Confinement of hydrogen atom with Dirac equation

The problem of a particle confined in a spherical cavity is studied with the Dirac equation. A hard confinement is obtained by forcing the large component to vanish at the cavity radius. It is shown that the small component cannot vanish simultaneously at this radius. In the case of a confined hydrogen atom, the energies are given by an implicit equation. For some values of the radius, explicit analytical expressions of the energy exist like in the nonrelativistic case. Very accurate energies and wave functions are obtained with the Lagrange-mesh method with few mesh points. To this end, two differently regularized Lagrange-Jacobi bases associated with the same mesh are used for the large and small components. The importance of relativistic effects is discussed for hydrogen-like ions. The validity of this definition of hard confinement is discussed with a soft-confinement model studied with the R-matrix method.     

Table of Feynman diagrams of the interacting Fermion Green's function

The Feynman diagrams of the Green's function expansion of fermions interacting with a nonrelativistic 2‐body interaction are displayed in first, second, and third perturbative order of the interaction as 2, 10, and 74 diagrams, respectively. A name convention for the diagrams is proposed and then used to tabulate the diagrams of fourth to seventh order. The Hartree–Fock approximation summons up 2, 8, 40, and 224 of them in first through fourth order. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Theoretical study of molecular conduction: I. Effective Green's function based on perturbation theory

There have been many experimental and theoretical studies on molecular conduction, as it is a fundamental parameter in the study of molecular‐scale electronics. We have investigated the features of molecular conduction using a Green's function method, which has often been used to solve problems in quantum transport and is also effective in elucidating electron transport in molecules. We have obtained the novel effective Green's functions, including the first‐order energy corrections, by accommodating the self‐energy of the electrodes as perturbation terms. Although these approximate Green's functions only provide information on the first‐order energy corrections, they can involve the elementary properties of molecular conduction. We propose a scheme for the analysis of the relations between molecular orbitals and their roles in molecular conduction and present analytical calculations for normal and cyclic polyenes. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

A scheme of numerical solution for three‐dimensional isoelectronic series of hydrogen atom using one‐dimensional basis functions

The solution of three‐dimensional Schrödinger wave equations of the hydrogen atoms and their isoelectronic ions (Z = 1 − 4) are obtained from the linear combination of one‐dimensional hydrogen wave functions. The use of one‐dimensional basis functions facilitates easy numerical integrations. An iteration technique is used to obtain accurate wave functions and energy levels. The obtained ground state energy level for the hydrogen atom converges stably to −0.498 a.u. The result shows that the novel approach is efficient for the three‐dimensional solution of the wave equation, extendable to the numerical solution of general many‐body problems, as has been demonstrated in this work with hydrogen anion.     

A finite element approach for the continuum spectrum of the Dirac radial equation

The Dirac radial functions are expanded in polynomial B-spline basis, transforming the Dirac equation in a generalized eigensystem matrix problem. Due to the locality nature of the B-spline functions the matrix representation of all the involved operators are highly sparse. Diagonalization of the matrix equations provides the bound and continuum eigenstates. Energies and oscillator strengths for Hydrogen and Rubidium are presented.AMS subject classification: 65705, 81V05     

Green's function calculation of through‐bond electronic coupling in donor–bridge–acceptor model systems II: Importance factors applied to atomic sites

Importance factors, associated with the Green's function formalism, are introduced. They are applied for the determination of the relative atomic site contribution to the electronic interaction propagation in a molecular system. The calculation is performed at the Hartree–Fock (self‐consistent) level, using ab initio STO‐3G, 4‐31G, and D95 basis sets. The results are compared with those obtained from the charge densities of the appropriate molecular orbitals at the ab initio STO‐3G level. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Carbon nanotube, graphene, nanowire, and molecule-based electron and spin transport phenomena using the nonequilibrium Green's function method at the level of first principles theory

