed-clean-surface approximation holds well for the derivation of the embedding potential. Additionally, we demonstrate that a merging approach, which involves generating ASs from the molecular fragments at each configuration, is preferable to a creeping approach, which utilizes ASs from adjacent structures as an initial guess, for the generation of consistent potential energy curves involving open-d-shell metal surfaces, and, finally, we show that it is essential to include bands of excited states in their entirety when simulating excited-state reaction pathways.

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1. Introduction

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Traditional correlated wavefunction (CW)-based methods aimed at resolving the electronic structure in strongly correlated systems commonly rely on complete active space self-consistent field (CASSCF) wavefunctions to describe a system’s multiconfigurational character. (1) The first step in such calculations, generally, is the selection of an orbital active space (AS), which, due to the limitations placed upon its size by the exponential scaling of the multiconfigurational wavefunction, commonly relies on approaches based upon chemical intuition and trial-and-error. While some progress has been made in recent years in the development of black box and automated AS selection schemes for CASSCF and other multireference methods, including the use of ASs based on natural orbital (NO) occupation numbers (NOONs), from unrestricted Hartree–Fock (UHF), (2,3) second-order Møller–Plesset perturbation theory, (4) partially converged density-matrix renormalization group, (5) and N-electron valence state theory wavefunctions, (6) as well as automated selection schemes, e.g., based on atomic valence AS, (7) orbital entanglement entropies, (8) or machine learning, (9,10) CASSCF calculations remain far from being black box. More specifically, difficulties arise when constructing ASs for transition metal clusters where meaningfully localized molecular orbitals (LMOs) cannot be obtained using localization schemes designed for molecules due to the large number of orbitals with significant partial occupancy (occupation numbers between 0.1 and 1.9).
Generating a consistent AS along reaction pathways (11,12) is also difficult as the optimal ASs of different atomic structures are not always the same; some active orbitals that are important in a reactant might be insignificant in the product and vice versa. To construct a consistent AS of common size for all structures along the reaction pathway, it is desirable to include all important orbitals for all structures. However, the expansion of the AS along the reaction pathway can quickly become intractable when the reaction is complicated, i.e., involving many degrees of freedom, or when the system is large. As an example, three approaches were used in ref (11), which looked at O–O bond formation from water, catalyzed by a ruthenium (Ru) complex using CASSCF: (i) AS construction for each structure based on their LMOs; (ii) all structures along the reaction path use the same initial guess of active orbitals derived from the transition state (TS); and (iii) from TS to the reactant and from TS to the product, the calculation of a structure along the path (frame) always takes the AS of the previous or next frame as the initial guess (henceforth referred to as the “creeping” strategy), with all three approaches yielding identical results. Examples of successful implementation of the “creeping” strategy include study of ground- and excited-state heterogeneous catalysis in the context of plasmonics for various chemistries and surfaces, e.g., H2 dissociation on Au(111), (13−15) N2 dissociation on Fe- and Mo-doped Au(111), (16,17) NH3 decomposition on Fe- and Ru-doped Cu(111), (18−20) CH4 decomposition on pure and Ru-doped Cu(111), (21) and H2 desorption from Pd(111). (22)
The approach taken in our group for embedded correlated wavefunction (ECW) theory simulations (23−25) of heterogeneously catalyzed reactions generally utilizes NOONs based on unrestricted Kohn–Sham (UKS) density functional theory (DFT) to select orbitals on metal clusters. Toth and Pulay discussed the advantages and disadvantages of the unrestricted NO criterion, where the NOs and their occupation numbers from UHF were used to select active orbitals. (26) Because HF can give qualitatively incorrect descriptions for metals, such as a spuriously vanishing density of states around the Fermi level (27) and incorrect spin and charge density waves, (28) we use the one-electron wavefunctions from UKS instead of UHF as the initial guess for CASSCF calculations. When considering reaction pathways, we obtain the CASSCF orbitals of the isolated adsorbate (closed or open shell) at its geometry as optimized along the reaction path. We then concatenate the UKS orbitals of the isolated metal cluster (structurally fixed in most cases) and the CASSCF NOs of the isolated adsorbate and use the merged orbitals as the initial guess for the embedded CASSCF (emb-CASSCF) wavefunction of metal–adsorbate complexes. This merging approach, which has previously been offered as an alternative to creeping when the latter fails, involves generation of the guess orbitals from the optimized orbitals of each fragment in the system. Merging can improve CASSCF convergence, avoid artificial symmetry breaking, enforce convergence toward the desired states, and prevent unwanted orbitals from entering the AS, among other advantages. For example, Klüner et al. first employed a merging approach to treat the excited states of CO absorbed on Pd(111), (29,30) Hirano and Nagashima used merging to enforce certain symmetry constraints in FeCO molecules, (31) and Cimpoesu et al. used merging to generate good initial guesses for lanthanide complexes. (32)
In this work, we explored why creeping fails to provide consistent ASs for a particularly challenging case, ammonia (NH3) decomposition on palladium(111) (hereafter denoted as Pd(111) + NH3), followed by presentation of guidelines for obtaining reliable energy landscapes using the ECW theory for surface reactions more generally. We initially studied the effects arising from basis set size, basis set superposition error (BSSE), embedded cluster size, and the manner in which the embedding potential is derived (whether from the clean Pd surface or from surfaces optimized in the presence of adsorbates) and find that these factors are not the reasons for discontinuous potential energy curves (PECs). We narrow down the source of error to the inconsistency in the CASSCF AS. We generated smooth PECs from embedded complete AS second-order perturbation theory (33) (emb-CASPT2) by constructing ASs using merging for the preceding emb-CASSCF, which provided the reference wavefunctions for emb-CASPT2. We then compare the NOs in the ASs between merging and creeping to understand why the latter fails for this particular system. Finally, in light of recent success in utilizing ECW theory simulations to resolve photocatalytically and plasmonically driven processes, (34) we also studied the effect of the number of states included in embedded state-averaged CASSCF (emb-SA-CASSCF) calculations on excited-state landscapes.

2. Methods

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2.1. Periodic DFT

We performed periodic Kohn–Sham DFT calculations using the Vienna Ab initio Simulation Package (VASP), version 5.4.4. (35−37) We employed standard VASP projector augmented-wave potentials, self-consistently solving for the valence electrons, namely, the H 1s, N 2s,2p, and Pd 4d,5s orbitals, with a kinetic energy cutoff of 700 eV for the plane-wave (PW) basis set, in combination with the Perdew–Burke–Ernzerhof (PBE) generalized gradient approximation exchange–correlation (XC) density functional. (38) Since DFT–PBE predicts a nonzero magnetization ground state for face-centered cubic (fcc) Pd, (39) we suppressed this unphysical excess electronic spin by using spin-restricted (S = SZ = 0) DFT. We utilized a self-consistent dipole-field energy and potential correction orthogonal to the surface to remove the spurious dipole-field interactions between periodic images along that direction. (40) We sampled the Brillouin zone with the Monkhorst–Pack method (41) utilizing Γ-point-centered 16 × 16 × 16, 7 × 7 × 1, and 5 × 5 × 1 k-point meshes for the bulk four-atom unit cell, and the five-layer 3 × 3 and four-layer 5 × 5 supercell periodic slabs, respectively (vide infra). We employed the Methfessel–Paxton method (42) with a width of 0.09 eV to facilitate electronic convergence by smearing the electronic states at and near the Fermi level. We chose these parameters to achieve a total energy convergence of 1.0 meV/atom or better.

2.2. Atomic Models

We optimized the bulk nonmagnetic four-atom fcc Pd unit cell (a = 3.940 Å, −1.3% error (43)) and then generated five-layer 3 × 3 and four-layer 5 × 5 periodic slabs of the most stable (111) surface using the optimized unit cell lattice vectors. A vacuum layer of ∼15 Å was added along the surface normal to separate the slab from its periodic images. The dimensions of the five-layer 3 × 3 and four-layer 5 × 5 supercells are 8.357 Å × 8.357 Å × 24 Å and 13.929 Å × 13.929 Å × 22 Å, respectively.

2.3. Geometry and Minimum Energy Path Optimization

For geometry optimization, we set the maximum absolute total force on each atom to 0.01 eV/Å as the convergence threshold. For five-layer 3 × 3 (Pd45) and four-layer 5 × 5 (Pd100) supercells, we fixed the lattice vectors along with the coordinates of the two bottom-most layers for the former and the very-bottom layer for the latter to emulate a semi-infinite bulk surface while relaxing the atoms in all other layers. We optimized the minimum energy paths (MEPs) for reactions via the climbing image nudged elastic band (CI-NEB) method, (44,45) setting the fictitious spring force constant to 5.0 eV/Å2 and the force convergence threshold to 0.03 eV/Å. We utilized the five-layer 3 × 3 supercells for the CI-NEB calculations involving the dehydrogenation of NH3, NH2, and NH. This supercell size is also adequate for the calculation of geometry-specific embedding potentials for studying the first N–H bond scission step (vide infra). On the other hand, we used the four-layer 5 × 5 supercells for the CI-NEB calculations involving the associative desorption of 2*N to N2 and all other embedding potential optimizations. We define the reaction coordinate (RC) as the cumulative distance between the images on the MEP
RC(n)=∑i=0n−1|Ri+1−Ri|
(1)
where RC(n) is the RC value for the nth image and Ri represents the Cartesian coordinate vectors of the atoms in image i.

2.4. Density Functional Embedding Theory with ECW Theory

In density functional embedding theory (DFET), (23,24,46) one divides the total system into a region (or regions) of interest (here, a cluster) and its (or their) environments (if one has multiple embedded subsystems) and derives an embedding potential Vemb to describe the interaction between them. We used the four-layer 5 × 5 supercell described above as the total system, carving out Pd10, Pd12, and Pd14 clusters (displayed in Figure 1; rationales for these choices are provided later) from the aforementioned Pd100 slab, and derived the Vembs using our in-house modified VASP 5.3.3 code, which is now also available for VASP 6. (47) In DFET, one optimizes Vemb by maximizing the extended Wu–Yang functional, (48) W[Vemb]
W[Vemb]=Ecl[ρcl,Vemb]+Eenv[ρenv,Vemb]−∫ρtotVembdr
(2)
where, for the simplest case of two subsystems, Ecl and Eenv are the DFT energies of the cluster and its environment in the presence of the Vemb, respectively, and ρcl, ρenv, and ρtot are the self-consistently optimized electron densities of the cluster, environment, and total system (the latter being simply the periodic DFT reference density), respectively. We determine the optimal Vemb by setting to zero the functional derivative of W with respect to Vemb. Because the functional derivative of the energy with respect to Vemb is formally equal to the density, Vemb convergence is reached when ρtot = ρcl + ρenv, which also makes sense physically: the sum of the embedded subsystem densities, given the physically correct embedding potential, should reproduce the total density at the same level of theory.

Figure 1

Figure 1. (a) Four-layer 5 × 5 Pd(111) (Pd100) periodic slab model. Top-down view of the (b) Pd10 cluster, (c) Pd12 cluster, and (d) Pd14 cluster carved out from the Pd100 periodic slab. Subsurface atoms are faded out. Isosurface plots (yellow: +1.2 V, cyan: −1.2 V) of the optimized embedding potentials generated for the (e) Pd10 cluster in its Pd90 environment, (f) Pd12 cluster in its Pd88 environment, and (g) Pd14 cluster in its Pd86 environment.

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To prepare for embedded cluster calculations, we transform Vemb from a real-space grid basis to an atomic Gaussian-type-orbital (GTO) basis using our in-house code. (49) Here, we performed embedded cluster calculations using MOLPRO 2021.2, (50−52) utilizing the matrix operation (MATROP) function as implemented in MOLPRO to add Vemb in the GTO basis to the original one-electron Hamiltonian H0. The following embedded DFT and ECW calculations, therefore, include the interactions between the cluster with its periodic environment at the PW-DFT level, here, within the PBE XC functional approximation.
We chose CASPT2 for carrying out ECW cluster calculations, a method that relies on CASSCF to generate the reference many-body wavefunction. We utilized a 12-electron in 12-orbital (12e, 12o) AS for the emb-CASSCF calculations of N–H bond dissociation in NH3, *NH2, and *NH. The AS comprises three N–X (X = H or Pd) σ-bond orbitals and their corresponding antibonding σ* orbitals and the three highest occupied molecular orbitals (HOMO, HOMO – 1, HOMO – 2) and the three lowest unoccupied orbitals (LUMO, LUMO + 1, LUMO + 2) originating from the Pd clusters. Ground-state emb-CASPT2 calculations were based on the ground-state emb-CASSCF wavefunctions. To obtain excited-state wavefunctions, we applied the emb-SA-CASSCF method, utilizing the ground-state emb-CASSCF wavefunctions as the initial guess. We assigned equal weights to all roots for a balanced treatment of each state and we computed single-state emb-CASPT2 energies for each state based on the emb-SA-CASSCF wavefunctions. Finally, we added the emb-CASPT2 excitation energies onto the emb-CASPT2 ground-state PECs for each image along MEP to get the excited-state PECs. (21,22) For emb-CASPT2 calculations labeled “without 4s4p dynamic correlation”, we included all excitations from all orbitals, except for the core orbitals, which comprise the 28 core electrons of each Pd subsumed into its effective core potential (ECP) and the semicore 4s4p orbitals of Pd and the core 1s orbital of N. For emb-CASPT2 labeled “with 4s4p dynamic correlation,” excitations from these outer core orbitals were included. We applied an IPEA shift (53) of 0.25 hartree to alleviate the intruder state problem in CASPT2 in combination with a level shift (54) of 0.3 hartree to facilitate CASPT2 energy convergence, which are especially critical for metal clusters with a high degree of degeneracies.
Finally, we express the total ECW energy (EtotalECW) as the energy of the total system in PW-DFT-PBE (EtotalPW‐DFT‐PBE) plus a correction obtained from the difference between the embedded cluster energy computed using GTO-DFT-PBE (Eemb,clusterGTO-DFT-PBE) and a CW method (Eemb,clusterGTO‐CW) in the presence of Vemb: (23,24,34,46)
EtotalECW=EtotalPW‐DFT‐PBE+(Eemb,clusterGTO‐CW−Eemb,clusterGTO‐DFT‐PBE)
(3)
To demonstrate convergence with respect to cluster size and explore the effects of using a fixed vs structure-specific Vemb, we replaced the CW method with a much simpler electronic structure theory, namely, DFT with local density approximation (LDA) for the XC functional. We simply require the “correcting” method to be distinct from the base method that one used to construct Vemb. Therefore, in Section 3.1, among the various tests we conducted, we calculated the emb-PW-DFT-LDA and emb-GTO-DFT-LDA energies using similar formulas
Etotalemb‐PW‐DFT‐LDA=EtotalPW‐DFT‐PBE+(Eemb,clusterPW‐DFT‐LDA−Eemb,clusterPW‐DFT‐PBE)
(4)
Etotalemb‐GTO‐DFT‐LDA=EtotalPW‐DFT‐PBE+(Eemb,clusterGTO‐DFT‐LDA−Eemb,clusterGTO‐DFT‐PBE)
(5)

3. Results and Discussion

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3.1. Effects of Cluster Size (Pd10, Pd12, and Pd14)

First, we established the effects arising from the size of the embedded metal cluster within embedded PW-DFT, i.e., embedded-DFT-in-DFT, which we achieved by performing emb-PW-DFT-LDA, henceforth further abbreviated as emb-LDA, for the dissociative adsorption reaction NH3(gas) → *NH3 → *NH2 + *H. We used Pd10, Pd12, and Pd14 clusters (the structures of these clusters are shown in Figure 1b–d) to investigate cluster size convergence. The structures of all images along the MEP are displayed in the Supporting Information (SI), Figure S1. Figure 2 shows that the emb-LDA energies of images 0–2 (gas-phase structures), 4 (adsorbed structure), and 7–10 (the NH2–H bond breaking) overlap very well with the full PW-DFT-LDA energies regardless of the cluster size, with differences in the cluster size leading to deviations in relative energies of less than 0.05 eV (compare the red line with the blue lines in Figure 2). Figure 2 also shows that PW-DFT-PBE (black line) and PW-DFT-LDA (red line) predict very distinct reaction curves, with the latter exhibiting the typical LDA overbinding. Note that we used the former electronic structure theory to derive the Vemb.

Figure 2

Figure 2. Effect of cluster size for NH3(gas) → *NH3 → *NH2 + *H. We calculate the emb-LDA energies (defined in eq 4) for Pd10, Pd12, and Pd14 clusters. Numerical indices correspond to structures shown in Figure S1.

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The minor discrepancies between emb-LDA energies and PW-DFT-LDA energies for images 3, 5, and 6 (respectively, NH3 close to the surface and the tilted adsorbed states) only slightly decrease as the cluster size increases. As it turns out, this energetic discrepancy between emb-LDA and PW-DFT-LDA originates from the fixed-surface approximation, i.e., the cluster atoms are fixed to their optimized positions in the absence of the adsorbate, introduced when constructing Vemb (see also Section 3.4). As proof, we find that emb-LDA PECs almost perfectly reproduce the PW-DFT-LDA PECs when the full reference system PW-DFT-LDA calculations also have all Pd atoms fixed to their clean-surface structure (PW-DFT-LDA-clean, Figure S2a). Figure S2b shows that PW-DFT-PBE is less sensitive to the fixed surface approximation between images 2 and 7. Unlike the total emb-LDA energies (Etotalemb-PW-DFT-LDA), however, the cluster-only emb-LDA energy terms (Eemb,clusterPW‐DFT‐LDA) change significantly with respect to the cluster size (Figure S3). This is because embedded cluster energies include the embedding energy contribution (∫Vembρcluster(r) dr), which describes the interaction between the cluster and adsorbate electron densities and Vemb. Changes in the size or shape of the cluster can cause significant changes in said interaction. However, by casting the embedding theory total energy (Etotalemb‐PW‐DFT‐LDA) as the total DFT energy plus a regional correction involving the difference between two terms containing the embedding potential, as done in our embedding (DFET/ECW) formalism, the cluster size/change sensitivity quickly damps out.
We also compared the emb-PW-DFT-LDA and emb-GTO-DFT-LDA energies (eqs 4 and 5 and Figure S4) and found that the relative energy discrepancies between the two basis set types are minor and originate from the differences in their size (completeness) and their core electron representation and from BSSEs when using atom-centered GTO basis sets (vide infra).
Our results demonstrate that (1) the fixed surface approximation in deriving Vemb and energy correction leads to negligible and tractable energetic discrepancies although such an approximation requires testing (see also Section 3.4); (2) the minimal Pd10 cluster is sufficiently large to obtain an accurate Vemb and moving to larger Pd12 and Pd14 clusters negligibly affects the energetics for the NH3(gas) → *NH3 → *NH2 + *H reaction, based on the embedding total energy expression; and that (3) transforming Vemb from a PW basis to a GTO basis retains accuracy, with minor discrepancies in part arising from BSSE (vide infra) and not from the ECW theory.

3.2. GTO Basis Set Size Effect on Ground-State PECs within Emb-CASPT2

Following the initial studies on cluster size dependence, we utilized the embedded Pd10 cluster to study the effects of the basis set size on the ground- and excited-state PECs for the NH3 (gas) → *NH3→ *NH2+ *H reaction. The semicore Pd 4s4p dynamic correlation was excluded from the CASPT2 calculations (see Methods). We calculated a DFT-corrected initial adsorption energy (Einit, expt) for NH3 on Pd(111) to serve as a benchmark for the energy of the molecularly adsorbed state of NH3 (image 4, RC ∼ 4.8 Å), using the equation Einit,expt = Einit,PW-DFT + ΔHinit,expt – ΔHinit,PW-DFT. Einit,PW-DFT and ΔHinit,PW-DFT are the PW-DFT-PBE reaction internal energy (−0.76 eV) and enthalpy (−0.67 eV, see ref (55) for details on how this term is calculated) changes at 305 K, respectively, at a coverage of θNH3 = 1/9. ΔHinit,expt (−0.77 eV) is the initial heat of adsorption measured at 305 K obtained from calorimetry and Pd powder. (56)
We tested the following Dunning basis sets: (57−60) cc-pVXZ-PP (hereafter, denoted VXZ), aug-cc-pVXZ-PP (AVXZ), cc-pwCVXZ-PP (WCVXZ), and aug-cc-pwCVXZ-PP (AWCVXZ) with X = D, T to determine the effects of adding higher-order polarization functions (pVX), diffuse functions (aug), and core–valence (CV) correlation. We retrieved all basis sets from the basis set exchange (61) and tested different basis set combinations for Pd and adsorbate (N and H), which are denoted as Pd/adsorbate, e.g., VDZ/VDZ if cc-pVDZ is utilized for Pd, N, and H. The energy curves for the surveyed basis set combinations are displayed in Figure 3. We found that all basis sets yield similar TS barriers, ranging from 1.57 to 1.74 eV, except for the two smallest basis set combinations, VDZ/VDZ and VDZ/AVDZ, which yield over- and under-estimations, with predicted barriers of 1.81 and 1.37 eV, respectively. The larger basis sets give molecular adsorption energies that are in good agreement with the DFT-corrected experimental benchmark Einit,expt (shown as the dashed black line in Figure 3), with AVTZ/AVTZ showing the largest deviation at −0.14 eV. Our data demonstrate that all basis sets for Pd larger than VDZ yield qualitatively similar results, agreeing to within 0.20 eV for the adsorption energy and to within 0.17 eV for the TS barrier.

Figure 3

Figure 3. Basis set effect for NH3(gas) → *NH3 → *NH2 + *H at the emb-CASPT2 level using an embedded Pd10 cluster. Basis sets for Pd and adsorbates are listed as Pd basis/adsorbate basis in the legend. DFT-corrected Einit,expt is the DFT-corrected electronic energy based on the experimental enthalpy of NH3 adsorption on Pd powder. See detailed explanation in the text.

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We also investigated the effect of including semicore Pd 4s4p dynamic correlation on the emb-CASPT2 energies (Figure S5; AVTZ/AVTZ results are not available due to prohibitively high memory requirements). Including the semicore Pd 4s4p dynamic correlation that reduced the TS barrier for all tested basis sets and for smaller basis sets without CV correlation, the difference in the TS barrier could be as large as ∼0.3 eV (Figure S5a,b,d). Adding CV correlation, specifically, VDZ to WCVDZ (compare Figure S5a,b vs c) and AVDZ to AWCVDZ (Figure S5d vs e) and higher-order angular momentum functions (VDZ to VTZ) (Figure S5b vs f,g) reduces the effect of including 4s4p dynamic correlation on the barrier to ∼0.1 eV. Consequently, a basis set suitable to capture the CV correlation (as in the WCV basis set family) is needed to properly include the semicore 4s4p dynamic correlation─otherwise it is preferable to not include it in the calculation. This is likely the result of an increase in the 4d95s1 → 4d85s2 excitation energy upon inclusion of 4s4p dynamic correlation, which was found by Peterson et al. (57) to be 3.6 kcal/mol in the complete basis set limit for the Pd atom. Although the inclusion of the semicore 4s4p dynamic correlation has a relatively small effect (∼0.1 eV) on the ground-state and excitation energies of Pd, CV correlation is important for alkali (group 1) and alkaline earth (group 2) metals (62) and first-row transition metals (63) and should be considered in these cases with the proper basis set to capture them. Additionally, Peterson et al. found that the inherent error within the ECP approximation for the Pd atom is small (ΔEECP < 1.1 kcal/mol) at the CCSD(T) level of theory when ignoring the semicore 4s4p dynamic correlation. (57) Here, ΔEECP was defined as the difference between the results using all-electron Douglas–Kroll–Hess (DKH) basis sets of triplet-ζ quality with the DKH Hamiltonian (64,65) and the results using aug-cc-pwCVTZ-PP quality basis sets with a nonrelativistic Hamiltonian. As the error within the ECP approximation for Pd is minimal, we did not consider it here.
We conclude that AVDZ/AVDZ basis sets without evaluating the semicore 4s4p dynamic correlation offer the best balance between computational cost and accuracy, given the considerable expense of adding CV correlation and the attendant 4s4p dynamic correlation. Diffuse functions, which are known to be critical for treating excited states, especially those with Rydberg character, (66,67) are included, providing significant improvements over VDZ/VDZ results, while the larger AVTZ/AVTZ basis set is too computationally intensive and requires prohibitively large amounts of memory. We utilized the AVDZ/AVDZ basis sets for the remainder of the calculations discussed in the following sections.

