Spherical Gaussian functions

I hi additional terms are spherical Gaussian functions with a width determined by the parameter L. It was found that the values of these parameters were not critical and many... 

Comparing the core-core repulsion of the above two equations with those in the MNDO method, it can be seen that the only difference is in the last term. The extra terms in the AMI core-core repulsion define spherical Gaussian functions — the a, b, and c are adjustable parameters. AMI has between two and four Gaussian functions per atom. 

We define a spherical Gaussian function (SGTF) centred at the origin... 

Additionally, from Fig. 29 one sees that, if, as proposed by Frost 42), a spherical gaussian function is a fair representation of the distribution of charge within an electride ion, there should he, as found by Slater 97>, a very good correlation, and in many cases practically an equality, between the atomic radii. . . and the calculated radius of maximum radial charge density in the outermost shell of the atom". 

For the expansion of the Hartree-Fock molecular orbitals we have used either Slater or Cartesian Gaussian functions. In addition to these basis functions we can also include spherical Gaussian functions in the initial scattering basis. A detailed discussion of the single-center expansion of Slater and Cartesian Gaussian functions has been given by Harris and Michels (19) and by Fliflet and McKoy (20), respectively. Spherical Gaussian functions, i.e.. 

The basis sets that we have considered thus far are sufficient for most calculations. However, for some high-level calculations a basis set that effectively enables the basis set limit to be achieved is required. The even-tempered basis set is designed to achieve this each function m this basis set is the product of a spherical harmonic and a Gaussian function multiplied... 

Here, n corresponds to the principal quantum number, the orbital exponent is termed and Ylm are the usual spherical harmonics that describe the angular part of the function. In fact as a rule of thumb one usually needs about three times as many GTO than STO functions to achieve a certain accuracy. Unfortunately, many-center integrals such as described in equations (7-16) and (7-18) are notoriously difficult to compute with STO basis sets since no analytical techniques are available and one has to resort to numerical methods. This explains why these functions, which were used in the early days of computational quantum chemistry, do not play any role in modem wave function based quantum chemical programs. Rather, in an attempt to have the cake and eat it too, one usually employs the so-called contracted GTO basis sets, in which several primitive Gaussian functions (typically between three and six and only seldom more than ten) as in equation (7-19) are combined in a fixed linear combination to give one contracted Gaussian function (CGF),... 

These permutations on coordinates are equivalent to operations on the basis functions. We will use shifted spherical Gaussians for this example (these functions will be discussed in a detailed way below in this chapter) ... 

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

One vexed question concerning polarization sets is the number of functions in a given shell, as discussed elsewhere. The early quantum chemistry codes employed Cartesian Gaussian functions, so that a d set actually comprises the five spherical harmonic d functions and a 3s function generally termed a contaminant. The reason for this emotive terminology is that with multiple polarization sets the contaminants,... 

Double Zeta + Polarization functions Extended Hartree-Fock Electron Spectroscopy for Chemical Analysis Floating Ellipsoidal Gaussian Orbital Floating Spherical Gaussian Orbital Generalized Atomic Effective Potential Gaussian Type Orbital... 

To obtain a closed expression for A2, suitable for all values of z, two types of theories have been developed by several authors in recent years. The first type of theory is based on the uniformly expanded chain model and on a spherically symmetrical distribution of segments about the molecular center of mass. The segment distribution is taken to be a spherical cloud of constant density in Flory s first theory 101), a Gaussian function about the center of mass in Fi.ory and Kkigbaum s (103 ) and in Orofino and Flory s (204) theories, and a sum of N different Gaussian functions in Isihara and Koyama s theory (132 ). All of these theories may be summarized in the following type of equation given by Orofino and Flory,... 

Each component of generally contracted with Gaussian type spherical harmonics functions. Contraction coefficients of the basis sets are determined by four-component atomic calculations [5],... 

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