Fock Matrix Research Articles

The interplay and strength of the π⋯HF, C⋯HF, F⋯HF and F⋯HC hydrogen bonds upon the formation of multimolecular complexes based on C2H2⋯HF and C2H4⋯HF small dimers.

The conception of this theoretical research was idealized aiming to unveil the intermolecular structures of complexes formed by acetylene or ethylene and hydrofluoric acid. At light of computational calculations by using the B3LYP/6-311++G(d,p) method, the geometries of the C2H2⋯(HF), C2H2⋯2(HF), C2H2⋯4(HF), C2H4⋯(HF), C2H4⋯2(HF) and C2H4⋯4(HF) hydrogen-bonded complexes were fully optimized. Moreover, the Post-Hartree-Fock calculations MP2/6-311++G(d,p), MP2/aug-cc-pVTZ, MP4(SDQ)/6-311++G(d,p) and CCSD/6-311++G(d,p) also were also used. The infrared spectra were analyzed in order to identify the new vibrational modes and frequencies of the proton donors shifted to red region. Through the modeling of charge-fluxes on the basis of the Quantum Theory of Atoms In Molecules (QTAIM) and, by contradicting the expectation of the hydrofluorination mechanisms of acetylene or ethylene, C⋯HF was recognized as a new type of hydrogen bond instead of the already well known π⋯H. The calculations of the Natural Bonding Orbital (NBO) and Charges derived from the Electrostatic Potential Grid-based (ChElPG) were also applied to interpret the shifting frequencies as well as measuring of the punctual charge-transfer after the formation of the complexes. Finally, the determination of the stabilization energy was carried out through the arguments of the Fock matrix in NBO basis and through the supermolecule approach. Also it is worthwhile to notice that some algebraic formulations were used for determining the electronic cooperative effect (CE).

Prediction of many-electron wavefunctions using atomic potentials: extended basis sets and molecular dissociation.

A one-electron Schrödinger equation based on special one-electron potentials for atoms is shown to exist that produces orbitals for an arbitrary molecule that are sufficiently accurate to be used without modification to construct single- and multi-determinant wavefunctions. The exact Hamiltonian is used to calculate the energy variationally and to generate configuration interaction expansions. Earlier work on equilibrium geometries is extended to larger basis sets and molecular dissociation. For a test set of molecules representing different bonding environments, a single set of invariant atomic potentials gives wavefunctions with energies that deviate from configuration interaction energies based on SCF orbitals by less than 0.04 eV per bond or valence electron pair. On a single diagonalization of the Fock matrix, the corresponding errors are reduced 0.01 eV. Atomization energies are also in good agreement with CI values based on canonical SCF orbitals. Configuration interaction applications to single bond dissociations of water and glycine, and multiple bond dissociations of ethylene and oxygen produce dissociation energy curves in close agreement with CI calculations based on canonical SCF orbitals for the entire range of internuclear distances.

Computational construction of the electronic Hamiltonian for photoinduced electron transfer and Redfield propagation.

The construction of open-system diabatic Hamiltonians relevant for investigation of electron transfer processes is a computational challenge. In this paper, we present how the full system Hamiltonian, as well as relevant system-bath coupling parameters can be computed from a purely computational starting point. We have investigated two methods for calculating electronic couplings, Generalized Mulliken Hush (GMH) and Fock Matrix Reconstruction (FMR). We apply these methods to calculate the couplings in a model molecular triad, thus constructing the system-Hamiltonian in a diabatic basis. The triad is constructed with a donor-antenna-acceptor type architecture, and a two-step photoinduced electron transfer is expected in this system. With the calculated electronic couplings in combination with Huang-Rhys type electron-phonon couplings, we are able to construct two open-system Hamiltonians from a computational bottom-up approach, where the phonon-reservoir is approximated as harmonic oscillators. Based on these Hamiltonians, two separate propagations of populations are performed using the Redfield formalism. Based on the dynamics, we observe small differences between the results from the GMH and FMR simulations. The overall picture is similar for the two methods. Thereby, we conclude that the FMR approach is suitable as an initial screening tool for identifying long-lived photoinduced charge separated states and that a GMH based Hamiltonian can then be constructed to scrutinize promising candidate molecules. Furthermore, either method can be used to construct all relevant operators needed for the Redfield tensor, without prior knowlegde from experimental data.