Based on density functional theory, we have developed a program code to investigate the electron transport characteristics for a variety of nanometer scaled devices in the presence of an external bias voltage. We employed basis sets comprised of linear combinations of numerical type atomic orbitals, particularly focusing on k-point sampling for the realistic modeling of the bulk electrode. The scheme coupled with the matrix version of the nonequilibrium Green's function method enables calculation of the transmission coefficients at a given energy and voltage in a self-consistent manner as well as the corresponding current-voltage (I-V) characteristics. This scheme has advantages because it is applicable to large systems, easily transportable to different types of quantum chemistry packages, and extendable to time-dependent phenomena or inelastic scatterings. It has been applied to diverse types of practical electronic devices such as carbon nanotubes, graphene nanoribbons, metallic nanowires, and molecular electronic devices. The quantum conductance phenomena for systems involving quantum point contacts and I-V curves for a single molecule in contact with metal electrodes using the k-point sampling method are described.     

Exact solution of multidimensional hyper‐radial Schrödinger equation for many‐electron quantum systems

In quantum theory, solving Schrödinger equation analytically for larger atomic and molecular systems with cluster of electrons and nuclei persists to be a tortuous challenge. Here, we consider, Schrödinger equation in arbitrary N‐dimensional space corresponding to inverse‐power law potential function originating from a multitude of interactions participating in a many‐electron quantum system for exact solution within the framework of Frobenius method via the formulation of an ansatz to the hyper‐radial wave function. Analytical expressions for energy spectra, and hyper‐radial wave functions in terms of known coefficients of inverse‐power potential function, and wave function parameters have been obtained. A generalized two‐term recurrence relation for power series expansion coefficients has been established. © 2016 Wiley Periodicals, Inc.     

A Green's function approach to the photoelectron spectrum of bis(π-allyl)nickel[1]

The vertical ionization potentials of bis(-allyl)nickel (see (1) in Fig. 1) are calculated by means of the Green's function approach within a semiempirical INDO extension to the first transition metal series. The computed ionization potentials are in good agreement with an experimentally deduced assignment. In contrast to earlier theoretical and experimental studies, the 7a _u () level is predicted on top of the levels corresponding to the Ni 3d orbitals. Our approach leads to a complete assignment of the PE spectrum of (1) in the outer valence region.     

A rigorous approach to the formulation of extended Born‐Oppenheimer equation for a three‐state system

If a coupled three‐state electronic manifold forms a sub‐Hilbert space, it is possible to express the non‐adiabatic coupling (NAC) elements in terms of adiabatic–diabatic transformation (ADT) angles. Consequently, we demonstrate: (a) Those explicit forms of the NAC terms satisfy the Curl conditions with non‐zero Divergences; (b) The formulation of extended Born‐Oppenheimer (EBO) equation for any three‐state BO system is possible only when there exists coordinate independent ratio of the gradients for each pair of ADT angles leading to zero Curls at and around the conical intersection(s). With these analytic advancements, we formulate a rigorous EBO equation and explore its validity as well as necessity with respect to the approximate one (Sarkar and Adhikari, J Chem Phys 2006, 124, 074101) by performing numerical calculations on two different models constructed with different chosen forms of the NAC elements. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

The Fukui function of an atom in a molecule: A criterion to characterize the reactive sites of chemical species

The principle of hard and soft acids and bases is interpreted as the result of two opposing tendencies, one related to the charge transfer process (chemical potential equalization principle), and the other one related to the reshuffling of the electronic density (maximum hardness or minimum softness principle). A local version of the principle is elucidated by assuming that these tendencies are dominated by the local properties rather than by the global properties of the molecule. This principle is used together with the Fukui function of the atoms in the molecule to characterize the reactive sites. The results presented for the nucleophilic addition to the pyridinium ion, and for the electrophilic substitution on pyridine oxide show the usefulness of these concepts in describing the inherent reactivity of chemical species.     

