Optimization techniques Newton-Raphson methods

Direct minimization techniques. The variational principle indicates that we want to minimize the energy as a function of the MO coefficients or the corresponding density matrix elements, as given by eq. (3.54). In this formulation, the problem is no different from other types of non-linear optimizations, and the same types of technique, such as steepest descent, conjugated gradient or Newton-Raphson methods can be used (see Chapter 12 for details). 

Although the calculation of the Hessian is time consuming, the effort is quickly compensated by the excellent convergence properties of the Newton-Raphson approach [293-295]. This optimization technique solved the convergence problems of first-order MCSCF methods, which optimized orbitals and Cl coefficients in an alternating manner (recall chapter 9). Even perturbative improvements of the four-component CASSCF wave function are feasible and have been implemented and investigated [527]. 

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