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The natural orbital functional theory of the bonding inCr2, Mo2 and W2

F. Ruiperez,*a M. Piris,ab J. M. Ugaldea and J. M. Matxaina

In this paper, we present for the first time a description based on the natural orbital functional theory

(NOFT) of the group VI dimers, namely, Cr2, Mo2 and W2. The PNOF5, Piris Natural Orbital Functional,

has been used throughout this work, and the results are compared to multireferential perturbation

theory (CASPT2) results. Both methods have been combined with effective core potentials to take into

account the scalar relativistic effects. In addition, for Cr2, an all-electron TZVP quality basis set has also

been used to recover the core-valence dynamical correlation. In all cases, PNOF5 shows better behavior

than CASPT2, which needs a larger basis set to recover comparable amounts of dynamical correlation.

PNOF5 is able to account for the non-dynamical electron correlation, which is responsible for the

multireferential nature of these dimers. However, it does not fully recover the dynamical correlation,

which is crucial for the accurate description of these challenging potential energy curves. Consequently,

PNOF5 predicts longer equilibrium distances and lower dissociation energies than the experimental

values. Unlike CASPT2, the PNOF5 results do not significantly improve by using larger basis sets. These

new findings represent a major step in the NOFT development, since PNOF5 is the first functional of

the natural orbitals reported to yield a chemically balanced and accurate description of these

challenging transition metal dimers.

1 Introduction

Natural Orbital Functional Theory (NOFT)1–3 appears as analternative formalism to both DFT and wavefunction-basedmethods, as it describes the electronic structure in terms ofthe natural orbitals and their occupation numbers. The NOFThas been reviewed in a recent paper4 where further details maybe found. Since then, a number of new NOFs have appeared inthe literature, and the applications of NOFT to realistically largeand complex chemical systems have increased significantly,especially in molecules containing first- and second-row elementsof the periodic table.5 Unfortunately, applications of NOFT totransition metal containing systems are not available in theliterature yet. Therefore, at the present stage of NOFT development,the correct treatment of these systems remains a challenging task.

It is well known that the research performed in smalltransition metal clusters is of foremost importance in scienceand technology because of their wide range of applications inoptics, biomedicine and catalysis, among others.6,7 The use of

small clusters facilitates the experimental and theoretical analysesand the study of transition metal dimers, in particular, allows anextensive understanding of their properties without the presence ofligands that may obscure the interpretation of the experiments andrelieves the computational effort as well.

Among the transition metals, the group VI dimers, namelyCr2, Mo2 and W2, have been the object of both experimental8–15

and theoretical16–30 studies in the last few years due to thepossibility to generate sextuple bonds involving the nd and (n + 1)sorbitals. However, from a theoretical point of view, the descriptionof transition metal dimers shows several difficulties: the presenceof a dense manifold of low-lying states due to the partial occupa-tion of the nd and (n + 1)s shells leads to ground states of intrinsicmultireferential character and thus a large number of configura-tions is needed to properly account for the static (or non-dynamical)correlation energy. The 1S+

g ground state of these dimers ishighly multiconfigurational and the Cr2, in particular, presentsthe following dominant configuration 4ssg

23dsg23dpu

43ddg4,

which weighs only 47% of the total wavefunction.18 Besides, theexperimental potential energy surface (PES) of this dimer showsone minimum at 1.68 Å approximately, which corresponds tothe 3d–3d interaction, and a shelf-like region around 2.5 Åwhere the 4s–4s interaction is prevalent.11 Finally, the dimerdissociates into two Cr atoms with a high-spin open-shell

a Kimika Fakultatea, Euskal Herriko Unibertsitatea (UPV/EHU), and Donostia

International Physics Center (DIPC), P.K. 1072, 20080 Donostia, Euskadi, Spain.

