Gaussian-type orbital definition

The set made by the three p-type functions in a Gaussian type orbital (GTO) framework is sufQciently simple to be used as an example but at the same time provides a yet unexplored context, which can show how QS techniques can handle quantum system states, even degenerate ones. The set of p-type Gaussian functions can be collected into a vector, which can be associated in turn to three degenerate model wavefunctions Ip) = qr exp(-alrP) where q is a normalization factor, r = (x,y,z), and a is an arbitrary positive-definite parameter. The set of attached DF can be also written as a vector Id) = q (r r) exp (-2alrP), where the position vector product symbol is defined employing the inward product (r r) = Thus, the components... 

The third step is the VB calculation itself. Hereafter, we only consider the 14 valence electrons and the 26 basis functions that are kept for the definition of the VB orbitals, in the order 2S, 2PX, 2PY, 2PZ, 3S, 3PX, and so on, as in the gaussian program (see Table 10.1). There are eight VB orbitals, the definitions of which are specified in the section orb . In this section, the first line indicates the number of basis functions over which each VB orbital is being expanded. These basis functions are specified in the following lines, for each VB orbital. The VB orbital 1 can be expanded on the basis functions 1, 4, 5, 8, 9, that is, the 2S, 2PZ, 3S, 3PZ, and DO basis functions of the Fa atom. The second VB orbital can span the basis functions 2PX, 3PX and D+1 on Fa, it is therefore an orbital of tt type, and so on. Note that the numbering of the basis functions, in this step, corresponds to the basis functions that are kept for the VB calculations. This numbering is therefore different from the one that appears in the Hartree—Fock output. 

By choosing the contracting scheme, it is possible to determine the contracted gaussian functions which best approximate Slater functions,88 Hartree-Fock atomic orbitals,89-90 etc. It should be noted that the size and balance (revealed, for example, by the optimum ratio of the number of s-type and p-type functions) of the basis are definite factors which control the quality of the final result (e.g., ref. 90). 

If the group is rotational or helical and ij> is not 5-type, then the />, on each site become linear combinations of basis functions related by the rotation matrix of the appropriate angular momentum and the appropriate rotational or helical step angle [27]. It is traditional to use Cartesian-Gaussian orbital basis sets in quantum-chemical calculations [28], but solid-spherical-harmonic Gaussians [29] are best for symmetry adaption and matrix element evaluation. Including an extra factor of (-)M in the definition of the solid spherical harmonics [30]... 

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