Information entropy as a measure of the correlation energy associated with the cumulant

A linear relation is found between Jaynes entropy and the correlation energy associated with the cumulant for a correlated molecular system, as demonstrated in ${\mathrm{H}}_2$ and a water molecule. The relation modifies the Collins conjecture, in which the correlation energy is defined as the total energy difference with reference to the Hartree-Fock counterpart. The linear behavior is system dependent and shifts slightly on a different basis set. This correlation energy is extensive. For a system whose wave function can be expressed as the antisymmetrized product of orthogonal groups, the correlation energy is just the sum of the individual groups. The finding offers a convenient way to construct an approximate correlation energy functional in reduced density-matrix-functional theory, which may be used in large-scale molecular dynamics simulations and potential energy surface calculations where the molecular geometry needs to be frequently updated.

Figure 1

Jaynes entropy of ${\mathrm{H}}_2$ is given as a function of internuclear separation $R$.

Figure 2

The correlation energy associated with the cumulant as a function of Jaynes entropy in ${\mathrm{H}}_2$. Also in display are the differences of two-electron energy between the HF and CI calculations, $U_{0\mathrm{ee}}-U_{\mathrm{ee}}$.

Figure 3

The comparison of a linear relation between the correlation energy associated with the cumulant and entropy for two basis sets in ${\mathrm{H}}_2$, aug-cc-pV6Z(s) and cc-pVDZ.

Figure 4

The correlation energy associated with the cumulant is plotted with respect to Jaynes entropy in ${\mathrm{H}}_2\mathrm{O}$. For comparison, the differences of two-electron energies between HF and CI calculations,$U_{0\mathrm{ee}}-U_{\mathrm{ee}}$, are also in display. The data are from Table 2.