Basis function, Gaussian

Dupuis M, Rys J and King H F 1976 Evaluation of molecular integrals over Gaussian basis functions J. Chem. Phys. 65 111-16... 

McMurchie L E and Davidson E R 1978 One-and two-electron integrals over Cartesian Gaussian functions J. Comp. Phys. 26 218-31 Gill P M W 1994 Molecular integrals over Gaussian basis functions Adv. Quantum Chem. 25 141-205... 

Next, we shall consider four kinds of integrals. The first is the expectation value of the Coulomb potential by one nucleus for one of the primitive basis function centered at that nucleus. The second is the expectation value of the Coulomb potential by one nucleus for one of the primitive basis function centered at a different point (usually another nucleus). Then, we will consider the matrix element of a Coulomb term between two primitive basis functions at different centers. The third case is when one basis function is centered at the nucleus considered. The fourth case is when both basis functions are not centered at that nucleus. By that we mean, for two Gaussian basis functions defined in Eqs. (73) and (74), we are calculating... 

Chapter 10 represented a wave function as a linear combination of Gaussian basis functions. Today, there are so many basis sets available that many researchers will never need to modify a basis set. However, there are occasionally times when it is desirable to extend an existing basis set in order to obtain more accurate results. The savvy researcher also needs to be able to understand the older literature, in which basis sets were customized routinely. 

Gaussian Basis Functions for use in Molecular Calculations III Contraction of (10s, 6p) Atomic Basis Sets for the First-Row Atoms T. FI. Dunning, Jr... 

The latter approach has the advantage that the exact J is approached strictly from above, however for technical reasons it is only applicable if Gaussian basis functions are employed (Dunlap, Connolly, and Sabin, 1979). Both schemes are of course subject to the constraint that the fitted density is normalized to the total number of electrons, i. e ... 

Of the several approaches that draw upon this general description, radial basis function networks (RBFNs) (Leonard and Kramer, 1991) are probably the best-known. RBFNs are similar in architecture to back propagation networks (BPNs) in that they consist of an input layer, a single hidden layer, and an output layer. The hidden layer makes use of Gaussian basis functions that result in inputs projected on a hypersphere instead of a hyperplane. RBFNs therefore generate spherical clusters in the input data space, as illustrated in Fig. 12. These clusters are generally referred to as receptive fields. 

Appendix B Expansion of Cartesian Gaussian Basis Functions Using Spherical Harmonics... 

APPENDIX B EXPANSION OF CARTESIAN GAUSSIAN BASIS FUNCTIONS USING SPHERICAL HARMONICS... 

Spherical harmonics, Cartesion Gaussian basis functions, 261... 

Fast methods for evaluating these integrals for the case of gaussian basis functions are known [12], Also, Hall has described how to get the symmetry operators (B) 1SjB, r, for any crystal space group [13]. The parameters account for thermal smearing of the charge density. In this work I use the form recommended by Stewart [14],... 

We have used various integrators (e.g., Runga-Kutta, velocity verlet, midpoint) to propagate the coupled set of first-order differential equations Eqs. (2.8) and (2.9) for the parameters of the Gaussian basis functions and Eq. (2.11) for the complex amplitudes. The specific choice is guided by the complexity of the problem and/or the stiffness of the differential equations. 

One can also ask about the relationship of the FMS method, as opposed to AIMS, with other wavepacket and semiclassical nonadiabatic dynamics methods. We first compare FMS to previous methods in cases where there is no spawning, and then proceed to compare with previous methods for nonadiabatic dynamics. We stress that we have always allowed for spawning in our applications of the method, and indeed the whole point of the FMS method is to address problems where localized nuclear quantum mechanical effects are important. Nevertheless, it is useful to place the method in context by asking how it relates to previous methods in the absence of its adaptive basis set character. There have been many attempts to use Gaussian basis functions in wavepacket dynamics, and we cannot mention all of these. Instead, we limit ourselves to those methods that we feel are most closely related to FMS, with apologies to those that are not included. A nice review that covers some of the... 

Since our Gaussian basis functions depend on the electronic (Xg) and nuclear (A ) coordinates only in the special combination ( ) = ( > Xe - Xj ), one can write... 

Gaussian Basis Set. A Basis Set made up of Gaussian Basis Functions. 

In all calculations, standard Gaussian basis functions are used to construct the wave function for each specific diabatic state. For comparisons purposes, basis sets ranging from 3-21G to aug-cc-pVTZ have been used. Specific details on the choice and definition ofdiabatic states are given below for each individual case. 

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