Hartree-Fock Orbitals Research Articles

An Improvement on Noninteger n-Generalized Exponential Type Orbitals with Hyperbolic Cosine in Atomic Calculations

Abstract An improvement on the total energy results reported in Ref. 9 in atomic calculations based on a modified type hyperbolic cosine function cosh(βrμ + γ) is presented. It is shown that the noninteger n-generalized exponential type orbitals rn* - 1e- ζ rμ with modified type hyperbolic cosine as radial basis functions are a much better approximation to the Hartree–Fock orbitals than a double-zeta basis set of Slater type functions. The efficiency of the new basis function is tested by application to some closed and open shell neutral atoms and their ions. A substantial improvement in both the total and orbital energies is obtained within the minimal basis sets framework. The total energy values obtained in this work are significantly close to the numerical Hartree–Fock results. These results supersede all previous minimal basis function total energies achieved in the literature.

How Important is Orbital Choice in Single-Determinant Diffusion Quantum Monte Carlo Calculations?

The accuracy of total electronic energies obtained using the fixed-node diffusion quantum Monte Carlo (FN-DMC) method is determined by the choice of the many-body nodal surface. Here, we perform a systematic comparison of the quality of FN-DMC energies for a selection of atoms and diatomic molecules using nodal surfaces defined by single determinants of Hartree-Fock, B3LYP, and LDA orbitals. Through comparison with experimental results, we show that the use of Kohn-Sham orbitals results in significantly improved FN-DMC atomization energies over those obtained using Hartree-Fock orbitals. We also discuss the effect of spin contamination in the orbitals.

Localization of molecular orbitals on fragments

A non-iterative algorithm for the localization of molecular orbitals (MOs) from complete active space self consistent field (CASSCF) and for single-determinantal wave functions on predefined moieties is given. The localized fragment orbitals can be used to analyze chemical reactions between fragments and also the binding of fragments in the product molecule with a fragments-in-molecules approach by using a valence bond expansion of the CASSCF wave function. The algorithm is an example of the orthogonal Procrustes problem, which is a matrix optimization problem using the singular value decomposition. It is based on the similarity of the set of MOs for the moieties to the localized MOs of the molecule; the similarity is expressed by overlap matrices between the original fragment MOs and the localized MOs. For CASSCF wave functions, localization is done independently in the space of occupied orbitals and active orbitals, whereas, the space of virtual orbitals is mostly uninteresting. Localization of Hartree-Fock or Kohn-Sham density functional theory orbitals is not straightforward; rather, it needs careful consideration, because in this case some virtual orbitals are needed but the space of virtual orbitals depends on the basis sets used and causes considerable problems due to the diffuse character of most virtual orbitals.

Multiplet ligand-field theory using Wannier orbitals

We demonstrate how ab initio cluster calculations including the full Coulomb vertex can be done in the basis of the localized Wannier orbitals which describe the low-energy density functional (local-density approximation) band structure of an infinite crystal, e.g., the transition-metal $3d$ and oxygen $2p$ orbitals. The spatial extent of our $3d$ Wannier orbitals (orthonormalized $N$th-order muffin-tin orbitals) is close to that found for atomic Hartree-Fock orbitals. We define ligand orbitals as those linear combinations of the O $2p$ Wannier orbitals which couple to the $3d$ orbitals for the chosen cluster. The use of ligand orbitals allows for a minimal Hilbert space in multiplet ligand-field theory calculations, thus reducing the computational costs substantially. The result is a fast and simple ab initio theory, which can provide useful information about local properties of correlated insulators. We compare results for NiO, MnO, and SrTiO${}_{3}$ with x-ray absorption, inelastic x-ray scattering, and photoemission experiments. The multiplet ligand-field theory parameters found by our ab initio method agree within $\ensuremath{\sim}10%$ with known experimental values.

Approaching chemical accuracy with quantum Monte Carlo

A quantum Monte Carlo study of the atomization energies for the G2 set of molecules is presented. Basis size dependence of diffusion Monte Carlo atomization energies is studied with a single determinant Slater-Jastrow trial wavefunction formed from Hartree-Fock orbitals. With the largest basis set, the mean absolute deviation from experimental atomization energies for the G2 set is 3.0 kcal/mol. Optimizing the orbitals within variational Monte Carlo improves the agreement between diffusion Monte Carlo and experiment, reducing the mean absolute deviation to 2.1 kcal/mol. Moving beyond a single determinant Slater-Jastrow trial wavefunction, diffusion Monte Carlo with a small complete active space Slater-Jastrow trial wavefunction results in near chemical accuracy. In this case, the mean absolute deviation from experimental atomization energies is 1.2 kcal/mol. It is shown from calculations on systems containing phosphorus that the accuracy can be further improved by employing a larger active space.

