Floating basis function

If the Hellmann-Feynman theorem is to be valid for forces on nuclei, the Coulomb cusp condition must be satisfied. However, if the nuclei are displaced, the orbital Hilbert space is modified. Hurley [179] noted this condition for finite basis sets, and introduced the idea of floating basis functions, with cusps that can shift away from the nuclei, in order to validate the theorem for such forces. 

An interesting approach for comparison with XPS experiments is a generalization of the Floating Spherical Gaussian Orbital (FSGO) technique. In this method, each electron pair is represented by a single Gaussian basis function whose exponent and position are obtained by a variational procedure or by reference... 

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. 

This section collects results obtained with the three exact-decoupling methods within the same implementation and follows the discussion in Ref. [647]. The number of matrix operations necessary for the implementation of different two-component approaches has been collected in Table 14.2. The multiplication of a general matrix with a diagonal matrix requires O(m ) multiplications of floating-point numbers, where m is the dimension of the matrix identical to the number of (scalar) basis functions in this context. The multiplication of two general matrices scales formally as If m is large, the cost of the... 

Specifically, floating spherical Gaussian orbitals (FSGO), developed from an extension of Frost s simple electron pair model of molecular electronic structure, are used as basis functions in SCF-MO and Cl calculations. As will be described in the following section, these functions, unlike atomic basis orbitals, generally are not confined to atoms, but are allowed to occupy electron-rich regions corresponding... 

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,... 

A second important generalization, first introduced by Gerratt (1967) and Gerratt and Mills (1968), allows the finite basis itself to depend on the perturbation in the presence of an electric field, for example, each ba function may acquire a polarization or if the nuclei are moved, the basis functions may follow them, or possibly float away from them. 

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... 

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