Nonrelativistic 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. 

For simplicity, only nondegenerate nonrelativistic ground states will be considered. The reference-state density function is p = , niPi = X/ lli4>i4>i-where the occupation numbers nt are one or zero and , ni — N. Because p has this orbital decomposition, any density functional F p] is also an orbital functional F[ j, ip. This implies a natural definition of an energy functional E = E0 + Ec,... 

Here E and E are the exact energies of the two individual molecules A and B when they are isolated, while E" is the exact energy of the supersystem (molecular complex, for example). Theoretically, these quantities can be obtained from the exact solution of the Schrodinger equation for the corresponding systems. (We remain within the nonrelativistic Born-Oppenheimer model.) This requires the definition of the Hamiltonians H", H and H" , and one feels challenged to handle these Hamiltonians in a common (e.g., perturbational) scheme. This point is not at all trivial especially if approximate model Hamiltonians are used. In what follows we shall consider this issue emphasizing the points where the second quantized approach can help to clarify the situation. 

The set of approximations leading to Eq. 3.33 is often summarized as the sudden approximation. The thumb rule described above concerns an independent particle description of the shake phenomenon. With this description one can thus associate each pair of occupied and unoccupied orbital indices to two specific shake-up states, a triplet and a singlet parent coupled state. Systems that can be described this way receive shake intensity through the effect of orbital relaxation — the more relaxation the more intensity. The effect of relaxation on intensities is clearly also seen from the overlap element in Eq. 3.33. However, in order to obtain quantitative prediction for shake-up intensities, more sophisticated models, including the effect of electron correlation, must be considered. With the definition of correlation energy as the difference between one-determinant, Hartree-Fock, energy and the exact expectation value of the electronic nonrelativistic Hamiltonian, final-state correlation is always... 

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