Floating-center basis functions

It can be shown that the basis of spherical explicitly correlated Gaussian functions with floating centers (FSECG) form a complete set. These functions... 

Hence, if we are to make bE = 0, we can avoid these terms. But to do so we have to have E optimum with respect to the location of the atomic basis functions, t (R) the MO coefficients, c(R) and the Cl coefficients, C(R). The first cannot be satisfied unless the atomic orbital basis set is floated off the atomic centers to an optimum location [105], while the second requires optimum MO coefficients, and the third optimum Cl coefficients. In practice, we will introduce atomic orbital derivatives explicitly, so the AOs can follow their atoms. Now focusing only on the MO and Cl coefficients, in SCF we have optimum MOs and no Cl term. In MCSCF, both terms would vanish, whUe in Cl, the MO derivatives would remain, but the Cl coefficients contribution would vanish. In the non-variational coupled-cluster theory, neither will vanish and this means that CC theory forces us into some new considerations for analytical forces. 

There are two important parameters for the present method to be numerically successful. One is the position of the boundary at which the electron flux is calculated, and the other is the number of basis functions. The theory should work better as the boundary radius (the distance between the boundary and the molecular center) is taken to be longer. In fact, exact photoelectron spectroscopic data should be obtained if a complete basis set (including plane waves or outgoing spherical waves) could be used within a very large boundary surface. On the other hand, the longer is the boimdary radius, the more diffused basis functions and/or floating functions localized at many places are required, and consequently the more difficult becomes the computation. Therefore we have to optimize performance in these two factors at the level of accuracy required. 

Furthermore, one could most probably use only two different basis sets, one for the large and one for the small component as a function of r, because the angular part of the functions is fixed in the case of an H atom. On the other hand, to maintain the possibility of basis functions not centered at the atomic nuclei (such as function centers at the midpoints of chemical bonds, or floating Gaussians), we prefer to work with four different basis sets. In a practical case the four basis sets could be equal,... 

When the size of the nuclear framework of the molecule or cluster is small compared to the extent of the diffuse basis functions, it probably does not matter much where the floating center is positioned. In applications to (H20) , for example, Sommerfeld et al. place a single set of diffuse functions on one oxygen atom, using an even-tempered progression out to a maximum FWHM of 80 A. It is reported that the VDE depends only weakly on which oxygen atom is chosen as the center of this expansion. 

In a large molecule or cluster, however, a single floating center cannot be expected to replace atom-centered diffuse basis functions. In the absence... 

The development of the basis sets with off-centered functions aimed at avoiding use of higher spherical harmonics (i e. d, f,. .. type functions) without losing the basis set flexibility. Actually, the lobe function and floating spherical function basis sets are constructed only from s-type functions. Computationally, this restriction is very advantageous..The evaluation of integrals over s-type functions is... 

Work by Frost and co-workers in the mid 1960s abandoned the idea of AO-based functions to arrive at an even more compact basis set. They suggested the use of s-type Gaussians that were not fixed at the atomic centers, but could float in space so as to optimally represent each localized pair of electrons. Because only one function was needed for each pair of electrons, the basis sets used in floating spherical Gaussian (FSGO) scheme were often referred to as subminimal. Extensions of Frost s idea to ellipsoidal Gaussians of the form... 

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