Orbital currents in the Colle-Salvetti correlation energy functional and
  the degeneracy problem
Orbital currents in the Colle-Salvetti correlation energy functional and
the degeneracy problem
S. Pittalis1, S. Kurth1, S. Sharma1,2 and E.K.U. Gross1
1 Institut für Theoretische Physik, Freie Universität Berlin,
Arnimallee 14, D-14195 Berlin, Germany and
2 Fritz Haber Institute of the Max Planck Society,
Faradayweg 4-6, D-14195 Berlin, Germany.
(Dated: October 30, 2018)
Abstract
Popular density functionals for the exchange-correlation energy typically fail to reproduce the
degeneracy of different ground states of open-shell atoms. As a remedy, functionals which explic-
itly depend on the current density have been suggested. We present an analysis of this problem
by investigating functionals that explicitly depend on the Kohn-Sham orbitals. Going beyond the
exact-exchange approximation by adding correlation in the form of the Colle-Salvetti functional
we show how current-dependent terms enter the Colle-Salvetti expression and their relevance is
evaluated. A very good description of the degeneracy of ground-states for atoms of the first and
second row of the periodic table is obtained.
http://arxiv.org/abs/0704.1593v2
I. INTRODUCTION
Common approximations to the exchange-correlation functional of density functional
theory (DFT) [1, 2] and spin-DFT (SDFT) [3] often fail to reproduce the degeneracy of dif-
ferent ground states. An illustrative example are ground states of open-shell atoms where
one usually and erroneously obtains different total energies for states with zero and non-
vanishing current density. The local spin density approximation (LSDA) gives rather
small splittings (of the order of 1 kcal/mole), but generalized gradient approximations
(GGAs) and meta-GGAs can introduce splittings of 10 kcal/mol [4, 5, 6, 7, 8, 9, 10, 11].
These spurious energy splittings would vanish if the exact exchange-correlation func-
tional could be used. The exchange-correlation energy functional can be represented
in terms of the exchange-correlation hole function. Considering current-carrying states,
Dobson showed how the expression for the exchange-hole curvature has to be changed
by including current-dependent terms [12]. Later, Becke found the same kind of terms
in the short-range behavior of the exchange-correlation hole, and observed that they
also enter the spin-like correlation-hole function [13]. For open-shell atoms, inclusion
of these current-dependent terms results in spurious energy splittings of less than 1
kcal/mole [9]. Along these lines, Maximoff at al. [10] worked out a correction for the
system-averaged exchange hole of the Perdew-Burke-Ernzerhof (PBE) GGA [14], which
improves the corresponding spurious splittings. Alternatively, Tao and Perdew [11, 15]
proposed a scheme for the extension of existing functionals using ideas of current density
functional theory (CDFT) [16, 17] which, again, improves the description of the degener-
The performance of the exact-exchange (EXX) energy functional - which, by definition,
describes the exchange hole correctly - has been evaluated for the spurious splittings in
DFT and SDFT [18]. In the EXX-DFT (i.e., spin-restricted calculations using one and the
same Kohn-Sham potential for spin-up and spin-down orbitals) the degeneracy is well re-
produced to within 0.6 kcal/mole but, surprisingly, in EXX-SDFT (i.e., spin-unrestricted
calculations using two Kohn-Sham potentials, one for spin-up and one for spin-down or-
bitals) spurious splittings up to 3 kcal/mole are obtained. In particular, current-carrying
states always have higher total energies than states without current. This observation
motivated the applications of the optimized-effective-potential (OEP) method [19, 20, 21]
generalized to current-spin-density functional theory (CSDFT) to these current-carrying
states [22]. As expected, EXX-CSDFT total energies for current-carrying states are lower
than those of EXX-SDFT. However, this lowering is too small to give a substantial im-
provement of the spurious energy splittings. These studies lead to the conclusion that
correlation is needed for any further improvement.
