On the global convergence of MCSCF wave function optimization: The method of trigonometric interpolation

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A new method of MCSCF wave function optimization is presented. This method is based on a nonlinear transformation of the wave function variation coordinates along with the construction of a global interpolating function. This interpolating function is constructed for each MCSCF iteration in such a way that it reproduces certain known behavior of the exact energy function. It reproduces exactly the energy, gradient, and hessian at the expansion point, at an infinite number of isolated points, and at points on the surfaces of an infinite number of nested multidimensional balls within the wave function variational space. The optimization of the wave function correction parameters on this interpolating function does not require integral transformations or density matrix constructions, although one-index transformation and transition density matrix techniques may be used if desired. The nonlinear coordinate transformations, along with the necessary derivatives, are computed with simple matrix operations, and require onlyO(Norb3) effort. The new method differs from previous optimization methods in several respects. (1) It reproduces certain behavior of the exact energy function that is not displayed by previous approaches. (2) The orbital-state coupling is included explicitly via the partitioned orbital hessian matrix. (3) The minimization of the approximate energy function is simpler than with previous similar approaches. (4) The treatment of redundant orbital rotations is straightforward, since the exact and approximate energy functions display the same qualitative behavior with respect to these wave function variations. (5) Finally, the present method may be implemented as a simple extension to essentially any existing second-order MCSCF code, the required changes being localized within a rather small part of the overall iterative procedure. Examples of the convergence of the new method are presented, along with numerical demonstrations of some of the relevant features of the exact and interpolated energy functions.