Consequences of approximating electron correlation effects

Figures & data

Figure 1. Coordinates utilised in this work (i) Inter-particle coordinates, $r_1,r_2$ and $r_{12}$ and (ii) Jacobi coordinates $r\equiv r_{12},R$ and θ.

Figure 2. (left) Cartesian coordinate system used in calculating dynamic Coulomb holes. The nucleus is fixed at the origin, electron 1 is fixed on the positive x axis and electron 2 is moved along the x axis. (right) The resulting dynamic Coulomb hole for the helium atom using the Cartesian schematic on the left.

Figure 3. The exact static Coulomb hole curve (solid red line) [23], compared to the difference between the intracule distribution functions $D_{\mathrm{FC}}(r)$ and $D_{\mathrm{CS}}(r)$ (solid orange line). The inset plots in (a,b) reveals the secondary Coulomb holes, and the inset plots in (c,d) show a zoom of the primary Coulomb hole region. (a) Li${}^{+}$ (b) He (c) H${}^{-}$ (d) Z${}_C^{\mathrm{FC}}$

Figure 4. The dynamic Coulomb holes for Li${}^{+}$ (left), He (center), and H${}^{-}$ (right); note the different y-axis scales. The nucleus is at the origin, electron 1 is fixed on the positive x axis (indicated by an arrow). Top row corresponds to electron 1 at the most probable distance from the nucleus $r_{max}$, middle row corresponds to the crossing point in the static Coulomb hole ${\mathrm{\Delta }}_1^{root}$, and row 3 corresponds to electron 1 at 2 $a_0$.

Table 1. Roots, areas and minima of the $D_{\mathrm{FC}}-D_{\mathrm{CS}}$ curves in the helium-like ions in atomic units.

system	${\mathrm{\Delta }}_1^{min}$	${\mathrm{\Delta }}_1^{root}$	${\mathrm{\Delta }}_2^{min}$	${\mathrm{\Delta }}_2^{root}$	${\mathrm{\Delta }}_1^{area}$	${\mathrm{\Delta }}_2^{area}$
Z${}_C^{\mathrm{FC}}$	2.02 (1.91, 5.8%)	4.87 (4.85, 0.4%)	–	–	2.08$\times {10}^{-1}$ (2.39$\times {10}^{-1}$ , −9.6%)	–
H${}^{-}$	1.58 (1.46, 8.2%)	3.21 (3.20, 0.3%)	–	–	1.12$\times {10}^{-1}$ (1.33$\times {10}^{-1}$, −15.8%)	–
He	0.59 (0.52, 13.5%)	1.07 (1.07, 0%)	4.08 (4.08, 0%)	3.56 (3.58, −0.6%)	3.26$\times {10}^{-2}$ (4.63$\times {10}^{-2}$, −29.6%)	6.28$\times {10}^{-4}$ (6.12$\times {10}^{-4}$, 2.6%)
Li${}^{+}$	0.37 (0.32, 15.6%)	0.66 (0.66, 0%)	2.50 (2.51, −0.4%)	2.19 (2.21, −0.9%)	1.99$\times {10}^{-2}$ (2.80$\times {10}^{-2}$, −28.9%)	3.63$\times {10}^{-4}$ (3.50$\times {10}^{-4}$, 3.7%)

Notes: ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$ refer to the primary and secondary Coulomb holes, respectively. Numbers in brackets are the Coulomb hole values using a HF reference [23] and the percentage difference between using CS and HF as reference.

Figure 5. (Left) exact DCH using (13(13) ${\rho }_{12}(r,r^{\mathrm{\prime }})=\underset{\mathrm{Fermi}\mathrm{Coulomb}\mathrm{hole}}{\underbrace{{\rho }_{xc}(r,r^{\mathrm{\prime }})}}-\underset{\mathrm{Fermi}\mathrm{hole}}{\underbrace{{\rho }_x(r,r^{\mathrm{\prime }})}},$(13) ) and (right) approximate DCH using (17(17) ${\rho }_c({\mathbf{r}}_1,{\mathbf{r}}_2)=\frac{{\mathrm{\Gamma }}_2^0({\mathbf{r}}_1,{\mathbf{r}}_2)[{\varphi }^2({\mathbf{r}}_1,{\mathbf{r}}_2)-2\varphi ({\mathbf{r}}_1,{\mathbf{r}}_2)]}{{\rho }^0({\mathbf{r}}_1)},$(17) ) for Li${}^{+}$, He and H${}^{-}$. The position of electron 1 is indicated by the circles along the base of each plot. The orange dashed line represents the dynamic Coulomb hole when electron 1 is at $r_{max}$.

