Accepted Manuscript

Study of Ne-core and He-core Pseudopotential Errors in the MnO Molecule:

Quantum Monte Carlo Benchmark

Minyi Zhu, Lubos Mitas

PII: S0009-2614(13)00457-0

DOI: http://dx.doi.org/10.1016/j.cplett.2013.04.006

Reference: CPLETT 31155

To appear in: Chemical Physics Letters

Received Date: 14 January 2013

Accepted Date: 4 April 2013

Please cite this article as: M. Zhu, L. Mitas, Study of Ne-core and He-core Pseudopotential Errors in the MnO

Molecule: Quantum Monte Carlo Benchmark, Chemical Physics Letters (2013), doi: http://dx.doi.org/10.1016/

j.cplett.2013.04.006

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Study of Ne-core and He-core Pseudopotential Errors in the MnO Molecule:Quantum Monte Carlo Benchmark

Minyi Zhu, Lubos Mitas

Department of Physics and ChiPS, North Carolina State University, Raleigh, NC 27695, USA

Abstract

Accuracy of effective core potential (ECP) is studied for two sizes of cores by Density Functional Theory, Hartree-Fock and quantum Monte Carlo (QMC) methods using the MnO molecule as a test system. We compare the energydifferences between high-spin and low-spin states that were previously found to be problematic for transition metaloxide solids calculations with ECPs. In order to disentangle errors caused by ECPs and by subsequent methods used incalculations, we construct a scalar-relativistic He-core ECP for Mn atom. We find that within high quality correlatedcalculations both Ne-core and He-core ECPs provide energy differences with comparable, high accuracy.

Keywords:effective core potential, quantum Monte Carlo, electron correlationPACS:

1. Introduction

For several decades transition metal oxide systemshave been one of the most challenging focal topics inchemical and condensed matter physics. Mainstreamapproaches such as density functional theory (DFT) andHartree-Fock (HF) have been applied to these systemsquite extensively, however, many of obtained results aremixed at best. For example, it is well-known that tra-ditional DFT approaches underestimate the band gap oftransition metal oxides very significantly and even failto predict correct ground states such as insulating anti-ferromagnets for FeO, CoO and other systems.

Several post-DFT approaches, such as hybrid func-tionals, on site Coulomb repulsion correction (LDA+U)or GW approximation have been applied in order tobetter capture the physics of these strongly correlatedsystems. Performance of some of these approaches fortransition metal oxide systems have been benchmarkedfor properties such as band structures, equations of stateand magnetic moments, see, for example, Ref. [1]. Al-though these more advanced DFT approaches showedmarked improvements, significant uncertainties are stillpresent, for example, in the prediction of pressure in-duced structural phase transitions [1, 2] and other prop-erties. Of particular interest are, for example, phasetransitions related to the collapse of local magnetic mo-ments such as the one observed in MnO [1, 2]. Similar

importance of high-spin vs low-spin state energy differ-ences can be found in physics and chemistry of molec-ular (nano) systems with potential use in spintronics orother applications [3, 4].

The electronic structure calculations for transitionmetal oxides are often carried out with the frozencore or with effective core potentials (pseudopotentials)(ECPs). It is assumed that such modification of the orig-inal Hamiltonian does not affect the valence propertieswithin the desired accuracy threshold. This assump-tion is well justified providing the core is sufficientlysmall and the corresponding ECPs are accurate enough.For 3d elements the Ne-core is considered to be deepand compact enough so that ECPs with 3s, 3p, 3d, 4sstates in the valence space faithfully capture the essen-tial physics of all valence properties.

Unfortunately, for some approaches the proper treat-ment of calculations with ECPs is less straightforwardthan one would expect and wish. We illustrate this issueby mentioning a few results of our recent study of MnO

Table 1: The discrepancies between all-electron and Ne-corepseudopotential for the energy difference (eV) between antiferromag-netic and nonmagnetic states in the MnO solid[2].

HF B3LYP PW91Eall

AFM→NM − EECPAFM→NM -0.55 0.18 0.57

Preprint submitted to Elsevier April 13, 2013

solid. In Table 1 [2] we show errors for the differncebetween the nonmagnetic (NM) and antiferromagnetic(AFM) states of MnO as obtained from all-electron andECP calculations.

