Virtual Orbitals Research Articles

Orbital-optimized density functional calculations of molecular Rydberg excited states with real space grid representation and self-interaction correction.

Density functional calculations of Rydberg excited states up to high energy are carried out for several molecules using an approach where the orbitals are variationally optimized by converging on saddle points on the electronic energy surface within a real space grid representation. Remarkably good agreement with experimental estimates of the excitation energy is obtained using the generalized gradient approximation (GGA) functional of Perdew, Burke, and Ernzerhof (PBE) when Perdew-Zunger self-interaction correction is applied in combination with complex-valued orbitals. Even without the correction, the PBE functional gives quite good results despite the fact that corresponding Rydberg virtual orbitals have positive energy in the ground state calculation. Results obtained using the Tao, Perdew, Staroverov, and Scuseria (TPSS) and r2SCAN meta-GGA functionals are also presented, but they do not provide a systematic improvement over the results from the uncorrected PBE functional. The grid representation combined with the projector augmented-wave approach gives a simpler and better representation of diffuse Rydberg orbitals than a linear combination of atomic orbitals with commonly used basis sets, the latter leading to an overestimation of the excitation energy due to confinement of the excited states.

Accelerating Coupled-Cluster Calculations with GPUs: An Implementation of the Density-Fitted CCSD(T) Approach for Heterogeneous Computing Architectures Using OpenMP Directives.

An algorithm is presented for the coupled-cluster singles, doubles, and perturbative triples correction [CCSD(T)] method based on the density fitting or the resolution-of-the-identity (RI) approximation for performing calculations on heterogeneous computing platforms composed of multicore CPUs and graphics processing units (GPUs). The directive-based approach to GPU offloading offered by the OpenMP application programming interface has been employed to adapt the most compute-intensive terms in the RI-CCSD amplitude equations with computational costs scaling as , , and (where NO and NV denote the numbers of correlated occupied and virtual orbitals, respectively) and the perturbative triples correction to execute on GPU architectures. The pertinent tensor contractions are performed using an accelerated math library such as cuBLAS or hipBLAS. Optimal strategies are discussed for splitting large data arrays into tiles to fit them into the relatively small memory space of the GPUs, while also minimizing the low-bandwidth CPU-GPU data transfers. The performance of the hybrid CPU-GPU RI-CCSD(T) code is demonstrated on pre-exascale supercomputers composed of heterogeneous nodes equipped with NVIDIA Tesla V100 and A100 GPUs and on the world's first exascale supercomputer named "Frontier", the nodes of which consist of AMD MI250X GPUs. Speedups within the range 4-8× relative to the recently reported CPU-only algorithm are obtained for the GPU-offloaded terms in the RI-CCSD amplitude equations. Applications to polycyclic aromatic hydrocarbons containing 16-66 carbon atoms demonstrate that the acceleration of the hybrid CPU-GPU code for the perturbative triples correction relative to the CPU-only code increases with the molecule size, attaining a speedup of 5.7× for the largest circumovalene molecule (C66H20). The GPU-offloaded code enables the computation of the perturbative triples correction for the C60 molecule using the cc-pVDZ/aug-cc-pVTZ-RI basis sets in 7 min on Frontier when using 12,288 AMD GPUs with a parallel efficiency of 83.1%.

Integrating Explainability into Graph Neural Network Models for the Prediction of X-ray Absorption Spectra.

The use of sophisticated machine learning (ML) models, such as graph neural networks (GNNs), to predict complex molecular properties or all kinds of spectra has grown rapidly. However, ensuring the interpretability of these models' predictions remains a challenge. For example, a rigorous understanding of the predicted X-ray absorption spectrum (XAS) generated by such ML models requires an in-depth investigation of the respective black-box ML model used. Here, this is done for different GNNs based on a comprehensive, custom-generated XAS data set for small organic molecules. We show that a thorough analysis of the different ML models with respect to the local and global environments considered in each ML model is essential for the selection of an appropriate ML model that allows a robust XAS prediction. Moreover, we employ feature attribution to determine the respective contributions of various atoms in the molecules to the peaks observed in the XAS spectrum. By comparing this peak assignment to the core and virtual orbitals from the quantum chemical calculations underlying our data set, we demonstrate that it is possible to relate the atomic contributions via these orbitals to the XAS spectrum.

