Comparative study of the Hartree–Fock and the Hartree–Fock–Slater method

The Hartree–Fock method (standard Roothaan closed-shell HF –LCAO theory) and the Hartree–Fock–Slater method (restricted HFS –LCAO –DV method developed by Baerends and Ros) have been compared with emphasis on the respective one-electron equations and on the matrix elements of the respective Fock operators. Using the same STO basis in the two cases, the matrix elements of the Fock operators and of their separate one-electron, Coulomb, and exchange contributions have been calculated for the same orbitals and density of the ground state of the diatomic molecule ZnO. The effects of methodical (exchange potential) and numerical (DV method, density fit) differences between the HF and HFS methods on the various matrix elements have been analyzed. As expected the methodical effect prevails and is responsible for the higher (less negative) values of the matrix elements of the HFS Fock operator compared to those of the HF Fock operator. Numerical effects are observable also and are caused by the difference in integration procedures (DV method), not by the density fit.     

Kramers' restricted hartree—fock method for polyatomic molecules using ab initio relativistic effective core potentials with spin—orbit operators

A two-component Kramers' restricted Hartree–Fock method (KRHF) has been developed for the polyatomic molecules with closed shell configurations. The present KRHF program utilizes the relativistic effective core potentials with spin–orbit operators at the Hartree–Fock (HF) level and produces molecular spinors obeying the double group symmetry. The KRHF program enables the variational calculation of spin–orbit interactions at the HF level. KRHF calculations have been performed for the HX, X₂, XY(X, Y = I, Br), and CH₃I molecules. It is demonstrated that the orbital energies from KRHF calculations are useful for the interpretation of spin-orbit splittings in photoelectron spectra. In all molecules studied, bond lengths are only slightly expanded, harmonic vibrational frequencies are reduced, and bond energies are significantly decreased by the spin–orbit interactions.     

Analytic techniques for the coupled multiconfiguration SCF perturbation theory

The perturbation theory based on the paired excitation multiconfiguration self-consistent field approach of Clementi and Veillard is considered. The coupled first-order perturbed orbital equations are analysed and an appropriate computational scheme for their solution is discussed. The proposed computational scheme is analogous to the technique employed for the solution of the coupled Hartree–Fock equations in the one-configuration approximation. However, because of the presence ofnondiagonal Lagrangian multipliers and the use of different one-electron operators for different orbitals, the present scheme raises some new computational problems. In this context a new technique for the solution of the unperturbed multiconfiguration self-consistent field equations is proposed. A simple illustration of the superiority of the multiconfiguration perturbation approach with respect to the ordinary coupled Hartree–Fock scheme is given. Also the validity of the variation formulation of the presented scheme and its relation to the finite-field approach are discussed.     

A comparison of Hartree—Fock,MP2, and DFT results for the HCN dimer and crystal

A number of hydrogen-bond related quantities—geometries, interaction energies, dipole moments, dipole moment derivatives, and harmonic vibrational frequencies—were calculated at the Hartree—Fock, MP2, and different DFT levels for the HCN dimer and the periodic HCN crystal. The crystal calculations were performed with the Hartree—Fock program CRYSTAL92, which routinely allows an a posteriori electron-correlation correction of the Hartree—Fock obtained lattice energy using different correlation-only functionals. Here, we have gone beyond this procedure by also calculating the electron-correlation energy correction during the structure optimization, i.e., after each CRYSTAL92 Hartree—Fock energy evaluation, the a posteriori density functional scheme was applied. In a similar manner, we optimized the crystal structure at the MP2 level, i.e., for each Hartree—Fock CRYSTAL92 energy evaluation, an MP2 correction was performed by summing the MP2 pair contributions from all HCN molecules within a specified cutoff distance. The crystal cell parameters are best reproduced at the Hartree—Fock and the nongradient-corrected HF + LDA and HF + VWN levels. The BSSE-corrected MP2 method and the HF + P91, HF + LDA, and HF + VWN methods give lattice energies in close agreement with the ZPE-corrected experimental lattice energy. The (HCN)₂ dimer properties are best reproduced at the MP2 level, at the gradient-corrected DFT levels, and with the B3LYP and BHHLYP methods. © 1996 John Wiley & Sons, Inc.     

