Geometrical linear responses and directional energy derivatives for energetically degenerate MCSCF electronic functions

For state-averaged multiconfigurational self consistent field (SA-MCSCF) wave functions second order geometrical response equations are derived that allow the determination of first-order configuration amplitude responses for equally weighted, energetically degenerate states. The first-order response equations obtained in earlier work do not suffice to determine these particular response parameters. To formulate such a derivation in a well defined manner, it is found that a specific linear combination of the degenerate states must be formed; this specific combination of states then defines how state energies and wave functions evolve as one passes through the surface intersection. The linear combination among the degenerate states is dependent upon the molecular distortion for which the responses are to be evaluated. Expressions for first- and second-order directional energy derivatives for these energetically degenerate wave functions are also derived. All the equations obtained are computationally tractable and expressed in terms of quantities that result from optimizing the SA-MCSCF wavefunctions and from solving the first- and part of the second-order geometrical response equations.