Energy electron correlation, definition

Let us turn to the electron correlation energy. According to its definition... 

These concepts, inherent in the TDHF formalism, generalize immediately to orbital functional theory, when electronic correlation energy is included in the model. Given some definition that determines a reference state for any... 

The term "electron correlation energy" is usually defined as the difference between the exact nonrelativistic energy and the energy provided by the simplest MO wave function, the mono-determinantal Hartree-Fock wave function. This latter model is based on the "independent particle" approximation, according to which each electron moves in an average potential provided by the other electrons [14]. Within this definition, it is customary to distinguish between non dynamical and dynamical electron correlation. 

In spectroscopy we may distinguish two types of process, adiabatic and vertical. Adiabatic excitation energies are by definition thermodynamic ones, and they are usually further defined to refer to at 0° K. In practice, at least for electronic spectroscopy, one is more likely to observe vertical processes, because of the Franck-Condon principle. The simplest principle for understandings solvation effects on vertical electronic transitions is the two-response-time model in which the solvent is assumed to have a fast response time associated with electronic polarization and a slow response time associated with translational, librational, and vibrational motions of the nuclei.92 One assumes that electronic excitation is slow compared with electronic response but fast compared with nuclear response. The latter assumption is quite reasonable, but the former is questionable since the time scale of electronic excitation is quite comparable to solvent electronic polarization (consider, e.g., the excitation of a 4.5 eV n — n carbonyl transition in a solvent whose frequency response is centered at 10 eV the corresponding time scales are 10 15 s and 2 x 10 15 s respectively). A theory that takes account of the similarity of these time scales would be very difficult, involving explicit electron correlation between the solute and the macroscopic solvent. One can, however, treat the limit where the solvent electronic response is fast compared to solute electronic transitions this is called the direct reaction field (DRF). 49,93 The accurate answer must lie somewhere between the SCRF and DRF limits 94 nevertheless one can obtain very useful results with a two-time-scale version of the more manageable SCRF limit, as illustrated by a very successful recent treatment... 

The use of the UHF method as the standard for the independent electron model also rationalises the definition of correlation energy (9). 

Light of definite energy and polarization has a selective power to exclusively excite dye molecules whose electronic transition energy and orientation match these parameters. Thus, if a dye is excited by polarized fight, its emission will also be highly polarized. Depolarization occurs only when the time correlation of these selectively excited species is lost due to their rotation or participation in some... 

The numerical determination of E grr by the use of many-body theory is a formidable task, and estimates of it based on E j and E p serve as important benchmarks for the development of methods for calculating electron correlation effects. The purpose of this work is to obtain improved estimates of Epp by combining the leading-order relativistic and many-body effects which have been omitted in Eq. (1) with experimentally determined values of the total electronic energy, and precise values of Epjp. We then obtain empirical estimates of E grr for the diatomic species N2, CO, BF, and NO using Epip and E p and the definition of E g in Eq. (1). 

Two key pieces of theoretical data are required to obtain an empirical estimate of the correlation energy from the experimental data collected in the preceding section the total molecular Hartree-Fock energy and the relativistic corrections . It is implicit in the definition of the correlation energy presented in Eq. (1) that the total electronic energy ofa given molecule, Ef, may be divided into three constituent parts,... 

The theorem holds if the exchange-correlation potential VXc equals the functional derivative of the exchange-correlation energy /iXc with respect to the electron density p - an operational definition, which is intrinsic to DFT. 

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