Analogy between real irreducible tensor operator and molecular two-particle operator

. Molecular matrix elements of a physical operator are expanded in terms of polycentric matrix elements in the atomic basis by multiplying each by a geometrical factor. The number of terms in the expansion can be minimized by using molecular symmetry. We have shown that irreducible tensor operators can be used to imitate the actual physical operators. The matrix elements of irreducible tensor operators are easily computed by choosing rational irreducible tensor operators and irreducible bases. A set of geometrical factors generated from the expansion of the matrix elements of irreducible tensor operator can be transferred to the expansion of the matrix elements of the physical operator to compute the molecular matrix elements of the physical operator. Two scalar product operators are employed to simulate molecular two-particle operators. Thus two equivalent approaches to generating the geometrical factors are provided, where real irreducible tensor sets with real bases are used. Received: 3 September 1996 / Accepted: 19 December 1996     

Conservation of connectivity of model-space effective interactions under a class of similarity transformation

Effective interaction operators usually act on a restricted model space and give the same energies (for Hamiltonian) and matrix elements (for transition operators, etc.) as those of the original operators between the corresponding true eigenstates. Various types of effective operators are possible. Those well defined effective operators have been shown to be related to each other by similarity transformation. Some of the effective operators have been shown to have connected-diagram expansions. It is shown in this paper that under a class of very general similarity transformations, the connectivity is conserved. The similarity transformation between Hermitian and non-Hermitian Rayleigh-Schrodinger perturbative effective operators is one of such transformations and hence the connectivity can be deducted from each other.     

Exact quantum mechanical vibration-rotation kinetic energy operator of four-particle system in various coordinates

The vibration-rotational kinetic energy operators of four-particle system in various coordinates are derived using a new and simple angular momentum method. The operators are respectively suitable for studying the systems described by scattering coordinate, valence coordinate, Radau coordinate, Radau/Jacobi and Jacobi/valence hybrid coordinates and so on. Certain properties of these operators and their possible applications are discussed.     

One-center expansion for pseudopotential matrix elements

Semilocal pseudopotential operators can be expressed as a linear combination of nonlocal (projection) operators. Pseudopotential operator integrals over a molecular basis set are therefore reduced to linear combinations of overlap integrals products. Molecular calculations indicate that sufficient precision can be achieved with a limited number of nonlocal operators. Analytic derivatives of pseudopotential integrals are easily deduced and implemented in a standard quantum chemistry program.     

Symmetrization of operator matrix elements

The method of Dupuis and King for generating matrix elements of a totally symmetric one-electron operator in terms of symmetry-distinct integrals only is generalized to the case of nontotally symmetric operators. For operators constructed from two-electron integrals, explicit reduction of integral processing to permutationally inequivalent symmetry-distinct integrals only is described, while for one-electron operators further reductions are derived using double coset decompositions. Finally, some computational consequences of this approach are briefly discussed.     

Stockholder projector analysis: a Hilbert-space partitioning of the molecular one-electron density matrix with orthogonal projectors

A previously introduced partitioning of the molecular one-electron density matrix over atoms and bonds [D. Vanfleteren et al., J. Chem. Phys. 133, 231103 (2010)] is investigated in detail. Orthogonal projection operators are used to define atomic subspaces, as in Natural Population Analysis. The orthogonal projection operators are constructed with a recursive scheme. These operators are chemically relevant and obey a stockholder principle, familiar from the Hirshfeld-I partitioning of the electron density. The stockholder principle is extended to density matrices, where the orthogonal projectors are considered to be atomic fractions of the summed contributions. All calculations are performed as matrix manipulations in one-electron Hilbert space. Mathematical proofs and numerical evidence concerning this recursive scheme are provided in the present paper. The advantages associated with the use of these stockholder projection operators are examined with respect to covalent bond orders, bond polarization, and transferability.     

Unitary transformations with the unitary operators depending on projection operators

Unitary transformations of operators and variables are considered for those cases when a Hermitian operator in the expression U = exp (i S ) explicitly depends on the projection operators. Some simple examples of such transformations are given and the equations obtained which define projectors (among special sets of them), thus bringing forth better estimations for upper and lower bounds. The unitary transformations are also applied to the reduced resolvent operator and equations are presented for the operators U that minimize the energy functional.     

Global optimization of atomic and molecular clusters using the space-fixed modified genetic algorithm method

A modified genetic algorithm approach has been applied to atomic Ar clusters and molecular water clusters up to (H₂O)₁₃. Several genetic operators are discussed which are suitable for real-valued space-fixed atomic coordinates and Euler angles. The performance of these operators has been systematically investigated. For atomic systems, it is found that a mix of operators containing a coordinate-averaging operator is optimal. For angular coordinates, the situation is less clear. It appears that inversion and two-point crossover operators are the best choice. © 1997 John Wiley & Sons, Inc. J Comput Chem 18: 1233–1244     

The Harmonious Dissipative Operators and the Completely Square Conservative Difference Scheme in an Explicit Way

In this paper, a new definition on harmonious dissipative operators is given and some important properties of theirs are shown. Especially, the relationship between a harmonious dissipative operator and the completely square conservative difference scheme in an explicit way is revealed. Kinds of 2-order, 3-order and 4-order harmonious dissipative operators are constructed by using the traditional Runge-Kutta method and a species of general m-order harmonious dissipative operators is established in the linear case. In addition, an efficiency parameter to appraise the time benefits of a harmonious dissipative operator is defined in this paper. It is testified in numerical tests that the harmonious dissipative operators are indeed able to improve the time-efficiency and computational effect of the completely square conservative difference scheme in an explicit way.     