Figure 6. Electron correlation energies as a function of nuclear charge Z. The black cross corresponds to $Z_C^{\mathrm{HF}}=1.031177528$ and the orange cross corresponds to $Z_C^{{\mathrm{HF}}_0}=0.828161008$. The grey shaded region represents unbound systems, i.e. with $Z\le 0.910$ [15,37]. The inset provides a zoomed view near the maximum in $E_{\mathrm{corr}}$.

Figure 7. FC and HF radial (intracule) energies, ${\epsilon }^{\mathrm{FC}}(r)$, ${\epsilon }^{\mathrm{HF}}(r)$ for Li${}^{+}$, He, H${}^{-}$ and $Z_C^{\mathrm{FC}}$. The difference between these two curves (shaded region) represents the electron correlation energy distribution, ${\epsilon }_{\mathrm{corr}}(r)$, where ${\int }_0^{\mathrm{\infty }}{\epsilon }_{\mathrm{corr}}(r)\mathrm{d}r=E_{\mathrm{corr}}$. (a) Li${}^{+}$ (b) He (c) H${}^{-}$ (d) Z${}_C^{\mathrm{FC}}$.

Figure 8. (a) The FC and HF radial (extracule) energies, ${\epsilon }^{\mathrm{FC}}(R)$, ${\epsilon }^{\mathrm{HF}}(R)$ and electron correlation energy distribution, ${\epsilon }_{\mathrm{corr}}(R)$, and (b) comparison of ${\epsilon }_{\mathrm{corr}}(R)$ with $E_{\mathrm{corr}}^{CS}(R)$ calculated using Equation (9(9) $\begin{array}{rl}{E}_{\text{corr}}^{\text{CS-LYP}}=-a\pi \int & \frac{\gamma (\mathbf{R})}{1+\frac{d}{q}{\rho }^{-1/3}(\mathbf{R})}\{\rho (\mathbf{R})+\frac{8b}{q^2}{\rho }^{-5/3}(\mathbf{R})\\ & [{\rho }_{\alpha }(\mathbf{R})t_{\text{HF}}^{\alpha }+{\rho }_{\beta }(\mathbf{R})t_{\text{HF}}^{\beta }-\rho (\mathbf{R})t_W(\mathbf{R})]\\ & e^{-\frac{c}{q}{\rho }^{-1/3}(\mathbf{R})}\}\text{d}\mathbf{R},\end{array}$(9) ). All data are for the helium atom.

Figure 9. Radial electron correlation energy distributions, $E_{\mathrm{corr}}(R)$, calculated using (a) the CS open-shell energy formula (9(9) $\begin{array}{rl}{E}_{\text{corr}}^{\text{CS-LYP}}=-a\pi \int & \frac{\gamma (\mathbf{R})}{1+\frac{d}{q}{\rho }^{-1/3}(\mathbf{R})}\{\rho (\mathbf{R})+\frac{8b}{q^2}{\rho }^{-5/3}(\mathbf{R})\\ & [{\rho }_{\alpha }(\mathbf{R})t_{\text{HF}}^{\alpha }+{\rho }_{\beta }(\mathbf{R})t_{\text{HF}}^{\beta }-\rho (\mathbf{R})t_W(\mathbf{R})]\\ & e^{-\frac{c}{q}{\rho }^{-1/3}(\mathbf{R})}\}\text{d}\mathbf{R},\end{array}$(9) ) and (b) the LYP functional (10(10) $\begin{array}{rl}{E}_{\mathrm{corr}}^{\mathrm{LYP}}& =-a\pi \int \frac{\gamma (\mathbf{R})}{1+\frac{d}{q}{\rho }^{-1/3}(\mathbf{R})}\{\rho (\mathbf{R})+\frac{8b}{q^2}{\rho }^{-5/3}(\mathbf{R})\\ & [\phantom{+\frac{1}{18}({\rho }_{\alpha }(\mathbf{R}){\mathrm{\nabla }}^2{\rho }_{\alpha }(\mathbf{R})+{\rho }_{\beta }(\mathbf{R}){\mathrm{\nabla }}^2{\rho }_{\beta }(\mathbf{R}))]{\mathrm{e}}^{-\frac{c}{q}{\rho }^{-1/3}(\mathbf{R})}\}}{2}^{2/3}{C}_F{\rho }_{\alpha }^{8/3}(\mathbf{R})+2^{2/3}{C}_F{\rho }_{\beta }^{8/3}(\mathbf{R})-\rho (\mathbf{R})t_W\\ & +\frac{1}{9}({\rho }_{\alpha }(\mathbf{R})t_W^{\alpha }(\mathbf{R})+{\rho }_{\beta }(\mathbf{R})t_W^{\beta }(\mathbf{R}))\\ & +\frac{1}{18}({\rho }_{\alpha }(\mathbf{R}){\mathrm{\nabla }}^2{\rho }_{\alpha }(\mathbf{R})+{\rho }_{\beta }(\mathbf{R}){\mathrm{\nabla }}^2{\rho }_{\beta }(\mathbf{R}))]\\ & {\mathrm{e}}^{-\frac{c}{q}{\rho }^{-1/3}(\mathbf{R})}\phantom{\frac{1}{18}}\}\mathrm{d}\mathbf{R},\end{array}$(10) ), for anionic (B${}^{-}$, F${}^{-}$), neutral (Li, O) and cationic (C${}^{+}$, N${}^{+}$) atoms. The inset plots show a zoomed view near the origin, highlighting the small positive regions. (a) CS (b) LYP.

