The connected‐moments polynomial approach for Hamiltonian eigenvalues calculation and its application to the one‐particle systems

The new connected‐moments polynomial approach (CMP) is developed for evaluation of Hamiltonian eigenvalues. It is based on properties of specially designed polynomial and does not use any basis set and variational procedure. Like all the methods based on hamiltonain moments knowledge, the CMP is conceptually simple but is less tedious and is usually convergent even for very “crude” trial functions. This method is applicable not only to the ground state energy calculation but also to the excited states. The formalism is presented in two modifications: non‐local (integral) and local (integral‐free) ones. An accuracy of both versions is illustrated by numerical examples of Hamiltonian eigenvalues calculations for harmonic and anharmonic oscillators. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

A numerical solution of the pair equation of a model two‐electron diatomic system

It has been well‐documented that about 90% of the total correlation energy of atomic systems can be obtained by solving so‐called pair equations. For atoms, this approach requires solving partial differential equations (PDE) in two variables. In case of a diatomic molecule, we face devising a method for treating PDEs in five variables. This article shows how a well‐established finite difference method used to solve Hartree–Fock equations for diatomic molecules can be extended to solve numerically a model two‐electron Schrödinger equation for such systems. We show that using less than 100 grid points in each variable, it is possible to obtain the total energy of the helium atom and hydrogen molecule with a chemical accuracy and the S energy of the helium atom and hydride ion as accurately as the best results available. © 2015 Wiley Periodicals, Inc.     

The Rayleigh‐Schrödinger perturbation series of quasi‐degenerate systems

We describe the first representation of the general term of the Rayleigh‐Schrödinger series for quasidegenerate systems. Each term of the series is represented by a tree and there is a straightforward relation between the tree and the analytical expression of the corresponding term. The combinatorial and graphical techniques used in the proof of the series expansion allow us to derive various resummation formulas of the series. A relation with several combinatorial objects used for special cases (degenerate or non‐degenerate systems) is established. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

A new size extensive multireference perturbation theory

A new multireference perturbation series is derived based on the Rayleigh–Schrödinger perturbation theory. It is orbitally invariant. Its computational cost is comparable to the single reference Møller–Plesset perturbation theory. It is demonstrated numerically that the present multireference second‐ and third‐order energies are size extensive by two types of supermolecules composed of H₂ and BH monomers. Spectroscopic constants of as well as the ground state energies of H₂O, NH₂, and CH₂ at three bond lengths have been calculated with the second multireference perturbation theory. The dissociation behaviors of CH₄ and HF have also been investigated. Comparisons with other approximate theoretical models as well as the experimental data have been carried out to show their relative performances. © 2013 Wiley Periodicals, Inc.     

Symplectic algorithm for use in computing the time‐independent Schrödinger equation

The structure of benzoic acid as monomer was studied by semiempirical, ab initio, and density functional methods using several basis sets. The performance of these methods in calculating and describing the vibrational frequencies of benzoic acid and several derivatives was determined. The cyclic dimer form of benzoic acid was also reproduced. Two new procedures of scaling the frequencies were presented. For the ring modes, specific scale equations and scale factors were used from benzene molecule. For the carboxylic group, scaling equations and specific scale factors at different levels were also determined to be used in benzoic acid derivatives. A reassignment of several bands was done. A comparison of the cost/effective method and procedure of scaling was carried out. A significant reduction of the error in the predicted frequencies was obtained over the one‐factor standard scaling procedure. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Multi‐state multi‐reference Møller–Plesset second‐order perturbation theory for molecular calculations

This work presents multi‐state multi‐reference Møller–Plesset second‐order perturbation theory as a variant of multi‐reference perturbation theory to treat electron correlation in molecules. An effective Hamiltonian is constructed from the first‐order wave operator to treat several strongly interacting electronic states simultaneously. The wave operator is obtained by solving the generalized Bloch equation within the first‐order interaction space using a multi‐partitioning of the Hamiltonian based on multi‐reference Møller–Plesset second‐order perturbation theory. The corresponding zeroth‐order Hamiltonians are nondiagonal. To reduce the computational effort that arises from the nondiagonal generalized Fock operator, a selection procedure is used that divides the configurations of the first‐order interaction space into two sets based on the strength of the interaction with the reference space. In the weaker interacting set, only the projected diagonal part of the zeroth‐order Hamiltonian is taken into account. The justification of the approach is demonstrated in two examples: the mixing of valence Rydberg states in ethylene, and the avoided crossing of neutral and ionic potential curves in LiF. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Pair density related to one‐electron information for the ground state of spin‐compensated two‐electron systems

The recent study by Joubert on effects of Coulomb repulsions in a many‐electron system has focused attention on an integral identity involving the pair density. This has motivated the derivation presented here of a vectorial differential form related to this integral result. Our differential identity is then illustrated explicitly by using (i) an exact ground‐state wave function for the so‐called Hookean atom having external potential energy (1/2)kr², with k = 1/4, and (ii) Moshinsky's model in which both the interparticle interaction and the external potential are of harmonic type. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Ritz variational calculation for the singly excited states of compressed two‐electron atoms

A detailed analysis on the effect of spherical impenetrable confinement on the structural properties of two‐electron ions in states has been performed. The energy values of 1sns [ ] ( ) states of helium‐like ions ( ) are estimated within the framework of Ritz variational method using explicitly correlated Hylleraas‐type basis sets. The correlated wave functions used here are consistent with the finite boundary conditions due to spherical confinement. A comparative study between the singlet and triplet states originating from a particular electronic configuration shows incidental degeneracy and the subsequent level‐crossing phenomena. The thermodynamic pressure felt by the ion inside the sphere pushes the energy levels toward continuum. Critical pressures for the transition to strong confinement regime (where the singly excited two‐electron energy levels cross the corresponding one‐electron threshold) as well as for the complete destabilization are also estimated.     

A two‐scale approach to electron correlation in multiconfigurational perturbation theory

We present a new approach for the calculation of dynamic electron correlation effects in large molecular systems using multiconfigurational second‐order perturbation theory (CASPT2). The method is restricted to cases where partitioning of the molecular system into an active site and an environment is meaningful. Only dynamic correlation effects derived from orbitals extending over the active site are included at the CASPT2 level of theory, whereas the correlation effects of the environment are retrieved at lower computational costs. For sufficiently large systems, the small errors introduced by this approximation are contrasted by the substantial savings in both storage and computational demands compared to the full CASPT2 calculation. Provided that static correlation effects are correctly taken into account for the whole system, the proposed scheme represent a hierarchical approach to the electron correlation problem, where two molecular scales are treated each by means of the most suitable level of theory. © 2014 Wiley Periodicals, Inc.     

Correlation energy,correlated electron density,and exchange‐correlation potential in some spherically confined atoms

We report correlation energies, electron densities, and exchange‐correlation potentials obtained from configuration interaction and density functional calculations on spherically confined He, Be, Be²⁺, and Ne atoms. The variation of the correlation energy with the confinement radius R_c is relatively small for the He, Be²⁺, and Ne systems. Curiously, the Lee–Yang–Parr (LYP) functional works well for weak confinements but fails completely for small R_c. However, in the neutral beryllium atom the CI correlation energy increases markedly with decreasing R_c. This effect is less pronounced at the density‐functional theory level. The LYP functional performs very well for the unconfined Be atom, but fails badly for small R_c. The standard exchange‐correlation potentials exhibit significant deviation from the “exact” potential obtained by inversion of Kohn–Sham equation. The LYP correlation potential behaves erratically at strong confinements. © 2016 Wiley Periodicals, Inc.