Newton-Raphson optimization

CASVB has been shown to be applicable to electronically excited states as well [18], by the adoption of a clever generalization [21] of the second-order stabilized Newton-Raphson optimization procedure [22] used by it, SC [4], and OBS-GMCSC [1][2]. It turns out that essentially the same procedure works for OBS-GMCSC as well [23], allowing it to exploit basis-set optimization where it can be especially advantageous, i.e. for excited states. 

The most popular optimization techniques are Newton-Raphson optimization, steepest ascent optimization, steepest descent optimization. Simplex optimization. Genetic Algorithm optimization, simulated annealing. - Variable reduction and - variable selection are also among the optimization techniques. 

All matrix elements in the Newton-Raphson methods may be constructed from the one- and two-particle density matrices and transition density matrices. The linear equation solutions may be found using either direct methods or iterative methods. For large CSF expansions, such micro-iterative procedures may be used to advantage. If a micro-iterative procedure is chosen that requires only matrix-vector products to be formed, expansion-vector-dependent effective Hamiltonian operators and transition density matrices may be constructed for the efficient computation of these products. Sufficient information is included in the Newton-Raphson optimization procedures, through the gradient and Hessian elements, to ensure second-order convergence in some neighborhood of the final solution. 

Approximating the real function by a second-order polynomial forms the basis for the Newton-Raphson optimization techniques described in Section 12.2. 

Moreover, the second-generation MCSCF parametrizes the wave function in a way that enables the simultaneous optimization of spinors and Cl coefficients, in this context then called orbital or spinor rotation parameters and state transfer parameters, respectively. Then, a Newton-Raphson optimization method is employed which also requires the second derivatives of the MCSCF electronic energy with respect to the molecular spinor coefficients (more precisely, to the orbital rotation parameters) and to the Cl coefficients. As we have seen, in Hartree-Fock theory the second derivatives are usually not calculated to confirm that a solution of the SCF procedure has indeed reached a minimum with respect to the large component and not a saddle point. Now, these general MCSCF methods could, in principle, provide such information, although it is often not needed in practice. 

A Newton-Raphson optimization shows that for 70[Pg.507]

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