On the universality of the long-/short-range separationin multiconfigurational density-functional theory. II.Investigating f0 actinide species

Emmanuel Fromager,1,a� Florent Réal,2,3 Pernilla Wåhlin,3 Ulf Wahlgren,4 andHans Jørgen Aa. Jensen5

1Laboratoire de Chimie Quantique, Institut de Chimie, CNRS/Université de Strasbourg, 4 rue Blaise Pascal,67000 Strasbourg, France and Department of Chemistry, Centre for Theoretical and ComputationalChemistry, University of Tromsø, 9037 Tromsø, Norway2Laboratoire PhLAM UMR8523-CNRS, Université Lille 1 (Sciences et Technologies),59655 Villeneuve d’Ascq Cedex, France3Department of Physics, AlbaNova University Centre, Stockholm University, SE-106 91 Stockholm, Sweden4NORDITA, AlbaNova University Center, SE-106 91 Stockholm, Sweden5Department of Physics and Chemistry, University of Southern Denmark, Campusvej 55,DK-5230 Odense M, Denmark

�Received 24 April 2009; accepted 6 July 2009; published online 4 August 2009�

In a previous paper �Fromager et al., J. Chem. Phys. 126, 074111 �2007��, some of the authorsproposed a recipe for choosing the optimal value of the � parameter that controls the long-range/short-range separation of the two-electron interaction in hybrid multiconfigurational self-consistentfield short-range density-functional theory �MC-srDFT� methods. For general modeling withMC-srDFT methods, it is clearly desirable that the same universal value of � can be used for anymolecule. Their calculations on neutral light element compounds all yielded �opt=0.4 a.u. In thiswork the authors investigate the universality of this value by considering “extreme” study cases,namely, neutral and charged isoelectronic f0 actinide compounds �ThO2, PaO2

+, UO22+, UN2, CUO,

and NpO23+�. We find for these compounds that �opt=0.3 a.u. but show that 0.4 a.u. is still

acceptable. This is a promising result in the investigation of a universal range separation. Theaccuracy of the currently best MC-srDFT ��=0.3 a.u.� approach has also been tested forequilibrium geometries. Though it performs as well as wave function theory and DFT forstatic-correlation-free systems, it fails in describing the neptunyl �VII� ion NpO2

3+ where staticcorrelation is significant; bending is preferred at the MC-srDFT ��=0.3 a.u.� level, whereas themolecule is known to be linear. This clearly shows the need for better short-range functionals,especially for the description of the short-range exchange. It also suggests that the bendingtendencies observed in DFT for NpO2

3+ cannot be fully explained by the bad description of staticcorrelation effects by standard functionals. A better description of the exchange seems to be essentialtoo. © 2009 American Institute of Physics. �DOI: 10.1063/1.3187032�

I. INTRODUCTION

Nowadays, density-functional theory �DFT� �Ref. 1� iswidely used for electronic structure calculations becauseDFT normally produces reasonable results at a relativelycheap computational cost and thus enables the investigationof large molecular systems. However, even though currentstandard approximate functionals in many cases provide formolecules a satisfactory description of the short-range dy-namic correlation �Coulomb hole�,2–4 static correlation ef-fects �multiconfigurational character of the wave function�are in general not treated adequately. On the other hand,those effects can be described in wave function theory�WFT�, for example, with a multiconfigurational self-consistent field �MCSCF� model, but the description of thedynamic correlation requires then a long configuration ex-pansion of the wave function as well as large basis sets, thus

preventing large scale calculations. It is therefore of interestto develop hybrid MCSCF-DFT models, which combine thebest of both MCSCF and DFT approaches with respect toaccuracy and computational cost.

This can be achieved rigorously by separating the regu-lar two-electron Coulomb interaction 1 /r12 into long-rangeand short-range parts, as initially proposed by Savin.2 Thelong-range interaction is then described within the MCSCFapproach and the short-range interaction in DFT. A com-monly used long-range interaction, the so-called “erf” inter-action, is based on the error function

weelr,��r12� =

erf��r12�r12

, �1�

where � is a free parameter in �0,+�� controlling the rangeseparation. The short-range interaction is thus �-dependentand equalsa�Electronic mail: emmanuel.fromager@chimie.u-strasbg.fr.

THE JOURNAL OF CHEMICAL PHYSICS 131, 054107 �2009�

0021-9606/2009/131�5�/054107/11/$25.00 © 2009 American Institute of Physics131, 054107-1

Downloaded 22 Jun 2010 to 193.49.225.25. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp

weesr,��r12� =

1 − erf��r12�r12

. �2�

As we have to use approximate wave functions and approxi-mate short-range functionals in practical calculations, an op-timal �opt parameter has to be defined. For general modelingwith such MCSCF-DFT hybrid methods, it is clearly desir-able that the same universal value of � can be used for anymolecule. Note that the use of a local ��r� parameter, asexplored by Krukau et al.5 for hybrid functionals where therange separation is used only for the exchange, is not perti-nent in this context since the range separation is also used forthe correlation energy. Indeed, in the core region of for ex-ample the argon atom,5 the local ��r� parameter is quitelarge �between 4 and 5 a.u.� meaning that most of the dy-namic correlation that we want to describe within DFTwould then have to be calculated within the MCSCF ap-proach, which is of course not desirable. As an attempt todefine a universal � parameter, some of the authors haverecently proposed a recipe, which aims at assigning staticand dynamic correlations, respectively, to the long-rangeMCSCF and short-range DFT �srDFT� parts of the energy;they define �opt as the largest � value for which the hybridMCSCF-DFT wave function is essentially a single determi-nant in systems without significant long-range �static or Lon-don dispersion interaction� correlation effects.6 With thischoice, the MCSCF-DFT wave function gets a short configu-ration expansion,7 indicating that most of the short-range dy-namic correlation is calculated within DFT and thus that theMCSCF is only used to describe static correlation. Note thatlong-range dynamic correlation effects such as London dis-persion interactions can be recovered, in addition to staticcorrelation, with multireference perturbation theory tech-niques using as zeroth-order wave function the MCSCF-DFTwave function.7 Since the multiconfigurational character ofthe wave function in srDFT is induced by the long-rangeinteraction, the definition of �opt is in principle independentof the approximate wave function. It should also not dependon the approximate short-range functional �short-range local-density approximation �srLDA�, short-range generalized gra-dient approximation �srGGA�, etc.� since all of those, evensrLDA, provide a reasonable description of the electron den-sity in the valence shell for long-range-correlation-freesystems.6 Therefore its universality only depends on the in-terval of the average dynamic correlation distances betweenthe valence electrons �which could be interpreted as the av-erage of the screening function of Krukau et al.5 in the va-lence region� in all systems without significant long-rangecorrelation effects. Is this interval basically the same for allmolecules? Calculations on neutral light element compoundsall yielded �=0.4 a.u.6 Interestingly, this value is in goodagreement with calibration studies performed with long-range corrected hybrid density-functionals based on othercriteria.8–14 At this stage an obvious question is if �=0.4 a.u. still is optimal for molecules containing heavierelements.

