Evaluation of a PPD

As shown in the last section, Hamiltonian and overlap matrix elements are expressed in terms of PPDs. A practical VB package highly depends on an efficient routine for the evaluation of a PPD. Although a PPD may be expressed in terms of sub-PPDs of any given order and their complementary minors, in the present version of Xiamen-99, an algorithm of 2x(V-2) expansion is used. This is because the 1-e and 2-e electron integrals may be built as effective 2x2 PPDs. 

A procedure for the evaluation involves two parts one being the numerical operations of matrix elements, the other being the index operations of the sub-PPDs. It is obvious that the index operation is independent of the system that is being studied. To save CPU time in VB applications, all index operations are pre-computed and stored in the file that accompanies the source code of the package. In addition, all sub-PPDs that are required in the evaluation are computed first and are labeled. This will enable one to avoid repeated computations of sub-PPDs and minimize the computational effort in the calculation. 

From Eq. (50), an overlap matrix element is exactly a PPD and can easily be evaluated from the routine for PPDs, while Hamiltonian matrix elements may be obtained by a similar routine to that for PPDs, where 2x2 sub-PPDs are replaced with effective sub-PPDs of one-electron and two-electron integrals. 

As mentioned in Section 1, in a traditional VB treatment, a VB wavefunction is expressed as the linear combination of 2m Slater determinants, where m is the number of covalent bonds in the system. For some applications in which only a few bonds are involved in the reaction, it is too luxurious to adopt the PPD algorithm, as the number of Slater determinants is still not too large to deal with. It would be more efficient to use a traditional Slater determinant expansion algorithm than the PPD algorithm. Therefore, as a complement, a Slater determinant expansion algorithm is also implemented in the package. 

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Generally, Eq. (45) is much more troublesome than Eq. (42) and it is not essential to the evaluation of a PPD. However, it is of great importance to the application in the VB approach. Using Eq. (45) successively, one can expand a PPD by taking a determinant with order 2 as a minor [39], which is also required in the VB approach. 

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