On the nodal structure of single-particle approximation based atomic wave functions

The nodal structures of atomic wave functions based on a product of spatial orbitals, namely, restricted, unrestricted, and generalized valence bond wave functions, are shown to be equivalent. This result is verified by fixed node-diffusion Monte Carlo simulations for atoms up to Ne. Also for a molecular system, Li(2) at the equilibrium geometry, a multideterminantal generalized valence bond wave function does not improve the nodal surfaces of a restricted Hartree-Fock wave function.     

The optimized-basis-set multiconfiguration spin-coupled method for the ab initio calculation of atomic and molecular electronic wave functions

An ab initio method for the calculation of atomic and molecular electronic wave functions is presented. The “Optimized-Basis-Set Multiconfiguration Spin-Coupled” (OBS –MCSC ) method may be viewed either as a multiconfiguration generalization of the spin-coupled approach or as a nonorthogonal variant of the MCSCF method. In addition, the OBS –MCSC method optimizes the basis-set exponential parameters simultaneously with all other variational parameters, through a second-derivative minimization procedure. Explicit analytic expressions for the required first and second derivatives of the energy with respect to all variational parameters are obtained. Test calculations prove the capability of the method to yield compact yet accurate electronic wave functions.© 1993 John Wiley & Sons, Inc.     

Density matrix averaged atomic natural orbital (ANO) basis sets for correlated molecular wave functions

Summary Generally contracted Basis sets for the atoms H-Kr have been constructed using the atomic natural orbital (ANO) approach, with modifications for allowing symmetry breaking and state averaging. The ANO's are constructed by averaging over the most significant electronic states, the ground state of the cation, the ground state of the anion for some atoms and the homonuclear diatomic molecule at equilibrium distance for some atoms. The contracted basis sets yield excellent results for properties of molecules such as bond-strengths and-lengths, vibrational frequencies, and good results for valence spectra, ionization potentials and electron affinities of the atoms, considering the small size of these sets. The basis sets presented in this article constitute a balanced sequence of basis sets suitable for larger systems, where economy in basis set size is of importance.     

Density matrix averaged atomic natural orbital (ANO) basis sets for correlated molecular wave functions

Generally contracted basis sets for the first row transition metal atoms Sc-Zn have been constructed using the atomic natural orbital (ANO) approach, with modifications for allowing symmetry breaking and state averaging. The ANOs are constructed by averaging over the three electronic configurationsd ⁿ ,d ^n–1 s, andd ^n–2 s ² for the neutral atom as well as the ground state for the cation and the ground state atom in an external electric field. The primitive sets are 21s15p10d6f4g. Contraction to 6s5p4d3f2g yields results that are virtually identical to those obtained with the corresponding uncontracted basis sets for the atomic properties, which they have been designed to reproduce. Slightly larger deviations are obtained with the 5s4p3d2f1g for the polarizability, while energetic properties still have only small errors. The design objective has been to describe the ionization potential, the polarizability and the valence spectrum as accurately as possible. The result is a set of well-balanced basis sets for molecular calculations, which can be used together with basis sets of the same quality for the first and second row atoms.     

Density matrix averaged atomic natural orbital (ANO) basis sets for correlated molecular wave functions

Summary Generally contracted basis sets for first row atoms have been constructed using the Atomic Natural Orbital (ANO) approach, with modifications for allowing symmetry breaking and state averaging. The ANOs are constructed by averaging over several atomic states, positive and negative ions, and atoms in an external electric field. The contracted basis sets give virtually identical results as the corresponding uncontracted sets for the atomic properties, which they have been designed to reproduce. The design objective has been to describe the ionization potential, the electron affinity, and the polarizability as accurately as possible. The result is a set of well-balanced basis sets for molecular calculations. The starting primitive sets are 8s4p3d for hydrogen, 9s4p3d for helium, and 14s9p4d3f for the heavier first row atoms.     

Density matrix averaged atomic natural orbital (ANO) basis sets for correlated molecular wave functions

Summary Generally contracted basis sets for second row atoms have been constructed using the Atomic Natural Orbital (ANO) approach, with modifications for allowing symmetry breaking and state averaging. The ANOs are constructed by averaging over several atomic states, positive and negative ions, and atoms in an external electric field. The contracted basis sets give virtually identical results as the corresponding uncontracted sets for the atomic properties, which they have been designed to reproduce. The design objective has been to describe the ionization potential, the electron affinity, and the polarizability as accurately as possible. The result is a set of well balanced basis sets for molecular calculations. The starting primitive sets are 17s12p5d4f for the second row atoms Na-Ar. Corresponding ANO basis sets for first row atoms have recently been published.     

The use of explicitly correlated,partially antisymmetric wave functions in atomic and molecular calculations

Trial wave functions, written as the sum of a configuration interaction expansion and an explicitly correlated term which is not antisymmetric, are proposed for use in calculating the electronic properties of atoms and molecules. A variational principle, modified to allow the use for such partially antisymmetric wave functions, is developed. It is shown that the consequences of partial antisymmetry on calculated expectation values can be estimated. The method avoids difficult three-electron integrals which arise in other theories.     

