• This chapter is, verbatim, the paper: Shubin Liu, Peter Süle, Roberto López-Boada and Ágnes Nagy, "Applications to atoms, ions, and molecules of a novel form of the correlation energy density functional", Chem. Phys, Lett., 257, 68(1996).
  

4.2 Theoretical formulation
In Refs. [15-16], it has been established that the correlation energy density functional Ec[r] can be expanded effectively in terms of homogeneous functionals An[r], n=1, 2, 3, ..., i.e.,

(1)
It was shown [15, 17] that An[r] is homogenous of degree (1-n) in coordinate scaling, that is,

(2)
with
(3)
This scaling of the density keeps it normalized for all g. Consequently, one has [18,15]

(4)
One of us [19] has proved that the exact form of An[r] should have the form,

(5)
where an([r]; r1, r2) is another unknown functional homogeneous of degree (-n) in coordinate scaling, i.e.,

(6)
As an approximation, at first, we suppose that all An[r] are local, that is,

(7)
With this assumption, one finds [20,21]

(8)
Combination of Eqs. (4) and (8) gives

(9)
which means that An[r] is homogeneous of degree (4-n)/3 in density scaling. With the uniqueness theorem in Appendix of Ref. [15], one finally obtains the analytical expressions for all An[]

(10)
where the Cn are constants to be determined. The consequence is that, under the local assumption, the correlation energy density functional Ec[r] (and similarly its kinetic component Tc[r]) are expressible as

(11)
where N is the total number of electrons of the system, and the constants C1 to C4 are to be determined.
It should be mentioned that the DFT-version correlation energy and the conventional correlation energy are different quantities in principle. The former contains two parts by definition: the potential part Vc[r] and the kinetic part Tc[r], while the latter is defined as the difference between the experimental total energy and the Hartree-Fock-limit total energy. We do not know the relationship between the two correlation energies at the present stage. But theoretical data for them for some atoms show that they are almost the same. We thus suspect that the conventional correlation energy can also be approximately expressed as in Eq. (11). Numerical data below will support this hypothesis.

4.3 Calculational method
The densities used in Eq. (11) for first 18 neutral atoms are both the Hartree-Fock (HF) and CI densities. The HF densities are calculated explicitly using the wave functions of Clementi and Roetti [22], and the CI densities are obtained using the ATOMCI program [23]. A full CI calculation was performed on He. Multireference CI calculations with single and double excitations were performed on the other atoms. Densities from the CI wave functions were scaled to satisfy the virial theorem. The CI densities are finally spherically averaged and spin-traced. The densities for positive ions, e.g., Be isoelectronic series and Ne ion series, are explicitly computed from HF wave functions of Clementi and Roetti [22]. And the HF densities for 21 diatomic and polyatomic molecules are from the GAMES [23] program package. We first perform HF calculations in a triple zeta Gaussian basis set with additional 3d-polarization functions (Duning's TZP basis [24,25]) to obtain the SCF and HF wave functions. A numerical integration is then carried out to obtain the densities and integrals of density to the 2/3 and 1/3 powers.
The coefficients that appear in Eq. (11) are obtained by the least square fit using the best theoretical values of the correlation energy of the first 18 atoms and their HF or CI densities. The best CI correlation values of the neutral atoms and ions, and molecules are from Refs. [26] and [27], respectively. Five schemes are employed in the fits, denoted by CI3, CI4, HF3, HF4 and EcTc, respectively. CI3 means using the CI densities of first 18 neutral atoms and their "best" correlation energies to fit the first three coefficients in Eq. (11), while CI4 means using the CI densities to fit all four parameters in their equation, and similarly with HF3 and HF4. Scheme EcTc, however, is from Ref. [15], in which Ec and Tc are fitted together to obtain the four coefficients by using CI densities. The coefficients for EcTc are C1=-0.0532, C2=0.0121, C3=-0.0003, and C4=-0.0070.
For comparison, we also present correlation energy values of other approximate functionals, for example, Wigner, PL, VWN, and ZLP local forms and the Lee-Yang-Parr (LYP) nonlocal form. Results of PL, VWN and LYP forms for atoms and ions are obtained from the Gaussian 92 package, and those of the Wigner form are computed using densities described above and Süle and Nagy's reparameterized Wigner form [28]. For molecules, Wigner and LYP values were taken from Refs.29 and 30, respectively, while VWN and PL values were computed from Gaussian 92. To obtain the results of the ZLP (Zhao-Levy-Parr) local form, we reparameterized the two parameters a0 and k appearing in their formula by using the HF densities and the exact values of conventional correlation energies of first 18 neutral atoms. We obtain a0 = 0.050587 and k = 0.226154. ZLP values are then calculated using the HF densities for above ions, and molecules.