E-mail: fernando.ruiperez@ehu.es; Fax: +34 943 015 270; Tel: +34 943 015 312b IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Euskadi, Spain

Received 9th October 2012,Accepted 29th November 2012

DOI: 10.1039/c2cp43559d

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ground state, 3d54s1 (7S3), that is, each atom has six unpairedelectrons, which implies that the electron correlation is greatlyimportant all along the PES. Moreover, the static correlationalone is not sufficient to provide a satisfactory description ofthe PES and the incorporation of the dynamical correlationenergy is essential. The multiconfigurational nature of thewavefunction prevents the use of post-Hartree–Fock (HF) methodsand poses important challenges to density functional theory (DFT).DFT-based studies carried out in Cr2 have revealed that, in general,exchange-correlation functionals inadequately describe thepotential surface17 and spin contamination is present, whichdetermines that a single-reference representation is somewhatsimplistic for the description of this molecule. Under thesecircumstances, the use of multireferential methods is appealingand, therefore, the most accurate theoretical results in Cr2 areobtained by using this methodology.18,21,31,32

NOFT is, in principle, an alternative to single-reference post-HFor DFT methods since it allows us to take into account the non-dynamical effects. Therefore, NOFT results could be, likely,competitive with high-level multireference methods. However,as mentioned above, up to now one cannot find any work in theliterature where NOFT has been applied to transition metals ingeneral, and the group VI dimers in particular.

Among the NOFT functionals available, PNOF3,33 PNOF434

and PNOF535 functionals are the most promising ones, withinthe Piris reconstruction formulation.36 Concretely, the PNOF3formulation showed an outstanding performance for atomsand molecules33,37 and in the description of potential energysurfaces.38 Unfortunately, an in-depth analysis of several diatomicmolecules dissociation curves34 as well as the description ofdiradicals and diradicaloids39 revealed that PNOF3 overestimatesthe amount of electron correlation when near-degeneracyeffects become important. This is related to the violation ofthe N-representability conditions, in particular to the violationof the G positivity condition.34 As a consequence, this functionalfails dramatically in predicting transition metal properties,where the non-dynamical effects are very important.

Accordingly, a more restricted functional was developed,PNOF4,34 that meets rigorously the known three necessaryN-representability conditions (D, Q and G) of the 2-RDM.Hence, PNOF4 correctly describes molecules with electrons inorbitals that become increasingly degenerated.34,39 However,the formulation of the PNOF4 functional contains a positivevariable SF which must be bound from above by one (seeeqn (7)–(11) of ref. 34), and several calculations have revealedthat these conditions do not meet when four or more orbitalsbecome degenerate.35 Note that, as mentioned above, in thegroup VI dimers, the singlet ground state dissociates into twosextuplets, and, consequently, this functional also fails indescribing correctly the PES of these dimers. Additionally, we havealso found that PNOF4 leads frequently to incorrect diatomicdissociation limits with fractional numbers of electrons on thedissociated atoms.

In order to solve these problems, the PNOF5 functional35

was introduced to deal with multiconfigurational states. Apreliminary performance test showed that PNOF5 predicts the

barrier for the ethylene torsion in an outstanding agreementwith highly accurate multireferential perturbation theory (CASPT2)calculations.40 Finally, the dissociation of the N2 isoelectronicseries, where molecules dissociate into two quartet atoms,has been studied41 and an integer number of electrons on theseparated atoms was obtained, therefore leading to physicallymeaningful solutions. Furthermore, equilibrium bond distances,dipole moments and dissociation energies were accurately repre-sented. Moreover, it has been shown that PNOF5 not only takesinto account non-dynamical effects, but also part of the dynamicalelectron correlation.42,43 Finally, it has recently been demonstratedthat PNOF5 provides an upper bound to the exact energy.44

Therefore, PNOF5 appears at the pole-position among naturalorbital functionals in order to describe, for the first time withinNOFT, transition metal containing systems.

Hence, we have performed an analysis of the potentialenergy curves of the group VI dimers in their ground state,namely Cr2, Mo2 and W2, using the PNOF5 functional, and theresults obtained are compared to ab initio wavefunction-basedCASSCF/CASPT2 calculations. In Section 2, these methods arebriefly described and the computational details are explained.In Section 3, the obtained results are presented and discussed.Finally, in Section 4 the main conclusions are given.

