Dynamical correlation energy definition

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. 

Conventionally, correlation is explained as dynamic fluctuations in the electronic density. By definition, it is the difference of the exact energy from the HF energy. Correlation energy can be separated into dynamical and static contributions. Notice that exchange repulsion occurs between electrons of equal spin. This implies some correlated motion between these electrons which is absent for electrons of opposite spin in HF. When this correlation is related to intramolecular interactions it is called dispersion, a more difficult quantity to calculate. 

In addition to MP2, MP3, and MP4 calculations, CCSD(T), CASSCF, FOCI (First-Order Configuration Interaction), and sometimes SOCI (Second-Order Configuration Interaction) approaches have been used to ensure the convergence of the results. The complete definitions of the variational spaces used are given in [61,62,63]. Electronically-excited states have been obtained by means of the MC/P method, recently developed in our group [64,65] it couples a variational treatment to deal with the nondynamic correlation effects and a perturbation treatment to account for the dynamic correlation effects as well as the non-dynamic effects not treated at the variational level becanse of their limited contributions to the phenomena investigated. All electronic transitions reported here are vertical transition energies. 

The HF picture does not account for dynamic electron correlation. In fact, within formal definition of 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. 

Experimental work providing information on reaction kinetics— the time dependence of reactants and products under defined conditions—served indispensably to correlate structure-reactivity data and to provide estimates of transition state energies. Theory-based definitions of transition structures gave some clues as to how reactions might actually take place. But the dynamic aspects of chemical reactions remained inaccessible, or only poorly accessible. 

The two previous secfions were devoted to modeling quantum resonances by means of effective Hamiltonians. From the mathematical point of view we have used two principal tools projection operators that permit to focus on a few states of interest and analytic continuation that allows to uncover the complex energies. Because the time-dependent Schrodinger equation is formally equivalent to the Liouville equation, it is attractive to try to solve the Liouville equation using the same tools and thus establishing a link between the dynamics and the nonequilibrium thermodynamics. For that purpose we will briefly recall the definition of the correlation functions which are similar to the survival and transition amplitudes of quantum mechanics. Then two models of regression of a fluctuation and of a chemical kinetic equation including a transition state will be presented. 

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