Perturbation treatment of the Hartree–Fock equations of 1s2p3s 4Pθ state of the lithium isoelectronic sequence

The first order Hartree–Fock equations of the 1s2p3s ⁴P⁰ state of the three-electron atomic systems have been solved exactly. These solutions are used to evaluate Hartree–Fock energy up to third order with high accuracy. The third order Hartree–Fock energies for Li to Ne⁷⁺ are compared with those derived from experiment and other theoretical calculations.     

Near-Dirac–Hartree–Fock results for first-row atoms calculated with GTO basis sets

Relativistic basis sets for first-row atoms have been constructed by using the near-Hartree–Fock (nonrelativistic) eigenvectors calculated by Partridge. These bases generate results of near-Dirac–Hartree–Fock quality. Relativistic total and orbital energies, relativistic corrections to the total energy, and magnetic interaction energies for the first-row atoms have been presented. The smallest Gaussian expansions (13s8 p expansions) yield Dirac–Hartree–Fock total energies accurate through six significant digits, while the largest expansions (18s13p expansions) give these energies accurate through seven significant digits. These results are more accurate than some of the results reported earlier, particularly for the open-shell atoms, indicating that the basis employed is reasonably economical for relativistic calculations. © 1995 John Wiley & Sons, Inc.     

Solution of the Hartree–Fock problem by expansion onto nested bases

A method for solving the Hartree–Fock problem in a finite basis set is derived, which permits each orbital to be expanded in a different basis. If the basis set for each orbital ?_i contains the basis functions for the preceding orbitals, ?_i?1, ?_i?2,… ?₁, then the ?_i form an orthonormal set. One advantage over the standard Hartree–Fock method is that a different long range behavior for each orbital, as for example is required in the Hartree–Fock-Slater method, can be forced. A calculation on the ground state of beryllium is performed using the nested procedure. Very little energy is lost because of nesting, and the node in the 1s orbital disappears.     

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     

Self-consistent calculation of 1s2s1S state wave function of helium

A simple self consistent variation perturbation method in the coupled Hartree–Fock scheme has been proposed to calculate 1s2s ¹S state of the He atom. The present paper deals with an 1s2s ¹S wave function in which all the relevant orthogonality conditions are imposed in successive stages. The resulting wave functions together with some interesting features are discussed.     

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     

Self-consistent calculation of 1s2s3S state of helium and heliumlike ions

In this paper we present a simple method within the coupled Hartree–Fock framework to calculate the 1s2s ³S state of helium and heliumlike ions. The results are in very good agreement with those obtained by the use of multiterm correlated wave functions. Some interesting observations associated with the wave function are presented.     

Some properties of the bivariational Hartree–Fock scheme for complex symmetric many-particle operators

The bivariational Hartree–Fock scheme for a general many-body operator T is discussed with particular reference to the complex symmetric case: T^? = T*. It shown that, even in the case when the complex symmetric operator T is real and hence also self-adjoint, the complex symmetric Hartree–Fock scheme does not reduce to the conventional real form, unless one introduces the constraint that the N-dimensional space spanned by the Hartree–Fock functions ? should be stable under complex conjugation, so that ?* = ?α. If one omits this constraint, one gets a complex symmetric formulation of the Hartree–Fock scheme for a real N-electron Hamiltonian having the properties H = H* = H^?, in which the effective Hamiltonian H_eff (1) may have complex eigenvalues ε_k. By using the method of complex scaling, it is indicated that these complex eigenvalues—at least for certain systems—may be related to the existence of so-called physical resonance states, and a simple example is given. Full details will be given elsewhere.     

Basis set quality versus size II. Approximate GTO wave functions for second row transition metal atoms

Basis sets ranging in size from (16, 10, 7) to (20, 14, 11) have been derived for the atoms Y–Cd. Separate sets represent the energy optimized wave functions for each of the s²dⁿ, s¹dⁿ⁺¹, and s⁰dⁿ⁺² configurations. The energies from the largest sets are within 3 mhartrees of the values obtained in numerical Hartree–Fock calculations. Reasonable Hartree–Fock s²dⁿ– s¹dⁿ⁺¹ and s²dⁿ– s⁰dⁿ⁺² excitation energies may be obtained either using the largest basis sets, or using d-orbitals optimized for the s⁰dⁿ⁺² configurations. The basis sets are slightly unbalanced in favor of the s-functions and in disfavor of the d-functions, but various alternative basis sets may be derived by combining parts of the five parent sets. The convergence of radial expectation values is discussed.     

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.     

Extended Hartree–Fock calculations for the helium ground state

The extended Hartree–Fock (EHF) wave function of an n-electron system is defined (Löwdin, Phys. Rev. 97 , 1509 (1955)) as the best Slater determinant built on one-electron spin orbitals having a complete flexibility and projected onto an appropriate symmetry subspace. The configuration interaction equivalent to such a wavefunction for the ¹S state of a two-electron atom is discussed. It is shown that there is in this case an infinite number of solutions to the variational problem with energies lower than that of the usual Hartree–Fock function, and with spin orbitals satisfying all the extremum conditions. Two procedures for obtaining EHF spin orbitals are presented. An application to the ground state of Helium within a basic set made up of 4(s), 3(p₀), 2(d₀) and 1 (f₀) Slater orbitals has produced 90% of the correlation energy.     

Structure and electronic structure of polyacene

It is shown that infinite long polyacene chains may have three energetically close but structurally distinct isomers (a symmetrical, sym, form and two lower symmetry forms: one with double bonds in a trans and another isomer with double bonds in a cis pattern). The energetics is based on solid state MNDO theory. We discuss that the symmetrical form has a substantial energy gap E_g in the Hartree–Fock approach owing to exact exchange terms, which are nonlocal. Broken symmetry Hartree–Fock (HF ) solutions for polyacene are also described. An angularly distorted structure suggested earlier on Jahn–Teller grounds is found to be energetically not favorable.     