3.3. BSSE and Counterpoise Correction

BSSEs often arise in weakly bound systems when treated with basis sets of small size as a consequence of borrowing basis functions from adjacent atoms at short interatomic distances. When these distances change, they lead to an apparent strengthening of short-range intermolecular interactions because in the separated case, the atoms can no longer borrow each other’s basis functions. (68−71) The BSSE may be removed by utilizing the so-called counterpoise (CP) correction, performing calculations on the individual fragments utilizing ghost functions (the other fragment’s basis set) to obtain corrections to the interaction energy. (72−74) The calculations of the BSSE and CP corrections are discussed in more detail in the Supporting Information (Note S1).
Our results, shown in Figure 4, demonstrate that BSSE increases as NH3 adsorbs and dissociates on the embedded Pd cluster (higher values of the RC), arising from the adsorbate’s ability to borrow the basis functions of Pd (and vice versa) as the molecule (or its fragments) approaches the surface. The results also confirm that it is essential to utilize a basis set with diffuse functions to minimize BSSE, with the smallest surveyed VDZ/VDZ (red) resulting in BSSEs of up to 1.0 eV in CASPT2 calculations for the NH3(gas) → *NH3 → *NH2 + *H reaction. CASPT2 calculations with AVDZ (yellow), AVTZ (purple), and AWCVDZ/AVDZ (green) basis sets suffer much less BSSE (less than 0.5 eV).

Figure 4

Figure 4. CP corrections for NH3(gas) → *NH3 → *NH2 + *H on embedded Pd10. Combined CP corrections of the adsorbates and the Pd10 cluster are plotted for emb-CASPT2 with various basis sets (denoted as Pd basis/NH3 basis in the legends) and DFT-PBE with the AVDZ basis set.

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It is well known that self-consistent mean-field theories, such as DFT, achieve basis set convergence (and hence lower BSSE) faster than non-self-consistent CW theories such as CASPT2. This is consistent with our findings, with CASPT2 AVTZ/AVTZ errors still exceeding those of DFT-PBE at the AVDZ/AVDZ level. Comparing results for different basis sets on the adsorbate, namely, VDZ/AVDZ (pink) to VDZ/VDZ (red) and VTZ/AVTZ (blue) to VTZ/VTZ (light blue), reveals that the addition of diffuse functions on only the adsorbate can worsen BSSE by ∼0.4 eV. These data illustrate the importance of using basis sets of equivalent quality on both adsorbate and the metal cluster. We also posit that AVDZ/AVDZ presents the optimal choice (yellow) of basis sets for emb-CASPT2 simulations, given that simulations utilizing AVTZ basis sets, especially at the SA-CASSCF and CASPT2 levels of theory, are too computationally expensive for routine use.
By comparing the implied needed CP corrections from the estimated BSSEs in Figure 4 and the PECs in Figure 3, both as functions of the basis set, it becomes clear that the CP corrections will be severe overestimations of the errors due to the basis set size. This behavior has been known in the literature, notably for small basis sets. (74−76) In Figure 3, all PECs agree to less than 0.5 eV except for the case of the VDZ/AVDZ basis set, whereas Figure 4 suggests CP corrections more than +0.75 eV relative to AVTZ/AVTZ. This implied correction is much higher in magnitude and in the opposite direction as Figure 3 suggests. Figure 3 shows that AVTZ/AVTZ is among the basis set with the most negative adsorption and dissociation energies and lowest effective barrier relative to NH3(g) with VDZ/AVDZ as an outlier. We therefore additionally recommend that basis set extrapolation to the complete basis set limit (77) should be performed in lieu of CP correction if calculation with multiple basis sets can be afforded, as we have done, e.g., in ref (78).

3.4. Effects of Geometry-Dependent Vemb

Next, we investigated the effect of utilizing geometry-dependent Vembs on emb-CASSCF PECs. Given that many reactions on metal surfaces do not result in significant structural changes of the metal surface, utilizing a fixed geometry for the metal surface is often a reasonable approximation (Section 3.1) where one assumes that the periodic DFT term in the embedding total energy already correctly captures the surface reorganization energy. However, the frozen surface approximation should be tested in cases where the surface metal atoms change significantly. (21) We carried out one such test here, where in the case of adsorbed nitrogen atoms recombining to form N2 gas on Pd(111), the optimized structures along the minimum energy path show fairly large structural changes on the Pd surface.
Up to this point, all results presented above utilized geometry-independent Vembs obtained from a clean Pd surface for all calculations. We now compare the reaction barrier for *N + *N → *[η2-N2] → N2(g) using a geometry-independent Vemb and geometry-dependent Vembs (Figure S6), where *[η2-N2] denotes a molecularly adsorbed N2 molecule in a bidentate (η2) configuration (reaction path discussed in ref (55)). For the geometry-independent Vemb, we obtained Vemb as usual from a clean Pd100 slab (Figure S6a). The metal–adsorbate complex structures then were obtained by translating the optimized reactant and TS structures of the adsorbed N atoms from CI-NEB calculations for the periodic slab onto the clean Pd10 cluster while fixing the internal coordinates of the adsorbates at their PW-DFT-PBE structures. Following this procedure, we projected the Vemb in the PW basis derived from the clean Pd10 cluster to various adsorbate configurations in the GTO basis. The geometry-dependent Vemb instead was derived using the optimized structure of the Pd100 + 2 *N (or *[η2-N2]) periodic slab instead of the clean Pd100 periodic slab, i.e., with the adsorbate present. Figures S6d,e and g,h illustrate the total system and cluster used to derive geometry-dependent Vemb for the reactant and TS, respectively.
We also performed “modified-geometry-dependent” Vemb calculations to determine Vemb from a clean slab without adsorbates and with the clean Pd100 at the geometry optimized with 2 *N or *[η2-N2], i.e., we optimized Vemb without the adsorbates (as we do in the fixed cluster approximation) but the surface Pd atoms are configured as if the adsorbates are present. Emb-LDA reaction barriers derived from geometry-independent, geometry-dependent, and modified-geometry-dependent Vembs are displayed in Table S1. The modified-geometry-dependent Vemb most closely reproduced the full periodic slab DFT-LDA prediction, with an error of only −0.15 eV. The geometry-independent Vemb was slightly less reliable, with an error of 0.36 eV. Note that the errors of geometry-independent Vemb are much smaller for the three dehydrogenation steps than for N2 associative desorption because the adsorbate-induced metal surface distortions are larger in the latter. Interestingly, the geometry-dependent Vemb exhibited the largest error of 0.85 eV, likely the result of significant changes in density upon adsorption, which are larger than those arising from changes in geometry. These findings suggest that it could be worthwhile to use the modified-geometry-dependent Vemb, i.e., still without the adsorbate, when the structure of the metal surface differs significantly from its clean-surface structure; however, optimization of Vemb corresponding to each adsorbate structure along the MEP is computationally demanding and improvements to the accuracy are minor. These results suggest that the use of the geometry-independent Vemb is in most cases sufficient, given the current cost to optimize Vemb. Ongoing work in our group endeavors to reduce this cost so that in the future, the modified-geometry-dependent Vemb might be worth the cost to generate along reaction paths.

3.5. Comparison of Creeping vs Merging for AS Selection in the Reference Emb-CASSCF

Next, we investigated the effects of AS selection on the PECs for the NH3 dehydrogenation steps. Here, we compare the PECs obtained from two commonly utilized approaches for generating initial AS orbital guesses when modeling catalytic pathways, namely, the creeping and merging approaches. The former refers to the strategy of constructing ASs using orbitals of adjacent or successive structures along the reaction path as the initial guess for the CASSCF calculation for the current structure, while the latter involves the generation of new guess orbitals for each structure from the separately optimized orbitals of each fragment in the system. The PECs are displayed in Figure 5 and the NOs comprising the ASs are shown in the Supporting Information (obtained with creeping in Figures S7–S9 and from merging in Figures S10–S12.)

Figure 5

Figure 5. Emb-CASPT2 results using creeping to obtain CASSCF orbitals for (a) NH3(gas) → *NH3 → *NH2 + *H, (b) *NH2 + *H → *NH+ 2*H, and (c) *NH → *N+ *H. The corresponding NOs in the ASs using creeping are given in Figures S7–S9, while the ones from merging are shown in Figures S10–S12 in the Supporting Information. All calculations utilize the Pd10 cluster model.

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There are two key indicators of a well-chosen AS: (1) the PEC from creeping is continuous during bond breaking/forming and (2) the orbitals in the AS correspond to the correct bonding pattern for all geometries. Conversely, indicators of a bad AS include: (1) a discontinuous PEC even without bond breaking/forming, (2) flipping of the energy ordering of ground and excited states going from SA-CASSCF to SS-CASPT2, and (3) nonoptimal orbital rotation from inactive to ASs or AS orbital occupations along the path, likely due to new degeneracies associated with different d-orbitals near the Fermi level.
Comparing the results of creeping and merging approaches for the NH3 dehydrogenation steps, we observe that discrepancies usually occur around bond breaking and forming. This is illustrated in Figure 5a (RC ∼ 7 Å), where *NH2 forms a new bond with another Pd atom in the top layer; Figure 5b (RC ∼ 0.5–1 Å), where a N–H bond breaks; Figure 5b (RC ∼ 1.5–2.8 Å), where a new N–Pd bond forms; and Figure 5c (RC ∼ 1.5 Å), where a N–H bond breaks; the largest discrepancies are generally observed around the product and the TS. While the failure of the creeping approach is less obvious for the first and third dehydrogenation steps (Figure 5a,c), where energetic deviations between PW-DFT-PBE and emb-CASPT2 obtained from merging are small, here, we encounter violation of rule (3), i.e., we observe the nonoptimal occupation of metal-based AS orbitals along the path. Comparisons of the AS NOs from creeping and merging approaches show that the (6e, 6o) orbitals primarily associated with the adsorbate are consistent across both, while the (6e, 6o) orbitals primarily from the metal cluster differ (e.g., compare Figures S7 vs S10). In creeping, the metal-based orbitals spatially localize on the same set of Pd atoms along MEPs (Figure S7), which is expected given that each calculation utilizes the orbital coefficients of the previous structure as the initial guess. On the other hand, in the merging approach, metal orbitals may localize on different Pd atoms along MEPs (Figure S10) as the bonding patterns and, consequently, the NOs change when the adsorbate geometry changes along the MEP. Creeping fails to incorporate these changes in the metal orbitals into the AS, falling into local minima during the CASSCF optimization.
Earlier studies by Bao and Carter on NH3 decomposition on Cu(111) (19) and Ru-doped Cu(111) (20) using the creeping strategy delivered consistent results. Comparing the AS NOs of Pd(111) with Cu(111) and Ru-doped Cu(111), we note that the orbitals on Pd are mostly of d-orbital character, while the orbitals on Cu are mostly of s-orbital character. d orbitals are localized and anisotropic, while s orbitals are delocalized and isotropic. This observation may explain why creeping presented a viable approach for pure and doped copper (Cu) (18−21,79) and pure and doped gold (Au) (13−17) in previous studies, while it failed for Pd. In the former examples, either the delocalized sp bands or the well-defined d orbitals of the surface dopants were involved in bonding with the adsorbate, while in the latter case, the nearly degenerate d bands dominated the bonding with the adsorbate, resulting in a plethora of local minima and leading to errors in the PEC when creeping is employed. Merging solves this problem.

3.6. Number of Excited States Included in Emb-SA-CASSCF Calculations

Finally, we investigated the sensitivity of excitation energies and excited-state PECs with respect to the number of states included in SA-CASSCF calculations. Such calculations play a key role in the modeling of plasmonically driven and photocatalytic processes, (25,34) and emb-NEVPT2 and emb-CASPT2 calculations have been utilized successfully to elucidate, e.g., the mechanism of plasmon-induced methane dry reforming to syngas on Ru-doped Cu nanoparticles (21) and the photocatalytic generation of H2 from NH3 utilizing Fe-doped Cu nanoparticles. (18) Here, we consider excited-state photocatalytic decomposition of NH3 via the reactions NH3 (gas) → *NH3 → * NH2 + *H (Figure 6), *NH2 + *H → *NH + 2*H, and *NH → *N + *H (Figures S13 and S14, respectively). We draw some general conclusions from this series of PECs next.

Figure 6

Figure 6. Ground- and excited-state MEP energetics as calculated with emb-CASPT2 for NH3 (gas) → *NH3 → * NH2 + *H. Red arrows represent vertical excitation of S0 → SN-1, where SN-1 represents the highest excited state. Green arrows represent reaction barriers for S0 or SN–1. Each panel represents different numbers of states included in the N-state emb-SA-CASSCF calculations: N = (a) 5, (b) 7, (c) 9, (d) 11, (e) 15, and (f) 17.

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First, in the case of distinguishable bands (characterized by a large energy gap between two consecutive states), the data highlight that the entire band must be included in the emb-SA-CASSCF calculations to avoid root flipping and abnormal excitation energies. For example, three distinguishable bands exist at the reactant configuration (RC ∼ 4.8 Å, Figure 6f) in the first N–H bond breaking when looking at the emb-CASPT2 PECs from the 17-state (the largest number of states we were able to include) emb-SA-CASSCF calculation. These bands of states are composed of the singlet ground state to the fourth excited state (S0–S4), fifth to 14th singlet excited states (S5–S14), and 15th to 16th singlet excited states (S15–S16). When the second band is only partially included in the emb-SA-CASSCF calculations, as in the seven-, nine-, and 11-state calculations (Figure 6b–d), the excited states S2–S4 are significantly different from the results when either only the first band is included, as in the five-state calculation (Figure 6a), or the entire first and second bands are included, as in the 15-state calculation (Figure 6e). Increasing the number of excited states by groups of bands generally preserves the overall shape of the lower band, although the excitation energies of the lower band will change slightly when higher bands are included. When including the third band, as in the 17-state calculation (Figure 6f), the lower bands of the three images with RC < 2.5 Å shift to much higher energies, in comparison to the results with only the first and second bands (e.g., 15-state, Figure 6e). Therefore, including these high-lying excited states, namely, S15 and S16, deteriorates the quality of lower excited states. With respect to NH3 adsorption (images 0–4), we conclude that inclusion of the entire first and second bands (15-state, Figure 6e) yields the appropriate description, while the effect of the excited states on the NH2–H bond breaking barrier is best described when the additional third band of excited states is also included (17-state, Figure 6f).
Comparing the above PECs for the first N–H bond scission to the subsequent H abstraction steps, it appears that the breaking of the third and final N–H bond (Figure S14) is less sensitive to the choice of the number of states included in the SA-CASSCF calculations than that of the first (Figure 6) and second (Figure S13) N–H bonds. Inspection of the dipole moments of all studied states reveals that they are small and of similar magnitude for *NH → *N + *H, while several states have significantly different or very large dipole moments for the NH3 (gas) → *NH3 → * NH2 + *H and *NH2 + *H → *NH + 2*H reactions (an Excel document containing tables of dipole moments for all states of each structure along each of the bond dissociation reaction paths may be found in the Supporting Information. Each sheet contains tables for the dipole moments of all distinct N-state SA-CASSCF calculations carried out). The dipole moments of the different states therefore can be used as a secondary method to group the excited states into different classes, serving as a convenient alternative indicator to determine if additional states added in the SA procedure will belong to the same already sampled band of states (without the need to calculate even more states to identify bands). As stated above, it is crucial to sample adequately a band to avoid root flipping. We found that both NH3 (gas) → *NH3 → * NH2 + *H and *NH2 + *H → *NH + 2*H (Supporting Information Excel document, tabs labeled as NH3 image# and NH2 image#, respectively) exhibit varying polarization profiles among different excited states, thus, explaining the sensitivity of the energy landscape with the number of states in SA-CASSCF. For certain reactions, one may encounter less distinct groups of excited states as indicated by their dipole moments, e.g., *NH → *N + *H (Supporting Information Excel document, sheets labeled as NH image#), and wherever that happens, the result will be less sensitive to the number of states being averaged in SA-CASSCF.
These results highlight the challenge presented to emb-SA-CASSCF simulations, which rely on a single set of compromise canonical orbitals used to expand the wavefunctions of all states being averaged in the SA-CASSCF procedure, when treating states with different characters. In view of the foregoing, we conclude that two factors prevent the study of higher-lying excited states with emb-SA-CASSCF-based methods. First, adding higher excited states deteriorates the quality of the SA-CASSCF wavefunction, especially when the characters of the excited states are very different (e.g., *NH2 + *H → *NH + 2*H) or a large energy gap exists between the higher and the lower excited states (e.g., NH3 (gas) → *NH3 → * NH2 + *H). This problem may be alleviated using variable weighted SA-CASSCF; however, this solution introduces more variables to optimize and potential additional sources of error. Second, higher-lying excited states become too dense to yield any additional insights into the effect of excitation on the effective barrier and will only make the calculation more expensive (as is seen in the *NH2 + *H → *NH + 2*H and *NH → *N + *H reactions). These issues potentially could be solved by using multistate CASPT2; (80) however, this method is significantly more costly and proved to be intractable for even just two states of the systems we considered here.

4. Summary and Conclusions

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In the present study, we used a Pd(111) + NH3 model system to systematically determine the optimal approximations for embedded (SA-)CASSCF/CASPT2 simulations conducted within DFET. Specifically, we considered how PECs are affected by cluster size, basis set, structure of the cluster when Vemb is determined, method utilized to generate initial MO guesses for the CASSCF AS, and the number of states included in the SA-CASSCF calculations of excited states. We conclude that the energetics converge quickly even with a small cluster of 10 atoms embedded in a fairly large periodic slab, as long as all direct metal–adsorbate interactions are included. The relative insensitivity to the cluster size is due to the DFET approach that damps out edge/finite-size effects rapidly and the DFET energy expression which limits the size-dependence due to the regional nature of the energy correction. Second, we found that the aug-cc-pVDZ basis set provides the best compromise between accuracy and computational cost, with the larger aug-cc-pVTZ basis set yielding only minor improvements while significantly reducing computational tractability. That said, we recommend complete basis set extrapolation rather than CP corrections, which overestimate the finite basis set errors. Third, we found that it is best to determine Vemb without the adsorbates but using a cluster structure reflecting the structure of the surface as if adsorbates are present, although a fixed-clean-surface approximation holds well even when the surface undergoes significant structural changes. Fourth, by comparing merging and creeping approaches to generate initial AS guesses for the three consecutive dehydrogenation steps of NH3 on Pd(111), we found significant shortcomings in the ability of a creeping approach to generate consistent ASs along reaction pathways when simulating surfaces consisting of highly localized and near-degenerate, partly filled d-orbital manifolds, such as in the case of Pd (which adopts an s1d9 valence electronic structure in its bulk phase), leading to discontinuities in the CASSCF/CASPT2 PECs. In such cases, we recommend utilizing a merging approach, generating ASs from the molecular fragments at each surveyed configuration. Finally, when aiming to study excited-state reaction pathways, e.g., in the study of photocatalytic or plasmonic processes, we demonstrated that it is essential to include bands of excited states in their entirety to avoid discontinuities in the PECs. This generally requires performing benchmarks that identify how excited states group into bands in SA-CASSCF calculations.
The results presented in this study provide important insights for future studies aiming to elucidate mechanisms of heterogeneously catalyzed reactions utilizing (SA-)CASSCF-based ECW theory simulations. Importantly, the guidelines derived here will significantly reduce the need to perform initial benchmarking calculations when commencing an ECW theory study of a new chemical system, which are generally necessary to choose the correct level of theory and often consume significant computational resources.

Supporting Information

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The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.4c00558.

  • Plots of all geometries, clean-surface approximation energetics, cluster-size effects, PW to GTO basis set effects, Pd 4s4p dynamic correlation effects, overview of BSSE and CP, plots and energetics of geometry effects on Vemb, NO plots of all structures surveyed, and SA-CASPT2 ground- and excited-state MEPs utilizing different state averaging (PDF)

  • Tables of SA-CASSCF dipole moments (XLSX)

Terms & Conditions

Most electronic Supporting Information files are available without a subscription to ACS Web Editions. Such files may be downloaded by article for research use (if there is a public use license linked to the relevant article, that license may permit other uses). Permission may be obtained from ACS for other uses through requests via the RightsLink permission system: http://pubs.acs.org/page/copyright/permissions.html.

Author Information

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  • Corresponding Author
    • Emily A. Carter - Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544-5263, United States;  Princeton Plasma Physics Laboratory, Princeton, New Jersey 08540-6655, United States;  Andlinger Center for Energy and the Environment, Princeton University, Princeton, New Jersey 08544-5263, United States;  Orcidhttps://orcid.org/0000-0001-7330-7554;  Email: [email protected]
  • Authors
    • Xuelan Wen - Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544-5263, United States;  Orcidhttps://orcid.org/0000-0002-7151-6157
    • Jan-Niklas Boyn - Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544-5263, United States;  Orcidhttps://orcid.org/0000-0002-6240-3759
    • John Mark P. Martirez - Princeton Plasma Physics Laboratory, Princeton, New Jersey 08540-6655, United States;  Orcidhttps://orcid.org/0000-0003-0566-6605
    • Qing Zhao - Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544-5263, United States;  Present Address: Department of Chemical Engineering, Northeastern University, Boston, Massachusetts 02115–5005, United States;  Orcidhttps://orcid.org/0000-0002-5003-9355
  • Author Contributions

    The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

  • Notes
    The authors declare no competing financial interest.

Acknowledgments

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The Air Force Office of Scientific Research via the Department of Defense Multidisciplinary University Research Initiative, under award no. FA9550-15-1-0022, provided the financial support for the majority of this work. E.A.C. and J.M.P.M. were also supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences (BES), under award no. DE-SC0023357. Princeton University’s Terascale Infrastructure for Groundbreaking Research in Engineering and Science (TIGRESS) and the High Performance Computing Modernization Program (HPCMP) of the US Department of Defense provided the computational resources.

