Mn Dimer can be Described Accurately with Density Functional Calculations when Self-interaction Correction is Applied
Aleksei V. Ivanov, Tushar K. Ghosh, Elvar ?. Jónsson, Hannes Jónsson
MMn Dimer can be Described Accurately withDensity Functional Calculations whenSelf-interaction Correction is Applied
Aleksei V. Ivanov, † , ‡ Tushar K. Ghosh, † , ¶ Elvar ¨O. J´onsson, † , ¶ and HannesJ´onsson ∗ , † , ¶ † Science Institute and Faculty of Physical Sciences, University of Iceland VR-III, 107Reykjav´ık, Iceland. ‡ St. Petersburg State University, 199034, St. Petersburg, Russia ¶ Department of Applied Physics, Aalto University, Espoo, FI-00076, Finland.
E-mail: [email protected] a r X i v : . [ phy s i c s . c h e m - ph ] F e b bstract Qualitatively incorrect results are obtained for the Mn dimer in density functionaltheory calculations using the generalized gradient approximation (GGA) and similarresults are obtained from local density and meta-GGA functionals. The bond betweenthe atoms is predicted to be an order of magnitude too strong and about an ˚Angstrømtoo short, and the coupling is predicted to be ferromagnetic rather than antiferromag-netic. Explicit, self-interaction correction (SIC) applied to a commonly used GGAenergy functional, however, provides close agreement with both experimental data andhigh-level, multi-reference wave function calculations. These results show that the fail-ure is not due to strong correlation but rather the single electron self-interaction that isnecessarily introduced in estimates of the classical Coulomb and exchange-correlationinteractions when only the total electron density is used as input. The correctedfunctional depends explicitly on the orbital densities and can, therefore, avoid the in-troduction of self-Coulomb interaction. The error arises because of over-stabilizationof bonding d -states in the minority spin channel resulting from an overestimate ofthe d -electron self-interaction in the GGA exchange-correlation functional. Since thecomputational effort in the self-interaction corrected calculations scales with systemsize in the same way as for regular semi-local functional calculations, this approachprovides a way to calculate properties of Mn nanoclusters as well as biomolecules andextended solids where Mn dimers and larger cluster are present, while multi-referencewave function calculations can only be applied to small systems. for example in the oxygen-evolving photo-system II. The properties of such systems are ofgreat interest and theoretical calculations could in principle provide valuable information tohelp gain an understanding of the role Mn atoms play. However, theoretical calculationsprove to be particularly challenging for these systems. The Mn dimer is the simplest man-ganese complex and it represents an important test system for theoretical methods that couldultimately be used for the larger and more complex systems. Its properties are quite wellknown from electron spin resonance and optical absorption measurements of dimers confinedin a rare gas matrix. The ground state is found to be antiferromagnetic with a small bondenergy of 0.13 ± . and a large bond length of 3.4 ˚A. Resonance Raman spectra givevibrational frequency of 76 cm − in a Kr matrix and 68 cm − in a Xe matrix. High level wave function calculations of an isolated Mn dimer give results in closeagreement with the experimental measurements. Both complete-active-space self-consistentfield (CAS-SCF) in combination with second-order perturbation theory, as well as multi-reference calculations have been carried out. They predict bond energy in the range 0.10- 0.14 eV and bond length of 3.3 - 3.8 ˚A in an antiferromagnetic ground state with a couplingconstant of J=-5.8 cm − . It is clear from the close agreement between these calculations andthe experimental measurements that the effect of the confining rare gas matrix is small andthe measurements indeed probe the properties of the Mn dimer. Such high-level, wave func-tion based calculations become, however, impractical for larger systems due to the strongscaling of the computational effort with system size.Kohn-Sham density functional theory (KS-DFT) can provide a valuable tool for the-oretical studies of large systems with up to several hundred atoms, free – in principle – ofadjustable parameters with unknown values. Unfortunately, the results of such calculationsfor the Mn dimer with commonly used energy functionals such as local density approximation(LDA) and generalized gradient approximation (GGA), are in strong disagreement with the3xperimental measurements and the high level, wave function calculations. The groundstate is predicted to be ferromagnetic rather than antiferromagnetic with the bond betweenthe Mn atoms being much too strong and too short. All electron calculations with variousGGA functionals give a binding energy of around 0.9 eV and bond length of 2.57-2.61 ˚A. Below, we present results using the elaborate meta-GGA functional, SCAN, where the Mndimer is also found to be poorly described, although the bond energy is not overestimated asmuch. We will from now on refer to calculations using LDA, GGA or meta-GGA functionalscollectively as semi-local DFT calculations.Similar failure in DFT calculations of magnetic coupling constants has also been reportedfor various manganese binuclear complexes. Shortcomings of DFT calculations are oftenascribed to strong correlation effects and systems where large errors are obtained are oftencharacterized as ‘highly correlated systems’. The Mn dimer is an example of such a system.It has, however, been shown that calculated results for the Mn dimer can be improved whenexact exchange is added to a semi-local DFT functional to form a hybrid functional, but theweight of the exact exchange in this blend needs to be significantly larger than the range of0.20 to 0.25 in commonly used functionals of this form.
