Generalization of the second quantization operators: Electron attachment and detachment operators

This paper proposes a method for the generalization of the second quantization operators. The method is based on two main hypotheses: (i) the same form of the correlation operator, which is good for closed-shell systems is chosen and (ii) a system of equations is given to calculate the differences (x_k0?x_k1), which characterize both attachment and detachment operators.     

Addition theorem and matrix elements of radiative multipole operators

The addition theorem for radiative multipole operators, i.e., electric-dipole, electric-quadropole, or magnetic-dipole, etc., is derived through a translational transformation. The addition theorem of μth component of the angular momentum operator, L _μ (r), is also derived as a simple expression that represents a general translation of the angular momentum operator along an arbitrary orientation of a displacement vector and when this displacement is along the Z-axis. The addition theorem of the multipole operators is then used to analytically evaluate the matrix elements of the electric and magnetic multipole operators over the basis functions, the spherical Laguerre Gaussian-type function (LGTF), . The explicit and simple formulas obtained for the matrix elements of these operators are in terms of vector-coupling coefficients and LGTFs of the internuclear coordinates. The matrix element of the magnetic multipole operator is shown to be a linear combination of the matrix element of the electric multipole operator     

Multiconfigurational expansions of density operators: equations of motion and their properties

 Multiconfigurational expansions of density operators which may be used in numerical treatments of the dynamics of closed and open quantum systems are introduced. The expansions of the density operators may be viewed as analogues of those used in the multiconfiguration time-dependent Hartree (MCTDH) method, which is a well-established and highly efficient method for propagating wavepackets in several dimensions. There is no unique multiconfigurational representation of a density operator and two sensible types of MCTDH-like expansions are studied. Equations of motion for these multiconfigurational expansions are presented by adopting the Dirac–Frenkel/McLachlan variational principle (or variants of thereof). Various properties of these sets of equations of motion are derived for closed and open system dynamics. The numerical and technical aspects of this approach have been recently discussed by us [(1999) J Chem Phys 111: 8759]. Here we discuss the formal aspects of the approach in a more general context. Received: 26 January 2000 / Accepted: 8 February 2000 / Published online: 12 May 2000     

On N -body Schr?dinger operators

In these notes, we study the estimates of the resolvent or the unitary group of theN-body Schr?dinger operator. The main strategy is to introduce an algebra of operators having nice commutation relations with the many-body Schr?dinger operator. These estimates are applied to derive the detailed properties of the S-matrices associated with the many-body collision process.     

Variation principle for the time dependence of density operators and its relation to linear decoupling procedures for propagators

A modified form of Frenkel's time-dependent variation principle, suggested by McLachlan for state vectors, is employed to discuss the optimal time evolution of a density operator ρ(t). An ansatz is made for this operator such that i(dρ/dt) = [S, ρ], where S(t) is a linear combination of operators belonging to a particular manifold of “basis operators.” The expansion coefficients are required to minimize the error Tr{[S – H, ρ]? [S – H, ρ]}. Linear response functions corresponding to the variationally determined density operator are compared to those derived by means of linear decoupling procedures for propagators based on the same operator manifold. The two approximation schemes are not equivalent, in general, and several consistency requirements must be fulfilled before it can be ascertained that a given linear decoupling procedure corresponds to an optimal time development of the density operator in the sense of McLachlan. Finally, the general applicability of the suggested variation principle is discussed.     

Unitary group approach to spin-adapted open-shell coupled cluster theory

We show that the irreducible tensor operators of the unitary group provide a natural operator basis for the exponential Ansatz which preserves the spin symmetry of the reference state, requires a minimal number of independent cluster amplitudes for each substitution order, and guarantees the invariance of the correlation energy under unitary transformations of core, open-shell, and virtual orbitals. When acting on the closed-shell reference state with n_c doubly occupied and n_v unoccupied (virtual) orbitals, the irreducible tensor operators of the group U(n_c) ? U(n_V) generate all Gelfand-Tsetlin (GT) states corresponding to appropriate irreducible representation of U(n_c + n_v). The tensor operators generating the M-tuply excited states are easily constructed by symmetrizing products of M unitary group generators with the Wigner operators of the symmetric group S_M. This provides an alternative to the Nagel-Moshinsky construction of the GT basis. Since the corresponding cluster amplitudes, which are also U(n_c) ? U(n_s) tensors, can be shown to be connected, the irreducible tensor operators of U(n_c) ? U(n_v) represent a convenient basis for a spin-adapted full coupled cluster calculation for closed-shell systems. For a high-spin reference determinant with n, singly occupied open-shell orbitals, the corresponding representation of U(n), n=n_c + n_v + n_s is not simply reducible under the group U(n_c) ? U(n_s) ? U(n_v). The multiplicity problem is resolved using the group chain U(n) ? U(n_c + n_v) ? U(n_s) ? U(n_c) ?U(n_s)? U(n_v) ? U(n_v). The labeling of the resulting configuration-state functions (which, in general, are not GT states when n_c > 1) by the irreducible representations of the intermediate group U(n_c + n_v) ?U(n_s) turns out to be equivalent to the classification based on the order of interaction with the reference state. The irreducible tensor operators defined by the above chain and corresponding to single, double, and triple substitutions from the first-, second-, and third-order interacting spaces are explicitly constructed from the U(n) generators. The connectedness of the corresponding cluster amplitudes and, consequently, the size extensivity of the resulting spin-adapted open-shell coupled cluster theory are proved using group theoretical arguments. The perturbation expansion of the resulting coupled cluster equations leads to an explicitly connected form of the spin-restricted open-shell many-body perturbation theory. Approximation schemes leading to manageable computational procedures are proposed and their relation to perturbation theory is discussed. © 1995 John Wiley & Sons, Inc.     