Linear Scaling Calculations of Excitation Energies with Active-Space Particle-Particle Random-Phase Approximation.

We developed an efficient active-space particle-particle random-phase approximation (ppRPA) approach to calculate accurate charge-neutral excitation energies of molecular systems. The active-space ppRPA approach constrains both indexes in particle and hole pairs in the ppRPA matrix, which only selects frontier orbitals with dominant contributions to low-lying excitation energies. It employs the truncation in both orbital indexes in the particle-particle and the hole-hole spaces. The resulting matrix, whose eigenvalues are excitation energies, has a dimension that is independent of the size of the systems. The computational effort for the excitation energy calculation, therefore, scales linearly with system size and is negligible compared with the ground-state calculation of the (N - 2)-electron system, where N is the electron number of the molecule. With the active space consisting of 30 occupied and 30 virtual orbitals, the active-space ppRPA approach predicts the excitation energies of valence, charge-transfer, Rydberg, double, and diradical excitations with the mean absolute errors (MAEs) smaller than 0.03 eV compared with the full-space ppRPA results. As a side product, we also applied the active-space ppRPA approach in the renormalized singles (RS) T-matrix approach. Combining the non-interacting pair approximation that approximates the contribution to the self-energy outside the active space, the active-space GRSTRS@PBE approach predicts accurate absolute and relative core-level binding energies with the MAEs around 1.58 and 0.3 eV, respectively. The developed linear scaling calculation of excitation energies is promising for applications to large and complex systems.

Four-Membered Bimetallic Carbon Complex [(MCp*)2(μ-NPh)(μ-C)] (M = Fe, Ru, Os) - An Ambiphilic Carbene with Lone Pair and σ-Hole Reactivity.

Singlet carbenes are extensively studied compounds capable of electrophilic, nucleophilic or ambiphillic behaviour. The ambiphilic reactivity of singlet carbenes have been conventionally observed in orthogonal planes. Here, we report a detailed bonding and reactivity study of a homobimetallic carbon complex [(MCp*)2(μ-NPh)(μ-C)] (1M, M = Fe, Ru, Os) that shows ambiphilicity in the same direction. The structure of this complex can be considered as two fused three-membered M-C-M and M-N-M rings. The bonding analysis suggests that these 17VE homobimetallic complexes have one formal M-M bond with a bridging carbene centre featuring a high lying spn-hybridised lone pair. Accordingly, the carbene centre shows high proton affinity and act as a good 2e- donor to Lewis acids and transition metal fragment. Apart from the transition metal non-bonding electrons, the π-framework of M-C-M and M-N-M arms can be best described as 3c-2e- bonds. The two transition metals in the four-membered skeleton generate many low-lying, virtual orbitals. These low-lying virtual orbitals induces electron excitation from the spn-hybrid orbital in presence of H- and other 2e- donor ligands such as PMe3, NHC and CO. Hence, the spn-hybrid lone pair orbital shows σ-hole reactivity in presence of Lewis bases.

Local N-electron valence state perturbation theory using pair-natural orbitals based on localized virtual molecular orbitals.

Second-order N-electron valence state perturbation theory (NEVPT2) is an exactly size-consistent and intruder-state-free multi-reference theory. To accelerate the NEVPT2 computation, Guo and Neese combined it with the local pair-natural orbital (PNO) method using the projected atomic orbitals (PAOs) as the underlying local basis [Guo et al., J. Chem. Phys. 144, 094111 (2016)]. In this paper, we report the further development of the PNO-NEVPT2 method using the orthonormal and non-redundant localized virtual molecular orbitals (LVMOs) instead of PAOs. The LVMOs were previously considered to perform relatively poor compared to PAOs because the resulting orbital domains were unacceptably large. Our prior work, however, showed that this drawback can be remedied by re-forming the domain construction scheme using differential overlap integrals [Saitow et al., J. Chem. Phys. 157, 084101 (2022)]. In this work, we develop further refinements to enhance the feasibility of using LVMOs. We first developed a two-level semi-local approach for screening out so-called weak-pairs to select or truncate the pairs for PNO constructions more flexibly. As a refinement specific to the Pipek-Mezey localization for LVMOs, we introduced an iterative scheme to truncate the Givens rotations using varying thresholds. We assessed the LVMO-based PNO-NEVPT2 method through benchmark calculations for linear phenyl alkanes, which demonstrate that it performs comparably well relative to the PAO-based approach. In addition, we evaluated the Co-C bond dissociation energies for the cobalamin derivatives composed of 200 or more atoms, which confirms that the LVMO-based method can recover more than 99.85% of the canonical NEVPT2 correlation energy.