Although highly accurate small Ne-core pseudopo-tentials [5, 6] were used, significant disagreements withall-electron calculation were found ranging from +0.57eV in generalized gradient DFT to -0.55 eV in HF.

In this work we show that these ”pseudopotential er-rors” originate in biases generated by the application ofapproximate methods such as DFT or HF to bonded sys-tems. Assuming that the valence space is sufficientlylarge, the net contribution of the ECPs to such errorsis marginal when compared to errors generated by theapproximate approaches applied to many-body systemswith bonds. In this work we explicitly demonstrate thatdeep cores have much smaller effect on the basic va-lence properties than the results from such approximatecalculations might suggest. The fact that the approx-imate theories can generate additional errors for ECPHamiltonians has been known in the DFT context fora long time. For example, the nonlinear core correc-tions for the DFT pseudopotentials have been devisedjust for this purpose, ie, to remove the differences be-tween all-electron and pseudopotential calculations [8].This has been originally justified by the DFT pseudopo-tential construction which is burdened by the nonlinear-ity of exchange-correlation functionals so that the core-valence partitioning is difficult when these two densi-ties overlap appreciably [8]. Very recently, these typesof corrections have been suggested not only for tran-sition elements but also for sp systems in order to de-crease the related errors [9]. However, the most accu-rate ECPs are generated from Dirac-Fock atomic cal-culations in energy-adjusted framework (ie, by repro-ducing not only one-particle norm conserving propertiesbut also excitation energies) and are designed to repro-duce the physical ion in a true ab initio sense, ie, foruse in calculations which can reach nearly exact eigen-states. Therefore any nonlinearity-like corrections areunusable in many-body wave function methods. Of thekey importance is the genuine many-body accuracy ofthe (effective) ECP Hamiltonian: its exact eigenstatesand eigenenergies. In fact, effective Hamiltonians of-ten come from various sources which are independentfrom subsequent methods used to solve the underlyingmany-body problem. Sometimes effective Hamiltoni-ans come from theory, in other cases they are generatedfrom experiments as it is common, for example, in nu-clear physics or in ultracold condensates. The key pointis that the effective Hamiltonian should accurately re-produce the system and its physics at the given scale of

interest. In many-body methods such as quantum MonteCarlo the key focus then becomes the accuracy and fea-sibility of such calculations.

What we present here is a more thorough analysis thatdemonstrates and quantifies this reasoning for an accu-rately solvable case of the MnO molecule. The calcula-tions illustrate the mentioned issues very clearly and fol-lows up our previous study where we originally raisedsome of these points [2].

Similarly to our previous study of the MnO solid,we carry out a series of calculations for high-spin andlow-spin states of the MnO molecule. We employDFT and HF methods together with quantum MonteCarlo (QMC) approach. Whenever feasible, the resultsfrom all-electron calculations are presented as well.QMC is an accurate albeit more expensive computa-tional approach which enables to treat the many-bodyeffects directly for systems which are too demanding formore traditional correlated wave function methods. Foranalysis of the pseudopotential vs the other errors wewill introduce He-core ECP for Mn which enables toclearly delineate discrepancies coming from the methodused to treat the bonded system (such as DFT). Us-ing He-core pseudopotential generated from Dirac-Fockcalculations, most of all-electron properties includingthe scalar-relativistic effects can be recovered in bothtraditional methods as well as in QMC. The removalof 1s electrons eliminates the dominant energy scale sothat the computational cost of the He-core QMC calcu-lations is not overpowering.

The rest of this paper is organized as follows. A briefsummary of relevant methodology is presented in Sec.II, including the construction of the He-core ECP. InSec. III, we show the performance of the constructedECP and primary results of DFT and QMC methods.We then discuss further aspects of ECPs and electronicstructure methods for similar systems.

2. Methodology

2.1. DFT and HF approaches

The practical implementation of the DFT methodscan be formulated through the following one-particleKohn-Sham Hamiltonian

HKS = −12∇2 + Vext(r) + VH(r, n) + Vxc(r, n) (1)

which contains kinetic energy, external potential,Hartree term generated by the one-particle charge dis-tribution and an exchange-correlation potential that is acomplicated nonlinear function of the charge density n.

