The perturbation theory of the extended Hartree–Fock approximation for two-electron atoms

The first-order 1/Z perturbation theory of the extended Hartree–Fock approximation for two-electron atoms is described. A number of unexpected features emerge: (a) it is proved that the orbitals must be expanded in powers of Z^?1/2, rather than in Z^?1 as expected; (b) it is shown that the restricted Hartree–Fock and correlation parts of the orbitals can be uncoupled to first order, so that second-order energies are additive; (c) the equation describing the first-order correlation orbital has an infinite number of solutions of all angular symmetries in general, rather than only one of a single symmetry as expected; (d) the first-order correlation equation is a homogeneous linear eigenvalue-type equation with a non-local potential. It involves a parameter μ and an eigenvalue ω(μ) which may be interpreted as the probability amplitude and energy of a virtual correlation state. The second-order correlation energy is 2μ²ω. Numerical solutions for the first-order correlation orbitals, obtained variationally, are presented. The approximate second-order correlation energy is nearly 90% of the exact value. The first-order 1/Z perturbation theory of the natural-orbital expansion is described, and the coupled first-order integro-differential perturbation equations are obtained. The close relationship between the first-order extended Hartree–Fock correlation orbitals and the first-order natural correlation orbitals is discussed. A comparison of the numerical results with those of Kutzelnigg confirms the similarity.     

Møller–Plesset theory for atomic ground state energies

Mø–Plesset theory, in which electron correlation energy is calculated by perturbation techniques, is used in second order to calculate energies of the ground states of atoms up to neon. The unrestricted Hartree–Fock (UHF) Hamiltonian is used as the unperturbed system and the technique is then described as unrestricted Mø–Plesset to second order (UMP2). Use of large Gaussian basis sets suggests that the limiting UMP2 energies with a complete basis of s, p, and d functions account for 75–84% of the correlation energy. Preliminary estimates of the contributions of basis functions with higher angular quantum numbers indicate that full UMP2 limits give even more accurate total energies.     

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.     

Extended Hartree–Fock self-consistent field with some results for berylliumlike atomic systems

The general methods of deriving the extended Hartree–Fock equations are described. The rules for going over from the energy expression in the ordinary method of calculation to that in an extended one are reformulated and illustrated. The extended Hartree—Fock equations for berylliumlike atomic systems based on the use of nonorthogonal radial orbitals are given and solved. The numerical values of overlap integrals and total energies are given and discussed.     

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.     

Construction of an effective hamiltonian for open-shell molecular systems

An effective Hamiltonian for open-shell molecular systems is constructed. The unrestricted Hartree–Fock orbitals are applied as a basis set of one-particle functions. This effective Hamiltonian is determined as a simple product of the original total Hamiltonian and the spin annihilator. The second-quantization formalism and the Feynman–Goldstone diagrammatic technique are used. The resulting effective hamiltonian is composed of zero- to four-particle terms. A possibility of applying the nondegenerate diagrammatic perturbation theory constructed over this effective Hamiltonian is discussed.     

Incomplete basis set problem. IV. Perturbative corrections to pair energies

A partitioning of the molecular Hamiltonian into the occupied and virtual orbital spaces and their orthogonal complement is introduced and used to develop a perturbation expansion of the exact ground-state energy relative to the Hartree–Fock energy computed using an incomplete basis set. The leading perturbation corrections to pair energies due to using the incomplete basis set are considered in detail. Summations of certain classes of pair contributions are discussed and a resummed correction is obtained.     

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.     

Application of the many-body perturbation theory by using localized orbitals

Diagrammatic formulation of the many-body perturbation theory is investigated when both the occupied orbitals and the virtual ones are localized, i.e., they are unitary transforms of the canonical Hartree–Fock orbitals. All diagrams representing ground state correlation energy can be generated through fifth order. For cyclic polyenes C₆H₆ and C₁₀H₁₀ as model systems, the energy corrections are calculated in the Pariser–Parr–Pople approximation for a wide range of the coupling constant β^?1, through fourth order including some fifth order terms. The results are compared to those obtained by other methods: perturbation theory by using canonical orbitals and full CI. The effect of neglecting contributions from orbitals localized into neighboring sites is also studied.     

A systematic preparation of new contracted Gaussian-type orbital sets. VI. Ab initio calculation on molecules containing Na through Cl

Compact contracted Gaussian basis sets introduced in the preceding article are tested for ab initio molecular calculations on molecules containing third-row atoms (Na through Cl). It is found that the effect of splitting valence orbitals is essential for these molecules and addition of polarization functions to split basis sets can yield computed geometries, spectroscopic constants, and atomization energies in close agreement with the result of near Hartree–Fock calculations.     

Evaluation of one‐electron molecular integrals over complete orthonormal sets of Ψα ‐ETO using auxiliary functions

By the use of expansion and one‐range addition theorems, the one‐electron molecular integrals over complete orthonormal sets of Ψ^α ‐exponential type orbitals arising in Hartree–Fock–Roothaan equations for molecules are evaluated. These integrals are expressed through the auxiliary functions in ellipsoidal coordinates. The comparison is made using Slater‐, Coulomb‐Sturmian‐, and Lambda‐type basis functions. Computation results are in good agreement with those obtained in the literature. The relationships obtained are valid for the arbitrary quantum numbers, screening constants, and location of orbitals. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Improved uncoupled Hartree–Fock (IUCHF) perturbation methods and bounds for the second-order energy in coupled Hartree–Fock perturbation theory

Different kinds of improved uncoupled Hartree–Fock methods are proposed for the calculation of second-order perturbation energies. Using these methods inequalities are derived for the error of the uncoupled procedure with geometric approximation.     

