Pierre Claverie

The exact multicenter multipolar part of a molecular charge distribution and its simplified representations

We study the problem of representing the molecular charge distribution in a convenient way for practical applications and we propose, instead of a single representation, a flexible procedure for building approximations with an arbitrary level of accuracy as concerns the long‐range part of the electrostatic potential. We first discuss the splitting of the total electrostatic potential into a multipolar part (long‐range) and a penetration part (shortrange) in connection with the usual one‐center multipole expansion: at large enough distances, this expansion precisely converges towards the multipolar part. However, this representation is not practically efficient as soon as the molecule departs from a spherical shape, and we therefore consider a so‐called multicenter multipole representation. In the MO‐LCAO framework, the use of a basis of Gaussian atomic orbitals {χα} generates such a representation in a natural way: indeed, each elementary distribution χ*αχβ then reduces to a one‐center distribution with a...

Development of a pure diffusion quantum Monte Carlo method using a full generalized Feynman–Kac formula. I. Formalism

This paper presents systematic developments in the previously initiated line of research concerning a quantum Monte Carlo (QMC) method based on the use of a pure diffusion process corresponding to some reference function and a generalized Feynman–Kac path integral formalism. Not only mean values of quantum observables, but also response properties are expressed using suitable path integrals involving the diffusion measure of the reference diffusion process. Moreover, by relying on the ergodic character of this process, path integrals may be evaluated as time‐averages along any sample trajectory of the process. This property is of crucial importance for the computer implementation of the method. As concerns the treatment of many‐fermion systems, where the Pauli principle must be taken into account, we can use the fixed‐node approximation, but we also discuss the potentially exact release‐node procedure, whereby some adequate symmetry is imposed on the integrand (of the generalized Feynman–Kac formula), ass...

Second‐order exchange effects in intermolecular interactions. The water dimer

A new method of deriving explicit formulas for the calculation of second‐order exchange contributions (induction as well as dispersion) within the framework of symmetry‐adapted perturbation theories is presented. It is shown how exchange contributions can be expressed as a combination of electrostatic interaction energies between suitably generalized charge distributions (overlap intermolecular charge distributions). Each of these contributions are derived within the Hartree–Fock approximation (neglect of all electron correlation effects within the noninteracting molecules) and by considering only single‐electron exchange between interacting molecules. Numerical calculations for the interaction of two water molecules are presented. In the region of the equilibrium geometry, it is found that the complete second‐order exchange contribution accounts for about 20% of the total intermolecular interaction energy. This contribution is essentially dominated by the exchange induction component which is found to re...

Development of a pure diffusion quantum Monte Carlo method using a full generalized Feynman–Kac formula. II. Applications to simple systems

We have described in part I of this work the theoretical basis of a quantum Monte Carlo method based on the use of a pure diffusion process and of the so‐called full generalized Feynman–Kac (FGFK) formula. In this second part, we present a set of applications (one‐dimensional oscillator, helium‐like systems, hydrogen molecule) with the purpose of illustrating in a systematic way the various aspects pertaining to the practical implementation of this method. We thus show how energy and other observables can be obtained, and we discuss the various sources of biases occurring in the different procedures (notably the so‐called short‐time approximation pertaining to the generation of the sample trajectories of the diffusion process, and the numerical integration pertaining to the evaluation of the ‘‘Feynman–Kac factor’’). After having thus considered the case of the genuine ‘‘bosonic’’ ground state, we illustrate the various proposals for dealing with some ‘‘relative’’ ground state (namely the lowest state belo...

Quantum Monte Carlo method for some model and realistic coupled anharmonic oscillators

A new quantum Monte Carlo (QMC) method of evaluating low lying vibrational levels for coupled modes is presented. We use a modified fixed‐node (FN) approach in which an extremum principle for energy levels is invoked. In this way, the nodal hypersurfaces of the nuclear wave function are parametrized and then optimized for each excited state. The method is tested on the fundamental excitations of some two‐dimensional model potentials and is applied to the case of realistic coupled modes of the CO molecule adsorbed on a palladium cluster. The effect of an external electric field is also examined. The quantum Monte Carlo results are compared with those obtained in the conventional variational treatment of the nuclear Schrodinger equation for coupled vibrations. The QMC results give the exact values with an error which is in general less than 1 cm−1 . In all cases (even in the case of strong coupling) the use of our procedure leads to ‘‘optimal’’ nodal lines (in the sense of the extremum principle used in thi...

