Modified Cartesian Gaussian functions and their use in quantum chemistry

The general formula for the electron-electron repulsion integral in the modified Cartesian Gaussian basis set derived in Ref. [1] is simplified. A general relation between the standard and modified CG functions is given. A possible use of the modified CG functions to quantum chemical calculations which include the correlation factor r_ij² is indicated.     

A mixed basis set of Slater and Gaussian functions. Integrals with modified Gaussian functions

The analytic expressions of the integral prototypes involving both Slater and s-type Gaussian functions, explicity derived in Ref. 1, are generalized to the case of higher order modified Gaussian functions [2].     

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.     

Integrals of quantum chemical interest in bases of laguerre functions. I. Functions of order 2l+2

Formulae are presented for the evaluation of some definite integrals in which the Laguerre orthogonal functions are contained. In some cases mechanical quadratures are to be preferred, and a scheme is presented for generating such a quadrature as it is needed.     

Integrals and derivatives for correlated Gaussian functions using matrix differential calculus

The matrix differential calculus is applied for the first time to a quantum chemical problem via new matrix derivations of integral formulas and gradients for Hamiltonian matrix elements in a basis of correlated Gaussian functions. Requisite mathematical background material on Kronecker products, Hadamard products, the vec and vech operators, linear structures, and matrix differential calculus is presented. New matrix forms for the kinetic and potential energy operators are presented. Integrals for overlap, kinetic energy, and potential energy matrix elements are derived in matrix form using matrix calculus. The gradient of the energy functional with respect to the correlated Gaussian exponent matrices is derived. Burdensome summation notation is entirely replaced with a compact matrix notation that is both theoretically and computationally insightful. © 1996 John Wiley & Sons, Inc.     

Symmetrizing broken symmetry wave functions in quantum chemistry

A general scheme to symmetrize a broken symmetry wave function is described. It offers great flexibility in the choice of the set of functions used to expand the symmetrized functions. The traditional symmetrization approaches are related to some particular choices of these functions. New choices are also considered. A postsymmetrization treatment is proposed. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 91–101, 1998     

On some problems in quantum pharmacology I. The partition functions

We performed an analysis of the drug-receptor equilibrium constant, K_i = {Q_DiRQ_Di^?1Q_R^?1} exp (?Δ_εi/kT). It is shown that, for a group of nonrotating molecules we may consider the product of the partition functions as constant if log(mass) = constant for all the molecules.     

A density functional representation of quantum chemistry. I. Motivation and general formalism

The molecular N-electron problem is formulated in terms of the particle density ??( r ) = ∑σ( r ? q _n) and its canonically conjugated field. Weyl's representation of the canonical commutation relations is used and a characteristic functional of the molecular system is introduced as the expectation value of Weyl's functional exp {i(α, Q) + i(β, P)}. The Hamiltonian equation of motion for the characteristic functional is derived and its cumulant approximation is discussed. The first paper is restricted to a general discussion, independent of a special choice of the Hamiltonian. The density functional representation may be considered as a step towards a formulation of a renormalized quantum chemistry. The lowest order cumulants of the characteristic functional are accessible to actual precision measurements; on the other hand, most experiments are correctly represented by a characteristic functional of the Gaussian type. The significance of the higher order correlations is neither theoretically nor experimentally disclosed yet, their possible importance for biological systems is mentioned, and the question is raised of how far quantum mechanics is empirically confirmed for extremely complex systems.     

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.     

Spin-free quantum chemistry. XIII. Spin waves

We have obtained the Bloch spin wave dispersion formula using the methods of spin-free quantum chemistry. The spin-free eigenvectors are waves in spin-free space. This development makes the point that Bloch spin waves are dynamically spin-free. The neutron diffraction transition moment for spin waves is calculated employing the antisymmetrized projections of vector products of spin-free eigenkets and spin kets and is found to be agreement with results of Moorhouse.     

Spin-free quantum chemistry. XVI. Spin correlation

We discuss the calculation of quasielastic critical neutron scattering for systems defined by a spin-free Hamiltonian. The system dependent property which determines the critical scattering is the thermal average of a spin-free operator called the general unpaired electron correlation operator. For localized models, we obtain the scattering as the Fourier transform of an unpaired site correlation function. The algebraic relationship between this correlation function and the usual spin correlation function is discussed as are its relationships to thermodynamic properties. Correlations in the infinite interaction range model, linear spin wave model, and infinite chain Heisenberg model are given. In the last case, an estimate of the low temperature correlation length is made.     

Models and paradoxes in quantum chemistry

Ten different, unexpected features of the classificatory results of quantum mechanics are discussed as well as the question whether the Schrödinger equation only applies to sufficiently small systems which can be indiscernibly reproduced and possess assembly porperties.Dedicated to Professor H. Hartmann on occasion of his sixtieth birthday 4. May 1974.     

Exponentially Modified Gaussian functions—A good model for chromatographic peaks in isocratic HPLC?

Summary The Exponentially Modified Gaussian (EMG) function is fitted to a variety of typical peaks from isocratic HPLC, to assess its validity for peak characterization. This model has previously been applied, but with simplifying assumptions, allowing estimation of peak parameters from simple front and rear half-width measurements. The full EMG function is employed in this work. The results of the simplified approach have been compared to the EMG fitted results. Comparison is also made to fitted Gaussian functions. Results show that the EMG function is a good model for typical LC peaks, under a range of conditions, and that the simplified approaches give close correlation for a range of peak assymetries.     

Spin-free quantum chemistry. XIX. Particle-hole and pairing symmetries

Particle–hole and pairing relationships are obtained within the framework of the unitary group formulation of the many-electron problem using the concept of particle–hole conjugation. Besides the familiar relationships for alternant hydrocarbons, relationships among various pericyclic reaction paths are obtained.     

Central moments in quantum chemistry

We define central moments of operators on finite‐dimensional vector spaces and study some of their basic aspects. Central moments may be viewed as generalizations of the dispersion of a Hermitian operator. We show how eigenvalues may be represented by central moments, and how central moments may be used to obtain Krylov subspace approximations for operators on inner product spaces. We show that central‐moments approximations are compatible with the concepts of size‐consistency in quantum chemistry, and we use this to suggest a foundation for central‐moments approximations in Coupled Cluster theory. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008