Semi-local DFT functionalstend to give errors of opposite sign to those of Hartree-Fock calculations, so some mix of thetwo can often be tuned to give the desired result.An alternative reason for the failure of semi-local DFT calculations is the self-interactionerror that is necessarily introduced in the estimate of the classical Coulomb interactionbetween the electrons when only the total electron density is used as input. This error ishighly non-local. In the exchange-correlation part of the semi-local functionals a correctionis estimated, i.e. a self-interaction contribution of opposite sign, but the cancellation canmathematically not be complete and, therefore, self-interaction error remains and can lead toerratic results. The question remains whether the large error in semi-local DFT calculationsof the Mn dimer is due to strong correlation or self-interaction, or possibly some other sourceof error. Previously it has been speculated that the self-interaction error is not responsible4ut rather strong correlation in the interaction between the Mn atoms.
In this letter, the results of variational, self-consistent calculations are presented where theself-interaction error is removed explicitly as proposed by Perdew and Zunger. As describedbelow, the results are found to be in remarkably good agreement with the high level wavefunction calculations. This shows that the problem in the semi-local DFT calculations ofthe Mn dimer is not strong correlation, but rather the self-interaction error. In addition toproviding important insight into the reason for the failure of DFT for the Mn dimer, thisopens up an avenue for accurate calculations of larger systems containing Mn complexesbecause the computational effort of the GGA calculations with self-interaction correctionscales with system size in the same way as regular GGA calculations.In Kohn-Sham density functional theory, the energy of an electronic system is estimatedfrom E KS [ ρ ] = T s + (cid:90) v ext ( r ) ρ ( r ) d r + E C [ ρ ] + E xc [ ρ ↑ , ρ ↓ ] (1)were, T s is the kinetic energy of an independent electron system described by spin-orbitals φ and the electron density ρ σ ( r ) = (cid:88) i σ ρ i σ ( r ) = (cid:88) i σ | φ i σ ( r ) | (2)corresponds to the ground state electron density of the interacting electron system for eachspin channel, σ = {↑ , ↓} . The energy due to the electron-nuclei interaction, described bythe external potential v ext , can be evaluated correctly from the total electron density, ρ ( r ) = ρ ↑ ( r ) + ρ ↓ ( r ), but the estimate of the classical Coulomb interaction between the electrons E C [ ρ ] = (cid:90) (cid:90) ρ ( r ) ρ ( r (cid:48) ) | r − r (cid:48) | d r d r (cid:48) (3)includes spurious interaction of each electron with itself. This is most clearly seen for a systemwith a single electron where non-zero interaction energy is obtained from this estimate.5 more accurate estimate can be obtained by using the spin-orbital densities, ρ i , whereinteraction of a spin-orbital with itself is avoided E SICC [ ρ , . . . , ρ N ] = E C [ ρ ] − (cid:88) i (cid:90) (cid:90) ρ i ( r ) ρ i ( r (cid:48) ) | r − r (cid:48) | d r d r (cid:48) = (cid:88) i (cid:88) j>i (cid:90) (cid:90) ρ i ( r ) ρ j ( r (cid:48) ) | r − r (cid:48) | d r d r (cid:48) (4)Here, the summation indices run over both spin channels, i = { i ↑ , i ↓ } . The exchange-correlation energy term in the KS-DFT functional, E xc , attempts to provide such a correc-tion but, because of the semi-local form, cannot accurately cancel out the non-local self-interaction error in E C [ ρ ].Perdew and Zunger proposed a procedure where the net self-interaction error is estimatedfor each spin-orbital and the sum subtracted from the Kohn-Sham functional E SIC [ ρ , . . . , ρ N ] = E KS [ ρ ] − (cid:88) i ( E C [ ρ i ] + E xc [ ρ i , . (5)This provides the correction to the classical Coulomb energy as in Eqn. (4) and also ad-dresses the extent to which the exchange-correlation functional is able to cancel out theself-interaction by evaluating the net self-interaction for each spin-orbital separately. For aone electron system, the corrected functional is guaranteed to be self-interaction free, butfor many electron systems, this correction procedure is approximate. While this approachwas originally proposed for the LDA functional, it can also be applied to GGA functionalsbut there it has been found to give an overcorrection and a scaling by 1/2 has been shown togive good results for a wide range of systems and properties such as atomization energy ofmolecules, band gaps of solids and the balance between localized and delocalized electronicstates. We choose here to use the PBE functional, a GGA functional approximationthat is commonly used in calculations of condensed matter and refer to the corrected func-tional as PBE-SIC/2. As discussed below, the scaling of 1/2 is not essential here, similarresults are obtained for the Mn dimer without the scaling.The