Variational -matrix theory

The Wigner-Eisenbud [428] 7 -matrix, or derivative matrix, is defined by the relationship between radial channel orbitals fps(r) and their derivatives on some sphere of radius r that surrounds a target system [214,33], Assuming spherical geometry, the dimensionless radial f -matrix Rpq at r is defined by 

The theory of the / -matrix was developed in nuclear physics. As usually presented, the theory makes use of a Green function to relate value and slope of the radial channel orbitals at r, expanding these functions for r r as linear combinations of basis functions that satisfy fixed boundary conditions at r. The true logarithmic derivative (or reciprocal of the / -matrix in multichannel formalism) 

The A-matrix can be matched at r to external channel orbitals, solutions in principle of external close-coupling equations, to determine scattering matrices. Radial channel orbital vectors, of standard asymptotic form for the A -matrix, 

The A-matrix relation between function value and gradient can be solved for the W-matrix. 

As written, these equations refer to open channels only. When external closed channels are considered, an external closed channel orbital that vanishes as r - oo must be included for each such channel. The indices p,q,s run over all channels, 

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VARIATIONAL TWO-ELECTRON REDUCED-DENSITY-MATRIX THEORY... 

G. Gidofalvi and D. A. Mazziotti, Variational reduced-density-matrix theory strength of Hamiltonian-dependent positivity conditions. Chem. Phys. Lett. 398, 434 (2004). 

J. R. Hammond and D. A. Mazziotti, Variational two-electron reduced-density-matrix theory partial 3-positivity conditions for A-representability. Phys. Rev. A 71, 062503 (2005). 

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... 

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