As an attempt to answer this question, we considered inthis work “extreme” study cases, namely, neutral andcharged isoelectronic f0 actinide �Th, Pa, and U� compounds

chosen to represent static-correlation-free heavy element sys-tems. In addition, we studied the neptunyl �VII� NpO2

3+ ion,chosen to represent systems with significant static correlationeffects,15 and for which standard DFT fails. The latter givesan equilibrium geometry, which is bent, whereas all WFT-based methods clearly indicate that it should be linear.15 Thepaper is organized as follows. We first introduce in Sec. IIthe hybrid range-separated MCSCF-DFT model, referred toas MC-srDFT in the following, and give the computationaldetails in Sec. III. We thereafter present in Sec. IV A 1 anumerical investigation of �opt on the selected set of f0 Th,Pa, and U compounds. The neptunyl �VII� ion is then con-sidered in Sec. IV A 2. The performance of the MC-srDFTmethod in terms of equilibrium geometries is finally investi-gated in Secs. IV B 1 and IV B 2. In the latter the “bendingproblem” of NpO2

3+ is also analyzed in the light of the MC-srDFT equilibrium structures obtained when varying the �parameter. Finally, conclusions are given in Sec. V.

II. THEORY

The hybrid MCSCF-DFT model used in this work isbased on the separation of the regular two-electron Coulombinteraction wee�r12�=1 /r12 into long-range and short-rangeparts,

wee�r12� = weelr,��r12� + wee

sr,��r12� . �3�

The long-range interaction is in principle arbitrary. In prac-tice, the so-called erf interaction, which is based on the errorfunction �see Eq. �1��, is commonly used. The universalfunctional defined as1

F�n� = min�→n

���T + Wee��� ,

Wee =1

2�i�j

wee�rij� �4�

can thus be decomposed into long-range and short-rangecontributions,

F�n� = Flr,��n� + EHxcsr,��n� , �5�

where the universal long-range functional Flr,��n� equals

Flr,��n� = min�→n

���T + Weelr,����

= ����n��T + Weelr,�����n�� ,

Weelr,� =

1

2�i�j

weelr,��rij� , �6�

and EHxcsr,��n� denotes the �-dependent short-range Hartree,

exchange, and correlation energy functional, which describesthe short-range interaction and the coupling between thelong-range and short-range correlations.16 The ground-stateenergy, which is obtained according to the variational prin-ciple by

054107-2 Fromager et al. J. Chem. Phys. 131, 054107 �2009�

Downloaded 22 Jun 2010 to 193.49.225.25. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp

E = minnF�n� + drvne�r�n�r�� , �7�

can then be expressed as

E = min�

����T + Weelr,� + Vne��� + EHxc

sr,��n��

= ����T + Weelr,� + Vne���� + EHxc

sr,��n��� , �8�

where n� is the density related to the wave function � and

Vne=�drvne�r�n�r� is the nuclear potential operator. Notethat standard Kohn–Sham models �KS-DFT� are recoveredin the �=0 limit, while pure WFT models are obtained in the�→+� limit as seen from Eqs. �1� and �8�. The minimizingwave function �� in Eq. �8� fulfills the following self-consistent equation:

�T + Weelr,� + Vne + VHxc

sr,��n�������� = E����� , �9�

where E� is the Lagrange multiplier related to the normaliza-tion of �� and where the short-range Hartree, exchange, and

correlation local potential operator VHxcsr,��n� for a given den-

sity n is defined as follows:

VHxcsr,��n� = drvHxc

sr,��n��r�n�r� ,

�10�

vHxcsr,��n��r� =

�EHxcsr,�

�n�r��n� .

Observe that the ground-state density of the fully inter-acting real system is obtained from a fictitious long-rangeinteracting system whose ground-state wave function �� canbe deduced from wave function-based correlated methods.We refer to this approach as srDFT since the density-functional does not describe the long-range part of the two-electron interaction. It follows that any standard wavefunction-based approximation, such as Hartree–Fock �HF�,configuration interaction �CI�, MCSCF, second-orderMøller–Plesset �MP2�, coupled cluster �CC�, etc., can beadapted to srDFT �see Refs. 6 and 17 and references therein�.In this work we have used the MC-srDFT approach, togetherwith HF-srDFT and MP2-srDFT methods, for analysis pur-poses. Note that as in standard KS-DFT, the exact short-range exchange and correlation functional Exc

sr,��n� is un-known. Therefore, an approximate form must be used inpractical calculations; results obtained with both srLDA andsrGGA functionals2,18–21 are presented in the following.

III. COMPUTATIONAL DETAILS

A. Hamiltonians and basis sets

Calculations have been performed mainly with smallcore �60 electrons� scalar relativistic effective core potentials�RECPs� on the actinides, thus treating explicitlyalso the 5s, 5p, 5d, 6s, and 6p orbitals in addition tothe 5f , 6d, 7s, and 7p orbitals. The corresponding�14s13p10d8f6g� / �6s6p5d4f3g� atomic natural orbital�ANO� basis sets22 were used. Concerning the lighter ele-ments �N, C, and O�, RECPs were also used �for the 1s

electrons� with a �4s5p1d� / �2s3p1d� basis set.23 For analysispurposes, all-electron calculations were also performed usingthe second-order Douglas–Kroll–Hess DKH�2,0�Hamiltonian24,25 with ANO-RCC triple-� basis sets con-tracted to �9s8p6d4f2g0h� for the actinides26 and to�4s3p2d� for the lighter elements �O and N�.27 Spin-orbitcoupling effects were neglected since f0 actinide compoundsare closed-shell systems.

B. Methods

HF-srDFT,28 MC-srDFT,28 and MP2-srDFT �Ref. 17�calculations were performed with a development version ofthe DALTON program package.29 Two approximate spin-independent short-range functionals were tested: the srLDAfunctional2,18 and a short-range GGA functional, referred toas PBEHSERI in Ref. 6 and simply denoted srPBE in thispaper, as an extension of the PBE functional.30 In the lattercase the short-range exchange PBE functional of the Heyd,Scuseria, and Ernzerhof hybrid functional19,20 was used inconjunction with a short-range correlation functional, basedon a rational interpolation between the standard PBE corre-lation functional at �=0 and the asymptotic expansion of thesrLDA correlation functional in the �→+� limit, as sug-gested by Savin and co-workers.16,21 Since all MC-srDFTcalculations were performed with complete active space�CAS� configuration spaces, they will be referred to as CAS-srDFT in the following. The CAS orbitals were chosen asfollows: in f0 actinyls the primary valence shells of the ac-tinide center, 5f and 6d, form bonds with the 2p orbitals ofthe oxygen.31 This gives six doubly occupied orbitals��g ,�u ,�u ,�g� and the corresponding antibonding orbitals��g

