Spin-Dependent operators in the spin-free quantum chemistry

The two-particle spatial density matrix components introduced by McWeeny are expressed in terms of the Fock coordinate wave function, which is constructed from an arbitrary function of N spatial coordinates. The integral relations for these components are verified. The necessary matrix elements of a standard representation of the S_N group are calculated.     

On geminals and symplectic bases in quantum chemistry

The construction of a symplectic basis set with >N electrons is exhibited by means of three kinds of units, the first kind geminal, the second kind geminal and the one‐particle operators. The optimization procedure of the variation method is extended to the coefficients in the linear sum of the symplectic bases, the parameters in the geminals, and the orbitals. For practical use, these bases are expanded explicitly as a linear sum of the Slater determinants. For illustration, the LiH molecule, which is taken as an example, is calculated by using some symplectic bases. This revised version was published online in July 2006 with corrections to the Cover Date.     

Haptic quantum chemistry

We present an implementation designed to physically experience quantum mechanical forces between reactants in chemical reactions. This allows one to screen the profile of potential energy surfaces for the study of reaction mechanisms. For this, we have developed a interface between the user and a virtual laboratory by means of a force‐feedback haptic device. Potential energy surfaces of chemical reactions can be explored efficiently by rendering in the haptic device the gradients calculated with first‐principles methods. The underlying potential energy surface is accurately fitted on the fly by the interpolating moving least‐squares (IMLS) scheme to a grid of quantum chemical electronic energies (and geometric gradients). In addition, we introduce a new IMLS‐based method to locate minimum‐energy paths between two points on a potential energy surface. © 2009 Wiley Periodicals, Inc. J Comput Chem 2009     

On the use of multipole expansion of the Coulomb potential in quantum chemistry

Some features of the multipole expansion of the Coulomb potential V for a system of point charges are studied. It is shown that multipole expansion is convergent both locally in L₂(R³) and weakly on some classes of functions. One-particle Hamiltonians H_n = H₀ + V_n, where H₀ is the kinetic energy operator and V_n is the n-th partial sum of the multipole expansion of V, are discussed, and the convergence of their eigenvalues to those of H = H₀ + V with increasing n is proved. It is also shown that the discrete spectrum eigenfunctions of H_n converge to those of H both in L₂(R³) (together with their first and second derivatives) and uniformly on R³. © 1996 John Wiley & Sons, Inc.     

On the validity of the mass–velocity operator in quantum chemistry

The problem of the validity of the mass–velocity operator in computational quantum chemistry is discussed. The opinion that the mass–velocity operator is incorrect is shown not to be well founded.     

On quantum groups and their potential use in mathematical chemistry

The quantum algebrasu _q (2) is introduced as a deformation of the ordinary Lie algebrasu(2). This is achieved in a simple way by making use ofq-bosons. In connection with the quantum algebrasu _q (2) we discuss theq-analogues of the harmonic oscillator and the angular momentum. We also introduceq-analogues of the hydrogen atom by means of aq-deformation of the Pauli equations and of the so-called Kustaanheimo Stiefel transformation.     

Emergence and quantum chemistry

This paper first queries what type of concept of emergence, if any, could be connected with the different chemical activities subsumed under the label ‘quantum chemistry’. In line with Roald Hoffmann, we propose a ‘rotation to research laboratory’ in order to point out how practitioners hold a molecular whole, its parts, and the surroundings together within their various methods when exploring chemical transformation. We then identify some requisite contents that a concept of emergence must incorporate in order to be coherent from the standpoint of the scientific practices involved. In this respect, we finally propose a relational form of emergence which pays attention to the constitutive role of the modes of intervention and to the co-definition of the levels of organization. No metaphysical distinction between the higher and basic levels of organization is supposed, but only a plurality of modes of access. Moreover, these modes of access are not construed as mere ways of revealing intrinsic patterns of organization but, on the contrary, are considered to be active elements on which the constitution of those patterns depends. What is at stake in this paper is therefore not an ontological form of emergence but an agnostic one which fits what chemists do in their daily work.     

Dynamic aspects of quantum chemistry

Some contributions of quantum chemistry to the field of atomic dynamics are reviewed and it is argued that, in return, the area of application of quantum chemistry has become considerably extended and has given rise to insight into processes closely related to bond formation.     

Semigroups and the density matrix formulation of quantum mechanics

A formulation of quantum mechanics is presented based on the theory of semigroups and the associated enveloping algebras of functions defined on countable subsemigroups. The existence of a unique *-involution is not assumed. The fundamental elements of a semigroup are identified with experimental precedures for the separation of subensembles from a given ensemble of experimental systems. Observables are represented as elements of enveloping algebras, and ensembles as density matrices within an enveloping algebra. The statistical properties of ensembles are expressed in terms of traces defined on the semigroup and its enveloping algebras. The elements and generators of the Poincaré group can be defined and interpreted in the usual way. A variety of applications is described, in which the theory of the density matrix plays an essential or effective role. Advantages associated with the resulting freedom from the limitations of Hilbert space are illustrated.     

Spin-unrestricted calculations in quantum chemistry

The unrestricted complete active space self-consistent field (UCASSCF ) function is defined, and a proof that a UCASSCF eigenfunction of the spin operator S ² is a CASSCF function is given. The spin-contamination for an unrestricted Hartree–Fock (UHF ) function is evaluated by using Araki angle operators, and the UHF function is then projected on the restricted open-shell Hartree–Fock (ROHF ) space. The present analysis has deep consequences since it implies that the only non-spin-contaminated UHF functions are the ROHF functions. This is illustrated in a calculation of the spin density of He. © 1993 John Wiley & Sons, Inc.     

High symmetries in quantum chemistry

A method of simplifying the solution of secular equations occurring in electronic structure, normal mode and nuclear spin state calculations for molecules possessing symmetry is illustrated by applying it to the π-electrons of the recently discovered C₆₀ cluster. In contrast to the usual procedure, the method of reduction to characters does not require actual matrix realizations of the irreducible representations of the symmetry group but only the information contained in the character and multiplication tables. In the case of C₆₀ it leads to explicit algebraic solutions for all the π-orbital energies.