Enabling Molecular-Level Computational Description of Redox and Proton-Coupled Electron Transfer Reactions of Samarium Diiodide.

Samarium diiodide (SmI2, Kagan's reagent) is a one-electron reductant with applications ranging from organic synthesis to nitrogen fixation. Highly inaccurate relative energies of redox and proton-coupled electron transfer (PCET) reactions of Kagan's reagent are predicted by pure and hybrid density functional approximations (DFAs) when only scalar relativistic effects are accounted for. Calculations including spin-orbit coupling (SOC) show that the SOC-induced differential stabilization of the Sm(III) versus the Sm(II) ground state is little affected by ligands and solvent, and a standard SOC correction derived from atomic energy levels is thus included in the reported relative energies. With this correction, selected meta-GGA and hybrid meta-GGA functionals predict Sm(III)/Sm(II) reduction free energies to within 5 kcal/mol of the experiment. Considerable discrepancies remain, however, in particular for the PCET-relevant O-H bond dissociation free energies, for which no regular DFA is within 10 kcal/mol of the experiment or CCSD(T). The main cause behind these discrepancies is the delocalization error, which leads to excess ligand-to-metal electron donation and destabilizes Sm(III) versus Sm(II). Fortunately, static correlation is unimportant for the present systems, and the error may be reduced by including information from virtual orbitals via perturbation theory. Contemporary, parametrized double-hybrid methods offer promise as companions to experimental campaigns in the further development of the chemistry of Kagan's reagent.

Natural virtual orbitals for the GW method in the random-phase approximation and beyond.

The increasingly popular GW method is becoming a convenient tool to determine vertical ionization energies in molecular systems. However, depending on the formalism used and the range of orbitals investigated, it may be hampered by a steep computational scaling. To alleviate this issue, correlated natural virtual orbitals (NVOs) based on second-order Møller-Plesset (MP2) and direct MP2 correlation energies are implemented, and the resulting correlated NVOs are tested on GW quasiparticle energies. Test cases include the popular GW variants G0W0 and evGW0 as well as more elaborate vertex corrections. We find that for increasingly larger molecular systems and basis sets, NVOs considerably improve efficiency. Furthermore, we test the performance of the truncated (frozen) NVO ansatz on the GW100 test set. For the latter, it is demonstrated that, using a carefully chosen truncation threshold, NVOs lead to a negligible loss in accuracy while providing speedups of one order of magnitude. Furthermore, we compare the resulting quasiparticle energies to very accurate vertical ionization energies obtained from coupled-cluster theory with singles, doubles, and noniterative triples [CCSD(T)], confirming that the loss in accuracy introduced by truncating the NVOs is negligible compared to the methodical errors in the GW approximation. It is also demonstrated that the choice of basis set impacts results far more than using a suitably truncated NVO space. Therefore, at the same computational expense, more accurate results can be obtained using NVOs. Finally, we provide improved reference CCSD(T) values for the GW100 test set, which have been obtained using the def2-QZVPP basis set.

Real-Time Equation-of-Motion Coupled-Cluster Cumulant Green's Function Method: Heterogeneous Parallel Implementation Based on the Tensor Algebra for Many-Body Methods Infrastructure.

We report the implementation of the real-time equation-of-motion coupled-cluster (RT-EOM-CC) cumulant Green's function method [ J. Chem. Phys. 2020, 152, 174113] within the Tensor Algebra for Many-body Methods (TAMM) infrastructure. TAMM is a massively parallel heterogeneous tensor library designed for utilizing forthcoming exascale computing resources. The two-body electron repulsion matrix elements are Cholesky-decomposed, and we imposed spin-explicit forms of the various operators when evaluating the tensor contractions. Unlike our previous real algebra Tensor Contraction Engine (TCE) implementation, the TAMM implementation supports fully complex algebra. The RT-EOM-CC singles (S) and doubles (D) time-dependent amplitudes are propagated using a first-order Adams-Moulton method. This new implementation shows excellent scalability tested up to 500 GPUs using the Zn-porphyrin molecule with 655 basis functions, with parallel efficiencies above 90% up to 400 GPUs. The TAMM RT-EOM-CCSD was used to study core photoemission spectra in the formaldehyde and ethyl trifluoroacetate (ESCA) molecules. Simulations of the latter involve as many as 71 occupied and 649 virtual orbitals. The relative quasiparticle ionization energies and overall spectral functions agree well with available experimental results.

