DaltonTransactions

PAPER

Cite this: Dalton Trans., 2014, 43,11145

Received 23rd January 2014,Accepted 7th March 2014

DOI: 10.1039/c4dt00251b

www.rsc.org/dalton

The use of localised orbitals for the bondingand mechanistic analysis of organometalliccompounds†

Pietro Vidossich* and Agustí Lledós*

Through a series of examples we show how, upon orbital localisation, the outcome of an electronic struc-

ture calculation reveals features, such as bonding and oxidation states, which are controversial to grasp by

alternative methods. The approach can also be applied to the analysis of reaction mechanisms. Because

of the insight it provides in a limited execution time, we believe that this approach, known since the early

developments of computational quantum chemistry, could find wider applications in the organometallic

community than it actually has and facilitate communication between computational and experimental

chemists.

Introduction

Is there a bond between atoms X and Y? What is the oxidationstate of atom X? Both are central questions in chemistry, towhich we can often answer on the basis of the knowledge webuilt on related systems. However, as chemical researchpushes its limits towards the synthesis of compounds withnovel properties, new situations arise for which it is difficult tomake an assignment. When this occurs, computationalquantum chemistry is asked to resolve the issue. Indeed, com-putational quantum chemistry is capable, within its own limit-ations, of providing a picture of the system of interest at aresolution which is in general not accessible by experimentalmeans. However, the solution of the quantum chemicalproblem comes in a form (a set of nuclear coordinates andelectronic orbitals) which has to be reconciled with acceptedchemical concepts. Concepts such as bonding and oxidationstates are deep-rooted in the chemist’s way of thinking and areso useful to guide the design of new compounds that we donot want to abandon them. And the problem lies there: how dowe reduce the N-body solution to a set of estimates of chemicalconcepts. Many methods have been devised for this purpose.Among these, we recall population analysis (Mulliken1 andvariants2,3), the theory of atoms in molecules,4 and molecularorbital analysis (including the natural bonding orbitals5). All

these methods are widely used in the theoretical community,each with its own advantages and limitations, such that someresearchers may prefer one over the other. In this communi-cation we want to call attention to the use of localised mole-cular orbitals to perform the analysis. The approach datesback to the work of Boys in the sixties,6,7 but is not much usedby the organometallic community. In contrast, it is muchmore employed (in the form of maximally localised Wannierfunctions8,9) in condensed matter research where it has impor-tant applications.10 Through a few examples drawn from ourcurrent research and cases discussed in the literature, we willshow how localised molecular orbitals may be convenientlyused to answer the opening questions. It is not our intentionto criticize previous work, nor to propose that the use of thisprocedure should substitute other analysis techniques. We justwant to share with the reader the clear cut picture of the elec-tronic structure which arises from the application of the local-isation procedure.

Methods

Orbital localisation consists of finding the unitary transform-ation U acting on the Kohn–Sham orbitals φKS

i ,

φlocn ¼

X

i

U inφKSi

which minimizes the spread functional Ω

Ω ¼X

n

kφlocn jr2jφloc

n l � kφlocn jrjφloc

n l2� �

The procedure consists of first performing a standarddensity functional theory (DFT) calculation within the Kohn–

†Electronic supplementary information (ESI) available: Supporting Fig. S1–5.Details of the calculation of localised orbitals and orbital centroids inGaussian09. Cartesian coordinates of atoms and orbital centroids in thecomplexes analyzed. See DOI: 10.1039/c4dt00251b

Departament de Química, Universitat Autònoma de Barcelona, 08193 Cerdanyola

del Vallès, Spain. E-mail: vido@klingon.uab.es, agusti@klingon.uab.es

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Sham approach, and then transforming the resulting orbitalsaccording to the above criterion. This localisation criterionwas originally proposed by Boys in the context of quantumchemical calculations of isolated systems.6 Alternative localis-ation criteria have been proposed.11–13 Methods have beendeveloped to apply Boys’ criterion also to periodic systems,and in this context the resulting orbitals, known as maximallylocalised Wannier functions,9 have important applications inthe theory of polarization and for the development of linearscaling electronic structure methods.10 In this communicationwe show the use of localised orbitals for the analysis of chemi-cal bonding in organometallic systems. Similar analyses havebeen performed by other authors on different systems.14–16

