Variational functional Schwinger

Variational methods are at present used extensively in the study of inelastic and reactive scattering involving atoms and diatomic molecules[l-5]. Three of the most commonly used variational methods are due to Kohn (the KVP)[6], Schwinger (the SVP)[7] and Newton (the NVP)[8]. In the applications of these methods, the wavefunction is typically expanded in a set of basis functions, parametrized by the expansion coefficients. These linear variational parameters are then determined so as to render the variational functional stationary. Unlike the variational methods in bound state calculations, the variational principles of scattering theory do not provide an upper or lower bound to the quantity of interest, except in certain special cases.[9] Neverthless, variational methods are useful because, the minimum basis size with which an acceptable level of accuracy can be achieved using a variational method is often much smaller than those required if nonvariational methods are used. The reason for this is generally explained by showing (as... 

In a true scattering problem, an incident wave is specified, and scattered wave components of ifr are varied. In MST or KKR theory, the fixed term x in the full Lippmann-Schwinger equation, f = x + / GqVms required to vanish, x is a solution of the Helmholtz equation. In each local atomic cell r of a space-filling cellular model, any variation of i// in the orbital Hilbert space induces an infinitesimal variation of the KR functional of the form 8 A = fr Govi/s) + he. This... 

Specializing the present derivation to the principal value Green function, the unsymmetrical expression tan = — 2(wo Av f) is exact for an exact solution of the Lippmann-Schwinger equation, but it is not stationary with respect to infinitesimal variations about such a solution. Since w0 = / + G Avf for such a solution, this can be substituted into the unsymmetrical formula to give an alternative, symmetrical expression tan r] = —2(/1 At> + AvG At> /), which is also not stationary. However, these expressions can be combined to define the Schwinger functional... 

In analogy to the Schwinger variational principle, consider the product functional... 

The use of a separable potential of the form of Equation 8 in Equation 6 to obtain solutions of the form of Equation 9 can be shown to be equivalent to using the functions ct (r) in the Schwinger variational principle for collisions (13). At this stage the functions ai(r) can be chosen to be entirely discrete basis functions such as Cartesian Gaussian (lA) or spherical Gaussian (15) functions. We note that with discrete basis functions alone the approximate solution satisfies the scattering boundary condition. Such basis... 

We are interested in molecules in stationary states and in this case Schwinger s principle takes on a particularly simple form. In this instance, it yields Schrodinger s equation for a stationary state, equation 11, and equation 14 for the variation in Schrodinger s energy functional G[T ],... 

As an example of the use of this functional, we now use it to obtain the Schwinger variational principle for the static-exchange potential. The total wavefunction is taken as... 

We can see that this condition will be satisfied if, and only t/the wavefimction/ satisfies the lippmann-Schwinger equation, so that (l-G V) f = lj>. This, we feel, is where the power of a variational method over a nonvariadond one originates. The requirement that the functional (5) be stationary with respect to small changes in/ sets a level of accuracy with which the equation (1-G V) > = ls> must be satisfied, which in turn forces an accurate solution for/ A comparison of Rg.l to Figs. 2 and 3 supports this argument,... 

The Kohn variational principle is perhaps the simplest of the three scattering variational principles mentioned above [9]. In particular, it requires that one calculate matrix elements only over the total Hamiltonian H of the system, and not over the Green s function Gq E) of some reference Hamiltonian Hq. While matrix elements of H between energy-independent basis functions are also energy-independent, all matrix elements of G E) have to be re-evaluated at each new scattering energy E. The Kohn variational principle is therefore somewhat easier to apply than the Schwinger and Newton... 

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