Visualizing and comparing quantum interference in the π-system and σ-system of organic molecules.

Quantum interference effects in conjugated molecules have been well-explored, with benzene frequently invoked as a pedagogical example. These interference effects have been understood through a quantum interference map in which the electronic transmission is separated into interfering and non-interfering terms, with a focus on the π-orbitals for conjugated molecules. Recently, saturated molecules have also been reported to exhibit destructive quantum interference effects; however, the very different σ-orbital character in these molecules means that it is not clear how orbital contributions manifest. Herein, we demonstrate that the quantum interference effects in conjugated molecules are quite different from those observed in saturated molecules, as demonstrated by the quantum interference map. While destructive interference at the Fermi energy in the π-system of benzene arises from interference terms between paired occupied and virtual orbitals, this is not the case at the Fermi energy in saturated systems. Instead, destructive interference is evident when contributions from a larger number of non-paired orbitals cancel, leading to more subtle and varied manifestations of destructive interference in saturated systems.

Further investigations into a Laplace MP2 method using range separated Coulomb potential and orbital selective virtuals: Multipole correction, OSV extrapolation, and critical assessment.

We report further investigations to aid the development of a Laplace MP2 (second-order Møller Plesset) method with a range separated Coulomb potential partitioned into short- and long-range parts. The implementation of the method extensively uses sparse matrix algebra, density fitting techniques for the short-range part, and a Fourier transformation in spherical coordinates for the long-range part of the potential. Localized molecular orbitals are employed for the occupied space, whereas virtual space is described by orbital specific virtual orbitals (OSVs) associated with localized molecular orbitals. The Fourier transform is deficient for very large distances between localized occupied orbitals, and a multipole expansion for widely separated pairs is introduced for the direct MP2 contribution, which is applicable also to non-Coulombic potentials that do not satisfy the Laplace equation. For the exchange contribution, an efficient screening of contributing localized occupied pairs is employed, which is discussed more completely here. To mitigate errors due to the truncation of OSVs, a simple and efficient extrapolation procedure is used to obtain results close to MP2 for the full basis set of atomic orbitals Using a suitable set of default parameters, the accuracy of the approach is demonstrated. The current implementation of the approach is not very efficient, and the aim of this paper is to introduce and critically discuss ideas that can have more general applicability beyond MP2 calculations for large molecules.

Theoretical survey on the electronic, linear and nonlinear optical properties of substituted benzenes and polycondensed π-systems. A density functional theory study

Herein dipole moments, static and dynamic (hyper)polarizabilities, delocalization energies, distortions of the phenyl rings of the substituted benzenes are computed using density functional theory procedure. It is revealed that the biphenyl molecule exhibits the largest frequency-dependent second order hyperpolarizabilities. Furthermore, the application of an external electric field along the direction parallel to the molecular plane induces a considerable decrease in the energy of the unoccupied orbitals of biphenyl, resulting in more red-shifted virtual orbitals compared with benzene and the highest static second-order hyperpolarizability for the molecule. It is also found that when the two adjacent rings share only one carbon atom, high σ and π-electronic contributions to the second-order hyperpolarizabilities are expected. Interactions of the type π* → π* with high delocalization energies are predicted for pyridine, nitrobenzene, chlorobenzene, bromobenzene, phenol, and benzoic acid. The quantum–mechanical calculations of the present work may provide theoretical insights into designing high (hyper)polarizability materials.

Local Potentials Reconstructed within Linearly Independent Product Basis Sets of Increasing Size.