Generalized Hartree-Fock with Nonperturbative Treatment of Strong Magnetic Fields: Application to Molecular Spin Phase Transitions.

In this work, we present a framework of an ab initio variational approach to effectively explore electronic spin phase transitions in molecular systems inside of a homogeneous magnetic field. In order to capture this phenomenon, the complex generalized Hartree-Fock ([Formula: see text]) method is used in the spinor formalism with London orbitals. Recursive algorithms for computing the one- and two-electron integrals of London orbitals are also provided. A Pauli matrix representation of the [Formula: see text] method is introduced to separate spin contributions from the scalar part of the Fock matrix. Next, spin phase transitions in two different molecular systems are investigated in the presence of a strong magnetic field. Noncollinear spin configurations are observed during the spin phase transitions in H2 and a dichromium complex, [(H3N)4Cr(OH)2Cr(NH3)4]4+, with an increase in magnetic field strength. The competing driving forces of exchange coupling and the spin Zeeman effect have been shown to govern the spin phase transition and its transition rate. Additionally, the energetic contributions of the spin Zeeman, orbital Zeeman, and diamagnetic terms to the potential energy surface are also analyzed.

Development of the fragment-based COHSEX method for large and complex molecular systems

We present a fragment-based many-body Green's function method suitable for treating large molecular systems in heterogeneous polarizable environments. The Green's function for a total system is approximated from fragment Green's functions and is expanded up to two-body terms. The screened Coulomb potential is approximated from the sum of intrafragment density-response functions, with interfragment polarization terms being neglected. The approximations for the Green's function and screened Coulomb potential lead to a many-body expansion of the self-energy. This expansion is essentially equivalent to the many-body expansion of the Fock matrix in the fragment molecular orbital method. To handle large molecular systems, the present implementation relies on the Coulomb hole plus screened exchange (COHSEX) approximation for the $GW$ self-energy. The accuracy of the FMO-COHSEX method was demonstrated in comparison to conventional COHSEX results for organic molecular aggregates. We confirmed that the present fragmentation approximation can provide reasonably accurate results, and mean absolute errors for quasiparticle energies of less than 0.1 eV have been achieved for valence orbitals. We also assessed the accuracy of the COHSEX approximation to describe the effects of molecular aggregation of electronic states, by comparing them with the $GW$ method. The COHSEX approximation has been shown to successfully describe the induced polarization and dispersion effects. As an illustrative application of the present method, we considered the electronic states of the pentacene thin film, which contains 1476 atoms. We investigated the impact of the induced polarization effect in the heterogeneous environment, highlighting the gap renormalization and the polarization-induced localization. This application shows that the present fragment-based method is useful for studying electronic structures of molecular aggregates in complex environments.

Quantum system partitioning at the single-particle level.

We discuss the partitioning of a quantum system through subsystem separation by unitary block-diagonalization (SSUB) applied to a Fock operator. For a one-particle Hilbert space, this separation can be formulated in a very general way. Therefore, it can be applied to very different partitionings ranging from those driven by features in the molecular structure (such as a solute surrounded by solvent molecules or an active site in an enzyme) to those that aim at an orbital separation (such as core-valence separation). Our framework embraces recent developments of Manby and Miller as well as the older ones of Huzinaga and Cantu. Projector-based embedding is simplified and accelerated by SSUB. Moreover, it directly relates to decoupling approaches for relativistic four-component many-electron theory. For a Fock operator based on the Dirac one-electron Hamiltonian, one would like to separate the so-called positronic (negative-energy) states from the electronic bound and continuum states. The exact two-component (X2C) approach developed for this purpose becomes a special case of the general SSUB framework and may therefore be viewed as a system-environment decoupling approach. Moreover, for SSUB, there exists no restriction with respect to the number of subsystems that are generated-in the limit, decoupling of all single-particle states is recovered, which represents exact diagonalization of the problem. The fact that a Fock operator depends on its eigenvectors poses challenges to all system-environment decoupling approaches and is discussed in terms of the SSUB framework. Apart from improved conceptual understanding, these relations bring about technical advances as developments in different fields can immediately cross-fertilize one another. As an important example, we discuss the atomic decomposition of the unitary block-diagonalization matrix in X2C-type approaches that can inspire approaches for the efficient partitioning of large total systems based on SSUB.

