The importance of Localized Molecular Orbitals in correlation treatments beyond mean-field calculation and in the illustration of chemical bonding (and antibonding) can hardly be overstated. However, generation of orthonormal localized occupied MOs is significantly more straightforward than obtaining orthonormal localized virtual MOs. Orthonormal molecular orbitals allow facile use of highly efficient group theoretical methods (e.g., graphical unitary group approach) for calculation of Hamiltonian matrix elements in multireference configuration interaction calculations (such as MRCISD) and in quasidegenerate perturbation treatments, such as the Generalized Van Vleck Perturbation Theory. Moreover, localized MOs can elucidate qualitative understanding of bonding in molecules, in addition to high accuracy quantitative descriptions. We explore algorithms based on various levels of approximation of second-order descriptions of the optimization on orthonormal Riemannian manifolds. Specifically, we adopt the powers of the fourth moment cost function introduced by Jørgensen and coworkers. Because the fourth moment cost functions are prone to having multiple negative Hessian eigenvalues when starting from easily available canonical (or near canonical) MOs, standard optimization algorithms can fail to obtain the orbitals of the virtual space. To overcome this drawback, we explored Riemannian Trust Region algorithms with approximate retractions from the tangent space built into first and second derivatives of the cost function. Moreover, the Riemannian Trust Region outer iterations were coupled to truncated conjugate gradient inner loops, which avoided any costly solutions of simultaneous linear equations or eigenvector/eigenvalue solutions. Numerical examples will be provided on model systems, including the high connectivity H<sub>10</sub> set.

Riemannian trust region method for minimization of fourth central moment

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