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By R. Sneider

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N be a running index such that each p stands for a particular triplet of integer powers {i, j, k} appearing in (19). Each term in the summation in (19) then corresponds to a member of a basis set of functions defined by k −αr1 −βr2 M e Yl1 l2 L (ˆr1 , rˆ 2 ). χ p (α, β) = r1i r2 r12 j (20) The central idea of the variational method is to regard the function Ψ (r1 , r2 ) as a trial solution Ψtr to the Schr¨odinger equation (18) with adjustable linear parameters ai jk ≡ a p and nonlinear parameters α and β.

Instead, some sort of iterative method must be used to locate the energy minimum on the two-dimensional energy surface as α and β are varied. This problem is further discussed in Sect. III. This much is standard text book material, and the proof can be found in any book on quantum mechanics. What is not so well known is the Hylleraas–Undheim–McDonald theorem (6, 7), which states that the variational bound property applies not just the lowest eigenvalue λmin , but also to all the higher-lying variational eigenvalues of (22).

ADVANCED CONCEPTS (Phase field model of an electrochemical system) This relatively new method (10, 11), employs a so-called phase field variable. , the electrode or electrolyte. The behavior of this variable is governed by a partial differential equation (PDE) that is itself coupled to the relevant transport equations for the material. The interface between the phases is thus described by smooth but highly localized changes of this variable. This approach avoids the mathematically difficult problem of applying boundary conditions at an interface whose location is part of the unknown solution.

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