Factorization of certain evolution operators using lie algebra: Formulation of the method

In this work, a new method to factorize certain evolution operators into an infinite product of simple evolution operators is presented. The method uses Lie operator algebra and the evolution operators are restricted to exponential form. The argument of these forms is a first-order linear partial differential operator. The method has broad applications, including the areas of sensitivity analysis, the solution of ordinary differential equations, and the solution of Liouville's equation. A sequence of -approximants is generated to represent the Lie operators. Under certain conditions, the convergence rate of the -approximant sequences is remarkably high. This work presents the general formulation of the scheme and some simple illustrative examples. Investigation of convergence properties is given in a companion paper.Supported by the Department of Energy.     

Relativistic electric and magnetic property operators for two‐component transformed Hamiltonians

A general procedure is presented for the derivation of property operators for electric and magnetic perturbations for Hamiltonians derived from the Dirac Hamiltonian by a partially block‐diagonalizing unitary transformation. The procedure involves a regularized expansion in powers of p ²/m²c². Property operators are expressed in terms of the solid spherical harmonics. Expressions for the free‐particle Foldy–Wouthuysen, Douglas–Kroll, and Barysz–Sadlej–Snijders transformations are compared with the well‐known Pauli results. Explicit examples of a constant electric field and a constant magnetic field are given. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 412–421, 2000     

Some comments on the properties of unitary transformations on linear spaces having an indefinite metric and the connection with the theory of spin

After a brief review of the history of the discovery of the spin, some fundamental properties of linear spaces having an indefinite metric are being discussed. The study starts with an elementary survey of the theory of matrices and their stability problem. It is emphasized that—by a similarity transformation—all matrices may be brought to classical canonical form characterized by the diagonal elements called eigenvalues, their multiplicities, their Jordan blocks, and their Segré characteristics. In connection with the reduced Cayley-Hamilton equation, the existence of the product projection operators and their main properties is briefly discussed. Particular attention is paid to the concept of a basis for the linear space and the associated metric matrix, which is self-adjoint and may be brought to diagonal form with the eigenvalues ± 1 by a unitary transformation, which reveals the indices of inertia, p and q. The Minkowski space having p = 3 and q = 1 is used as an example. After this introduction, some properties of linear operators defined on an indefinite space are discussed, and it is pointed out that self-adjoint operators and unitary operators may now have a rather peculiar and unexpected behavior, and the special Lorentz transformations are used as an example. It is then shown that these features are of essential importance in studying rotations as special cases of unitary transformations defined on an indefinite space. The rotations are here defined by means of their reduced Cayley-Hamilton equation, and their properties are studied by means of the associated product projection operators, which are idempotent, mutually exclusive, and form a resolution of the identity. In a previous article, it was shown that, in a positive definite space, there is a close connection between the requirement that all rotations around an external axis form a group and the existence of an anticommutator algebra leading to the concept of spinors. The rotations are expressed in the exponential form U=exp(iO), where O is a self-adjoint operator which is independent of any choice basis, coordinate system, etc., and which is, hence, a true invariant. It is shown that this approach may now be extended also to indefinite spaces and may lead to operators O which are both rotationally and relativistically invariant. In this connection, the full Lorentz transformations are given a particularly simple form. The article may be considered as a simple exercise in linear algebra, in which the mathematical connection between the concept of rotations and the existence of spinors is strongly emphasized. © John Wiley & Sons, Inc.     

