Electronic correlation energy

The LSDA approximation in general underestimates the exchange energy by 10%, thereby creating errors which are larger tlian the whole correlation energy. Electron correlation is furthermore overestimated, often by a factor close to 2, and bond strengths are as a consequence overestimated. Despite the simplicity of the fundamental assumptions, LSDA methods are often found to provide results with an accuracy similar to that obtained by wave mechanics HE methods. 

Table 3.1.1 Pair correlation energies for the four electrons in Be.

The orbitals from which electrons are removed can be restricted to focus attention on the correlations among certain orbitals. For example, if the excitations from the core electrons are excluded, one computes the total energy that contains no core correlation energy. The number of CSFs included in the Cl calculation can be far in excess of the number considered in typical MCSCF calculations. Cl wavefimctions including 5000 to 50 000 CSFs are routine, and fimctions with one to several billion CSFs are within the realm of practicality [53]. 

Krishnan R and Pople J A 1978 Approximate fourth-order perturbation theory of the electron correlation energy Int. J. Quantum Chem. 14 91-100... 

MP2 correlation energy calculations may increase the computational lime because a tw o-electron integral Iran sfonnalion from atomic orbitals (.40 s) to molecular orbitals (MO s) is ret]uired. HyperClicrn rnayalso need additional main memory arul/orcxtra disk space to store the two-eleetron integrals of the MO s. 

HyperChem supports MP2 (second order Mdllcr-l Icsset) correlation energy calcu latiou s u sin g any available basis set. lu order to save main memory and disk space, the HyperChem MP2 electron correlation calculation normally uses a so called frozen-core approximation, i.e. th e in n er sh el I (core) orbitals are omitted. A sett in g in CHHM.IX I allows excitation s from th e core orbitals to be include if necessary (melted core). Only the single poin t calcula-tion is available for this option. 

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 ... 

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. 

Size-extensivity is of importance when one wishes to compare several similar systems with different numbers of atoms (i.e., methanol, ethanol, etc.). In all cases, the amount of correlation energy will increase as the number of atoms increases. However, methods that are not size-extensive will give less correlation energy for the larger system when considered in proportion to the number of electrons. Size-extensive methods should be used in order to compare the results of calculations on different-size systems. Methods can be approximately size-extensive. The size-extensivity and size-consistency of various methods are summarized in Table 26.1. 

Numerous m.o.-theoretical calculations have been made on quinoline and quinolinium. Comparisons of the experimental results with the theoretical predictions reveals that, as expected (see 7.2), localisation energies give the best correlation. jr-Electron densities are a poor criterion of reactivity in electrophilic substitution the most reactive sites for both the quinolinium ion and the neutral molecule are predicted to be the 3-, 6- and 8-positions. ... 

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. 

Specifies the calculation of electron correlation energy using the Mpller-Plesset second order perturbation theory (MP2). This option can only be applied to Single Point calculations. 

Electron correlation. Electrons in an atom or molecule do not move entirely independently of each other but their movements are correlated. The associated correlation energy is often neglected in SCF calculations. 

We can compute all of the results except those in the first row by running just three jobs QCISD(T,E4T] calculations on HF and fluorine and a Hartree-Fock calculation on hydrogen (with only one electron, the electron correlation energy is zero). Note that the E4T option to the QCISDfT) keyword requests that the triples computation be included in the component MP4 calculation as well as in the QCISD calculation (they are not needed or computed by default). 

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