Calculation of open p-shell atoms in the algebraic approach of the Hartree–Fock method

Highly precise calculations of analytical Hartree–Fock orbitals and energies have been performed within the limits of the Roothaan–Hartree–Fock atomic theory (Roothaan–Bagus method) for all open p-shell atoms of the Periodic Table. They were calculated in an algebraic approach using Slater-type atomic orbitals (AOs) as basis functions. Nonlinear parameters (orbital exponents) of AOs were optimized with exceptional accuracy by second-order methods. As a result, it was possible to satisfy exactly the virial relation (10–14–10–17) with calculated atomic term energies being close to the Hartree–Fock limit.

The optimized orbital coupled cluster doubles method and optical rotation

The impact of variational optimization of the molecular orbitals used in coupled cluster response theory is compared to the use of conventional Hartree–Fock orbitals for the computation of optical rotations in a number of chiral compounds. For molecular structures near equilibrium, where the coupled cluster method is relatively insensitive to the choice of reference wave function, the two approaches yield similar results for test cases such as (S)-methyloxirane, (S)-methylthiirane, (S)-2-chloropropionitrile, and (1S,4S)-norbornenone. However, small distortions away from equilibrium affect the optimized-orbital coupled cluster approach to a much greater extent than its Hartree–Fock counterpart. This effect is tested for methyloxirane in which the C–O bond associated with the stereogenic carbon is stretched and for a pyramidalized fluoroformaldehyde where the corresponding C=O bond is similarly extended. The potential repercussions of this strong geometry dependence on the magnitude of vibrational corrections are discussed.

Charge-consistent redefinition of Fock integrals

The Fock integrals in the Hartree–Fock (HF) theory have been redefined such that every orbital pair density of an electron appearing in the conventional definition is replaced by a net neutral density that is the sum of the orbital pair density and an appropriate portion of the nuclear charge density. These charge-consistent Fock integrals in the canonical HF orbitals are shown to differ from the conventional ones only in the diagonal elements and by merely a constant, thus not altering the HF energy, orbitals, correlation energies, etc. They are shown numerically to converge much more rapidly with respect to the number of unit cells included in the lattice sums for one-dimensional solids because they contain no charge-multipole interactions in their definition unlike the conventional Fock integrals. The multipole expansion of the long-range lattice sums in the charge-consistent Fock integrals is also formulated and implemented for one-dimensional solids.

Synthesis, Crystal Structure and ab Initio Studies of 5-Phenylamino-3-phenylimino-3H[1, 2]dithiole-4-carboxylic acid ethyl ester

Phenylamino-3-phenylimino-3H(1, 2)dithiole- 4-carboxylic acid ethyl ester, C18H16N2O2S2, (I), has been synthesized and the structure has been solved by X-ray diffraction. The crystals are triclinic, space group P 1, with a = 5.7127(2) A u , b = 12.1757(5) A u , c = 14.0734(5) A u , a =112.1217(16), b = 97.786(2), c = 100.694(2), Mr = 356.45, V = 868.11(6) A u 3 , Z = 2 and R = 0.0373. In the title compound there is an intramolecular N-HO hydrogen bond. The crystal structure is stabilized only by weak C-Hp and pp interactions as well as by van der Waals forces. The geometry of the isolated molecule was optimized by ab initio quantum mechanical calculations, employing both molecular orbital Hartree-Fock (HF) and density functional theory (DFT) methods. In the DFT cal- culation the minimum energy was achieved for a confor- mation very similar to that of the solid-state molecule.

Statistical correlation between atomic electron pairs

The statistical correlation between a pair of electrons in Hartree–Fock orbitals is measured by mutual information and studied in position and in momentum space. We show that there are same- and opposite-spin orbital pairs where the correlation is larger in momentum space. Among these are opposite-spin valence shell pairs where the correlation arises from the indistinguishability of electron spins.

Conformational Preferences of Modified Nucleoside N2-methylguanosine (m2G) and Its Derivative N2, N2-dimethylguanosine (m 2 2 G) Occur at 26th Position (Hinge Region) in tRNA

Conformational preferences of the modified nucleosides N(2)-methylguanosine (m(2)G) and N(2), N(2)-dimethylguanosine (m(2)(2)G) have been studied theoretically by using quantum chemical perturbative configuration interaction with localized orbitals (PCILO) method. Automated complete geometry optimization using semiempirical quantum chemical RM1, along with ab initio molecular orbital Hartree-Fock (HF-SCF), and density functional theory (DFT) calculations has also been made to compare the salient features. Single-point energy calculation studies have been made on various models of m(2)G26:C/A/U44 and m(2)(2)G26:C/A/U44. The glycosyl torsion angle prefers "syn" (χ = 286°) conformation for m(2)G and m(2)(2)G molecules. These conformations are stabilized by N(3)-HC2' and N(3)-HC3' by replacing weak interaction between O5'-HC(8). The N(2)-methyl substituent of (m(2)G26) prefers "proximal" or s-trans conformation. It may also prefer "distal" or s-cis conformation that allows base pairing with A/U44 instead of C at the hinge region. Thus, N(2)-methyl group of m(2)G may have energetically two stable s-trans m(2)G:C/A/U or s-cis m(2)G:A/U rotamers. This could be because of free rotations around C-N bond. Similarly, N(2), N(2)-dimethyl substituent of (m(2)(2)G) prefers "distal" conformation that may allow base pairing with A/U instead of C at 44th position. Such orientations of m(2)G and m(2)(2)G could play an important role in base-stacking interactions at the hinge region of tRNA during protein biosynthesis process.