The construction of a correlation energy functional compatible with EXX is a difficult
task [23, 24, 25], but for spherical atoms it was found that EXX combined with the Colle-
Salvetti (CS) functional for correlation [26, 27, 28] leads to very accurate total energies
[29]. The CS functional has been used to derive the popular Lee-Yang-Parr (LYP) func-
tional [30], which is most commonly used together with Becke’s exchange functional [31]
(BLYP) and in hybrid schemes such as B3LYP [32, 33]. On the other hand, the CS correla-
tion energy functional also has its limitations [34, 35, 36]. In particular, while short-range
correlations are well described [35], very important long-range correlations are missing.
These correlations often cannot be ignored in molecules and solids, but are negligible in
atoms. This fact, together with the encouraging results for spherical atoms [29], indicates
that it is appropriate to employ the CS functional to analyze the degeneracy problem for
open-shell atoms beyond EXX. Furthermore, the expression of the CS functional also al-
lows a reconsideration of the relevance of the orbital currents as ingredient of correlation
functionals.
Although the general density functional formalism to deal with degenerate ground
states includes densities which can only be obtained by a weighted sum of several deter-
minantal densities [37, 38], as in many previous investigations [8, 9, 10, 11] we only con-
sider densities which may be represented by a single Slater determinant of Kohn-Sham
orbitals.
II. THEORY
Going beyond the EXX approximation, we here consider the correlation-energy func-
tional of Colle and Salvetti [26]. This expression relies on the assumption that the corre-
lated two-body reduced density matrix may be approximated by the Hartree-Fock (HF)
two-body reduced density matrix ρHF2 (r1, r2), multiplied by a Jastrow-type correlation
factor. After a series of approximations, the following expression is obtained for the cor-
relation energy
Ec = −4a
ρHF2 (r, r)
1 + bρ−
3 (r)
, r− s
3 (r)
1 + dρ−
3 (r)
where ρHF2 (r, s) is expressed in terms of the average and relative coordinates r =
(r1+r2)
and s = r1 − r2. Here, ρ(r) is the electron density and the constants a = 0.049, b = 0.132,
c = 0.2533, d = 0.349 are determined by a fitting procedure using the Hartree-Fock (HF)
orbitals for the Helium atom.
Following Lee, Yang and Parr, this expression can be restated as a formula involving
only the total charge-density, the charge-density of each Hartree-Fock orbital and their
gradient and Laplacian [30]. In this derivation, the single-particle orbitals are tacitly as-
sumed to be real. We denote the resulting expression as CSLYP. In the following, we
relax this restriction and consider complex orbitals. We then proceed in analogy to the
inclusion of current-dependent terms in the Fermi-hole curvature [9, 12] and in the ex-
tension of the electron-localization-function (ELF) [39] for time-dependent states [40]. As
a consequence, in addition to the term already present in CSLYP expression, the current
densities of the single-particle orbitals appear in the final formula. In order to obtain this
expression, which in the following will be denoted as JCSLYP, we rewrite the Laplacian of
the Hartree-Fock (HF) two-body reduced density matrix in Eq.(1) in terms of the original
particle coordinates
∇21 +
∇22 −
∇1 · ∇2
2 (r1, r2)|r1=r2 . (2)
where
2 (r1, r2) =
ρ(r1)ρ(r2)−
1,σ (r1, r2)ρ
1,σ (r1, r2) . (3)
Here,
1,σ (r1, r2) =
ψk,σ(r1)ψ
k,σ(r2), (4)
is the first-order HF density matrix (for a single Slater determinant) expressed in terms
of the single-particle orbitals ψk,σ(r). The corresponding spin-density is simply given by
