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.     

Kramers' unrestricted Hartree–Fock and second-order Møller–Plesset perturbation methods using relativistic effective core potentials with spin–orbit operators: Test calculations for HI and CH3I

The Kramers' restricted Hartree–Fock (KRHF) and second-order Møller–Plesset perturbation (KRMP2) methods using relativistic effective core potentials (RECP) with spin–orbit operators and two-component spinors are extended to the unrestricted forms, KUHF and KUMP2. As in the conventional unrestricted methods, the KUHF and KUMP2 methods are capable of qualitatively describing the bond breaking for a single bond. As a result, it is possible to estimate spin–orbit effects along the dissociation curve at the HF and MP2 levels of theory as is demonstrated by the test calculations on the ground states of HI and CH₃I. Since the energy lowering due to spin–orbit interactions is larger for the I atom than for the closed-shell molecules, dissociation energies are reduced and bond lengths are slightly elongated by the inclusion of the spin–orbit interactions. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 66 : 91–98, 1998     

Development of spin‐dependent relativistic open‐shell Hartree–Fock theory with time‐reversal symmetry (II): The restricted open‐shell approach

An open‐shell Hartree–Fock (HF) theory for spin‐dependent two‐component relativistic calculations, termed the Kramers‐restricted open‐shell HF (KROHF) method, is developed. The present KROHF method is defined as a relativistic analogue of ROHF using time‐reversal symmetry and quaternion algebra, based on the Kramers‐unrestricted HF (KUHF) theory reported in our previous study (Int. J. Quantum Chem., doi: 10.1002/qua.25356 ). As seen in the nonrelativistic ROHF theory, the ambiguity of the KROHF Fock operator gives physically meaningless spinor energies. To avoid this problem, the canonical parametrization of KROHF to satisfy Koopmans' theorem is also discussed based on the procedure proposed by Plakhutin et al. (J. Chem. Phys. 2006 , 125, 204110). Numerical assessments confirmed that KROHF using Plakhutin's canonicalization procedure correctly gives physical spinor energies within the frozen‐orbital approximation under spin–orbit interactions.     

Coulomb-Hole–Hartree–Fock functional

We have extended to molecules a density functional previously parametrized for atomic computations. The Coulomb-hole–Hartree–Fock functional, introduced by Clementi in 1963, estimates the dynamical correlation energy by the computations of a Hartree–Fock-type single-determinant wave function, where the Hartree–Fock potential was augmented with an effective potential term, related to a hard Coulomb hole enclosing each electron. The method was later revisited by S. Chakravorty and E. Clementi [Phys. Rev. A 39 , 2290 (1989)], where a Yukawa-type soft Coulomb hole replaced the previous hard hole; atomic correlation energies, computed for atoms with Z = 2 to Z = 54 as well as for a number of excited states, validated the method. In this article, we parametrized a function, which controls the width of the soft Coulomb hole, by fitting the first and second atomic ionization potentials of the atoms with 1 ? Z ? 18. The parametrization has been preliminarily validated by computing the dissociation energy for a number of molecules. A few-determinant version of the Coulomb-hole–Hartree–Fock method, necessary to account for the nondynamic correlation corrections, is briefly discussed. © 1994 John Wiley & Sons, Inc.     

Computational linear dependence in molecular electronic structure calculations using universal basis sets

Distributed universal even‐tempered basis sets have been developed over recent years that are capable of supporting Hartree–Fock energies to an accuracy approaching the sub‐μHartree level. These basis sets have also been exploited in correlation studies, in applications to polyatomic molecules, and in the calculation of electric properties, such as multipole moments, polarizabilities, and hyperpolarizabilities. Jorge and coworkers have also developed universal basis sets and have recently reported applications to diatomic molecular systems. In this article, we compare the molecular calculations reported by Jorge and coworkers with our previous studies. Particular attention is given to the degree of computational linear dependence associated with the various basis sets employed and the consequential effects of the accuracy of the calculated energies. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Dyson-corrected orbital energies for the perturbative treatment of electron correlation

The effect of replacing the Hartree–Fock one-particle energies with ionization potentials obtained from inverse Dyson equation when calculating electron correlation energies perturbatively is investigated. Though the energy shifts vary from system to system, the slight decrease of the resulting excitation energies at around equilibrium geometries leads to a slight increase of the correlation energies in most cases. In the dissociation limit the inverse Dyson equation opens the gap, thus nondiverging potential curves emerge even at the restricted Hartree–Fock (RHF)+RS2 level. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 713–719, 1998     