References

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    Journal of Computational Chemistry (2019), 40 (25), 2216-2226CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)
    We present our implementation AUTOCAS for fully automated multiconfigurational calcns., which we also make available free of charge on our webpages. The graphical user interface of AUTOCAS connects a general electronic structure program with a d.-matrix renormalization group program to carry out our recently introduced automated active space selection protocol for multiconfigurational calcns. (Stein and Reiher, J. Chem. Theory Comput., 2016, 12, 1760). Next to this active space selection, AUTOCAS carries out several steps of multiconfigurational calcns. so that only a minimal input is required to start them, comparable to that of a std. Kohn-Sham d.-functional theory calcn., so that black-box multiconfigurational calcns. become feasible. Furthermore, we introduce a new extension to the selection algorithm that facilitates automated selections for mols. with large valence orbital spaces consisting of several hundred orbitals. © 2019 Wiley Periodicals, Inc.
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  9. 9
    Jeong, W.; Stoneburner, S. J.; King, D.; Li, R.; Walker, A.; Lindh, R.; Gagliardi, L. Automation of Active Space Selection for Multireference Methods via Machine Learning on Chemical Bond Dissociation. J. Chem. Theory Comput. 2020, 16 (4), 2389– 2399,  DOI: 10.1021/acs.jctc.9b01297
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    Golub, P.; Antalik, A.; Veis, L.; Brabec, J. Machine Learning-Assisted Selection of Active Spaces for Strongly Correlated Transition Metal Systems. J. Chem. Theory Comput. 2021, 17 (10), 6053– 6072,  DOI: 10.1021/acs.jctc.1c00235
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    Han, R.; Luber, S. Complete active space analysis of a reaction pathway: Investigation of the oxygen–oxygen bond formation. J. Comput. Chem. 2020, 41 (17), 1586– 1597,  DOI: 10.1002/jcc.26201
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    Stein, C. J.; Reiher, M. Automated Identification of Relevant Frontier Orbitals for Chemical Compounds and Processes. Chimia 2017, 71 (4), 170,  DOI: 10.2533/chimia.2017.170
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    12
    Automated identification of relevant frontier orbitals for chemical compounds and processes
    Stein, Christopher J.; Reiher, Markus
    Chimia (2017), 71 (4), 170-176CODEN: CHIMAD ISSN:. (Swiss Chemical Society)
    Quantum-chem. multi-configurational methods are required for a proper description of static electron correlation, a phenomenon inherent to the electronic structure of mols. with multiple (near-)degenerate frontier orbitals. Here, we review how a property of these frontier orbitals, namely the entanglement entropy is related to static electron correlation. A subset of orbitals, the so-called active orbital space is an essential ingredient for all multi-configurational methods. We proposed an automated selection of this active orbital space, that would otherwise be a tedious and error prone manual procedure, based on entanglement measures. Here, we extend this scheme to demonstrate its capability for the selection of consistent active spaces for several excited states and along reaction coordinates.
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  13. 13
    Libisch, F.; Cheng, J.; Carter, E. A. Electron-Transfer-Induced Dissociation of H2 on Gold Nanoparticles: Excited-State Potential Energy Surfaces via Embedded Correlated Wavefunction Theory. Z. Phys. Chem. 2013, 227 (11), 1455– 1466,  DOI: 10.1524/zpch.2013.0406
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    Libisch, F.; Krauter, C. M.; Carter, E. A. Corrigendum to: Plasmon-Driven Dissociation of H2 on Gold Nanoclusters. Z. Phys. Chem. 2016, 230 (1), 131– 132,  DOI: 10.1515/zpch-2015-5001
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    Mukherjee, S.; Libisch, F.; Large, N.; Neumann, O.; Brown, L. V.; Cheng, J.; Lassiter, J. B.; Carter, E. A.; Nordlander, P.; Halas, N. J. Hot Electrons Do the Impossible: Plasmon-Induced Dissociation of H2 on Au. Nano Lett. 2013, 13 (1), 240– 247,  DOI: 10.1021/nl303940z
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    15
    Hot Electrons Do the Impossible: Plasmon-Induced Dissociation of H2 on Au
    Mukherjee, Shaunak; Libisch, Florian; Large, Nicolas; Neumann, Oara; Brown, Lisa V.; Cheng, Jin; Lassiter, J. Britt; Carter, Emily A.; Nordlander, Peter; Halas, Naomi J.
    Nano Letters (2013), 13 (1), 240-247CODEN: NALEFD; ISSN:1530-6984. (American Chemical Society)
    Heterogeneous catalysis is of paramount importance in chem. and energy applications. Catalysts that couple light energy into chem. reactions in a directed, orbital-specific manner would greatly reduce the energy input requirements of chem. transformations, revolutionizing catalysis-driven chem. Here we report the room temp. dissocn. of H2 on gold nanoparticles using visible light. Surface plasmons excited in the Au nanoparticle decay into hot electrons with energies between the vacuum level and the work function of the metal. In this transient state, hot electrons can transfer into a Feshbach resonance of an H2 mol. adsorbed on the Au nanoparticle surface, triggering dissocn. We probe this process by detecting the formation of HD mols. from the dissocns. of H2 and D2 and investigate the effect of Au nanoparticle size and wavelength of incident light on the rate of HD formation. This work opens a new pathway for controlling chem. reactions on metallic catalysts.
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  16. 16
    Martirez, J. M. P.; Carter, E. A. Excited-State N2 Dissociation Pathway on Fe-Functionalized Au. J. Am. Chem. Soc. 2017, 139 (12), 4390– 4398,  DOI: 10.1021/jacs.6b12301
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  17. 17
    Martirez, J. M. P.; Carter, E. A. Prediction of a low-temperature N2 dissociation catalyst exploiting near-IR–to–visible light nanoplasmonics. Sci. Adv. 2017, 3 (12), eaao4710  DOI: 10.1126/sciadv.aao4710
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    Yuan, Y.; Zhou, L.; Robatjazi, H.; Bao, J. L.; Zhou, J.; Bayles, A.; Yuan, L.; Lou, M.; Lou, M.; Khatiwada, S. Earth-abundant photocatalyst for H2 generation from NH3 with light-emitting diode illumination. Science 2022, 378 (6622), 889– 893,  DOI: 10.1126/science.abn5636
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    18
    Earth-abundant photocatalyst for H2 generation from NH3 with light-emitting diode illumination
    Yuan, Yigao; Zhou, Linan; Robatjazi, Hossein; Bao, Junwei Lucas; Zhou, Jingyi; Bayles, Aaron; Yuan, Lin; Lou, Minghe; Lou, Minhan; Khatiwada, Suman; Carter, Emily A.; Nordlander, Peter; Halas, Naomi J.
    Science (Washington, DC, United States) (2022), 378 (6622), 889-893CODEN: SCIEAS; ISSN:1095-9203. (American Association for the Advancement of Science)
    Catalysts based on platinum group metals have been a major focus of the chem. industry for decades. We show that plasmonic photocatalysis can transform a thermally unreactive, earth-abundant transition metal into a catalytically active site under illumination. Fe active sites in a Cu-Fe antenna-reactor complex achieve efficiencies very similar to Ru for the photocatalytic decompn. of ammonia under ultrafast pulsed illumination. When illuminated with light-emitting diodes rather than lasers, the photocatalytic efficiencies remain comparable, even when the scale of reaction increases by nearly three orders of magnitude. This result demonstrates the potential for highly efficient, elec. driven prodn. of hydrogen from an ammonia carrier with earth-abundant transition metals.
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  19. 19
    Bao, J. L.; Carter, E. A. Surface-Plasmon-Induced Ammonia Decomposition on Copper: Excited-State Reaction Pathways Revealed by Embedded Correlated Wavefunction Theory. ACS Nano 2019, 13 (9), 9944– 9957,  DOI: 10.1021/acsnano.9b05030
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    19
    Surface-Plasmon-Induced Ammonia Decomposition on Copper: Excited-State Reaction Pathways Revealed by Embedded Correlated Wavefunction Theory
    Bao, Junwei Lucas; Carter, Emily A.
    ACS Nano (2019), 13 (9), 9944-9957CODEN: ANCAC3; ISSN:1936-0851. (American Chemical Society)
    NH3 is a promising H storage medium; however, its decompn. via conventional thermal catalysis requires a significant amt. of thermal energy input to overcome the reaction barriers. Embedded correlated wavefunction (ECW) theory was used to quantify reaction pathways and energetics for NH3 decompn. (N-H bond dissocn. and N2 and H2 associative desorption) on Cu nanoparticles using a Cu (111) surface model. The authors predict that surface plasmon excitations will be able to facilitate NH3 decompn. by substantially reducing the effective barriers along excited-state pathways. The authors est. the redns. in reaction barriers for breaking the 1st N-H bond and for recombinative desorption of surface-bound N and H atoms to be ∼1.7, 0.8, and 0.5 eV, resp. Further, by using the exptl. N2 desorption barrier as a ref., the authors compare the accuracy of various theor. methods, including plane-wave Kohn-Sham d. functional theory calcns. with commonly used exchange-correlation functionals, embedded complete active space 2nd-order perturbation theory, and embedded multiconfiguration pair-d. functional theory. This work offers further confirmation that the ECW theor. framework is the most robust for treating highly correlated local electronic structures of solids.
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  20. 20
    Bao, J. L.; Carter, E. A. Rationalizing the Hot-Carrier-Mediated Reaction Mechanisms and Kinetics for Ammonia Decomposition on Ruthenium-Doped Copper Nanoparticles. J. Am. Chem. Soc. 2019, 141 (34), 13320– 13323,  DOI: 10.1021/jacs.9b06804
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    20
    Rationalizing the Hot-Carrier-Mediated Reaction Mechanisms and Kinetics for Ammonia Decomposition on Ruthenium-Doped Copper Nanoparticles
    Bao, Junwei Lucas; Carter, Emily A.
    Journal of the American Chemical Society (2019), 141 (34), 13320-13323CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)
    Localized surface plasmons generated on metallic nanostructures provide an efficient driving force for catalyzing chem. reactions, the kinetics of which cannot be understood properly by d. functional theory, despite its wide use in simulating heterogeneous catalytic reaction mechanisms. Reaction pathways for the NH3 decompn. reaction on Ru-doped Cu studied by the embedded correlated wave function method are reported. The computations provide a qual. explanation for the exptl. obsd. change in the reaction order from thermal catalysis to hot-carrier-mediated photocatalysis, as reported recently in Zhou, L., et al. (2018).
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  21. 21
    Zhou, L.; Martirez, J. M. P.; Finzel, J.; Zhang, C.; Swearer, D. F.; Tian, S.; Robatjazi, H.; Lou, M.; Dong, L.; Henderson, L. Light-driven methane dry reforming with single atomic site antenna-reactor plasmonic photocatalysts. Nat. Energy 2020, 5 (1), 61– 70,  DOI: 10.1038/s41560-019-0517-9
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    21
    Light-driven methane dry reforming with single atomic site antenna-reactor plasmonic photocatalysts
    Zhou, Linan; Martirez, John Mark P.; Finzel, Jordan; Zhang, Chao; Swearer, Dayne F.; Tian, Shu; Robatjazi, Hossein; Lou, Minhan; Dong, Liangliang; Henderson, Luke; Christopher, Phillip; Carter, Emily A.; Nordlander, Peter; Halas, Naomi J.
    Nature Energy (2020), 5 (1), 61-70CODEN: NEANFD; ISSN:2058-7546. (Nature Research)
    Syngas, an extremely important chem. feedstock composed of carbon monoxide and hydrogen, can be generated through methane (CH4) dry reforming with CO2. However, traditional thermocatalytic processes require high temps. and suffer from coke-induced instability. Here, we report a plasmonic photocatalyst consisting of a Cu nanoparticle antenna with single-Ru at. reactor sites on the nanoparticle surface, ideal for low-temp., light-driven methane dry reforming. This catalyst provides high light energy efficiency when illuminated at room temp. In contrast to thermocatalysis, long-term stability (50 h) and high selectivity (>99%) were achieved in photocatalysis. We propose that light-excited hot carriers, together with single-atom active sites, cause the obsd. performance. Quantum mech. modeling suggests that single-atom doping of Ru on the Cu(111) surface, coupled with excited-state activation, results in a substantial redn. in the barrier for CH4 activation. This photocatalyst design could be relevant for future energy-efficient industrial processes.
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  22. 22
    Spata, V. A.; Carter, E. A. Mechanistic Insights into Photocatalyzed Hydrogen Desorption from Palladium Surfaces Assisted by Localized Surface Plasmon Resonances. ACS Nano 2018, 12 (4), 3512– 3522,  DOI: 10.1021/acsnano.8b00352
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    22
    Mechanistic Insights into Photocatalyzed Hydrogen Desorption from Palladium Surfaces Assisted by Localized Surface Plasmon Resonances
    Spata, Vincent A.; Carter, Emily A.
    ACS Nano (2018), 12 (4), 3512-3522CODEN: ANCAC3; ISSN:1936-0851. (American Chemical Society)
    Nanoparticles synthesized from plasmonic metals can absorb low-energy light, producing an oscillation/excitation of their valence electron d. that can be utilized in chem. conversions. For example, heterogeneous photocatalysis can be achieved within heterometallic antenna-reactor complexes (HMARCs), by coupling a reactive center at which a chem. reaction occurs to a plasmonic nanoparticle that acts as a light-absorbing antenna. For example, HMARCs composed of aluminum antennae and palladium (Pd) reactive centers have been demonstrated recently to catalyze selective hydrogenation of acetylene to ethylene. Here, we explore within a theor. framework the rate-limiting step of hydrogen photodesorption from a Pd surface-crucial to achieving partial rather than full hydrogenation of acetylene-to understand the mechanism behind the photodesorption process within the HMARC assembly. To properly describe electronic excited states of the metal-mol. system, we employ embedded complete active space SCF and n-electron valence state perturbation theory to second order within d. functional embedding theory. The results of these calcns. reveal that the photodesorption mechanism does not create a frequently invoked transient neg. ion species but instead enhances population of available excited-state, low-barrier pathways that exhibit negligible charge-transfer character.
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  23. 23
    Huang, C.; Pavone, M.; Carter, E. A. Quantum mechanical embedding theory based on a unique embedding potential. J. Chem. Phys. 2011, 134 (15), 154110,  DOI: 10.1063/1.3577516
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    23
    Quantum mechanical embedding theory based on a unique embedding potential
    Huang, Chen; Pavone, Michele; Carter, Emily A.
    Journal of Chemical Physics (2011), 134 (15), 154110/1-154110/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    We remove the nonuniqueness of the embedding potential that exists in most previous quantum mech. embedding schemes by letting the environment and embedded region share a common embedding (interaction) potential. To efficiently solve for the embedding potential, an optimized effective potential method is derived. This embedding potential, which eschews use of approx. kinetic energy d. functionals, is then used to describe the environment while a correlated wavefunction (CW) treatment of the embedded region is employed. We first demonstrate the accuracy of this new embedded CW (ECW) method by calcg. the van der Waals binding energy curve between a hydrogen mol. and a hydrogen chain. We then examine the prototypical adsorption of CO on a metal surface, here the Cu(111) surface. In addn. to obtaining proper site ordering (top site most stable) and binding energies within this theory, the ECW exhibits dramatic changes in the p-character of the CO 4σ and 5σ orbitals upon adsorption that agree very well with x-ray emission spectra, providing further validation of the theory. Finally, we generalize our embedding theory to spin-polarized quantum systems and discuss the connection between our theory and partition d. functional theory. (c) 2011 American Institute of Physics.
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    Yu, K.; Libisch, F.; Carter, E. A. Implementation of density functional embedding theory within the projector-augmented-wave method and applications to semiconductor defect states. J. Chem. Phys. 2015, 143 (10), 102806,  DOI: 10.1063/1.4922260
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    24
    Implementation of density functional embedding theory within the projector-augmented-wave method and applications to semiconductor defect states
    Yu, Kuang; Libisch, Florian; Carter, Emily A.
    Journal of Chemical Physics (2015), 143 (10), 102806/1-102806/15CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    We report a new implementation of the d. functional embedding theory (DFET) in the VASP code, using the projector-augmented-wave (PAW) formalism. Newly developed algorithms allow us to efficiently perform optimized effective potential optimizations within PAW. The new algorithm generates robust and phys. correct embedding potentials, as we verified using several test systems including a covalently bound mol., a metal surface, and bulk semiconductors. We show that with the resulting embedding potential, embedded cluster models can reproduce the electronic structure of point defects in bulk semiconductors, thereby demonstrating the validity of DFET in semiconductors for the first time. Compared to our previous version, the new implementation of DFET within VASP affords use of all features of VASP (e.g., a systematic PAW library, a wide selection of functionals, a more flexible choice of U correction formalisms, and faster computational speed) with DFET. Furthermore, our results are fairly robust with respect to both plane-wave and Gaussian type orbital basis sets in the embedded cluster calcns. This suggests that the d. functional embedding method is potentially an accurate and efficient way to study properties of isolated defects in semiconductors. (c) 2015 American Institute of Physics.
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  25. 25
    Libisch, F.; Huang, C.; Carter, E. A. Embedded Correlated Wavefunction Schemes: Theory and Applications. Acc. Chem. Res. 2014, 47 (9), 2768– 2775,  DOI: 10.1021/ar500086h
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    25
    Embedded Correlated Wavefunction Schemes: Theory and Applications
    Libisch, Florian; Huang, Chen; Carter, Emily A.
    Accounts of Chemical Research (2014), 47 (9), 2768-2775CODEN: ACHRE4; ISSN:0001-4842. (American Chemical Society)
    A review. Ab initio modeling of matter has become a pillar of chem. research: with ever-increasing computational power, simulations can be used to accurately predict, for example, chem. reaction rates, electronic and mech. properties of materials, and dynamical properties of liqs. Many competing quantum mech. methods have been developed over the years that vary in computational cost, accuracy, and scalability: d. functional theory (DFT), the workhorse of solid-state electronic structure calcns., features a good compromise between accuracy and speed. However, approx. exchange-correlation functionals limit DFT's ability to treat certain phenomena or states of matter, such as charge-transfer processes or strongly correlated materials. Furthermore, conventional DFT is purely a ground-state theory: electronic excitations are beyond its scope. Excitations in mols. are routinely calcd. using time-dependent DFT linear response; however applications to condensed matter are still limited. By contrast, many-electron wavefunction methods aim for a very accurate treatment of electronic exchange and correlation. Unfortunately, the assocd. computational cost renders treatment of more than a handful of heavy atoms challenging. On the other side of the accuracy spectrum, parametrized approaches like tight-binding can treat millions of atoms. In view of the different (dis-)advantages of each method, the simulation of complex systems seems to force a compromise: one is limited to the most accurate method that can still handle the problem size. For many interesting problems, however, compromise proves insufficient. A possible soln. is to break up the system into manageable subsystems that may be treated by different computational methods. The interaction between subsystems may be handled by an embedding formalism. In this Account, we review embedded correlated wavefunction (CW) approaches and some applications. We first discuss our d. functional embedding theory, which is formally exact. We show how to det. the embedding potential, which replaces the interaction between subsystems, at the DFT level. CW calcns. are performed using a fixed embedding potential, i.e., a non-self-consistent embedding scheme. We demonstrate this embedding theory for two challenging electron transfer phenomena: (1) initial oxidn. of an aluminum surface and (2) hot-electron-mediated dissocn. of hydrogen mols. on a gold surface. In both cases, the interaction between gas mols. and metal surfaces were treated by sophisticated CW techniques, with the remainder of the extended metal surface being treated by DFT. Our embedding approach overcomes the limitations of conventional Kohn-Sham DFT in describing charge transfer, multiconfigurational character, and excited states. From these embedding simulations, we gained important insights into fundamental processes that are crucial aspects of fuel cell catalysis (i.e., O2 redn. at metal surfaces) and plasmon-mediated photocatalysis by metal nanoparticles. Moreover, our findings agree very well with exptl. observations, while offering new views into the chem. We finally discuss our recently formulated potential-functional embedding theory that provides a seamless, first-principles way to include back-action onto the environment from the embedded region.
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    Tóth, Z.; Pulay, P. Comparison of Methods for Active Orbital Selection in Multiconfigurational Calculations. J. Chem. Theory Comput. 2020, 16 (12), 7328– 7341,  DOI: 10.1021/acs.jctc.0c00123
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    Comparison of Methods for Active Orbital Selection in Multiconfigurational Calculations
    Toth, Zsuzsanna; Pulay, Peter
    Journal of Chemical Theory and Computation (2020), 16 (12), 7328-7341CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    Several methods of constructing the active orbital space for multiconfigurational wave functions are compared on typical moderately strongly or strongly correlated ground-state mols. The relative merits of these methods and problems inherent in multiconfigurational calcns. are discussed. Strong correlation in the ground electronic state is found typically in larger conjugated and in antiarom. systems, transition states which involve bond breaking or formation, and transition metal complexes. Our examples include polyenes, polyacenes, the reactant, product and transition state of the Bergman cyclization, and two transition metal complexes: Hieber's anion [(CO)3FeNO]- and ferrocene. For the systems investigated, the simplest and oldest selection method, based on the fractional occupancy of UHF natural orbitals (the UNO criterion), yields the same active space as much more expensive approx. full CI methods. A disadvantage of this method used to be the difficulty of finding broken spin symmetry UHF solns. However, our anal. method, accurate to fourth order in the orbital rotation angles (Toth and Pulay, J. Chem. Phys., 2016, 145, 164102), has solved this problem. Two further advantages of the UNO criterion are that, unlike most other methods, it measures not only the energetic proximity to the Fermi level but also the magnitude of the exchange interaction with strongly occupied orbitals and therefore allows the estn. of the correlation strength for orbital selection in Restricted Active Space methods.
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    Klüner, T.; Govind, N.; Wang, Y. A.; Carter, E. A. Prediction of Electronic Excited States of Adsorbates on Metal Surfaces from First Principles. Phys. Rev. Lett. 2001, 86 (26), 5954– 5957,  DOI: 10.1103/PhysRevLett.86.5954
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    Klüner, T.; Govind, N.; Wang, Y. A.; Carter, E. A. Periodic density functional embedding theory for complete active space self-consistent field and configuration interaction calculations: Ground and excited states. J. Chem. Phys. 2002, 116 (1), 42– 54,  DOI: 10.1063/1.1420748
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    Second-order perturbation theory with a complete active space self-consistent field reference function
    Andersson, Kerstin; Malmqvist, Per Aake; Roos, Bjoern O.
    Journal of Chemical Physics (1992), 96 (2), 1218-26CODEN: JCPSA6; ISSN:0021-9606.
    The recently implemented second-order perturbation theory based on a complete active space SCF ref. function has been extended by allowing the Fock-type one-electron operator, which defines the zeroth-order Hamiltonian to have nonzero elements also in nondiagonal matrix blocks. The computer implementation is now less straightforward and more computer time will be needed in obtaining the second-order energy. The method is illustrated in a series of calcns. on N2, NO, O2, CH3, CH2, and F-.
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    Martirez, J. M. P.; Bao, J. L.; Carter, E. A. First-Principles Insights into Plasmon-Induced Catalysis. Annu. Rev. Phys. Chem. 2021, 72 (1), 99– 119,  DOI: 10.1146/annurev-physchem-061020-053501
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    Kresse, G.; Hafner, J. Ab initio molecular dynamics for liquid metals. Phys. Rev. B: Condens. Matter Mater. Phys. 1993, 47 (1), 558– 561,  DOI: 10.1103/PhysRevB.47.558
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    Ab initio molecular dynamics of liquid metals
    Kresse, G.; Hafner, J.
    Physical Review B: Condensed Matter and Materials Physics (1993), 47 (1), 558-61CODEN: PRBMDO; ISSN:0163-1829.
    The authors present ab initio quantum-mech. mol.-dynamics calcns. based on the calcn. of the electronic ground state and of the Hellmann-Feynman forces in the local-d. approxn. at each mol.-dynamics step. This is possible using conjugate-gradient techniques for energy minimization, and predicting the wave functions for new ionic positions using sub-space alignment. This approach avoids the instabilities inherent in quantum-mech. mol.-dynamics calcns. for metals based on the use of a factitious Newtonian dynamics for the electronic degrees of freedom. This method gives perfect control of the adiabaticity and allows one to perform simulations over several picoseconds.
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    Kresse, G.; Furthmuller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B: Condens. Matter Mater. Phys. 1996, 54 (16), 11169– 11186,  DOI: 10.1103/PhysRevB.54.11169
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    Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set
    Kresse, G.; Furthmueller, J.
    Physical Review B: Condensed Matter (1996), 54 (16), 11169-11186CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
    The authors present an efficient scheme for calcg. the Kohn-Sham ground state of metallic systems using pseudopotentials and a plane-wave basis set. In the first part the application of Pulay's DIIS method (direct inversion in the iterative subspace) to the iterative diagonalization of large matrixes will be discussed. This approach is stable, reliable, and minimizes the no. of order Natoms3 operations. In the second part, we will discuss an efficient mixing scheme also based on Pulay's scheme. A special "metric" and a special "preconditioning" optimized for a plane-wave basis set will be introduced. Scaling of the method will be discussed in detail for non-self-consistent and self-consistent calcns. It will be shown that the no. of iterations required to obtain a specific precision is almost independent of the system size. Altogether an order Natoms2 scaling is found for systems contg. up to 1000 electrons. If we take into account that the no. of k points can be decreased linearly with the system size, the overall scaling can approach Natoms. They have implemented these algorithms within a powerful package called VASP (Vienna ab initio simulation package). The program and the techniques have been used successfully for a large no. of different systems (liq. and amorphous semiconductors, liq. simple and transition metals, metallic and semiconducting surfaces, phonons in simple metals, transition metals, and semiconductors) and turned out to be very reliable.
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    Kresse, G.; Furthmüller, J. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mater. Sci. 1996, 6 (1), 15– 50,  DOI: 10.1016/0927-0256(96)00008-0
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    Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set
    Kresse, G.; Furthmuller, J.
    Computational Materials Science (1996), 6 (1), 15-50CODEN: CMMSEM; ISSN:0927-0256. (Elsevier)
    The authors present a detailed description and comparison of algorithms for performing ab-initio quantum-mech. calcns. using pseudopotentials and a plane-wave basis set. The authors will discuss: (a) partial occupancies within the framework of the linear tetrahedron method and the finite temp. d.-functional theory, (b) iterative methods for the diagonalization of the Kohn-Sham Hamiltonian and a discussion of an efficient iterative method based on the ideas of Pulay's residual minimization, which is close to an order N2atoms scaling even for relatively large systems, (c) efficient Broyden-like and Pulay-like mixing methods for the charge d. including a new special preconditioning optimized for a plane-wave basis set, (d) conjugate gradient methods for minimizing the electronic free energy with respect to all degrees of freedom simultaneously. The authors have implemented these algorithms within a powerful package called VAMP (Vienna ab-initio mol.-dynamics package). The program and the techniques have been used successfully for a large no. of different systems (liq. and amorphous semiconductors, liq. simple and transition metals, metallic and semi-conducting surfaces, phonons in simple metals, transition metals and semiconductors) and turned out to be very reliable.
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    Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 1996, 77 (18), 3865– 3868,  DOI: 10.1103/PhysRevLett.77.3865
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    Generalized gradient approximation made simple
    Perdew, John P.; Burke, Kieron; Ernzerhof, Matthias
    Physical Review Letters (1996), 77 (18), 3865-3868CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
    Generalized gradient approxns. (GGA's) for the exchange-correlation energy improve upon the local spin d. (LSD) description of atoms, mols., and solids. We present a simple derivation of a simple GGA, in which all parameters (other than those in LSD) are fundamental consts. Only general features of the detailed construction underlying the Perdew-Wang 1991 (PW91) GGA are invoked. Improvements over PW91 include an accurate description of the linear response of the uniform electron gas, correct behavior under uniform scaling, and a smoother potential.
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    Singh-Miller, N. E.; Marzari, N. Surface energies, work functions, and surface relaxations of low-index metallic surfaces from first principles. Phys. Rev. B: Condens. Matter Mater. Phys. 2009, 80 (23), 235407,  DOI: 10.1103/PhysRevB.80.235407
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    Surface energies, work functions, and surface relaxations of low-index metallic surfaces from first principles
    Singh-Miller, Nicholas E.; Marzari, Nicola
    Physical Review B: Condensed Matter and Materials Physics (2009), 80 (23), 235407/1-235407/9CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
    We study the relaxations, surface energies, and work functions of low-index metallic surfaces using pseudopotential plane-wave d.-functional calcns. within the generalized gradient approxn. We study here the (100), (110), and (111) surfaces of Al, Pd, Pt, and Au and the (0001) surface of Ti, chosen for their use as contact or lead materials in nanoscale devices. We consider clean, mostly nonreconstructed surfaces in the slab-supercell approxn. Particular attention is paid to the convergence of these quantities with respect to slab thickness; furthermore, different methodologies for the calcn. of work functions and surfaces energies are compared. The use of bulk refs. for calcns. of surface energies and work functions can be detrimental to convergence unless numerical grids are closely matched, esp. when surface relaxations are being considered. Calcd. values often do not quant. match exptl. values. This may be understandable for the surface relaxations and surface energies, where exptl. values can have large error but even for the work functions, neither local nor semilocal functionals emerge as an accurate choice for every case.
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    Makov, G.; Payne, M. C. Periodic boundary conditions in ab initio calculations. Phys. Rev. B: Condens. Matter Mater. Phys. 1995, 51 (7), 4014– 4022,  DOI: 10.1103/PhysRevB.51.4014
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    Physical Review B: Condensed Matter (1995), 51 (7), 4014-22CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
    The convergence of the electrostatic energy in calcns. using periodic boundary conditions is considered in the context of periodic solids and localized aperiodic systems in the gas and condensed phases. Conditions for the abs. convergence of the total energy in periodic boundary conditions are obtained, and their implications for calcns. of the properties of polarized solids under the zero-field assumption are discussed. For aperiodic systems the exact electrostatic energy functional in periodic boundary conditions is obtained. The convergence in such systems is considered in the limit of large supercells, where, in the gas phase, the computational effort is proportional to the vol. It is shown that for neutral localized aperiodic systems in either the gas or condensed phases, the energy can always be made to converge as O (L-5) where L is the linear dimension of the supercell. For charged systems, convergence at this rate can be achieved after adding correction terms to the energy to account for spurious interactions induced by the periodic boundary conditions. These terms are derived exactly for the gas phase and heuristically for the condensed phase.
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    A sampling method is given for Brillouin-zone integration in metals which converges exponentially with the no. of sampling points, without the loss of precision of normal broadening techniques. The scheme is based on smooth approximants to the δ and step functions which are constructed to give the exact result when integrating polynomials of a prescribed degree. In applications to the simple-cubic tight-binding band as well as to band structures of simple and transition metals, significant improvement over existing methods was shown. The method promises general applicability in the fields of total-energy calcns. and many-body physics.
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    Henkelman, Graeme; Uberuaga, Blas P.; Jonsson, Hannes
    Journal of Chemical Physics (2000), 113 (22), 9901-9904CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    A modification of the nudged elastic band method for finding min. energy paths is presented. One of the images is made to climb up along the elastic band to converge rigorously on the highest saddle point. Also, variable spring consts. are used to increase the d. of images near the top of the energy barrier to get an improved est. of the reaction coordinate near the saddle point. Applications to CH4 dissociative adsorption on Ir(111) and H2 on Si(100) using plane wave based d. functional theory are presented.
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    Werner, H.-J.; Knowles, P. J.; Knizia, G.; Manby, F. R.; Schütz, M. Molpro: a general-purpose quantum chemistry program package. Wiley Interdiscip. Rev. Comput. Mol. Sci. 2012, 2 (2), 242– 253,  DOI: 10.1002/wcms.82
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    Wiley Interdisciplinary Reviews: Computational Molecular Science (2012), 2 (2), 242-253CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell)
    Molpro is a general-purpose quantum chem. program. The original focus was on high-accuracy wave function calcns. for small mols., but using local approxns. combined with explicit correlation treatments, highly accurate coupled-cluster calcns. are now possible for mols. with up to approx. 100 atoms. Recently, multireference correlation treatments were also made applicable to larger mols. Furthermore, an efficient implementation of d. functional theory is available.