The thermodynamic functions of molecular liquids in the interaction site model

The excess Helmholtz Free Energy (H.F.E.) of the hard molecule liquid with respect to the perfect gas (point-like non-interacting molecule) is obtained; the Equation of State (E.S.) is then derived. The site-site and molecular correlation functions involved in both the excess H.F.E. and the E.S. are determined in the framework of a modified Reference Interaction Site Model. For a symmetric diatomic molecule the exact value g 11(r = σ+) = 1/4 of the site-site correlation function and the exact second virial coefficient is obtained.

Identification des modèles à fonction de transfert: la méthode Padé-transformée en Z

We present and discuss the pade z-transform method for the identification of a transfer function model; an application is given with the geometric lag model, the ratios of which can be real and/or complex numbers. The method consists first in considering the z-transform of the transfer function model, which is numerically calculated by a Taylor expansion about a point ?? which can be equal to, or different from, zero. By contrast, the Box and Jenkins, Corner, and Lii methods rely on the implicit ??. An advantage of the method is this new degree of freedom which may improve the accuracy of the results. Second, to identify and estimate the model, this z-transform is analyzed by the use of the pade approximant technique: namely the search of a bloc of stable pade approximants. If the transfer function is rational, then the degrees and the coefficients of the polynomials are obtained. In the case of a lag distribution which is a linear combination of elementary geometric distributions, of the method gives estimates of the number of elementary distributions, of the corresponding ratios and coefficients: the functional form of the lag distribution is fully identified and estimated. Monte Carlo simulations are run to check the limits and qualities of the method : satisfactory results are obtained, with a better accuracy for choice different from zero.

Treatment of the Schrödinger equation through a Monte Carlo method based upon the generalized Feynman-Kac formula

We present a new Monte Carlo method based upon the theoretical proposal of Claverie and Soto. (1) By contrast with other Quantum Monte Carlo methods used so far, (2 8) the present approach uses a pure diffusion process without any branching. The many-fermion problem (with the specific constraint due to the Pauli principle) receives a natural solution in the framework of this method: in particular, there is neither the fixed-node approximation not the nodal release problem which occur in other approaches (see, e.g., Ref. 8 for a recent account). We give some numerical results concerning simple systems in order to illustrate the numerical feasibility of the proposed algorithm. OUTLINE OF THE METHOD ~1) Following previous authors, (9) a rather arbitrary Schr6dinger Hamiltonian H ~~ may be changed through a simple transformation into the infinitesimal generator L ~~ of a diffusion process, namely Ll~176162176 0 H~~ b~m), where E(o ~ and ~b(o ~ denote, respectively, the energy and eigenfunction of the (mathematical) ground state of H m) (this process was also introduced by Nelson, ~1~ but in a quite different perspective). The operators L ~~ and E(o~ m) are similar; consequently, the quantum-mechanical Greens function of H ~~ is closely related with the Greens function of L m), namely the transition probability density of the

Deterministic macroeconomic dynamic equations from a stochastic microeconomic description: The role of time scales

Abstract In the present paper, we develop the ‘stochastic’ microeconomic approach to obtained a deterministic macrodynamic equation. Three time scales have to be introduced. To be valid, the dynamic equation has to be written on the meso-scopic time scale. ‘Discrete’ and ‘continuous’ time versions are given. On the other hand, the comparison between the ‘stochastic’ and the ‘average’ (all the agents have the same characteristics, the ‘average’ ones) approaches is presented. Both methods give the same form for the dynamic equation, nevertheless the ‘average’ approach gives conditions of validity which are not strict enough. An advantage of the ‘stochastic’ approach is to inforce this point and to propose a better framework for writing down deterministic macrodynamic equations and for making further developments if necessary.