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     

Soft Coulomb hole method applied to molecules

The soft Coulomb hole method introduces a perturbation operator, defined by ?e/r₁₂ to take into account electron correlation effects, where ω represents the width of the Coulomb hole. A new parametrization for the soft Coulomb hole operator is presented with the purpose of obtaining better molecular geometries than those resulting from Hartree–Fock calculations, as well as correlation energies. The 12 parameters included in ω were determined for a reference set of 12 molecules and applied to a large set of molecules (38 homo‐ and heteronuclear diatomic molecules, and 37 small and medium‐size molecules). For these systems, the optimized geometries were compared with experimental values; correlation energies were compared with results of the MP2, B3LYP, and Gaussian 3 approach. On average, molecular geometries are better than the Hartree–Fock values, and correlation energies yield results halfway between MP2 and B3LYP. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Application of the Stabilized Koopmans’ Theorem to the Temporary Anion States of Chlorosilanes in Long‐Range Corrected Density Functional Theory

The stabilized Koopmans’ theorem (SKT) is very successful in predicting relative vertical electron attachment energies in the Hartree‐Fock theory. It is mainly accomplished by systematically scaling the most diffuse functions in the basis set. Recently, the SKT has been extended to the temporary anion states (TASs) of polyatomic molecules in the long‐range corrected density functional theory. In this paper, this method will be further applied to chlorosilanes for their importance in the chemical processes of the semiconductor industry. Their resonance energies and lifetimes are determined by computing the density of resonance states via SKT. The detailed characteristics of resonance orbitals are then analyzed. It turns out that the lowest unfilled orbitals of chlorosilanes are essentially σ*_Si‐Cl in character. Moreover, several TASs with strong Si/Cl “d” character have been identified. These results, definitely, will help us in understanding the peculiar bonding and chemical properties of chlorosilanes.     

Geometry,basis set,and correlation energy dependence of computed protonation energies of imino bases

The nitrogen protonation energies of the imino bases HN?CHR, where R is H, CH₃, NH₂, OH, and F, have been evaluated to determine the dependence of absolute and relative protonation energies on geometry, basis set, and correlation effects. Reliable absolute protonation energies require a basis set larger than a split-valence plus polarization basis, the inclusion of correlation, and optimized geometries of at least Hartree–Fock 4-31G quality. Consistent relative protonation energies can be obtained at the Hartree–Fock level with smaller basis sets. Extending the split-valence basis set by the addition of polarization functions on all atoms decreases the computed absolute Hartree–Fock nitrogen protonation energies of the imino bases HN?CHR except when R is F, but increases the oxygen protonation energies of the carbonyl bases O?CHR.     

Accurate atomic Gaussian basis functions for second-row atoms: Small split-valence 3-21SP and 4-22SP basis sets

Small split-valence Gaussian 3-21SP and 4-22SP basis sets, previously reported for the first-row atoms [Chem. Phys. Lett., 229 , 151 (1996)], have been extended for the second-row elements of the Periodic Table. The total energies of the ground states of the second-row atoms calculated with the new basis sets are significantly lower than those obtained with the well-known 3-21G (J. Am. Chem. Soc., 104 , 2797 (1982)] and 4-31G [J. Chem. Phys., 56 , 5255 (1972)] basis sets. This is because, as first noted in our previous work for first-row atoms, that the 3-21G and 4-31G basis sets only correspond to a local minimum of the Hartree–Fock energy functional, which is relatively far from its global minimum. The proposed basis sets have been tested by performing geometry optimizations and calculations of normal frequencies in the harmonic approximation of some diatomic and polyatomic molecules at the Hartree–Fock level. © 1997 John Wiley & Sons, Inc. J Comput Chem 18: 1200–1210     

Adapted Gaussian basis sets for atoms Cs to Lr based on the generator coordinate Hartree–Fock method

We have applied a discretized version of the generator coordinate Hartree–Fock method to generate adapted Gaussian basis sets for atoms Cs (Z=55) to Lr (Z=103). Our Hartree–Fock total energy results, for all atoms studied, are better than the corresponding Hartree–Fock energy results attained with previous Gaussian basis sets. For the atoms Cs to Lr we have obtained an energy value within the accuracy of 10⁻⁴ to 10⁻³ hartree when compared with the corresponding numerical Hartree–Fock total energy results. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 858–865, 1998     

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.     

On the calculation of bonding energies by the Hartree Fock Slater method

A transition state method has been proposed for the calculation of bonding energies and bond distances within the Hartree Fock Slater Method. Calculations on a number of diatomic molecules and a few transition metal complexes show better agreement with experiment than corresponding Hartree Fock results. The proposed transition state method gives a direct connection between bond orders and bonding energies.     

Explaining the “large numbers” via a unified (classical) theory of strong and gravitational interactions

An advantage of modified virtual orbitals of Hartree–Fock method is discussed in the calculation of the second-order perturbation energy. All the modified virtual orbitals can be fitted for the intermediate virtual states in the perturbation expansion, only if the molecular orbitals are expanded in terms of infinite basis functions and the set of molecular orbitals is infinite and complete. If the molecular orbitals are expanded in terms of finite basis functions, only the modified virtual orbitals with lower energies are appropriate to describe the intermediate virtual states, but the modified virtual orbitals with higher energies become inadequate. To explain the usefulness of the modified virtual orbitals, the calculation by the modified Hartree–Fock method without CI are compared with the calculation by the traditional Hartree–Fock method with complete CI .     