Given a matrix representation of a local potential v(r) within a one-electron basis set of functions that form linearly independent products (LIP), it is possible to construct a well-defined local potential that is equivalent to v(r) within that basis set and has the form of an expansion in basis function products. Recently, we showed that for exchange-correlation potentials vXC(r) defined on the infinite-dimensional Hilbert space, the potentials reconstructed from matrices of vXC(r) within minimal LIP basis sets of occupied Kohn-Sham orbitals bear only qualitative resemblance to the originals. Here, we show that if the LIP basis set is enlarged by including low-lying virtual Kohn-Sham orbitals, the agreement between and vXC(r) improves to the extent that the basis function products are appropriate as a basis for vXC(r). These findings validate the LIP technology as a rigorous potential reconstruction method.

Theoretical Investigation of the X-ray Stark Effect in Small Molecules.

We have studied the Stark effect in the soft x-ray region for various small molecules by calculating the field-dependent x-ray absorption spectra. This effect is explained in terms of the response of molecular orbitals (core and valence), the molecular dipole moment, and the molecular geometry to the applied electric field. A number of consistent trends are observed linking the computed shifts in absorption energies and intensities with specific features of the molecular electronic structure. We find that both the virtual molecular orbitals (valence and/or Rydberg) as well as the core orbitals contribute to observed trends in a complementary fashion. This initial study highlights the potential impact of x-ray Stark spectroscopy as a tool to study electronic structure and environmental perturbations at a submolecular scale.

Sparsity of the electron repulsion integral tensor using different localized virtual orbital representations in local second-order Møller-Plesset theory.

Utilizing localized orbitals, local correlation theory can reduce the unphysically high system-size scaling of post-Hartree-Fock (post-HF) methods to linear scaling in insulating molecules. The sparsity of the four-index electron repulsion integral (ERI) tensor is central to achieving this reduction. For second-order Møller-Plesset theory (MP2), one of the simplest post-HF methods, only the (ia|jb) ERIs are needed, coupling occupied orbitals i, j and virtuals a, b. In this paper, we compare the numerical sparsity (called the "ragged list") and two other approaches revealing the low-rank sparsity of the ERI. The ragged list requires only one set of (localized) virtual orbitals, and we find that the orthogonal valence virtual-hard virtual set of virtuals originally proposed by Subotnik et al. gives the sparsest ERI tensor. To further compress the ERI tensor, the pair natural orbital (PNO) type representation uses different sets of virtual orbitals for different occupied orbital pairs, while the occupied-specific virtual (OSV) approach uses different virtuals for each occupied orbital. Our results indicate that while the low-rank PNO representation achieves significant rank reduction, it also requires more memory than the ragged list. The OSV approach requires similar memory to that of the ragged list, but it involves greater algorithmic complexity. An approximation (called the "fixed sparsity pattern") for solving the local MP2 equations using the numerically sparse ERI tensor is proposed and tested to be sufficiently accurate and to have highly controllable error. A low-scaling local MP2 algorithm based on the ragged list and the fixed sparsity pattern is therefore promising.

The potential of pure and atom-decorated AlP nano-sheet as a drug delivery system for procarbazine: A DFT approach

The impact of decorating the Y (Y = Ti or Sc) atom on the procarbazine (PA) drug delivery performance of an AlP nano-sheet (AlP-NS) was investigated by performing DFT calculations. It was found that the pristine AlP-NS was not a good choice for the drug delivery of PA. Decorating Ti and Sc into the AlP-NS surface increased the adsorption energy of PA from −6.2 to −29.7 and −31.9 kcal/mol, respectively. The Y atom considerably contributed to the generation of virtual orbitals of Y-decorated AlP-NS, which showed that these atoms were more suitable for a nucleophilic attack compared to Al. Finally, the adsorption performance and capacity of PA increased after the decoration of the Y atom, which made the nano-sheet more suitable for the drug delivery of PA. In cancerous cells with a low pH, PA was significantly protonated, which led to the separation of PA from the Y-decorated AlP-NS surface.

Particle-Breaking Hartree-Fock Theory for Open Molecular Systems.

In this work we present the particle-breaking Hartree-Fock (PBHF) model which is a mean-field approach to open molecular systems. The interaction of a system with the environment is parametrized through a particle-breaking term in the molecular Hamiltonian. The PBHF wave function is constructed through an exponential unitary transformation of a Slater determinant with a given number of electrons. We consider only the closed-shell formalism. The parametrization results in a linear combination of Slater determinants with different numbers of electrons, i.e., the PBHF wave function is not an eigenfunction of the number operator. As a result, the density matrix may have fractional occupations which are, because of the unitary parametrization, always between 0.0 and 2.0. The occupations are optimized simultaneously with the orbitals, using the trust-region optimization procedure. In the limit of a particle-conserving Hamiltonian, the PBHF optimization will converge to a standard Hartree-Fock wave function. We show that the average number of electrons may be decreased or increased depending on whether the particle-breaking term affects occupied or virtual orbitals.