ρσ(r) = ρ
1,σ (r, r). (5)
Allowing the single-particle orbitals ψk,σ(r) to be complex, a given orbital not only gives
the contribution ρk,σ(r) = |ψk,σ(r)|
2 to the density, but also the contribution jp k,σ(r) =
ψk,σ(r)∇ψ
k,σ(r)
to the paramagnetic current density which is given by
jp,σ(r) =
jp k,σ(r) . (6)
After some straightforward algebra, the Laplacian of the second-order HF reduced den-
sity matrix takes the final form
ρ(r)∇2ρ(r)−
(∇ρ(r))
ρσ(r)∇
ρσ(r) (7)
ρσ(r)
(∇ρk,σ(r))
ρk,σ(r)
+ J(r)
where
J(r) =
ρσ(r)
j2pσ(r)
ρσ(r)
j2p k,σ(r)
ρk,σ(r)
contains all the current-dependent terms. Alternatively, Eq. (7) may also be expressed in
terms of the non-interacting kinetic energy density
τσ(r) =
|∇ψk,σ(r)|
(∇ρk,σ(r))
ρk,σ(r)
j2p k,σ(r)
ρk,σ(r)
ρ(r)∇2ρ(r)−
(∇ρ(r))
ρσ(r)∇
ρσ(r)
2ρσ(r)τσ(r)− jpσ(r)
. (10)
Comparison of Eq.(7) and Eq.(10) shows that J(r), as defined in Eq.(8), also contains
current-dependent terms coming from the kinetic energy density. Thus, this would also
suggest to reconsider the gradient expansion of τ for current-carrying states. Impor-
tant consequences maybe expected for all approximated exchange-correlation functional
involving the kinetic energy density as ingredient, such as the CS functional, and meta-
GGAs. This issue will be specifically considered in a future work. In the next section, we
assess the performance of the CS functional, and, in particular, the relevance of J(r), in
reproducing the degeneracy of atomic states.
III. RESULTS AND DISCUSSION
We consider ground states of open-shell atoms having densities that can be repre-
sented by a single Slater determinant of Kohn-Sham orbitals. Due to the symmetry of the
problem, these Slater determinants are eigenstates of the z-component of both spin and
orbital angular momentum. The single-particle orbitals are generally complex-valued
and states with different total magnetic quantum numbers, ML, correspond to differ-
ent current densities. By means of an accurate exchange-correlation functional the same
total energies would be obtained. In the previous section, we have shown how current-
dependent terms enter the expression of the CS functional when complex-valued orbitals
are considered. Here, we evaluate the performance of EXX plus the CS functional in re-
producing the degeneracy and study the effect of the current-dependent terms, in SDFT
and DFT calculations.
We consider atoms of the first and second row of the periodic table: these are the ref-
erence cases for which a vast amount of numerical data is available [8, 9, 10, 11, 18, 22].
In analogy to the procedure where Hartree-Fock orbitals are used as input to the CS for-
mula, we have evaluated the correlation energies in a post-hoc fashion using Kohn-Sham
(KS) orbitals. We expand the KS orbitals in Slater-type basis functions (QZ4P of Ref. [42])
for the radial part, multiplied with spherical harmonics for the angular part. We obtain
the KS orbitals from self-consistent EXX-only calculations employing the approximation
of Krieger, Li, and Iafrate (KLI) [41], which has been shown to be extremely good at least
for small systems [21]. In principle a functional should be evaluated with KS orbitals
obtained from self-consistent calculations, and this is certainly possible for the CS func-
tional [29]. However, thanks to the variational nature of DFT, it is common experience to
observe only minor quantitative differences between post-hoc and self-consistent evalu-
ation of total energies. This is the reason why the functionals designed for solving the
degeneracy problem are typically evaluated in a post-hoc manner [9, 10, 11].
