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 explicitly 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

Baerends E. J.

E. K. U. Gross

S. Kurth

S. Pittalis

S. Sharma

English

arXiv

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 explicitly 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 rows of the Periodic Table is obtained

Pittalis, S.

Kurth, S.

Sharma, S.

Gross, E.

MPG.PuRe

arXiv:0704.1593v2  [cond-mat.mtrl-sci]  31 Jul 2007Orbital currents in the Colle-Salvetti correlation energy functional andthe degeneracy problemS. Pittalis1, S. Kurth1, S. Sharma1,2 and E.K.U. Gross11 Institut für Theoretische Physik, Freie Universität Berlin,Arnimallee 14, D-14195 Berlin, Germany and2 Fritz Haber Institute of the Max Planck Society,Faradayweg 4-6, D-14195 Berlin, Germany.(Dated: January 21, 2008)AbstractPopular density functionals for the exchange-correlation energy typically fail to reproduce thedegeneracy 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 problemby investigating functionals that explicitly depend on the Kohn-Sham orbitals. Going beyond theexact-exchange approximation by adding correlation in the form of the Colle-Salvetti functionalwe show how current-dependent terms enter the Colle-Salvetti expression and their relevance isevaluated. A very good description of the degeneracy of ground-states for atoms of the first andsecond row of the periodic table is obtained.1I. INTRODUCTIONCommon approximations to the exchange-correlation functional of density functionaltheory (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 whereone usually and erroneously obtains different total energies for states with zero and non-vanishing current density. The local spin density approximation (LSDA) gives rathersmall 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 representedin terms of the exchange-correlation hole function. Considering current-carrying states,Dobson showed how the expression for the exchange-hole curvature has to be changedby including current-dependent terms [12]. Later, Becke found the same kind of termsin the short-range behavior of the exchange-correlation hole, and observed that theyalso enter the spin-like correlation-hole function [13]. For open-shell atoms, inclusionof these current-dependent terms results in spurious energy splittings of less than 1kcal/mole [9]. Along these lines, Maximoff at al. [10] worked out a correction for thesystem-averaged exchange hole of the Perdew-Burke-Ernzerhof (PBE) GGA [14], whichimproves the corresponding spurious splittings. Alternatively, Tao and Perdew [11, 15]proposed a scheme for the extension of existing functionals using ideas of current densityfunctional theory (CDFT) [16, 17] which, again, improves the description of the degener-acy.The performance of the exact-exchange (EXX) energy functional - which, by definition,describes the exchange hole correctly - has been evaluated for the spurious splittings inDFT and SDFT [18]. In the EXX-DFT (i.e., spin-restricted calculations using one and thesame 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-unrestrictedcalculations 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-carryingstates always have higher total energies than states without current. This observationmotivated the applications of the optimized-effective-potential (OEP) method [19, 20, 21]2generalized to current-spin-density functional theory (CSDFT) to these current-carryingstates [22]. As expected, EXX-CSDFT total energies for current-carrying states are lowerthan 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 thatcorrelation is needed for any further improvement.The construction of a correlation energy functional compatible with EXX is a difficulttask [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-rangecorrelations are well described [35], very important long-range correlations are missing.These correlations often cannot be ignored in molecules and solids, but are negligible inatoms. This fact, together with the encouraging results for spherical atoms [29], indicatesthat it is appropriate to employ the CS functional to analyze the degeneracy problem foropen-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 correlationfunctionals.Although the general density functional formalism to deal with degenerate groundstates 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-Shamorbitals.II. THEORYGoing 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 correlationfactor. After a series of approximations, the following expression is obtained for the cor-3relation energyEc = −4a∫drρHF2 (r, r)ρ(r)1 + bρ−83 (r)[∇2sρHF2(r + s2, r − s2)∣∣∣∣s=0]e−cρ−13 (r)1 + dρ−13 (r)(1)where ρHF2 (r, s) is expressed in terms of the average and relative coordinates r =12(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 involvingonly the total charge-density, the charge-density of each Hartree-Fock orbital and theirgradient 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, werelax this restriction and consider complex orbitals. We then proceed in analogy to theinclusion 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]. Asa consequence, in addition to the term already present in CSLYP expression, the currentdensities of the single-particle orbitals appear in the final formula. In order to obtain thisexpression, which in the following will be denoted as JCSLYP, we rewrite the Laplacian ofthe Hartree-Fock (HF) two-body reduced density matrix in Eq.