� ,�u� ,�u

� ,�g��. The nonbonding f� and f� orbitals remain

mainly atomic. Therefore, the minimal CAS consists of 12electrons in 12 orbitals if dissociation processes should bedescribed.31 The employed CAS choices are denoted Ne /Nac

in the following, Ne being the number of electrons and Nac

the number of active orbitals. For analysis purposes otherCAS choices, deduced from the MP2-srDFT natural orbitalsoccupancies17,32 of NpO2

3+ for �=0.4 a.u., have been se-lected: six electrons in the �g, �g, �u

�, and �u� orbitals �6/6�,

ten electrons in the �g, �u, �g, �u�, �u

�, and �g� orbitals �10/

10�, and finally 12 electrons in 12 orbitals plus three corre-lating orbitals �12/15�. In the cases of CUO and UN2, a CAS12/12 identical to the one used for the f0 actinyls was used.For analysis purposes, standard WFT and DFT calculationswere also performed with RECPs using the MOLCAS �Ref.33� and MOLPRO �Ref. 34� program packages. At the WFTlevel, the following approximations were considered: HF andCASSCF �with the 12/12 active space� as well as MP2,CASPT2,35,36 and CCSD�T�37–40 without correlating the 5s,5p, and 5d orbitals in geometry optimizations. For DFT cal-culations, the pure LDA �Ref. 41� and PBE as well as thehybrid B3LYP �Ref. 42� functionals were used.

IV. RESULTS AND DISCUSSIONS

A. Optimal � parameter for early actinide compounds

The � parameter that defines the range separation of thetwo-electron Coulomb interaction in Eqs. �1�–�3� is, in prin-

054107-3 Long-/short-range separation. II J. Chem. Phys. 131, 054107 �2009�

Downloaded 22 Jun 2010 to 193.49.225.25. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp

ciple, arbitrary. However, since approximate wave functionsand short-range functionals are used in practice, an optimal�opt parameter has to be defined for practical srDFT calcula-tions. As mentioned in Sec. I, some of the authors have re-cently proposed to define �opt as the largest � value forwhich the srDFT wave function, simply referred to as wavefunction in the following, is a single determinant in systemswith no significant long-range correlation effects.6 Thisrecipe aims at assigning static and dynamic correlations, re-spectively, to the long-range MCSCF and srDFT parts of theenergy. In order to verify if the numerical value of�opt=0.4 a.u. obtained for molecules with only lightelements6 is still optimal for heavier elements, we considerin Sec. IV A 1 five f0 early actinide species, which areknown to be static-correlation-free. We then focus in Sec.IV A 2 on the neptunyl �VII� ion NpO2

3+, chosen to representf0 actinide compounds with significant static correlationeffects.15 Calculations are performed at the CCSD�T� equi-librium geometries given by Straka et al.15 for the actinylsand at the CASPT2 equilibrium geometries given byGagliardi and Roos43 for the other compounds.

1. Thorium, protactinium, and uranium compounds

In this section we consider a test set consisting of five f0

static-correlation-free15,44,45 actinide compounds �ThO2,PaO2

+, UO22+, UN2, and CUO� and investigate the multicon-

figurational character of their wave function with respect tothe � parameter, comparing first their HF-srDFT and CAS-srDFT energies obtained with RECPs. Using the srLDAfunctional, the HF-srDFT and CAS-srDFT energy curvessplit �within an accuracy of 10−3 a.u.� in the vicinity of �=0.3 a.u. for the five systems �see Fig. 1 and Fig. A in thesupplementary materials46�. As expected, a similar result isobtained using the more accurate srPBE functional. We canthus conclude that the optimal �opt parameter should be

around 0.3 a.u. As shown in Fig. B in the supplementarymaterials,46 all-electron calculations lead to the same conclu-sion, in agreement with that the definition we use for �opt

depends only on the electron density in the valence region.As a result the same numerical value of �opt can be used inboth RECPs and all-electron calculations. On the other hand,the occupation numbers of the natural active orbitals in theCAS-srDFT calculations, using either the srLDA or srPBEfunctionals with RECPs, show that the wave function is asingle determinant for �0.3 a.u. within an accuracy of10−3 and for �0.4 a.u. within an accuracy of 10−2 forThO2, PaO2

+, and UO22+ �see Fig. 2�. This confirms that

choosing �opt=0.3 a.u. is optimal for the early actinides. Italso shows that the optimal value of 0.4 a.u. obtained forlight elements is still acceptable since, in that case, a rathersmall part of the dynamic correlation is assigned to the long-range interaction and thus treated by the CASSCF approach.

2. Neptunyl „VII… ion

We now consider the neptunyl �VII� ion NpO23+, which is

isoelectronic to the compounds studied above and wherestatic correlation effects become significant.15 As shown inFig. 1 and Fig. B in the supplementary materials,46 the HF-srDFT and CAS-srDFT energies �at both RECP and all-electron levels� differ for �=0.3 a.u. by 10−2 a.u., which isten times larger than what was found for the static-correlation-free systems studied in Sec. IV A 1. In this re-spect, the wave function of NpO2

3+ can be considered asslightly multiconfigurational when �=0.3 a.u., whereas forthe other compounds it is essentially a single determinant. Itis not, however, as striking as it can be for the stretchedwater molecule, for example.6 In the f0 actinyl compounds,the long-range effects are induced by the actinide centerwhich becomes “hungry” in f electrons as its chargeincreases.15 Static correlation effects in NpO2

3+ are therefore

FIG. 1. Total ground-state energies off0 early actinyl compounds calculatedat the RECP level by the HF-srDFT�uppermost curve� and CAS-srDFT�lower curves� methods with respect tothe � parameter, with the srLDA andsrPBE functionals. The CAS-srDFTactive spaces are given in parentheses.

054107-4 Fromager et al. J. Chem. Phys. 131, 054107 �2009�

Downloaded 22 Jun 2010 to 193.49.225.25. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp

not expected to be as strong as in a stretched molecule. Wemay state that they are in fact on the same order of magni-tude as the dynamic correlation. That appears clearly in thestandard CASSCF calculation where, using the 12/12 activespace, the wave functions of UO2

2+ and NpO23+ mainly differ

in one configuration, which corresponds to double excita-tions from the bonding �u orbitals to the antibonding �u

�

orbitals and whose largest coefficient is respectively 0.07 and0.10. As a result, the HF-srDFT configuration remains thedominant configuration in the CAS-srDFT wave function ex-pansion as it is for the UO2

2+ ion. Nevertheless, the occupa-tion numbers of the natural CAS-srDFT active orbitals plot-ted in Fig. 2 show that for �=0.3 a.u., the wave function ofNpO2

3+ starts getting a multiconfigurational character, whileit essentially remains a single determinant for ThO2, PaO2

+,and UO2

2+. Note that this analysis supports the works of Val-let et al.,44,45 which state that UO2

2+ has no significant staticcorrelation effects, in contrast to Rotzinger’s statement.47,48

Returning to the NpO23+ ion, static correlation seems to

be related at this stage to double excitations from the �u

orbitals. However, if we compare the 6/6 and 10/10 CAS-srDFT active spaces �see Sec. III B�, it is noteworthy in Fig.1 that for 0.3�0.4 a.u., they give different energies forThO2, PaO2

+, and UO22+ but rather similar energies for NpO2

3+.It means that introducing the �u orbitals in the active spacedoes not change the CAS-srDFT energy of the latter signifi-

cantly. Ignoring the HF-srDFT determinant, the configura-tions corresponding to double excitations from the �g and �g

orbitals to the �u� and �u

� orbitals are in this case dominant inthe CAS-srDFT wave function. As shown in Fig. 3, the �g

natural orbitals are in fact less occupied than the �u naturalorbitals, whereas it is the opposite for the standard CASSCFwave function with the same active space �� →+� limit�.