2

The density is expanded in one-particle orbitals whichare self-consistent eigenstates of HKS as given by

HKSϕi = εiϕi (2)

with eigenvalues εi. The Hartree-Fock method canbe formulated in a similar manner except that theexchange-correlation potential includes only the Fockexchange and instead of density only it explicitly de-pends on the full set of occupied eigenstates {ϕi}.

2.2. Quantum Monte Carlo method

QMC is one of the many-body alternatives to moretraditional methods such as DFT, HF or post-HF meth-ods. It has become an important tool for exploration ofchallenging electronic structure problems in chemicaland condensed matter physics.

The most commonly used QMC approach is the diffu-sion Monte Carlo (DMC) which is a stochastic solutionof the stationary Schrodinger equation based on evolu-tion of the trial wave function in the imaginary time.Consider a variational/trial wave function ΨT . Projec-tor operator exp[−τ(H − ET )] will filter out any mixedhigher excitations so that for τ → ∞ one obtains theground state within the given symmetry :

Ψ0(R) = Ψ(R,∞) = limτ→∞

exp[−τ(H − ET )]ΨT (R) (3)

where R is the set of electron coordinates and ET is anenergy offset that keeps the normalization of the wavefunction asymptotically constant. H is the many-bodyelectron-ion Hamiltonian in the Born-Oppenheimer ap-proximation

H = −12

∑

i

∇2i +

∑

iI

VI(ri) +∑

i< j

1ri j

+ Vion−ion (4)

where i, j denote electrons and I ions. The trial func-tions which we employ are of the commomly usedSlater-Jastrow form

ΨT =∑

n

dnD↑n(ϕα)D↓n(ϕβ) exp

∑

i, j,I

U(ri j, riI , , r jI)

(5)

where φα, φβ are spin-up and spin-down orbitals and theexponential factor describes the correlations betweenthe particles (see, eg, Refs. [11, 12] for more details).

The imaginary time evolution of ΨT eliminates mostof the variational bias and the ground state energy can

be obtained by

EDMC = limτ→∞

⟨ΨT e−τH

∣∣∣ H |ΨT 〉⟨ΨT e−τH

∣∣∣ ΨT 〉=〈Ψ0|H |ΨT 〉〈Ψ0| ΨT 〉

=1M

M∑

i=1

HΨT (Rm)ΨT (Rm)

(6)

Here {Rm} is set of samples of electron configurationsof the product f (R, τ) = lim

τ→∞Ψ(R, τ)ΨT (R). The practi-

cal implementation of the DMC method requires the so-called fixed-node approximation which makes the nodesof the solution Ψ(R, τ) identical to the nodes of ΨT (R)avoiding thus the fermion sign problem. Therefore thefixed-node approximation causes the energy to dependon the node of trial wavefunction.

Another approximation in DMC comes from thetreatment of the nonlocal ECP in the fixed-node DMC.In order to avoid nonlocal ”hops” in the stochastic evo-lution, the nonlocal ECP is localized by a projectiononto the trial function [10]. This transforms the conven-tional nonlocal pseudopotential operator into a many-body local potential which forms a new effective Hamil-tonian [10]. It has been shown that this locality approx-imation is of the second order in the trial function de-viation from the exact one, similarly to the fixed-nodeerror. However, a number of calculations indicate thatthe localization error is overshadowed by the fixed-nodebias which appears to be dominant [11, 12, 13]. This istrue in particular for ECPs with small cores which arespatially rather confined such as the Ne-core ECPs forthe third row transition elements.

The errors of the fixed-node approximation and ECPlocalization in DMC are usually tackled via construc-tion of sufficiently accurate trial wave functions. For theoptimized Slater-Jastrow wave functions given abovethe accuracy of the DMC results is remarkably high.In most cases, the fixed-node DMC calculations recover90-95% of the correlation energy for both molecular andsolid systems with hundreds of electrons as has beendemonstrated on many applications [11, 12, 13].

2.3. Ne-core and He-core ECPs

In order to eliminate unnecessary computational costECPs are widely employed in calculations with heavyelements. The basic assumption is that the core statesare both energetically and spatially separated enoughthat it is possible to partition an atom into core and va-lence spaces and to construct effective operators which

3

mimic the action of the core on valence states. Typi-cally, the valence space includes only electronic stateswhich enter chemical bonds. For some elements, how-ever, this is too crude of an approximation.