4.4 Results
At first, in Table 1 we present our fitted results. One sees that CI3, CI4, HF3 and HF4 give almost the same results. This justifies the use of HF densities in calculating correlation energies of atoms. Besides, since there is no essential difference between three and four parameter fits, we prefer to use the simpler three-parameter formula. In what follows, we will only present HF3 results for ions and molecules. A feature of these fits is that the successive coefficients become smaller very rapidly in the series except for the constant term. In Column 7, what we show is that using the CI densities and the coefficients for the DFT-version correlation energy [15], we can fairly well predict the conventional correlation energies, especially for heavy atoms. In Column 8, best theoretical DFT Ec values are shown. Compared with the conventional values, one finds that they follow at the same trends, semiquantitatively.
In Table 2, results of application of Schemes HF3 and EcTc, together with those of Wigner, VWN, PL, ZLP and LYP, for 21 diatomic and polyatomic molecules are shown. Curves of the exact values vs. approximate ones are plotted in Fig. 1. For clarity, only one of our results, Scheme EcTc, is shown in this and later figures. It is seen that among VWN, PL, Wigner, ZLP and Eq. (11), the new formula gives the best prediction for these molecules. The results are comparable to those of the LYP formula for these species, as shown in the Figure. In order to see if the same conclusion is true for ions, results of Be isoelectronic (4-electron) ion series up to Ar+14 are shown in Table 3; corresponding curves are plotted in Fig. 2. In Table 4 and Fig. 3, the same procedure is applied to Ne positive ion series from 0 to +7. It is found that the VWN and Perdew'81 (PL) forms always overestimate the correlation energy, while the Wigner form underestimates it in most cases. Our formula, however, gives values that lie just between them in these three plots, and thus give better results than other local forms. In addition in some cases, Eq. (11) even gives better results than LYP, which is nonlocal in nature.
It is mentioned that the ZLP (Zhao-Levy-Parr) form was originally obtained from the Wigner form by the adiabatic connection formulation. Its results, however, vary quite substantially from case to case. It is seen from Tables 3 and 4, and Figs. 2 and 3 that the ZLP form gives very good results for ions we considered here, and is comparable with LYP and Eq. (11) and sometimes even better than them. For molecules, however, as shown in Table 2 and Fig. 1, the results are poor. It is worse than its parent Wigner form.

4.5 Discussion
It is interesting to discuss the relationship of our formula with the Wigner form. Recall that the Wigner form was resulted from the analytical parameterization of high- and low-density limits of the homogeneous electron gas for the intermediate density range, while Eq. (11) was obtained analytically, under the locality assumption, from adiabatic connection and functional expansion formulations of a coupled Hamiltonian based on the constrained-search procedure. Consider the Wigner form,

(12)
where a and k are constants to be determined. Recently, Süle and Nagy [29] reparamerized it to reproduce the conventional Ec values of eight closed-shell atomic species, giving a=-0.1247 and k=1.0910. In the high-density limit, Eq. (12) can be expanded up to the third order as

(13)
which is exactly the same series as we obtained in Eq. (11) except for different coefficients. This shows that in the high-density regime, Eqs. (11) and (12) may be parameterized to produce almost the same results. For the low-density limit, however, one has

(14)
All terms on the right-hand side of Eq. (14) are not allowed in the local density approximation of the correlation energy, as proved in Ref. [15]. It is therefore clear that the Wigner form takes good account of the high-density regime, but it does poorly in the low-density one. It is believed that the electron dynamic correlation is a short-range behavior, and for atoms and molecules it becomes important only when the electron density is high, i.e., not far from nuclei. This fact somehow compensates the weakness of the Wigner form, and thus it turns out to be one of the commonly-used approximate forms for the correlation energy density functional we have so far.
Owing to the simple, general, and relatively accurate performance of the local correlation energy functional of Eq. (11), we expect that it would be readily applicable to other systems.