The use of partitioning technique in the solution of operator equations for ionizations and excitations

An inhomogeneous operator equation is associated with the equations of motion for ionization- and excitation operators. The partitioning technique is applied to solve the equation. Formal expressions for the ionization- and excitation operators are obtained and examples of approximations to these are given.     

Ladder operators for the Morse potential

A realization of the raising and lowering operators for the Morse potential is presented. It is shown that these operators satisfy the commutation relations for the SU(2) group. Closed analytical expressions are obtained for the matrix elements of different operators such as 1/y and d/dy. The harmonic limit of the SU(2) operators is also studied and an approach previously proposed to calculate the Franck–Condon factors is discussed. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001     

Method of recurrent construction of Löuwdin spin-adapted wave functions. II. Local representation of creation–annihilation operators

Some compositions of the addition and subtraction operators and recurrence relations for the Sanibel-type coefficients c_{u, v} (n, s, M) generated by these compositions are studied. A local representation of the fermion creation–annihilation operators via the addition and subtraction operators is obtained. Operators of single excitations, coupling, and decoupling operators, in terms of which the unitary group generators can be expressed are defined. The resulting representation of the nonelementary unitary group generators is much more simple than in the Gelfand–Tzetlin basis and in the most general case contains only six logically different terms, each of them possessing quite transparent physical significance.     

Quantum dynamics of the CH3 fragment: a curvilinear coordinate system and kinetic energy operators

A curvilinear coordinate system for AB(3) fragments is given. The corresponding exact kinetic energy operator is derived and a series of simpler, progressively more approximate kinetic energy operators are suggested. The operators are tailored for quantum dynamics simulations using the multiconfigurational time-dependent Hartree approach. It is outlined how these fragment coordinates can be utilized to set up coordinate systems for larger systems such as AB(3)C or AB(3)CD. Calculations of the vibrational levels of CH(3) and quantum dynamics studies investigate the accuracy of the different kinetic energy operators suggested.     

Harmonic oscillator tensors. II. angular momentum expressions of matrix elements of vibrational operators

The spatial symmetries of the harmonic oscillator and the recently found irreducible tensors constructed from the associated annihilation and creation operators are exploited to obtain new expressions for the elements of the matrix representatives of several examples of vibrational operators. Since all vibrational operators are expressible in terms of the irreducible tensors, their matrix elements reflect the angular momentum symmetry inherent in them, for the results derived here are in terms of the Clebsch–Gordan coefficients and the isoscalar factors that arise from the couplinig rule of the irreducible tensors. Familiarity with the mathematical properties of these quantities derived from the elementary theory of angular momentum facilitates the evaluation of many vibrational operators that may be of importance in the study of potentials in this basis. In particular, it is shown that the nonvanishing of matrix elements is governed by a law of conservation of angular momentum along the axis of quantization of the nondegenerate harmonic oscillator. © 1993 John Wiley & Sons, Inc.     

A definition for the covalent and ionic bond index in a molecule

Formulae for hermitian operators representing covalent, ionic, and total bond indices are derived. The eigenstates of these operators come in pairs, and can be considered as bonding, anti-bonding and lone-pair orbitals. The form of these operators is derived by generalising the rule that the bond order be defined as the net number of bonding electron pairs. The percentage of covalency and ionicity of a chemical bond may be obtained, and bond indices can also be defined between groups of atoms. The calculation of the bond indices depends only on the electron density operator, and certain projection operators used to represent each atom in the molecule. Bond indices are presented for a series of first and second row hydrides and fluorides, hydrocarbons, a metal complex, a Diels–Alder reaction and a dissociative reaction. In general the agreement between the bond indices is in accord with chemical intuition. The bond indices are shown to be stable to basis set expansion.     

On the acceleration of the convergence of singular operators in Gaussian basis sets

Gaussian type wave functions do not reproduce the interparticle cusps which result in a slow convergence of the expectation values of the operators involved in calculations of the relativistic and QED energy corrections. Methods correcting this deficiency are the main topic discussed in this paper. Benchmark expectation values of the singular operators for several few-electron systems are presented.     

Monomer basis-set truncation effects in calculations of interaction energies: A model study

Supermolecular interaction energies are analyzed in terms of the symmetry-adapted perturbation theory and operators defining the inaccuracy of the monomer wave functions. The basis set truncation effects are shown to be of first order in the monomer inaccuracy operators. On the contrary, the usual counterpoise correction schemes are of second order in these operators. Recognition of this difference is used to suggest an approach to corrections for basis-set truncation effects. Also earlier claims--that dimer-centered basis sets may lead to interaction energies free of basis-set superposition effects--are shown to be misleading. According to the present study the basis-set truncation contributions, evaluated by means of the symmetry-adapted perturbation theory with monomer-centered basis sets, provide physically and mathematically justified corrections to supermolecular results for interaction energies.     

Usage de l'Algèbre de Lie su (n) dans l'Etude des Systèmes Quantiques à n Etats. II. Transformation de l'Espace des Observables,Problèmes d'Evolution

In the previous paper we examined, for a quantum system, the relation between its n-dimensional state space and the su (n) Lie algebra. The present paper is devoted to relations between unitary transformations in the state space and orthogonal transformations in Lie's algebra. Two cases can happen. First, the transformations are independently chosen in the two spaces; this amounts to changing the former relation. On the other hand, the relation is maintained and the unitary operators are then related to some of the orthogonal operators. This second case is used to study the evolution operators.