Table 2. Inter-particle expectation values in atomic units, nucleus-electron cusp ${\nu }_{31}$ (exact value $=-Z$) and electron-electron cusp ${\nu }_{12}$ (exact value $=0.5$), for $Z_C^{\mathrm{FC}}$, H${}^{-}$, He and Li${}^{+}$ using either the fully-correlated (FC), Hartree-Fock (HF) or Colle-Salvetti (CS) wavefunctions.

	Z${{}_{\mathbf{C}}}^{\mathrm{FC}}=0.911028224$			H${}^{-}$
Property	FC	HF	CS	FC	HF	CS
〈r1〉	4.146	2.989 00	2.924 668 660	2.710 178 278	2.503 959	2.452 502 536
〈r12〉	7.083	4.493 9	4.483 972 863	4.412 694 49	3.739 273	3.730 877 063
〈δ(r1)〉	0.119 094	0.108 4	0.108 370 481	0.164 552 8	0.154	0.154 432 600
〈δ(r12)〉	0.001 114	0.008 601 2	0.002 932 080	0.002 738	0.012 983 476 3	0.004 870 980
〈1/r1〉	0.578 108	0.595 799 1	0.587 185 657	0.683 261 767 65	0.685 672 15	0.676 064 953
〈1/r12〉	0.223 374	0.337 767	0.313 387 449	0.311 021 502 2	0.395 484 84	0.368 658 025
${\nu }_{31}$	−0.911 0	−0.911 1	−0.911 058 578	−0.999 999 91	−1.000 04	−1.000 005
${\nu }_{12}$	0.499 4	n/a	0.5	0.499 98	n/a	0.5
	He			Li${}^{+}$
	FC	HF	CS	FC	HF	CS
〈r1〉	0.929 472 294 87	0.927 273 404 7	0.914 050 347	0.572 774 149 97	0.572 366 815 001	0.566 098 399
〈r12〉	1.422 070 255 56	1.362 124 383 6	1.358 856 514	0.862 315 375 4	0.838 314 780 311	0.836 578 885
〈δ(r1)〉	1.810 429	1.797 959	1.795 389 539	6.852 00	6.836 07	6.828 714 314
〈δ(r12)〉	0.106 345	0.190 603 997 806	0.116 862 484	0.533 72	0.770 240 340 922	0.555 670 985
〈1/r1〉	1.688 316 800 71	1.687 282 215 281	1.671 771 326	2.687 924 397	2.687 419 466 644	2.669 333 872
〈1/r12〉	0.945 818 448	1.025 768 869	0.983 541 121	1.567 719 559 1	1.651 686 396 960	1.602 611 500
${\nu }_{31}$	−1.999 996	−1.999 999 4	−1.999 999 8	−2.999 999 1	−2.999 999 2	−2.999 999 96
${\nu }_{12}$	0.499 95	n/a	0.5	0.499 998	n/a	0.5

Note: FC and HF values taken from our previous work [23].

A.L. Baskerville, A.W. King and H. Cox, R. Soc. Open. Sci. 6, 181357 (2019). doi:10.1098/rsos.181357

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Data availability statement

The wavefunctions for each system (Li+, He, H- and ZC), along with all the particle density, energy density data, and correlation energy data presented in the paper, are available at http://dx.doi.org/10.25377/sussex.21566013.