2 Methods2.1 PNOF5

The PNOF5 energy for a singlet state of an N-electron systemcan be expressed in the following way:35

EPNOF5 ¼XN

p¼1½npð2Hpp þ JppÞ �

ffiffiffiffiffiffiffiffiffiffin~pnpp

Kp~p�

þXN

p;q¼1

00nqnpð2Jpq � KpqÞ

ð1Þ

where p states for the spatial natural orbital and np its occupa-tion. Hpp is the matrix element of the kinetic energy andnuclear attraction terms, whereas Jpq = hpq|pqi and Kpq =hpq|qpi are the Coulomb and exchange integrals, respectively.The p state defines the coupled natural orbital to the orbital p,that is, p = N � p + 1, N being the number of particles ofthe system. It is worth noting that we look for the pairs ofcoupled orbitals (p,p) which yield the minimum energy for thefunctional defined above. The actual p and p paired orbitals arenot constrained to remain fixed along the orbital optimizationprocess. Consequently, the pairing scheme of the orbitals isallowed to vary along the optimization process until the mostfavorable orbital interactions are found. In accordance withthis assumption, all occupancies vanish for p > N. The doubleprime in eqn (1) indicates that both the q = p term and thecoupled one-particle state term p = p are omitted from the lastsummation. Additionally, the bounds on the off-diagonal elementsof D that stem from imposing the N-representability D and Qpositivity necessary conditions of the 2-RDM, as well as the sum

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rule that D must fulfill, imply that the occupation of the p levelmust coincide with that of the hole of its coupled state p, namely,

np = hp, np + np = 1 (2)

where hp denotes the hole 1 � np in the spatial orbital p.The solution in NOFT is established by optimizing the

energy functional with respect to the occupation numbersand to the natural orbitals, separately. The orbital optimizationconstitutes the bottleneck of this algorithm since direct mini-mization of the orbitals has been proven to be a costlymethod.45 Thus, in this work we have employed a recentsuccessful implementation of an iterative diagonalizationprocedure.46

The number of particles is always conserved (N = 2P

pnp)due to the relation of eqn (2) to the occupation numbers of thecoupled one-particle states. Eqn (2) and the N-representabilitybounds (0 r np r 1) of the 1-RDM are easily enforced by settingnp = cos2gp and np = sin2gp. Note that PNOF5 allows constraint-free minimization with respect to the auxiliary variables {gp},which yield substantial savings for the computational time.

This functional, as implemented in the PNOFID program,47

has been used to calculate the PES of these dimers. Single-pointcalculations were carried out at different distances to obtain thecurves up to 10 Å. Once the potential energy surfaces werecharacterized, the LEVEL 8.0 program48 was used to determinethe spectroscopic constants.

2.2 CASSCF/CASPT2

Spin-free wavefunction-based calculations were performed in atwo-step procedure. Firstly, we carried out complete activespace self-consistent field calculations (CASSCF),49–51 wherethe active space is defined by the distribution of 12 electronsin the following 12 molecular orbitals: ssg, dsg, dpu, ddg, ss�u,ds�u, dp�g and dd�u, composed of linear combinations of the nd

and (n + 1)s atomic orbitals. Secondly, in order to include thedynamical correlation effects, complete active space second-order perturbation theory calculations (CASPT2)52,53 are performed,where the ns2 and np6 atomic shells are correlated dynamicallyas well. The imaginary shift technique54 was used to avoid thepresence of intruder states, with a shift value of 0.20 au. Finally,the potential energy surfaces are scanned using the VIBROTprogram55 in order to gather the spectroscopic constants. Allcalculations were performed using the MOLCAS 7.6 suite ofprograms.55

2.3 Effective core potentials and basis sets

Effective core potentials (ECP) have been used to represent theinner electrons (valence ns, np, nd and (n + 1)s shells), including thescalar relativistic effects, together with the following valence basisset (8s7p6d)/[6s5p3d] for Cr,56 Mo57 and W.57 This combination ofECP and basis functions will be named ECP hereafter for the sakeof clarity. In Cr2, the importance of core-valence correlation,compared to the relativistic effects, is very large. Therefore, we havealso carried out all-electron calculations in Cr2 using the basis set(14s11p6d3f)/[8s6p4d1f], developed by Wachters.58 This basis

set will be denoted AE from now on. In this way, it is possible tostudy both the importance of the use of all-electron basis sets inCr2 and the scalar relativistic effects, which are known to bevery important in Mo2 and W2. Note that, at this moment,relativistic effects can only be considered via effective corepotentials, since the relativistic Hamiltonian is not implemen-ted in PNOFID yet.