FIG. 2. Occupancies of the natural CAS-srLDA bonding orbitals calculated with respect to the � parameter for ThO2, PaO2+, UO2

2+, and NpO23+. Active spaces

are 12/12.

FIG. 3. Comparison of the natural CAS-srLDA bonding orbitals occupan-cies of NpO2

3+ plotted with respect to the � parameter. Active space is 12/12.

054107-5 Long-/short-range separation. II J. Chem. Phys. 131, 054107 �2009�

Downloaded 22 Jun 2010 to 193.49.225.25. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp

This indicates that essential parts of the static correlationdo not fully originate in the long-range interaction for�=0.3 a.u. Can we thus conclude that �=0.3 a.u. does notensure a proper separation of static and dynamic correlationsin this case? Keeping in mind that for any range separation,the exact wave functions of the real fully interacting systemand the fictitious long-range interacting system should havethe same density, it is worth wondering if the density at theCASSCF 12/12 level �where only valence correlation istaken into account� is accurate enough. For analysis pur-poses, we computed at both RECP and all-electron levels theMP2 natural orbitals occupation numbers49,50 of NpO2

3+ andUO2

2+. Concerning NpO23+, when including core-core and

core-valence correlations, the occupancies of the �g orbitalsbecome lower than the occupancies of the �u orbitals, as forthe CAS-srDFT wave function. This may simply indicatethat the CAS-srDFT approach using �=0.3 a.u. describesnot only the static correlation �within the long-rangeCASSCF� but also core-core and core-valence correlations�within the srDFT�. In this respect it calculates much moreelectron correlation than a standard CASSCF calculation us-ing the same minimal 12/12 active space. Let us howevermention that Knecht51 also performed all-electron four-component spinfree CISD calculations on NpO2

3+ using theparallelized CI module of the DIRAC program package52,53

with Dyall basis sets.54 He found that when correlating innershells, the �g and �u natural orbital occupancies get closer,but no inversion in their ordering was observed even whenusing a rather large active space �34 electrons and a 16 a.u.cut-off for the virtuals�. The discrepancy with the MP2 oc-cupancies is still not clear—second-order perturbation theorymight not be accurate enough—but both methods indicatethat core-core and core-valence correlations tend to move �g

and �u natural orbital occupancies closer, which is qualita-tively in agreement with the CAS-srDFT ��=0.3� resultsdiscussed above. For that reason, �=0.3 a.u. is clearly opti-mal in terms of computational cost. As long as the long-range correlation is correctly described within WFT, accu-racy depends then on the quality of the approximate short-range functional, as discussed in details in Sec. IV B 2.

Note that for � values larger than 0.6 a.u., the �g naturalorbitals become more occupied than the �u natural orbitals�see Fig. 3� as for the standard CASSCF 12/12 wave func-tion, which indicates that most of the electron correlation is,in this case, described by the long-range CASSCF. In thecase of UO2

2+, the occupancies of the natural �u orbitals re-main at the MP2 level �including core-core and core-valencecorrelations� lower than the occupancy of the �g orbitals.This could explain why the 6/6 and 10/10 energy curves inFig. 1 are split in the vicinity of �=0.4 a.u. for UO2

2+ but notfor NpO2

3+.As a conclusion, choosing �=0.3 a.u. seems to be opti-

mal for separating static and dynamic correlations into long-range and short-range correlations in the early actinide com-pounds considered in this work. The slightly larger �=0.4 a.u. value obtained for light elements is still acceptablethough. In Sec. IV B we investigate the influence of the �parameter on equilibrium geometries and thus evaluate theaccuracy of the srLDA and srPBE functionals.

B. Influence of the � parameter on bondingand bending

We argued above that choosing �=0.3 a.u. is optimalfor early actinide compounds since it ensures the assignmentof the dynamic and static correlations, respectively, to thesrDFT and long-range CASSCF parts of the energy. We in-vestigate in this section the performance of the CAS-srDFTapproach for equilibrium geometries, testing both srLDA andsrPBE functionals and varying the � parameter around itsoptimal value. The first section �Sec. IV B 1� is devoted tothe thorium, protactinium, and uranium compounds. We thenfocus in Sec. IV B 2 on the neptunyl �VII� ion and proposean analysis of the bending problem, introduced in Sec. I froma consideration of the natural orbital occupancies discussedin Sec. IV A 2.

1. Thorium, protactinium, and uranium compounds

Equilibrium geometries for ThO2, PaO2+, UO2

2+, UN2,and CUO have been calculated at the HF-srDFT and CAS-srDFT �with the 12/12 active space� levels using both srLDAand srPBE functionals with RECPs. Four different values forthe � parameter have been used: 0.3, 0.4, 1.0, and 2.0 a.u.,together with the DFT and WFT limits ��=0 and � →+��.As shown in Tables I and II, HF-srDFT and CAS-srDFTgeometries are very similar for the optimal �opt=0.3 a.u.value. The largest difference in terms of bond distance �0.007Å� is obtained for the uranium compounds. This was ex-pected since in all those systems, static correlation effects arenot significant. As the � parameter increases, the deviation ofthe HF-srDFT bond distance from its standard DFT value��=0 limit� gets larger when going from thorium to uraniumcompounds. It can be interpreted as follows: increasing �leads to a transfer of the electron correlation from the short-range to the long-range correlation, which is not described atthe HF-srDFT level. We therefore illustrate the fact that dy-namic correlation has an increasing influence when goingfrom thorium to uranium compounds, as already noticed byStraka et al.15 Note that for ThO2, the bond angle at theHF-srDFT ��=0.3 a.u.� level is slightly overestimated incomparison to values obtained with standard methods. Weshould however mention that the potential curve is rather flataround the equilibrium angle; the HF-srLDA ��=0.3 a.u.�energy increases, for example, by 1.15 kJ/mol as the angledecreases from 125° to 119° with a fixed bond distance of1.890 Å. As shown in Fig. 4�d� the standard LDA angle�118°� is correctly recovered in the �=0 limit and the HFangle �120°� in the � →+� limit.