DFT calculations and orbital localisation were performedwith the CP2K code.17 Calculations were based on the PBEexchange-correlation functional.18 Only valence electrons wereincluded in the calculations, representing the valence–coreinteractions by means of GTH-type pseudopotentials.19–21 TheQuickstep22 algorithm was used to solve the electronic struc-ture problem using a double-zeta plus polarization (DZVP)23

basis set to represent the orbitals and plane waves (up to 300Ry) for the electron density. Wave function optimization wasachieved through an orbital transformation method.24 Modelswere treated as isolated25 and optimized (until gradients were<5 × 10−4 a.u.). Full details on the localisation algorithm asimplemented in CP2K may be found in ref. 26. Orbital localis-ation is also available in other electronic structure codes (seeESI†). Atomic electron populations have been computed inte-grating the orbital density within the atomic basins deter-mined according to Bader’s Atom in Molecules theory.4

ResultsAn illustrative example: PdCl4

2−

First we will give an illustrative example to outline the pro-cedure and point out the advantages resulting from the localis-ation of the molecular orbitals.

The complex PdCl42− has 24 doubly occupied orbitals (in

our computational setup the Pd semi-core electrons 4s and 4pare treated explicitly). Table 1 shows the orbital electron popu-lation of Pd for both the Kohn–Sham (KS) and localised (loc)representations. There are 15 KS orbitals with an appreciableelectron population (>0.1 e), whereas there are 12 in the loca-lised representation. Thus, the first thing to note is that thelocalisation simplifies the molecular orbital analysis in that itreduces the number of orbitals we have to inspect. Further-more, in the localised representation 8 orbitals have a Pd elec-tron population close to 2. These are clearly the metal4s24p64d8 electrons. In contrast, in the KS representation only5 orbitals have Pd electron population close to 2, thus requir-ing the user to inspect the remaining orbitals in search for themetal d electrons. Similarly, the bonding orbitals may easily berecognized in the localised representation. These are the fourorbitals with Pd population of 0.37. The low Pd electron popu-lation is indicative of a dative bond, as confirmed by visual

inspection which reveals pear-shaped orbitals centered on theCl ligands (Fig. 1).

As the size of the system under investigation increases, visu-alization of molecular orbitals to decide on their naturebecomes time intensive. The localisation procedure we employprovides the centroid of charge and the spread of each orbital.This is of great help in the analysis of complex systems.Indeed, by visualizing the molecular structure together withthe orbital centroids we readily have a picture of “where” elec-trons are and who they belong to (Fig. 1). Recently, Sit et al.16

suggested to use the position of the centroid and the shape ofthe orbital (whether it is covalent or dative), to assign a formaloxidation state to the metal, just as we would do looking at theLewis structure of the compound and taking into account theelectronegativities of the constituent atoms. Accordingly, in

Table 1 Orbital electron populationa of Pd in PdCl42− in the Kohn–

Sham (KS) and localised (loc) representations

Orbital KS loc Orbital KS loc

1 2.00 1.97 13 0.11 0.372 2.00 1.98 14 0.61 0.023 2.00 1.97 15 0.61 0.024 2.00 1.98 16 0.05 0.025 0.03 0.02 17 0.01 1.976 0.01 0.02 18 0.01 0.027 0.01 0.02 19 0.01 0.028 0.02 0.02 20 0.00 0.029 0.95 0.37 21 1.10 1.9710 0.44 0.37 22 1.36 1.9811 0.85 0.37 23 1.36 0.0212 0.11 0.02 24 1.87 1.98

a Atomic electron populations have been computed integrating theorbital density within the atomic basins determined according toBader’s Atoms in Molecules theory.4

Fig. 1 Square planar PdCl42− complex. Centroids of the localised orbi-

tals are shown as small red dots (each centroid accounts for two elec-trons; some centroids are hidden below the atom’s spheres). Anisosurface of a localised orbital corresponding to one Pd–Cl bond isalso shown.