(1) in terms of the originalparticle coordinates∇2sρHF2(r +s2, r −s2)∣∣∣∣s=0=(14∇21 +14∇22 −12∇1 · ∇2)ρHF2 (r1, r2)|r1=r2 . (2)whereρHF2 (r1, r2) =12ρ(r1)ρ(r2) −12∑σρHF1,σ (r1, r2)ρHF,∗1,σ (r1, r2) . (3)Here,ρHF1,σ (r1, r2) =Nσ∑k=1ψk,σ(r1)ψ∗k,σ(r2), (4)is the first-order HF density matrix (for a single Slater determinant) expressed in termsof the single-particle orbitals ψk,σ(r). The corresponding spin-density is simply given byρσ(r) = ρHF1,σ (r, r). (5)4Allowing the single-particle orbitals ψk,σ(r) to be complex, a given orbital not only givesthe contribution ρk,σ(r) = |ψk,σ(r)|2 to the density, but also the contribution jp k,σ(r) =Im(ψk,σ(r)∇ψ∗k,σ(r))to the paramagnetic current density which is given byjp,σ(r) =Nσ∑k=1jp k,σ(r) . (6)After some straightforward algebra, the Laplacian of the second-order HF reduced den-sity matrix takes the final form∇2sρHF2(r +s2, r −s2)∣∣∣∣s=0=14ρ(r)∇2ρ(r) −14(∇ρ(r))2 −14∑σρσ(r)∇2ρσ(r) (7)+14∑σρσ(r)[Nσ∑k=1(∇ρk,σ(r))2ρk,σ(r)]+ J(r)whereJ(r) =∑σρσ(r)[−j2pσ(r)ρσ(r)+Nσ∑k=1j2p k,σ(r)ρk,σ(r)](8)contains all the current-dependent terms. Alternatively, Eq. (7) may also be expressed interms of the non-interacting kinetic energy densityτσ(r) =12Nσ∑k=1|∇ψk,σ(r)|2 =18Nσ∑k=1(∇ρk,σ(r))2ρk,σ(r)+12Nσ∑k=1j2p k,σ(r)ρk,σ(r)(9)as∇2sρHF2(r +s2, r −s2)∣∣∣∣s=0=14ρ(r)∇2ρ(r) −14(∇ρ(r))2 −14∑σρσ(r)∇2ρσ(r)+∑σ(2ρσ(r)τσ(r) − jpσ(r)2). (10)Comparison of Eq.(7) and Eq.(10) shows that J(r), as defined in Eq.(8), also containscurrent-dependent terms coming from the kinetic energy density. Thus, this would alsosuggest to reconsider the gradient expansion of τ for current-carrying states. Impor-tant consequences maybe expected for all approximated exchange-correlation functionalinvolving 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, weassess the performance of the CS functional, and, in particular, the relevance of J(r), inreproducing the degeneracy of atomic states.5III. RESULTS AND DISCUSSIONWe 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 theproblem, these Slater determinants are eigenstates of the z-component of both spin andorbital angular momentum. The single-particle orbitals are generally complex-valuedand states with different total magnetic quantum numbers, ML, correspond to differ-ent current densities. By means of an accurate exchange-correlation functional the sametotal 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 orbitalsare 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 SDFTand 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 obtainthe KS orbitals from self-consistent EXX-only calculations employing the approximationof Krieger, Li, and Iafrate (KLI) [41], which has been shown to be extremely good at leastfor small systems [21]. In principle a functional should be evaluated with KS orbitalsobtained 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 toobserve 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 thedegeneracy problem are typically evaluated in a post-hoc manner [9, 10, 11].Table I shows the spurious energy splittings (difference in the total energies) betweenKohn-Sham Slater determinants with total magnetic quantum number |ML| = 1 andML = 0, from our SDFT and DFT calculations. The deviation of these total energies fromthe exact values is plotted in Fig. (1), where again the spurious energy splittings are visi-6SDFT (DFT)Atom ∆JCSLY P ∆CSLY PB 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-shellatoms, computed in SDFT (DFT results in parenthesis for comparison). Correlation energy hasbeen added to the KLI-EXX energies including (JCSLYP) and neglecting (CSLYP) the current termsof Eq. (7). The last row shows the mean error (me) and mean absolute (mae) of the spurioussplittings.ble. These results highlight the importance of including J(r) in Eq. (7). In particular, it isremarkable to observe that SDFT splittings are within 0.9 kcal/mol (with a mean error of0.3 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 isworthwhile to note that in several cases inclusion of correlation leads to current-carryingstates (|ML| = 1) with lower total energy than zero-current states (ML = 0). These resultsare in contrast to EXX-only [18] cases where: (a) the zero-current states are always lowestin 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 showthe deviations from exact total energies for the states with different magnetic quantum7B C O F Al Si S Cl-1001020304050E(J)CSLYP - Eexact (kcal/mol)CSLYP   ML = 0 JCSLYP    ML = 0CSLYP   |ML|= 1 JCSLYP |ML|= 1B C O F Al Si S ClSDFT DFTFIG. 1: Deviation from exact total energies for SDFT and DFT calculations employing the CSfunctional, 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 takenfrom Ref. [29] and references therein.numbers, i.e. different current-carrying states. This further emphasizes the importanceof proper inclusion of J(r) in Eq. (7).IV. CONCLUSIONSWe have shown that going beyond the exact-exchange approximation by includingcorrelation 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, onaverage 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 theexpression of the Colle-Salvetti functional. If these terms are neglected, the degeneracyis not well described and the total energies are also less accurate. 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Orbital currents in the Colle-Salvetti correlation energy functional and the degeneracy problem

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Orbital currents in the Colle-Salvetti correlation energy functional and
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