Concerning the CAS-srDFT geometries, it is noteworthyfor protactinium and uranium compounds �see Tables I and IIand Fig. 4�b�� that the �-dependence of the bond distances ismuch smaller than it is at the HF-srDFT level. It can beinterpreted as the transfer of the dynamic correlation fromthe srDFT to the long-range CASSCF energy contribution. Inaddition, when comparing in Figs. 4�a� and 4�b� the HF-srLDA and CAS-srLDA distance curves with respect to �,for both ThO2 and UO2

2+, we notice that they become parallelfor �2.0 a.u. A possible interpretation is that for such val-ues of �, most of the electron correlation is described by the

054107-6 Fromager et al. J. Chem. Phys. 131, 054107 �2009�

Downloaded 22 Jun 2010 to 193.49.225.25. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp

CASSCF approach. This is supported by the fact that thesrLDA and srPBE functionals give, at either the HF- or CAS-srDFT level, identical geometries for all compounds when�=2.0 a.u.

With respect to accuracy, the CAS-srDFT ��=0.3 a.u.�bond distances, using either the srLDA or srPBE functionals,are in very good agreement with both standard DFT �LDA,PBE, and B3LYP� and CCSD�T� values �see Tables I and II�.Note that it still holds for the slightly larger optimal value�=0.4 a.u. obtained for light elements.6 We could thus statethat srLDA and srPBE functionals give a good description ofthe dynamic correlation for those static-correlation-free sys-tems. However as shown in Fig. 5, the CAS-srLDA��=0.3 a.u.� bending frequency of UO2

2+ is too low com-

pared to its CASSCF value. This highlights a first problemwith the approximate short-range functionals we use thatcould partly explain the bending problem of NpO2

3+ dis-cussed further in Sec. IV B 2. Note however that the CAS-srLDA method improves the pure LDA bending frequency ofUO2

2+. It is consistent with the fact that hybrid functionalsgive for UO2

2+ better bending frequencies than LDA and pureGGA functionals.55 Concerning ThO2, its bond angle isslightly overestimated at the CAS-srDFT ��=0.3 a.u.� levelcompared to standard methods. However, as already ob-served at the HF-srDFT level, the potential curve around theequilibrium bond angle is rather flat; the CAS-srLDA��=0.3 a.u.� energy increases, for example, by 0.5 kJ/mol asthe angle decreases from 123° to 119° with a fixed bond

TABLE I. Equilibrium geometries for the ground state of ThO2, PaO2+, UO2

2+, and UN2 calculated by HF-srDFTand CAS-srDFT methods �using both srLDA and srPBE functionals with �=0.3, 0.4, 1.0, and 2.0 a.u.� andcompared to standard WFT and DFT equilibrium structures. The CASSCF and CAS-srDFT active spaces are12/12. The 5s, 5p, and 5d orbitals were not correlated at the MP2 and CCSD�T� levels. Bond distances are inangstrom. For ThO2 the bond angle �in degrees� is given in parentheses.

Method �

ThO2 PaO2+ UO2

2+ UN2

srLDA/srPBE srLDA/srPBE srLDA/srPBE srLDA/srPBE

HF-srDFT 0.3 1.877�125�/1.885�125� 1.750/1.757 1.678/1.683 1.703/1.7070.4 1.873�126�/1.878�126� 1.741/1.746 1.666/1.669 1.692/1.6971.0 1.866�123�/1.866�123� 1.715/1.717 1.631/1.631 1.664/1.6652.0 1.869�122�/1.869�121� 1.713/1.713 1.624/1.625 1.664/1.664

CAS-srDFT 0.3 1.882�123�/1.890�123� 1.755/1.762 1.685/1.690 1.710/1.7140.4 1.882�123�/1.888�123� 1.753/1.757 1.680/1.684 1.708/1.7101.0 1.896�121�/1.897�120� 1.755/1.756 1.677/1.677 1.710/1.7102.0 1.907�119�/1.907�119� 1.760/1.761 1.679/1.679 1.717/1.717

DFT-LDA ¯ 1.885�118� 1.764 1.698 1.719DFT-PBE ¯ 1.898�118� 1.774 1.707 1.720DFT-B3LYP ¯ 1.899�119� 1.766 1.691 1.718HF ¯ 1.887�120� 1.729 1.637 1.640CASSCF ¯ 1.931�117� 1.780 1.698 1.736MP2 ¯ 1.914�113� 1.782 1.718 1.721CCSD�T� ¯ 1.911�116� 1.771 1.696 1.722

TABLE II. Equilibrium geometries for the ground state of CUO calculated by HF-srDFT and CAS-srDFTmethods �using both srLDA and srPBE functionals with �=0.3, 0.4, 1.0, and 2.0 a.u.� and compared to standardWFT and DFT equilibrium structures. The CASSCF and CAS-srDFT active spaces are 12/12. The 5s, 5p, and5d orbitals were not correlated at the MP2, CASPT2, and CCSD�T� levels. Bond distances are in angstrom.

Method �

HF-srDFT

CUO

CAS-srDFT

LDA PBE LDA PBEU-O/U-C U-O/U-C U-O/U-C U-O/U-C

0.3 1.775/1.730 1.789/1.730 1.780/1.740 1.789/1.7400.4 1.773/1.715 1.778/1.716 1.779/1.737 1.789/1.7401.0 1.749/1.683 1.749/1.686 1.787/1.738 1.789/1.7402.0 1.749/1.683 1.749/1.686 1.796/1.745 1.797/1.750

DFT-LDA ¯ 1.782/1.750DFT-PBE ¯ 1.799/1.756DFT-B3LYP ¯ 1.792/1.743HF ¯ 1.753/1.701CASSCF ¯ 1.816/1.763CASPT2 ¯ 1.786/1.755MP2 ¯ 1.792/1.759CCSD�T� ¯ 1.792/1.749

054107-7 Long-/short-range separation. II J. Chem. Phys. 131, 054107 �2009�

Downloaded 22 Jun 2010 to 193.49.225.25. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp

distance of 1.890 Å. Note that the CCSD�T� angle we found�116°� is smaller by 6° than the one given by Straka et al.15

We first checked that using a RECP for the oxygen atom �aswe did� or not �as they did� has no influence on the equilib-rium angle. Though their basis set differs from ours �see Sec.III A�, both are expected to give similar equilibriumgeometries.22 The difference in CCSD�T� bond angles forThO2 is then probably related to the flatness of the potentialcurve; increasing the angle from 116° to 122° changes theenergy by less than 2 kJ/mol.

In conclusion, the CAS-srDFT approach, using either thesrLDA or srPBE functional with �=0.3 a.u., gives for theTh, Pa, and U f0 compounds equilibrium geometries in fairlygood agreement with both standard DFT and WFT results.For those static-correlation-free systems, HF-srDFT andCAS-srDFT geometries are very similar as expected.

