Xavier Chapuisat

Momentum, Quasi-momentum and Hamiltonian Operators in Terms of Arbitrary Curvilinear Coordinates, With Special Emphasis On Molecular Hamiltonians

Since the conformation of physical systems is often advantageously described with the help of generalized (i.e. curvilinear) coordinates, the following questions are raised and general answers given to them: (i) What are the components of the momentum operators, those of the adjoint momentum operators and those of the hermitian momentum operators? (ii) What is the general form of the hamiltonian operator? (iii) What do the general forms of the momentum and hamiltonian operators become when expressed in terms of quasi-momentum operators (e.g. angular momentum component operators)? (iv) How are the answers to the above questions affected by an arbitrary choice of normalization convention for the total wave-function? A complete set of formulae (whatever the independent choices of coordinates, quasi-momentum operators and normalization) is given and various ‘historical’ formulae are shown to be particular instances of the general formulae proposed.

Dynamical study of non adiabatic unimolecular reactions: The conical intersection between the B 2B2 and à 2A1 states of H2O+.

The conical intersection connecting the B 2A′ and A 2A′ states of the H2O+ ion is studied. The two potential energy surfaces are calculated ab initio by the SCF/CI method within the CS point group. The nonadiabatic coupling matrix elements 〈A‖∂/∂q‖B〉 are computed for several cross sections throughout the potential energy surfaces. A transformation to the diabatic representation is performed. The linear model is found to be a good approximation in the region close to the apex of the cone. The global functions t(s) and T(S) governing the nonadiabatic transition probability are calculated; their shapes are those predicted by the Landau–Zener model (in the Nikitin bidimensional version). A dynamical study is undertaken by means of classical trajectory calculations on the upper adiabatic potential energy surface. An averaged transition probability Ptr is derived. Excitation of rotation or of the bending mode of H2O before photon impact has no influence on Ptr. Excitation of the symmetrical or antisymmetr...

An ab initio description of the excited states of the reaction O(3P, 1D) + H2 → OH(2Π, 2Σ+) + H. An attempt to describe several potential energy surfaces with constant accuracy

Abstract A set of potential energy surfaces describing all the possible pathways for the reaction O( 3 P, 1 D)+H 2 (X 1 Σ 8 + ) → OH(X 2 Π, A 2 Σ + )+H( 2 S) are calculated and studied. The energy range is between the ground state H 2 O(X 1 A 1 ) and the dissociation continuum O( 3 P)+2H( 2 S), i.e. ≈10 eV. Special attention is paid to the non-adiabatic effects at surface crossings, in particular those giving rise to conical intersections. After a summary of former calculations, our calculational method is briefly described. The results are reported in cross-section form, generally in two dimensions for geometries within high-symmetry point groups (C 2v , C xv and, in between, D xb ). The general three-dimensional situation (within the framework of point group C ? ) is systematically sketched.

Exact quantum molecular Hamiltonians

A mathematical procedure to derive in a systematic manner exact quantum mechanical Hamiltonians for N-particle systems is presented. As an application, exact quantum mechanical Hamiltonians are obtained for the motion of three atoms, with the help of various sets of curvilinear internal coordinates and for various body-fixed frames of reference.

Hamiltonians for constrained N-particle systems

Abstract The efficiency of the theoretical methods to obtain exact expressions of classical and quantum-mechanical Hamiltonians for N -particle systems described in terms of curvilinear coordinates ( q ), is shown to depend strongly on whether the system is subjected to constraints or not. If it is free, the method based upon the use of the relations q = q ( x ), where x denotes moving-frame Cartesian coordinates of the particles, is preferable. If it is constrained, the method making use of x = x ( q ) can become more efficient.

A Method To Obtain the Eckart Hamiltonian and the Equations of Motion of a Highly Deformable Polyatomic System in Terms of Generalized Coordinates

Abstract A general method that makes it possible to derive the Eckart hamiltonian and the equations of motion of a highly deformable polyatomic system in terms of generalized coordinates is presented. The overall rotation is taken into account and the method is applied to various non-reactive collisional tetratomic systems. namely (i) diatom-diatom, (ii) atom-linear symmetric triatomic and (iii) atom-bent symmetric triatomic.

Anharmonicity effects in the collinear collision of two diatomic molecules

Abstract The collinear collision of two diatomic molecules is studied. Six different systems - namely N 2 + N 2 , N 2 + CO, N 2 + OC, N 2 + O 2 , H 2 + H 2 , and H 2 + HBr - are selected as samples and compared to each other. In addition, the effect of anharmonicity on the vibrational excitation is investigated. Exact quantum-mechanical transition probabilities are calculated, the Morse and harmornic binding potentials being considered in each case. It is shown that anharmonicity often decreases the TV transition probabilities by a large factor, whereas it has only a small effect on VV transition probabilities. The results are interpreted on the basis of previous ideas on atom-diatom and diatom-diatom collinear collisions.

A quantum-mechanical collinear model study of the collision N2 + O2

Abstract A model close-coupling study of the translation-vibration-vibration (TVV) energy transfer in the collinear collision of two non-identical harmonic diatomic molecules is presented. The numerical application are for N 2 + O 2 driven by an exponential repulsion. Numerous transition probabilities are given, and the various dynamical processes are classified according to their importance. The classification is true whatever the collision energy. A resonable interpretation of the dynamics based on (i) our knowledge of the simpler atom-diatom collision, (ii) the results of auxiliary calculations using various truncated expansions of the interaction potential, and (iii) classical trajectory calculations for the same system, is finally presented. The main results are: (i) The TV transition probabilities of O 2 are greater than that of N 2 by several orders of magnitude; this is due to O 2 being the more excitable molecule and being acted on by the more exciting field; (ii) The secondary processes are interfering sequences of the two principal processes which are, respectively, the TV one-quantum jump in O 2 and VV one-quantum exchange between N 2 and O 2 . (iii) In the case of the TV one-quantum jump in N 2 the mechanism is not unique according as O 2 is vibrationally excited - or not - prior to collision.

Exact quantum molecular hamiltonians. II, On the choice of the moving frame of reference : the principal axis system

It is demonstrated that, for an N-atom molecular system, a body-fixed (BF) frame of reference such that the rotational kinetic energy considered a quadratic form of the components of the total angular momentum measured in the BF frame: where q collectively denotes the internal coordinates and μ is a 3 × 3 symmetric matrix, can always be taken as instantaneously block-diagonal. Next, it is shown that it is not much complication to use the principal axis system (PAS) as the BF frame of reference. An explicit solution of this problem is proposed. Using the PAS to treat the dynamics of a molecular system can be very helpful, in particular when the system does not fragment into more than two pieces. Applications are presented for the three-body problem, using various parameterizations of the internal deformation motion.

To what extent can an intrinsic definition of a reaction path be given

Abstract Starting from the mathematical coordinate-free definition of a path of steepest descent in an euclidean space and in a nemannian manifold imbedded in it, an intrinsic definition of the reaction path for a molecular system is given. The remaining arbitrariness is closely linked to the unavoidable arbitrariness in the choice of configuration space and of some constraints.