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.     

Excitation energies,oscillator strengths,and frequency dependent polarizabilities of Be: Comparison of TDHF,EOM (second order), and MCTDHF

Low-lying excitation energies from the ground state of Be were calculated using a basis set of 61 Cartesian Gaussian functions. Three approximations were employed: the time-dependent Hartree–Fock (TDHF ), second-order equations-of-motion (EOM ), and multiconfigurational time-dependent Hartree–Fock (MCTDHF ). The TDHF excitation energies are 0.5–1.1 eV lower than experiment, and the EOM values are 0.3–1.2 eV lower than experiment, whereas the MCTDHF excitation energies deviate on the absolute average from experiment by only 0.03 eV. We found that in an MCTDHF calculation, any proper MCSCF stationary point is a good reference (i.e., initial) state, not just the ground state. Experimental values for oscillator strength are accurately known only for the 2s²X¹S → 2s2p¹P⁰ transition. The TDHF value and the MCTDHF value agree with experiment, but the EOM value does not. The agreement of the TDHF value with experiment seems to be coincidental, because for higher lying transitions the TDHF values differ by approximately a factor of two or more from the more accurate MCTDHF . Frequency independent polarizabilities, α(0), were also calculated with the TDHF , HRPA , and MCTDHF and frequency dependent polarizabilities, β(ω), were calculated with the MCTDHF . No experimental data for Be polarizabilities exist, but we expect the MCTDHF values to be among the most accurate calculations available.     

Electronic structure of long cumulene chains

The analysis of the equations of the unrestricted Hartree–Fock (UHF) method for polyenes C_NH_N+2 with even and odd N » 1 is carried out. The equations of the UHF method are shown to be the same in both cases. The comparison of the UHF method with the extended Hartree–Fock (EHF) method applied to large systems is performed. The ground state and π-electron spectra of long cumulene chains C_NH₄ are treated by the EHF Method. The end effects are taken into consideration. It is shown that the EHF method gives a finite value of the first optical transition frequency and, at the same time, zero value of torsion barrier of end CH₂–groups in long cumulene chains (N → ) in contrast to previous calculations of cumulenes by the Huckel method and the restricted Hartree–Fock method.     

Computation of Fourier transform quantities in Hartree–Fock calculations for simple crystals

An efficient method is presented to compute the Fourier transforms of lattice sums over Slater-type orbital products that arise in crystal Hartree–Fock calculations. Introduction of one-dimensional integral representations for the Fourier transforms of the STO 'S enables a separation of the three infinite sums under the (two-dimensional) integral sign. This, in turn, makes it possible to calculate and store quantities associated with the three Fourier transform vector components separately. The orbital transform integrations are then performed numerically. The method is very advantageous when lattice sum transforms are needed for a large number of transform vectors. Formulas and results are presented for simple-cubic crystals when 1s, 2s, and 2p Slater orbitals are involved.     

On the role of symmetry in the ab initio hartree-fock linear-combination-of-atomic-orbitals treatment of periodic systems

The symmetry properties of the mono- and bielectronic terms contributing to the Fock matrix in the ab initio Hartree–Fock treatment of periodic systems are discussed. A computational scheme which takes full advantage of the point symmetry is presented; in this respect, it represents a generalization of the scheme proposed in Int. J. Quantum Chem. 17 , 501 (1980). Computational details and numerical examples are reported; it is shown that with respect to two of the bottlenecks of this kind of calculation, namely, computer time and backing storage required for the bielectronic integrals, it is possible to obtain saving factors as large as h and h², respectively, where h is the order of the point group. Preliminary tests are reported which indicate that the study of relatively complicated systems, like quartz or corundum (9 and 10 atoms in the unit cell, respectively) at an ab initio Hartree–Fock level is now within reach.     

Examination of several density functionals in numerical Kohn–Sham calculations for atoms

We studied several exchange‐only and exchange–correlation energy density functionals in numerical, i.e., basis‐set‐free, nonrelativistic Kohn–Sham calculations for closed‐shell ¹S states of atoms and atomic ions with N electrons, where 2≤N≤120. Accurate total energies are presented to serve as reference data for algebraic approaches, as do the numerical Hartree–Fock results, which are also provided. Gradient‐corrected exchange‐only functionals considerably improve the total energies obtained from the usual local density approximation, when compared to the Hartree–Fock results. Such an improvement due to gradient corrections is not seen in general for highest orbital energies, neither for exchange‐only results (to be compared with Hartree–Fock results), nor for exchange–correlation results (to be compared with experimental ionization energies). © 2001 John Wiley & Sons, Inc. Int J Quant Chem 82: 227–241, 2001     

Generator coordinate Hartree–Fock method applied to the choice of a contracted Gaussian basis for the second-row atoms

The generator coordinate Hartree–Fock (GCHF) method is employed as a criterion for the selection of a 18s12p Gaussian basis for the atoms Na–Ar. The role of the weight functions in the assessment of the numerical integration range of the GCHF equations is shown. The extended basis is then contracted to (10s6p) by a standard procedure and in combination with the previously contracted (7s5p) Gaussian basis for the atoms Li–Ne is enriched with polarization functions. This basis is tested for AlF, SiO, PN, BCl, and P₂. The properties of interest were HF total energies, MP2 dipolar moments, bond distances, and dissociation energies. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 63: 927–934, 1997