Gaussian functional integrals

McMurchie L E and Davidson E R 1978 One-and two-electron integrals over Cartesian Gaussian functions J. Comp. Phys. 26 218-31 Gill P M W 1994 Molecular integrals over Gaussian basis functions Adv. Quantum Chem. 25 141-205... 

Thus, in a two-electron integral of the form p,v a), the product < (1)< (1) (where 0 and may be on different centres) can be replaced by a single Gaussian function that is centred at the appropriate point C. For Cartesian Gaussian functions the calculation is more complicated than for the example we have stated above, due to the presence of the Cartesian functions, but even so, efficient methods for performing the integrals have been devised. 

Equation (1-15) is an analytical form that has a closed integral. The Gaussian function... 

The integral of the Gaussian function over the interval [a, b in a onedimensional probability space z is... 

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),... 

McMurchie LE, Davidson ER (1978) One- and two-electron integrals over cartesian gaussian functions. J Comp Phys 26 218... 

In order to obtain the vibrational populations, Gaussian functions were fitted to each vibrational peak and then integrated to determine the relative population of that peak relative to the whole curve. We recognize that the Gaussians sometimes do not accurately represent the vibrational populations due to the tail on the high energy side of the vibrational peak... 

The dependence of the proton resonance integral J for the unexcited vibrational states on the vibrations of the crystal lattice was taken into account recently in Ref. 47 for proton transfer reactions in solids. The dependence of J on the nuclear coordinates was chosen phenomenologically as an exponential Gaussian function. 

The various MO calculations use different basis sets and have different ways of calculating multicenter coulomb and exchange integrals. The current trend in MO is to expand as a linear combination of atomic orbitals (LCAO). The atomic orbitals are represented by Slater functions with expansion in gaussian functions, taking advantage of the additive rule. When the calculation is performed in this... 

Representation of each molecular orbital as a linear combination of atomic orbitals (atomic basis sets). Atomic basis sets are usually represented as Slater type orbitals or as combinations of Gaussian functions. The latter is very popular, due to a very fast algorithm for the computation of bielectronic integrals. 

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... 

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