An Efficient Hartree-Fock Implementation Based on the Contraction of Integrals in the Primitive Basis.

A new approach to implement the restricted closed-shell Hartree-Fock equation is proposed. In the ansatz presented, the explicit transformation of integrals from the primitive to the atomic-orbital basis is omitted. Instead, the density matrix is transformed to the primitive basis, in which it is contracted with the untransformed integrals. Obtained is the two-electron part of the Fock matrix, which is transformed back to the atomic orbital basis. The remaining steps of the self-consistent field algorithm are then performed as usual. The program presented here incorporates the most important standard techniques, such as integral prescreening, convergence acceleration (via the direct inversion of the iterative subspace ansatz), and the differential density scheme. Test calculations on standard Hartree-Fock problems were compared to the commercially available MOLPRO and TURBOMOLE program packages. Except in a few special cases, the performance of the program presented here is superior, in comparison to those two programs. Accelerations by up to a factor of 5 were found, with respect to MOLPRO calculations, and up to 3 for TURBOMOLE (in the latter case, up to 55 for generalized contracted basis sets). The program structure is independent of the type of radial contraction; however, the best results are obtained for generalized radial contraction basis sets of low contraction. The program is written in C++ and utilizes code generation engines to automatically generate the routines for the integration and density contraction. Streaming SIMD extensions are used explicitly.

Comprehensive Analysis of the Neglect of Diatomic Differential Overlap Approximation.

Many modern semiempirical molecular orbital models are built on the neglect of diatomic differential overlap (NDDO) approximation. An in-depth understanding of this approximation is therefore indispensable to rationalize the success of these semiempirical molecular orbital models and to develop further improvements on them. The NDDO approximation provides a recipe to approximate electron-electron repulsion integrals (ERIs) in a symmetrically orthogonalized basis based on a far smaller number of ERIs in a locally orthogonalized basis. We first analyze the NDDO approximation by comparing ERIs in both bases for a selection of molecules and for a selection of basis sets. We find that the errors in Hartree-Fock and second-order Møller-Plesset perturbation theory energies grow roughly linearly with the number of basis functions. We then examine different approaches to correct for the errors caused by the NDDO approximation and propose a strategy to directly correct for them in the two-electron matrices that enter the Fock operator.

Direct inversion of the iterative subspace with contracted planewave basis functions.

Ways to reduce the computational cost of periodic electronic structure calculations by using basis functions corresponding to linear combinations of planewaves have been examined recently. These contracted planewave (CPW) basis functions correspond to Fourier series representations of atom-centered basis functions, and thus provide access to some beneficial properties of planewave (PW) and localized basis functions. This study reports the development and assessment of a direct inversion of the iterative subspace (DIIS) method that employs unique properties of CPW basis functions to efficiently converge electronic wavefunctions. This method relies on access to a PW-based representation of the electronic structure to provide a means of efficiently evaluating matrix-vector products involving the application of the Fock matrix to the occupied molecular orbitals. These matrix-vector products are transformed into a form permitting the use of direct diagonalization techniques and DIIS methods typically employed with atom-centered basis sets. The abilities of this method are assessed through periodic Hartree-Fock calculations of a range of molecules and solid-state systems. The results show that the method reported in this study is approximately five times faster than CPW-based calculations in which the entire Fock matrix is calculated. This method is also found to be weakly dependent upon the size of the basis set, thus permitting the use of larger CPW basis sets to increase variational flexibility with a minor impact on computational performance. © 2018 Wiley Periodicals, Inc.

