Correlation cusp

Kutzelnigg, W., Klopper, W. Wave functions with terms linear in the interelectronic coordinates to take care of the correlation cusp. I. General theory. J. Chem. Phys. 1991, 94, 1985-2001. 

Noga, J., Kutzelnigg, W., Klopper, W. CC-R12, a correlation cusp corrected coupled-cluster method with a pilot application to the Be2 potential curve. Chem. Phys. Lett. 1992, 199, 497-504. 

To improve on a single determinant reference we must develop a superior treatment of the two-electron interaaion. Logically, this would involve explicit use of the two-particle operator in a trial wavefunaion, and such Hylleraas-type trial wavefunctions have been used to obtain the best results (for, e.g.. He, Li, Be, H2, and LiH). Analysis of certain other analytic properties of the correlation cusp l/( r,- — r, ) have also been exploited to develop better descriptions without having to use wavefunctions that are explicitly dependent on r j, but such methods also have many computational restriaions. 

W. Kutzelnigg and W. Klopper, ]. Chem. Phys., 94,1985 (1991). Wavefunctions with Terms Linear in the Interelectronic Coordinates to Take Care of the Correlation Cusp. I. General Theory. 

If one does not care to describe the correlation cusp correctly, but uses a Cl-like expansion of the wave function, the just-mentioned singularity would not show up, so it is not surprising that so far it has not plagued any numerical calculation. In particular, everything remains regular in the framework of Hartree-Fock or MC-SCF theory. 

This means, that if one truncates at some L = Imax, one makes an error 0 [L + 1] ) of the energy. The origin of this slow convergence is the correlation cusp [85], i.e. the fact that... 

The history and the present state of the treatment of electron correlation is reviewed. For very small atoms or molecules calculations of higher than spectroscopic accuracy are possible. A detailed account for many-electron methods in terms of one-electron basis sets is given with particular attention to the scaling of computer requirements with the size of the molecule. The problems related to the correlation cusp, especially the slow convergence of a basis expansion, as well as their solutions are discussed. The unphysical scaling with the particle number may be overcome by localized-correlation methods. Finally density functional methods as an alternative to traditional ab-initio methods are reviewed. 

For He 12-figure accuracy was reported [36, 16]. This is surprising since this ansatz neither fulfills the nuclear cusp nor the correlation cusp conditions. Although it is not yet fully understood why this works, some preliminary comments can be made. 

This is an extremely slow convergence and is related to the fact that the exact wave function has a correlation cusp [20], i.e. a discontinuous first derivative at ri2 — 0, such that... 

As to problem (a) it is at least understood that the basic difficulty is caused by the correlation cusp and that much better convergence is obtained if one uses wave functions depending explicitly on the interelectronic coordinates. The problem of difficult integrals can be avoided in the R12-methods, or possibly by using Gaussian geminals. One may also think of improved extrapolation techniques based on the known behaviour of the wave function for r,j —> 0. 

The main challenge as to an improved theory of electron correlation as a basis of accurate numerical quantum chemistry have been mentioned in this review, namely (a) the explicit treatment of the correlation cusp, (b) the formulation of methods that scale with a low power of the number of particles, (c) the consistent combination of MC-SCF-theory for the nondynamic and coupled-cluster methods for the dynamic correlation. 

The higher NO s may be given similar though more complicated pictorial interpretations, but their role is rather to represent — as far as this is possible — the correlation cusp. Usually very few configurations convey the bulk effect of electron correlation and very many are needed to get the remaining few per cent. 

W. Klopper and W. Kutzelni, Chem. Phys. Lett., 134, 17 (1987). Moller-Plesset Calculations Talcing Care of the Correlation Cusp. 

W. Kuuelnigg and W. Klopper, J. Chem. Phys., 94,1985 (1991). Wave Functions with Terms Linear in the Interelectronic Coordinates to Take Care of the Correlation Cusp. 1. General Theory. V. Termath, W. Klopper, and W. Kutzelnigg,/. Chem. Phys., 94,2002 (1991). Wave Functions with Terms Linear in the Interelectronic Coordinates to Take Care of the Correlation Cusp. II. Second-Order Maller-Plesset (MP2-R12) Calculations on Closed-Shell Atoms. W. Klopper and W. Kutzelnigg /. Chem. Phys., 94, 2020 (1991). Wave Functions with Terms Linear in the Interelectronic Coordinates to Take Care of the Correlation Cusp. III. Second-Order Moller-Plesset (MP2-R12) Calculations on Molecules of First Row Atoms. 

It has been shown [29,21] that, as a consequence of the correlation cusp condition [30] in the coalescence region (ri = r ), is approximately known and takes the form... 

The Gaussian geminals do not satisfy the correlation cusp condition (p. 587). because of factor exp —brfj). It is required (for simplicity, we write = r) that... 

The Coulomb hole that wavefunctions are predicted to have for close anti-parallel-spin electrons is also called a correlation hole. As a condition for wavefunctions containing correlation holes, Kato proposed a correlation cusp condition (Kato 1957),... 

The Hartree-Fock wavefunction violates this condition, because it gives zero for the left-hand side of this equation. As shown in Fig. 3.1, a wavefunction satisfying this condition contains a correlation hole, which contains a sharp dip, called a cusp, near ri2 = 0. This correlation hole causes anti-parallel-spin electrons to be further apart, and therefore reduces Coulomb interactions, thus lowering the total electronic energies. Sinanoglu named this electron correlation in the correlation cusp condition as dynamical correlation (Sinanoglu 1964). 

Fig. 3.1 Correlation cusp in the wavefunction of the hydrogen molecule (Frye et al. 1990). The wavefunctions of the Hartree-Fock and configuration interaction (Cl) (see Sect. 3.3) methods are compared to the wavefunction of the Hylleraas Cl (HCl) method (see Sect. 3.5), which is close to the exact wavefunction. Since X2 is set to zero in this figure, X corresponds to Note that the wavefunction is not zero even for fn = 0, because there remtiins one electron at = 0...

The slow convergence of configuration interaction (Cl) expansions in orbital basis sets is linked to the presence of the correlation cusp in the wave function. Within the molecular Hamiltonian the interelectronic Coulomb operator scales like the reciprocal of the distance between the electrons and, for the part of the configuration space where the electrons are close to each other, the Coulomb interaction diverges. However, the local energy defined as... 

Klopper W, Kutzelnigg W (1987) Mpller-Plesset calculations taking care of the correlation cusp. Chem Phys Lett 134 17-22... 

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