2. Bending in the neptunyl „VII… ion

We investigate in this section the influence of the � pa-rameter on bonding and bending in the neptunyl �VII� NpO2

3+

ion at both HF-srDFT and CAS-srDFT levels. RECPs andcorresponding basis sets have been used with either srLDAor srPBE functionals, and the CAS-srDFT active space is12/12 �see Sec. III B�. As shown in Table III, standard purefunctionals �LDA and PBE� give for NpO2

3+ a bent equilib-rium geometry, as the hybrid B3LYP functional does,15

whereas all WFT-based calculations indicate that the neptu-nyl �VII� is linear. According to Straka et al.,15 static corre-lation effects �induced by the increase in the actinide center’scharge when going from ThO2 to NpO2

3+� may explain thebending tendencies observed in standard DFT calculations.In the light of the HF- and CAS-srDFT results presented inTables III and IV, we propose in the following a more de-tailed analysis of this bending problem. When using theCAS-srLDA or CAS-srPBE methods with the optimal

FIG. 4. Equilibrium Re�An-O� bonddistances and O-An-O angles calcu-lated with respect to the � parameterby the HF-srLDA and CAS-srLDAmethods for f0 AnO2

n+ species �An=Th, U, and Np�. In �a� the top curvesare for ThO2 and the bottom ones arefor NpO2

3+. In �a� and �b� the straightlines correspond to the CCSD�T� equi-librium bond distances. RECPs andcorresponding basis sets have beenused. The CAS-srLDA active space is12/12.

FIG. 5. Total ground-state energies of UO22+ and NpO2

3+ calculated withrespect to the bond angle O-An-O �the An-O bond distance is fixed to 1.680� by the DFT-LDA, CASSCF, HF-srLDA, and CAS-srLDA methods.RECPs and corresponding basis sets have been used. The CASSCF andCAS-srLDA active spaces are 12/12.

054107-8 Fromager et al. J. Chem. Phys. 131, 054107 �2009�

Downloaded 22 Jun 2010 to 193.49.225.25. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp

�opt=0.3 a.u. value, the equilibrium bond angle is improvedcompared to standard LDA or PBE angles ��=0 limit� butonly by 7°. The bent structure is still preferred, although anon-negligible part of static correlation is now assigned tothe long-range interaction and thus described by theCASSCF part �see occupancies in Table IV�. As � increases,bending gets weaker. This was expected since the linearCASSCF equilibrium geometry must be recovered in the�→+� limit. Linearity is in fact preferred from �=0.6 a.u.,which is consistent with the fact that for �0.6 a.u. most ofthe electron correlation is described by the long-rangeCASSCF, as concluded above from the natural orbitals occu-pation numbers �see Sec. IV A 2�. Given that for �=0.3 a.u.the dynamic correlation is assigned to the srDFT part of theenergy, the minimal 12/12 active space should be sufficientfor describing the �static� long-range correlation. As a result,the bending tendency can only be inherent to the inaccuracyof the approximate short-range exchange and correlationfunctionals we use. Note that as shown in Table III, the

srLDA and srPBE functionals give �at both HF- and CAS-srDFT levels� very similar equilibrium geometries. We thusfocus on HF-srLDA and CAS-srLDA results in the follow-ing.

In order to analyze the bending problem observed at theCAS-srLDA level, let us first investigate the �-dependenceof the HF-srLDA equilibrium geometry �see Table III, Figs.4�a� and 4�c��. When increasing the � parameter, the Np–Obond distance is shortened significantly compared to its DFT-LDA ��=0 limit� value. It means that electron correlation,which is transferred from the short-range �described withinDFT� to the long-range correlation �not described at the HF-srDFT level� as � increases, has a strong effect on the bonddistance of the neptunyl �VII� ion. This is in agreement withthe work of Straka et al.15 On the other hand, the O–Np–Oangle increases with �. Linearity is in fact preferred when �becomes larger than 0.45 a.u. Note, however, that for �=0.45 a.u. the HF-srLDA potential curve is extremely flat�see Fig. 5�. In addition, as shown in Table IV, 5f-6d hybrid-

TABLE III. Equilibrium geometries for the ground state of NpO23+ calculated by HF-srDFT and CAS-srDFT

methods �using both srLDA and srPBE functionals and varying � between 0.3 and 2.0 a.u.� and compared tostandard WFT and DFT equilibrium structures. The CASSCF and CAS-srDFT active spaces are 12/12. The 5s,5p, and 5d orbitals were not correlated at the MP2, CASPT2, and CCSD�T� levels. Bond distances are inangstrom. For the bent structures the angle �in degrees� is given in parentheses.

Method �

HF-srDFT

NpO23+

CAS-srDFT

srLDA/srPBE srLDA/srPBE

0.30 1.653�157�/1.658�157� 1.666�156�/1.672�155�0.40 1.634�168�/1.637�168� 1.659�160�/1.663�160�0.45 1.626/1.628 1.657�163�/1.660�163�0.50 1.620/1.621 1.655�166�/1.657�167�0.55 1.614/1.615 1.654�170�/1.656�172�0.60 1.609/1.610 1.653�179�/1.6540.70 1.601/1.601 1.652/1.6531.00 1.585/1.586 1.650/1.6512.00 1.575/1.576 1.652/1.652

DFT-LDA ¯ 1.687�149�DFT-PBE ¯ 1.700�148�DFT-B3LYP ¯ 1.666�165�HF ¯ 1.585MP2 ¯ 1.756CASSCF ¯ 1.673CASPT2 ¯ 1.685CCSD�T� ¯ 1.675

TABLE IV. Occupation numbers of the �uy��ux� and �gy��gx� CAS-srLDA natural orbitals calculated at theCAS-srLDA equilibrium geometry for different � values �see Table III�. In addition the largest �in absolutevalue� �u and �g HF-srLDA orbital coefficients on the 5f and 6d atomic orbitals are given for each geometry.

� Geometry

NpO23+

�uy��ux� �gy��gx�Occ. 5f 6d Occ. 5f 6d

0.00 1.687�149� 2.0000�2.0000� 0.57�0.62� 0.07�0.06� 2.0000�2.0000� 0.30�0.34� 0.25�0.20�0.30 1.666�156� 1.9942�1.9944� 0.58�0.60� 0.06�0.06� 1.9912�1.9911� 0.23�0.45� 0.26�0.18�0.45 1.657�163� 1.9825�1.9826� 0.58�0.60� 0.04�0.04� 1.9801�1.9800� 0.16�0.50� 0.27�0.24�0.60 1.653�179� 1.9714�1.9714� 0.59�0.59� 0.00�0.00� 1.9715�1.9715� 0.00�0.04� 0.28�0.28�0.70 1.652�180� 1.9642 0.58 0.00 1.9670 0.00 0.282.00 1.652�180� 1.9359 0.54 0.00 1.9487 0.00 0.27

054107-9 Long-/short-range separation. II J. Chem. Phys. 131, 054107 �2009�

Downloaded 22 Jun 2010 to 193.49.225.25. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp

ization occurs when the structure is bent. In that case, it islargely the gerade orbitals in the linear symmetry ��g� thatform hybrids with a fairly equal mix of 5f and 6d orbitalsunlike in the bent ThO2 compound, where it is largely theungerade orbitals ��u� that get hybridized.56 For �=0.45 a.u., the hybridization is not facile anymore, at leastfor the �gy orbital, but is forced by the bending.