How to make continuum solvation incredibly fast in a few simple steps: A practical guide to the domain decomposition paradigm for the conductor‐like screening model

AbstractWe illustrate the domain decomposition Conductor‐like Screening Model (ddCOSMO) implementation and how to couple it with an existing classical or quantum mechanical (QM) code. We review in detail what input needs to be provided to ddCOSMO and how to assemble it, describe how the ddCOSMO equations are solved and how to process the results to assemble the required quantities, such as Fock matrix contributions for the QM case, or forces for the classical one. Throughout the article, we will make explicit references to the ddCOSMO module, which is an open source, Fortran 90 implementation of ddCOSMO that can be downloaded and distributed under the LGPL license.

Communication: Random-phase approximation excitation energies from approximate equation-of-motion coupled-cluster doubles.

The ground-state correlation energy calculated in the random-phase approximation (RPA) is known to be identical to that calculated using a subset of terms appearing in coupled-cluster theory with double excitations (CCD). In particular, for particle-hole (ph) RPA this equivalence requires keeping only those terms that generate time-independent ring diagrams, and for particle-particle (pp) RPA it requires keeping only those terms that generate ladder diagrams. Here I show that these identities extend to excitation energies, for which those calculated in each RPA are identical to those calculated using approximations to equation-of-motion coupled-cluster theory with double excitations (EOM-CCD). The equivalence requires three approximations to EOM-CCD: first, the ground-state CCD amplitudes are obtained from the ring-CCD or ladder-CCD equations (the same as for the correlation energy); second, the EOM eigenvalue problem is truncated to the minimal subspace, which is one particle + one hole for ph-RPA and two particles or two holes for pp-RPA; third, the similarity transformation of the Fock operator must be neglected, as it corresponds to a Brueckner-like dressing of the single-particle propagator, which is not present in the conventional RPA.

Transferability in Machine Learning for Electronic Structure via the Molecular Orbital Basis.

We present a machine learning (ML) method for predicting electronic structure correlation energies using Hartree-Fock input. The total correlation energy is expressed in terms of individual and pair contributions from occupied molecular orbitals, and Gaussian process regression is used to predict these contributions from a feature set that is based on molecular orbital properties, such as Fock, Coulomb, and exchange matrix elements. With the aim of maximizing transferability across chemical systems and compactness of the feature set, we avoid the usual specification of ML features in terms of atom- or geometry-specific information, such atom/element-types, bond-types, or local molecular structure. ML predictions of MP2 and CCSD energies are presented for a range of systems, demonstrating that the method maintains accuracy while providing transferability both within and across chemical families; this includes predictions for molecules with atom-types and elements that are not included in the training set. The method holds promise both in its current form and as a proof-of-principle for the use of ML in the design of generalized density-matrix functionals.

Charge Transfer Excitations with Range Separated Functionals Using Improved Virtual Orbitals.

We present an implementation of range separated functionals utilizing the Slater function on grids in real space in the projector augmented waves method. The screened Poisson equation is solved to evaluate the necessary screened exchange integrals on Cartesian grids. The implementation is verified against existing literature and applied to the description of charge transfer excitations. We find very slow convergence for calculations within linear response time-dependent density functional theory and unoccupied orbitals of the canonical Fock operator. Convergence can be severely improved by using Huzinaga's virtual orbitals instead. This combination furthermore enables an accurate determination of long-range charge transfer excitations by means of ground-state calculations.

Quantum chemical approach for positron annihilation spectra of atoms and molecules beyond plane-wave approximation.