Table 2: Parameters of the He-core ECP for Mn, the conventionalECP representation in most quantum chemistry programs is used:Vl(r) = r−2 ∑

k Aklrnkl e−Bklr2

Akl nkl Bkl

Vloc 142.035 2 184.8089169.403 3 791.537

23.000 1 398.671Vs 1129.804 2 154.000

-1714.430 2 360.33136.313 2 228.902

Vp -699.747 2 246.034182.536 2 206.325-102.771 2 458.19649.5482 2 533.693

In particular, for 3d transition metal elements it isnecessary to include more states into the valence spacebecause 3d orbitals occupy almost the same region ofspace as 3s and 3p semi-core subshells. Since thereare significant exchange and correlation effects betweenthese orbitals one needs to decrease the size of the coreso that the resulting ECP is transferable and repoducesMn atom with high fidelity in all bonded systems. Inthis paper, we employ two types of norm-conservingNe-core pseudopotentials in order to compare their per-formance. One is the energy-consistent pseudopotentialfrom the Stuttgart ECP table and it is labeled as STU[6]. The other is constructed by using Troullier-Martinsscheme and it is labeled as YN [5]. In addition, we con-struct a new He-core pseudopotential for the purpose todistinguish between the biases which are coming gen-uinely from ECPs vs biases from subsequent electronicstructure methods. Clearly, a straightforward test wouldbe a comparison with all-electron results. However, thisis more difficult for QMC than for other approaches be-cause all-electron QMC with relativistic effect is cur-rently unavailable. The best substitute is therefore thesmallest core pseudopotential (He-core for Mn) since1s states are almost completely inert and affected onlyby the scalar relativistic corrections. The recipe for con-structing the He-core pseudopotential is set forth belowusing Mn as an example.

We have chosen to represent the He-core nonlocal

pseudopotential using the following simple form:

V(r) =Vloc(r) + |s〉 ξ exp(−ηr2) 〈s|+ |p〉 µ exp(−υr2) 〈p| (7)

The parameters ξ, η, µ, υ are found by minimizationof the following measure of discrepancy between all-electron Dirac-Fock (DF) and self-consistent ECP cal-culations

W =∑

i

εAE

i − εECPi

εAEi

2

+∑

m

(εAE

m − εECPm

εAEm

)2

(8)

where the first term represents the error of valenceeigenvalues in the ground state and second term isthe error of eigenvalues for several atomic excitations.Therefore within the given functional form the ECP re-produces both ground and energetically close excitedstates, in the spirit of energy-adjusted ECPs [14] . Theparameters of this ECP are listed in Table 2.

-1000

-800

-600

-400

-200

0

200

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

V(r

)

r(a.u)

Vd (local)Vs (non-local)Vp (non-local)

Zeff/r

Figure 1: He-core pseudopotentials for the Mn atom.

Table 3: Errors (in eV) of the self-consistent excitation energies ofthe Mn atom for the two types of ECPs with regard to the all-electronDirac-Fock calculationState He-core YN/Ne-coreMn (4s13d6) 0.009 0.018Mn1+ (4s13d5) 0.002 0.004Mn1+ (4s03d6) 0.004 0.011Mn2+ (4s03d5) 0.001 0.061Mn2+ (4s13d4) 0.033 0.003Mn3+ (4s03d4) 0.030 0.042

In conventional ECPs Vloc(r) is simply −Ze f f /r, whereZe f f is the effective core charge. However, in QMC the

4

singularity of 1/r is inconvenient and unnecessarily in-creases the energy fluctuations as well as forces to usemuch smaller time steps [7, 15]. Additional ECP termsare therefore introduced in order to smoothly eliminatethe singularity as discussed in Ref.[7]

Vloc(r) = −Ze f f

r+

Ze f f

rexp(−αr2)

+ Ze f fαr exp(−βr2) + γ exp(−δr2)(9)

where α, β, δ become variational parameters which arefound in the optimization setup described above.

