Further developments for simple systems

The former solution for the hydrogen atom illustrates both the power of the GCM and the mathematieal diffieulty of its application. Thus, most nuclear, atomic and molecular applications relied on either numerical techniques, approximations, or both. StiU, simple systems have always been models for developing new techniques which, when successful, may show new routes for more complicated cases. In this section, we introduce some formal refinements and we continue to rely on the hydrogen atom and the Gaussian function to explore some numerical experiments. 

Once the kernels (4) and (5) are calculated, the HW equation (3) becomes an integral equation, from which we should be able to obtain both the weight function and the energy E. Discretization techniques (see Section 5) could be efficient, but alternative mathematical tools should be tested. 

Let us first examine where integration by parts could lead. We take that in equation (3) H a, jS) and S a, fi) are Hermitian and the former as well as /(/3) belong to class (meaning basically continuous and that the first derivative exists). Integration by parts of equation (3) gives 

Equation (16) may be regarded as the continuous generalization of the secular equation. In fact, in the past we already advocated an intrinsic continuous character of the Roothaan expansion (see Ref. [16b]). Then we may write 

In what follows, we perform a numerical exercise with the same model case described in Section 3. 

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