We report theoretical calculations of positron-electron annihilation spectra of noble gas atoms and small molecules using the nuclear orbital plus molecular orbital method. Instead of a nuclear wavefunction, the positronic wavefunction is obtained as the solution of the coupled Hartree-Fock or Kohn-Sham equation for a positron and the electrons. The molecular field is included in the positronic Fock operator, which allows an appropriate treatment of the positron-molecule repulsion. The present treatment succeeds in reproducing the Doppler shift, i.e., full width at half maximum (FWHM) of experimentally measured annihilation (γ-ray) spectra for molecules with a mean absolute error less than 10%. The numerical results indicate that the interpretation of the FWHM in terms of a specific molecular orbital is not appropriate.

Kohn–Sham approach for fast hybrid density functional calculations in real-space numerical grid methods

Real-space methods have not been suitable for hybrid density functional calculations due to high cost coming from the nonlocality of Fock operator. Here we propose a practical approach for fast computation. The key is to use a strictly local Kohn–Sham potential that can be deduced from any hybrid functional using the optimized effective potential method. This new approach improved the computation speed of the global and range-separated hybrid methods up to 30 and 80 times, respectively. As a result, accurate prediction of the size-dependent excitonic spectra of Si quantum dots became feasible.

Accuracy of electron densities obtained via Koopmans-compliant hybrid functionals

We evaluate the accuracy of electron densities and quasiparticle energy gaps given by hybrid functionals by directly comparing these to the exact quantities obtained from solving the many-electron Schrodinger equation. We determine the admixture of Hartree-Fock exchange to approximate exchange-correlation in our hybrid functional via one of several physically justified constraints, including the generalized Koopmans' theorem. We find that hybrid functionals yield strikingly accurate electron densities and gaps in both exchange-dominated and correlated systems. We also discuss the role of the screened Fock operator in the success of hybrid functionals.

A Benchmark Study of Electronic Couplings in Donor-Bridge-Acceptor Systems with the FMR-B Method.

We present a benchmarking study for the evaluation of electronic couplings in donor-bridge-acceptor systems with the Fock matrix reconstruction-bridge (FMR-B) method. We compile a data set for the benchmark that contains 29 molecules for which reliable experimental coupling values are available: DBA29. This data set is general and includes different types of donor, acceptor, and bridge units as well as different bridge lengths, and it spans a range of couplings from 0.1 to 0.8 eV. We use DBA29 to test FMR-B with 11 density functionals belonging to different classes (pure, global hybrid, and range-separated) and the Hartree-Fock (HF) method. We also test a subset of these methods with nine basis sets from the Pople and Dunning families, which include a varying number of polarization and diffuse functions. We find that the best accuracy and lowest computational cost is obtained with range-separated functionals and compact basis sets. Global hybrids with a large amount of HF exchange also work well because of error cancellation between the approximate exchange-correlation kernel and the HF part. Pure functionals, although less accurate, still provide reasonable results with a consistent underestimation of the experimental values, and they can be used for larger and more computationally demanding systems.

Energy decomposition analysis of the intermolecular interaction energy between different gas molecules (H2, O2, H2O, N2, CO2, H2S, and CO) and selected Li+-doped graphitic molecules: DF-SAPT (DFT) calculations

In the present study, the energy decomposition analysis of the intermolecular interaction energy (Eint) between different Li+-doped graphitic molecules (Li+–CxHy) and the gases such as H2, O2, N2, CO, H2S, and H2O was performed using the density fitting–density functional theory–symmetry-adapted perturbation theory [DF-SAPT (DFT)] method. The size effect of the graphitic molecule and its aromaticity on the interaction energy (Eint) and its decomposed components [electrostatic (Eelec), exchange (Eexch), induction (Eind), and dispersion energy (Edisp)] were studied. Also, the second-order perturbation energy analysis of the Fock matrix in NBO basis was performed on all desired dimers, consisting of Li+–C x H y molecules and the selected gases, as well as the Li+–C x H y substrate alone. The aim was to investigate the relation between the charge transfer and (charge) delocalization from the graphitic substrate to the Li+ with the size of the graphitic structures in the presence and absence of the interaction. The calculations showed that the Eint between the Li+-doped graphitic molecules and the gases does not show considerable changes with the size of the graphitic substrate for all selected gas molecules. Also, the value of the Eelec was nearly insensitive to the size of the graphitic substrate, especially for the H2, CO, and N2 molecules, while for the H2O showed a small dependency. Similarly, the Eexch was nearly size independent for the H2 and CO gases; however, Eexch showed some dependencies on the size of the graphitic substrate for the H2O and N2 molecules. This later dependency was higher than that of the Eelec. The variation in the Eind with the size of the desired systems for the H2O was more evident than that of the H2, N2, and CO. The dependency of the Edisp on the size of the model was less than that of the other energy components, in particular for the H2O molecule.

