Gaussian integral evaluations

The CRAY FORTRAN compiler, CFT, sometimes produces nearly optimal code (as for example in Gaussian integrals evaluation) sometimes an increase in speed by a factor of 3 is observed on proceeding to the assembler, CAL. Hiis factor of 3 is often present in rather simple applications such as matrix multiply. 

One of the major selling points of Q-Chem is its use of a continuous fast multipole method (CFMM) for linear scaling DFT calculations. Our tests comparing Gaussian FMM and Q-Chem CFMM indicated some calculations where Gaussian used less CPU time by as much as 6% and other cases where Q-Chem ran faster by as much as 43%. Q-Chem also required more memory to run. Both direct and semidirect integral evaluation routines are available in Q-Chem. 

To some other experts the meaning of the term ab initio is rather clear cut. Their response is that "ab initio" simply means that all atomic/molecular integrals are computed analytically, without recourse to empirical parametrization. They insist that it does not mean that the method is exact nor that the basis set contraction coefficients were obtained without recourse to parametrization. Yet others point out that even the integrals need not be evaluated exactly for a method to be called ab initio, given that, for instance, Gaussian employs several asymptotic and other cutoffs to approximate integral evaluation. 

With our model the evaluation of PtJ(r n) is an exercise in doing Gaussian integrals. (See Appendix A 3.3.) Wo only note here the results, leaving the explicit calculation to Sect. 3.3. We find... 

The Gaussian integral over is easily carried out. Using Stirling s formula to evaluate Mnl after a little calculation we find an expression for the density of the free energy in the large volume limit ... 

Integral Evaluation. We here confine our attention to the evaluation of integrals over Cartesian Gaussians (V7.) A seen in Table IV, the essentially unmodified FORTRAN source of the ATMOL3 Gaussian Integrals program compiled on the CRAY runs at approximately 6.2 times faster than the IBM 370/165 (circa. 2.4 times... 

Figure 2. Orthodox procedure for evaluating Gaussian integrals.

The integral over Ry can be evaluated by converting it into a Gaussian integral (by writing the complex representation of sine and completing the square in the exponent) ... 

This Gaussian integral can be evaluated in a straightforward way. To get some physical insight consider the result obtained from the initial wavepacket with xo = po = Ao = 0. In this case (2.111) yields... 

---