3 Results and discussion

In this work, we have performed a study of the ground state ofCr2, Mo2 and W2 dimers using the PNOF5 (Piris Natural OrbitalFunctional) as well as the CASSCF/CASPT2 method, in order tocompare their performance with respect to the benchmarkexperimental data. We have calculated the potential energysurfaces up to 10.0 Å to get the spectroscopic constants:equilibrium bond length (Re), vibrational frequency (oe) anddissociation energy (De). Besides, we have carried out ananalysis of the occupation of the natural orbitals and multi-plicity of the bond by using the effective bond order (EBO)definition of Roos and co-workers,22,59 which is described asP

(Zb � Zab)/2, where Zb and Zab state for the natural orbitaloccupation of the bonding and antibonding orbitals, respec-tively. In this way, the bond order is calculated for each pair ofbonding–antibonding orbitals and the multiplicity of the bondis given by the sum of all individual EBO contributions. ThisEBO is a non-integer number and, in order to state the multi-plicity of the bond, the lowest integer value larger than the EBOcalculated should be used. The effect of the basis set is studiedin the Cr2 dimer, where both effective core potential (ECP) andall-electron (AE) calculations are performed as well.

3.1 Cr2 dimer

The ground state of the chromium atom is 7S3, arising from the3d54s1 electronic configuration.60 The molecular ground stateof the Cr2 dimer is a singlet, 1S+

g, with the following dominantelectronic configuration: jC1Sþg

i ¼ j4ssg23dsg

23dpu 43ddg 4i,whose weight is only 47% of the total wavefunction, whichhighlights its large multiconfigurational character.18 As it wasmentioned in the Introduction, Cr2 is one of the most challengingdimers and is considered a benchmark molecule for theoreticalmethods due to its complicated electronic structure.

The multireferential nature of the molecule is reflected inthe occupations of the molecular orbitals, which are collectedin Table 1, calculated at both PNOF5/ECP and CASSCF/ECPlevels of theory. Although this is a closed-shell singlet state, it isremarkable how the occupation of the bonding orbitals is farfrom being 2, with a sizeable charge transfer to the antibonding

Table 1 Occupation of the molecular orbitals and effective bond order (EBO) ofCr2 calculated at the experimental equilibrium bond length, Rexp = 1.68 Å8

ssg dsg dpu ddg ss�u ds�u dp�g dd�u EBO

PNOF5 1.91 1.76 1.81 1.42 0.09 0.24 0.19 0.58 4.10CASSCF 1.92 1.77 1.80 1.58 0.08 0.23 0.20 0.42 4.44

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orbitals, which indicates that the gap between the bonding andantibonding orbitals is small. Note that the occupation of thebonding ddg is only 1.42 calculated with PNOF5, and 1.58 calcu-lated with CASSCF, and the occupation of the correspondingantibonding dd�u is 0.58 and 0.42, obtained with PNOF5 andCASSCF, respectively. It is worth noticing the outstanding agree-ment between the PNOF5 and the CASSCF results, with only a tinydiscrepancy in the occupation of the dd and dd* orbitals.

The variation of the occupation of the PNOF5 molecularorbitals along the potential energy surface is represented inFig. 1, which reveals that, at longer distances, the occupationchanges smoothly, decreasing for the bonding and increasingfor the antibonding orbitals, until the twelve orbitals haveoccupation 1. This behavior demonstrates that PNOF5 is ableto properly reproduce the dissociation of the Cr2 dimer, inwhich each chromium atom achieves six unpaired electrons.

From the occupation numbers at the experimental equilibriumbond length (1.68 Å) we perform an analysis of the multiplicity ofthe bond by means of the effective bond order (EBO) concept,which is calculated to be 4.10 with PNOF5 and 4.44 with CASSCF.Thus, both methods predict a formal bond order of 5, that is, aquintuple bond, which is consistent with previous high-leveltheoretical results available in the literature.61 Therefore, wecan conclude that PNOF5 is capable to correctly represent thenon-dynamical correlation, that is, the multiconfigurationalnature of the bond, as well as the electron density rearrangementat dissociation.

3.1.1 Effective core potential vs. all-electron calculations.In this section we analyze the effect of substituting the effectivecore potential (ECP) by an all-electron (AE) basis set on theshape of the PES and the spectroscopic constants. This analysishas been motivated by the unexpected low-quality results

provided by the CASSCF/CASPT2 approach together with theECP. The calculated spectroscopic constants at PNOF5, CASSCFand CASPT2 levels of theory using both the ECP and the AEbasis set are collected in Table 2, together with the experi-mental results.

