Colle-Salvetti functional

A second orbital-dependent expression, originally introduced for use with the Hartree-Fock scheme, is the Colle-Salvetti (CS) correlation functional [23]. The starting point for the derivation of the CS functional is an approximation for the correlated wavefunction F(ricri. rjvajv). The ansatz for. .. r aA consists of a product of the HF Slater determinant and Jastrow factors, 

CS then used a model for the correlation functions y (rj,rj) which satisfies the electron-electron cusp condition at r = Vj [75]. The free parameter in 

This functional depends on the spin-density and the kinetic energy of spin T,, Ok V4 k(r In the DFT context, this latter dependence makes an implicit functional for which the OPM has to be utilized. Therefore, the CS correlation functional has been suggested as a first candidate for going 

However, like the correlation part of the SIC-LDA, this CS functional is rather local. Its nonlocality is restricted to the first gradients of the KS orbitals. In this respect it is very similar to the Meta-GGA [77]. In the case of the Meta-GGA higher gradients of the density, i.e. its Laplacian, are effectively included into the GGA via the kinetic energy expression 0k 4 k(T [ However, in contrast to the CS-functional, the Meta-GGA has not yet been applied within the OPM (although, in principle, it should be treated within the OPM). In any case, neither the SIC-LDA nor the CS and Meta-GGA functionals are sufficiently nonlocal to deal with dispersion forces The argument given in Sect. 4 for the LDA applies equally well to these types of functionals. 

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Ec = E c - Ex have been employed. On the one hand, LDA and GGA type correlation functionals have been used [14], However, the success of the LDA (and, to a lesser extent, also the GGA) partially depends on an error cancellation between the exchange and correlation contributions, which is lost as soon as the exact Ex is used. On the other hand, the semiempirical orbital-dependent Colle-Salvetti functional [22] has been investigated [15]. Although the corresponding atomic correlation energies compare well [15] with the exact data extracted from experiment [23], the Colle-Salvetti correlation potential deviates substantially from the exact t)c = 8Ecl5n [24] in the case of closed subshell atoms [25]. 

For the spin-unpolarized uniform electron gas, the LYP correlation energy functional reduces to that of Colie and Salvetti [58], on which LYP is based. McWeeny s work [59] may have contributed to the widespread misimpression that the Colle-Salvetti functional is accurate in this limit. Note however the McWeeny tested Eq. (9) of Ref. [58], and not the further-approximated Eq. (19) of Ref. [58], which is the basis of LYP and other Colle-Salvetti applications, and which is shown in Table 2. 

The corrected potential energy curves show an improvement over the original GVB-pp curves, being generally the Colle-Salvetti functional the one that yields better results. 

The application of the OEP methodology to the Fock term (47), either with or without the JCLl approximation, is also known as the EXX method. The OEP-EXX equations have been solved for atoms " and solids, with very encouraging results. Other orbital-dependent functionals that have been treated within the OEP scheme are the PZ-SIC and the Colle-Salvetti functional. A detailed review of the OEP and its JCLl approximation is given in Ref. [135]. 

Lee C, W Yang and R G Parr 1988. Development of the Colle-Salvetti Correlation Energy Formula into a Functional of the Electron Density. Physical Review B37 785-789. 

C. Lee, W. Yang and R. G. Parr, Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density, Physical Review B, 37, 785 (1988). 

Colle-Salvetti Correlation-Energy Formula into a Functional of the Electron Density. 

Lee, C. Yang, W. Parr, R.G. Development of the Colle-Salvetti Correlation-Energy Formula into a Functional of the Electron Density. Phys. Rev., 1988, B37,785 Reed, A. E. Weinstock, R.B. Weinhold, F. Natural Population Analysis. J. Chem. Phys. 1985, 83 (2), 735... 

It was obtained by a modified Colle-Salvetti approach. At first, an analytic expression of the kinetic contribution to the correlation energy per electron was determined. Then, the total correlation energy was derived by means of the DFT virial theorem. The value of the only parameter entering in this approach was fixed by applying the resulting expression to the uniform electron gas (UEG) in the low and high density limit cases. Thus, in spite of the constants entering in (1.8), the RC correlation functional is parameter-free. 

The orbital functional derivative 8 c/( i8< ) = vc<(> is derived taking variations of unoccupied orbitals induced by unitarity into account. This gives an explicit formula for any orbital-functional model of Exc (e.g. LDA [4] or Colle-Salvetti. [38]) In an... 

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