Rotational basis functions

Similar to Eq. (2), the time-dependent wavefunction is expanded in terms of the BF parity-adapted rotational basis functions ... 

We had earlier vindicated treating one of the OH bonds in the H2O molecule as a spectator bond in studying the abstraction reaction. Another key assumption that needed to be checked was the centrifugal sudden (CS) approximation which was invoked to reduce the number of rotational basis functions used in the computations.28 Under the CS approximation and using only the K = 0 rotational basis functions, there was a total of 220 million basis functions for J = 15 alone. Relaxing the CS approximation, for example, with K = 0,1 and J = 15 led to 650 million basis functions. To approach the fully coupled-channel (CC) results, i.e. without... 

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 rotational basis functions used in this work are very similar to those used in the initial state wave packet dynamics study of H+H2O. The rotational basis function has the following explicit form... 

The time-dependent wavefunction is expanded in the parity-adapted rotational basis functions as. 

For a linear molecule, the position of the symmetry axis (the molecule-fixed. z-axis) in space is specified by only two Euler angles, / and 7, which are respectively identical to the spherical polar coordinates 6 and (see Fig. 2.4). The third Euler angle, a, which specifies the orientation of the molecule-fixed x- and y-axes, is unaffected by molecular rotation but appears explicitly as an O- dependent phase factor in the rotational basis functions [Eq. (2.3.41)]. Cartesian coordinates in space- and molecule-fixed systems are related by the geometrical transformation represented by the 3x3 direction cosine matrix (Wilson et al., 1980, p. 286). The direction cosine matrix a given by Hougen (1970, p. 18) is obtained by setting a = 7t/2 (notation of Wilson et al, 1980 6 fi,4)=, x = oi 7t/2). The direction cosine matrix is expressed in terms of sines and cosines of 9 and 4>. Matrix elements (J M O la JMQ), evaluated in the JMQ) basis, of the direction cosines, are expressed in terms of the J, M, and quantum numbers. The direction cosine matrix elements of Hougen (1970, p. 31), Townes and Schawlow (1955, p. 96), and Table 2.1 assume the basis set definition derived from Eq. (2.3.40) and the phase choice a = 7t/2 ... 

Mathematically, the conservation of the J quantum number on perturbation can be clearly shown, since no perturbation operator contains the 9 and angular coordinates of the nuclei and consequently none can act on the rotational basis functions. As the rotational functions JMQ.) are orthogonal for different values of J, a matrix element of any perturbation operator, H, between two rotational functions is given by... 

Nonadiabatic transitions among these states are induced by either the first term of Eq. (43b) or Hcor The states of the same A are coupled by Tad given by Eq. (4). Transitions between the states of different electronic symmetries (different A ) are induced by the Coriolis coupling Hmi and have quite different properties from the radially induced transitions. In order to look into this in more detail, let us introduce the electronic-rotational basis functions defined as... 

In propagation, the wave packet is expanded in terms of a body-fixed translational-vibrational-rotational basis functions as follows ... 

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