Coupling-Constant Integration

The definitions (1.65) and (1.68) are formal ones, and do not provide much intuitive or physical insight into the exchange and correlation energies, or much guidance for the approximation of their density functionals. These insights are provided by the coupling-constant integration [30,31,32,33] to be derived below. 

The Hellmann-Feynman theorem of Sect. 1.2.4 allows us to simplify (1.74) 

Equation (1.75) looks like a potential energy the kinetic energy contribution to Tlxc has been subsumed by the couphng-constant integration. We should remember, of course, that only A = 1 is real or physical. The Kohn-Sham system at A = 0, and all the intermediate values of A, are convenient mathematical fictions. 

To make further progress, we need to know how to evaluate the AT-electron expectation value of a sum of one-body operators like T, or a sum of two-body operators like V. For this purpose, we introduce one-electron (pi) and two-electron (p2) reduced density matrices [34]  

We interpret the positive number p2(l r)d r d r as the joint probability of finding an electron in volume element d r at r, and an electron in d r at r. By standard probability theory, this is the product of the probability of finding an electron in d r (n(r)d r) and the conditional probability of finding an electron in d r, given that there is one at r (ri2(r,r )d r )  

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We define a normal system as one in which the hole density at the weakly-interacting end of the coupling-constant integration is close to that of a single Slater determinant. In such a system, the local and gradient-corrected holes, evaluated for the exact spin-densities, are nearly exact near the position of the... 

We further define the coupling constant integrated pair correlation function [31,32] by... 

This expression shows explicitly how the coupling constant integration introduces the kinetic part of We can further split up g into an exchange part and a correlation part g ... 

Note that the exchange hole is not affected by the coupling constant integration as the density and therefore the Kohn-Sham orbitals also remain the same during the integration. The exchange and correlation holes satisfy the important sum rule properties... 

Eq. (14) for the exchange-correlation energy in terms of the coupling constant integrated pair-correlation function. If we take the functional derivative of this equation we find that we can write as [66]... 

We will discuss in detail in the next section this potential, which incorporates the main features of the atomic shell structure. The screening potential is just the potential of the coupling constant integrated exchange-correlation hole. Due to the fact that this hole integrates to one electron, the screening potential has... 

The corresponding energy shift can be evaluated by a coupling constant integration with respect to the external potential. Scaling the external potential Hamiltonian by A,... 

ID spectrum ID Information from chemical shifts, coupling constants, integrals... 

The coupling constant integration can be easily performed to yield... 

The actually interesting energy, corresponding to A = 1, can be obtained by a coupling constant integration approach (using a normalized ground state 4 (A)) for all A),... 

Coupling Constant Integration Analysis of Density Functionals for Subsystems 219... 

The present work focuses on the coupling constant integration (CCI) amalysis (20-22) of such nonadditive... 

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