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PdCl42− we have 8 centroids close to Pd (the 4s24p64d8 metal

electrons, as mentioned above), as expected for a PdII oxi-dation state. Assignment of oxidation states from populationanalysis is in general problematic, as has been pointed out byother authors.27 An example comparing the localized orbitalapproach to population analysis for oxidation state assignmentin PdIV and PdII complexes is discussed in the ESI.†

Bonding and oxidation states

We will now report on the results of applying the localisedorbital analysis to few systems which were debated in therecent literature.

[Pt(NHC(Dip)2)(SiMe2Ph)2] (NHC = N-heterocyclic carbene;Dip = 2,6-diisopropylphenyl). This unique tricoordinatecomplex has been recently isolated and structurally character-ized.28 The compound displays a Y-shaped geometry, and isregarded as informative of the reaction pathway for the reduc-tive elimination of disilane from [Pt(SiR3)2]. This compoundwas the subject of a recent theoretical study aimed at elucidat-ing the oxidation state of Pt and the bonding interactionsbetween the Si atoms.29 On the basis of bond indices, mole-cular orbital populations and calculated Pt chemical shifts, itwas argued that the compound “should be understood as aσ-disilane complex of Pt(0) rather than a disilyl complex ofPt(II)”.29 Upon transformation of the molecular orbitals into alocalised set, we observe that the number of electrons localisedon Pt is consistent with a Pt(II) assignment (see Fig. 2). Fur-thermore, the centroids of two localised orbitals lie along the

Pt–Si axes, indicative of bonding. No centroid localises alongthe Si–Si axis. Thus, from the localised orbital approach, thecompound should be described as a bis(silyl)platinum(II)–NHC complex. The unusual Y-shaped geometry should beattributed to the steric effect of the bulky substituent on thecarbene ligand.29

[Pd3(dmpe)2(1-(SiH2),2-(SiH)-C6H4)2(1,2-(SiH)2C6H4)] (dmpe =bis-(dimethylphosphanyl)ethane). The structure of this com-pound, obtained upon thermal condensation of [Pd(dmpe)2-(1,2-(SiH2)2C6H4)], was determined by X-ray analysis.30

The authors, on the basis of the analysis of interatomic dis-tances, were not able to conclude on the oxidation state of thecentral Pd atom and its bonding interactions. Specifically, thequestion remained open on whether the compound should beconsidered as a Pd(VI) species or rather as a bis σ-complex ofPd(II) with Si–Si bonds. The issue of the best bonding model todescribe the compound was addressed by theoreticians, whoconcluded in favor of the Pd(II) species.31,32 The localisedorbital approach supports this assignment (see Fig. 3). Thenumber of orbital centroids around the central Pd is consist-ent with Pd(II). Furthermore, localised orbital centroids arefound lying along two Pd–Si axes (out of six at the central Pd)and along two Si–Si axes consistently with the presence of twoSi–Si bonds. To note that the centroids of the Si–Si bonds areslightly displaced from the Si–Si axes, suggestive of σ-bondinteractions with the central Pd.

[(L2Cu)3S2]3+(L = Me2NCH2CH2NMe2). The nature of the

bonding interactions in this cluster has been the subject ofdispute in the literature, recently reviewed in ref. 33. Thecentral question was whether the two S atoms at 2.70 Å dis-tance were bonded or not. In summary, Alvarez supported theview of two S2− ligands with the metal atoms in the con-figuration Cu2

IICuIII.34 Mealli and Hoffmann proposed the

Fig. 2 [Pt(NHC(Dip)2)(SiMe2Ph)2] complex. Centroids of the localisedorbitals are shown as small green dots (each centroid accounts for twoelectrons). Centroids lying on the dotted lines are indicative of thebonding interactions at Pt.