corrected functional is not unitary invariant as it depends explicitly on the orbital6ensities, as indicated in Eqs. (4-5), and it turns out that the optimal spin-orbitals thatminimize the energy of the system are hybrid orbitals, i.e. linear combinations of the canon-ical orbitals that are eigenfunctions of the Hamiltonian. The calculations need to make useof complex-valued functions to represent the optimal spin-orbitals. Here, a real-space gridhas been used combined with projector augmented wave (PAW) to represent the effect ofthe frozen core electron of the atoms. A localized atomic orbital basis set is used includ-ing primitive Gaussians from the def2-TZVPD basis set augmented with a single-zetabasis. Tests against calculations using full flexibility of the real-space grid with mesh sizeof 0.15 ˚A give nearly identical results, showing that the atomic basis set is sufficient. Theenergy is variationally minimized using an exponential transform direct optimization methoddescribed elsewhere.
Fig. 1 shows the energy of the three lowest energy electronic states of the Mn dimer asa function of the distance between the atoms, calculated with the PBE functional with andwithout self-interaction correction. Three relevant, low lying electronic states are found inthe calculations: two ferromagnetic states, Σ + u and Π u , and an antiferromagnetic Σ + g state. The self-interaction corrected functional predicts the antiferromagnetic state to bethe ground state, and gives a binding energy of 0.18 eV and bond length of 3.32 ˚A, in closeagreement with the experimental results as well as the high level wave function calculations.The ferromagnetic Σ + u state is only 0.04 eV higher in energy at the optimal bond length.A vibrational frequency of 69 cm − is obtained from the ground state energy curve, ingood agreement with the experimental measurements. The magnetic coupling constant iscalculated using the relationship J = E AF − E F (cid:104) S F (cid:105) − (cid:104) S AF (cid:105) , (6)where E AF and E F are the energy of the antiferromagnetic and ferromagnetic states, and < S > is evaluated taking into account spin contamination. This gives J= -5.8 cm − ,7he same value as was obtained in the MCQDPT2 calculations. The scaling of the self-interaction correction is not important in this case. When the full correction is used as inEqn. (5), similar results are obtained, namely bond energy of 0.12 eV, bond length of 3.5˚A and vibrational frequency of 56 cm − . Most importantly, the self-interaction correction,scaled or not scaled, gives the right antiferromagnetic ground state.Different results are obtained with the uncorrected PBE functional, as can be seen fromFig. 1. There, the ferromagnetic state Π u is lowest in energy and even the other ferro-magnetic state, Σ + u , turns out to be lower than the antiferromagnetic, Σ + g state at theexperimental bond length. The calculated binding energy in the ground state is 0.90 eVwith a bond length of 2.6 ˚A in strong disagreement with best estimates. These results areconsistent with previously reported calculations using semi-local functionals. At a largedistance between the Mn atoms, the Π u state becomes higher in energy than the othersas it dissociates into a d s configuration for one of the Mn atoms. Calculations with themeta-GGA SCAN functional were also carried out using the VASP software and theresults are qualitatively similar to the PBE results, in that they also give Π u as the groundelectronic state, but the binding energy is smaller than for PBE, 0.46 eV and bond lengthlonger, 2.58 ˚A, but still far from the correct values.In order to analyse this failure of the semi-local functional calculations, a molecularorbital diagram is shown in Fig. 2. Interestingly, the occupation of minority-spin molecularorbitals turns out to play an important role here. In the PBE calculation, the bondingminority spin orbital π (3 d ) is lower in energy than the anti-bonding σ ∗ (4 s ), and this leadsto d - d bond formation in the minority spin states. When the self-interaction correction isapplied, the relative energy of π (3 d ) and σ ∗ (4 s ) is reversed and the anti-bonding spin-orbitalis occupied instead. The relative energy of the d and s atomic orbitals is an important issuehere.It is well known that semi-local DFT functionals do not describe well the energy balancebetween localized and delocalized electrons, and this is reflected in the relative energy of d and8 .4 2.6 2.8 3.0 3.2 3.4 3.6 3.8R Mn Mn (Å)1.00.80.60.40.20.00.20.40.60.8 E n e r g y ( e V ) E x p . b o n d l e n g t h Exp. bond energyPBE,
11 + u PBE, g PBE, u PBE-SIC/2, u PBE-SIC/2, g PBE-SIC/2,
11 + u MCQDPT2 CASPT2