Returning to the CAS-srLDA results, the equilibrium ge-ometry is bent when �=0.45 a.u. Surprisingly, for 0.45�0.6 a.u., the CAS-srLDA method favors bending,while the HF-srLDA method favors linearity. As expected,the former �which describes the long-range correlation� givesa significantly larger bond distance than the latter. Note thatwhen freezing the bond distance to its equilibrium HF-srLDA value �1.626 Å�, we still obtain a bent structure withan equilibrium angle of 165° at the CAS-srLDA��=0.45 a.u.� level. Moreover, linearity is still preferred atthe HF-srLDA ��=0.45 a.u.� level when the bond distanceis frozen to its CAS-srLDA value �1.682 Å� as shown in Fig.5. We can thus conclude that the bending problem does notdepend on the bond distance. In the light of this analysis,different hypotheses can be raised for explaining why theCAS-srDFT ��=0.3 a.u.� approach favors the bent geom-etry. The first one is the bad description of the short-rangeexchange by the approximate short-range functionals we use;as � increases, more HF exchange is introduced into theHF-srDFT energy, increasing thus the O–Np–O angle untillinearity is reached �for �=0.45 a.u.�. The larger equilib-rium angle obtained at the hybrid DFT-B3LYP level com-pared to pure DFT-LDA and DFT-PBE values supports thishypothesis. It is still not clear whether or not linearity andbending observed for 0.45�0.6 a.u. at the HF-srDFTand CAS-srDFT levels, respectively, are only inherent to theapproximate short-range exchange functional used. If not,the approximate short-range correlation functional mightalso need improvements in describing the short-range corre-lation and its coupling to the long-range correlation. In con-clusion, this analysis suggests that the bending problem ob-served for NpO2

3+ in standard DFT is not only related to abad description of long-range static correlation effects bystandard functionals as previously suggested;15 otherwise wewould expect the CAS-srDFT model to favor linearity when�=0.3 a.u. A correct description of both long-range andshort-range exchange seems to be essential too.

V. CONCLUSIONS

The optimal �opt value for the � parameter that controlsthe range separation in hybrid MC-srDFT calculations hasbeen investigated on a test set consisting of isoelectronic f0

early actinide compounds. Using the recipe proposed re-cently by some of the authors,6 which aims at assigning staticand dynamic correlations to the long-range and short-rangeinteractions, respectively, we found �opt=0.3 a.u. Neverthe-less, the slightly larger value �0.4 a.u.� obtained previouslyfor light elements is still acceptable since, in that case, onlya very small part of the dynamic correlation is assigned tothe long-range interaction. This is a promising result in ourinvestigations to see if a universal range separation can be

used for all molecules. However, we still need to perform asimilar analysis on open-shell systems where static correla-tion effects are sometimes very strong such as in actinides�due the near-degeneracy of the 5f orbitals�. It is also impor-tant to check if choosing �=0.4 a.u. is still optimal for cal-culating excited states properties within MC-srDFT responsetheory. Work is currently in progress in these directions.

The accuracy of the MC-srDFT method for equilibriumgeometries has then been tested using �=0.3 a.u. with boththe srLDA and srPBE functionals �they actually give verysimilar bond distances and angles�. For static-correlation-freecompounds, geometries are in very good agreement withboth WFT and DFT results. For the neptunyl �VII� ionNpO2

3+, which has significant static correlation effects, bend-ing is preferred at the MC-srDFT level, whereas all WFT-based methods favor linearity. This clearly shows the needfor better short-range functionals. A good description of theshort-range exchange in particular seems to be important. Italso suggests that the bending tendencies of NpO2

3+ observedin DFT by Straka et al.15 may not only relate to the baddescription of static correlation effects by standard function-als.

For completeness we want to emphasize that this studyis for molecules. Some issues are different for extended sys-tems �periodic systems�, in particular, problems with long-range exact exchange. See, for example, two recent paperson properties of bulk actinide oxides.57,58 For bulk systemswith significant static correlation effects, we imagine that theso-called middle-range functionals59 will be optimal.

ACKNOWLEDGMENTS

The authors are grateful to Stefan Knecht for his all-electron four-component spinfree CISD calculations on theneptunyl �VII� ion, as well as for his comments on this work.The authors wish to thank Dr. Valérie Vallet, Dr. Jean-PierreFlament, and Dr. Trond Saue for very stimulating discus-sions. This research was supported by the Danish NaturalScience Research Council under Grant No. 272-06-0620, theNorwegian Center for Theoretical and Computational Chem-istry �CTCC�, the EC supported ACTINET Network of Ex-cellence �Project No. JRP 01-12�, grants from SKB, theSwedish Research Council, the Carl Trygger Foundation, andLaboratoire de Physique des Lasers, Atomes et Molecules.This work is also part of the project WADEMECOM funded bythe Agence Nationale de la Recherche �ANR�, France. Com-putational resources have been provided by the Danish Cen-ter for Scientific Computing.

1 P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 �1964�.2 A. Savin, Recent Developments and Applications of Modern DensityFunctional Theory �Elsevier, Amsterdam, 1996�, p. 327.

3 H. Stoll and A. Savin, Density Functional Methods in Physics �Plenum,New York, 1985�, p. 177.

4 A. Savin, Int. J. Quantum Chem. S22, 59 �1988�.5 A. V. Krukau, G. E. Scuseria, J. P. Perdew, and A. Savin, J. Chem. Phys.

129, 124103 �2008�.6 E. Fromager, J. Toulouse, and H. J. Aa. Jensen, J. Chem. Phys. 126,074111 �2007�.

7 E. Fromager, R. Cimiraglia, and H. J. Aa. Jensen, “Merging multi-reference perturbation and density-functional theories by means of rangeseparation” �to be published�.