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    Werner, H.-J.; Knowles, P. J.; Manby, F. R.; Black, J. A.; Doll, K.; Heßelmann, A.; Kats, D.; Köhn, A.; Korona, T.; Kreplin, D. A. The Molpro quantum chemistry package. J. Chem. Phys. 2020, 152 (14), 144107,  DOI: 10.1063/5.0005081
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    The Molpro quantum chemistry package
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    Journal of Chemical Physics (2020), 152 (14), 144107CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    Molpro is a general purpose quantum chem. software package with a long development history. It was originally focused on accurate wavefunction calcns. for small mols. but now has many addnl. distinctive capabilities that include, inter alia, local correlation approxns. combined with explicit correlation, highly efficient implementations of single-ref. correlation methods, robust and efficient multireference methods for large mols., projection embedding, and anharmonic vibrational spectra. In addn. to conventional input-file specification of calcns., Molpro calcns. can now be specified and analyzed via a new graphical user interface and through a Python framework. (c) 2020 American Institute of Physics.
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    Werner, H.-J.; Knowles, P. J.; Knizia, G.; Manby, F. R.; Schütz, M.; Celani, P.; Györffy, W.; Kats, D.; Korona, T.; Lindh, R.; MOLPRO , version 2021.2, a package of ab initio programs, 2021. https://www.molpro.net.
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    Ghigo, G.; Roos, B. O.; Malmqvist, P.-Å. A modified definition of the zeroth-order Hamiltonian in multiconfigurational perturbation theory (CASPT2). Chem. Phys. Lett. 2004, 396 (1–3), 142– 149,  DOI: 10.1016/j.cplett.2004.08.032
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    A modified definition of the zeroth-order Hamiltonian in multiconfigurational perturbation theory (CASPT2)
    Ghigo, Giovanni; Roos, Bjoern O.; Malmqvist, Per-Ake
    Chemical Physics Letters (2004), 396 (1-3), 142-149CODEN: CHPLBC; ISSN:0009-2614. (Elsevier B.V.)
    A new shifted zeroth-order Hamiltonian is presented, which will be used in second-order multiconfigurational perturbation theory (CASPT2). The new approxn. corrects for the systematic error of the original formulation, which led to an relative overestimate of the correlation energy for open shell system, resulting in too small dissocn. and excitation energies. Errors in the De values for 49 diat. mols. were reduced with more than 50%. Calcns. on excited states of the N2 and benzene mols. give a similar improvement.
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    Roos, B. O.; Andersson, K. Multiconfigurational perturbation theory with level shift ─ the Cr2 potential revisited. Chem. Phys. Lett. 1995, 245 (2–3), 215– 223,  DOI: 10.1016/0009-2614(95)01010-7
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    Multiconfigurational perturbation theory with level shift - the Cr2 potential revisited
    Roos, Bjoern O.; Andersson, Kerstin
    Chemical Physics Letters (1995), 245 (2,3), 215-23CODEN: CHPLBC; ISSN:0009-2614. (Elsevier)
    A level shift technique is suggested for removal of intruder states in multiconfigurational second-order perturbation theory (CASPT2). The first-order wavefunction is first calcd. with a level shift parameter large enough to remove the intruder states. The effect of the level shift on the second-order energy is removed by a back correction technique (the LS correction). It is shown that intruder states are removed with little effect on the remaining part of the correlation energy. New potential curves have been computed for the X 1Σ+g and the a' 3Σ+u states of Cr2 using large basis sets (ANO: 8s7p6d4f2g) and accounting for relativistic effects, 3s and 3p correlation, and basis set superposition effects. The computed spectroscopic consts. (exptl. values in parentheses) for the X 1Σ+g state are re = 1.69(1.68) Å, ΔG1/2 = 535(452) cm-1, D0 = 1.54(1.44) eV. The corresponding values for a' 3Σ+u are re = 1.64(1.65) Å, ΔG1/2 = 667(574) cm-1, Te = 1.79(1.76) eV.
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    Wen, X.; Martirez, J. M. P.; Carter, E. A. Plasmon-driven ammonia decomposition on Pd(111): Hole transfer’s role in changing rate-limiting steps. ACS Catal. 2024, 14, 9539– 9553,  DOI: 10.1021/acscatal.4c01869
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    Peterson, K. A.; Figgen, D.; Dolg, M.; Stoll, H. Energy-consistent relativistic pseudopotentials and correlation consistent basis sets for the 4d elements Y–Pd. J. Chem. Phys. 2007, 126 (12), 124101,  DOI: 10.1063/1.2647019
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    Energy-consistent relativistic pseudopotentials and correlation consistent basis sets for the 4d elements Y-Pd
    Peterson, Kirk A.; Figgen, Detlev; Dolg, Michael; Stoll, Hermann
    Journal of Chemical Physics (2007), 126 (12), 124101/1-124101/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    Scalar-relativistic pseudopotentials and corresponding spin-orbit potentials of the energy-consistent variety have been adjusted for the simulation of the [Ar]3d10 cores of the 4d transition metal elements Y-Pd. These potentials have been detd. in a one-step procedure using numerical two-component calcns. so as to reproduce at. valence spectra from four-component all-electron calcns. The latter have been performed at the multi-configuration Dirac-Hartree-Fock level, using the Dirac-Coulomb Hamiltonian and perturbatively including the Breit interaction. The derived pseudopotentials reproduce the all-electron ref. data with an av. accuracy of 0.03 eV for configurational avs. over nonrelativistic orbital configurations and 0.1 eV for individual relativistic states. Basis sets following a correlation consistent prescription have also been developed to accompany the new pseudopotentials. These range in size from cc-pVDZ-PP to cc-pV5Z-PP and also include sets for 4s4p correlation (cc-pwCVDZ-PP through cc-pwCV5Z-PP), as well as those with extra diffuse functions (aug-cc-pVDZ-PP, etc.). In order to accurately assess the impact of the pseudopotential approxn., all-electron basis sets of triple-zeta quality have also been developed using the Douglas-Kroll-Hess Hamiltonian (cc-pVTZ-DK, cc-pwCVTZ-DK, and aug-cc-pVTZ-DK). Benchmark calcns. of at. ionization potentials and 4dm-25s2 → 4dm-15s1 electronic excitation energies are reported at the coupled cluster level of theory with extrapolations to the complete basis set limit.
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    Dunning, T. H., Jr. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. J. Chem. Phys. 1989, 90 (2), 1007– 1023,  DOI: 10.1063/1.456153
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    Journal of Chemical Physics (1989), 90 (2), 1007-23CODEN: JCPSA6; ISSN:0021-9606.
    Guided by the calcns. on oxygen in the literature, basis sets for use in correlated at. and mol. calcns. were developed for all of the first row atoms from boron through neon, and for hydrogen. As in the oxygen atom calcns., the incremental energy lowerings, due to the addn. of correlating functions, fall into distinct groups. This leads to the concept of correlation-consistent basis sets, i.e., sets which include all functions in a given group as well as all functions in any higher groups. Correlation-consistent sets are given for all of the atoms considered. The most accurate sets detd. in this way, [5s4p3d2f1g], consistently yield 99% of the correlation energy obtained with the corresponding at.-natural-orbital sets, even though the latter contains 50% more primitive functions and twice as many primitive polarization functions. It is estd. that this set yields 94-97% of the total (HF + 1 + 2) correlation energy for the atoms neon through boron.
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    Kendall, R. A.; Dunning, T. H., Jr.; Harrison, R. J. Electron affinities of the first-row atoms revisited. Systematic basis sets and wave functions. J. Chem. Phys. 1992, 96 (9), 6796– 6806,  DOI: 10.1063/1.462569
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    Kendall, Rick A.; Dunning, Thom H., Jr.; Harrison, Robert J.
    Journal of Chemical Physics (1992), 96 (9), 6796-806CODEN: JCPSA6; ISSN:0021-9606.
    The authors describe a reliable procedure for calcg. the electron affinity of an atom and present results for H, B, C, O, and F (H is included for completeness). This procedure involves the use of the recently proposed correlation-consistent basis sets augmented with functions to describe the more diffuse character of the at. anion coupled with a straightforward, uniform expansion of the ref. space for multireference singles and doubles configuration-interaction (MRSD-CI) calcns. A comparison is given with previous results and with corresponding full CI calcns. The most accurate EAs obtained from the MRSD-CI calcns. are (with exptl. values in parentheses): H 0.740 eV (0.754), B 0.258 (0.277), C 1.245 (1.263), O 1.384 (1.461), and F 3.337 (3.401). The EAs obtained from the MR-SDCI calcns. differ by less than 0.03 eV from those predicted by the full CI calcns.
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    Peterson, K. A.; Dunning, T. H., Jr. Accurate correlation consistent basis sets for molecular core–valence correlation effects: The second row atoms Al–Ar, and the first row atoms B–Ne revisited. J. Chem. Phys. 2002, 117 (23), 10548– 10560,  DOI: 10.1063/1.1520138
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    Accurate correlation consistent basis sets for molecular core-valence correlation effects: The second row atoms Al-Ar, and the first row atoms B-Ne revisited
    Peterson, Kirk A.; Dunning, Thom H., Jr.
    Journal of Chemical Physics (2002), 117 (23), 10548-10560CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    Correlation consistent basis sets for accurately describing core-core and core-valence correlation effects in atoms and mols. have been developed for the second row atoms Al-Ar. Two different optimization strategies were investigated, which led to two families of core-valence basis sets when the optimized functions were added to the std. correlation consistent basis sets (cc-pVnZ). In the first case, the exponents of the augmenting primitive Gaussian functions were optimized with respect to the difference between all-electron and valence-electron correlated calcns., i.e., for the core-core plus core-valence correlation energy. This yielded the cc-pCVnZ family of basis sets, which are analogous to the sets developed previously for the first row atoms [D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys. 103, 4572 (1995)]. Although the cc-pCVnZ sets exhibit systematic convergence to the all-electron correlation energy at the complete basis set limit, the intershell (core-valence) correlation energy converges more slowly than the intrashell (core-core) correlation energy. Since the effect of including the core electrons on the calcn. of mol. properties tends to be dominated by core-valence correlation effects, a second scheme for detg. the augmenting functions was investigated. In this approach, the exponents of the functions to be added to the cc-pVnZ sets were optimized with respect to just the core-valence (intershell) correlation energy, except that a small amt. of core-core correlation energy was included in order to ensure systematic convergence to the complete basis set limit. These new sets, denoted weighted core-valence basis sets (cc-pwCVnZ), significantly improve the convergence of many mol. properties with n. Optimum cc-pwCVnZ sets for the first-row atoms were also developed and show similar advantages. Both the cc-pCVnZ and cc-pwCVnZ basis sets were benchmarked in coupled cluster [CCSD(T)] calcns. on a series of second row homonuclear diat. mols. (Al2, Si2, P2, S2, and Cl2), as well as on selected diat. mols. involving first row atoms (CO, SiO, PN, and BCl). For the calcn. of core correlation effects on energetic and spectroscopic properties, the cc-pwCVnZ basis sets are recommended over the cc-pCVnZ ones.
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    Pritchard, B. P.; Altarawy, D.; Didier, B.; Gibson, T. D.; Windus, T. L. New Basis Set Exchange: An Open, Up-to-Date Resource for the Molecular Sciences Community. J. Chem. Inf. Model. 2019, 59 (11), 4814– 4820,  DOI: 10.1021/acs.jcim.9b00725
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    New Basis Set Exchange: An Open, Up-to-Date Resource for the Molecular Sciences Community
    Pritchard, Benjamin P.; Altarawy, Doaa; Didier, Brett; Gibson, Tara D.; Windus, Theresa L.
    Journal of Chemical Information and Modeling (2019), 59 (11), 4814-4820CODEN: JCISD8; ISSN:1549-9596. (American Chemical Society)
    A review. The Basis Set Exchange (BSE) has been a prominent fixture in the quantum chem. community. First publicly available in 2007, it is recognized by both users and basis set creators as the de facto source for information related to basis sets. This popular resource has been rewritten, utilizing modern software design and best practices. The basis set data has been sepd. into a stand-alone library with an accessible API, and the Web site has been updated to use the current generation of web development libraries. The general layout and workflow of the Web site is preserved, while helpful features requested by the user community have been added. Overall, this design should increase adaptability and lend itself well into the future as a dependable resource for the computational chem. community. This article will discuss the decision to rewrite the BSE, the new architecture and design, and the new features that have been added.
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    Hill, J. G.; Peterson, K. A. Gaussian basis sets for use in correlated molecular calculations. XI. Pseudopotential-based and all-electron relativistic basis sets for alkali metal (K–Fr) and alkaline earth (Ca–Ra) elements. J. Chem. Phys. 2017, 147 (24), 244106,  DOI: 10.1063/1.5010587
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    Gaussian basis sets for use in correlated molecular calculations. XI. Pseudopotential-based and all-electron relativistic basis sets for alkali metal (K-Fr) and alkaline earth (Ca-Ra) elements
    Hill, J. Grant; Peterson, Kirk A.
    Journal of Chemical Physics (2017), 147 (24), 244106/1-244106/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    New correlation consistent basis sets based on pseudopotential (PP) Hamiltonians have been developed from double- to quintuple-zeta quality for the late alkali (K-Fr) and alk. earth (Ca-Ra) metals. These are accompanied by new all-electron basis sets of double- to quadruple-zeta quality that have been contracted for use with both Douglas-Kroll-Hess (DKH) and eXact 2-Component (X2C) scalar relativistic Hamiltonians. Sets for valence correlation (ms), cc-pVnZ-PP and cc-pVnZ-(DK,DK3/X2C), in addn. to outer-core correlation [valence + (m-1)sp], cc-p(w)CVnZ-PP and cc-pwCVnZ-(DK,DK3/X2C), are reported. The -PP sets have been developed for use with small-core PPs [I. S. Lim et al., J. Chem. Phys. 122, 104103 (2005) and I. S. Lim et al., J. Chem. Phys. 124, 034107 (2006)], while the all-electron sets utilized second-order DKH Hamiltonians for 4s and 5s elements and third-order DKH for 6s and 7s. The accuracy of the basis sets is assessed through benchmark calcns. at the coupled-cluster level of theory for both at. and mol. properties. Not surprisingly, it is found that outer-core correlation is vital for accurate calcn. of the thermodn. and spectroscopic properties of diat. mols. contg. these elements. (c) 2017 American Institute of Physics.
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    Balabanov, N. B.; Peterson, K. A. Systematically convergent basis sets for transition metals. I. All-electron correlation consistent basis sets for the 3d elements Sc–Zn. J. Chem. Phys. 2005, 123 (6), 064107,  DOI: 10.1063/1.1998907
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    Systematically convergent basis sets for transition metals. I. All-electron correlation consistent basis sets for the 3d elements Sc-Zn
    Balabanov, Nikolai B.; Peterson, Kirk A.
    Journal of Chemical Physics (2005), 123 (6), 064107/1-064107/15CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    Sequences of basis sets that systematically converge towards the complete basis set (CBS) limit have been developed for the first-row transition metal elements Sc-Zn. Two families of basis sets, nonrelativistic and Douglas-Kroll-Hess (-DK) relativistic, are presented that range in quality from triple-ζ to quintuple-ζ. Sep. sets are developed for the description of valence (3d4s) electron correlation (cc-pVnZ and cc-pVnZ-DK; n=T,Q, 5) and valence plus outer-core (3s3p3d4s) correlation (cc-pwCVnZ and cc-pwCVnZ-DK; n=T,Q, 5), as well as these sets augmented by addnl. diffuse functions for the description of neg. ions and weak interactions (aug-cc-pVnZ and aug-cc-pVnZ-DK). Extensive benchmark calcns. at the coupled cluster level of theory are presented for at. excitation energies, ionization potentials, and electron affinities, as well as mol. calcns. on selected hydrides (TiH, MnH, CuH) and other diatomics (TiF, Cu2). In addn. to observing systematic convergence towards the CBS limits, both 3s3p electron correlation and scalar relativity are calcd. to strongly impact many of the at. and mol. properties investigated for these first-row transition metal species. A (A=Sc-Zn).
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    Revision of the Douglas-Kroll transformation
    Jansen; Hess
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    Douglas, M.; Kroll, N. M. Quantum electrodynamical corrections to the fine structure of helium. Ann. Phys. 1974, 82 (1), 89– 155,  DOI: 10.1016/0003-4916(74)90333-9
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    Quantum electrodynamical corrections to the fine structure of helium
    Douglas, Marvin; Kroll, Norman M.
    Annals of Physics (San Diego, CA, United States) (1974), 82 (1), 89-155CODEN: APNYA6; ISSN:0003-4916.
    Corrections of order α6mc2 (α = fine structure const., mc2 = the electron rest energy) to the fine-structure splitting of the deepest lying triplet P state (23P0,1,2) of the 4He atom were investigated. The investigation is based on the covariant Bethe-Salpeter equation including an external potential to take account of the nuclear Coulomb field. All order α6mc2 corrections that arise from Feynman diagrams involving the exchange of 1, 2, and 3 photons, as well as radiative corrections to the electron magnetic moment were found. The results are presented in a form suitable for computerized numerical evaluation.
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    Silva-Junior, M. R.; Schreiber, M.; Sauer, S. P. A.; Thiel, W. Benchmarks of electronically excited states: Basis set effects on CASPT2 results. J. Chem. Phys. 2010, 133 (17), 174318,  DOI: 10.1063/1.3499598
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    Benchmarks of electronically excited states: Basis set effects on CASPT2 results
    Silva-Junior, Mario R.; Schreiber, Marko; Sauer, Stephan P. A.; Thiel, Walter
    Journal of Chemical Physics (2010), 133 (17), 174318/1-174318/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    Vertical excitation energies and one-electron properties are computed for the valence excited states of 28 medium-sized org. benchmark mols. using multistate multiconfigurational second-order perturbation theory (MS-CASPT2) and the augmented correlation-consistent aug-cc-pVTZ basis set. They are compared with previously reported MS-CASPT2 results obtained with the smaller TZVP basis. The basis set extension from TZVP to aug-cc-pVTZ causes rather minor and systematic shifts in the vertical excitation energies that are normally slightly reduced (on av. by 0.11 eV for the singlets and by 0.09 eV for the triplets), whereas the changes in the calcd. oscillator strengths and dipole moments are somewhat more pronounced on a relative scale. These basis set effects at the MS-CASPT2 level are qual. and quant. similar to those found at the coupled cluster level for the same set of benchmark mols. The previously proposed theor. best ests. (TBE-1) for the vertical excitation energies for 104 singlet and 63 triplet excited states of the benchmark mols. are upgraded by replacing TZVP with aug-cc-pVTZ data that yields a new ref. set (TBE-2). Statistical evaluations of the performance of d. functional theory (DFT) and semiempirical methods lead to the same ranking and very similar quant. results for TBE-1 and TBE-2, with slightly better performance measures with respect to TBE-2. DFT/MRCI is most accurate among the investigated DFT-based approaches, while the OMx methods with orthogonalization corrections perform best at the semiempirical level. (c) 2010 American Institute of Physics.
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    Kánnár, D.; Tajti, A.; Szalay, P. G. Accuracy of Coupled Cluster Excitation Energies in Diffuse Basis Sets. J. Chem. Theory Comput. 2017, 13 (1), 202– 209,  DOI: 10.1021/acs.jctc.6b00875
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    Jansen, H. B.; Ros, P. Non-empirical molecular orbital calculations on the protonation of carbon monoxide. Chem. Phys. Lett. 1969, 3 (3), 140– 143,  DOI: 10.1016/0009-2614(69)80118-1
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    Jansen, H. B.; Ros, P.
    Chemical Physics Letters (1969), 3 (3), 140-3CODEN: CHPLBC; ISSN:0009-2614.
    Single configurational L.C.A.O. Hartree-Fock M.O.-self-consistent field calcns. with gaussian functions were carried out for several configurations of protonated C monoxide in order to gain some insights about the geometry and stability of protonated CO. The most stable configuration is a linear [HCO]+ structure. The C-O bond distance is 0.02 A. smaller than in CO itself. The interaction of the proton with the π electrons of CO stabilizes the structure. The energy of protonation is 152 kcal./mole and agrees reasonably well with a value of 133 kcal./mole which was calcd. from thermochem. data for CO and H+.
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    Accurate calculation of the attractive interaction of two ground state helium atoms
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    Journal of Chemical Physics (1973), 59 (8), 4557-8CODEN: JCPSA6; ISSN:0021-9606.
    A configuration interaction (CI) calcn. is given of the van der Waals interaction between 2 ground state He atoms. The CI calcn. converges on the exact clamped nuclei result within an accuracy of ∼0.1°K = 3 × 10-7 at. units for the interaction energy. The results agree with accurate perturbation theory results to within 0.02°K.
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    Gutowski, M.; Van Lenthe, J. H.; Verbeek, J.; Van Duijneveldt, F. B.; Chałasinski, G. The basis set superposition error in correlated electronic structure calculations. Chem. Phys. Lett. 1986, 124 (4), 370– 375,  DOI: 10.1016/0009-2614(86)85036-9
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    Simon, S.; Duran, M.; Dannenberg, J. J. How does basis set superposition error change the potential surfaces for hydrogen-bonded dimers?. J. Chem. Phys. 1996, 105 (24), 11024– 11031,  DOI: 10.1063/1.472902
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    How does basis set superposition error change the potential surfaces for hydrogen-bonded dimers?
    Simon, Silvia; Duran, Miquel; Dannenberg, J. J.
    Journal of Chemical Physics (1996), 105 (24), 11024-11031CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    We describe a simple method to automate the geometric optimization of MO calcns. of supermols. on potential surfaces that are cor. for basis set superposition error using the counterpoise (CP) method. This method is applied to the H-bonding complexes HF/HCN, HF/H2O, and HCCH/H2O using the 6-31G(d,p) and D95++(d,p) basis sets at both the Hartree-Fock and second-order Moeller-Plesset levels. We report the interaction energies, geometries, and vibrational frequencies of these complexes on the CP-optimized surfaces; and compare them with similar values calcd. using traditional methods, including the (more traditional) single point CP correction. Upon optimization on the CP-cor. surface, the interaction energies become more neg. (before vibrational corrections) and the H-bonding stretching vibrations decrease in all cases. The extent of the effect vary from extremely small to quite large depending on the complex and the calculational method. The relative magnitudes of the vibrational corrections cannot be predicted from the H-bond stretching frequencies alone.
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    Boys, S. F.; Bernardi, F. The calculation of small molecular interactions by the differences of separate total energies. Some procedures with reduced errors. Mol. Phys. 1970, 19 (4), 553– 566,  DOI: 10.1080/00268977000101561
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    The calculation of small molecular interactions by the differences of separate total energies. Some procedures with reduced errors
    Boys, S. F.; Bernardi, F.
    Molecular Physics (1970), 19 (4), 553-566CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)
    A new direct difference method for the computation of mol. interactions has been based on a bivariational transcorrelated treatment, together with special methods for the balancing of other errors. It appears that these new features can give a strong redn. in the error of the interaction energy, and they seem to be particularly suitable for computations in the important region near the min. energy. It has been generally accepted that this problem is dominated by unresolved difficulties and the relation of the new methods of these apparent difficulties is analyzed here.
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    Gray, M.; Bowling, P. E.; Herbert, J. M. Systematic Evaluation of Counterpoise Correction in Density Functional Theory. J. Chem. Theory Comput. 2022, 18 (11), 6742– 6756,  DOI: 10.1021/acs.jctc.2c00883
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    Systematic Evaluation of Counterpoise Correction in Density Functional Theory
    Gray, Montgomery; Bowling, Paige E.; Herbert, John M.
    Journal of Chemical Theory and Computation (2022), 18 (11), 6742-6756CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    A widespread belief persists that the Boys-Bernardi function counterpoise (CP) procedure "overcorrects" supramol. interaction energies for the effects of basis-set superposition error. To the extent that this is true for correlated wave function methods, it is usually an artifact of low-quality basis sets. The question has not been considered systematically in the context of d. functional theory, however, where basis-set convergence is generally less problematic. We present a systematic assessment of the CP procedure for a representative set of functionals and basis sets, considering both benchmark data sets of small dimers and larger supramol. complexes. The latter include layered composite polymers with ~ 150 atoms and ligand-protein models with ~ 300 atoms. Provided that CP correction is used, we find that intermol. interaction energies of nearly complete-basis quality can be obtained using only double-ζ basis sets. This is less expensive as compared to triple-ζ basis sets without CP correction. CP-cor. interaction energies are less sensitive to the presence of diffuse basis functions as compared to uncorrected energies, which is important because diffuse functions are expensive and often numerically problematic for large systems. Our results upend the conventional wisdom that CP "overcorrects" for basis-set incompleteness. In small basis sets, CP correction is mandatory in order to demonstrate that the results do not rest on error cancellation.
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    Mentel, Ł. M.; Baerends, E. J. Can the Counterpoise Correction for Basis Set Superposition Effect Be Justified?. J. Chem. Theory Comput. 2014, 10 (1), 252– 267,  DOI: 10.1021/ct400990u
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    Can the Counterpoise Correction for Basis Set Superposition Effect Be Justified?
    Mentel, L. M.; Baerends, E. J.
    Journal of Chemical Theory and Computation (2014), 10 (1), 252-267CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    The basis set superposition effect (BSSE) is a simple concept, and its validity is almost universally accepted. So is the counterpoise method to correct for it. The idea is that the basis set is biased toward the dimer because each monomer in the dimer can "use" the basis functions on the other monomer, which it cannot in a simple monomer calcn. This hypothesis can only be tested if basis set free benchmark nos. are available for monomers and dimer. We are testing the hypothesis on a few systems (in this paper Be2) that are small enough that sufficiently accurate benchmark nos. (basis set free, or close to basis set limit; full CI or close to full CI) are available or can be obtained. We find that the answer to the title question is neg.: the std. basis sets of quantum chem. appear to be biased toward the atom in the sense that basis set errors are larger for the dimer than the monomer. Applying the counterpoise correction increases the imbalance by reducing the already smaller basis set error of the monomer even further. Counterpoise cor. bond energies then deviate more from the basis set limit nos. than uncorrected bond energies. These conclusions hold both at the Hartree-Fock level and (much stronger) at the correlated (CCSD-(T), full CI) levels. So the answer to the title question is No.
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    van Duijneveldt, F. B.; van Duijneveldt-van de Rijdt, J. G. C. M.; van Lenthe, J. H. State of the Art in Counterpoise Theory. Chem. Rev. 1994, 94 (7), 1873– 1885,  DOI: 10.1021/cr00031a007
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    State of the Art in Counterpoise Theory
    van Duijneveldt, Frans B.; van Duijneveldt-van de Rijdt, Jeanne G. C. M.; van Lenthe, Joop H.
    Chemical Reviews (Washington, DC, United States) (1994), 94 (7), 1873-85CODEN: CHREAY; ISSN:0009-2665.
    A review with 95 refs.
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    Burns, L. A.; Marshall, M. S.; Sherrill, C. D. Comparing Counterpoise-Corrected, Uncorrected, and Averaged Binding Energies for Benchmarking Noncovalent Interactions. J. Chem. Theory Comput. 2014, 10 (1), 49– 57,  DOI: 10.1021/ct400149j
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    Comparing Counterpoise-Corrected, Uncorrected, and Averaged Binding Energies for Benchmarking Noncovalent Interactions
    Burns, Lori A.; Marshall, Michael S.; Sherrill, C. David
    Journal of Chemical Theory and Computation (2014), 10 (1), 49-57CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    High-quality benchmark computations are crit. for the development and assessment of approx. methods to describe noncovalent interactions. Recent advances in the treatment of dispersion by d. functional theory and also the development of more efficient wave function techniques to reliably address noncovalent interactions motivate new benchmark computations of increasing accuracy. This work considers focal point approxns. to est. the complete basis set limit of coupled-cluster theory through perturbative triples [CCSD-(T)/CBS] and evaluates how this approach is affected by the use or absence of counterpoise (CP) correction or, as has recently gained traction, the av. of those values. Current benchmark protocols for interaction energies are computed with all CP procedures and assessed against the A24 and S22B databases and also to highly converged results for formic acid, cyanogen, and benzene dimers. Whether CP correction, no correction, or the av. is favored depends upon the theor. method, basis set, and binding motif. In recent high-quality benchmark studies, interaction energies often use second-order perturbation theory with extrapolated aug-cc-pVTZ (aTZ) and aug-cc-pVQZ (aQZ) basis sets [MP2/aTQZ] combined with a "coupled-cluster correction," δMP2CCSD(T), evaluated in an aug-cc-pVDZ basis. For such an approach, averaging CP-cor. and uncorrected values for the MP2 component and using CP-cor. δMP2CCSD(T) values offers errors more balanced among binding motifs and generally more favorable overall. Other combinations of counterpoise correction are not quite as accurate. When employing MP2/aQ5Z extrapolations and an aTZ basis for δMP2CCSD(T), using CP-cor. or averaged MP2 ests. are about equally effective (and slightly superior to uncorrected MP2 values), but the counterpoise treatment of δMP2CCSD(T) makes little difference. Focal point ests. at this level achieve benchmark quality results otherwise accessible only with CCSD-(T)/aQZ or better.
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    Helgaker, T.; Klopper, W.; Koch, H.; Noga, J. Basis-set convergence of correlated calculations on water. J. Chem. Phys. 1997, 106 (23), 9639– 9646,  DOI: 10.1063/1.473863
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    Basis-set convergence of correlated calculations on water
    Helgaker, Trygve; Klopper, Wim; Koch, Henrik; Noga, Jozef
    Journal of Chemical Physics (1997), 106 (23), 9639-9646CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    The basis-set convergence of the electronic correlation energy in the water mol. is investigated at the second-order Moller-Plesset level and at the coupled-cluster singles-and-doubles level with and without perturbative triples corrections applied. The basis-set limits of the correlation energy are established to within 2mEh by means of (1) extrapolations from sequences of calcns. using correlation-consistent basis sets and (2) from explicitly correlated calcns. employing terms linear in the inter-electronic distances rij. For the extrapolations to the basis-set limit of the correlation energies, fits of the form a + bX-3 (where X is two for double-zeta sets, three for triple-zeta sets, etc.) are found to be useful. CCSD(T) calcns. involving as many as 492 AOs are reported.
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    Wei, Z.; Martirez, J. M. P.; Carter, E. A. Introducing the embedded random phase approximation: H2 dissociative adsorption on Cu(111) as an exemplar. J. Chem. Phys. 2023, 159 (19), 194108,  DOI: 10.1063/5.0181229
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    Zhou, L.; Swearer, D. F.; Zhang, C.; Robatjazi, H.; Zhao, H.; Henderson, L.; Dong, L.; Christopher, P.; Carter, E. A.; Nordlander, P. Quantifying hot carrier and thermal contributions in plasmonic photocatalysis. Science 2018, 362 (6410), 69– 72,  DOI: 10.1126/science.aat6967
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    Quantifying hot carrier and thermal contributions in plasmonic photocatalysis
    Zhou, Linan; Swearer, Dayne F.; Zhang, Chao; Robatjazi, Hossein; Zhao, Hangqi; Henderson, Luke; Dong, Liangliang; Christopher, Phillip; Carter, Emily A.; Nordlander, Peter; Halas, Naomi J.
    Science (Washington, DC, United States) (2018), 362 (6410), 69-72CODEN: SCIEAS; ISSN:0036-8075. (American Association for the Advancement of Science)
    Photocatalysis based on optically active, "plasmonic" metal nanoparticles has emerged as a promising approach to facilitate light-driven chem. conversions under far milder conditions than thermal catalysis. However, an understanding of the relation between thermal and electronic excitations has been lacking. We report the substantial light-induced redn. of the thermal activation barrier for ammonia decompn. on a plasmonic photocatalyst. We introduce the concept of a light-dependent activation barrier to account for the effect of light illumination on electronic and thermal excitations in a single unified picture. This framework provides insight into the specific role of hot carriers in plasmon-mediated photochem., which is critically important for designing energy-efficient plasmonic photocatalysts.
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    Finley, J.; Malmqvist, P.-Å.; Roos, B. O.; Serrano-Andrés, L. The multi-state CASPT2 method. Chem. Phys. Lett. 1998, 288 (2–4), 299– 306,  DOI: 10.1016/S0009-2614(98)00252-8
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    80
    The multi-state CASPT2 method
    Finley, James; Malmqvist, Per-Ake; Roos, Bjorn O.; Serrano-Andres, Luis
    Chemical Physics Letters (1998), 288 (2,3,4), 299-306CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)
    An extension of the multiconfigurational second-order perturbation approach CASPT2 is suggested, where several electronic states are coupled at second order via an effective-Hamiltonian approach. The method has been implemented into the MOLCAS-4 program system, where it will replace the single-state CASPT2 program. The accuracy of the method is illustrated through calcns. of the ionic-neutral avoided crossing in the potential curves for LiF and of the valence-Rydberg mixing in the V-state of the ethylene mol.
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  • Abstract

