Integrals over basis functions

The total energy (3.32) in term of integrals over basis functions is given as... 

In the scheme which we label (a), the same basis set is employed for both the ground and excited state. Therefore, the same integrals over basis functions are used for both states. The values of the even-tempered parameters a and /3 for M = 6 are those which were optimized for the ground state of the atom as reported by Schmidt and Ruedenberg [9]. These values are given in Table 1 of reference [9j. Larger basis sets were generated by means of the recursion formulae [10] ... 

The derivation of the second line of equation (A.81) follows the same reasoning as was used to obtain the one-electron part of the electronic energy [equation (A.21)], since both fi and h are sums of single-particle operators. The dipole moment integrals over basis functions in the last line of equation (A.81) are easily evaluated. Within the HF approximation, dipole moments may be calculated to about 10% accuracy provided a large basis set is used. 

Thus the integral ij kl) can be written as a sum of similar primitive integrals over basis functions which we can write... 

The primitive integrals over basis functions are, of course, just numbers which reflect the spinor structure of the primitive basis functions used in their determination. We shall reduce these atomic integrals still further in Section 4.3. The molecular case is discussed in Sections 5.3 and 5.4. 

We now have a collection of integrals over basis functions which must be evaluated in order to construct the Fock matrix. For S-spinors, these can be deduced from formulae given by Kim [95,37]. The main difference is that Kim was not aware of the importance of kinetic matching adaptation of his formulae is routine. The integrals involved are all related to the gamma function or the error function [83, Chapters 6 and 7]. We therefore concentrate here on G-spinors, which can be applied both to atomic and to molecular calculations. 

The notation involving the generic symbol <) is a little redundant when discussing electron-repulsion integrals only over the basis functions and there is no danger of confusion with integrals over other functions, so we drop the symbol 0 and refer to the integrals over basis functions by the simpler notation rs, tu). In this notation the above equalities become simply... 

The h and quantities are (first) derivatives of one- and two-electron integrals over basis functions. 

For any of the levels of theory considered in this Chapter, the total electronic energy corresponding to the Hamiltonian given in eq 2 and a wavefunction such as eq 4 or 5, can be written in terms of integrals over molecular orbitals or directly in terms of integrals over basis functions... 

Since each configuration D contains products of MOs, each of which is typically a sum of AOs, the integrals Hij and Sij can result in very large numbers of integrals over basis functions when they are expanded. This is the sort of situation where a computer is essential, and Cl on atoms and molecules, while still expensive compared to SCF, have become routine on modem computers. 

For integrals over basis functions i to / on atomic centers A to D, the following two-electron integrals are calculated analytically ... 

By suitably grouping closed- and open-shell terms, and expressing all integrals in terms of integrals over basis functions, we obtain an expression of the desired form but now every open-shell orbital must be regarded as an independent shell in itself. Thus... 

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