Quantum chemistry of many-electron systems: high-priority aspects

The review generalizes the studies devoted to the development of a new quantum chemistry method representing an alternative to the Hartree–Fock approximation. Based on the hypothesis of prohibition of equipotential surfaces, which clarifies the physical sense of the Pauli exclusion principle, and taking account of the condition for antisymmetrical wave function of the triplet state (3S) of He atom, the Hartree–Fock approximation is inappropriate for a priori determination of the nodal surfaces of many-electron wave functions (MWFs) for the test systems traditionally used in quantum chemistry, namely, excited triplet state of H₂ molecule and the ground electronic states of Li atom and LiH molecule. The nodal surfaces of the wave functions corresponding to the minimum basis set of Slater orbitals in the Hartree–Fock approximation are constructed and analyzed. An alternative to the Hartree–Fock approximation is provided by the MWF quantum chemical method being developed by the authors. In the MWF method, the nodal surfaces for H₂(³Σ _u ^v ) and Li(²S) are specified a priori. Some aspects of geometric interpretation of the Pauli exclusion principle are discussed. Unlike the MWF method, the Hartree–Fock approximation is unsuitable for taking account of the dependence of the MWF nodal surfaces on the nuclear charges and on correlation effects related to the motion of electrons with antiparallel spins because such nodal surfaces are predefined by the mathematical properties of Slater determinants rather than by physically clear and more practically valuable algebraic products of electrostatic potential differences.     

Electronic structure and bonding of the pentasulfur hexanitride (S5N6) molecule

An ab initio version of the Hartree–Fock–Slater method is applied to obtain molecular orbitals and eigenvalues for S₅N₆. The electronic structure, bonding, stability, and electronic spectrum are discussed.     

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.     

Calculations of Isoelectronic Series of He Using Noninteger n‐Slater Type Orbitals in Single and Double Zeta Approximations

Using integer and noninteger n-Slater type orbitals in single- and double-zeta approximations, the Hartree-Fock-Roothaan calculations were performed for the ground states of first ten cationic members of the isoelectronic series of He atom. All the noninteger parameters and orbital exponents were fully optimized. In the case of noninteger n-Slater type orbitals in double zeta basis sets, the results of calculations obtained are more close to the numerical Hatree-Fock values and the average deviations of our ground state energies do not exceed 2×10^－6 hartrees of their numerical results.     

New derivation of the Waller–Hartree–Fock spatial wave function

A new derivation is given for the Waller–Hartree–Fock double-determinantal spatial wave function. One starts from the single-determinant wave function in which a orbitals are doubly occupied, and decomposes it into a sum of products of spatial and spin functions. The spatial product of the first genealogical spin eigenfunction is a double-determinantal function. The derivation is based on the simple form of U_1?(P) when the representation matrix is obtained from the genealogical spin eigenfunction.     

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.     

X-ray and electron scattering factors for many-electron atomic system

Analytical expressions are developed for the x-ray and electron scattering factors for a many-electron atomic system when the single configuration wave function of the system is written as a sum of Slater determinants of spin orbitals. The radial part of the orbital is expanded in terms of Slater-type orbitals (STO 's). The expressions so developed have been used to calculate the coherent and incoherent x-ray and electron scattering factors and intensities for all the neutral atoms up to krypton (Z = 36) and for some positive and negative ions of chemical interest. The results obtained are used to test the value of Hartree–Fock wave functions for the evaluation of “one-electron properties” of many-electron atomic systems.     

Cluster calculations of the H2O/Pt(111) system

The electronic interaction between water and a Pt(111) surface as evaluated for different Pt_x(H₂O)_y clusters is discussed. Hartree–Fock–Slater (HFS ) one-electron ground state energies, ionization potentials, partial densities of states, and Mulliken occupation numbers are related to bonding shifts, as well as initial and final state screening for different orientations of the molecule. The formation of Pt? H₂O bonds are sensitive to the orientation since surface oriented H atoms bridge the spatial separation between O 2p and Pt 5d orbitals and thus increase the intermixing of metal and adsorbate orbitals. The dipole moment and the net charge of the H₂O molecule is also discussed. Finally, approximations of the metal–H₂O potential for use in statistical models of the liquid–metal interface are suggested.     

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.     

EHF Description of superexchange: Relations to Anderson's model

It is shown that the treatment of the superexchange in insulators, based on the spin projected extended Hartree–Fock (EHF ) method may be considered as a direct generalization of Anderson's ligand field (or kinetic exchange) theory: the EHF methods permits to construct explicitly the effective orbitals for the magnetic electrons and to determine all the parameters occurring in Anderson's theory from only one variational problem.     