3. Results

The He-core pseudopotential (Fig. 1) we have con-structed turned out to be very accurate when com-pared to relativistic all-electron Dirac-Fock calcula-tions. Comparison of the excitation and ionization en-ergies shows the mean absolute deviation (MAD) fromall-electron relativistic results for lowest states being ≈0.01 eV, see Tab. 3. It appears also marginally moreaccurate than the other two Ne-core ECPs.

We list the excitation energy results of MnO mole-cule obtained with all-electron, Ne-core pseudopoten-tials, frozen ([Ne]) core approximation and He-corepseudopotential in Table 4. The two Ne-core pseudopo-tentials are labeled as STU and YN. An electronic struc-ture program GAMESS [16] was used to carry out thefollowing calculations: HF, DFT(PW91) and DFT hy-brid functionals (B3LYP and PBE0w=0.2 ). The excita-tion energies from the high-spin state (σ1δ1δ1π1π1) tothe low-spin state (σ1δ2δ

2) of the MnO molecule are

shown, as well as their errors. The error in the tableis defined as the difference between all-electron Dirac-Fock and pseudopotential self-consistent results.

In the previous study [2], we have reported that theexplicit presence of the core electrons does matter inthe MnO solid when approximate, effective functionalmethods such as DFT are used. As observed before,there is a significant discrepancy between the Ne-coreand all-electron calculations for the DFT methods. Notethat He-core ECPs are consistents with all-electron andfrozen core results showing that the key issue is the re-moval of the 2s and 2p states.

The different results between Ne-core pseudopoten-tial and frozen [Ne] core calculations can be understoodby the non-linear density dependence of Vxc. In thepseudopotential calculation,

V ppxc (r) = Vxc(r, nv) (10)

Table 4: Energy difference ∆ = Ehs − Els between high-spin (2S +

1 = 6) and low-spin (2S + 1 = 2) states of the MnO molecule andcorresponding errors quantified as the disagreement between ∆all and∆ECP for different treatment of the cores and methods.

XC functional Core ∆(eV) Error(eV)PW91 All-electron -3.11

Frozen core[Ne] -3.11 <0.01He-core -3.15 0.04STU[Ne] -3.51 0.39YN [Ne] -3.45 0.34

B3LYP All-electron -3.49Frozen core[Ne] -3.49 <0.01

He-core -3.52 0.03STU [Ne] -3.79 0.3YN [Ne] -3.74 0.25

PBE0w=0.2 All-electron -3.81Frozen core [Ne] -3.81 <0.01

He-core -3.85 0.04STU [Ne] -4.16 0.35YN [Ne] -4.10 0.30

HF/DF All-electron -6.61Frozen core[Ne] -6.62 0.01

He-core -6.62 0.01STU [Ne] -6.75 0.14YN [Ne] -6.69 0.08

5

the exchange-correlation functional depends only on thevalence electron density. On the other hand, in the all-electron calculation we have

Vaexc (r) = Vxc(r, nv + nc) , Vxc(r, nc) + Vxc(r, nv)

(11)so that due to the non-linear character of the xc-functional this decomposition is only approximate un-less there is no significant overlap between the valenceand core wavefunction (He-core case).

The key question is now: what is the true accuracy ofthe Ne-core ECPs ? In order to answer this question weuse the DMC method which is applicable to both He-core and Ne-core ECP Hamiltonians (but is problem-atic in all-electron setting due to difficulties in the sim-ulaneous treatment of relativistic and electron correla-tion effects [19]). The He-core pseudopotential we con-structed is therefore very convenient and suitable sub-stitute for all-electron QMC approach.

Table 5: Comparison of QMC excitation energies (eV) and the dif-ferences with regard to the He-core ECP and differences between thethree pseudopotentials. In the brackets are types of trial functionswhich are either single reference built from DFT orbitals or in the lastrow, multi-reference from CI.

