Independent particle theory with electron correlation

We formulate an effective independent particle model where the effective Hamiltonian is composed of the Fock operator and a correlation potential. Within the model the kinetic energy and the exchange energy can be expressed exactly leaving the correlation energy functional as the remaining unknown. Our efforts concentrate on finding a correlation potential such that exact ionization potentials and electron affinities can be reproduced as orbital energies. The equation-of-motion coupled-cluster approach enables us to define an effective Hamiltonian from which a correlation potential can be extracted. We also make the connection to electron propagator theory. The disadvantage of the latter is the inherit energy dependence of the potential resulting in a different Hamiltonian for each orbital. Alternatively, the Fock space coupled-cluster approach employs an effective Hamiltonian which is energy independent and universal for all orbitals. A correlation potential is extracted which yields the exact ionization potentials and electron affinities and a set of associated molecular orbitals. We also describe the close relationship to Brueckner theory.     

A simple model for the Slater exchange potential and its performance for solids

A simple local model for the Slater exchange potential is determined by least square fit procedure from Hartree–Fock (HF) atomic data. Since the Slater potential is the exact exchange potential yielding HF electron density from Levy‐Perdew‐Sahni density functional formalism (Levy et al., Phys. Rev. A 1984, 30, 2745), the derived local potential is significantly more negative than the conventional local density approximation. On the set of 22 ionic, covalent and van der Waals solids including strongly correlated transition metal oxides, it has been demonstrated, that this simple model potential is capable of reproducing the band gaps nearly as good as popular meta GGA potentials in close agreement with experimental values.     

The exact Fermi potential yielding the Hartree–Fock electron density from orbital‐free density functional theory

The exact expression for the Fermi potential yielding the Hartree–Fock electron density within an orbital‐free density functional formalism is derived. The Fermi potential, which is defined as that part of the potential that depends on the particles’ nature, is in this context given as the sum of the Pauli potential and the exchange potential. The exact exchange potential for an orbital‐free density functional formalism is shown to be the Slater potential.     

One- and two-center physical space partitioning of the energy in the density functional theory

A conceptually new approach is introduced for the decomposition of the molecular energy calculated at the density functional theory level of theory into sum of one- and two-atomic energy components, and is realized in the "fuzzy atoms" framework. (Fuzzy atoms mean that the three-dimensional physical space is divided into atomic regions having no sharp boundaries but exhibiting a continuous transition from one to another.) The new scheme uses the new concept of "bond order density" to calculate the diatomic exchange energy components and gives them unexpectedly close to the values calculated by the exact (Hartree-Fock) exchange for the same Kohn-Sham orbitals.     

Accurate orbital amplitudes at nuclei frog gaussian basis sets

A simple extrapolation procedure combining wavefunctions obtained from gaussian basis sets with exact solutions of the nuclear cusp equations is proposed for computing orbital amplitudes at nuclei. Comparison with exact results for atoms and diatomic molecules indicates that the procedure is capable of giving Hartree—Fock amplitudes with errors of at most 10^?2 for the low amplitude outer orbitals and errors of less than 10^?3 for the important inner orbitals. The resulting errors in the total densities are around 10^?2. These accuracies are comparable with those obtained with energy-optimized Slater basis sets.     

Unified treatment of complete orthonormal sets of nonrelativistic,quasirelativistic and relativistic sets of spinor wave functions,and Slater spinor orbitals in coordinate,momentum and four-dimensional spaces

Using the properties of tensor spherical harmonics introduced by the author in previous paper (Guseinov, Phys Lett A 372:44, 2007) and complete orthonormal scalar basis sets of nonrelativistic -exponential type orbitals ( -ETO), - momentum space orbitals ( -MSO) and z ^α-hyperspherical harmonics (z ^α-HSH) for particles with spin s = 0 the new analytical relations for the quasirelativistic and relativistic spinor wave functions and Slater spinor orbitals in coordinate, momentum and four-dimensional spaces are derived, where α = 1, 0, −1, −2,.... The 2-component quasirelativistic and 4-component relativistic spinor wave functions obtained are complete without the inclusion of the continuum. The relativistic spinor wave function sets and Slater spinor orbitals are expressed through the corresponding quasirelativistic spinor wave functions and Slater spinor orbitals, respectively. The analytical formulas for overlap integrals over quasirelativistic and relativistic Slater spinor orbitals with the same screening constants in coordinate space are also derived.     