From Hartree–Fock and Heitler–London to chemical orbitals

For chemistry the theoretical representation of the forces connecting atoms in molecules was and is a central problem. The Atomic Orbital and the Molecular Orbital are basic building blocks in the Heitler–London (HL) and in the Linear Combination of Atomic Orbitals–Molecular Orbital (LCAO-MO) methods, which have lead to the construction of modern Valence Bond and Hartree–Fock methods (and related extensions). However, accurate predictions from non semi-empirical methods often require enormous amount of computer power, if applied to molecules of reasonable size and current chemical interest. We have critically re-examined the two basic methods and suggested a few extensions. Merging of the Hartree–Fock with the Heitler–London algorithms, as recently proposed in the Hartree–Fock–Heitler–London (HF–HL) method, reduces the length of the expansions needed in AO or MO ab initio models in the computation of binding energy; this simplification allows easy interpretation of the resulting wave function. The HF–HL method is exemplified with systematic computations on ground and excited state of the hydrides and homonuclear diatomic molecules with atoms of the first and second period of the periodic table. Further, we show that the HF–HL method is derivable from a wave function constructed with a new type of orbital, the Chemical orbital (CO), which embodies the characterization of MO near equilibrium, AO at dissociation and at the united atom. Preliminary computations with CO are included. The new method provides the conceptual origin of both the HF and VB approaches, thus the foundation of an 80 years effort in variational quantum chemistry.     

Formulation of unrestricted and restricted Hartree–Fock–Bogoliubov equations

The Hartree–Fock–Bogoliubov (HFB) method, dealing with Bogoliubov orbitals, which consist of particle and hole part, can provide states with pair correlations associated with Cooper pairs. The dimension of HFB Fock matrices can be reduced by restrictions of spin states of Bogoliubov orbitals similarly to ordinary Hartree–Fock (HF) equations such as restricted HF (RHF), unrestricted HF (UHF), and generalized HF (GHF). However, there are few studies of moderate restricted HFB equations such as UHF‐based HFB equations. In this article, formulation and calculations of restricted HFB equations are described. The solutions of general and restricted HFB equations are compared. Pair correlations taking account of restricted and general HFB equations are discussed. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

The modified Hartree–Fock self-consistent field equation

A one-electron correlation operator is introduced into the Hartree–Fock self-consistent field equation. The correlation operator is derived from the second-order perturbation theory. Energies of atomic and molecular systems calculated from this modified Hartree–Fock equation are equal to that from second-order perturbation of Hartree–Fock equation. The modified equation can also be solved self-consistently by the LCAO approximation. We also presented the modified expressions for other operators.     

Simple derivation of conditions for instability in the Hartree–Fock and projected Hartree–Fock schemes

The conditions for instability of solutions of Hartree–Fock and projected Hartree–Fock equations are derived in a form involving finite real symmetric matrices. These conditions are also expressed in terms of the Fock–Dirac density matrix, both at the spin–orbital and at the orbital level. The particular variations which give rise to the so-called singlet and triplet instabilities are described.     

Elongation cutoff technique armed with quantum fast multipole method for linear scaling

A linear‐scaling implementation of the elongation cutoff technique (ELG/C) that speeds up Hartree‐Fock (HF) self‐consistent field calculations is presented. The cutoff method avoids the known bottleneck of the conventional HF scheme, that is, diagonalization, because it operates within the low dimension subspace of the whole atomic orbital space. The efficiency of ELG/C is illustrated for two model systems. The obtained results indicate that the ELG/C is a very efficient sparse matrix algebra scheme. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2009     

HF and MP2 calculations on CN−, N2, AlF,SiO, PN,SC, ClB,and P2 using correlated molecular wave functions

Contracted basis sets of double zeta valence quality plus polarization functions (DZP) and augmented DZP basis sets, which were recently constructed for the first‐ and second‐row atoms, are applied to study the electronic ground states of the diatomic molecules CN^?, N₂, AlF, SiO, PN, SC, ClB, and P₂. At the Hartree–Fock (HF) and/or Møller–Plesset second‐order (MP2) levels, total and molecular orbital energies, dissociation energies, bond lengths, harmonic vibrational frequencies, and dipole moments are calculated and compared with available experimental data and with the results obtained from correlation consistent polarized valence basis sets of Dunning's group. For N₂, calculations of polarizabilities at the HF and MP2 levels with the sets presented above are also done and compared with results reported in the literature. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Ab initio study of organic mixed valency