Method Core ∆ ∆He − ∆Ne

DMC(PW91) He-core -4.24STU -4.28 0.04YN -4.23 -0.01

DMC(B3LYP) He-core -4.22STU -4.23 0.01YN -4.27 0.05

DMC(PBE0w=0.2) He-core -4.20STU -4.21 -0.01YN -4.20 <0.01

DMC(HF) He-core -4.74STU -4.88 0.14YN -4.98 0.24

DMC(multireference, CI) YN -4.23

In this study, we have utilized QWalk[17] packageto perform accurate DMC calculation with both He andNe-core ECPs. In earlier works [3, 18], DMC has beenused to calculate a number of properties transition metaloxygen systems in very good agreement with experi-ments. The results for the difference between the high-spin and low-spin states and the discrepancies for this

quantity between He-core and Ne-core ECPs are sum-marized in Table 5.

We show the DMC results for different trial functionsdepending on which method was employed to gener-ate the orbitals for the Slater part. As we have shownelsewhere, this makes a significant difference for the ac-curacy of the nodal surface. Using DMC with hybridfunctional PBE0 orbital provides the smallest discrep-ancies ≈ 0.01 eV. DMC with other DFT orbitals alsogive rather small errors of the order of 0.05 eV. In orderto probe for the genuine accuracy of the nodal surfaceswe have carried calculation with trial function based onthe Configuration Interaction anti-symmetric part with≈ 900 determinants. The result indicates that DMC withDFT orbitals provides very consistent estimations of thehigh-spin low-spin difference. The excitation energiesfrom the trial function with HF orbitals are systemati-cally higher when compared with DMC/CI result and,in addition, the difference between different ECPs showmoderate biases up to 0.2 eV or so. The more significanterror indicates lower accuracy of the HF wave functionnodal surfaces as observed also in other cases of sys-tems with transition elements [4]. The final and key ob-servation is that the He-core and Ne-core ECPs providecomparable accuracy and are reliable within ≈ 0.1 eV.It is not too far fetched to expect that with more elabo-rated and better tuned ECPs one can reach accuracy of0.01 eV or possibly better. In this respect,The new set offirst-row transition metal ECPs by M. Dolg[20] as wellas ECPs by M. Burkatzki et al[7] will be of high interestin the future.

Additional comment can be made with regard to com-putational demands of calculations with He-core ECPsin the QMC framework. Although the total energyis much larger when compared with the Ne-core ECP,some time saving comes from the fact that the non-local terms (which are very expensive in QMC) havemuch smaller radial extent. That means that the nonlo-cal terms become less prominent in the overall compu-tational budget and such calculations, albeit longer, arefeasible for systems which are not too large. This willtherefore enable to test the ECPs for many transition el-ements and could eventually lead to ECPs with muchhigher overall accuracy, say, of the order of 0.01 eV forenergy differences.

4. Conclusion

In conclusion, we have carried study of ECPs accu-racy by comparing all-electron, He-core and Ne-coreECPs in DFT, HF and QMC approaches applied tothe MnO molecule. In agreement with our previous

6

study, we reaffirmed the finding which relates discrep-ancies between the all-electron and ECP calculations inDFT to the problems with nonlinearity of the exchange-correlation functionals. This makes such calculationsproblematic unless these errors are explicitly addressedby additional adjustments such as nonlinear core cor-rections. Such corrections are, however, not usableor applicable in other methods. High accuracy diffu-sion Monte Carlo calculation of the MnO molecule con-firmed that the Ne-core and He-core ECPs are of com-parable quality and therefore enable to reproduce en-ergy differences within 0.1 eV or better accuracy mar-gin. In addition, we have corroborated previous resultson nodal surfaces which are more most accurate whenusing trial functions based on orbitals from hybrid func-tionals.

5. Acknowledgements

Supports by NSF grant OCI-0904794, AROW911NF-04-D-0003-0012 and DOE DE-AC52-06NA25396 through subcontract from LANL as wellas computer time allocations at TACC under XSEDEprogram are gratefully acknowledged.

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Highlights: 

Accuracy of ECP is studied by quantum Monte Carlo method using MnO as a test system 

An accurate scalar‐relativistic He‐core ECP for the Mn atom is constructed.  

Density Functional Theory shows problem in treating nonlinearity of exchange correlation functional 

DMC calculation of MnO indicates Ne‐core and He‐core ECPs are of comparable quality 

-1000

-800

-600

-400

-200

0

200

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

V(r

)

r(a.u)

He-core pseudopotential for Mn atom

Vd (local)Vs (non-local)Vp (non-local)

Zeff/r