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    Figure 1

    Figure 1. (a) Four-layer 5 × 5 Pd(111) (Pd100) periodic slab model. Top-down view of the (b) Pd10 cluster, (c) Pd12 cluster, and (d) Pd14 cluster carved out from the Pd100 periodic slab. Subsurface atoms are faded out. Isosurface plots (yellow: +1.2 V, cyan: −1.2 V) of the optimized embedding potentials generated for the (e) Pd10 cluster in its Pd90 environment, (f) Pd12 cluster in its Pd88 environment, and (g) Pd14 cluster in its Pd86 environment.

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    Figure 2

    Figure 2. Effect of cluster size for NH3(gas) → *NH3 → *NH2 + *H. We calculate the emb-LDA energies (defined in eq 4) for Pd10, Pd12, and Pd14 clusters. Numerical indices correspond to structures shown in Figure S1.

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    Figure 3

    Figure 3. Basis set effect for NH3(gas) → *NH3 → *NH2 + *H at the emb-CASPT2 level using an embedded Pd10 cluster. Basis sets for Pd and adsorbates are listed as Pd basis/adsorbate basis in the legend. DFT-corrected Einit,expt is the DFT-corrected electronic energy based on the experimental enthalpy of NH3 adsorption on Pd powder. See detailed explanation in the text.

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    Figure 4

    Figure 4. CP corrections for NH3(gas) → *NH3 → *NH2 + *H on embedded Pd10. Combined CP corrections of the adsorbates and the Pd10 cluster are plotted for emb-CASPT2 with various basis sets (denoted as Pd basis/NH3 basis in the legends) and DFT-PBE with the AVDZ basis set.

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    Figure 5

    Figure 5. Emb-CASPT2 results using creeping to obtain CASSCF orbitals for (a) NH3(gas) → *NH3 → *NH2 + *H, (b) *NH2 + *H → *NH+ 2*H, and (c) *NH → *N+ *H. The corresponding NOs in the ASs using creeping are given in Figures S7–S9, while the ones from merging are shown in Figures S10–S12 in the Supporting Information. All calculations utilize the Pd10 cluster model.

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    Figure 6

    Figure 6. Ground- and excited-state MEP energetics as calculated with emb-CASPT2 for NH3 (gas) → *NH3 → * NH2 + *H. Red arrows represent vertical excitation of S0 → SN-1, where SN-1 represents the highest excited state. Green arrows represent reaction barriers for S0 or SN–1. Each panel represents different numbers of states included in the N-state emb-SA-CASSCF calculations: N = (a) 5, (b) 7, (c) 9, (d) 11, (e) 15, and (f) 17.

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    This article references 80 other publications.

