The correlation energy

The correlation energy is usually defined relative to the energy of the RHF (rather than the UHF) wave function. Note that this definition comprises both the dynamical and the nondynamical correlation discussed in Section 5.2.10 but not the Fermi correlation. 

The correlation energy in (5.4.9) is defined in terms of a complete one-electron basis. In practice, however, an incomplete basis must be used for the calculation of the correlation energy. The term correlation energy is then used more loosely to denote the energy obtained from (5.4.9) in a given one-electron basis. As we shall see for example in Section 8.4.2, the correlation energy usually increases in magnitude with the size of the orbital basis, since a small basis does not have the flexibility required for an accurate representation of correlation effects. 

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Although it is now somewhat dated, this book provides one of the best treatments of the Hartree-Fock approximation and the basic ideas involved in evaluating the correlation energy. An especially valuable feature of this book is that much attention is given to how these methods are actually implemented. 

This result applies when the number of up spins equals the number of down spins and so is not applicable to systems with an odd number of electrons. The correlation energy functional was also considered by Vosko, Wdk and Nusarr [Vosko et al. 1980], whose expression is ... 

Cl calculations can be used to improve the quality of the wave-function and state energies. Self-consistent field (SCF) level calculations are based on the one-electron model, wherein each electron moves in the average field created by the other n-1 electrons in the molecule. Actually, electrons interact instantaneously and therefore have a natural tendency to avoid each other beyond the requirements of the Exclusion Principle. This correlation results in a lower average interelectronic repulsion and thus a lower state energy. The difference between electronic energies calculated at the SCF level versus the exact nonrelativistic energies is the correlation energy. 

Configuration interaction (Cl) is a systematic procedure for going beyond the Hartree-Fock approximation. A different systematic approach for finding the correlation energy is perturbation theory... 

Because the interelectronic cusp is difficult to describe well with one-electron basis functions, pair correlation energies converge much more slowly (as N" ) than SCF energies (which converge as f ). This fact makes the use of CBS extrapolations of the correlation energy very beneficial in terms of both accuracy and computational cost. 

In Part 2 of their paper, Hohenberg and Kohn go on to investigate the form of the functional F[P(r)] in the special cases of certain limiting charge densities. They find that F[P(r)] can be expressed in terms of the correlation energy and electric polarizabilities. 

The principal deficiency of CISD is the lack of the TI term, which is the main reason for CISD not being size extensive. Furthermore, this term becomes more and more important as the number of electrons increases, and CISD therefore recovers a smaller and smaller percentage of the correlation energy as the system increases. There are various approximate corrections for this lack of size extensivity which can be added to standard CISD. The most widely known of these is the Davidson correction, sometimes denoted CISD - - Q(Davidson), where the quadruples contribution is approximated as... 

The main difference between the G1/G2 and CBS methods is the way in which they try to extrapolate the correlation energy, as described below. Both tire G1/G2 and CBS methods come in different flavours, depending on the exact combinations of metliods used for obtaining the above four contributions. 

The acronym SEC refers to the case where the reference wave function is of the MCSCF type and tire correlation energy is calculated by an MR-CISD procedure. When the reference is a single determinant (HE) the SAC nomenclature is used. In the latter case the correlation energy may be calculated for example by MP2, MP4 or CCSD, producing acronyms like MP2-SAC, MP4-SAC and CCSD-SAC. In the SEC/SAC procedure the scale factor F is assumed constant over the whole surface. If more than one dissociation channel is important, a suitable average F may be used. 

The Parameterized Configuration Interaction (PCI-X) method simply takes the correlation energy and scales it by a constant factor X (typical value 1.2), i.e. it is assumed that the given combination of method and basis set recovers a constant fraction of the correlation energy. 

The most difficult part in calculating absolute stabilities (heat of formation) is the correlation energy. For calculating energies relative to isolated atoms, the goal of tire... 

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