Brueckner doubles coupled cluster method with the polarizable continuum model of solvation

We present the theory and implementation for computing the (free) energy and its analytical gradients with the Brueckner doubles (BD) coupled cluster method in solution, in combination with the polarizable continuum model of solvation (PCM). The complete model, called PTED, and an efficient approximation, called PTE, are introduced and tested with numerical examples. Implementation details are also discussed. A comparison with the coupled-cluster singles and doubles CCSD-PCM-PTED and CCSD-PCM-PTE schemes, which use Hartree-Fock (HF) orbitals, is presented. The results show that the two PTED approaches are mostly equivalent, while BD-PCM-PTE is shown to be superior to the corresponding CCSD scheme when the HF reference wave function is unstable. The BD-PCM-PTE scheme, whose computational cost is equivalent to gas phase BD, is therefore a promising approach to study molecular systems with complicated electronic structure in solution.

Sulfur 1s near-edge x-ray absorption fine structure (NEXAFS) of thiol and thioether compounds

The speciation and quantification of sulfur species based on sulfur K-edge x-ray absorption spectroscopy is of wide interest, particularly for biological and petroleum science. These tasks require a firm understanding of the sulfur 1s near-edge x-ray absorption fine structure (NEXAFS) spectra of relevant species. To this end, we have examined the gas phase sulfur 1s NEXAFS spectra of a group of simple thiol and thioether compounds. These high-resolution gas phase spectra are free of solid-state broadening, charging, and saturation effects common in the NEXAFS spectra of solids. These experimental data have been further analyzed with the aid of improved virtual orbital Hartree-Fock ab initio calculations. The experimental sulfur 1s NEXAFS spectra show fine features predicted by calculation, and the combination of experiment and calculation has been used to improve assignment of spectroscopic features relevant for the speciation and quantification of the sulfur compounds.

Local orbitals by minimizing powers of the orbital variance

It is demonstrated that a set of local orthonormal Hartree-Fock (HF) molecular orbitals can be obtained for both the occupied and virtual orbital spaces by minimizing powers of the orbital variance using the trust-region algorithm. For a power exponent equal to one, the Boys localization function is obtained. For increasing power exponents, the penalty for delocalized orbitals is increased and smaller maximum orbital spreads are encountered. Calculations on superbenzene, C(60), and a fragment of the titin protein show that for a power exponent equal to one, delocalized outlier orbitals may be encountered. These disappear when the exponent is larger than one. For a small penalty, the occupied orbitals are more local than the virtual ones. When the penalty is increased, the locality of the occupied and virtual orbitals becomes similar. In fact, when increasing the cardinal number for Dunning's correlation consistent basis sets, it is seen that for larger penalties, the virtual orbitals become more local than the occupied ones. We also show that the local virtual HF orbitals are significantly more local than the redundant projected atomic orbitals, which often have been used to span the virtual orbital space in local correlated wave function calculations. Our local molecular orbitals thus appear to be a good candidate for local correlation methods.

A Locality Analysis of the Divide-Expand-Consolidate Coupled Cluster Amplitude Equations.

We present a thorough locality analysis of the divide-expand-consolidate amplitude equations for second-order Møller-Plesset perturbation theory and the coupled cluster singles doubles (CCSD) model, which demonstrates that the amplitude equations are local when expressed in terms of a set of local occupied and local unoccupied Hartree-Fock orbitals, such as the least-change molecular basis. The locality analysis thus shows that a CC calculation on a large molecular system may be carried out in terms of CC calculations on small orbital fragments of the total molecular system, where the sizes of the orbital fragment spaces are determined in a black box manner to ensure that the CC correlation energy is calculated to a preset energy threshold. A practical implementation of the locality analysis is described, and numerical results are presented, which demonstrate that both the orbital fragment sizes and the relative energy error compared to a full CC calculation are independent of the molecular system size.