054107-10 Fromager et al. J. Chem. Phys. 131, 054107 �2009�

Downloaded 22 Jun 2010 to 193.49.225.25. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp

8 H. Iikura, T. Tsuneda, T. Yanai, and K. Hirao, J. Chem. Phys. 115, 3540�2001�.

9 J. W. Song, T. Hirosawa, T. Tsuneda, and K. Hirao, J. Chem. Phys. 126,154105 �2007�.

10 O. A. Vydrov, J. Heyd, A. V. Krukau, and G. E. Scuseria, J. Chem. Phys.125, 074106 �2006�.

11 O. A. Vydrov and G. E. Scuseria, J. Chem. Phys. 125, 234109 �2006�.12 I. C. Gerber and J. G. Ángyán, Chem. Phys. Lett. 415, 100 �2005�.13 J. D. Chai and M. Head-Gordon, J. Chem. Phys. 128, 084106 �2008�.14 M. A. Rohrdanz, K. M. Martins, and J. M. Herbert, J. Chem. Phys. 130,

054112 �2009�.15 M. Straka, K. G. Dyall, and P. Pyykkö, Theor. Chem. Acc. 106, 393

�2001�.16 J. Toulouse, F. Colonna, and A. Savin, Phys. Rev. A 70, 062505 �2004�.17 E. Fromager and H. J. Aa. Jensen, Phys. Rev. A 78, 022504 �2008�.18 J. Toulouse, A. Savin, and H. J. Flad, Int. J. Quantum Chem. 100, 1047

�2004�.19 J. Heyd, G. E. Scuseria, and M. Ernzerhof, J. Chem. Phys. 118, 8207

�2003�.20 J. Heyd and G. E. Scuseria, J. Chem. Phys. 120, 7274 �2004�.21 P. Gori-Giorgi and A. Savin, Phys. Rev. A 73, 032506 �2006�.22 X. Cao, M. Dolg, and H. Stoll, J. Chem. Phys. 118, 487 �2003�.23 A. Bergner, M. Dolg, W. Kuechle, H. Stoll, and H. Preuss, Mol. Phys.

80, 1431 �1993�.24 G. Jansen and B. A. Heß, Phys. Rev. A 39, 6016 �1989�.25 M. Douglas and N. M. Kroll, Ann. Phys. 82, 89 �1974�.26 B. O. Roos, R. Lindh, P.-Å. Malmqvist, V. Veryazov, and P.-O. Widmark,

Chem. Phys. Lett. 409, 295 �2005�.27 B. O. Roos, R. Lindh, P.-Å. Malmqvist, V. Veryazov, and P.-O. Widmark,

J. Phys. Chem. A 108, 2851 �2004�.28 J. K. Pedersen, “Description of correlation and relativistic effects in cal-

culations of molecular properties,” Ph.D. thesis, University of SouthernDenmark, 2004.

29DALTON, Release 2.0, An ab initio electronic structure program �2005�.See www.kjemi.uio.no/software/dalton/dalton.html.

30 J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865�1996�.

31 R. G. Denning, J. Phys. Chem. A 111, 4125 �2007�.32 E. Fromager and H. J. Aa. Jensen, “Self-consistency effects in multicon-

figurational range-separated density-functional theory: a many-body-perturbation-theory-based investigation” �to be published�.

33 K. Andersson, M. Barysz, A. Bernhardsson, M. R. A. Blomberg, D. L.Cooper, M. P. Fülscher, C. de Graaf, B. A. Heß, G. Karlström, R. Lindhet al., MOLCAS Version 7.0, Lund University, Sweden �2006�.

34MOLPRO, a package of ab initio programs designed by H.-J. Werner and P.J. Knowles, version 2002.6, R. D. Amos, A. Bernhardsson, A. Berninget al.

35 K. Andersson, P.-Å. Malmquist, B. Roos, A. Sadlej, and K. Wolinski, J.Phys. Chem. 94, 5483 �1990�.

36 K. Andersson, P.-Å. Malmquist, and B. Roos, J. Chem. Phys. 96, 1218�1992�.

37 C. Hampel, K. A. Peterson, and H.-J. Werner, Chem. Phys. Lett. 190, 1�1992�.

38 P. J. Knowles, C. Hampel, and H.-J. Werner, J. Chem. Phys. 99, 5219�1993�.

39 P. J. Knowles, C. Hampel, and H.-J. Werner, J. Chem. Phys. 112, 3106�2000�.

40 M. J. O. Deegan and P. J. Knowles, Chem. Phys. Lett. 227, 321 �1994�.41 S. H. Vosko, L. Wilk, and M. Nusair, Can. J. Phys. 58, 1200 �1980�.42 A. D. Becke, J. Chem. Phys. 98, 5648 �1993�.43 L. Gagliardi and B. O. Roos, Chem. Phys. Lett. 331, 229 �2000�.44 V. Vallet, P. Macak, U. Wahlgren, and I. Grenthe, Theor. Chem. Acc. 115,

145 �2006�.45 V. Vallet, U. Wahlgren, and I. Grenthe, Chem.-Eur. J. 13, 10294 �2007�.46 See EPAPS document number E-JCPSA6-131-003930 for the total HF-

and CAS-srDFT ground-state energies plotted with respect to the � pa-rameter at the RECP level for CUO and UN2 and at the all-electron levelfor PaO2

+, UO22+, UN2, and NpO2

3+. For more information on EPAPS, seehttp://www.aip.org/pubservs/epaps.html.

47 F. P. Rotzinger, Chem. Rev. �Washington, D.C.� 105, 2003 �2005�.48 F. P. Rotzinger, Chem.-Eur. J. 13, 10298 �2007�.49 H. J. Aa. Jensen, P. Jørgensen, H. Ågren, and J. Olsen, J. Chem. Phys.

88, 3834 �1988�.50 H. J. Aa. Jensen, P. Jørgensen, H. Ågren, and J. Olsen, J. Chem. Phys.

89, 5354 �1988�.51 S. Knecht, private communication �2009�.52 S. Knecht, H. J. A. Jensen, and T. Fleig, J. Chem. Phys. 128, 014108

�2008�.53

DIRAC, a relativistic ab initio electronic structure program, ReleaseDIRAC08 �2008�, written by L. Visscher, H. J. Aa. Jensen, and T. Saue,with new contributions from R. Bast, S. Dubillard, K. G. Dyall, U. Ek-ström, E. Eliav, T. Fleig, A. S. P. Gomes, T. U. Helgaker, J. Henriksson,M. Iliaš, Ch. R. Jacob, S. Knecht, P. Norman, J. Olsen, M. Pernpointner,K. Ruud, P. Sałek, and J. Sikkema �see http://dirac.chem.sdu.dk�.

54 K. G. Dyall, Theor. Chem. Acc. 117, 491 �2007� �available from theDirac web site: http://dirac.chem.sdu.dk�.

55 Y.-K. Han and K. Hirao, J. Chem. Phys. 113, 7345 �2000�.56 K. G. Dyall, Mol. Phys. 96, 511 �1999�.57 I. D. Prodan, G. E. Scuseria, and R. L. Martin, Phys. Rev. B 73, 045104

�2006�.58 I. D. Prodan, G. E. Scuseria, and R. L. Martin, Phys. Rev. B 76, 033101

�2007�.59 T. H. Henderson, A. F. Izmaylov, G. E. Scuseria, and A. Savin, J. Chem.

Phys. 127, 221103 �2007�.

054107-11 Long-/short-range separation. II J. Chem. Phys. 131, 054107 �2009�

Downloaded 22 Jun 2010 to 193.49.225.25. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp