Different perspectives on the correlation energy

For the correlation energy no general explicit expression is known, neither in terms of orbitals nor densities. Different ways to understand correlations are described below. 

Correlation energy probabilistic approach. Recalling the quantum mechanical interpretation of the wave function as a probability amplitude, we see that a product form of the many-body wave function corresponds to treating the probability amplitude of the many-electron system as a product of the probability amplitudes of individual electrons (the orbitals). Mathematically, the probability of a composed event is only equal to the probability of the individual events if the individual events are independent (i.e., uncorrelated). Physically, this means that the electrons described by the product wave function are independent.30 Such wave functions thus neglect the fact that, as a consequence of the Coulomb interaction, the electrons try to avoid 

28The Hartree-Fock and the Kohn-Sham Slater determinants are not identical, since they are composed of different single-particle orbitals, and thus the definition of exchange and correlation energy in DFT and in conventional quantum chemistry is slightly different [52]. 

29A lower bound is provided by the Lieb-Oxford formula, given in Eq. (64). 

30Correlation is a mathematical concept describing the fact that certain events are not independent. It can be defined also in classical physics, and in applications of statistics to other fields than physics. Exchange is due to the indistinguishability of particles, and a true quantum phenomenon, without any analogue in classical physics. 

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