Body-fixed functions

This equation corresponds to a unitary transformation from the space-fixed basis functions jf(R,r) to the body-fixed functions J,K,M,p)QfK Q)-The inverse transforamtion is... 

For complexes with very small anisotropies and small reduced masses, such as those containing He or H2, a space-fixed quantisation scheme is appropriate. However, for most other Van der Waals complexes, the anisotropy is strong enough for the diatom angular momentum to be quantised along the R vector, and the total wavefunction is most conveniently expanded in body-fixed functions. 

The total wavefunction is again most conveniently expanded in body-fixed functions,... 

The theory behind body-fixed representations and the associated angular momentum function expansions of the wavefunction (or wave packet) in terms of bases parameterized by the relevant constants of the motion and approximate constants of the motion is highly technical. Some pertinent results will simply be stated. The two good constants of the motion are total angular momentum, J, and parity, p = +1 or 1. An approximate constant of the motion is K, the body-fixed projection of total angular momentum on the body-fixed axis. For simplicity, we will restrict attention to the helicity-decoupled or centrifugal sudden (CS) approximation in which K can be assumed to be a constant of the motion. In terms of aU its components, and the iteration number k, the real wave packet is taken to be [21]... 

As discussed in Section II. A, the adiabatic electronic wave functions, a and / 1,ad depend on the nuclear coordinates R> only through the subset (which in the triatomic case consists of a nuclear coordinate hyperradius p and a set of two internal hyperangles this permits one to relate the 6D vector W(1)ad(Rx) to another one w(1 ad(q J that is 3D. For a triatomic system, let aIX = (a1 -. blk, crx) be the Euler angles that rotate the space-fixed Cartesian frame into the body-fixed principal axis of inertia frame IX, and let be the 6D gradient vector in this rotated frame. The relation between the space-fixed VRi and is given by... 

In the present work, we must carry out transformations of the dipole moment functions analogous to those descrihed for triatomic molecules in Refs. [18,19]. Our approach to this problem is completely different from that made in Refs. [18,19]. We do not transform analytical expressions for the body-fixed dipole moment components (/Zy, fiy, fi ). Instead we obtain, at each calculated ab initio point, discrete values of the dipole moment components fi, fiy, fif) in the xyz axis system, and we fit parameterized, analytical functions of our chosen vibrational coordinates (see below) through these values. This approach has the disadvantage that we must carry out a separate fitting for each isotopomer of a molecule Different isotopomers with the same geometrical structure have different xyz axis systems (because the Eckart and Sayvetz conditions depend on the nuclear masses) and therefore different dipole moment components (/Z, fiy, fij. We resort to the approach of transforming the dipole moment at each ab initio point because the direct transformation of analytical expressions for the body-fixed dipole moment components (/Zy, fiyi, fi i) is not practicable for a four-atomic molecule. The fact that the four-atomic molecule has six vibrational coordinates causes a huge increase in the complexity of the transformations relative to that encountered for the triatomic molecules (with three vibrational coordinates) treated in Refs. [18,19]. 

The form of the scattering wavefunction has been represented in a very general manner, specifically, as a product of a function in body-fixed coordinates the parity-adapted total angular momenrnm... 

The solution to this problem is to transform, or half-transform, the S matrix from the body-fixed to the space-fixed axis system then to use the known analytic properties of the spherical Bessel functions, which are the solutions to the potential-free scattering problem in the space-fixed axes and finally to transform back to the body-fixed axes and then to use Eq. (4.46) to calculate the differential cross section. 

After making these adjustments to allow for the fact that the analysis line cannot be located in the region of space where the centrifugal coupling in the body-fixed coordinates is negligible, and also for the fact that the analysis of Ref. 75 did not account for the long-range analytic form of the spherical Bessel functions, the space-fixed S matrix of Eq. (4.47) must be transformed back to the body-fixed axes and Eq. (4.46) must be used to compute the state-to-state differential cross sections [136,160]. 

This section of the appendix is based on Appendix B of Ref 80. It outlines the transformation of the space-fixed form of the continuum wavefunction, Eq. (4.3), to a body-fixed form. It differs from the previous development in that the angular functions used in the final equations are all parity-adapted. 

In order to transform to the body-fixed representation, we will need to relate the angular functions Wj (R,r) to angular functions defined relative to the body-fixed axes [L., J,K,M,p)QjK ), where J,K,M,p) are the parity-adapted total angular momentum eigenfunctions of Eq. (4.5) and x(0) normalized associated Legendre polynomials of the body-fixed Jacobi angle]. 

If we now rotate the axis system for the two functions on the RHS of the equation so that they are referred to the body-fixed, rather than space-fixed, z axes, we can write... 

An extra summation over p has been added as Eq. (A.5) contains a summation over both parities (i.e., Yhi f)- We need to transform the radial functions R) to the body-fixed basis (i.e., from using IXoK). To accomplish this, we... 

The identity operator within the space of functions with a hxed value of J and the parity (denoted by p), and that are associated assymptotically with a quantum number K of the body-fixed z component of the total angular momentum, is... 

M is the projection of J on the space-fixed z-axis, 0 its projection on the body-fixed z-axis, which is chosen here along the r vector. The D Ijq are Wigner matrices and are angular functions in the coupled BF representation. 

The projection of the electronic orbital angular momentum is neglected in this adiabatic representation, and the parity of the electronic function under reflection through the x — z body-fixed plane, (Txz, is given by... 

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. 

Equations (1-124) and (1-133) are valid in an arbitrary space-fixed coordinate system. However, since the angular functions A A (a)A, coB, R) are invariant with respect to any frame rotation162, a specific choice of the coordinate system may considerably simplify Eq. (1-125). In particular, in the body-fixed coordinate system with the z axis along the vector R the polar angles R = (/ , a) are zero. Using the fact that (r = (0,0)) = 8Mfi 14S, one gets,... 

Using the theory given in Refs. [33] (see especially appendix A) and [43]. we can write the final state wave function in the body-fixed form (see also Refs. [4] and [39] for details of the space-fixed bodj -fixed transformation) ... 

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 two most obvious possible choices of angular basis functions for use in this approach are the space-fixed and body-fixed repre-... 

---