Schlosser-Marcus variational principle

The Schlosser-Marcus variational principle is derived for a single surface a that subdivides coordinate space 9i3 into two subvolumes rm and rout. This generalizes immediately to a model of space-filling atomic cells, enclosed for a molecule by an external cell extending to infinity. The continuity conditions for the orbital Hilbert space require i out =a i in This implies a vanishing Wronskian surface integral 

Schlosser and Marcus [359] showed that for variations about such a continuous trial function, the induced first-order variations of Zr and Za exactly cancel, even if the orbital variations are discontinuous at a or have discontinuous normal gradients. After integration by parts, the variation of Zr about an exact solution is a surface Wronskian integral 

Because of this cancellation, variation about an arbitrary trial function gives 

This can vanish only if (H — r) b = 0 in both rin and rout. Moreover, this requires that both ( ifrin Wa if/in — fout) and (S j/0Ut Wa i//in — j/out) must vanish when ijrin and fs out are varied independently. By an extension of the surface matching theorem, both these Wronskian integrals must vanish in order to eliminate the value and normal gradient of tf/m — i// r on o. Practical applications of this formalism use independent truncated orbital basis expansions in adjacent atomic cells, so that the continuity conditions cannot generally be satisfied exactly. 

In a space-filling cellular model, the SM variational functional can be expanded in a local basis in each atomic cell. Variation of the expansion coefficients of the trial orbital function ifr = J2l lYl in ceH T/x induces the variation 

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The Schlosser-Marcus variational principle [359] provides an alternative that does not use structure constants. On substituting the expansion of an energy-independent ACO into the SM variational functional, the variational equations indexed by in cell r/( are... 

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