Variational energy convergence

The inclusion of the arcsin and the double summation in this formula unfortunately comphcates these odd power terms compared to the even power case. The implementation of odd powers nik requires significantly more computer time due to the complexity of this formula. Furthermore, we found that variation of near optimal by plus or minus one had negligeable effect on energy convergence. Therefore, in our calculations utilizing gradient formulas for energy optimization, we excluded the odd power case. 

Table 5 shows that such quantities as the total energy and the highest occupied orbital energy converge rather rapidly in the case of two intermolecular complexes C2H4-CH4 and CH4-CH4. Results of non-variational calculation (step 0) are also... 

So far, we have presented variational calculations with straightforward optimization by explicitly varying the parameters and increasing the basis. One may now try to elaborate a little on the convergence pattern. For instance, one may study the variational energy E N) as a function of the maximum number of states introduced into the h.o. expansion, and try to guess what E(p°) should be. To this end, E(N) can be written as a rational function... 

The calculation proceeds as illustrated in Table 2.2, which shows the variation in the coefficients of the atomic orbitals in the lowest-energy wavefunction and the energy for the first four SCF iterations. The energy is converged to six decimal places after six iterations and the charge density matrix after nine iterations. 

Quadralically Convergent or Second-Order SCF. As mentioned in Section 3.6, the variational procedure can be formulated in terms of an exponential transformation of the MOs, with the (independent) variational parameters contained in an X matrix. Note that the X variables are preferred over the MO coefficients in eq. (3.48) for optimization, since the latter are not independent (the MOs must be orthonormal). The exponential may be written as a series expansion, and the energy expanded in terms of the X variables describing the occupied-virtual mixing of the orbitals. 

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