An efficient method for calculating maxima of homogeneous functions of orthogonal matrices: Applications to localized occupied orbitals

Abstract

We present here three new algorithms (one purely iterative and two DIIS-like [Direct Inversion in the Iteractive Subspace]) to compute maxima of homogeneous functions of orthogonal matrices. These algorithms revolve around the mathematical lemma that, given an invertible matrix A, the function f(U)=Tr(AU) has exactly one local (and global) maximum for U special orthogonal [i.e., UUT=1 and det(U)=1]. This is proved in the Appendix. One application of these algorithms is the computation of localized orbitals, including, for example, Boys and Edmiston-Ruedenberg (ER) orbitals. The Boys orbitals are defined as the set of orthonormal orbitals which, for a given vector space of orbitals, maximize the sum of the distances between orbital centers. The ER orbitals maximize total self-interaction energy. The algorithm presented here computes Boys orbitals roughly as fast as the traditional method (Jacobi sweeps), while, for large systems, it finds ER orbitals potentially much more quickly than traditional Jacobi sweeps. In fact, the required time for convergence of our algorithm scales quadratically in the region of a few hundred basis functions (though cubicly asymptotically), while Jacobi sweeps for the ER orbitals traditionally scale as the number of occupied orbitals to the fifth power. As an example of the utility of the method, we provide below the ER orbitals of nitrated and nitrosated benzene, and we discuss the chemical implications.

The initial and final states of electron and energy transfer processes: Diabatization as motivated by system-solvent interactions

“…Nevertheless, if one uses the sparsity of the atomic orbital basis and one works in a subspace W with fewer than twenty adiabatic states ͑as would be expected usually͒, an optimal implementation of the ER algorithm should be fast, with only a marginal increase in computational cost after the necessary electronic structure calculations. Recent advances with resolution of the identity ͑RI͒ methods have made ER localization for orbitals relatively inexpensive, 64,65 and the same should be true for ER localization of diabatic states. See Appendix A for more details.…”

“…47,48,64 Because Boys localization requires the storage of only a quadratic number of variables, the algorithm runs almost instantaneously compared to any prerequisite electronic structure method no matter how big the system size. The ER algorithm scales worse than the Boys localization because one must work with the twoelectron matrix elements of 1 / r 12 , i.e., ͑ ͉ ͒ ͑see Appendix A͒.…”

“…All standard localization algorithms are quartic functions of the unitary matrix U and can be solved with standard "Jacobi sweeps" 48 or with other iterative approaches. 64,65 Because we do not anticipate having too many states to localize ͑usually, under 20͒ and computing an element of the tensor R JKLM will be expensive, Jacobi sweeps should be the most attractive approach for ER localization of states. In order to have a fast implementation of ER diabatization, the key bottleneck to overcome will be computing the tensor R JKLM in Eq.…”

Seniority number in spin-adapted spaces and compactness of configuration interaction wave functions

“…39 in which, starting with the canonical molecular orbital basis set, the sum μ 2 D μμ μμ = μ ij kl U * μi U μj U * μk U μl 2 D ik jl in the final transformed basis set is maximized. These mathematical frameworks for searching a basis set transformation maximizing/minimizing a determined quantity have been widely utilized in procedures of orbital localization, where other tensors different from the 2 D ik jl one have been manipulated.…”

“…39, and the CMO sets have been used as initial bases of that iteration. However, we have seen that identical final orbital sets are obtained when other initial orthonormal basis sets, like the NO ones, are utilized.…”