Generalization of the Optimized-Basis-Set Multi-Configuration Spin-Coupled method for the ab initio calculation of atomic and molecular electronic wave functions

An ab initio procedure for the calculation of atomic and molecular electronic wave functions, the Optimized-Basis-Set Multi-Configuration Spin-Coupled (OBS-MCSC) method, is generalized by introducing a separate linear combination of spin functions for each configuration, turning it into the OBS-GMCSC method. The ability to use a second-order minimization procedure in the computation of the wave function is maintained through appropriate generalization of the analytic expressions for the first and second derivatives of the energy with respect to the optimization parameters, as is the optional inclusion among the latter of the basis-function exponential parameters. The generalization, a variational improvement of the wave function, strengthens the connection with classical VB theory, of which the method can now be considered an optimized-orbitals variant, while maintaining the link with single-configuration Spin-Coupled theory, of which it may still be considered a multiconfiguration extension. The method can also be viewed as a nonorthogonal variant of the MCSCF approach. To demonstrate its practical feasibility and usefulness, the OBS-GMCSC method is applied to a study of the electronic structure and electron affinity of boron. © 1996 John Wiley & Sons, Inc.     

Factored wave function for bound S-type states of two-electron atomic systems

A simple and accurate variational wave function in which the dependence in the interelectronic distance is factored is proposed to describe S-type states of two-electron atomic systems. We introduce a parameterization which generalizes the previous ones used in this same framework and which allows us to obtain in a simple way the wave function of both symmetric and antisymmetric excited states. We performed a systematic analysis of some exact properties such as the virial theorem and the cusp conditions and a study of both the one- and two-body densities. Finally, a comparison among the different correlation functions for these states was performed for helium. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 405–413, 1998     

A minimum principle for atomic systems allowing the use of discontinuous wave functions

In this paper a variational principle proposed by Hall [1] is shown to be a minimum principle for coulombic systems. Into this principle it is possible to admit a larger class of trial wave functions than is possible in the conventional variational treatment, including wave functions with discontinuities. It is further shown that the upper bounds given by this treatment are always at least as good as that given by the Rayleigh–Ritz method. The theory is then applied to the hydrogen atom and upper bounds to the energy are calculated for various “cutoff” wave functions. It is usually possible to define an optimum “cut off” distance which minimizes the upper bound.     

Calculation of molecular systems via coordinate wave functions

The system of charges is in a state with a given total spin S, which is described by a configuration of one-electron orbitals with arbitrary filling (subject to the Pauli principle). Expressions are derived for the matrix elements of operators F and G that are independent of the spin. The energy of the interaction between the completely filled orbitals and the singly filled ones is found to be independent of the spin of the latter. The formulas may be used with the tables of [2] to derive directly the expressions for the matrix elements of a configuration having an arbitrary number of completely filled orbitals and up to six singly filled ones.     

Second order coalescence conditions of molecular wave functions

Kato's cusp condition gives the exact first order dependence of molecular wave functions on interparticle separation near the coalescence of two charged particles. We derive conditions correct to second order in interparticle separation, which concern second order derivatives of the wave function at the coalescence point. For identical particle coalescence, we give equations correct to third order. In addition to a universal, particle dependent term, a system and state dependent term arises in the higher order conditions, which we interpret as an effect of Coulombic screening. We apply our analysis to the standard orbital-based methods of quantum chemistry and discuss the implications for Jastrow- and R12-type correlation factors.     

Second order charge overlap effects and damping functions for isotropic atomic and molecular interactions

Second order charge overlap effects and the related dispersion energy damping functions have been evaluated for the H-(1s)-H(1s) interaction through partial wave indices l_a+l_b such that (l_a+l_b) ? 13. These results are a substantial improvement on, or addition to, previous literature results and are important since they can be used, following several available scaling approaches, to construct damping functions for other interactions. They also indicate that the “spherical” energies are not an insignificant part of the total second order Coulomb energy until R becomes reasonably large. The various approaches for evaluating the non-expanded second order Coulomb energy are compared and the difficulties associated with the accurate determination of these energies, and the related damping functions, for general interactions are discussed in some detail.     

Computation of molecular systems via coordinate wave functions

The total wave function is represented as the product of the coordinate and spin wave functions in order to formulate quantum chemistry problems in a form free from spin wave functions. The permutation symmetry of the coordinate function remembers the spin. Coordinate functions of a given symmetry are constructed via Young operators. The relation of Young's schemes to quantum-chemical structures is discussed.Read at the Symposium on Quantum Chemistry, Palanga, June 1965.     

Computation of molecular systems via coordinate wave functions

A method is presented for constructing the basis functions of the irreducible representations of the molecular point-group symmetry in terms of coordinate wave functions corresponding to a given total spin S of the molecule. The resulting functions are the eigenfunctions of the 2S+1 molecular multiplets and lead to quasi-diagonalization of the secular equation. Examples are given of calculations for the energy levels of a system of four electrons in a tetrahedral field and for those of the six electrons of benzene in the state with S=0.     

Calculation of molecular systems via coordinate wave functions

Expressions are derived for the matrix elements of the Coulomb interaction of a system of charges consisting of two subsystems, each in a state with a given spin. It is found that the interaction energy of the two subsystems is independent of the spin S₂ of the second subsystem to quantities of the order of the square of the overlap integral for the one-electron orbitals if the first subsystem has spin S[₁=0. Examples are given of calculations by this method. An appendix gives tables of the matrices needed to calculate the interaction energy of two subsystems with a total number of particles from 3 to 6.     

A physical interpretation of the cusp conditions for molecular wave functions

The cusp conditions for the Coulomb- and Correlation-cusp of molecular wave functions are derived directly in integrated form from the Schrödinger equation. For the Coulomb-cusp the slope of the wave function at nucleus α is given by the directional derivative

$$ - \left( {\frac{1}{\psi }\frac{{\partial \psi }}{{\partial r1\alpha }}} \right)_{r1\alpha = 0} = Z_\alpha + d \cdot \cos \vartheta _1 .$$