Figure 1: Energy of the Mn dimer in the three lowest lying electronic states as a function ofdistance between the atoms calculated with the PBE functional (filled diamonds) and self-interaction corrected PBE-SIC/2 (filled circles). The zero of energy is twice the energy of anMn atom in the S electron configuration. The ferromagnetic states Σ + u and Π u are shownin blue and green, respectively, and the antiferromagnetic state Σ + g in red. The ground statein the PBE calculations is ferromagnetic with a binding energy of nearly 0.9 eV and bondlength of 2.6 ˚A while the antiferromagnetic state is the ground state in the self-interactioncorrected PBE-SIC/2 calculations, with binding energy of 0.18 eV and bond length of 3.32 ˚A.Experimental estimates of the bond length and bond energy (shaded grey area indicatingthe estimated error bar) are shown with dashed lines. The triangles show results of highlevel calculations using MCQDPT2 (filled) and CASPT2 (open). The self-interactioncorrected PBE-SIC/2 calculation is in close agreement with experimental measurements aswell as the high level calculations, while the PBE results are qualitatively incorrect. Opendiamonds show results of calculations where the PBE energy is evaluated for the PBE-SIC/2electron density, showing that the dominant error is in the self-interaction rather than theelectron density. s atomic orbitals. The repulsive self-interaction error in the classical Coulomb interaction islarger the more localized the electrons are. One might, therefore predict that d -electrons arepredicted to be artificially high in energy compared with s -electrons. This, however, wouldgive a trend in the opposite direction to the results shown in the molecular orbital diagram9igure 2: Molecular orbital diagram for the lowest energy ferromagnetic state of the Mndimer calculated with the PBE functional and with the self-interaction corrected PBE-SIC/2functional, based on orbital energy of the canonical orbitals. In the PBE calculations, thebonding π (3 d ) minority spin orbital has lower energy than the antibonding σ ∗ (4 s ) orbital, andbecomes populated, resulting in a Π u ground state for the Mn dimer. When self-interactioncorrection is applied, in the PBE-SIC/2 functional, the relative energy of these two molecularspin-orbitals is reversed, and σ ∗ (4 s ) becomes populated resulting in a Σ + u state of the dimer.The magnetic coupling then makes the Σ g state slightly lower in energy than the Σ + u state,by 0.04 eV, resulting in an antiferromagnetic ground state in the PBE-SIC/2 calculation.The surfaces illustrating the molecular spin-orbitals in the insets correspond to a value of ± − / , where different colors indicate the sign for the PBE orbitals, but for PBE-SIC/2the amplitude of the orbitals is shown as they are complex.in Fig. 2, where the PBE functional ends up placing electrons in a molecular spin-orbitalformed from the d atomic orbitals rather than the one formed from the s atomic orbitals.The answer lies in the estimate of the self-interaction in the exchange-correlation part ofthe PBE functional, a contribution that should cancel out the repulsive self-interaction in theclassical Coulomb term. This estimate is based on an analysis of smoothly varying electrongas. The larger the deviation is from the uniform electron gas, the larger an error can beexpected in the exchange-correlation functional. The extent to which the self-interaction iscancelled out by the two contributions for the various atomic orbitals in the Mn atom as well10s a few other atoms is shown in Fig. 3. The cancellation of the self-interaction energy forthe 3 s atomic orbital is quite good, the net self-interaction being only 0.03 eV. But, the netself-interaction has a larger magnitude and is negative for the 3 d orbitals of the Mn atom,-2.8 eV. Similar trend is observed for orbitals of other atoms, as shown in the figure. Theself-interaction correction in the semi-local exchange-correlation functional is, therefore, anovercorrection and makes the d atomic orbitals too low in energy as compared with the s atomic orbitals. This leads to the population of a bonding π (3 d ) molecular spin-orbital inthe Mn dimer instead of an anti-bonding σ ∗ (4 s ) spin-orbital. As a result, the Mn dimer isoverbound in an incorrect electronic ground state in the semi-local DFT calculations.There are two aspects of the error in DFT calculations: (1) an error in the self-consistentelectron density, and (2) an error in the energy obtained for a given, possibly correct, electrondensity. A calculation using the PBE functional with the PBE-SIC/2 electron density asinput for the Mn dimer is also shown in Fig. 1. The binding energy for both ferromagneticstates is reduced, but the Π u state is still the lowest energy state and there is still largeover binding of the dimer. This shows that the self-interaction is the main source of errorrather than the electron density.To analyze this further, the electron interaction terms are evaluated using PBE for eachof the Mn molecular spin-orbitals separately using the optimal PBE-SIC/2 orbital densities(see Table 1). The net self-interaction energy estimated for the majority-spin Π u and Σ + u states differs only by ca.