Delocalization of Dyson orbitals in F−(H2O) and Cl−(H2O)

AbstractElectron propagator calculations accurately predict the vertical electron detachment energies of the title complexes. Corresponding Dyson orbitals closely resemble canonical Hartree–Fock orbitals. Approximations that use diagonal self‐energies in the latter basis, therefore, show promise for applications on larger anionic clusters. Neutral final states correspond to Cl 3p or water 1b1 holes, in accord with recent interpretations of Cl−(H2O) photoelectron spectra. However, for the F case, the Dyson orbitals are delocalized over the F and O centers and compromise interpretations that ascribe F(H2O) or F−(H2O+) charge‐transfer character to the final states. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2010

Picture change error correction in the radial distributions of canonical orbital densities and total electron density of radon atom: the effect of the size of nucleus and the basis set limit

The 2nd order Douglas-Kroll-Hess (DKH2) and the Infinite Order Two Component (IOTC) radial distributions of electron density of canonical Hartree-Fock (HF) orbitals of radon atom are presented. Furthermore, the total electron density is revisited. The picture change error (PCE) correction is investigated by analytical means. The point charge model of nucleus and the Gaussian nucleus model are employed. The basis set is extrapolated by means of including tight s and also p Gaussians within the original triple zeta basis set. It is found that the DKH1 PCE corrected DKH2 total electron and s orbital contact densities are negative for the point charge model of nucleus if tight enough s Gaussians are included in the basis set. It is shown that this failure is caused due to the missing terms of the second order Douglas-Kroll transformation for the DKH2 electron density. PCE is found the most striking in the DKH2/IOTC electron density of s orbitals close to the nucleus. The radial distributions of the 2-component p 1/2 orbital densities are considerably affected by PCE at the nucleus as well. Furthermore, the PCE corrected DKH2/IOTC scalar p orbital densities have a non-zero value of electron density at nucleus and can be considered as an spin-orbit (SO) average of the p 1/2 and p 3/2 orbitals. The d and f orbitals are affected by PCE in the vicinity of the nucleus only little. The PCE corrected DKH2 and IOTC radial distributions of orbital densities are nodeless, which is completely in agreement with the radial distribution of the analytic or numeric DCH orbital densities.

Spin-polarized self-consistent-field equations for paired orbitals

Unrestricted Hartree-Fock-like equations are proposed to find multiple spin-symmetry-broken states of the molecular systems. Developed equations are pseudo-eigenvalue-type equations for the Fock-type operators constructed in such a way to include an effective field which makes different-spin orbitals biorthogonal. The eigenvectors of these operators are noncanonical Hartree-Fock orbitals becoming L\"owdin-Amos-Hall paired (corresponding) orbitals after self-consistency is achieved. The eigenvalues of the modified Fock operators appear to be the energies of the paired orbitals. Because the paired orbitals do not follow the spatial symmetry of the molecular nuclear core, the equations allow one to obtain the broken symmetry states with relative ease as demonstrated for the model ${\mathrm{H}}_{6}$ hexagon molecule. For this molecule, the \ifmmode \check{C}\else \v{C}\fi{}\'{\i}\ifmmode \check{z}\else \v{z}\fi{}ek-Paldus instability matrix analysis predicts the existence of three spin-symmetry-broken states. All these solutions are systematically achieved by the paired equations, unlike the standard unrestricted equations which basically converge to a single solution. The proposed approach is also valid for the density functional theory in which the spin-polarized Kohn-Sham equations might be transformed to paired equations.

Higher order singular value decomposition in quantum chemistry

Higher order singular value decomposition is studied in the context of quantum chemistry, with particular focus on the decomposition of the 𝒯 2 amplitudes obtained from second order Møller Plesset perturbation (MP2) theory calculations. Our test calculations reveal that HOSVD transformed amplitudes yield considerably faster convergence in MP2 correlation energy, both in terms of amplitude and orbital truncation. Also, HOSVD orbitals display increased bonding/antibonding character compared to Hartree–Fock orbitals. In contrast to canonical MP2 theory, the leading amplitudes are those between corresponding occupied–virtual orbital pairs. The HOSVD orbitals are paired up automatically around the Fermi level in decreasing importance, so that the strongest occupied virtual–pair are the highest occupied molecular orbital and lowest unoccupied molecular orbital. We show that in the case of MP2 amplitudes, the HOSVD orbitals are equivalent to the unrelaxed MP2 natural orbitals. The least squares higher order orthogonal iteration algorithm yields only minor improvements over the sub-optimal truncated orbital space obtained from the HOSVD.

Optimizing Hartree-Fock orbitals by the density-matrix renormalization group

We have proposed a density-matrix renormalization group (DMRG) scheme to optimize the one-electron basis states of molecules. It improves significantly the accuracy and efficiency of the DMRG in the study of quantum chemistry or other many-fermion system with nonlocal interactions. For a water molecule, we find that the ground state energy obtained by the DMRG with only 61 optimized orbitals already reaches the accuracy of best quantum Monte Carlo calculation with 92 orbitals.