The rotation–vibration coupling equations for polyatomic molecules in internal coordinates

An exact vibration–rotation kinetic energy operator for polyatomic molecules has been obtained on the basis of Sutcliffe's method, in terms of curvilinear internal coordinates and rotational angular moment operators. This operator is derived from the kinetic energy operator in Cartesian coordinates by the successive transformations using the chain rule. This kinetic energy operator can be used not only for the system of any triatomic and tetraatomic molecules and common polyatomic molecules in chemistry, but also for the investigation of the collision problems between two molecules after some modifications. Finally, using this Hamiltonian, the rotation–vibration coupling equations of polyatomic molecules have been derived and discussed. © 1992 John Wiley & Sons, Inc.     

Convergence studies in quantum perturbation theory

Summary Wave operator equations associated with the determination of energies are generally solved by perturbation methods [1]. However, it is well known that for most actual systems the standard Rayleigh-Schrödinger and Brillouin-Wigner series have slow convergence properties. We suggest that one way how to deal with this problem is to modify or to renormalize the standard wave equations. For that purpose we introduce newderivative andconvergence operators associated with the basic Rayleigh-Schrödinger and Brillouin-Wigner formalisms. Since the direct use of these operators would imply difficult operator inversions, we investigate the efficiency of various approximations of the derivative operator. It is shown that these approximations can overcome convergence difficulties and also open the way to systematic derivations of infinite partial summation schemes. Our approach is also discussed with respect to the standard diagonalization procedure of Davidson. Three simple model systems are investigated numerically.On leave from J. Heyrovský Institute, Prague     

Reduced hamiltonian orbitals. III. Unitarily invariant decomposition of hermitian operators

A decomposition of an N-particle operator as a sum of N + 1 components is defined such that, in the case of a model system employing a finite one-particle basis set, the decomposition is invariant under unitary transformations of the basis set. Applied to a two-particle Hamiltonian, this decomposition gives rise to the distinction between the independent-particle energy and the coupling energy defined in previous papers. Applied to the reduced density operator for a quantum state, the decomposition corresponds to partitioning the density into irreducible components. This partitioning is illustrated by graphs of electron density for the water molecule.     

General quantum mechanical operators. An open-ended approach for one-electron integrals with Gaussian bases

A universal computational approach for evaluating integrals over gaussian basis functions for general operators of the form is presented. The implementation is open-ended with respect to the types of basis functions (s, p, d, f, g, h…) and with respect to the integers that specify the operator. These one-electron integrals comprise operators associated with electrical and magnetic properties of molecules and include those needed to find multipole polarizabilities, multipole susceptibilities, chemical shifts, and so on. The scheme also generates the usual kinetic, nuclear attraction, and overlap operators.     

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     

Average fock operators

Average procedures of SCF methods are discussed. Although average operators are known from early Hartree–Fock times, some remarks against indiscriminate use of such operator forms may still be useful. In general, unless a very particular set of average structures is chosen, a dependence of final results on the SCF starting vectors is a troublesome fact. Consequently, energies, populations and molecular parameters in general may show unpredictable behavior under various trial vectors. It is shown that variationally coherent operator forms are given by state parameters with equal values for all the active MO 's used in the construction of Coulomb and exchange terms throughout the SCF cycles.     

Products of class operators of the symmetric group

A combinatorial derivation of the product of the class of three cycles, [(1)^N?3(3)]_N with an arbitrary class operator of the symmetric group S_N is presented. The form of this result suggests a conjecture concerning the expression of the general class operator product in terms of a relatively small number of reduced class coefficients. The conjecture is applied to the determination of the products of [(1)^N?4(4)]_N, [(1)^N?4(2)²]_N, and [(1)^N?5(5)]_N with arbitrary class operators. General expressions for the reduced class coefficients of the simplest type are obtained.     

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.     

On formulas in closed form for Van Vleck expansions

Canonical transformations have been widely used to simplify Hamiltonians and other operators. In molecular and in solid state theory, the so-called Van Vieck expansion is usually employed for this purpose while in theories of particles interacting with fields a combination of canonical transformations in closed form with Van Vleck type expansions has been found effective. For some of the transformations used in applications formulas in closed form are well known. It will be shown here that such formulas can be derived whenever the transformation function is bilinear in the canonical variables, and further that the use of matrix operators makes it possible to simplify these derivations substantially. The Cayley-Hamilton theorem is then used to express the expansions for the matrix operators in closed form. The number of separate operator terms appearing in the formulas thus obtained is the same as the rank of the matrices used. To calculate the coefficients of these operator terms a new type of special functions is introduced. The resulting linear canonical transformations include generalized rotations in both ordinary and phase-space. Explicit results have been obtained for several two- to four-dimensional problems.