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      Roos, B. O. The Complete Active Space Self-Consistent Field Method and its Applications in Electronic Structure Calculations. Adv. Chem. Phys. 1987, 69, 399– 445,  DOI: 10.1002/9780470142943.ch7
      1
      The complete active space self-consistent field method and its applications in electronic structure calculations
      Roos, Bjoern
      Advances in Chemical Physics (1987), 69 (Ab Initio Methods Quantum Chem.--2), 399-445CODEN: ADCPAA; ISSN:0065-2385.
      A review with 121 refs.
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      Bofill, J. M.; Pulay, P. The unrestricted natural orbital–complete active space (UNO–CAS) method: An inexpensive alternative to the complete active space–self-consistent-field (CAS–SCF) method. J. Chem. Phys. 1989, 90 (7), 3637– 3646,  DOI: 10.1063/1.455822
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      Veryazov, V.; Malmqvist, P. Å.; Roos, B. O. How to select active space for multiconfigurational quantum chemistry?. Int. J. Quantum Chem. 2011, 111 (13), 3329– 3338,  DOI: 10.1002/qua.23068
      3
      How to select active space for multiconfigurational quantum chemistry?
      Veryazov, Valera; Malmqvist, Per Aake; Roos, Bjoern O.
      International Journal of Quantum Chemistry (2011), 111 (13), 3329-3338CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)
      A review. Bjoern Roos is one of the pioneers in the development and usage of multiconfigurational methods, in particular, the complete active space SCF method and the perturbational complete active space perturbation theory through second order. To perform multiconfigurational calcns. using these methods, a set of active orbitals must be selected, and the success of the methods depends on the choice of this set. This is not only sometimes easy but also sometimes difficult, esp. for use of the more recent RASSCF and RASPT2 methods (which use a "restricted active space" rather than the complete one). Although an automated procedure for selecting the active orbitals would be a preferable soln., this does not seem feasible yet. An account of the problem is given, with examples and some approaches that usually work. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem 111:3329-3338, 2011.
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      Jensen, H. J. A.; Jørgensen, P.; Ågren, H.; Olsen, J. Second-order Møller–Plesset perturbation theory as a configuration and orbital generator in multiconfiguration self-consistent field calculations. J. Chem. Phys. 1988, 88 (6), 3834– 3839,  DOI: 10.1063/1.453884
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      Wouters, S.; Bogaerts, T.; Van Der Voort, P.; Van Speybroeck, V.; Van Neck, D. Communication: DMRG-SCF study of the singlet, triplet, and quintet states of oxo-Mn(Salen). J. Chem. Phys. 2014, 140 (24), 241103,  DOI: 10.1063/1.4885815
      5
      Communication: DMRG-SCF study of the singlet, triplet, and quintet states of oxo-Mn(Salen)
      Wouters, Sebastian; Bogaerts, Thomas; Van Der Voort, Pascal; Van Speybroeck, Veronique; Van Neck, Dimitri
      Journal of Chemical Physics (2014), 140 (24), 241103/1-241103/5CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      We use CHEMPS2, our free open-source spin-adapted implementation of the d. matrix renormalization group (DMRG) [S. Wouters, W. Poelmans, P. W. Ayers, and D. Van Neck, Comput. Phys. Commun.185, 1501 (2014)], to study the lowest singlet, triplet, and quintet states of the oxo-Mn(Salen) complex. We describe how an initial approx. DMRG calcn. in a large active space around the Fermi level can be used to obtain a good set of starting orbitals for subsequent complete-active-space or DMRG SCF calcns. This procedure mitigates the need for a localization procedure, followed by a manual selection of the active space. Per multiplicity, the same active space of 28 electrons in 22 orbitals (28e, 22o) is obtained with the 6-31G*, cc-pVDZ, and ANO-RCC-VDZP basis sets (the latter with DKH2 scalar relativistic corrections). Our calcns. provide new insight into the electronic structure of the quintet. (c) 2014 American Institute of Physics.
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    6. 6
      Khedkar, A.; Roemelt, M. Active Space Selection Based on Natural Orbital Occupation Numbers from n-Electron Valence Perturbation Theory. J. Chem. Theory Comput. 2019, 15 (6), 3522– 3536,  DOI: 10.1021/acs.jctc.8b01293
      6
      Active Space Selection Based on Natural Orbital Occupation Numbers from n-Electron Valence Perturbation Theory
      Khedkar, Abhishek; Roemelt, Michael
      Journal of Chemical Theory and Computation (2019), 15 (6), 3522-3536CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      Efficient and robust approxns. to the full CI (full-CI) method such as the d. matrix renormalization group (DMRG) and the full-CI quantum Monte Carlo (FCIQMC) algorithm allow for MC-SCF calcns. of mols. with many strongly correlated electrons. This opens up the possibility to treat large and complex systems that were previously untractable, but at the same time it calls for an efficient and reliable active space selection as the choice of how many electrons and orbitals enter the active space is crit. for any multireference calcn. In this work we propose an Active Space Selection based on 1st order perturbation theory (ASS1ST) that follows a "bottom-up" strategy and utilizes a set of quasi-natural orbitals together with sensible thresholds for their occupation nos. The required quasi-natural orbitals are generated by diagonalizing the virtual and internal part of the one-electron reduced d. matrix that is obtained from strongly contracted n-electron valence perturbation theory (SC-NEVPT) on top of a minimal active space calcn. Self-consistent results can be obtained when the proposed selection scheme is applied iteratively. Initial applications on four chem. relevant benchmark systems indicate the capabilities of ASS1ST. Eventually, the strengths and limitations are critically discussed.
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      Sayfutyarova, E. R.; Sun, Q.; Chan, G. K.-L.; Knizia, G. Automated Construction of Molecular Active Spaces from Atomic Valence Orbitals. J. Chem. Theory Comput. 2017, 13 (9), 4063– 4078,  DOI: 10.1021/acs.jctc.7b00128
      7
      Automated Construction of Molecular Active Spaces from Atomic Valence Orbitals
      Sayfutyarova, Elvira R.; Sun, Qiming; Chan, Garnet Kin-Lic; Knizia, Gerald
      Journal of Chemical Theory and Computation (2017), 13 (9), 4063-4078CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      We introduce the at. valence active space (AVAS), a simple and well-defined automated technique for constructing active orbital spaces for use in multi-configuration and multi-ref. (MR) electronic structure calcns. Concretely, the technique constructs active MOs capable of describing all relevant electronic configurations emerging from a targeted set of at. valence orbitals (e.g., the metal d orbitals in a redcoordination complex). This is achieved via a linear transformation of the occupied and unoccupied orbital spaces from an easily obtainable single-ref. wavefunction (such as from a Hartree-Fock or Kohn-Sham calcns.) based on projectors to targeted at. valence orbitals. We discuss the premises, theory, and implementation of the idea, and several of its variations are tested. To investigate the performance and accuracy, we calc. the excitation energies for various transition metal complexes in typical application scenarios. Addnl., we follow the homolytic bond breaking process of a Fenton reaction along its reaction coordinate. While the described AVAS technique is not an universal soln. to the active space problem, its premises are fulfilled in many application scenarios of transition metal chem. and bond dissocn. processes. In these cases the technique makes MR calcns. easier to execute, easier to reproduce by any user, and simplifies the detn. of the appropriate size of the active space required for accurate results.
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    8. 8
      Stein, C. J.; Reiher, M. autoCAS: A Program for Fully Automated Multiconfigurational Calculations. J. Comput. Chem. 2019, 40, 2216– 2226,  DOI: 10.1002/jcc.25869
      8
      AUTOCAS: A Program for Fully Automated Multiconfigurational Calculations
      Stein, Christopher J.; Reiher, Markus
      Journal of Computational Chemistry (2019), 40 (25), 2216-2226CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)
      We present our implementation AUTOCAS for fully automated multiconfigurational calcns., which we also make available free of charge on our webpages. The graphical user interface of AUTOCAS connects a general electronic structure program with a d.-matrix renormalization group program to carry out our recently introduced automated active space selection protocol for multiconfigurational calcns. (Stein and Reiher, J. Chem. Theory Comput., 2016, 12, 1760). Next to this active space selection, AUTOCAS carries out several steps of multiconfigurational calcns. so that only a minimal input is required to start them, comparable to that of a std. Kohn-Sham d.-functional theory calcn., so that black-box multiconfigurational calcns. become feasible. Furthermore, we introduce a new extension to the selection algorithm that facilitates automated selections for mols. with large valence orbital spaces consisting of several hundred orbitals. © 2019 Wiley Periodicals, Inc.
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    9. 9
      Jeong, W.; Stoneburner, S. J.; King, D.; Li, R.; Walker, A.; Lindh, R.; Gagliardi, L. Automation of Active Space Selection for Multireference Methods via Machine Learning on Chemical Bond Dissociation. J. Chem. Theory Comput. 2020, 16 (4), 2389– 2399,  DOI: 10.1021/acs.jctc.9b01297
      There is no corresponding record for this reference.
    10. 10
      Golub, P.; Antalik, A.; Veis, L.; Brabec, J. Machine Learning-Assisted Selection of Active Spaces for Strongly Correlated Transition Metal Systems. J. Chem. Theory Comput. 2021, 17 (10), 6053– 6072,  DOI: 10.1021/acs.jctc.1c00235
      There is no corresponding record for this reference.
    11. 11
      Han, R.; Luber, S. Complete active space analysis of a reaction pathway: Investigation of the oxygen–oxygen bond formation. J. Comput. Chem. 2020, 41 (17), 1586– 1597,  DOI: 10.1002/jcc.26201
      There is no corresponding record for this reference.
    12. 12
      Stein, C. J.; Reiher, M. Automated Identification of Relevant Frontier Orbitals for Chemical Compounds and Processes. Chimia 2017, 71 (4), 170,  DOI: 10.2533/chimia.2017.170
      12
      Automated identification of relevant frontier orbitals for chemical compounds and processes
      Stein, Christopher J.; Reiher, Markus
      Chimia (2017), 71 (4), 170-176CODEN: CHIMAD ISSN:. (Swiss Chemical Society)
      Quantum-chem. multi-configurational methods are required for a proper description of static electron correlation, a phenomenon inherent to the electronic structure of mols. with multiple (near-)degenerate frontier orbitals. Here, we review how a property of these frontier orbitals, namely the entanglement entropy is related to static electron correlation. A subset of orbitals, the so-called active orbital space is an essential ingredient for all multi-configurational methods. We proposed an automated selection of this active orbital space, that would otherwise be a tedious and error prone manual procedure, based on entanglement measures. Here, we extend this scheme to demonstrate its capability for the selection of consistent active spaces for several excited states and along reaction coordinates.
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    13. 13
      Libisch, F.; Cheng, J.; Carter, E. A. Electron-Transfer-Induced Dissociation of H2 on Gold Nanoparticles: Excited-State Potential Energy Surfaces via Embedded Correlated Wavefunction Theory. Z. Phys. Chem. 2013, 227 (11), 1455– 1466,  DOI: 10.1524/zpch.2013.0406
      There is no corresponding record for this reference.
    14. 14
      Libisch, F.; Krauter, C. M.; Carter, E. A. Corrigendum to: Plasmon-Driven Dissociation of H2 on Gold Nanoclusters. Z. Phys. Chem. 2016, 230 (1), 131– 132,  DOI: 10.1515/zpch-2015-5001
      There is no corresponding record for this reference.
    15. 15
      Mukherjee, S.; Libisch, F.; Large, N.; Neumann, O.; Brown, L. V.; Cheng, J.; Lassiter, J. B.; Carter, E. A.; Nordlander, P.; Halas, N. J. Hot Electrons Do the Impossible: Plasmon-Induced Dissociation of H2 on Au. Nano Lett. 2013, 13 (1), 240– 247,  DOI: 10.1021/nl303940z
      15
      Hot Electrons Do the Impossible: Plasmon-Induced Dissociation of H2 on Au
      Mukherjee, Shaunak; Libisch, Florian; Large, Nicolas; Neumann, Oara; Brown, Lisa V.; Cheng, Jin; Lassiter, J. Britt; Carter, Emily A.; Nordlander, Peter; Halas, Naomi J.
      Nano Letters (2013), 13 (1), 240-247CODEN: NALEFD; ISSN:1530-6984. (American Chemical Society)
      Heterogeneous catalysis is of paramount importance in chem. and energy applications. Catalysts that couple light energy into chem. reactions in a directed, orbital-specific manner would greatly reduce the energy input requirements of chem. transformations, revolutionizing catalysis-driven chem. Here we report the room temp. dissocn. of H2 on gold nanoparticles using visible light. Surface plasmons excited in the Au nanoparticle decay into hot electrons with energies between the vacuum level and the work function of the metal. In this transient state, hot electrons can transfer into a Feshbach resonance of an H2 mol. adsorbed on the Au nanoparticle surface, triggering dissocn. We probe this process by detecting the formation of HD mols. from the dissocns. of H2 and D2 and investigate the effect of Au nanoparticle size and wavelength of incident light on the rate of HD formation. This work opens a new pathway for controlling chem. reactions on metallic catalysts.
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    16. 16
      Martirez, J. M. P.; Carter, E. A. Excited-State N2 Dissociation Pathway on Fe-Functionalized Au. J. Am. Chem. Soc. 2017, 139 (12), 4390– 4398,  DOI: 10.1021/jacs.6b12301
      There is no corresponding record for this reference.
    17. 17
      Martirez, J. M. P.; Carter, E. A. Prediction of a low-temperature N2 dissociation catalyst exploiting near-IR–to–visible light nanoplasmonics. Sci. Adv. 2017, 3 (12), eaao4710  DOI: 10.1126/sciadv.aao4710
      There is no corresponding record for this reference.
    18. 18
      Yuan, Y.; Zhou, L.; Robatjazi, H.; Bao, J. L.; Zhou, J.; Bayles, A.; Yuan, L.; Lou, M.; Lou, M.; Khatiwada, S. Earth-abundant photocatalyst for H2 generation from NH3 with light-emitting diode illumination. Science 2022, 378 (6622), 889– 893,  DOI: 10.1126/science.abn5636
      18
      Earth-abundant photocatalyst for H2 generation from NH3 with light-emitting diode illumination
      Yuan, Yigao; Zhou, Linan; Robatjazi, Hossein; Bao, Junwei Lucas; Zhou, Jingyi; Bayles, Aaron; Yuan, Lin; Lou, Minghe; Lou, Minhan; Khatiwada, Suman; Carter, Emily A.; Nordlander, Peter; Halas, Naomi J.
      Science (Washington, DC, United States) (2022), 378 (6622), 889-893CODEN: SCIEAS; ISSN:1095-9203. (American Association for the Advancement of Science)
      Catalysts based on platinum group metals have been a major focus of the chem. industry for decades. We show that plasmonic photocatalysis can transform a thermally unreactive, earth-abundant transition metal into a catalytically active site under illumination. Fe active sites in a Cu-Fe antenna-reactor complex achieve efficiencies very similar to Ru for the photocatalytic decompn. of ammonia under ultrafast pulsed illumination. When illuminated with light-emitting diodes rather than lasers, the photocatalytic efficiencies remain comparable, even when the scale of reaction increases by nearly three orders of magnitude. This result demonstrates the potential for highly efficient, elec. driven prodn. of hydrogen from an ammonia carrier with earth-abundant transition metals.
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    19. 19
      Bao, J. L.; Carter, E. A. Surface-Plasmon-Induced Ammonia Decomposition on Copper: Excited-State Reaction Pathways Revealed by Embedded Correlated Wavefunction Theory. ACS Nano 2019, 13 (9), 9944– 9957,  DOI: 10.1021/acsnano.9b05030
      19
      Surface-Plasmon-Induced Ammonia Decomposition on Copper: Excited-State Reaction Pathways Revealed by Embedded Correlated Wavefunction Theory
      Bao, Junwei Lucas; Carter, Emily A.
      ACS Nano (2019), 13 (9), 9944-9957CODEN: ANCAC3; ISSN:1936-0851. (American Chemical Society)
      NH3 is a promising H storage medium; however, its decompn. via conventional thermal catalysis requires a significant amt. of thermal energy input to overcome the reaction barriers. Embedded correlated wavefunction (ECW) theory was used to quantify reaction pathways and energetics for NH3 decompn. (N-H bond dissocn. and N2 and H2 associative desorption) on Cu nanoparticles using a Cu (111) surface model. The authors predict that surface plasmon excitations will be able to facilitate NH3 decompn. by substantially reducing the effective barriers along excited-state pathways. The authors est. the redns. in reaction barriers for breaking the 1st N-H bond and for recombinative desorption of surface-bound N and H atoms to be ∼1.7, 0.8, and 0.5 eV, resp. Further, by using the exptl. N2 desorption barrier as a ref., the authors compare the accuracy of various theor. methods, including plane-wave Kohn-Sham d. functional theory calcns. with commonly used exchange-correlation functionals, embedded complete active space 2nd-order perturbation theory, and embedded multiconfiguration pair-d. functional theory. This work offers further confirmation that the ECW theor. framework is the most robust for treating highly correlated local electronic structures of solids.
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    20. 20
      Bao, J. L.; Carter, E. A. Rationalizing the Hot-Carrier-Mediated Reaction Mechanisms and Kinetics for Ammonia Decomposition on Ruthenium-Doped Copper Nanoparticles. J. Am. Chem. Soc. 2019, 141 (34), 13320– 13323,  DOI: 10.1021/jacs.9b06804
      20
      Rationalizing the Hot-Carrier-Mediated Reaction Mechanisms and Kinetics for Ammonia Decomposition on Ruthenium-Doped Copper Nanoparticles
      Bao, Junwei Lucas; Carter, Emily A.
      Journal of the American Chemical Society (2019), 141 (34), 13320-13323CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)
      Localized surface plasmons generated on metallic nanostructures provide an efficient driving force for catalyzing chem. reactions, the kinetics of which cannot be understood properly by d. functional theory, despite its wide use in simulating heterogeneous catalytic reaction mechanisms. Reaction pathways for the NH3 decompn. reaction on Ru-doped Cu studied by the embedded correlated wave function method are reported. The computations provide a qual. explanation for the exptl. obsd. change in the reaction order from thermal catalysis to hot-carrier-mediated photocatalysis, as reported recently in Zhou, L., et al. (2018).
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    21. 21
      Zhou, L.; Martirez, J. M. P.; Finzel, J.; Zhang, C.; Swearer, D. F.; Tian, S.; Robatjazi, H.; Lou, M.; Dong, L.; Henderson, L. Light-driven methane dry reforming with single atomic site antenna-reactor plasmonic photocatalysts. Nat. Energy 2020, 5 (1), 61– 70,  DOI: 10.1038/s41560-019-0517-9
      21
      Light-driven methane dry reforming with single atomic site antenna-reactor plasmonic photocatalysts
      Zhou, Linan; Martirez, John Mark P.; Finzel, Jordan; Zhang, Chao; Swearer, Dayne F.; Tian, Shu; Robatjazi, Hossein; Lou, Minhan; Dong, Liangliang; Henderson, Luke; Christopher, Phillip; Carter, Emily A.; Nordlander, Peter; Halas, Naomi J.
      Nature Energy (2020), 5 (1), 61-70CODEN: NEANFD; ISSN:2058-7546. (Nature Research)
      Syngas, an extremely important chem. feedstock composed of carbon monoxide and hydrogen, can be generated through methane (CH4) dry reforming with CO2. However, traditional thermocatalytic processes require high temps. and suffer from coke-induced instability. Here, we report a plasmonic photocatalyst consisting of a Cu nanoparticle antenna with single-Ru at. reactor sites on the nanoparticle surface, ideal for low-temp., light-driven methane dry reforming. This catalyst provides high light energy efficiency when illuminated at room temp. In contrast to thermocatalysis, long-term stability (50 h) and high selectivity (>99%) were achieved in photocatalysis. We propose that light-excited hot carriers, together with single-atom active sites, cause the obsd. performance. Quantum mech. modeling suggests that single-atom doping of Ru on the Cu(111) surface, coupled with excited-state activation, results in a substantial redn. in the barrier for CH4 activation. This photocatalyst design could be relevant for future energy-efficient industrial processes.
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    22. 22
      Spata, V. A.; Carter, E. A. Mechanistic Insights into Photocatalyzed Hydrogen Desorption from Palladium Surfaces Assisted by Localized Surface Plasmon Resonances. ACS Nano 2018, 12 (4), 3512– 3522,  DOI: 10.1021/acsnano.8b00352
      22
      Mechanistic Insights into Photocatalyzed Hydrogen Desorption from Palladium Surfaces Assisted by Localized Surface Plasmon Resonances
      Spata, Vincent A.; Carter, Emily A.
      ACS Nano (2018), 12 (4), 3512-3522CODEN: ANCAC3; ISSN:1936-0851. (American Chemical Society)
      Nanoparticles synthesized from plasmonic metals can absorb low-energy light, producing an oscillation/excitation of their valence electron d. that can be utilized in chem. conversions. For example, heterogeneous photocatalysis can be achieved within heterometallic antenna-reactor complexes (HMARCs), by coupling a reactive center at which a chem. reaction occurs to a plasmonic nanoparticle that acts as a light-absorbing antenna. For example, HMARCs composed of aluminum antennae and palladium (Pd) reactive centers have been demonstrated recently to catalyze selective hydrogenation of acetylene to ethylene. Here, we explore within a theor. framework the rate-limiting step of hydrogen photodesorption from a Pd surface-crucial to achieving partial rather than full hydrogenation of acetylene-to understand the mechanism behind the photodesorption process within the HMARC assembly. To properly describe electronic excited states of the metal-mol. system, we employ embedded complete active space SCF and n-electron valence state perturbation theory to second order within d. functional embedding theory. The results of these calcns. reveal that the photodesorption mechanism does not create a frequently invoked transient neg. ion species but instead enhances population of available excited-state, low-barrier pathways that exhibit negligible charge-transfer character.
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    23. 23
      Huang, C.; Pavone, M.; Carter, E. A. Quantum mechanical embedding theory based on a unique embedding potential. J. Chem. Phys. 2011, 134 (15), 154110,  DOI: 10.1063/1.3577516
      23
      Quantum mechanical embedding theory based on a unique embedding potential
      Huang, Chen; Pavone, Michele; Carter, Emily A.
      Journal of Chemical Physics (2011), 134 (15), 154110/1-154110/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      We remove the nonuniqueness of the embedding potential that exists in most previous quantum mech. embedding schemes by letting the environment and embedded region share a common embedding (interaction) potential. To efficiently solve for the embedding potential, an optimized effective potential method is derived. This embedding potential, which eschews use of approx. kinetic energy d. functionals, is then used to describe the environment while a correlated wavefunction (CW) treatment of the embedded region is employed. We first demonstrate the accuracy of this new embedded CW (ECW) method by calcg. the van der Waals binding energy curve between a hydrogen mol. and a hydrogen chain. We then examine the prototypical adsorption of CO on a metal surface, here the Cu(111) surface. In addn. to obtaining proper site ordering (top site most stable) and binding energies within this theory, the ECW exhibits dramatic changes in the p-character of the CO 4σ and 5σ orbitals upon adsorption that agree very well with x-ray emission spectra, providing further validation of the theory. Finally, we generalize our embedding theory to spin-polarized quantum systems and discuss the connection between our theory and partition d. functional theory. (c) 2011 American Institute of Physics.
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    24. 24
      Yu, K.; Libisch, F.; Carter, E. A. Implementation of density functional embedding theory within the projector-augmented-wave method and applications to semiconductor defect states. J. Chem. Phys. 2015, 143 (10), 102806,  DOI: 10.1063/1.4922260
      24
      Implementation of density functional embedding theory within the projector-augmented-wave method and applications to semiconductor defect states
      Yu, Kuang; Libisch, Florian; Carter, Emily A.
      Journal of Chemical Physics (2015), 143 (10), 102806/1-102806/15CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      We report a new implementation of the d. functional embedding theory (DFET) in the VASP code, using the projector-augmented-wave (PAW) formalism. Newly developed algorithms allow us to efficiently perform optimized effective potential optimizations within PAW. The new algorithm generates robust and phys. correct embedding potentials, as we verified using several test systems including a covalently bound mol., a metal surface, and bulk semiconductors. We show that with the resulting embedding potential, embedded cluster models can reproduce the electronic structure of point defects in bulk semiconductors, thereby demonstrating the validity of DFET in semiconductors for the first time. Compared to our previous version, the new implementation of DFET within VASP affords use of all features of VASP (e.g., a systematic PAW library, a wide selection of functionals, a more flexible choice of U correction formalisms, and faster computational speed) with DFET. Furthermore, our results are fairly robust with respect to both plane-wave and Gaussian type orbital basis sets in the embedded cluster calcns. This suggests that the d. functional embedding method is potentially an accurate and efficient way to study properties of isolated defects in semiconductors. (c) 2015 American Institute of Physics.
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    25. 25
      Libisch, F.; Huang, C.; Carter, E. A. Embedded Correlated Wavefunction Schemes: Theory and Applications. Acc. Chem. Res. 2014, 47 (9), 2768– 2775,  DOI: 10.1021/ar500086h
      25
      Embedded Correlated Wavefunction Schemes: Theory and Applications
      Libisch, Florian; Huang, Chen; Carter, Emily A.
      Accounts of Chemical Research (2014), 47 (9), 2768-2775CODEN: ACHRE4; ISSN:0001-4842. (American Chemical Society)
      A review. Ab initio modeling of matter has become a pillar of chem. research: with ever-increasing computational power, simulations can be used to accurately predict, for example, chem. reaction rates, electronic and mech. properties of materials, and dynamical properties of liqs. Many competing quantum mech. methods have been developed over the years that vary in computational cost, accuracy, and scalability: d. functional theory (DFT), the workhorse of solid-state electronic structure calcns., features a good compromise between accuracy and speed. However, approx. exchange-correlation functionals limit DFT's ability to treat certain phenomena or states of matter, such as charge-transfer processes or strongly correlated materials. Furthermore, conventional DFT is purely a ground-state theory: electronic excitations are beyond its scope. Excitations in mols. are routinely calcd. using time-dependent DFT linear response; however applications to condensed matter are still limited. By contrast, many-electron wavefunction methods aim for a very accurate treatment of electronic exchange and correlation. Unfortunately, the assocd. computational cost renders treatment of more than a handful of heavy atoms challenging. On the other side of the accuracy spectrum, parametrized approaches like tight-binding can treat millions of atoms. In view of the different (dis-)advantages of each method, the simulation of complex systems seems to force a compromise: one is limited to the most accurate method that can still handle the problem size. For many interesting problems, however, compromise proves insufficient. A possible soln. is to break up the system into manageable subsystems that may be treated by different computational methods. The interaction between subsystems may be handled by an embedding formalism. In this Account, we review embedded correlated wavefunction (CW) approaches and some applications. We first discuss our d. functional embedding theory, which is formally exact. We show how to det. the embedding potential, which replaces the interaction between subsystems, at the DFT level. CW calcns. are performed using a fixed embedding potential, i.e., a non-self-consistent embedding scheme. We demonstrate this embedding theory for two challenging electron transfer phenomena: (1) initial oxidn. of an aluminum surface and (2) hot-electron-mediated dissocn. of hydrogen mols. on a gold surface. In both cases, the interaction between gas mols. and metal surfaces were treated by sophisticated CW techniques, with the remainder of the extended metal surface being treated by DFT. Our embedding approach overcomes the limitations of conventional Kohn-Sham DFT in describing charge transfer, multiconfigurational character, and excited states. From these embedding simulations, we gained important insights into fundamental processes that are crucial aspects of fuel cell catalysis (i.e., O2 redn. at metal surfaces) and plasmon-mediated photocatalysis by metal nanoparticles. Moreover, our findings agree very well with exptl. observations, while offering new views into the chem. We finally discuss our recently formulated potential-functional embedding theory that provides a seamless, first-principles way to include back-action onto the environment from the embedded region.