Table 1. Exact, fitted and predicted correlation energies for the first and second-row neutral atoms. See text. Atomic units.**

Atom	Exact	CI3	CI4	HF3	HF4	EcTc	DFT Ec *
H	0.0	-0.004	0.003	-0.005	0.003	-0.018	0.0
He	-0.043	-0.049	-0.042	-0.049	-0.041	-0.063	-0.042
Li	-0.045	-0.058	-0.056	-0.060	-0.057	-0.070	-0.051
B	-0.094	-0.070	-0.071	-0.072	-0.073	-0.094	-0.093
B	-0.125	-0.113	-0.115	-0.112	-0.114	-0.137	-0.129
C	-0.156	-0.163	-0.164	-0.162	-0.163	-0.185	-0.161
N	-0.188	-0.218	-0.218	-0.218	-0.217	-0.237	-0.188
O	-0.258	-0.274	-0.272	-0.275	-0.273	-0.289	-0.261
F	-0.325	-0.332	-0.328	-0.334	-0.330	-0.342	-0.322
Ne	-0.390	-0.392	-0.387	-0.394	-0.388	-0.397	-0.376
Na	-0.396	-0.410	-0.408	-0.413	-0.411	-0.410	-0.401
Mg	-0.438	-0.421	-0.423	-0.424	-0.426	-0.432	-0.452
Al	-0.470	-0.454	-0.457	-0.456	-0.460	-0.466	-0.491
Si	-0.505	-0.511	-0.513	-0.499	-0.502	-0.521	-0.527
P	-0.540	-0.550	-0.552	-0.549	-0.552	-0.560	-0.559
S	-0.605	-0.599	-0.601	-0.599	-0.601	-0.607	-0.629
Cl	-0.666	-0.658	-0.658	-0.658	-0.658	-0.662	-0.689
Ar	-0.722	-0.716	-0.714	-0.717	-0.715	-0.715	-0.744

• From Ref. [31]
• The fitted parameters:
  

Mols.	VWN	PL	LYP	Wigner	ZLP	HF3	EcTc	Exact
H2	-0.131	-0.096	-0.038	-0.029	-0.030	-0.024	-0.043	-0.041
LiH	-0.291	-0.217	-0.089	-0.083	-0.082	-0.076	-0.099	-0.083
Li2	-0.441	-0.329	-0.133	-0.134	-0.129	-0.080	-0.114	-0.122
Be2	-0.613	-0.462	-0.200	-0.193	-0.185	-0.152	-0.191	-0.205
CH4			-0.294	-0.241	-0.233	-0.253	-0.295	-0.293
B2	-0.807	-0.614	-0.289	-0.265	-0.254	-0.246	-0.286	-0.330
NH3			-0.318	-0.268	-0.258	-0.286	-0.320	-0.338
H2O			-0.340	-0.314	-0.286	-0.321	-0.345	-0.367
HF	-0.906	-0.701	-0.363	-0.335	-0.318	-0.357	-0.372	-0.387
LiF			-0.418	-0.343	-0.375	-0.431	-0.445	-0.447
C2H2			-0.443	-0.386	-0.365	-0.397	-0.442	-0.476
C2	-1.010	-0.773	-0.384	-0.344	-0.332	-0.355	-0.390	-0.514
HCN			-0.464	-0.410	-0.394	-0.430	-0.466	-0.527
C2H4			-0.497	-0.417	-0.403	-0.438	-0.493	-0.528
N2	-1.224	-0.942	-0.483	-0.435	-0.414	-0.461	-0.489	-0.546
CO	-1.228	-0.946	-0.484	-0.440	-0.419	-0.463	-0.490	-0.550
C2H6			-0.551	-0.426	-0.423	-0.479	-0.547	-0.553
O2	-1.429	-1.103	-0.583	-0.533	-0.505	-0.572	-0.592	-0.657
H2O2			-0.638	-0.569	-0.542	-0.611	-0.642	-0.691
F2	-1.667	-1.295	-0.675	-0.633	-0.597	-0.676	-0.689	-0.746
CO2			-0.791	-0.720	-0.680	-0.766	-0.797	-0.829

