Second-order optimization quadratic convergence

In complete active space self-consistent field (CASSCF) calculations with long configuration expansions the most expensive part is often the optimization of the Cl coefficients. It is, therefore, particularly important to minimize the number of Cl iterations. In conventional direct second-order MCSCF procedures , the Cl coefficients are updated together with the orbital parameters in each micro-iteration. Since the optimization requires typically 100-150 micro-iterations, such calculations with many configurations can be rather expensive. A possible remedy to this problem is to decouple the orbital and Cl optimizations , but this causes the loss of quadratic convergence. The following method allows one to update the Cl coefficients much fewer times than the orbital parameters. This saves considerable time without loss of the quadratic convergence behaviour. 

The optimization of the total energy might alternatively be expressed in terms of the variation parameters P [in exp(iS)]. The energy function E(S) would not be quadratic in these parameters P but would contain cubic, quartic, etc. terms in P. An explicit solution from which to determine a SP of the energy function when this unitary exp(iS) operator is used is very difficult to establish hence an iterative procedure is required to determine SPs of the energy hypersurface. One iterative scheme that is quadratically convergent is obtained if the terms that refer to the orbital optimization [exp(iA)] are neglected in the MCSCF derivation performed in Section B. The second-order Eq. (2.33) then would read... 

Quadratically 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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