The use of the ECP provides with very poor spectroscopicconstants, regardless of the method. In this manner, all of themlargely overestimate the experimental bond length (1.68 Å).Nevertheless, it is worth noting that PNOF5 is the method thatprovides with the best estimate (2.10 Å) followed by CASPT2(2.64 Å) and finally CASSCF (3.19 Å). Regarding the vibrationalfrequencies, CASSCF (73 cm�1) shows a significant discrepancywith the experimental value of oexp = 470 cm�1. The incorpora-tion of the dynamical correlation improves the result (160 cm�1

using CASPT2) however this value is still quite far from theexperimental mark. PNOF5 produces a vibrational frequency ofoe = 163 cm�1, similar to CASPT2. Finally, the best evaluationof the dissociation energy is given by PNOF5, De = 0.95 eVcompared to Dexp = 1.53 eV, although the result is not satisfactoryyet. CASSCF and CASPT2 give worse results, of 0.14 and 0.75 eVrespectively.

When the AE basis set is used, the improvement is negligibleat the CASSCF level of theory. In such a way, the equilibriumbond length remains almost unchanged (3.18 Å compared to3.19 Å using the ECP), while the vibrational frequency anddissociation energy are slightly increased (84 cm�1 and 0.34 eVvs. 73 cm�1 and 0.14 eV with the ECP). However, these resultsstill present a great discrepancy with the experimental data.The effect of the basis set is expected to be manifest in theCASPT2 calculations, since in this method the second-orderperturbational treatment makes use of the whole basis set. Thiseffect was previously observed by Roos and Andersson62 as wellas by Scuseria, which showed that the inclusion of g typefunctions has a considerable impact on spectroscopic constants.63

Hence, it is observed that the bond length is moderately shortened(from 2.64 Å to 2.43 Å), the vibrational frequency exhibits a notableimprovement (from 160 cm�1 to 326 cm�1) and the dissociationenergy is increased as well (from 0.75 eV to 1.07 eV). Nevertheless,the comparison with the experimental value is not satisfactory yet.If we analyze now the PNOF5 results, we find a similar correction tothat in CASPT2. That is, the bond length is shortened from 2.10 Åto 1.90 Å, the vibrational frequency is 264 cm�1, which is notablybetter than the result obtained with the ECP (163 cm�1), and thedissociation energy is calculated to be 1.06 eV, almost the samevalue as the one obtained with CASPT2. Anyhow, since the results

Fig. 1 Occupation of the Cr2 molecular orbitals along the potential energysurface, calculated at the PNOF5/ECP level of theory.

Table 2 Spectroscopic constants of the Cr2 dimer at PNOF5, CASSCF and CASPT2levels of theory using ECP and AE. Bond lengths in Å, vibrational frequencies incm�1 and dissociation energies in eV

PNOF5 CASSCF CASPT2

Exp.ECP AE ECP AE ECP AE

Re 2.10 1.90 3.19 3.18 2.64 2.43 1.68a

oe 163 264 73 84 160 326 470a

De 0.95 1.06 0.14 0.34 0.75 1.07 1.53b

a From ref. 8. b From ref. 13.

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provided by PNOF5 with the ECP were already better, theimprovement accomplished by the AE basis set makes thePNOF5 results to be the best among the three methods.

The potential energy surfaces obtained using the ECP andthe AE basis set are displayed in Fig. 2 and 3, respectively,together with the experimental PES.11 The discrepancy of theCASSCF and CASPT2 results is obvious in both cases, never-theless, although the PNOF5 curve calculated with the ECP stillshows a clear discrepancy, in Fig. 3 is observed how the use ofthe AE basis set in PNOF5 provides with a greatly improved PES,where it is possible to perceive a small shoulder that resemblesthe shelf-like region present in the experimental curve, afeature which is very difficult to reproduce even using high-level theoretical methods. In this case very large basis sets mustbe used in order to take into account correctly the dynamicalcorrelation. It should be pointed out that, although PNOF5suffers from a lack of dynamical correlation effects by construc-tion, it is able to recover the important amount to describequalitatively the shoulder of Cr2. In the shelf-like region, the4s–4s interaction is prevalent, a characteristic that is clearlyobserved in the distinct behavior of the occupation of the 4ssg

and 4ss�u molecular orbitals, see Fig. 1. This figure reveals that

the occupation of the rest of the orbitals is already close to one,retaining only the interaction due to the 4s-like molecularorbitals.