A simple approach to treat anisotropic elastic collisions in Monte Carlo calculations of the electron energy distribution function in cold plasmas

A simplified method allowing one to treat anisotropic electron heavy species elastic scattering in MonteCarlo models of gas discharges with the proper value for collision frequency is proposed The method is applied to an electric discharge in a Ne · Xe/HCl mixture, and the results are compared with the solution of the two-term expansion of the Boltzmann equation under the same conditions. Methods for reduction of computational time in Monte Carlo codes and the use of the Monte Carlo flux method are also discussed.     

Solution of periodic Poisson's equation and the Hartree–Fock approach for solids with extended electron states: Application to linear augmented plane wave method

We formulate a Hartree–Fock‐LAPW method for electronic band structure calculations. The method is based on the Hartree–Fock–Roothaan approach for solids with extended electron states and closed core shells where the basis functions of itinerant electrons are linear augmented plane waves. All interactions within the restricted Hartree–Fock approach are analyzed and in principle can be taken into account. In particular, we obtained the matrix elements for the exchange interactions of extended states and the crystal electric field effects. To calculate the matrix elements of exchange for extended states, we first introduce an auxiliary potential and then integrate it with an effective charge density corresponding to the electron exchange transition under consideration. The problem of finding the auxiliary potential is solved by using the strategy of the full potential LAPW approach, which is based on the general solution of periodic Poisson's equation. Here, we use an original technique for the general solution of periodic Poisson's equation and multipole expansions of electron densities. We apply the technique to obtain periodic potentials of the face‐centered cubic lattice and discuss its accuracy and convergence in comparison with other methods. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

A mechanism of the 1,3‐dipolar cycloaddition between the hydrogen nitryl HNO2 and acetylene HCCH: The electron localization function study on evolution of the chemical bonds

The reaction between the simplest nitro compound HNO₂ (hydrogen nitryl) and acetylene HCCH ‐ formally proceeding via 1,3‐dipolar cycloaddition ‐ has been studied by means of the B3LYP, MPW1K and MP2 methods. The energy barrier of 20.74 ÷ 32.91 kcal/mol is similar to ΔE_a of the NNO + HCCH process but is essentially larger than computed for the reactions of HCCH with fulminic acid (HCNO) and NNCH₂. Whole process is exothermic with the reaction energy: ?10.87 ÷ ?17.94 kcal/mol. An evolution of the chemical bonding has been analyzed by means of the Bonding Evolution Theory (BET) at the B3LYP/6‐31+G(d) and B3LYP/cc‐pVTZ levels. Two approximations of the reaction path have been considered, namely: the IRC and pseudo‐reaction paths. The reaction requires five steps and seven catastrophes of the fold and cusp type. A different effect of first fold catastrophe has been noticed. At the B3LYP/6‐31+G(d) level one of two nonbonding V_i=1,2(N) attractors is annihilated (F), meanwhile at B3LYP/cc‐pVTZ new V(N) attractor is created (F^?). The chemical bonds are not formed/broken in TS. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

A new approach to the exact and approximate anharmonic vibrational partition function of diatomic and polyatomic molecules utilizing Morse and Rosen–Morse oscillators

Exact closed forms of the equilibrium partition functions in terms Jacobi elliptic functions are derived for a particle in a box and Rosen–Morse (Poschl–Teller) oscillator (perfect for modeling bending vibrational modes). An exact form of the equilibrium partition function of Morse oscillator is reported. Three other approximate forms of Morse partition function are presented. Having an exact closed‐form for the vibrational partition function can be very helpful in evaluating thermodynamic state functions, e.g., entropy, internal energy, enthalpy, and heat capacity. Moreover, the herein presented closed forms of the vibrational partition function can be used for obtaining spectroscopic and dynamical information through evaluating the two‐ and four‐point dipole moment time correlation functions in anharmonic media. Finally, a closed exact form of the rotational partition function of a particle on a ring in terms of the first kind of complete elliptic integral is derived. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011