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There is also the question of how big are the derivative couplings arising from the GMH method, and we are unaware of any computational studies estimating them. Conversely, however, there is an analytical expression by Kryachko, specific to the two-state problem, showing how big the off-diagonal dipole element ought to be (rather than zero, which is the GMH ansatz) in order for the derivative couplings to vanish. See Refs..
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There is also the question of how big are the derivative couplings arising from the GMH method, and we are unaware of any computational studies estimating them. Conversely, however, there is an analytical expression by Kryachko, specific to the two-state problem, showing how big the off-diagonal dipole element ought to be (rather than zero, which is the GMH ansatz) in order for the derivative couplings to vanish. See Refs..
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Confusing as it must sound, credit for the "Boys" orbitals should actually be attributed both to Edmiston and Ruedenberg, as well as Foster and Boys. Foster and Boys suggested in 1960 (Ref.) the criterion of maximizing the (nonlinear) product of the distances between orbital centroids, while Edmiston and Ruedenberg improved on this criterion in 1963 (Ref.), proposing to maximize the (linear) sum of the distances between orbital centroids. The latter definition is the criterion used today for "Boys" orbitals. Furthermore, Edmiston and Ruedenberg were the first to suggest a formal technique for computing these orbitals exactly (rather than approximately, as Foster and Boys had proposed).
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Confusing as it must sound, credit for the "Boys" orbitals should actually be attributed both to Edmiston and Ruedenberg, as well as Foster and Boys. Foster and Boys suggested in 1960 (Ref.) the criterion of maximizing the (nonlinear) product of the distances between orbital centroids, while Edmiston and Ruedenberg improved on this criterion in 1963 (Ref.), proposing to maximize the (linear) sum of the distances between orbital centroids. The latter definition is the criterion used today for "Boys" orbitals. Furthermore, Edmiston and Ruedenberg were the first to suggest a formal technique for computing these orbitals exactly (rather than approximately, as Foster and Boys had proposed).
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58149259705
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This derivative can be done without too much difficulty by following the analogous calculations in Refs..
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This derivative can be done without too much difficulty by following the analogous calculations in Refs..
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more..
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44
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If we apply the Pipek-Mezey approach to states, rather than orbitals, the resulting algorithm will construct diabatic states by maximizing the sum of the squares of atomic populations. Notably, construction of diabatic states by maximizing the squares of natural spin-orbital populations was suggested 20 years ago in Ref.. These two approaches are different, however, because by maximizing the squares of atomic (rather than orbital) populations, the PM algorithm forces localization of the diabatic states; there is no need for such localization in the approach of Ref..
-
If we apply the Pipek-Mezey approach to states, rather than orbitals, the resulting algorithm will construct diabatic states by maximizing the sum of the squares of atomic populations. Notably, construction of diabatic states by maximizing the squares of natural spin-orbital populations was suggested 20 years ago in Ref.. These two approaches are different, however, because by maximizing the squares of atomic (rather than orbital) populations, the PM algorithm forces localization of the diabatic states; there is no need for such localization in the approach of Ref..
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