In summary, the results show that PNOF5 recovers thenon-dynamical correlation energy and represents correctlythe multiconfigurational nature of the Cr2 bond. Besides, thespectroscopic constants and potential energy surface are betterrepresented with an AE basis set than with an ECP andthe accuracy is at the same level of CASPT2. Inclusion ofthe remaining dynamical correlation would improve all thecalculated features, decreasing bond-lengths, and increasingboth vibrational frequencies and dissociation energies, sincedynamical correlation is larger at the equilibrium region.

3.2 Mo2 and W2 dimers

The study of these molecules has been performed using onlyECP, since accounting for the scalar relativistic effects ismandatory for the correct description of molecules containingheavy transition metals.

3.2.1 Mo2. The ground state of Mo2 is a singlet 1S+g, arising

from the 4d55s1 (7S3) ground state of the molybdenum atoms,60

with the following dominant electronic configuration:jC1Sþg

i ¼ j5ssg24dsg

24dpu 44ddg 4i. The CASSCF calculation

reveals that this configuration weights 64% of the total wave-function, demonstrating the clear multireferential nature of theground state. However, it is less than that in Cr2.

The occupation number of the molecular orbitals, collectedin Table 3, notably deviates from a closed-shell picture and isprominent in the case of the dd orbitals. The comparison ofPNOF5 with CASSCF is excellent, with the exception of the ddorbitals, and the variation of these occupation numbers alongthe PES is similar to that in Cr2, demonstrating the capability ofthis method to represent properly the multiconfigurationalnature of the Mo2 ground state as well as its dissociation.

The effective bond order is calculated to be 4.77 with PNOF5,that is, a formal quintuple bond, and 5.08, a very weak formalsextuple bond, with CASSCF. This discrepancy mainly derivesfrom the difference in the population of the ddg (1.58 and1.75 with PNOF5 and CASSCF, respectively) and dd�u orbitals(0.42 with PNOF5 and 0.25 with CASSCF). Previous high-leveltheoretical calculations23 yielded an EBO of 5.2, which meansthat the molybdenum dimer is well described as a sextuply-bonded molecule. Therefore, PNOF5 overestimates to someextent the multireferential character of this dimer, as it wasobserved in the Cr2 dimer, however, in this case, since the EBOvalue is close to 5.0, PNOF5 remains below this value while

Fig. 2 Potential energy surfaces calculated at the PNOF5, CASSCF and CASPT2levels of theory, with ECP, together with the experimental curve.

Fig. 3 Potential energy surfaces calculated at the PNOF5, CASSCF and CASPT2levels of theory, with AE basis set, together with the experimental curve.

Table 3 Occupation of the molecular orbitals and effective bond order (EBO) ofMo2 calculated at the experimental equilibrium bond length, Rexp = 1.94 Åa

ssg dsg dpu ddg ss�u ds�u dp�g dd�u EBO

PNOF5 1.90 1.89 1.91 1.58 0.10 0.11 0.09 0.42 4.77CASSCF 1.93 1.88 1.89 1.75 0.07 0.12 0.11 0.25 5.08

a From ref. 9.

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CASSCF is above, producing a different value of the bondmultiplicity.

The calculated and experimental spectroscopic constants ofMo2 are collected in Table 4, where it is observed that theagreement with the experiment is better than that in the case ofCr2, which emphasizes the difficulties inherent to this mole-cule. In this manner, the three methods yield roughly the samebond length (2.10 Å with PNOF5 and CASSCF, and 2.09 Å withCASPT2) not very far from the experimental value of 1.94 Å.

The vibrational frequencies calculated with PNOF5(368 cm�1), CASSCF (306 cm�1) and CASPT2 (358 cm�1) showan important discrepancy with the experimental value of oexp =477 cm�1. Finally, regarding the dissociation energy, it isobserved that the CASSCF performance is quite poor (De =0.55 eV) and, when the dynamical correlation energy is takeninto account, the improvement is noteworthy, and so, theCASPT2 value is De = 2.14. However, the best estimate isprovided by PNOF5, which predicts a dissociation energy ofDe = 3.26 eV, compared to the experimental value of Dexp =4.28 eV. In Fig. 4 are displayed the potential energy surfacesobtained with the three methods.

