Gaussian function approximation

In a more detailed study of technique validation and of the comparison of different similarity evaluation methods [1065], several Carbo indices (also using Gaussian function approximations), the Hodgkin index (eq. 213),... 

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]. 

While it is not essential to the method, frozen Gaussians have been used in all applications to date, that is, the width is kept fixed in the equation for the phase evolution. The widths of the Gaussian functions are then a further parameter to be chosen, although it appears that the method is relatively insensitive to the choice. One possibility is to use the width taken from the harmonic approximation to the ground-state potential surface [221]. 

As usual there is the question of the initial conditions. In general, more than one frozen Gaussian function will be required in the initial set. In keeping with the frozen Gaussian approximation, these basis functions can be chosen by selecting the Gaussian momenta and positions from a Wigner, or other appropriate phase space, distribution. The initial expansion coefficients are then defined by the equation... 

To calculate the matrix elements for H2 in the minimal basis set, we approximate the Slater Is orbital with a Gaussian function. That is, we replace the Is radial wave function... 

Having obtained a mediocre solution to the problem, we now seek to improve it. The next step is to take two Gaussian functions parameterized so that one fits the STO close to the nucleus and the other contributes to the part of the orbital approximation that was too thin in the STO-IG case, the part away from the nucleus. We now have a function... 

Minimal basis sets use fixed-size atomic-type orbitals. The STO-3G basis set is a minimal basis set (although it is not the smallest possible basis set). It uses three gaussian primitives per basis function, which accounts for the 3G in its name. STO stands for Slater-type orbitals, and the STO-3G basis set approximates Slater orbitals with gaussian functions. ... 

The method presented here allows, starting with trial gaussian functions, a partial analytical treatment which we have used to improve the LCAO-GTO orbitals (trial functions) essentially obtained from all ab initio quantum chemistry programs. As in r-representation, trial functions (t>i( Hp) (Eq. 21) are conveniently expressed as linear combinations of m functions Xi(P) themselves written as linear combinations of Gt gaussian functions (LCAO-GTO approximation) gta(P). 

At first sight, we are foreed to solve this equation numerically, but its overall form allows a qualitative insight into the number of solutions and their approximate values. For example, one easily see that S represents a sum of two identical quasi-atomic (onedimensional) functions each centered on the corresponding hydrogen nucleus. The functions are quite similar to 2pz Gaussian functions, but they differ by their one-dimensionality and by a different radial dependence. Indeed, instead of the usual... 

In fact, we have already used a modeling strategy when Po(AU) was approximated as a Gaussian. This led to the second-order perturbation theory, which is only of limited accuracy. A simple extension of this approach is to represent Pq(AU) as a linear combination of n Gaussian functions, p, (AU), with different mean values and variances [40]... 

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