Relating correlation measures: The importance of the energy gap

The concept of correlation is central to all approaches that attempt the description of many-body effects in electronic systems. Multipartite correlation is a quantum information theoretical property that is attributed to quantum states independent of the underlying physics. In quantum chemistry, however, the correlation energy (the energy not seized by the Hartree-Fock ansatz) plays a more prominent role. We show that these two different viewpoints on electron correlation are closely related. The key ingredient turns out to be the energy gap within the symmetry-adapted subspace. We then use a few-site Hubbard model and the stretched ${\mathrm{H}}_2$ to illustrate this connection and to show how the corresponding measures of correlation compare.

Figure 1

For the Hubbard model for three fermions on three sites we present several correlation measures as functions of the dimensionless coupling $U/t$. These are the distance $D({\mathrm{\Psi }}_{\mathrm{HF}},{\mathrm{\Psi }}_{\mathrm{GS}})$ of the GS to the HF state, $|E_{\mathrm{corr}}|/E_{\mathrm{gap}}$ and the one-particle correlation measures $S, \delta$ (see text).

Figure 2

For the stretched ${\mathrm{H}}_2$ we present several correlation measures as functions of the bond length. These are the distance $D({\mathrm{\Psi }}_{\mathrm{HF}},{\mathrm{\Psi }}_{\mathrm{GS}})$ of the GS to the HF state, $|E_{\mathrm{corr}}|/E_{\mathrm{gap}}$, and the one-particle correlation measures $S, \delta$ (see text).

Figure 3

Energy spectrum of the three-site three-fermion Hubbard model restricted to the Hilbert subspace where the ground state lies. The Hartree-Fock energy is also shown.