An investigation of the quantum chemical description of the ethylenic double bond in reactions: II. Insertion of ethylene into a titanium–carbon bond

Insertion of ethylene into the Ti–methyl bond in TiH₂CH⁺₃ is chosen as a model reaction for investigating the performance of a range of contemporary quantum chemical models in polymerization studies. Basis set effects are investigated at the self-consistent-field level, covering Hartree–Fock, pure DFT, and hybrid DFT. In agreement with findings in part I of this study, the basis set sensitivity of ethylene is shown to introduce a bias in computed energetics, amounting to 2–3 kcal/mol when DZP bases are used to compute the overall heat of monomer insertion. The geometry of stationary points relevant to the insertion reaction is determined using hybrid density functional theory. Based on these structures, the energy profile of the insertion reaction is computed using a range of popular quantum chemical approximations. The methods include Hartree–Fock and Møller–Plesset (MP) perturbation theory up through the fourth order in spin-restricted, spin-unrestricted, and spin-projected formalisms. Furthermore, configuration-interaction-based methods are included, of which the top level method is singly and doubly excited coupled clusters with a perturbative estimate of the contribution from triply excited configurations added [CCSD(T)]. The performance of the methods just mentioned, as well as three pure density functional and three hybrid density functional methods, are compared with respect to “best” relative energies, defined through extrapolation of CCSD(T) correlation energies according to the PCI scheme of Siegbahn and coworkers. Even though the MP series show poor convergence, spin-projected MP2, as well as two pure DFT methods (BPW91, BP86) and PCI-78 based on the MCPF method, show similar and very good agreement with best relative energies for the insertion reaction. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 947–960, 1998     

Molecular symmetry. IV. The coupled perturbed Hartree–Fock method

Symmetry methods employed in the ab initio polyatomic program HONDO are extended to the coupled perturbed Hartree–Fock (CPHF) formalism, a key step in the analytical computation of energy first derivatives for configuration interaction (CI) wavefunctions, and energy second derivatives for Hartree–Fock (HF) wavefunctions. One possible computational strategy is to construct Fock-like matrices for each nuclear coordinate in which the one- and two-electron integrals of the usual Fock matrix are replaced by the integral first derivatives. “Skeleton” matrices are constructed from the unique blocks of electron-repulsion integral derivatives. The correct matrices are generated by applying a symmetrization operator. The analysis is valid for many wavefunctions, including closed- or open-shell spin-restricted and spin-unrestricted HF wavefunctions. To illustrate the method, we compare the computer time required for setting up the coupled perturbed HF equations for eclipsed ethane using D_3h symmetry point group and various subgroups of D_3h. Computational times are roughly inversely proportional to the order of the point group.     

Bond length alternation in cyclic polyenes. I. Restricted Hartree–Fock method

The problem of bond length alternation in cyclic polyene models as described by the Pariser–Parr–Pople π-electron Hamiltonian and its relationship to the singlet stability problem for symmetry adapted Hartree–Fock solutions for these systems is investigated using the restricted Hartree–Fock method. The σ-energy contribution is approximated by a quasiharmonic empirical potential. It is shown that the restricted Hartree–Fock energies favor the cyclic polyene distortion and an estimate of the distortion and of the stabilization energy for infinite linear polyenes is obtained.     

Atomic Xα calculations based on EHFS(α) = Eexp

The total energies and one-electron energies for first- and second-row atoms were calculated by using the Hartree–Fock and the Hartree–Fock-Slater Hamiltonian with X_α orbitals, u_i(α_exp); α was parametrized from E^HFS (α_exp) = E^exp. The E^HF (α_exp) total energies are always higher than the Hartree–Fock energies for the atoms. The relation of the calculated ionization potential to the experimental ionization potential depends on the α used to define u_i(α), α_exp, or α_HF.     

Nonrelativistic and quasirelativistic model potential calculations on AgH and Ag2

The major relativistic effects are included into the model potential (MP) method of Bonifacic and Huzinaga. The effects are incorporated on the level of Cowan and Griffin's relativistic Hartree–Fock (RHF) method. The model potential parameters are determined using the results of nonrelativistic Hartree–Fock (NHF) and RHF calculations. A new scheme of selection of the basis functions for use in atomic and molecular MP calculations is proposed. To obtain agreement with the Hartree–Fock calculations on AgH and Ag₂, the 4p shell has to be included explicitly in the MP calculations. The explicit treatment of the 4p electrons and the resulting reduction of the core size are necessary in order to overcome difficulties with approximate representation of the large 4p–4d core-valence interactions on the MP level.