20 meV so the relative energy of these states is not affected stronglyby the self-interaction error. The reason is that they have similar type of bonding. However,for the minority-spin orbitals, the magnitude of the self-interaction energy of the Π u stateis 1.2 eV larger than that of the Σ + u state. As a result, the subtraction of the self-interactionerror from the PBE energy has large effect and reverses the relative energy of these states.11 ( E C + E x )[] ( e V ) MnNa Al ScMn(a) s p d0.200.150.10 E c o r [] ( e V ) Na Al MnMn Sc(b)
Figure 3: (a) Net self-interaction, E C [ ρ i ]+ E x [ ρ i , s orbitals,the negative self-interaction energy in the exchange part of the PBE functional compensateswell the positive self-interaction energy in the classical Coulomb part, but for d orbitals itovercorrects. The d orbitals in the Mn atom are, therefore, too low in energy compared tothe s orbitals and this leads to incorrect ordering of the bonding π (3 d ) molecular spin-orbitaland the anti-bonding σ ∗ (4 s ) spin-orbital in the minority-spin channel, as illustrated in Fig.2. (b) Self-interaction in the correlation part of the PBE functional evaluated for the atomicorbitals as in (a). Here, the largest contribution is obtained for the s orbitals, but thiscontribution is an order of magnitude smaller than the one in (a).Table 1: The sum of the classical Coulomb and exchange-correlation self-interaction energyevaluated with PBE for each molecular spin-orbital separately, using optimal spin-orbitalsfrom the PBE-SIC/2 calculation. Only the 14 valence electrons are included. For themajority-spin orbitals, the difference is just 0.02 eV, but for the minority-spin the self-interaction in the Π u state is 1.2 eV larger than for the Σ + u state and it thereby reversesthe relative stability of these states.Majority spin Minority spin (cid:80) E C [ ρ i ] (cid:80) E xc [ ρ i ] total maj (cid:80) E C [ ρ i ] (cid:80) E xc [ ρ i ] total min Π u Σ + u d -electrons and artificially lowers their energy with respect to that of s -electrons be-cause of an overestimate of the self-interaction in the exchange correlation functionals. As aresult, an electronic state that dissociates into an Mn atom with a d s electron configurationbecomes the ground state and much too strong bonding is obtained between constructiveoverlap of the d -orbitals in the minority spin channel. The results presented here show thatan explicit self-interaction correction for each spin-orbital applied to the PBE functionalaccurately describes the Mn dimer, predicting bond distance, binding energy, vibrationalfrequency and magnetic coupling constant in close agreement with experimental measure-ments and high-level quantum chemistry calculations. Since the computational effort in theself-interaction corrected calculations scales with system size in the same way as DFT cal-culations with semi-local functionals, i.e. as the system size to the third power, the methodcan be applied to large systems including extended solids. Acknowledgement
This work was supported by the Icelandic Research Fund and the Academy of Finland. AVIis supported by a doctoral fellowship from the University of Iceland and thanks TuomasRossi and Valery Uzdin for helpful discussions.13 eferences (1) Hanson, G., Berliner, L., Eds.
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GGA, u Self-interaction corrected g uu