A series of six radical cations of the type (D L D)⁺ was investigated at the ab initio unrestricted Hartree–Fock level. One localized and one delocalized conformation were systematically searched by full geometry optimization. At both nuclear arrangements, mostly found as being minima in the symmetry‐restrained Hartree–Fock framework, excitation energies were calculated through the expansion of the wave function on single electronic excitations of the Hartree–Fock fundamental determinant and at the unrestricted Hartree–Fock or at the multiconfigurational self consistent field levels. Few calculations were also performed by taking into account some part of the electronic correlation. Except for N,N,N′,N′‐tetramethyl p‐phenylenediamine, all the studied compounds are localized stable cations, at the symmetry‐restrained Hartree–Fock level. However, the reoptimization of their wave function changes this observation since only three of them seem to conserve a localized stable conformation. Most of the studied systems are characterized by one or two excited electronic states very close to the fundamental one and should thus present an unresolved broadened first absorption band in the near‐infrared region. These features are in agreement with the available experimental data. Strong Hartree–Fock instabilities are found for the delocalized structure and put in relation with the existence of the large nonadiabatic coupling in this conformational region. The solvent influence is discussed in the Onsager dipolar reaction field framework. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 76: 552–573, 2000     

Hartree–Fock instabilities and electronic properties

Hartree–Fock instabilities are investigated for about 80 compounds, from acetylene to mivazerol (27 atoms) and a cluster of 18 water molecules, within a double ζ basis set. For most conjugated systems, the restricted Hartree–Fock wave function of the singlet fundamental state presents an external or so‐called triplet instability. This behavior is studied in relation with the electronic correlation, the vicinity of the triplet and singlet excited states, the electronic delocalization linked with resonance, the nature of eventual heteroatoms, and the size of the systems. The case of antiaromatic systems is different, because they may present a very large internal Hartree–Fock instability. Furthermore, the violation of Hund's rule, observed for these compounds, is put in relation with the fact that the high symmetry structure in its singlet state has no feature of a diradical‐like species. It appears that the triplet Hartree–Fock instability is directly related with the spin properties of nonnull orbital angular momentum electronic systems. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 483–504, 2000     

Comparative study of unscreened and screened molecular static linear polarizability in the Hartree–Fock,hybrid‐density functional,and density functional models

The sum‐over‐states (SOS) polarizabilities are calculated within approximate mean‐field electron theories such as the Hartree–Fock approximation and density functional models using the eigenvalues and orbitals obtained from the self‐consistent solution of the single‐particle equations. The SOS polarizabilities are then compared with those calculated using the finite‐field (FF) method. Three widely used mean‐field models are as follows: (1) the Hartree–Fock (HF) method, (2) the three parameter hybrid generalized gradient approximation (GGA) (B3LYP), and (3) the parameter‐free generalized gradient approximation due to Perdew–Burke–Ernzerhof (PBE). The comparison is carried out for polarizabilities of 142 molecules calculated using the 6‐311++G(d,p) basis set at the geometries optimized at the B3LYP/6‐311G** level. The results show that the SOS method almost always overestimates the FF polarizabilities in the PBE and B3LYP models. This trend is reversed in the HF method. A few exceptions to these trends are found. The mean absolute errors (MAE) in the screened (FF) and unscreened (SOS) polarizability are 0.78, 1.87, and 3.44 ų for the HF, B3LYP, and PBE‐GGA methods, respectively. Finally, a simple scheme is devised to obtain FF quality polarizability from the SOS polarizability. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Variation of parameters in Becke‐3 hybrid exchange‐correlation functional

