Gaussian primitives, linear combination

The d-type functions that are added to a 6-31G basis to form a 6-3IG basis are a single set of uncontracted 3d primitive Gaussians. For computational convenience there are six 3d functions per atom—3dyy, 3d , 3d,y, 3dy2, and 3d x- These six, the Cartesian Gaussians, are linear combinations of the usual five 3d functions—3d y, 3d 2 y2, 3dy, 3d, and 3d i and a 3s function + z ). The 6-3IG basis, in addition to adding... 

The contracted Gaussian functions are a linear combination of the primitive Gaussian functions. That is,... 

The actual basis functions are formed as linear combinations of such primitive gaussians ... 

The columns to the right of the first vertical line of asterisks hold the exponents (a above) and the coefficients (the d p s) for each primitive gaussian. For example, basis function 1, an s function, is a linear combination of six primitives, constructed with the exponents and coefficients (the latter are in the column labeled S-COEF ) listed in the table. Basis function 2 is another s function, comprised of three primitives using the exponents and S-COEF coefficients from the section of the table corresponding to functions 2-5. Basis function 3 is a p function also made up or three primitives constructed from the exponents and P-COEF coefficients in the same section of the table ... 

Linear combinations of primitive gaussians like these are used to form the actual basis functions the latter are called contracted gaussians and have the form ... 

The next step was to represent each Slater atomic orbital as a fixed linear combination of Gaussian orbitals so a Slater-type orbital with exponent f is written as a sum of GTOs with exponents a, q 2, and so on. For example, in the case of three primitive GTOs we might write... 

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),... 

IG and 6-3IG. These are commonly used split-valence plus polarization basis sets. These basis sets contain inner-shell functions, written as a linear combination of six Gaussians, and two valence shells, represented by three and one Gaussian primitives, respectively (noted as 6-3IG). When a set of six d-type Gaussian primitives is added to each heavy atom and a single set of Gaussian p-type functions to each hydrogen atom, this is noted as and... 

However, most wave function based calculations also contain a semiempirical component. For example, the primitive Gaussian functions in all commonly used basis sets (e.g., the six Gaussian functions used to represent a li orbital on each first row atom in the 6-3IG basis set) are contracted into sums of Gaussians with fixed coefficients and each of these linear combinations of Gaussians is used to represent one of the independent basis functions that contribute to each AO. The sizes of the primitive Gaussians (compact versus diffuse) and the coefficient of each Gaussian in the contracted basis functions, are obtained by optimizing the basis set in calculations on free atoms or on small molecules." ... 

The correct limiting radial behavior of the hydrogen-like atom orbital is as a simple exponential, as in (A.62). Orbitals based on this radial dependence are called Slater-type orbitals (STOs). Gaussian functions are rounded at the nucleus and decrease faster than desirable (Figure 2.2b). Therefore, the actual basis functions are constructed by taking fixed linear combinations of the primitive Gaussian functions in such a way as to mimic exponential behavior, that is, resemble atomic orbitals. Thus... 

Another interesting possibility, which will not be discussed here, may consist into expanding the radial factors e and re as a linear combination of primitive gaussian functions [69]. Studies about this simpler approximation is imder way in our Laboratory. 

The Gaussian lobe function method was introduced by Preuss and developed for routine calculations by Whitten. The contraction of Gaussian lobe function (GLF) basis sets is made along the same lines as with Cartesian GTF s. As regards the primitives, the s-type functions are expressed in the usual way. But the primitives of p, d, f,. .. types are expressed as linear combinations of s-Gaussians (lobe functions) placed at different points so as to retain the proper symmetry (see Fig. 2,4 ). Thus, a p-type function on nucleus A may be... 

For computational reasons only (particularly for ab initio calculations), the atomic orbitals are usually expressed as a linear combination of primitive functions centered on the atom (Eq. [3]). These primitives are typically Gaussian-type functions, but Slater-type functions have also been used. 

For a given molecular structure characterised by initial nuclear coordinates the basis set functions are attributed to each atom. These simulate the behaviour of atomic orbitals and are given in terms of a (fixed) linear combination of the primitive (usually Gaussian type) functions p as follows... 

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