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    26. 26
      Tóth, Z.; Pulay, P. Comparison of Methods for Active Orbital Selection in Multiconfigurational Calculations. J. Chem. Theory Comput. 2020, 16 (12), 7328– 7341,  DOI: 10.1021/acs.jctc.0c00123
      26
      Comparison of Methods for Active Orbital Selection in Multiconfigurational Calculations
      Toth, Zsuzsanna; Pulay, Peter
      Journal of Chemical Theory and Computation (2020), 16 (12), 7328-7341CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      Several methods of constructing the active orbital space for multiconfigurational wave functions are compared on typical moderately strongly or strongly correlated ground-state mols. The relative merits of these methods and problems inherent in multiconfigurational calcns. are discussed. Strong correlation in the ground electronic state is found typically in larger conjugated and in antiarom. systems, transition states which involve bond breaking or formation, and transition metal complexes. Our examples include polyenes, polyacenes, the reactant, product and transition state of the Bergman cyclization, and two transition metal complexes: Hieber's anion [(CO)3FeNO]- and ferrocene. For the systems investigated, the simplest and oldest selection method, based on the fractional occupancy of UHF natural orbitals (the UNO criterion), yields the same active space as much more expensive approx. full CI methods. A disadvantage of this method used to be the difficulty of finding broken spin symmetry UHF solns. However, our anal. method, accurate to fourth order in the orbital rotation angles (Toth and Pulay, J. Chem. Phys., 2016, 145, 164102), has solved this problem. Two further advantages of the UNO criterion are that, unlike most other methods, it measures not only the energetic proximity to the Fermi level but also the magnitude of the exchange interaction with strongly occupied orbitals and therefore allows the estn. of the correlation strength for orbital selection in Restricted Active Space methods.
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    27. 27
      Monkhorst, H. J. Hartree-Fock density of states for extended systems. Phys. Rev. B: Condens. Matter Mater. Phys. 1979, 20 (4), 1504– 1513,  DOI: 10.1103/PhysRevB.20.1504
      There is no corresponding record for this reference.
    28. 28
      McAdon, M. H.; Goddard, W. A. Charge density waves, spin density waves, and Peierls distortions in one-dimensional metals. I. Hartree–Fock studies of Cu, Ag, Au, Li, and Na. J. Chem. Phys. 1988, 88 (1), 277– 302,  DOI: 10.1063/1.454654
      There is no corresponding record for this reference.
    29. 29
      Klüner, T.; Govind, N.; Wang, Y. A.; Carter, E. A. Prediction of Electronic Excited States of Adsorbates on Metal Surfaces from First Principles. Phys. Rev. Lett. 2001, 86 (26), 5954– 5957,  DOI: 10.1103/PhysRevLett.86.5954
      There is no corresponding record for this reference.
    30. 30
      Klüner, T.; Govind, N.; Wang, Y. A.; Carter, E. A. Periodic density functional embedding theory for complete active space self-consistent field and configuration interaction calculations: Ground and excited states. J. Chem. Phys. 2002, 116 (1), 42– 54,  DOI: 10.1063/1.1420748
      There is no corresponding record for this reference.
    31. 31
      Hirano, T.; Nagashima, U. Ro-vibrational properties of FeCO in the X̃ 3Σ- and ã 5Σ- electronic states: A computational molecular spectroscopy study. J. Mol. Spectrosc. 2015, 314, 35– 47,  DOI: 10.1016/j.jms.2015.05.007
      There is no corresponding record for this reference.
    32. 32
      Cimpoesu, F.; Dahan, F.; Ladeira, S.; Ferbinteanu, M.; Costes, J.-P. Chiral Crystallization of a Heterodinuclear Ni-Ln Series: Comprehensive Analysis of the Magnetic Properties. Inorg. Chem. 2012, 51 (21), 11279– 11293,  DOI: 10.1021/ic3001784
      There is no corresponding record for this reference.
    33. 33
      Andersson, K.; Malmqvist, P. Å.; Roos, B. O. Second-order perturbation theory with a complete active space self-consistent field reference function. J. Chem. Phys. 1992, 96 (2), 1218– 1226,  DOI: 10.1063/1.462209
      33
      Second-order perturbation theory with a complete active space self-consistent field reference function
      Andersson, Kerstin; Malmqvist, Per Aake; Roos, Bjoern O.
      Journal of Chemical Physics (1992), 96 (2), 1218-26CODEN: JCPSA6; ISSN:0021-9606.
      The recently implemented second-order perturbation theory based on a complete active space SCF ref. function has been extended by allowing the Fock-type one-electron operator, which defines the zeroth-order Hamiltonian to have nonzero elements also in nondiagonal matrix blocks. The computer implementation is now less straightforward and more computer time will be needed in obtaining the second-order energy. The method is illustrated in a series of calcns. on N2, NO, O2, CH3, CH2, and F-.
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    34. 34
      Martirez, J. M. P.; Bao, J. L.; Carter, E. A. First-Principles Insights into Plasmon-Induced Catalysis. Annu. Rev. Phys. Chem. 2021, 72 (1), 99– 119,  DOI: 10.1146/annurev-physchem-061020-053501
      There is no corresponding record for this reference.
    35. 35
      Kresse, G.; Hafner, J. Ab initio molecular dynamics for liquid metals. Phys. Rev. B: Condens. Matter Mater. Phys. 1993, 47 (1), 558– 561,  DOI: 10.1103/PhysRevB.47.558
      35
      Ab initio molecular dynamics of liquid metals
      Kresse, G.; Hafner, J.
      Physical Review B: Condensed Matter and Materials Physics (1993), 47 (1), 558-61CODEN: PRBMDO; ISSN:0163-1829.
      The authors present ab initio quantum-mech. mol.-dynamics calcns. based on the calcn. of the electronic ground state and of the Hellmann-Feynman forces in the local-d. approxn. at each mol.-dynamics step. This is possible using conjugate-gradient techniques for energy minimization, and predicting the wave functions for new ionic positions using sub-space alignment. This approach avoids the instabilities inherent in quantum-mech. mol.-dynamics calcns. for metals based on the use of a factitious Newtonian dynamics for the electronic degrees of freedom. This method gives perfect control of the adiabaticity and allows one to perform simulations over several picoseconds.
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    36. 36
      Kresse, G.; Furthmuller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B: Condens. Matter Mater. Phys. 1996, 54 (16), 11169– 11186,  DOI: 10.1103/PhysRevB.54.11169
      36
      Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set
      Kresse, G.; Furthmueller, J.
      Physical Review B: Condensed Matter (1996), 54 (16), 11169-11186CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
      The authors present an efficient scheme for calcg. the Kohn-Sham ground state of metallic systems using pseudopotentials and a plane-wave basis set. In the first part the application of Pulay's DIIS method (direct inversion in the iterative subspace) to the iterative diagonalization of large matrixes will be discussed. This approach is stable, reliable, and minimizes the no. of order Natoms3 operations. In the second part, we will discuss an efficient mixing scheme also based on Pulay's scheme. A special "metric" and a special "preconditioning" optimized for a plane-wave basis set will be introduced. Scaling of the method will be discussed in detail for non-self-consistent and self-consistent calcns. It will be shown that the no. of iterations required to obtain a specific precision is almost independent of the system size. Altogether an order Natoms2 scaling is found for systems contg. up to 1000 electrons. If we take into account that the no. of k points can be decreased linearly with the system size, the overall scaling can approach Natoms. They have implemented these algorithms within a powerful package called VASP (Vienna ab initio simulation package). The program and the techniques have been used successfully for a large no. of different systems (liq. and amorphous semiconductors, liq. simple and transition metals, metallic and semiconducting surfaces, phonons in simple metals, transition metals, and semiconductors) and turned out to be very reliable.
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    37. 37
      Kresse, G.; Furthmüller, J. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mater. Sci. 1996, 6 (1), 15– 50,  DOI: 10.1016/0927-0256(96)00008-0
      37
      Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set
      Kresse, G.; Furthmuller, J.
      Computational Materials Science (1996), 6 (1), 15-50CODEN: CMMSEM; ISSN:0927-0256. (Elsevier)
      The authors present a detailed description and comparison of algorithms for performing ab-initio quantum-mech. calcns. using pseudopotentials and a plane-wave basis set. The authors will discuss: (a) partial occupancies within the framework of the linear tetrahedron method and the finite temp. d.-functional theory, (b) iterative methods for the diagonalization of the Kohn-Sham Hamiltonian and a discussion of an efficient iterative method based on the ideas of Pulay's residual minimization, which is close to an order N2atoms scaling even for relatively large systems, (c) efficient Broyden-like and Pulay-like mixing methods for the charge d. including a new special preconditioning optimized for a plane-wave basis set, (d) conjugate gradient methods for minimizing the electronic free energy with respect to all degrees of freedom simultaneously. The authors have implemented these algorithms within a powerful package called VAMP (Vienna ab-initio mol.-dynamics package). The program and the techniques have been used successfully for a large no. of different systems (liq. and amorphous semiconductors, liq. simple and transition metals, metallic and semi-conducting surfaces, phonons in simple metals, transition metals and semiconductors) and turned out to be very reliable.
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    38. 38
      Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 1996, 77 (18), 3865– 3868,  DOI: 10.1103/PhysRevLett.77.3865
      38
      Generalized gradient approximation made simple
      Perdew, John P.; Burke, Kieron; Ernzerhof, Matthias
      Physical Review Letters (1996), 77 (18), 3865-3868CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      Generalized gradient approxns. (GGA's) for the exchange-correlation energy improve upon the local spin d. (LSD) description of atoms, mols., and solids. We present a simple derivation of a simple GGA, in which all parameters (other than those in LSD) are fundamental consts. Only general features of the detailed construction underlying the Perdew-Wang 1991 (PW91) GGA are invoked. Improvements over PW91 include an accurate description of the linear response of the uniform electron gas, correct behavior under uniform scaling, and a smoother potential.
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    39. 39
      Singh-Miller, N. E.; Marzari, N. Surface energies, work functions, and surface relaxations of low-index metallic surfaces from first principles. Phys. Rev. B: Condens. Matter Mater. Phys. 2009, 80 (23), 235407,  DOI: 10.1103/PhysRevB.80.235407
      39
      Surface energies, work functions, and surface relaxations of low-index metallic surfaces from first principles
      Singh-Miller, Nicholas E.; Marzari, Nicola
      Physical Review B: Condensed Matter and Materials Physics (2009), 80 (23), 235407/1-235407/9CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
      We study the relaxations, surface energies, and work functions of low-index metallic surfaces using pseudopotential plane-wave d.-functional calcns. within the generalized gradient approxn. We study here the (100), (110), and (111) surfaces of Al, Pd, Pt, and Au and the (0001) surface of Ti, chosen for their use as contact or lead materials in nanoscale devices. We consider clean, mostly nonreconstructed surfaces in the slab-supercell approxn. Particular attention is paid to the convergence of these quantities with respect to slab thickness; furthermore, different methodologies for the calcn. of work functions and surfaces energies are compared. The use of bulk refs. for calcns. of surface energies and work functions can be detrimental to convergence unless numerical grids are closely matched, esp. when surface relaxations are being considered. Calcd. values often do not quant. match exptl. values. This may be understandable for the surface relaxations and surface energies, where exptl. values can have large error but even for the work functions, neither local nor semilocal functionals emerge as an accurate choice for every case.
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    40. 40
      Makov, G.; Payne, M. C. Periodic boundary conditions in ab initio calculations. Phys. Rev. B: Condens. Matter Mater. Phys. 1995, 51 (7), 4014– 4022,  DOI: 10.1103/PhysRevB.51.4014
      40
      Periodic boundary conditions in ab initio calculations
      Makov, G.; Payne, M. C.
      Physical Review B: Condensed Matter (1995), 51 (7), 4014-22CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
      The convergence of the electrostatic energy in calcns. using periodic boundary conditions is considered in the context of periodic solids and localized aperiodic systems in the gas and condensed phases. Conditions for the abs. convergence of the total energy in periodic boundary conditions are obtained, and their implications for calcns. of the properties of polarized solids under the zero-field assumption are discussed. For aperiodic systems the exact electrostatic energy functional in periodic boundary conditions is obtained. The convergence in such systems is considered in the limit of large supercells, where, in the gas phase, the computational effort is proportional to the vol. It is shown that for neutral localized aperiodic systems in either the gas or condensed phases, the energy can always be made to converge as O (L-5) where L is the linear dimension of the supercell. For charged systems, convergence at this rate can be achieved after adding correction terms to the energy to account for spurious interactions induced by the periodic boundary conditions. These terms are derived exactly for the gas phase and heuristically for the condensed phase.
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    41. 41
      Monkhorst, H. J.; Pack, J. D. Special points for Brillouin-zone integrations. Phys. Rev. B: Solid State 1976, 13 (12), 5188– 5192,  DOI: 10.1103/PhysRevB.13.5188
      There is no corresponding record for this reference.
    42. 42
      Methfessel, M.; Paxton, A. T. High-precision sampling for Brillouin-zone integration in metals. Phys. Rev. B: Condens. Matter Mater. Phys. 1989, 40 (6), 3616– 3621,  DOI: 10.1103/PhysRevB.40.3616
      42
      High-precision sampling for Brillouin-zone integration in metals
      Methfessel, M.; Paxton, A. T.
      Physical Review B: Condensed Matter and Materials Physics (1989), 40 (6), 3616-21CODEN: PRBMDO; ISSN:0163-1829.
      A sampling method is given for Brillouin-zone integration in metals which converges exponentially with the no. of sampling points, without the loss of precision of normal broadening techniques. The scheme is based on smooth approximants to the δ and step functions which are constructed to give the exact result when integrating polynomials of a prescribed degree. In applications to the simple-cubic tight-binding band as well as to band structures of simple and transition metals, significant improvement over existing methods was shown. The method promises general applicability in the fields of total-energy calcns. and many-body physics.
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    43. 43
      Dutta, B. N.; Dayal, B. Lattice Constants and Thermal Expansion of Palladium and Tungsten up to 878 °C by X-Ray Method. Phys. Status Solidi B 1963, 3 (12), 2253– 2259,  DOI: 10.1002/pssb.19630031207
      There is no corresponding record for this reference.
    44. 44
      Jonsson, H.; Mills, G.; Jacobsen, K. W. Nudged elastic band method for finding minimum energy paths of transitions. In Classical and Quantum Dynamics in Condensed Phase Simulations; World Scientific, 1998; pp 385– 404.
      There is no corresponding record for this reference.
    45. 45
      Henkelman, G.; Uberuaga, B. P.; Jónsson, H. A climbing image nudged elastic band method for finding saddle points and minimum energy paths. J. Chem. Phys. 2000, 113 (22), 9901– 9904,  DOI: 10.1063/1.1329672
      45
      A climbing image nudged elastic band method for finding saddle points and minimum energy paths
      Henkelman, Graeme; Uberuaga, Blas P.; Jonsson, Hannes
      Journal of Chemical Physics (2000), 113 (22), 9901-9904CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      A modification of the nudged elastic band method for finding min. energy paths is presented. One of the images is made to climb up along the elastic band to converge rigorously on the highest saddle point. Also, variable spring consts. are used to increase the d. of images near the top of the energy barrier to get an improved est. of the reaction coordinate near the saddle point. Applications to CH4 dissociative adsorption on Ir(111) and H2 on Si(100) using plane wave based d. functional theory are presented.
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      https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3cXosFagurc%253D&md5=3899b9e2e9e3eb74009987d96623f018
    46. 46
      Yu, K.; Krauter, C. M.; Dieterich, J. M.; Carter, E. A. Density and Potential Functional Embedding: Theory and Practice. In Fragmentation: Toward Accurate Calculations on Complex Molecular Systems; Gordon, M., Ed.; John Wiley & Sons, 2017; pp 81– 117.
      There is no corresponding record for this reference.
    47. 47
      Yu, K.; Carter, E. A. VASP density functional embedding theory. https://github.com/EACcodes/VASPEmbedding (accessed May 16, 2022).
      There is no corresponding record for this reference.
    48. 48
      Wu, Q.; Yang, W. A direct optimization method for calculating density functionals and exchange–correlation potentials from electron densities. J. Chem. Phys. 2003, 118 (6), 2498– 2509,  DOI: 10.1063/1.1535422
      There is no corresponding record for this reference.
    49. 49
      Krauter, C. M.; Carter, E. A. Embedding Integral Generator. https://github.com/EACcodes/EmbeddingIntegralGenerator (accessed May 16, 2022).
      There is no corresponding record for this reference.
    50. 50
      Werner, H.-J.; Knowles, P. J.; Knizia, G.; Manby, F. R.; Schütz, M. Molpro: a general-purpose quantum chemistry program package. Wiley Interdiscip. Rev. Comput. Mol. Sci. 2012, 2 (2), 242– 253,  DOI: 10.1002/wcms.82
      50
      Molpro: a general-purpose quantum chemistry program package
      Werner, Hans-Joachim; Knowles, Peter J.; Knizia, Gerald; Manby, Frederick R.; Schuetz, Martin
      Wiley Interdisciplinary Reviews: Computational Molecular Science (2012), 2 (2), 242-253CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell)
      Molpro is a general-purpose quantum chem. program. The original focus was on high-accuracy wave function calcns. for small mols., but using local approxns. combined with explicit correlation treatments, highly accurate coupled-cluster calcns. are now possible for mols. with up to approx. 100 atoms. Recently, multireference correlation treatments were also made applicable to larger mols. Furthermore, an efficient implementation of d. functional theory is available.
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    51. 51
      Werner, H.-J.; Knowles, P. J.; Manby, F. R.; Black, J. A.; Doll, K.; Heßelmann, A.; Kats, D.; Köhn, A.; Korona, T.; Kreplin, D. A. The Molpro quantum chemistry package. J. Chem. Phys. 2020, 152 (14), 144107,  DOI: 10.1063/5.0005081
      51
      The Molpro quantum chemistry package
      Werner, Hans-Joachim; Knowles, Peter J.; Manby, Frederick R.; Black, Joshua A.; Doll, Klaus; Hesselmann, Andreas; Kats, Daniel; Koehn, Andreas; Korona, Tatiana; Kreplin, David A.; Ma, Qianli; Miller, Thomas F.; Mitrushchenkov, Alexander; Peterson, Kirk A.; Polyak, Iakov; Rauhut, Guntram; Sibaev, Marat
      Journal of Chemical Physics (2020), 152 (14), 144107CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      Molpro is a general purpose quantum chem. software package with a long development history. It was originally focused on accurate wavefunction calcns. for small mols. but now has many addnl. distinctive capabilities that include, inter alia, local correlation approxns. combined with explicit correlation, highly efficient implementations of single-ref. correlation methods, robust and efficient multireference methods for large mols., projection embedding, and anharmonic vibrational spectra. In addn. to conventional input-file specification of calcns., Molpro calcns. can now be specified and analyzed via a new graphical user interface and through a Python framework. (c) 2020 American Institute of Physics.
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    52. 52
      Werner, H.-J.; Knowles, P. J.; Knizia, G.; Manby, F. R.; Schütz, M.; Celani, P.; Györffy, W.; Kats, D.; Korona, T.; Lindh, R.; MOLPRO , version 2021.2, a package of ab initio programs, 2021. https://www.molpro.net.
      There is no corresponding record for this reference.
    53. 53
      Ghigo, G.; Roos, B. O.; Malmqvist, P.-Å. A modified definition of the zeroth-order Hamiltonian in multiconfigurational perturbation theory (CASPT2). Chem. Phys. Lett. 2004, 396 (1–3), 142– 149,  DOI: 10.1016/j.cplett.2004.08.032
      53
      A modified definition of the zeroth-order Hamiltonian in multiconfigurational perturbation theory (CASPT2)
      Ghigo, Giovanni; Roos, Bjoern O.; Malmqvist, Per-Ake
      Chemical Physics Letters (2004), 396 (1-3), 142-149CODEN: CHPLBC; ISSN:0009-2614. (Elsevier B.V.)
      A new shifted zeroth-order Hamiltonian is presented, which will be used in second-order multiconfigurational perturbation theory (CASPT2). The new approxn. corrects for the systematic error of the original formulation, which led to an relative overestimate of the correlation energy for open shell system, resulting in too small dissocn. and excitation energies. Errors in the De values for 49 diat. mols. were reduced with more than 50%. Calcns. on excited states of the N2 and benzene mols. give a similar improvement.
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    54. 54
      Roos, B. O.; Andersson, K. Multiconfigurational perturbation theory with level shift ─ the Cr2 potential revisited. Chem. Phys. Lett. 1995, 245 (2–3), 215– 223,  DOI: 10.1016/0009-2614(95)01010-7
      54
      Multiconfigurational perturbation theory with level shift - the Cr2 potential revisited
      Roos, Bjoern O.; Andersson, Kerstin
      Chemical Physics Letters (1995), 245 (2,3), 215-23CODEN: CHPLBC; ISSN:0009-2614. (Elsevier)
      A level shift technique is suggested for removal of intruder states in multiconfigurational second-order perturbation theory (CASPT2). The first-order wavefunction is first calcd. with a level shift parameter large enough to remove the intruder states. The effect of the level shift on the second-order energy is removed by a back correction technique (the LS correction). It is shown that intruder states are removed with little effect on the remaining part of the correlation energy. New potential curves have been computed for the X 1Σ+g and the a' 3Σ+u states of Cr2 using large basis sets (ANO: 8s7p6d4f2g) and accounting for relativistic effects, 3s and 3p correlation, and basis set superposition effects. The computed spectroscopic consts. (exptl. values in parentheses) for the X 1Σ+g state are re = 1.69(1.68) Å, ΔG1/2 = 535(452) cm-1, D0 = 1.54(1.44) eV. The corresponding values for a' 3Σ+u are re = 1.64(1.65) Å, ΔG1/2 = 667(574) cm-1, Te = 1.79(1.76) eV.
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    55. 55
      Wen, X.; Martirez, J. M. P.; Carter, E. A. Plasmon-driven ammonia decomposition on Pd(111): Hole transfer’s role in changing rate-limiting steps. ACS Catal. 2024, 14, 9539– 9553,  DOI: 10.1021/acscatal.4c01869
      There is no corresponding record for this reference.
    56. 56
      Stradella, L. Heats of adsorption of different gases on polycrystalline transition metals. Adsorpt. Sci. Technol. 1992, 9 (3), 190– 198,  DOI: 10.1177/026361749200900304
      There is no corresponding record for this reference.
    57. 57
      Peterson, K. A.; Figgen, D.; Dolg, M.; Stoll, H. Energy-consistent relativistic pseudopotentials and correlation consistent basis sets for the 4d elements Y–Pd. J. Chem. Phys. 2007, 126 (12), 124101,  DOI: 10.1063/1.2647019
      57
      Energy-consistent relativistic pseudopotentials and correlation consistent basis sets for the 4d elements Y-Pd
      Peterson, Kirk A.; Figgen, Detlev; Dolg, Michael; Stoll, Hermann
      Journal of Chemical Physics (2007), 126 (12), 124101/1-124101/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      Scalar-relativistic pseudopotentials and corresponding spin-orbit potentials of the energy-consistent variety have been adjusted for the simulation of the [Ar]3d10 cores of the 4d transition metal elements Y-Pd. These potentials have been detd. in a one-step procedure using numerical two-component calcns. so as to reproduce at. valence spectra from four-component all-electron calcns. The latter have been performed at the multi-configuration Dirac-Hartree-Fock level, using the Dirac-Coulomb Hamiltonian and perturbatively including the Breit interaction. The derived pseudopotentials reproduce the all-electron ref. data with an av. accuracy of 0.03 eV for configurational avs. over nonrelativistic orbital configurations and 0.1 eV for individual relativistic states. Basis sets following a correlation consistent prescription have also been developed to accompany the new pseudopotentials. These range in size from cc-pVDZ-PP to cc-pV5Z-PP and also include sets for 4s4p correlation (cc-pwCVDZ-PP through cc-pwCV5Z-PP), as well as those with extra diffuse functions (aug-cc-pVDZ-PP, etc.). In order to accurately assess the impact of the pseudopotential approxn., all-electron basis sets of triple-zeta quality have also been developed using the Douglas-Kroll-Hess Hamiltonian (cc-pVTZ-DK, cc-pwCVTZ-DK, and aug-cc-pVTZ-DK). Benchmark calcns. of at. ionization potentials and 4dm-25s2 → 4dm-15s1 electronic excitation energies are reported at the coupled cluster level of theory with extrapolations to the complete basis set limit.
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    58. 58
      Dunning, T. H., Jr. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. J. Chem. Phys. 1989, 90 (2), 1007– 1023,  DOI: 10.1063/1.456153
      58
      Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen
      Dunning, Thom H., Jr.
      Journal of Chemical Physics (1989), 90 (2), 1007-23CODEN: JCPSA6; ISSN:0021-9606.
      Guided by the calcns. on oxygen in the literature, basis sets for use in correlated at. and mol. calcns. were developed for all of the first row atoms from boron through neon, and for hydrogen. As in the oxygen atom calcns., the incremental energy lowerings, due to the addn. of correlating functions, fall into distinct groups. This leads to the concept of correlation-consistent basis sets, i.e., sets which include all functions in a given group as well as all functions in any higher groups. Correlation-consistent sets are given for all of the atoms considered. The most accurate sets detd. in this way, [5s4p3d2f1g], consistently yield 99% of the correlation energy obtained with the corresponding at.-natural-orbital sets, even though the latter contains 50% more primitive functions and twice as many primitive polarization functions. It is estd. that this set yields 94-97% of the total (HF + 1 + 2) correlation energy for the atoms neon through boron.
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    59. 59
      Kendall, R. A.; Dunning, T. H., Jr.; Harrison, R. J. Electron affinities of the first-row atoms revisited. Systematic basis sets and wave functions. J. Chem. Phys. 1992, 96 (9), 6796– 6806,  DOI: 10.1063/1.462569
      59
      Electron affinities of the first-row atoms revisited. Systematic basis sets and wave functions
      Kendall, Rick A.; Dunning, Thom H., Jr.; Harrison, Robert J.
      Journal of Chemical Physics (1992), 96 (9), 6796-806CODEN: JCPSA6; ISSN:0021-9606.