The results achieved by PNOF5 concerning the geometry ofthe Mo2 dimer are encouraging, in line with CASSCF andCASPT2 results. Regarding the dissociation energy, PNOF5 isthe method that gives a closer result to the experiment, thoughthe discrepancy is still large (1.02 eV). This discrepancy is dueto a lack of dynamical correlation in PNOF5 calculations. Theeffect of the inclusion of all dynamical correlation woulddecrease the predicted bond length, and increase the

dissociation energy, since dynamical correlation is larger nearthe equilibrium region.

3.2.2 W2. The ground state of the tungsten atom is 5D0

(5d46s2),60 and the W2 dimer is described, as well as the otherdimers considered in this work, as a singlet 1S+

g ground state,with the dominant electronic configuration jC1Sþg

i ¼j6ssg

25dsg25dpu 45ddg 4i weighing 68% of the total wavefunc-

tion. The multireferential character is quite pronounced, simi-lar to Mo2 (64%) and smaller than Cr2 (47%).

The occupations of the molecular orbitals are gathered inTable 5, calculated at the PNOF5/ECP and CASSCF/ECP levels oftheory. The deviation of a closed-shell reference is evident fromthe occupation of the dd orbitals, though less pronounced thanin the other dimers considered in this work (see Tables 1 and 3).

The calculated effective bond order using these occupationnumbers is 5.15 with PNOF5 and 5.18 with CASSCF. Thus, bothmethods predict a sextuply-bonded molecule, as it was previouslyfound by high-level ab initio calculations.25 The good agreementbetween the two methods is noticeable. Finally, the analysis of theoccupation numbers along the potential energy surface shows that,in the dissociation limit, all the orbitals are occupied by oneelectron, and, therefore, this demonstrates the capability of PNOF5to accurately represent the dissociation process of W2.

The calculated spectroscopic constants are collected inTable 6, together with the available experimental results. Theequilibrium bond length calculated with PNOF5 is Re = 2.30 Å,with CASSCF is Re = 2.16 Å and with CASPT2 is Re = 2.15 Å. Noexperimental result is available for the bond length, to ourknowledge, but we can compare with theoretical calculationsfrom the literature. Borin et al. have obtained a bond distanceof 2.01 Å using the CASSCF/CASPT2 approach together with aquadruple-z quality ANO-RCC basis set.25 The comparison withPNOF5 calculations is satisfactory, however, both CASSCF andCASPT2 results are closer to the reference value.

The vibrational frequencies calculated with PNOF5 (262 cm�1),CASSCF (266 cm�1) and CASPT2 (291 cm�1) underestimate theexperimental value of 337 cm�1 and again CASPT2 provideswith the best value. Finally, the dissociation energy calculated

Table 4 Spectroscopic constants of the Mo2 dimer at PNOF5, CASSCF and CASPT2levels of theory. Bond lengths in Å, vibrational frequencies in cm�1 anddissociation energies in eV

PNOF5 CASSCF CASPT2 Exp.

Re 2.10 2.10 2.09 1.94a

oe 368 306 358 477b

De 3.26 0.55 2.14 4.28b

a From ref. 9. b From ref. 10.

Fig. 4 Potential energy surface of the Mo2 dimer at PNOF5, CASSCF and CASPT2levels of theory, referred to the dissociation energy.

Table 5 Occupation of the molecular orbitals and effective bond order (EBO) ofW2 calculated at Re = 2.01 Åa

ssg dsg dpu ddg ss�u ds�u dp�g dd�u EBO

PNOF5 1.94 1.91 1.94 1.71 0.06 0.09 0.06 0.29 5.15CASSCF 1.93 1.88 1.91 1.78 0.07 0.13 0.09 0.22 5.18

a Theoretical value, from ref. 25.

Table 6 Spectroscopic constants of the W2 dimer at PNOF5, CASSCF and CASPT2levels of theory. Bond lengths in Å, vibrational frequencies in cm�1 anddissociation energies in eV

PNOF5 CASSCF CASPT2 Exp.

