Scaling in the correlation energies of atomic ions

We show through numerical investigations that the ground-state correlation energies of atomic ions follow an unexpectedly simple scaling relation, $E_c\approx Z^{4/3}{f}_c(Z/N)$, where $N$ is the number of electrons, $Z$ is the atomic number, and $f_c$ is a universal function, for which an analytic expression with a one-parameter fit can be provided. The relation agrees well with several sets of correlation energies obtained from different methods for atomic ions with $N=2,...,18$ and $Z=2,...,28$. Moreover, our relation gives a good agreement with neutral atoms up to $N\approx 90$. Our main result is readily applicable to estimating correlation energies of heavy elements, for which there are no available data in the literature. The simplicity of the relation may also have implications in the development of correlation functionals within density-functional theory.

Figure 1

The rms deviation of the scaling relation from the set of data for atomic ions in Refs. [28, 29, 30] as a function of the parameter $\alpha$ in Eq. (2). The best fit is obtained with $\alpha \approx 1.32\approx 4/3$.

Figure 2

Sign-reversed scaled correlation energies for atomic ions with $N=2,1\le Z\le 20$ [28] and $7\le N\le 18,N-1\le Z\le 28$ [30]. The solid line represent the function $f_c(Z/N)$, and the vertical gray line indicates the position of neutral atoms.

Figure 3

Sign-reversed scaled correlation energies for atomic ions computed with different methods (see the main text for details). The solid line shows $f_c(Z/N)$ from the scaling relation in Eq. (3), and the vertical gray line indicates the position of neutral atoms.

Figure 4

Sign-reversed correlation energies of neutral atoms as a function of the electron number $N$ (symbols) computed with different methods (see the text). The horizontal solid line shows our scaling relation in Eq. (3), i.e., $f_c(Z/N=1)=0.0165663$. The vertical gray lines indicate the positions of noble atoms with filled shells.