Product basis functions

The calculation method presented here also provides a possible extension of the single spin vector model. This extension is performed in two steps first to weakly coupled spin systems, then to strongly coupled ones. In the first case, the introduction of the well-known product basis functions and their coherences is sufficient while in the latter one the solution is not so trivial. The crucial point is the interpretation of the linear transformation between the basis functions and the eigenfunctions (or coherences) during the detection and exchange processes. These two processes can be described by the population changes of single quantum... 

Finally, we then use well-known mathematical procedures to determine the energy levels of the system and to describe the true wave functions in terms of linear combinations of the basis functions. As we shall see, this means that we set up the Hamiltonian as a matrix in terms of the spin product basis functions, then manipulate it into a form in which it is diagonal. The functions used to represent the Hamiltonian in diagonal form are, then, the true eigenfunctions, which are linear combinations of the basis functions. 

We now apply the concepts developed in the preceding section to the system of just two nuclei, first considering the case in which there is no spin coupling between them. We digress from our usual notation to call the two spins A and B, rather than A and X, because we later wish to use some of the present results in treating the coupled AB system. The four product basis functions are given in Eq. 6.1. We now compute the matrix elements needed for the secular determinant. Because there are four basis functions, the determinant is 4 X 4 in size, with 16 matrix elements. Many of these will turn out to be zero. For <3CU we have, from... 

This section deals with various types of nonzero off-diagonal matrix elements of H between approximate BO product basis functions. In order to go beyond the BO approximation, to try to obtain an exact solution, it is necessary to use a BO representation. In other words, the exact eigenvalues and eigenfunctions, which can be compared to observed energy levels, are expressed in terms of the BO representation, specifically as a linear combination of BO product functions. Presently, ip50 is the only available type of complete, rigorously definable basis set. 

The interpretation of the time evolution of l (r) is facilitated (Baggott et al., 1986) by using a zero-order product basis which includes the initially prepared state P(O). This product basis function can be identified as < )s. Then, according to Eq. (4.16), the probability amplitude of finding the system in (0) = c ) versus time is given by... 

Czako, G., Szalay, V, Csaszar, A.G. Finite basis representations with nondirect product basis functions having structure similar to that of spherical harmonics, J. Chem. Phys. 2006,124, 014110. 

Mendive-Tapia D, Lasome B, Worth GA, Robb MA, Bearpark Ml (2012) Towards converging non-adiabatic direct dynamics calculations using frozen-width variational Gaussian product basis functions. J Chem Phys 137 22A548... 

Finally, some comments on two recent studies on methods for quantum dynamics simulations. In the first study, by Mendive-Tapia et al., the convergence of non-adiabatic direct dynamics in conjunction with frozen-width variational Gaussian product basis functions is evaluated. The simulation of non-adiabatic dynamics can be subdivided into two groups semi-classical methods (like the trajectory surface hopping approach) and wavepacket methods (for example, the... 

To alleviate the problems associated with large matrices one may either (i) devise good basis functions which rep- resent wavefunctions compactly to minimize the number of basis functions (and the size of the matrix) required to obtain converged energy levels (ii) use simple product basis functions and calculate eigenvalues with an iterative method which exploits the simplicity of the basis (iii) use good (and therefore not simple) basis functions and an iterative method to calculate eigenvalues or (iv) use an iterative time-dependent method with either simple product or better basis functions. 

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