# Is the correlation energy of a system always negative?

The correlation energy of a system is defined as the difference between the exactly energy and the energy in the Hartree-Fock method: $$E_\mathrm{cor} = E - E_\mathrm{HF}$$. In the case of an atom or a molecule, is it possible say that $$E_\mathrm{cor} < 0$$, since the interaction between the electrons is positive?

• Being nit-picky, there are systems in which HF provides the exact answer and thus $E_{\text{cor}}$ is zero and thus not negative. – TAR86 Sep 30 at 10:46

## 1 Answer

The answer to this question rests on the fact that Hartree-Fock is a variational method. A variational method is one where you supply a guess wavefunction ($$|\Psi\rangle$$) with some parameter(s) ($$\alpha$$) and then use those parameter(s) to minimize the energy. Doing this, you are guaranteed that your resulting energy is an upper bound to the exact ground state energy. $$E_{\text{exact}} \leq \frac{\langle\Psi_{\alpha}|\hat{H}|\Psi_{\alpha}\rangle}{\langle\Psi_{\alpha}|\Psi_{\alpha}\rangle}$$

So, if we are talking about the exact correlation energy (such as one given by full CI) then the answer is that the correlation energy must always be negative. This is because Hartree-Fock always provides an upper bound to the exact ground state energy. However, if we use a non-variational method to approximate the correlation energy (such as coupled cluster) then we could find cases where we calculate a positive correlation energy, but this is distinct from the exact correlation energy.

(For an authoritative description of this in the context of quantum chemistry with examples and practice problems, I recommend the first chapter of Szabo and Ostlund's Modern Quantum Chemistry)