Basis function primitive

To incorporate the angular dependence of a basis function into Gaussian orbitals, either spherical haimonics or integer powers of the Cartesian coordinates have to be included. We shall discuss the latter case, in which a primitive basis function takes the form... 

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... 

Figure 4-7 The SCF energy of the neon atom converges exponentially with the number of Gaussian primitive basis functions.

This requires that the eigenfunctions of the Hamiltonian are simultaneously eigenfunctions of both the Hamiltonian and the symmetric group. This may be accomplished by taking the basis functions used in the calculations, which may be called primitive basis functions, and projecting them onto the appropriate irreducible representation of the symmetric group. After this treatment, we may call the basis functions symmetry-projected basis functions. 

The above operators apply only to primitive basis functions that have the spin degree of freedom included. In the current work we follow the work of Matsen and use a spin-free Hamiltonian and spin-free basis functions. This approach is valid for systems wherein spin-orbit type perturbations are not considered. In this case we must come up with a different way of obtaining the Young tableaux, and thus the correct projection operators. 

It is possible to use as the basis functions symmetry-adapted combinations of primitive basis functions. This affords a decomposition of the orbital interaction energy of Eq. [21] according to irreducible representations of the point group... 

It is also possible to perform a basis set transformation from primitive basis functions to symmetry combinations of the KS MOs of the atoms or larger fragments that constitute a system. In that case the population matrix elements P v become more meaningful, because they reflect the involvement of the fragment MOs in the orbitals of the total system. A Mulliken population analysis in... 

The BLW method can be considered as an extension of the orbital deletion procedure (ODP) (51,52), a simpler method that can only be applied to carbocations (52) and boranes (51). The ODP consists of representing a resonance structure displaying an electronic vacancy (Lewis acid character) by deleting the primitive basis functions corresponding to the empty site before launching the SCF calculation. As a typical example, the ODP has been applied to calculate the resonance energy of the allyl cation (52). 

Noting from (8.353) that each primitive basis function in case (a) is of the form... 

It is now a simple matter to use the above results for the primitive basis functions to generate matrix elements for the parity-conserved basis. For the positive-parity states the hyperfine matrix is as follows. 

E Matrix elements of the truncated harmonic potential F Matrix elements in Gaussian primitive basis functions... 

We turn next to examine the general structure of the matrix elements of the Hamiltonian operator H in terms of the basis functions of the first zone. A discussion of the matrix elements of H in terms of the primitive basis functions is found in Appendix F. 

Matrix elements of the other primitive basis functions are readily obtained by differentiation. It is possible to demonstrate that the form of the matrix elements is very much the same as for the simpler matrix elements that arise with the Morse potential. Thus, one finds for... 

Va . Then, in terms of the ground state primitive basis functions, the single center matrix element is given by... 

The cell that contains the origin of coordinates for the whole system is used as the reference cell. The two-center matrix element that involves only the ground state primitive basis functions < >0 is... 

The two-center matrix elements in the second primitive basis functions are... 

F Matrix elements in Gaussian primitive basis functions... 

An expansion of the Morse potential, for example, in a set of Gaussian functions is given by eq (C4) in Appendix C. Matrix elements of the Morse potential in terms of the Gaussian primitive basis functions are therefore simply three center overlap integrals [49], These matrix elements can be evaluated for each term in the sum and then converted to the final expression in a straightforward manner. 

A set of primitive basis functions with the same center and exponent are known as a primitive shell. For example, a set of p-functions [px, py,pz] on an atom is termed a primitive p-shell and, if an s-function (with the same exponent) is added, the shell becomes a primitive ip-shell. The most commonly occuring shells in modern computational chemistry are s, p, sp, d and/. 

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