Variational principles complex Kohn

In this / -matrix theory, open and closed channels are not distinguished, but the eventual transformation to a A -matrix requires setting the coefficients of exponentially increasing closed-channel functions to zero. Since the channel functions satisfy the unit matrix Wronskian condition, a generalized Kohn variational principle is established [195], as in the complex Kohn theory. In this case the canonical form of the multichannel coefficient matrices is... 

Hoheriberg and Kohn showed secondly that the energy functional (7.7) assumes its minimum value, the ground-state energy, for the correct ground-state density. Hence, if the universal functional F[n] = < T + U > were known it would be relatively simple to use this variational principle to determine the ground-state energy and density for any specified external potential. Unfortunately, the functional is not known, and the full complexity of the many-electron problem is associated with its determination. 

Equation (15) is the key equation of the Kohn variational principle for the -matrix (21). For small problems, when the spectral representation of ft can be obtained, both methods are essentially equivalent. If the linear equations are to be solved iteratively, the present method, Eq. (14), effectively requires to solve half the number of sets of simultaneous linear equations as the basis and xT can chosen real making Eq. (14) real while (15) remains complex. 

In this article we focus on one of these approaches, the complex Kohn (CK) method, both because it is a singularly successful example of the variational methods and because it provides a particularly clear view of the central problem of electron scattering, namely the consistent treatment of electronic correlation in the target molecule and correlation involving the scattered electrons. The CK method has its origins in Kohs s 1948 paper on variational principles, and some connections between this method and various other approaches have been discussed by McCurdy, Rescigno, and Schneider. ... 

The Kohn variational principle is a variational expression for the T-matrix, which is linearly related to the scattering amplitude. The variational expression is a stationary principle (not a minimum or maximum because the T-matrix and scattering amplitudes are intrinsically complex valued) for the T-matrix in terms of trial scattering wave functions with incident waves in the initial and final channels respectively ... 

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