Re 2.30 2.16 2.15 2.01a

oe 262 266 291 337b

De 3.48 1.74 3.14 5 � 1c

a Theoretical value from ref. 25. b From ref. 15. c From ref. 10.

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with PNOF5 (3.48 eV) is far from the experimental value of 5 �1 eV. Note, however, the large uncertainty of this experiment. Ineither way, PNOF5 is the method that provides with betterresults, since CASSCF considerably underestimates the experi-mental value (1.74 eV), and CASPT2 gives a smaller dissociationenergy (3.14 eV). Note, as it has been previously mentioned,that these values are a consequence of the lack of part of theelectron correlation in PNOF5.

Fig. 5 shows the potential energy surfaces corresponding tothe three methods referred to the dissociation energy.

To summarize, it is safe to say that the results obtained inW2 are very promising. The occupation numbers reflect theability of PNOF5 to recover the non-dynamical correlationenergy, as it was previously found for Cr2 and Mo2. Moreover,the spectroscopic constants compare rather well with the experi-ment in a molecule where the relativistic effects play a crucialrole, not only the scalar but also the spin–orbit coupling effects.

4 Conclusions

The PNOF5 (Piris Natural Orbital Functional) has beenemployed to investigate the electronic structure and bondingof the group VI homonuclear dimers, Cr2, Mo2 and W2, togetherwith the CASSCF/CASPT2 approach, in order to compare with awell-established wavefunction-based method. The resultsdemonstrate that PNOF5 is able to properly account for thenon-dynamical, or static, correlation energy, which is reflectedin the occupation numbers of the molecular orbitals. Theseoccupation numbers show an outstanding agreement with theCASSCF results, with small discrepancies in the occupation ofthe dd orbitals. This means that this functional correctly takesinto account the multiconfigurational nature of the groundstate of the considered dimers, which is outstanding in the caseof Cr2, known as a benchmark molecule for quantum chemicalmethods due to the extremely challenging electronic structureof the ground state and potential energy surface.

However, PNOF5 is not able to fully recover the dynamicalcorrelation energy and discrepancies in the spectroscopic

constants are observed, compared to the experimental results,especially in Cr2. The importance of the dynamical correlationin this dimer is remarkable, which is also reflected in the effectof the basis set. Thus, the use of an all-electron basis setaccomplishes an improvement in PNOF5 and CASPT2, whilein CASSCF it is hardly observed. This improvement is not onlynoticed in the spectroscopic constants but also in the shape ofthe potential energy surface, where the experimental shelf-likeregion is now distinguished on the PNOF5 curve.

The results obtained for Mo2 and W2 are particularly better,mainly in the equilibrium bond length, which highlightsthe difficulty inherent to the Cr2 molecule. However, somedisagreements in the spectroscopic constants calculated withPNOF5 still remain (as well as in CASSCF and CASPT2) as aconsequence of the previously mentioned lack of dynamicalcorrelation. The discrepancy in the W2 dissociation energy islarger, probably due to the spin–orbit coupling relativisticeffects, which are important for this heavy element.

Summarizing, it is safe to say that the accuracy of the PNOF5results lies between CASSCF and CASPT2. It recovers not onlythe non-dynamical correlation energy, as CASSCF, but alsosome of the dynamical, however not as much as CASPT2 does.It is important to mention that the results can hardly beimproved by using larger basis sets, as in CASPT2. Furtherinclusion of the remaining electron correlation is mandatory.

Finally, it is worth mentioning that these results compriseone of the greatest steps in NOFT development, since PNOF5is the first functional that can be properly used in the study ofthe electronic structure and properties of transition metalcomplexes, providing qualitatively good and quantitativelyreasonable results. Hence, future improvements of PNOF5appear to be very promising in order to achieve higher accuracyin transition metal calculations as well.

Acknowledgements

This research was funded by Eusko Jaurlaritza (the Basque Govern-ment) (GIC 07/85 IT-330-07) and the Spanish Office for ScientificResearch (CTQ2011-27374). Technical and human support pro-vided by IZO-SGI, SGIker (UPV/EHU, MICINN, GV/EJ, ERDF andESF) is gratefully acknowledged for assistance and generous alloca-tion of computational resources. JMM would like to thank SpanishMinistry of Science and Innovation for funding through a Ramon yCajal fellow position (RYC 2008-03216).

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