Table I shows the spurious energy splittings (difference in the total energies) between
Kohn-Sham Slater determinants with total magnetic quantum number |ML| = 1 and
ML = 0, from our SDFT and DFT calculations. The deviation of these total energies from
the exact values is plotted in Fig. (1), where again the spurious energy splittings are visi-
SDFT (DFT)
Atom ∆JCSLY P ∆CSLY P
B 0.8 (-0.3) 2.4 (1.4)
C 0.9 (-0.1) -3.2 (-4.3)
O -0.6 (-1.9) 0.9 (-0.4)
F -0.1 (-1.5) -3.5 (-5.1)
Al 0.4 (-0.5) 1.1 (0.2)
Si 0.5 (-0.4) -1.2 (-2.2)
S 0.1 (-1.6) 1.1 (-0.7)
Cl 0.7 (-1.3) -1.1 (-3.2)
me 0.3 (-1.0) -0.4 (-1.8)
mae 0.5 (1.0) 1.8 (2.2)
TABLE I: Spurios energy splittings, ∆ = E(|ML| = 1) − E(ML = 0) in kcal/mol for open-shell
atoms, computed in SDFT (DFT results in parenthesis for comparison). Correlation energy has
been added to the KLI-EXX energies including (JCSLYP) and neglecting (CSLYP) the current terms
of Eq. (7). The last row shows the mean error (me) and mean absolute (mae) of the spurious
splittings.
ble. These results highlight the importance of including J(r) in Eq. (7). In particular, it is
remarkable to observe that SDFT splittings are within 0.9 kcal/mol (with a mean error of
0.3 kcal/mol and a mean absolute error of 0.5 kcal/mol), and the corresponding DFT spu-
rious energy splittings are less than 1.9 kcal/mol (with mean errors of 1.0 kcal/mol). It is
worthwhile to note that in several cases inclusion of correlation leads to current-carrying
states (|ML| = 1) with lower total energy than zero-current states (ML = 0). These results
are in contrast to EXX-only [18] cases where: (a) the zero-current states are always lowest
in energy and (b) the spurious energy splittings are always smaller in DFT than in SDFT.
Going beyond EXX by including correlation in the form of CS functional accurate to-
tal energies can be obtained within the OEP method [29]. Figure (1) and Table II show
the deviations from exact total energies for the states with different magnetic quantum
B C O F Al Si S Cl
CSLYP   M
 = 0 
JCSLYP    M
CSLYP   |M
|= 1 
JCSLYP |M
B C O F Al Si S Cl
SDFT DFT
FIG. 1: Deviation from exact total energies for SDFT and DFT calculations employing the CS
functional, including (JCSLYP) and not including (CSLYP) the current-dependent term J in Eq. (7).
States with different magnetic quantum numbers ML are plotted. Exact total energies are taken
from Ref. [29] and references therein.
numbers, i.e. different current-carrying states. This further emphasizes the importance
of proper inclusion of J(r) in Eq. (7).
IV. CONCLUSIONS
We have shown that going beyond the exact-exchange approximation by including
correlation energy in the form of the Colle-Salvetti functional leads to a very good de-
scription of the degeneracy of open-shell atoms in both SDFT and DFT calculations.
Comparing DFT and SDFT results for the first and second row of the periodic table, on
average we observe a reduction of the spurious energy splittings and better total ener-
gies for SDFT. Furthermore, we have also shown how current-dependent terms enter the
expression of the Colle-Salvetti functional. If these terms are neglected, the degeneracy
is not well described and the total energies are also less accurate. Thus, this analysis re-
confirms the advantage of properly including the orbital current dependent terms as an
JCSLYP (CSLYP)
SDFT DFT SDFT DFT
|ML| 0 0 1 1
me 2.3 (16.1) 4.7 (18.5) 2.6 (15.7) 3.7 (16.8)
mae 4.8 (16.4) 5.4 (18.5) 4.7 (15.7) 4.7 (16.8)
TABLE II: Mean error (me) and mean absolute error (mae) in the total energies for the CS
functional, including (JCSLYP) and not including (CSLYP results in parenthesis) the current-
dependent term J in Eq. (7), in kcal/mol. Exact total energies are taken from Ref. [29] and ref-
erences therein.
ingredient in correlation functionals.
Acknowledgements
We acknowledge the Deutsche Forschungsgemeinschaft (SPP-1145) and NoE
NANOQUANTA Network (NMP4-CT-2004-50019) for financial support.
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	Introduction
	Theory
	Results and discussion
	Conclusions
	Acknowledgements
	References
