Gaussian integration

Because of the work involved in solving large systems of simultaneous linear equations it is desirable that only a small number of us be computed. Thus the gaussian integration formulas are useful because of the economy they offer. See references on numerical solutions of integral equations. 

A more vexing issue is that one of the e in (3.37) equals zero. To see this, note that the function Xins(t) [where Xjns is the instanton solution to (3.34)] can be readily shown to satisfy (3.37) with o = 0. Since the instanton trajectory is closed, it can be considered to start arbitrarily from one of its points. It is this zero mode which is responsible for the time-shift invariance of the instanton solution. Therefore, the non-Gaussian integration over Cq is expected to be the integration over... 

A similar expansion can be written in the vicinity of Q = 0. Path integration amounts to the Gaussian integration over the Q , whereas the integration over the unstable mode Qq is understood as described in section 3.3. In that section we also justified the correction factor (f) = T /T = X l2n which should multiply the Im F result in order to reproduce the correct high-temperature behavior. Direct use of the Im F formula finally yields... 

INPUFF is a Gaussian integrated puff model which is captible of addressing the accidental reletisc of a substance over several minutes or of modeling the more typical continuous plume from a stack. 

A more detailed calculation requires performing the Gaussian integration over the disorder realizations close to the saddle-point configuration Eq. (3.25). One then finds the following expression for the average density of electron states per unit length,... 

The integral can be further simplified using the following property of Gaussian integrals for any matrix A and B ... 

An alternative set of functions used to build up atomic orbitals are Gaussian functions that have a radical dependence expf-ftr2). A linear combination of these functions gives a reasonable representation of an atomic orbital. The functions are used for computational expediency in ab initio calculations, because four-center Gaussian integrals can be reduced to two-center integrals, which are relatively easy to calculate on a computer (21). 

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... 

This again amounts to a Gaussian integral which yields... 

The r.h.s. of Eqs. (A3.10), (A3.12) are well defined also foT nonintegelr d and can be taken to give a meaning to the Gaussian integral in nonintegcr dimensions. It will turn out to be important to define the theory for arbitrary... 

In the functional integral (A 5.12) we may shift y (r) as in normal Gaussian integrals. Substituting... 

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 ... 

The (VS)2 term was transformed by partial integration, assuming S(r) to vanish at the boundaries of the integration volume. Equation (A 7.21) has the form of a functional Gaussian integral yielding... 

The integral over q is a gaussian integral that yields to the final result... 

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.

In the CRAY-1 a Gaussian integrals program may be driven at 35 Mflops, a large closed shell SCF with a sparse list of integrals or Supermatrix at 10 Mflops, while a smaller SCF with a non-sparse Supermatrix may be driven at 135 Mflops. 

---