EFFICIENT USE OF JACOBI ROTATIONS FOR ORBITAL OPTIMIZATION AND LOCALIZATION

Quantum chemical orbital optimizations can be accomplished by Newton-type iterations, where all orbitals are improved at each step; or by a succession of Jacobi rotations, where only two orbitals are improved at any one step. In both schemes, the iterative updating of the four-index two-electron integrals requires a large computational effort. We show that the four-index transformation due to a Jacobi rotation can be simplified to such a degree that the successive execution of the four-index transformations of N(N - 1)/2 different Jacobi rotations requires no greater computational effort than that required by the one full orthogonal transformation which is the product of all N(N - 1)/2 Jacobi rotations. The four-index updating has therefore no bearing on the relative merit of the Newton approach versus the Jacobi approach. The Jacobi approach has, however, an advantage if the optimization of each Jacobi rotation angle is simple and if the effectiveness of the individual Jacobi rotations can be assessed without the execution of four-index transformations. For, in that case, all ineffectual rotations are easily deleted from the iterative sequence. Whether convergence can be guaranteed for one or the other approach is also relevant. Our conclusions are illustrated by application to the problem of intrinsic orbital localization where the succession of Jacobi rotations is the more effective strategy.