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     

Hartree—Fock—Slater functions as basis for one-electron perturbation calculations for molecules: I. Calculation of ground state electric dipole polarizabilities

The analysis of the decoupling of Hartree—Fock—Slater SCF perturbation equations for an external field is undertaken. The points of departure from the corresponding Hartree—Fock perturbation equations are stressed. Both formal and numerical results suggest that the fully uncoupled Hartree—Fock—Slater expression is a less drastic approximation than the same Hartree—Fock one. The uncoupled expression for the ground state electric dipole polarizability is calculated for CO, N₂, ethylene, acethylene and trans-butadiene in the dipole length—dipole length, dipole velocity—dipole length and dipole velocity—dipole velocity alternative formulations with an ab initio Hartree—Fock—Slater SCF basis set. The results compare well with other non-empirical results and the dipole velocity-dipole length results are in remarkably good agreement with experiments.     

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.     

A correlation study of Li ground state by the MCHF procedure

Results are reported for multiconfiguration Hartree–Fock studies of correlation in the lithium ground state, which maintain orthogonality of orbitals within a configuration. It is shown that when the 1s- and 2s-orbitals are fixed at their Hartree–Fock value, configurations for which Brillouin's theorem holds may be important, particularly for atomic properties other than energy. The Fermi contact term is considered as an example.     

Spin–other–orbit and spin–orbit interactions in ƒ2 and ƒ3 electron configurations

Expressions of the matrix elements of the spin–other–orbit and spin–orbit interactions for the various multiplets of all the states of ?²- and ?³-electron configurations are reported and used to evaluate the Hartree–Fock values of these interactions in the neutral atoms Ce(4?²), Pr(4?³), Ho(4?¹¹) and Er(4?¹²). The required values of the spin–spin parameters M, and the spin-orbit parameter ζ for these atoms were obtained using numerical Hartree–Fock wave functions.     

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.     

Ab initio Hartree–Fock–Slater calculation of Tetrathiafulvalene (TTF) and the TTF1+ and TTF2+ ions

The electronic structures of TTF, TTF¹⁺, and TTF²⁺ are described by means of an ab initio Hartree–Fock–Slater procedure with a double-zeta STO basis. Electronic and photoemission spectra, bonding, and charge distributions are discussed and compared to experiments and previous calculations.     

Spin-projected EHF method. IV. Comparison of potential curves given by different one-electron methods

Some aspects of the computer realization of the spin-projected extended Hartree–Fock (EHF ) method at the ab initio level are briefly discussed for the algorithm of solution developed in the previous papers of this series. Calculations have been performed for the BH molecule by using a small basis of contracted Gaussian lobes with the purpose of comparing the potential curves given by the different one-electron methods RHF , UHF , UHF with subsequent spin projection, and EHF . It is concluded that the UHF and, in particular, the EHF methods give a qualitatively correct shape of the potential curve; the RHF method shows the known incorrect dissociation behavior while the potential curve obtained by subsequent spin projection of the UHF wave function exhibits spurious extrema at intermediate internuclear separations.     

A partially projected wave function for radicals

A partially projected wave function for odd electron systems with quantum number M=1/2, containing μu spin functions α and μ spin functions α, with fractional spin component αS_z=1/2 and 3/2 are derived from the totally projected wave function. To obtain these wave functions new symmetry relations between Sanibel coefficients for the odd electron case have been found, as well as the relations between primitive spin functions and their spin permutations. The wave function for the doublet state is shown not to contain contamination of the quadruplet state, and the wave function for the quadruplet does not have contamination of the duplet. Both wave functions exhibit equal forms except in the signs of their summation terms. The number of primitive spin functions depends on the number of electrons (n_s), it grows linearly as n_s=(N+3)/2. It can be considered as a generalization of the half projected Hartree–Fock wave function to the odd electron case. The HPHF wave function is defined for even electron systems and consists of only two Slater determinants, it has been shown to introduce some correlation effects and it has been successfully applied to calculate the low-lying excited states of molecules. Therefore, this investigation is the first step to propose a method to calculate the excited states of radicals when other methods are impracticable. This revised version was published online in July 2006 with corrections to the Cover Date.     

PROJECTION OF SINGLE-DETERMINANT SOLUTIONS OF SPIN-POLARIZED SELF-CONSISTENT FIELD EQUATIONS IN THE BASIS SET OF SPIN-PAIRED ORBITALS

Journal of Structural Chemistry - A new scheme is presented for obtaining a pure spin state for the one-determinant wave function built on Hartree–Fock or Kohn–Sham orbitals. In this...