Optimized linear combinations of simple exponentials for atomic systems

Hartree-Fock wave functions for the He and Be isoelectronic sequences of ions are calculated using orbitals which are linear combinations of simple exponential functions. By a full optimization of the exponents and coefficients close approximations to the HartreeFock energies were obtained. To the same order of accuracy the resulting Hartree–Fock orbitals require fewer basis functions than used previously. A number of difficulties which arise in the numerical procedures as the size of the basis set is increased are analysed in detail. Similar results are obtained for the Li sequence using the Unrestricted HartreeFock method with and without projection.     

Highly correlated particle densities and idempotent one-densities

The problem of determining idempotent one-densities which integrate to the exact or to a highly correlated particle density is considered. A method for obtaining the minimum energy idempotent one-density integrating to a given correlated particle density within a finite basis is described. The implications of this are twofold. First, Hartree–Fock accuracy can be exceeded in describing the electron density with an idempotent one-density; this is particularly relevant to the problem of constructing orbitals from experimental x-ray scattering data. Second, electron densities from analytic CI or MCSCF wave functions can be made available in a form as compact as the Hartree–Fock density by reporting the orbitals which define the correlated density via an idempotent one-density. A numerical example of the new method is given in which an accurate correlated density for He is “fitted” by an idempotent one-density represented in a finite (near Hartree–Fock) basis. Considering the deficiencies of the basis for this purpose, a technique is suggested for constructing basis sets optimized for prediction of one-electron properties rather than for energy.     

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. © 2012 Wiley Periodicals, Inc.     

Vibrational spectra of molecules in the Hartree–Fock dielectric screening approach

Traditionally, the calculation of the vibrational spectra of molecules involves at one point or another a numerical differentiation procedure. Such a method has some serious drawbacks both in efficiency and in accuracy. In this paper, an alternative method based on linear response theory is presented. The second derivative of the ground-state energy is expressed in terms of the electron density response matrix by means of perturbation theory. The unperturbed wave functions are obtained from the Hartree–Fock equation. First-order perturbation theory applied to this equation leads to the Hartree–Fock linear response. As an illustration of this method the vibrational frequency of a H₂ molecule is calculated. The result is 1.348 × 10¹⁴ Hz as compared to the experimental value of 1.319 × 10¹⁴ Hz. This method is also applicable in the calculation of the phonon dispersion curves of solids.     

Local Hartree–Fock orbitals using a three‐level optimization strategy for the energy

Using the three‐level energy optimization procedure combined with a refined version of the least‐change strategy for the orbitals—where an explicit localization is performed at the valence basis level—it is shown how to more efficiently determine a set of local Hartree–Fock orbitals. Further, a core–valence separation of the least‐change occupied orbital space is introduced. Numerical results comparing valence basis localized orbitals and canonical molecular orbitals as starting guesses for the full basis localization are presented. The results show that the localization of the occupied orbitals may be performed at a small computational cost if valence basis localized orbitals are used as a starting guess. For the unoccupied space, about half the number of iterations are required if valence localized orbitals are used as a starting guess compared to a canonical set of unoccupied Hartree–Fock orbitals. Different local minima may be obtained when different starting guesses are used. However, the different minima all correspond to orbitals with approximately the same locality. © 2013 Wiley Periodicals, Inc.     

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.     

Multiplicity,instability, and SCF convergence problems in Hartree–Fock solutions

We present a study of the instability and convergence of Hartree–Fock (HF) ab initio solutions for the diatomic systems H₂, LiH, CH, C₂, and N₂. In our study, we consider real molecular orbitals (MOs) and analyze the classes of single‐determinant functions associated to Hartree–Fock–Roothaan (HFR) and Hartree–Fock–Pople–Nesbet (HFPN) equations. To determine the multiple HF solutions, we used either an SCF iterative procedure with aufbau and non‐aufbau ordering rules or the algebraic method (AM). Stability conditions were determined using TICS and ASDW stability matrices, derived from the maximum and minimum method of functions (MMF). We examined the relationship between pure SCF convergence criterion with the aufbau ordering rule, and the classification of the HF solution as an extremum point in its respective class of functions. Our results show that (i) in a pure converged SCF calculation, with the aufbau ordering rule, the solutions are not necessarily classified as a minimum of the HF functional with respect to the TICS or ASDW classes of solutions, and (ii) for all studied systems, we obtained local minimum points associated only with the aufbau rule and the solutions of lower energies. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 76: 600–610, 2000     

On the use of symmetry in first-order perturbed Hartree–Fock theory

A method for the determination of the symmetry of first-order vectors in Hartree–Fock perturbation theory is developed. This leads to the definition of symmetry-adapted basis vectors to be employed at first order in the perturbation. It is shown that computer time can be saved, to some extent, in the calculation of second-order properties, by exploiting molecular symmetry. Specific examples are given for methane, ammonia, and water.