Hamiltonian body-fixed

This paper draws a parallel between the (full) six-dimensional H + H2O —> H2 -I- OH and the (reduced) seven-dimensional H -l- CH4 —> H2 + CH3 abstraction reactions. In Sec. 2, we briefly present the initial state TD quantum wave packet approach for the A -I- BCD and X + YCZ3 reactions. The Hamiltonians, body-fixed (BE) parity-adapted rotational basis functions, initial state construction and wave packet propagation, and extraction of reaction probabilities, reaction cross sections, and thermal rate coefficients from the propagated wave packet to compare with experiments are discussed. In Sec. 3 we briefly outline the potential energy surfaces used in the calculations. Some... 

The electronic Hamiltonian and the comesponding eigenfunctions and eigenvalues are independent of the orientation of the nuclear body-fixed frame with respect to the space-fixed one, and hence depend only on m. The index i in Eq. (9) can span both discrete and continuous values. The q ) form... 

No one wants to work with 18 -y symbols. There are two work-arounds for the numerical implementation of the coupled channel theory in the total angular basis. First - as suggested by Tscherbul and Dalgarno [21] - one can use a basis of angular momentum states defined in the body-fixed coordinate frame. For example, for two molecules in a S electronic state with nonzero electron spin, the eigenstates of the full Hamiltonian can be written as... 

This work introduced the concept of a vibronic R-matrix, defined on a hypersurface in the joint coordinate space of electrons and intemuclear coordinates. In considering the vibronic problem, it is assumed that a matrix representation of the Schrodinger equation for N+1 electrons has been partitioned to produce an equivalent set of multichannel one-electron equations coupled by a matrix array of nonlocal optical potential operators [270], In the body-fixed reference frame, partial wave functions in the separate channels have the form p(q xN)YL(0, radial channel orbital function i/(q r) and antisymmetrized in the electronic coordinates. Here 0 is a fixed-nuclei A-electron target state or pseudostate and Y] is a spherical harmonic function. Both and i r are parametric functions of the intemuclear coordinate q. It is assumed that the target states 0 for each value of q diagonalize the A-electron Hamiltonian matrix and are orthonormal. 

Here, Aftot is the total mass of the nuclei. Note that when eliminating the motion of the center-of-mass we arbitrarily eliminated the first nuclear coordinate (we could eliminate any single coordinate y). Equation (1-13) can further be transformed to the body-fixed frame. When applying the so-called Eckart conditions35 one would get the standard Watson s Hamiltonian describing the nuclear motions in molecules36. 

Equation (1-13) or its body-fixed equivalent is of little use for Van der Waals complexes, as it discriminates one nuclear coordinate, e.g. y = 1. Specific mathematical forms of Hamiltonians describing the nuclear motions in Van der Waals dimers have been developed (7). This point will be discussed in more details in Section 12.4. Here we only want to stress that whatever the mathematical form of the Hamiltonian is used to solve the problem of nuclear motions, the results will be the same, if the Schrodinger equation is solved exactly. However, in weakly bound complexes there is a hierarchy of motions due to the strong intramolecular forces which determine the internal vibrations of the molecules, and to much weaker intermolecular forces which determine their relative translations and rotations. This hierarchy allows to make a separation between the intramolecular vibrations with high frequencies and the intermolecular modes with much lower frequencies. Such a separation of the fast intramolecular vibrations and slow rotation-vibration-tunneling motions can be performed if a suitable form of the Hamiltonian for the nuclear motions in Van der Waals molecules is used. 

In this contribution, within the asymptotic approach, we have elaborated the basis for the calculation either of adiabatic channel potentials (diagonalization of the full Hamiltonian in a body-fixed frame at given interfragment distances) or of axially-nonadiabatic channel potentials (diagonalization of the full Hamiltonian in a space-fixed frame at given interfragment distances). As a by-product, we have compared our asymptotic PES on different levels of approximations with available local ab initio data. In subsequent work, we envisage the calculation of low temperature rate constants for complex-formation of the title reactions. 

To obtain the scattering path hamiltonian we would begin with the general body-fixed hamiltonian in the variables (R.v.r)(13) and simply re-express it in terms of the variables (t.n.r) using the above transformation. For simplicity we consider the zero partial wave, J 0, and we obtain... 

Hamiltonian and Complex-Coordinate Coupled-Channel Formulation in Body-Fixed Coordinates (23). In the BF fr jne, the Hamiltonian is identical to equation 3 except that R and r are expressed relative to the unprlmed axes of figure 1 and the angular momentum operator of the rotation of R (i.e. X) is written as X = J-j. The operator (J-J)2 may be expressed as... 

The eigenfunction P(t) can be written as an eigenfunction of the body-fixed Hamiltonian in the form... 

It is thus possible to eliminate the angular motion from the problem and to write down an effective body-fixed Hamiltonian within any (J, M, k) rotational manifold, that depends only on the internal coordinates. The transformation from the laboratory-fixed to the translation-free frame is linear and so its jacobian is simply a constant that can be ignored and since the transformation from the t to the zi is essentially a constant orthogonal one, it has a unit jacobian. The transformation from the translation-free nuclear coordinates to the body-fixed ones is, however, non-linear and has a jacobian J -1 where J is the matrix constructed from the nuclear first derivative terms. 

Let us start with the Hamiltonian H of a diatomic molecule, which is given in the body-fixed coordinate system as (the mass polarization and relativistic effects are disregarded)... 

---