Gaussian Product Theorem

In reference [19] a General Gaussian Product Theorem is described, which allows us to obtain a convenient algorithm in order to contract the exponential part of each of the GTO functions involved... 

Use the GTF product. Use the Gaussian product theorem to reduce the two basis-function products to linear combinations of GTFs centred on just two points... 

Evaluation of the primitive ERIs, which are six-dimensional integrals, commences with application of the Gaussian product theorem, which yields... 

The Gaussian product theorem is a particularly useful relation because it demonstrates that the product of two Gaussians can be expressed as a new Gaussian (see Figure 1). It is worthwhile to note that a similar relation does not exist for the Slater-type basis sets. This is the most important reason Gaussian-type basis sets are more useful than Slater-type basis sets in computational chemistry when applied to polyatomic systems (some codes continue to use Slaters for diatomic systems). The Gaussian product theorem states that the product of two Gaussians can be expressed as... 

If two Gaussian functions are convolved, the result is a gaussian with variance equal to the sum of the variances of the components. Even when two functions are not Gaussian, their convolution product will have variance equal to the sum of the variances of the component functions. Furthermore, the second moment of the convolution product is given by the sum of the second moments of the components. The horizontal displacement of the centroid is given by the sum of the component centroid displacements. Kendall and Stuart (1963) and Martin (1971) provide helpful additional discussions of the central-limit theorem and attendant considerations. 

We also note that the derivatives dr /ds constitute a Gaussian random set (see Appendix A). Therefore the mean value of any product of these dr>(s)/ds can be calculated from (2.1.25) by applying Wick s theorem (see Appendix A). 

In the framework of the Gaussian approximation, the quantities A(n) determine the properties of the chain completely. Indeed, all mean values of products of components of the uj can be expressed in terms of the A(n) by means of Wick s theorem (see Appendix A). In particular, we have... 

However, for Fermi superoperators life is more complicated. The anticommutator corresponding to only the left or the right Fermi superoperators are numbers but that for the left and right superoperators, in general, is not a number. Thus, the Fermi superoperators are not Gaussian. However, since the left and right superoperators always commute, the following Wick s theorem [49] can be applied to the time-ordered product... 

Gaussian elimination can be used to efficiently evaluate a determinant, as follows. Divide each element of row 1 of the determinant (8.20) by an and place the factor Un in front of the determinant (Theorem IV of Section 8.3). Then subtract the appropriate multiples of the row 1 elements from row 2, row 3,..., row n to make fl2i. < 3i. . < ni zero (Theorem V). Then divide the second-row elements by the current value of 022 and insert the factor 022 in front of the determinant, and so on. Ultimately, we get a determinant all of whose elements below the principal diagonal are zero. From Problem 8.16, this determinant equals the product of its diagonal elements. Use this procedure to evaluate the determinant in Problem 8.17. 

Here we use the easily proven theorem that a product of two Gaussians can always be written as a single Gaussian multiplied by a constant. The exponent in the integrand of Equation A.7 may thus be written as ... 

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