Primitive Gaussians

The remainder of the input file gives the basis set. The line, 1 0, specifies the atom center 1 (the only atom in this case) and is terminated by 0. The next line contains a shell type, S for the Is orbital, tells the system that there is 1 primitive Gaussian, and gives the scale factor as 1.0 (unsealed). The next line gives Y = 0.282942 for the Gaussian function and a contiaction coefficient. This is the value of Y, the Gaussian exponential parameter that we found in Computer Project 6-1, Part B. [The precise value for y comes from the closed solution for this problem S/Oir (McWeeny, 1979).] There is only one function, so the contiaction coefficient is 1.0. The line of asterisks tells the system that the input is complete. 

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 ... 

Note that functions 1 through 9 form the heart of the 6-31G basis set three sets of functions formed from six, three and one primitive gaussian. 

The coefficients specified for the component primitive gaussians are chosen so that the resulting constructed basis functions are normalized. This means that one coefficient in each set is effectively constrained so that this condition is fulfilled. 

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 ... 

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 primitive Gaussians form the actual basis functions which are called contracted Gaussians. The atomic orbitals are then expressed as ... 

Gaussian functions are of the type exp(—ar2) and since they produce integrals that are easier to evaluate they are often preferred to STO s. In some applications Gaussian functions are used to approximate STO s, e.g. in the STO-3G method, three primitive Gaussians are used to approximate... 

On the convergence of the many-body perturbation theory second-order energy component for negative ions using systematically constructed basis sets of primitive Gaussian-type functions... 

Using the F ion as a prototype, the convergence of the many-body perturbation theory second-order energy component for negative ions is studied when a systematic procedure for the construction of even-tempered btisis sets of primitive Gaussian type functions is employed. Calculations are reported for sequences of even-tempered basis sets originally developed for neutral atoms and for basis sets containing supplementary diffuse functions. 

The Hartree-Fock ground state of the F anion is described by orbitals of s Emd of p symmetry. In the first part of this study, attention was restricted to the convergence of the second order many-body perturbation theory component of the correlation energy for stematically constructed even-tempered basis sets of primitive Gaussian-typ>e functions of s and p symmetry. 

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