  The importance of Localized Molecular Orbitals in correlation treatments
beyond mean-field calculation and in the illustration of chemical bonding 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 for calculation of Hamiltonian matrix
elements in multireference configuration interaction calculations and in
quasidegenerate perturbation treatments, such as the GVVPT2 Theory. Moreover,
localized MOs can elucidate qualitative understanding of bonding in molecules,
in addition to high accuracy quantitative descriptions. We adopt the powers of
the fourth moment cost function introduced by J{\o}rgensen and coworkers.
Because the fourth moment cost functions are prone to having multiple negative
Hessian eigenvalues when starting from easily available canonical MOs, standard
optimization algorithms can fail to obtain the orbitals of the virtual or
partially occupied spaces. To overcome this drawback, we applied a Trust Region
algorithm on an orthonormal Riemannian manifold with an approximate retraction
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 are provided on model systems, including the high connectivity H10 set
in 1-, 2- and 3-dimensional arrangements, and on a chemically realistic
description of cyclobutadiene, and the propargyl radical. In addition to
demonstrating the algorithm on occupied and virtual blocks of orbitals, the
method is also shown to work on the active space at the MCSCF level of theory.