Electron Propagator Theory of Vertical Electron Detachment Energies of Anions: Benchmarks and Applications to Nucleotides

A new generation of ab initio electron-propagator self-energy approximations that are free of adjustable parameters is tested on a benchmark set of 55 vertical electron detachment energies of closed-shell anions. Comparisons with older self-energy approximations indicate that several new methods that make the diagonal self-energy approximation in the canonical Hartree-Fock orbital basis provide superior accuracy and computational efficiency. These methods and their acronyms, mean absolute errors (in eV), and arithmetic bottlenecks expressed in terms of occupied (O) and virtual (V) orbitals are the opposite-spin, non-Dyson, diagonal second-order method (os-nD-D2, 0.2, OV2), the approximately renormalized quasiparticle third-order method (Q3+, 0.15, O2V3) and the approximately renormalized, non-Dyson, linear, third-order method (nD-L3+, 0.1, OV4). The Brueckner doubles with triple field operators (BD-T1) nondiagonal electron-propagator method provides such close agreement with coupled-cluster single, double, and perturbative triple replacement total energy differences that it may be used as an alternative means of obtaining standard data. The new methods with diagonal self-energy matrices are the foundation of a composite procedure for estimating basis-set effects. This model produces accurate predictions and clear interpretations based on Dyson orbitals for the photoelectron spectra of the nucleotides found in DNA.

Studying Ga and Ge-doped AlP nanotube as a drug carrier for ciclopirox anticancer drug using DFT

The impact of Ga and Ge-doping on the ciclopirox (CPX) drug delivery performance of an AlP nanotube (AlPNT) was scrutinized computationally. It was found that the pristine AlPNT was not a good choice for this drug delivery. Doping of the Ga and Ge into the AlPNT increased the adsorption energy of CPX from −8.0 to −26.8 and −29.2 kcal/mol, respectively. The Ge atom considerably contributed to the generation of the virtual orbitals of the Ge-doped AlPNT, which showed that this atom was suitable for nucleophilic attack compared to the Al atoms. Finally, the adsorption performance and capacity of CPX increased after doping Ga and Ge, which made the AlPNT more suitable for drug delivery. A mechanism for drug release was introduced in cancerous tissues, which showed that in cancerous cells with low pH, CPX was significantly protonated, which led to the separation of CPX from the Ge-doped AlPNT.

Periodic plane-wave electronic structure calculations on quantum computers

A procedure for defining virtual spaces, and the periodic one-electron and two-electron integrals, for plane-wave second quantized Hamiltonians has been developed, and it was validated using full configuration interaction (FCI) calculations, as well as executions of variational quantum eigensolver (VQE) circuits on Quantinuum’s ion trap quantum computers accessed through Microsoft’s Azure Quantum service. This work is an extension to periodic systems of a new class of algorithms in which the virtual spaces were generated by optimizing orbitals from small pairwise CI Hamiltonians, which we term as correlation optimized virtual orbitals with the abbreviation COVOs. In this extension, the integration of the first Brillouin zone is automatically incorporated into the two-electron integrals. With these procedures, we have been able to derive virtual spaces, containing only a few orbitals, that were able to capture a significant amount of correlation. The focus in this manuscript is on comparing the simulations of small molecules calculated with plane-wave basis sets with large periodic unit cells at the Gamma-point, including images, to results for plane-wave basis sets with aperiodic unit cells. The results for this approach were promising, as we were able to obtain good agreement between periodic and aperiodic results for an LiH molecule. Calculations performed on the Quantinuum H1-1 quantum computer produced surprisingly good energies, in which the error mitigation played a small role in the quantum hardware calculations and the (noisy) quantum simulator results. Using a modest number of circuit runs (500 shots), we reproduced the FCI values for the 1 COVO Hamiltonian with an error of 11 milliHartree, which is expected to improve with a larger number of circuit runs.