      The authors describe a reliable procedure for calcg. the electron affinity of an atom and present results for H, B, C, O, and F (H is included for completeness). This procedure involves the use of the recently proposed correlation-consistent basis sets augmented with functions to describe the more diffuse character of the at. anion coupled with a straightforward, uniform expansion of the ref. space for multireference singles and doubles configuration-interaction (MRSD-CI) calcns. A comparison is given with previous results and with corresponding full CI calcns. The most accurate EAs obtained from the MRSD-CI calcns. are (with exptl. values in parentheses): H 0.740 eV (0.754), B 0.258 (0.277), C 1.245 (1.263), O 1.384 (1.461), and F 3.337 (3.401). The EAs obtained from the MR-SDCI calcns. differ by less than 0.03 eV from those predicted by the full CI calcns.
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    60. 60
      Peterson, K. A.; Dunning, T. H., Jr. Accurate correlation consistent basis sets for molecular core–valence correlation effects: The second row atoms Al–Ar, and the first row atoms B–Ne revisited. J. Chem. Phys. 2002, 117 (23), 10548– 10560,  DOI: 10.1063/1.1520138
      60
      Accurate correlation consistent basis sets for molecular core-valence correlation effects: The second row atoms Al-Ar, and the first row atoms B-Ne revisited
      Peterson, Kirk A.; Dunning, Thom H., Jr.
      Journal of Chemical Physics (2002), 117 (23), 10548-10560CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      Correlation consistent basis sets for accurately describing core-core and core-valence correlation effects in atoms and mols. have been developed for the second row atoms Al-Ar. Two different optimization strategies were investigated, which led to two families of core-valence basis sets when the optimized functions were added to the std. correlation consistent basis sets (cc-pVnZ). In the first case, the exponents of the augmenting primitive Gaussian functions were optimized with respect to the difference between all-electron and valence-electron correlated calcns., i.e., for the core-core plus core-valence correlation energy. This yielded the cc-pCVnZ family of basis sets, which are analogous to the sets developed previously for the first row atoms [D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys. 103, 4572 (1995)]. Although the cc-pCVnZ sets exhibit systematic convergence to the all-electron correlation energy at the complete basis set limit, the intershell (core-valence) correlation energy converges more slowly than the intrashell (core-core) correlation energy. Since the effect of including the core electrons on the calcn. of mol. properties tends to be dominated by core-valence correlation effects, a second scheme for detg. the augmenting functions was investigated. In this approach, the exponents of the functions to be added to the cc-pVnZ sets were optimized with respect to just the core-valence (intershell) correlation energy, except that a small amt. of core-core correlation energy was included in order to ensure systematic convergence to the complete basis set limit. These new sets, denoted weighted core-valence basis sets (cc-pwCVnZ), significantly improve the convergence of many mol. properties with n. Optimum cc-pwCVnZ sets for the first-row atoms were also developed and show similar advantages. Both the cc-pCVnZ and cc-pwCVnZ basis sets were benchmarked in coupled cluster [CCSD(T)] calcns. on a series of second row homonuclear diat. mols. (Al2, Si2, P2, S2, and Cl2), as well as on selected diat. mols. involving first row atoms (CO, SiO, PN, and BCl). For the calcn. of core correlation effects on energetic and spectroscopic properties, the cc-pwCVnZ basis sets are recommended over the cc-pCVnZ ones.
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    61. 61
      Pritchard, B. P.; Altarawy, D.; Didier, B.; Gibson, T. D.; Windus, T. L. New Basis Set Exchange: An Open, Up-to-Date Resource for the Molecular Sciences Community. J. Chem. Inf. Model. 2019, 59 (11), 4814– 4820,  DOI: 10.1021/acs.jcim.9b00725
      61
      New Basis Set Exchange: An Open, Up-to-Date Resource for the Molecular Sciences Community
      Pritchard, Benjamin P.; Altarawy, Doaa; Didier, Brett; Gibson, Tara D.; Windus, Theresa L.
      Journal of Chemical Information and Modeling (2019), 59 (11), 4814-4820CODEN: JCISD8; ISSN:1549-9596. (American Chemical Society)
      A review. The Basis Set Exchange (BSE) has been a prominent fixture in the quantum chem. community. First publicly available in 2007, it is recognized by both users and basis set creators as the de facto source for information related to basis sets. This popular resource has been rewritten, utilizing modern software design and best practices. The basis set data has been sepd. into a stand-alone library with an accessible API, and the Web site has been updated to use the current generation of web development libraries. The general layout and workflow of the Web site is preserved, while helpful features requested by the user community have been added. Overall, this design should increase adaptability and lend itself well into the future as a dependable resource for the computational chem. community. This article will discuss the decision to rewrite the BSE, the new architecture and design, and the new features that have been added.
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    62. 62
      Hill, J. G.; Peterson, K. A. Gaussian basis sets for use in correlated molecular calculations. XI. Pseudopotential-based and all-electron relativistic basis sets for alkali metal (K–Fr) and alkaline earth (Ca–Ra) elements. J. Chem. Phys. 2017, 147 (24), 244106,  DOI: 10.1063/1.5010587
      62
      Gaussian basis sets for use in correlated molecular calculations. XI. Pseudopotential-based and all-electron relativistic basis sets for alkali metal (K-Fr) and alkaline earth (Ca-Ra) elements
      Hill, J. Grant; Peterson, Kirk A.
      Journal of Chemical Physics (2017), 147 (24), 244106/1-244106/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      New correlation consistent basis sets based on pseudopotential (PP) Hamiltonians have been developed from double- to quintuple-zeta quality for the late alkali (K-Fr) and alk. earth (Ca-Ra) metals. These are accompanied by new all-electron basis sets of double- to quadruple-zeta quality that have been contracted for use with both Douglas-Kroll-Hess (DKH) and eXact 2-Component (X2C) scalar relativistic Hamiltonians. Sets for valence correlation (ms), cc-pVnZ-PP and cc-pVnZ-(DK,DK3/X2C), in addn. to outer-core correlation [valence + (m-1)sp], cc-p(w)CVnZ-PP and cc-pwCVnZ-(DK,DK3/X2C), are reported. The -PP sets have been developed for use with small-core PPs [I. S. Lim et al., J. Chem. Phys. 122, 104103 (2005) and I. S. Lim et al., J. Chem. Phys. 124, 034107 (2006)], while the all-electron sets utilized second-order DKH Hamiltonians for 4s and 5s elements and third-order DKH for 6s and 7s. The accuracy of the basis sets is assessed through benchmark calcns. at the coupled-cluster level of theory for both at. and mol. properties. Not surprisingly, it is found that outer-core correlation is vital for accurate calcn. of the thermodn. and spectroscopic properties of diat. mols. contg. these elements. (c) 2017 American Institute of Physics.
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    63. 63
      Balabanov, N. B.; Peterson, K. A. Systematically convergent basis sets for transition metals. I. All-electron correlation consistent basis sets for the 3d elements Sc–Zn. J. Chem. Phys. 2005, 123 (6), 064107,  DOI: 10.1063/1.1998907
      63
      Systematically convergent basis sets for transition metals. I. All-electron correlation consistent basis sets for the 3d elements Sc-Zn
      Balabanov, Nikolai B.; Peterson, Kirk A.
      Journal of Chemical Physics (2005), 123 (6), 064107/1-064107/15CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      Sequences of basis sets that systematically converge towards the complete basis set (CBS) limit have been developed for the first-row transition metal elements Sc-Zn. Two families of basis sets, nonrelativistic and Douglas-Kroll-Hess (-DK) relativistic, are presented that range in quality from triple-ζ to quintuple-ζ. Sep. sets are developed for the description of valence (3d4s) electron correlation (cc-pVnZ and cc-pVnZ-DK; n=T,Q, 5) and valence plus outer-core (3s3p3d4s) correlation (cc-pwCVnZ and cc-pwCVnZ-DK; n=T,Q, 5), as well as these sets augmented by addnl. diffuse functions for the description of neg. ions and weak interactions (aug-cc-pVnZ and aug-cc-pVnZ-DK). Extensive benchmark calcns. at the coupled cluster level of theory are presented for at. excitation energies, ionization potentials, and electron affinities, as well as mol. calcns. on selected hydrides (TiH, MnH, CuH) and other diatomics (TiF, Cu2). In addn. to observing systematic convergence towards the CBS limits, both 3s3p electron correlation and scalar relativity are calcd. to strongly impact many of the at. and mol. properties investigated for these first-row transition metal species. A (A=Sc-Zn).
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    64. 64
      Jansen, G.; Hess, B. A. Revision of the Douglas-Kroll transformation. Phys. Rev. A 1989, 39 (11), 6016– 6017,  DOI: 10.1103/PhysRevA.39.6016
      64
      Revision of the Douglas-Kroll transformation
      Jansen; Hess
      Physical review. A, General physics (1989), 39 (11), 6016-6017 ISSN:0556-2791.
      There is no expanded citation for this reference.
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    65. 65
      Douglas, M.; Kroll, N. M. Quantum electrodynamical corrections to the fine structure of helium. Ann. Phys. 1974, 82 (1), 89– 155,  DOI: 10.1016/0003-4916(74)90333-9
      65
      Quantum electrodynamical corrections to the fine structure of helium
      Douglas, Marvin; Kroll, Norman M.
      Annals of Physics (San Diego, CA, United States) (1974), 82 (1), 89-155CODEN: APNYA6; ISSN:0003-4916.
      Corrections of order α6mc2 (α = fine structure const., mc2 = the electron rest energy) to the fine-structure splitting of the deepest lying triplet P state (23P0,1,2) of the 4He atom were investigated. The investigation is based on the covariant Bethe-Salpeter equation including an external potential to take account of the nuclear Coulomb field. All order α6mc2 corrections that arise from Feynman diagrams involving the exchange of 1, 2, and 3 photons, as well as radiative corrections to the electron magnetic moment were found. The results are presented in a form suitable for computerized numerical evaluation.
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    66. 66
      Silva-Junior, M. R.; Schreiber, M.; Sauer, S. P. A.; Thiel, W. Benchmarks of electronically excited states: Basis set effects on CASPT2 results. J. Chem. Phys. 2010, 133 (17), 174318,  DOI: 10.1063/1.3499598
      66
      Benchmarks of electronically excited states: Basis set effects on CASPT2 results
      Silva-Junior, Mario R.; Schreiber, Marko; Sauer, Stephan P. A.; Thiel, Walter
      Journal of Chemical Physics (2010), 133 (17), 174318/1-174318/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      Vertical excitation energies and one-electron properties are computed for the valence excited states of 28 medium-sized org. benchmark mols. using multistate multiconfigurational second-order perturbation theory (MS-CASPT2) and the augmented correlation-consistent aug-cc-pVTZ basis set. They are compared with previously reported MS-CASPT2 results obtained with the smaller TZVP basis. The basis set extension from TZVP to aug-cc-pVTZ causes rather minor and systematic shifts in the vertical excitation energies that are normally slightly reduced (on av. by 0.11 eV for the singlets and by 0.09 eV for the triplets), whereas the changes in the calcd. oscillator strengths and dipole moments are somewhat more pronounced on a relative scale. These basis set effects at the MS-CASPT2 level are qual. and quant. similar to those found at the coupled cluster level for the same set of benchmark mols. The previously proposed theor. best ests. (TBE-1) for the vertical excitation energies for 104 singlet and 63 triplet excited states of the benchmark mols. are upgraded by replacing TZVP with aug-cc-pVTZ data that yields a new ref. set (TBE-2). Statistical evaluations of the performance of d. functional theory (DFT) and semiempirical methods lead to the same ranking and very similar quant. results for TBE-1 and TBE-2, with slightly better performance measures with respect to TBE-2. DFT/MRCI is most accurate among the investigated DFT-based approaches, while the OMx methods with orthogonalization corrections perform best at the semiempirical level. (c) 2010 American Institute of Physics.
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    67. 67
      Kánnár, D.; Tajti, A.; Szalay, P. G. Accuracy of Coupled Cluster Excitation Energies in Diffuse Basis Sets. J. Chem. Theory Comput. 2017, 13 (1), 202– 209,  DOI: 10.1021/acs.jctc.6b00875
      There is no corresponding record for this reference.
    68. 68
      Jansen, H. B.; Ros, P. Non-empirical molecular orbital calculations on the protonation of carbon monoxide. Chem. Phys. Lett. 1969, 3 (3), 140– 143,  DOI: 10.1016/0009-2614(69)80118-1
      68
      Nonempirical molecular orbital calculations on the protonation of carbon monoxide
      Jansen, H. B.; Ros, P.
      Chemical Physics Letters (1969), 3 (3), 140-3CODEN: CHPLBC; ISSN:0009-2614.
      Single configurational L.C.A.O. Hartree-Fock M.O.-self-consistent field calcns. with gaussian functions were carried out for several configurations of protonated C monoxide in order to gain some insights about the geometry and stability of protonated CO. The most stable configuration is a linear [HCO]+ structure. The C-O bond distance is 0.02 A. smaller than in CO itself. The interaction of the proton with the π electrons of CO stabilizes the structure. The energy of protonation is 152 kcal./mole and agrees reasonably well with a value of 133 kcal./mole which was calcd. from thermochem. data for CO and H+.
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    69. 69
      Liu, B.; McLean, A. D. Accurate calculation of the attractive interaction of two ground state helium atoms. J. Chem. Phys. 1973, 59 (8), 4557– 4558,  DOI: 10.1063/1.1680654
      69
      Accurate calculation of the attractive interaction of two ground state helium atoms
      Liu, B.; McLean A. D.
      Journal of Chemical Physics (1973), 59 (8), 4557-8CODEN: JCPSA6; ISSN:0021-9606.
      A configuration interaction (CI) calcn. is given of the van der Waals interaction between 2 ground state He atoms. The CI calcn. converges on the exact clamped nuclei result within an accuracy of ∼0.1°K = 3 × 10-7 at. units for the interaction energy. The results agree with accurate perturbation theory results to within 0.02°K.
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    70. 70
      Gutowski, M.; Van Lenthe, J. H.; Verbeek, J.; Van Duijneveldt, F. B.; Chałasinski, G. The basis set superposition error in correlated electronic structure calculations. Chem. Phys. Lett. 1986, 124 (4), 370– 375,  DOI: 10.1016/0009-2614(86)85036-9
      There is no corresponding record for this reference.
    71. 71
      Simon, S.; Duran, M.; Dannenberg, J. J. How does basis set superposition error change the potential surfaces for hydrogen-bonded dimers?. J. Chem. Phys. 1996, 105 (24), 11024– 11031,  DOI: 10.1063/1.472902
      71
      How does basis set superposition error change the potential surfaces for hydrogen-bonded dimers?
      Simon, Silvia; Duran, Miquel; Dannenberg, J. J.
      Journal of Chemical Physics (1996), 105 (24), 11024-11031CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      We describe a simple method to automate the geometric optimization of MO calcns. of supermols. on potential surfaces that are cor. for basis set superposition error using the counterpoise (CP) method. This method is applied to the H-bonding complexes HF/HCN, HF/H2O, and HCCH/H2O using the 6-31G(d,p) and D95++(d,p) basis sets at both the Hartree-Fock and second-order Moeller-Plesset levels. We report the interaction energies, geometries, and vibrational frequencies of these complexes on the CP-optimized surfaces; and compare them with similar values calcd. using traditional methods, including the (more traditional) single point CP correction. Upon optimization on the CP-cor. surface, the interaction energies become more neg. (before vibrational corrections) and the H-bonding stretching vibrations decrease in all cases. The extent of the effect vary from extremely small to quite large depending on the complex and the calculational method. The relative magnitudes of the vibrational corrections cannot be predicted from the H-bond stretching frequencies alone.
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    72. 72
      Boys, S. F.; Bernardi, F. The calculation of small molecular interactions by the differences of separate total energies. Some procedures with reduced errors. Mol. Phys. 1970, 19 (4), 553– 566,  DOI: 10.1080/00268977000101561
      72
      The calculation of small molecular interactions by the differences of separate total energies. Some procedures with reduced errors
      Boys, S. F.; Bernardi, F.
      Molecular Physics (1970), 19 (4), 553-566CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)
      A new direct difference method for the computation of mol. interactions has been based on a bivariational transcorrelated treatment, together with special methods for the balancing of other errors. It appears that these new features can give a strong redn. in the error of the interaction energy, and they seem to be particularly suitable for computations in the important region near the min. energy. It has been generally accepted that this problem is dominated by unresolved difficulties and the relation of the new methods of these apparent difficulties is analyzed here.
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    73. 73
      Gray, M.; Bowling, P. E.; Herbert, J. M. Systematic Evaluation of Counterpoise Correction in Density Functional Theory. J. Chem. Theory Comput. 2022, 18 (11), 6742– 6756,  DOI: 10.1021/acs.jctc.2c00883
      73
      Systematic Evaluation of Counterpoise Correction in Density Functional Theory
      Gray, Montgomery; Bowling, Paige E.; Herbert, John M.
      Journal of Chemical Theory and Computation (2022), 18 (11), 6742-6756CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      A widespread belief persists that the Boys-Bernardi function counterpoise (CP) procedure "overcorrects" supramol. interaction energies for the effects of basis-set superposition error. To the extent that this is true for correlated wave function methods, it is usually an artifact of low-quality basis sets. The question has not been considered systematically in the context of d. functional theory, however, where basis-set convergence is generally less problematic. We present a systematic assessment of the CP procedure for a representative set of functionals and basis sets, considering both benchmark data sets of small dimers and larger supramol. complexes. The latter include layered composite polymers with ~ 150 atoms and ligand-protein models with ~ 300 atoms. Provided that CP correction is used, we find that intermol. interaction energies of nearly complete-basis quality can be obtained using only double-ζ basis sets. This is less expensive as compared to triple-ζ basis sets without CP correction. CP-cor. interaction energies are less sensitive to the presence of diffuse basis functions as compared to uncorrected energies, which is important because diffuse functions are expensive and often numerically problematic for large systems. Our results upend the conventional wisdom that CP "overcorrects" for basis-set incompleteness. In small basis sets, CP correction is mandatory in order to demonstrate that the results do not rest on error cancellation.
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    74. 74
      Mentel, Ł. M.; Baerends, E. J. Can the Counterpoise Correction for Basis Set Superposition Effect Be Justified?. J. Chem. Theory Comput. 2014, 10 (1), 252– 267,  DOI: 10.1021/ct400990u
      74
      Can the Counterpoise Correction for Basis Set Superposition Effect Be Justified?
      Mentel, L. M.; Baerends, E. J.
      Journal of Chemical Theory and Computation (2014), 10 (1), 252-267CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      The basis set superposition effect (BSSE) is a simple concept, and its validity is almost universally accepted. So is the counterpoise method to correct for it. The idea is that the basis set is biased toward the dimer because each monomer in the dimer can "use" the basis functions on the other monomer, which it cannot in a simple monomer calcn. This hypothesis can only be tested if basis set free benchmark nos. are available for monomers and dimer. We are testing the hypothesis on a few systems (in this paper Be2) that are small enough that sufficiently accurate benchmark nos. (basis set free, or close to basis set limit; full CI or close to full CI) are available or can be obtained. We find that the answer to the title question is neg.: the std. basis sets of quantum chem. appear to be biased toward the atom in the sense that basis set errors are larger for the dimer than the monomer. Applying the counterpoise correction increases the imbalance by reducing the already smaller basis set error of the monomer even further. Counterpoise cor. bond energies then deviate more from the basis set limit nos. than uncorrected bond energies. These conclusions hold both at the Hartree-Fock level and (much stronger) at the correlated (CCSD-(T), full CI) levels. So the answer to the title question is No.
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    75. 75
      van Duijneveldt, F. B.; van Duijneveldt-van de Rijdt, J. G. C. M.; van Lenthe, J. H. State of the Art in Counterpoise Theory. Chem. Rev. 1994, 94 (7), 1873– 1885,  DOI: 10.1021/cr00031a007
      75
      State of the Art in Counterpoise Theory
      van Duijneveldt, Frans B.; van Duijneveldt-van de Rijdt, Jeanne G. C. M.; van Lenthe, Joop H.
      Chemical Reviews (Washington, DC, United States) (1994), 94 (7), 1873-85CODEN: CHREAY; ISSN:0009-2665.
      A review with 95 refs.
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      https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2cXmtlymsbk%253D&md5=0fc94deb94b36f76be21fad52ed11691
    76. 76
      Burns, L. A.; Marshall, M. S.; Sherrill, C. D. Comparing Counterpoise-Corrected, Uncorrected, and Averaged Binding Energies for Benchmarking Noncovalent Interactions. J. Chem. Theory Comput. 2014, 10 (1), 49– 57,  DOI: 10.1021/ct400149j
      76
      Comparing Counterpoise-Corrected, Uncorrected, and Averaged Binding Energies for Benchmarking Noncovalent Interactions
      Burns, Lori A.; Marshall, Michael S.; Sherrill, C. David
      Journal of Chemical Theory and Computation (2014), 10 (1), 49-57CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      High-quality benchmark computations are crit. for the development and assessment of approx. methods to describe noncovalent interactions. Recent advances in the treatment of dispersion by d. functional theory and also the development of more efficient wave function techniques to reliably address noncovalent interactions motivate new benchmark computations of increasing accuracy. This work considers focal point approxns. to est. the complete basis set limit of coupled-cluster theory through perturbative triples [CCSD-(T)/CBS] and evaluates how this approach is affected by the use or absence of counterpoise (CP) correction or, as has recently gained traction, the av. of those values. Current benchmark protocols for interaction energies are computed with all CP procedures and assessed against the A24 and S22B databases and also to highly converged results for formic acid, cyanogen, and benzene dimers. Whether CP correction, no correction, or the av. is favored depends upon the theor. method, basis set, and binding motif. In recent high-quality benchmark studies, interaction energies often use second-order perturbation theory with extrapolated aug-cc-pVTZ (aTZ) and aug-cc-pVQZ (aQZ) basis sets [MP2/aTQZ] combined with a "coupled-cluster correction," δMP2CCSD(T), evaluated in an aug-cc-pVDZ basis. For such an approach, averaging CP-cor. and uncorrected values for the MP2 component and using CP-cor. δMP2CCSD(T) values offers errors more balanced among binding motifs and generally more favorable overall. Other combinations of counterpoise correction are not quite as accurate. When employing MP2/aQ5Z extrapolations and an aTZ basis for δMP2CCSD(T), using CP-cor. or averaged MP2 ests. are about equally effective (and slightly superior to uncorrected MP2 values), but the counterpoise treatment of δMP2CCSD(T) makes little difference. Focal point ests. at this level achieve benchmark quality results otherwise accessible only with CCSD-(T)/aQZ or better.
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    77. 77
      Helgaker, T.; Klopper, W.; Koch, H.; Noga, J. Basis-set convergence of correlated calculations on water. J. Chem. Phys. 1997, 106 (23), 9639– 9646,  DOI: 10.1063/1.473863
      77
      Basis-set convergence of correlated calculations on water
      Helgaker, Trygve; Klopper, Wim; Koch, Henrik; Noga, Jozef
      Journal of Chemical Physics (1997), 106 (23), 9639-9646CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      The basis-set convergence of the electronic correlation energy in the water mol. is investigated at the second-order Moller-Plesset level and at the coupled-cluster singles-and-doubles level with and without perturbative triples corrections applied. The basis-set limits of the correlation energy are established to within 2mEh by means of (1) extrapolations from sequences of calcns. using correlation-consistent basis sets and (2) from explicitly correlated calcns. employing terms linear in the inter-electronic distances rij. For the extrapolations to the basis-set limit of the correlation energies, fits of the form a + bX-3 (where X is two for double-zeta sets, three for triple-zeta sets, etc.) are found to be useful. CCSD(T) calcns. involving as many as 492 AOs are reported.
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    78. 78
      Wei, Z.; Martirez, J. M. P.; Carter, E. A. Introducing the embedded random phase approximation: H2 dissociative adsorption on Cu(111) as an exemplar. J. Chem. Phys. 2023, 159 (19), 194108,  DOI: 10.1063/5.0181229
      There is no corresponding record for this reference.
    79. 79
      Zhou, L.; Swearer, D. F.; Zhang, C.; Robatjazi, H.; Zhao, H.; Henderson, L.; Dong, L.; Christopher, P.; Carter, E. A.; Nordlander, P. Quantifying hot carrier and thermal contributions in plasmonic photocatalysis. Science 2018, 362 (6410), 69– 72,  DOI: 10.1126/science.aat6967
      79
      Quantifying hot carrier and thermal contributions in plasmonic photocatalysis
      Zhou, Linan; Swearer, Dayne F.; Zhang, Chao; Robatjazi, Hossein; Zhao, Hangqi; Henderson, Luke; Dong, Liangliang; Christopher, Phillip; Carter, Emily A.; Nordlander, Peter; Halas, Naomi J.
      Science (Washington, DC, United States) (2018), 362 (6410), 69-72CODEN: SCIEAS; ISSN:0036-8075. (American Association for the Advancement of Science)
      Photocatalysis based on optically active, "plasmonic" metal nanoparticles has emerged as a promising approach to facilitate light-driven chem. conversions under far milder conditions than thermal catalysis. However, an understanding of the relation between thermal and electronic excitations has been lacking. We report the substantial light-induced redn. of the thermal activation barrier for ammonia decompn. on a plasmonic photocatalyst. We introduce the concept of a light-dependent activation barrier to account for the effect of light illumination on electronic and thermal excitations in a single unified picture. This framework provides insight into the specific role of hot carriers in plasmon-mediated photochem., which is critically important for designing energy-efficient plasmonic photocatalysts.
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    80. 80
      Finley, J.; Malmqvist, P.-Å.; Roos, B. O.; Serrano-Andrés, L. The multi-state CASPT2 method. Chem. Phys. Lett. 1998, 288 (2–4), 299– 306,  DOI: 10.1016/S0009-2614(98)00252-8
      80
      The multi-state CASPT2 method
      Finley, James; Malmqvist, Per-Ake; Roos, Bjorn O.; Serrano-Andres, Luis
      Chemical Physics Letters (1998), 288 (2,3,4), 299-306CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)
      An extension of the multiconfigurational second-order perturbation approach CASPT2 is suggested, where several electronic states are coupled at second order via an effective-Hamiltonian approach. The method has been implemented into the MOLCAS-4 program system, where it will replace the single-state CASPT2 program. The accuracy of the method is illustrated through calcns. of the ionic-neutral avoided crossing in the potential curves for LiF and of the valence-Rydberg mixing in the V-state of the ethylene mol.
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    • Plots of all geometries, clean-surface approximation energetics, cluster-size effects, PW to GTO basis set effects, Pd 4s4p dynamic correlation effects, overview of BSSE and CP, plots and energetics of geometry effects on Vemb, NO plots of all structures surveyed, and SA-CASPT2 ground- and excited-state MEPs utilizing different state averaging (PDF)

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