Local harmonic approximation

A different approach is to represent the wavepacket by one or more Gaussian functions. When using a local harmonic approximation to the trae PES, that is, expanding the PES to second-order around the center of the function, the parameters for the Gaussians are found to evolve using classical equations of motion [22-26], Detailed reviews of Gaussian wavepacket methods are found in [27-29]. 

A technical difference from other Gaussian wavepacket based methods is that the local harmonic approximation has not been used to evaluate any integrals, but instead Martinez et al. use what they term a saddle-point approximation. This uses the localization of the functions to evaluate the integrals by... 

In this section, we describe wave packet dynamics within a (time-dependent) local harmonic approximation to the potential, since this enables us to write down relatively simple expressions for the time evolution of the wave packet. This provides a valuable insight into quantum dynamics and the approximation may be used, for example, to... 

We consider now the dynamics of the Gaussian wave packet within the framework of a time-dependent local harmonic approximation (LHA) to the exact potential V(x) around xt. ... 

The LHA is exact for potentials that contain, at most, quadratic terms but obviously an approximation for anharmonic potentials. Thus, a single Gaussian wave packet within a local harmonic approximation can, e.g., not tunnel or bifurcate, i.e., there will be no simultaneously reflected and transmitted part in scattering off barriers. 

It is important to notice that the energy threshold that enters the expression in Eq. (6.8) is Eq and not the classical threshold Ec, see Fig. 6.1.1. Within the normalmode description, that is, the local harmonic approximations to the potential energy... 

Using intersection-adapted coordinates, the quadratic approximation, in other words the local harmonic approximation, of the adiabatic energy difference for a finite displacement around Qo reads thus... 

A second, very important, approximation is the local harmonic approximation which states that all of the atoms in the system can be... 

Lastly, computational efficiency needs to be discussed. However complete the formulation of the coarse-grained alternative to MD methodology is up to here, additional approximations are required to make it computationally efficient. To begin with, it is assumed that the thermally averaged positions of the constrained atoms can be expressed as a finite-element interpolation of the positions of the representative atoms, i.e., using finite-element shape functions. This is analogous to the procedure followed in the standard QC method to determine the instantaneous positions of the nonrepresentative atoms. Moreover, the computation of Vcg is noticibly expedited when both the local harmonic approximation and the Cauchy-Born rule are taken into account. Under such circumstances, Vcg becomes... 

The assumption of a local harmonic approximation at an arbitrary point, go, on the potential energy surface, allows for a quadratic expansion of the surface in a small region around go, i-e.. 

The last term vanishes because of the orthonormality of the electronic wave functions, while the second term can be shown to be of higher order in the non-adiabatic coupling. The remaining term involves a transfer of momentum between electrons and nuclei. The expression is usually treated in the local harmonic approximation leading to a greatly simplified expression ... 

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