On the theory of electron correlation in atoms and molecules. II. General cluster expansion theory and the general correlated wave functions method

An exact cluster expansion of many electron wave functions is derived, beginning with a finite linear combination of Slater determinants rather than the more usual single determinant. This general cluster expansion is found to apply both in the case where all possible Slater determinants from a finite set of spin orbitals are included in the linear combination, and in the case where the number of determinants is restricted. The special properties of that finite linear combination of determinants closest to the exact wave function in the least squares sense are studied. These properties lead to the derivation of a general correlated wave functions method, illustrating again the close relationship between methods of this type and cluster expansion theory. Additional approximations, necessary for practical calculations, are set out.     

A sparse algorithm for the evaluation of the local energy in quantum Monte Carlo

A new algorithm is presented for the sparse representation and evaluation of Slater determinants in the quantum Monte Carlo (QMC) method. The approach, combined with the use of localized orbitals in a Slater-type orbital basis set, significantly extends the size molecule that can be treated with the QMC method. Application of the algorithm to systems containing up to 390 electrons confirms that the cost of evaluating the Slater determinant scales linearly with system size.     

Use of noninteger n‐Slater type orbitals in combined Hartree–Fock–Roothaan theory for calculation of isoelectronic series of atoms Be to Ne

The ground state calculations in the combined Hartree–Fock–Roothaan approach are performed for the neutral and the first 20 cationic members of the isoelectronic series of atoms from Be to Ne using noninteger n‐Slater type orbitals. For the total energies obtained, only a small deviation has been found. At the same time, the size of the present noninteger n‐Slater type orbitals is smaller than that of the usual extended integer n‐Slater functions in literature. All of the nonlinear parameters are fully optimized. The relationship between optimized parameters and atomic number Z is also investigated. For each atom, the total energies are given in tables. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

Hulthén and slater type 2d functions in some excited configurations of sulphur and phosphorus

It is shown that a substantial energy improvement is gained by the variational use of Hulthén orbitals, instead of single Slater orbitals, in the 3d shells of some excited configurations of sulphur and phosphorus. The energies obtained are close to those attained with two-term Slater functions. In some cases the radial distribution functions from Hulthén orbitals are as good an approximation of SCF radial distributions as those from two-term Slater orbitals. Single term 2d functions with only one parameter are found to give almost identical energies and radial distribution functions as those obtained from two-parameter Hulthén orbitals. It is shown that the relationship between one-term 2d orbitals and Hulthén orbitals gives a method of enforcing nuclear cusp conditions on the former with little effect on the energy.     

Theory of complete orthonormal relativistic vector wave function sets and Slater type relativistic vector orbitals in coordinate,momentum and four-dimensional spaces

The new analytical relations for the relativistic vector wave functions and Slater type relativistic vector orbitals in coordinate, momentum and four-dimensional spaces are derived using the properties of quasirelativistic vector spherical harmonics introduced by the author in previous paper (I.I. Guseinov, J. Math. Chem., 44, 197 (2008)) and complete orthonormal scalar basis sets of nonrelativistic ψ ^α -exponential type orbitals (ψ ^α -ETO), -momentum space orbitals (-MSO) and z ^α -hyperspherical harmonics (z ^α -HSH). The 6-component relativistic vector wave functions obtained are complete without the inclusion of the continuum. The relativistic vector wave function sets and Slater type relativistic vector orbitals are expressed through the corresponding quasirelativistic vector wave functions and Slater vector orbitals, respectively. The analytical formulas are also derived for overlap integrals over Slater type relativistic vector orbitals with the same screening constants in coordinate space.     

On the calculation of molecular dipole moments

The approximation made in the calculation of molecular dipole moments by including only the point charges and the atomic dipoles is evaluated in different all-valence (or all)-electrons MO procedures. In the CNDO method, the use of the exact formula after retransformation of the atomic basis into Slater orbitals gives poorer values than the Pople-Segal's procedure.     

Some efforts beyond the local density approximation

The exchange-correlation density functional can be expressed as a many-body perturbation series in terms of the Coulomb interaction using the exact Kohn-Sham orbitals as the basis. A self-consistent equation is derived for the exact exchangecorrelation potential. This perturbation approach forms a basis for going beyond the local density approximation (LDA ). The discontinuity in the exchange-correlation potential for semiconductors calculated by the perturbative approach gives a good account of the discrepancy of the band gap calculated in LDA . The discontinuity also plays an important role in the interface band diagrams. A theory to account for the interaction effects of localized d or f orbitals is reviewed and the physics of the applications to a model test, to some 3d transition metals, and to heavy fermions is discussed. The perturbative approach to improvement beyond LDA tends to be computation-intensive and to be system-specific. © 1995 John Wiley & Sons, Inc.     