Fig. 3 [Pd3(dmpe)2(1-(SiH2),2-(SiH)-C6H4)2(1,2-(SiH)2C6H4)] complex.Centroids of the localised orbitals are shown as small green dots (eachcentroid accounts for two electrons). Dotted lines highlight the Si–Siσ-interactions with the central Pd.

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existence of an S–S bond (i.e. a S22− moiety) interacting with a

Cu2IICuI cluster.35 Later on, Berry suggested that the cluster

should be better described by a Cu3IIS2

3−configuration. Arecent combined experimental and high level computationalstudy concluded that the cluster is “best described as havingall copper centers close to but more oxidized than CuII, whilethe charge on the S2 fragment is between that of a sulfide (S2−)and a subsulfide (S2

3−) species”.36 It is worth noting that theEPR spectrum exhibits a hyperfine pattern distinctive of threeequivalent Cu ions.34,36

Application of the orbital localisation procedure shows anorbital (singly occupied) coupling the S atoms with its centerof gravity at the midpoint of the S–S axis (Fig. 4). Applying theelectron counting technique proposed by Sit et al.16 points tothe electronic configuration Cu3

IIS23−. A recent theoretical ana-

lysis also favors this assignment.37

As a control calculation, the same analysis was performedon the related compound [(L2Cu)3O2]

3+ (see Fig. S1 of theESI†). Here, there is no electron localisation along the O–Oaxis and the orbital centroids are distributed according to theconfiguration Cu2

II CuIII(O2−)2, in line with the experimentalfindings.38

[MesDPBPh]NiH2 (MesDPBPh = MesB(o-Ph2PC6H4)2; Mes =mesityl). In a recent communication, Harman and Peters39,40

showed that addition of H2 to [MesDPBPh]Ni resulted in aspecies which was assigned as a metal hydride–borohydridecomplex. Such an assignment implies that the complex doesnot operate as a frustrated Lewis pair, with the Ni acting as abase (accepting a proton) and boron as a Lewis acid (acceptinga hydride). A detailed mechanistic analysis of H2 activation by[MesDPBPh]Ni has recently been reported.41 The optimizedstructure of [MesDPBPh]NiH2 shows one of the two H atomsbridging the B and Ni centers, with the Ni–H distance slightlylonger (0.1 Å) than the other Ni–H bond. Do we face a metalhydride–borohydride complex or an unusual nickel dihydride

species? Application of the orbital localisation procedure showsan orbital (doubly occupied) coupling the Ni, B and H atomswith its center of gravity displaced from both the B–H andNi–H axes (Fig. 5), suggestive of a three centers two electroninteraction. However, the density of such orbital increasesalong the B–H axis, thus we conclude that the configurationshould be regarded as a borohydrido, as originally proposed.39

Electron counting further indicates that the Ni is in a d8 elec-tron configuration as for NiII (Ni oxidized upon H2 addition).

[1.1.1.]Propellane. Yet another example (though not anorganometallic compound) of a molecule whose bondingpattern kept chemists intrigued.42 For some time it was notpossible to establish whether the bridgehead carbon atomswere bonded or not. Only recently, Wu et al.,43 employing thesophisticate ab initio valence bond approach, were able to con-clude that there is indeed a bond between the C atoms, morespecifically a bond of the “charge shift” type. Application ofthe orbital localisation procedure shows an orbital (doublyoccupied) coupling the bridgehead C atoms with its center ofgravity at the midpoint of the C–C axis (Fig. 6). As a control cal-culation, the same analysis was performed on the reducedcounterpart bicyclo[1.1.1]pentane. In this complex, at thebridgehead atoms, two C–H covalent bonds pointing outwardsare observed (see Fig. S2 of the ESI†).