We have investigated the consequences of varying the three parameters in Becke's hybrid exchange‐correlation functional, which includes five contributions: Hartree–Fock exchange, local exchange, Becke's gradient exchange correction, local correlation, and some form of gradient correlation correction. Our primary focus was upon obtaining orbital energies with magnitudes that are reasonable approximations to the electronic ionization potentials; however, we also looked at the effects on molecular geometries and atomization enthalpies. A total of 12 parameter combinations was considered for each of three different gradient correlation corrections: the Lee–Yang–Parr, the Perdew‐86, and the Perdew–Wang 91. Five molecules were included in the study: HCN, N₂, N₂O, F₂O, and H₂O. For comparison, a Hartree–Fock calculation was also carried out for each of these. The 6‐31+G** basis set was used throughout this work. We found that the ionization potential estimates can be greatly improved (to much better than Hartree–Fock levels) by increasing the Hartree–Fock exchange contribution at the expense of local exchange. In itself, this also introduces major errors in the atomization enthalpies. However, this can be largely or even completely counteracted by reducing or eliminating the role of the gradient exchange correction. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 227–238, 2000     

A study of dynamic quadrupolar and octupolar excitations and calculations on excited d and f states of atom and ions: He-sequence

The quadrupolar and octupolar distortion of the ions in the He-sequence caused by an external electro–magnetic field has been studied by a variation–perturbation method in the Hartree–Fock scheme. For certain frequencies singularities appear in the response of the system to the perturbation. Approximate representations for the excited d and f states have been obtained from a study of these resonances. Such a perturbation calculation has the advantage that representations of the different excited states are obtained independently. The orthogonality to all the lower lying levels of the same symmetry is not required. The only source of inaccuracy implicit in the procedure lies in the improper consideration of the inter-electronic interaction. This is corrected for by an independent calculation, which is again formulated in terms of a perturbation treatment. The resulting wave functions for the excited states are accurate in the Hartree–Fock model. Expectation values of several operators have been calculated with these corrected wave functions.     

The relationship between the unrestricted and projected Hartree–Fock methods in a simple three-electron model system

Various forms of the unrestricted Hartree–Fock (UHF ) scheme are studied for a simple three-electron model system, represented by the PPP (Pariser–Parr–Pople) π-electron model of the allyl radical. Both spin and space symmetry are violated in the UHF trial wave function, either individually or simultaneously. A comparison with the projected Hartree–Fock (PHF ) schemes studied earlier is made and the effect of the order in which various symmetries are broken in both UHF and PHF schemes is studied. The effectiveness of various schemes follows from a comparison of the correlation energy and the wave function is obtained by various UHF (or projected UHF ) and PHF schemes, in the whole range of the coupling constant, with the corresponding quantities given by the exact solution of the model. Finally, the implications of the stability of the restricted HF solutions for the behavior of various single- and multiparameter UHF and PHF schemes are briefly outlined and exemplified on the studied model.     

On the use of symmetry in first-order perturbed HF theory. II

A method is described whereby molecular symmetry is employed to reduce the number of two-electron integrals in perturbed Hartree–Fock calculations of second-order properties. The method is a generalization of the Dacre–Elder procedure. First- and second-rank perturbing tensor operators are examined in the coupled HF approach to electric dipole polarizabilities, magnetic susceptibilities, quadrupole polarizabilities, and spin-dopolar contributions to spin–spin coupling constants. The procedure sketched here permits a large saving of computer efforts, which is shown by some illustrative examples.     

Accurate local spin-polarized exchange potential: Reconciliation of generalized Slater and Kohn–Sham methods

Using simple physical arguments, a local spin-polarized exchange potential, V_xσ, is constructed from the single-particle Hartree–Fock (HF ) potentials (generalized Slater method) that reduces to the usual Kohn–Sham (KS ) result in the uniform gas limit. Numerical results for 10 closed subshell atoms demonstrate that the total energy calculated employing this V_xσ is closer to the exact KS results than those of other standard exchange approximations with electron densities and highest occupied orbital eigenvalues that closely approximate the HF results.     

Electronic structures of newly designed two-dimensional high-spin organic polymers

Practical two-dimensional high-spin organic polymers are newly designed and the electronic structures are examined on the basis of the ab initio two-dimensional unrestricted Hartree–Fock crystal orbital (UHF-CO) method. The present polymers can be oxidized up to six electrons per unit cell, and it is predicted that the three-electron and six-electron oxidations per unit cell lead to the high-spin organic polymers having superdegenerate band structures originating from the characteristic non-bonding crystal orbital patterns.