Non-integer Slater orbital calculations

In order to calculate the one- and two-electron, two-center integrals over non-integer n Slater type orbitals, use is made of elliptical coordinates for the monoelectronic, hybrid, and Coulomb integrals. For the exchange integrals, the atomic orbitals are translated to a common center. The final integration is performed by Gaussian quadrature.As an example, an SCF ab initio calculation is performed for the LiH molecule, both with integer and non-integer principal quantum number.     

New translation method for STOs and its application to calculation of overlap integrals

A new translation method for Slater‐type orbitals (STOs) is proposed involving exact translation of the regular solid harmonic part of the orbital followed by the series expansion of the residual spherical part in powers of the radial variable. The method is positively tested in the case of the overlap integral, showing good rate of convergence and great numerical stability under wide changes in the relevant molecular parameters. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 73: 333–340, 1999     

Molecule intrinsic minimal basis sets. I. Exact resolution of ab initio optimized molecular orbitals in terms of deformed atomic minimal-basis orbitals

A method is presented for expressing the occupied self-consistent-field (SCF) orbitals of a molecule exactly in terms of chemically deformed atomic minimal-basis-set orbitals that deviate as little as possible from free-atom SCF minimal-basis orbitals. The molecular orbitals referred to are the exact SCF orbitals, the free-atom orbitals referred to are the exact atomic SCF orbitals, and the formulation of the deformed "quasiatomic minimal-basis-sets" is independent of the calculational atomic orbital basis used. The resulting resolution of molecular orbitals in terms of quasiatomic minimal basis set orbitals is therefore intrinsic to the exact molecular wave functions. The deformations are analyzed in terms of interatomic contributions. The Mulliken population analysis is formulated in terms of the quasiatomic minimal-basis orbitals. In the virtual SCF orbital space the method leads to a quantitative ab initio formulation of the qualitative model of virtual valence orbitals, which are useful for calculating electron correlation and the interpretation of reactions. The method is applicable to Kohn-Sham density functional theory orbitals and is easily generalized to valence MCSCF orbitals.     

Calculation of Negative Ions of B,C, N,O and F Using Noninteger n Slater Type Orbitals

Combined Hartree‐Fock‐Roothaan calculations have been performed using noninteger n Slater type orbitals for the ground states of the lowest electron configurations 1s²2s²2pⁿ (2 ≤ n ≤ 6) for negative ions of B, C, N, O and F. These results are compared with the corresponding results obtained from the use of integer n Slater type orbitals. All of the nonlinear parameters are fully optimized. The results of calculation of coupling‐projection coefficients, orbital and total energies and virial ratios are presented. It is shown that the noninteger n Slater type orbitals, in general, improve the orbital energies.     

Self‐consistent shielding in atoms and molecules

The orbital exponents of trial wave functions for simple systems can be found from the potential energy terms alone. Shielding of the nuclear charge by one electron on another is determined by the relative values of the nuclear–electron attraction and the electron–electron repulsion. For two electrons in the same orbital, the shielding is divided equally. For different orbitals, only the inner electron shields the outer. The systems tested are first‐row atoms, using Slater orbitals. It appears that if this approach can be generalized, it may not be necessary to calculate kinetic energies in chemical systems, since they will be determined by the orbital exponents. This would be useful if trial wave functions were not available, but trial electron density functions were. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Structure of the optimized effective Kohn—Sham exchange potential and its gradient approximations

An analysis of the structure of the optimized effective Kohn-Sham exchange potential v_x and its gradient approximations is presented. The potential is decomposed into the Slater potential v_s and the response of v_s to density variations, v_resp. The latter exhibits peaks that reflect the atomic shell structure. Kohn—Sham exchange potentials derived from current gradient approaches for the exchange energy are shown to be quite reasonable for the Slater potential, but they fail to approximate the response part, which leads to poor overall potentials. Improved potentials are constructed by a direct fit of v_x with a gradient-dependent Padé approximant form. The potentials obtained possess proper asymptotic and scaling properties and reproduce the shell structure of the exact v_x. © 1996 John Wiley & Sons, Inc.     

Evaluation of Orbital- and Ground State Energies of Some Open- and Closed-Shell Atoms over Integer and Noninteger Slater Type Orbitals

Ab initio calculations of the orbital and the ground state energies of some open- and closed-shell atoms over Slater type orbitals with quantum numbers integer and Slater type orbitals with quantum numbers noninteger have been performed. In order to increase the efficiency of these calculations the atomic two-electron integrals were expressed in terms of incomplete beta function. Results were observed to be in good agreement with the literature.