Table 3. Prediction of the coventional correlation energy for Be isoelectronic (4-electron) ion series. Atomic units.

Ions	VWN	PL	LYP	Wigner	ZLP	HF3	EcTc	Exact*
B+	-0.330	-0.252	-0.107	-0.056	-0.109	-0.103	-0.122	-0.111
C+2	-0.353	-0.273	-0.115	-0.067	-0.125	-0.127	-0.141	-0.126
N+3	-0.372	-0.291	-0.119	-0.077	-0.139	-0.144	-0.154	-0.141
O+4	-0.389	-0.306	-0.123	-0.086	-0.152	-0.157	-0.163	-0.154
F+5	-0.404	-0.320	-0.125	-0.095	-0.163	-0.166	-0.170	-0.167
Ne+6	-0.417	-0.332	-0.127	-0.104	-0.173	-0.174	-0.176	-0.180
Na+7	-0.418	-0.333	-0.123	-0.111	-0.183	-0.180	-0.180	-0.193
Mg+8	-0.433	-0.347	-0.128	-0.119	-0.191	-0.185	-0.184	-0.205
Al+9	-0.445	-0.358	-0.129	-0.126	-0.199	-0.189	-0.187	-0.217
Si+10	-0.452	-0.368	-0.130	-0.132	-0.206	-0.193	-0.189	-0.230
P+11	-0.464	-0.377	-0.131	-0.139	-0.213	-0.196	-0.191	-0.242
S+12	-0.473	-0.385	-0.131	-0.145	-0.220	-0.199	-0.193	-0.254
Cl+13	-0.480	-0.392	-0.131	-0.151	-0.226	-0.201	-0.195	-0.266
Ar+14	-0.488	-0.399	-0.132	-0.156	-0.231	-0.203	-0.196	-0.278

• From Ref. [27]

	VWN	PL	LYP	Wigner	ZLP	HF3	EcTc	Exact*
Ne	-0.950	-0.741	-0.383	-0.191	-0.352	-0.394	-0.398	-0.392
Ne+1	-0.860	-0.672	-0.338	-0.180	-0.328	-0.365	-0.367	-0.327
Ne+2	-0.761	-0.592	-0.286	-0.167	-0.302	-0.332	-0.333	-0.266
Ne+3	-0.649	-0.500	-0.227	-0.153	-0.272	-0.296	-0.296	-0.203
Ne+4	-0.577	-0.450	-0.199	-0.137	-0.240	-0.256	-0.257	-0.187
Ne+5	-0.500	-0.394	-0.165	-0.120	-0.207	-0.215	-0.216	-0.179
Ne+6	-0.417	-0.332	-0.127	-0.104	-0.173	-0.174	-0.176	-0.180
Ne+7	-0.304	-0.239	-0.080	-0.089	-0.143	-0.135	-0.137	-0.051

• From Ref. [27].

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Captions of figures:

Fig. 1. Curves of the exact conventional correlation energies vs. approximate ones for 21 molecules. The dashed line is the exact values, and the dotted line is the LYP nonlocal values. Four local forms, VWN, Perdew'81 (PL), Wigner, and this work (Scheme EcTc) are plotted in solid lines. Another local form, ZLP, is plotted in a centered line to distinguish itself from the Wigner form.

Fig. 2. Curves of the exact conventional correlation energies vs. approximate ones for Be isoelectronic ion series from B+ to Ar+14.

Fig. 3. Curves of the exact conventional correlation energies vs. approximate ones for Ne positive ion series from Ne0 to Ne+7.