Reaction mechanisms: arrows

The discussion of the reactivity of a chemical system is gener-ally based on reaction schemes in which arrows highlight themovement of electrons accompanying the rearrangements ofatoms.44 As we have shown in the previous section, the cen-troids of the localised orbitals provide a representation ofbonds and lone pairs, and it would be tempting to associatethe displacement of the centroids to the arrows drawn in

Fig. 4 (TMEDA)3Cu3S23+ complex. Only core atoms and the centroids

of the localised orbitals are shown for the sake of clarity (each centroidaccounts for one electron, spin-up and -down in green and yellow,respectively; some centroids are hidden below the atom’s spheres, inparticular those of Cu).

Fig. 5 [MesDPBPh]NiH2 chemical diagram (top left) and relevant atoms(colored) shown together with the centroids (small red dots, eachaccounting for two electrons) of the localised orbitals involving the twohydride atoms. An isosurface of the localised orbital involving the brid-ging hydride is also shown.

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reaction schemes. Indeed, this approach had been pursued inthe past to describe organic reactions,45,46 and we applied ithere to a series of organometallic complexes capable of C–Hbond activation. We have computed the intrinsic reaction co-ordinate (IRC) of a few prototype reactions47 and localised theorbitals for each configuration along the IRC. Fig. 7 shows thesuperposition of each frame along the IRC of a given reaction.It may be appreciated that indeed the centroids of the orbitalsinvolved in the transformation travel along the IRC. The pathsfollowed by the centroids may be considered as the arrowsdrawn in reaction schemes. Interestingly, these paths reflectthe accepted mechanisms for these reactions. In Fig. 7a, ametal (M)–C bond transforms into a C–H bond as a C–H bondchanges into an M–C, clearly an σ-bond metathesis. In Fig. 7b,an example for an oxidative addition, the C–H bond breaks toform an M–C bond and a metal orbital picks up the proton toform the M–hydride (resulting in the oxidation of the metal).In Fig. 7c, the σ-bond metathesis features the assistance of ametal orbital along the path (in Fig. 7c, note the displacementof an Ir centroid accompanying the proton movement). Thecentroids of the localised orbital in reactants, transition statesand products are given in the ESI† (Fig. S3–5) for comparisonwith the analysis by Vastine and Hall using Bader’s atoms inmolecules approach.47

Conclusions

Orbital localisation provides insight into the electronic struc-ture of organometallic compounds. The approach is easy touse and requires a limited execution time, even for large com-plexes, making it a valuable tool in the hands of the compu-tational chemist. The centroids of the localised orbitals appearparticularly useful, as by just their visualization details of theelectron distribution within the system are provided. The posi-tion of the centroids may be related to bonding and oxidationstates,16 as the reported examples show. Furthermore, the dis-placement of the orbital centroids has been used to investigate

Fig. 6 [1.1.1.]Propellane. Centroids of the localised orbitals are shownas small green dots (each centroid accounts for two electrons). An iso-surface of the localised orbital corresponding to the bridgehead C–Cbond is also shown.

Fig. 7 Superposition of configurations from the IRC of C–H bond acti-vation for (a) Cp2Sc(CH4)(CH3); (b) Cp

*Ir(PMe3)(CH4)(CH3)+; (c) (acac)2Ir-

(CH2CH2Ph)(C6H6). The dotted lines highlight breaking/forming bonds.

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environmental effects48 and may be used to build ionicityscales.49 As we have shown, the displacement of the orbitalcentroids may be used to follow the movement of electronsduring chemical reactions and can thus aid in the classifi-cation of reaction mechanisms. In recent years, other authorshave been developing analysis tools based on localisationprocedures.50,51 In fact, localised orbitals appear to be moreeasily reconcilable with accepted chemical concepts, such asbonding and oxidation states, than delocalised molecularorbitals.52

Acknowledgements

Financial support from Spanish Ministerio de Economía yCompetitividad (DGICYT, CTQ2011-23336 and ORFEO Consoli-der-Ingenio 2010 CSD2007-00006) is acknowledged. Weacknowledge the generous allocation of computer time by theBarcelona Supercomputing Center, Red Española de Super-computación (BSC-RES).

Notes and references

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