Gaussian product

The systematic reduction of multi-centre integrals relies on the well-known relation 

Using either elementary algebra or Taylor s theorem we can expand this about any point P, giving 

However, following [109], it proves more convenient to expand a CGTF product in HGTF rather than CGTF. For simplicity, we first consider the one dimensional case, 

Similar equations can be written down for the other two Cartesian directions, so that 

The small number of coefficients needed can be pre-tabulated and held in memory, and we retain the computational simplicity of the Cartesian formulation along with the vital transformation properties of the spherical Gaus-sians. The coefficients [A, B p, P , i, j,k , i, j, kf ,s,t, ] are simple to construct, and the accumulation of sums like (208) can mostly be done in integer aritb-metic. The extensive cancellation which occurs for higher angular momentum spinors can therefore be done exactly without rounding error. The computational bottlenecks encountered in our preliminary work with the complex recurrence relations for direct constraction of Eg[A,B ,p,P ,n,l,m ,r, 1, m s,t,u] given by [107] are completely eliminated. The calculation of these coefficients and the spinor coefficients of the next section now constitutes a trivial part of the computational load. 

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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... 

Figure 1.13 Demonstration of the validity of equation 1.16 for the Gaussian product of column D formed by multiplication of the projections of the Gaussians in columns B and C following equations 1.16 to 1.19. Note, the cells bordered in black in figl-13.xls can be changed even with the spreadsheet locked .

The next instructions introduce the general simplification of Is Gaussian products of equations 1.16 to 1.19. 

Form the required product of the coefficients needed in the Gaussians products in row 12 again, for example... 

Now, establish the primary row and column of the two-variable table. Repeat the labels for the individual Gaussian products in row 15 and then enter the summed exponents for each of these in the next row, for example... 

Provide for a tw o-vahable table calculation in rows 10 to 31 of the spreadsheet, with the input row data, the sums of the exponents for the integrations over the primitive Gaussian products and the input column data, the products of the coefficients in the primitive Gaussian products. Make cells H 4 and I 4 the master cells for the macro. 

Pair-wise reduction of Gaussian products following equation 1.16 to 1.19 about the intermediate positions P and Q, leads to... 

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... 

Mendive-Tapia D, Lasome B, Worth GA, Robb MA, Bearpark Ml (2012) Towards converging non-adiabatic direct dynamics calculations using frozen-width variational Gaussian product basis functions. J Chem Phys 137 22A548... 

Finally, some comments on two recent studies on methods for quantum dynamics simulations. In the first study, by Mendive-Tapia et al., the convergence of non-adiabatic direct dynamics in conjunction with frozen-width variational Gaussian product basis functions is evaluated. The simulation of non-adiabatic dynamics can be subdivided into two groups semi-classical methods (like the trajectory surface hopping approach) and wavepacket methods (for example, the... 

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... 

The transfer equation can also shift angular momentum from centers A and B to the center of the Gaussian generated by the Gaussian product, P, that is... 

Let us finally consider the Gaussian product rule for a simple product of two s functions... 

Fig. 9.2. The Gaussian product rule. The shaded areas represent the product of the two individual Gaussians. In the upper plots, the exponents are equal to 1 and the two Gaussians are centred at 5 and 1 in the lower plots, the exponents are (for the Gaussian at the origin in both plots) and 25 (for the Gaussian centred at 0 and 1).

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