Coordination Behavior of Ni2+, Cu2+, and Zn2+ in Tetrahedral 1-Methylimidazole Complexes: A DFT/CSD Study.

The interaction between nickel (Ni2+), copper (Cu2+), and zinc (Zn2+) ions and 1-methylimidazole has been studied by exploring the geometries of eleven crystal structures in the Cambridge Structural Database (CSD). The coordination behavior of the respective ions was further investigated by means of density functional theory (DFT) methods. The gas-phase complexes were fully optimized using B3LYP/GENECP functionals with 6-31G∗ and LANL2DZ basis sets. The Ni2+ and Cu2+ complexes show distorted tetrahedral geometries around the central ions, with Zn2+ being a perfect tetrahedron. Natural bond orbital (NBO) analysis and natural population analysis (NPA) show substantial reduction in the formal charge on the respective ions. The interaction between metal d-orbitals (donor) and ligand orbitals (acceptor) was also explored using second-order perturbation of the Fock matrix. These interactions followed the order Ni2+ > Cu2+ > Zn2+ with Zn2+ having the least interaction with the ligand orbitals. Examination of the frontier orbitals shows the stability of the complexes in the order Ni2+ > Cu2+ < Zn2+ which is consistent with the Irving–Williams series.

Scalable Electron Correlation Methods. 5. Parallel Perturbative Triples Correction for Explicitly Correlated Local Coupled Cluster with Pair Natural Orbitals.

A well-parallelized perturbative triples correction implementation for the pair natural orbital based coupled cluster method PNO-LCCSD(T)-F12 is presented. A composite approach is adopted in addressing the coupling due to off-diagonal Fock matrix elements, in which the local triples amplitudes are iteratively solved using small domains of triples natural orbitals, and a semicanonical (T0) domain correction with larger domains is applied to reduce the domain errors. This treatment adds only about 20% to the computational cost of (T0) calculations with large domains, and the errors due to the use of small domains in the iterations are very small. In addition, a two-step triple list selection method is applied: First, an initial triple list is generated using LCCSD-F12 pair energy criteria, and the (T0) triples energies are computed using small domains. Second, this list is reduced by neglecting triples with small energy contributions, and the final calculation with large domains is only carried out for the reduced list. The cost of the (T) calculation scales asymptotically linear with the molecular size and shows excellent parallelization efficiency up to hundreds of CPU cores. The convergence of the (T) contribution to the relative energies of large molecular systems is carefully tested, and for most of the cases the results obtained with our default thresholds agree within ∼0.5 kcal mol-1 with those computed with very tight thresholds. For all tested molecular systems where canonical calculations are still feasible, the PNO-LCCSD(T)-F12 relative energies also agree within 0.5 kcal mol-1 with the canonical CCSD(T)-F12 results using the same F12 treatment. The (T) calculation generally takes 3-5 times the cost of the preceding PNO-LCCSD-F12 calculation, primarily due to the large number of triples required in obtaining accurate relative energies. We find that for large molecular systems the triple selection criteria used in previous local triples methods are insufficient, and a much larger number of triples are required than it was assumed so far. Still, using a commodity computer cluster, the PNO-LCCSD(T)-F12 calculation of molecules with ∼100 atoms using augmented triple-ζ basis sets can be carried out in a few hours of elapsed time.