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CHAPTER 6

Transition Metal- andActinide-Containing SystemsStudied with MulticonfigurationalQuantum Chemical Methods

Laura Gagliardi

University of Geneva, Geneva, Switzerland

INTRODUCTION

Ab initio quantum chemistry has advanced so far in the last 40 yearsthat it now allows the study of molecular systems containing any atom in thePeriodic Table. Transition metal and actinide compounds can be treatedroutinely, provided that electron correlation1 and relativistic effects2 areproperly taken into account. Computational quantum chemical methodscan be employed in combination with experiment, to predict a priori, toconfirm, or eventually, to refine experimental results. These methods canalso predict the existence of new species, which may eventually be madeby experimentalists. This latter use of computational quantum chemistryis especially important when one considers experiments that are not easyto handle in a laboratory, as, for example, explosive or radioactive species.It is clear that a good understanding of the chemistry of such species can beuseful in several areas of scientific and technological exploration. Quantumchemistry can model molecular properties and transformations, and in

Reviews in Computational Chemistry, Volume 25edited by Kenny B. Lipkowitz and Thomas R. Cundari

Copyright � 2007 Wiley-VCH, John Wiley & Sons, Inc.

249

combination with experiment, it can lead to an improved understanding ofprocesses such as, for example, nuclear waste extraction and storage proce-dures for radioactive materials.

Quantum chemists have developed considerable experience over theyears in inventing new molecules by quantum chemical methods, which insome cases have been subsequently characterized by experimentalists(see, for example, Refs. 3 and 4). The general philosophy is to explore thePeriodic Table and to attempt to understand the analogies between the beha-vior of different elements. It is known that for first row atoms chemicalbonding usually follows the octet rule. In transition metals, this rule isreplaced by the 18-electron rule. Upon going to lanthanides and actinides,the valence f shells are expected to play a role. In lanthanide chemistry,the 4f shell is contracted and usually does not directly participate in the che-mical bonding. In actinide chemistry, on the other hand, the 5f shell is morediffuse and participates actively in the bonding.

Actinide chemistry presents a challenge for quantum chemistry mainlybecause of the complexity of the electronic structure of actinide atoms. Theground state of the uranium atom is, for example, (5f)3(6d)(7s)2, 5L6. Theground level is thus 13-fold degenerate and is described using7þ 5þ 1 ¼ 13 atomic orbitals. The challenge for actinide quantum chemis-try is to be able to handle systems with a high density of states involvingmany active orbitals along with including relativistic effects. It is true thatmuch actinide chemistry involves highly oxidized actinide ions with fewatomic valence electrons usually occupying the 5f shells. A good exampleis the uranium chemistry involving the U6þ ion (in the uranyl ion UO2þ

2 ).Such compounds are often closed-shell species and can be treated usingwell-established quantum chemical tools where only scalar relativisitc effectsare taken into account.

However, an extensive actinide chemistry involves ions of lower valencyand even atoms. Also, in some chemical processes, we find situations where theoxidation number may change from zero to a large positive number, an exam-ple being the small molecule NUN that will be discussed in this review. Theformal oxidation number of the uranium ion is six, and the UN bonds arestrongly covalent. But consider the formation of this molecule, which isdone by colliding uranium atoms with N2 : UþN2 ! NUN.5 Here, the oxi-dation number of U changes from zero to six along the reaction path, and thespin quantum number changes from two to zero. The quantum chemicaldescription of the reaction path requires methods that can handle complexelectronic structures involving several changes of the spin state as well asmany close lying electronic states.

Another issue involving actinide complexes in the zero formal oxidationstate is the possible formation of actinide–actinide bonds. For example, themolecule U2 has recently been described theoretically,6 in which the electronicstructure is characterized by the existence of a large number of nearly

250 Transition Metal- and Actinide-Containing Systems

degenerate electronic states and wave functions composed of multiple electro-nic configurations.

The methods used to describe the electronic structure of actinide com-pounds must, therefore, be relativistic and must also have the capability todescribe complex electronic structures. Such methods will be described inthe next section. The main characteristic of successful quantum calculationsfor such systems is the use of multiconfigurational wave functions that includerelativistic effects. These methods have been applied for a large number ofmolecular systems containing transition metals or actinides, and we shallgive several examples from recent studies of such systems.

We first describe some recent advances in transition metal chemistry,e.g., the study of Re2Cl2�8 , the inorganic chemistry of the Cr2 unit, and the the-oretical characterization of the end-on and side-on peroxide coordination inligated Cu2O2 models.

The second part of this chapter focuses on actinide chemistry, where westart by describing some triatomic molecules containing a uranium atom,which have been studied both in the gas phase and in rare gas matrices.Most of actinide chemistry occurs, however, in solution, so we then describeactinide ions in solution. The extensive study of the multiple bond betweentwo uranium atoms in the U2 molecule and in other diactinides is thenreported. Finally, several examples of inorganic compounds that include U2

as a central unit are presented.

THE MULTICONFIGURATIONAL APPROACH

We describe here the methods that have been used in quantum chemicalapplications to transition metal-and actinide-containing molecules. Thesemethods are available in the computer software package MOLCAS-6,7

which has been employed in all reported calculations. Many such systemscannot be well described using single configurational methods like Har-tree–Fock (HF), density functional theory (DFT), or coupled cluster (CC)theory. Accordingly, a multiconfigurational approach is needed, where thewave function is described as a combination of different electronic configura-tions. A three-step procedure is used to accomplish this approach. In the firststep, a multiconfigurational wave function is defined using the completeactive space (CAS) SCF method. This wave function is employed in the sec-ond step to estimate remaining (dynamic) correlation effects using multicon-figurational second-order perturbation theory. Scalar relativistic effects areincluded in both of these steps, but not spin-orbit coupling (SOC), whichis included in a third step where a set of CASSCF wave functions are usedas basis functions to set up a spin-orbit Hamiltonian that is diagonalizedto obtain the final energies and wave functions. We describe each of thesesteps in more detail below.

The Multiconfigurational Approach 251

The Complete Active Space SCF Method

The CASSCF method was developed almost 30 years ago. It wasinspired by the development of the Graphical Unitary Group approach(GUGA) to the full CI problem by Shavitt,8 making it possible to solve largefull CI problems with full control of spin and space symmetry. The GUGAapproach is in itself not very helpful because it can only be used with verysmall basis sets and few electrons. It was known, however, that the impor-tant configurations (those with coefficients appreciably different from zero)in a full CI expansion used only a limited set of molecular orbitals. The fol-lowing idea emerged, especially the concept of a fully optimized reactionspace (FORS) introduced by Ruedenberg and Sundberg in 1976:9 The mole-cular orbital space is divided into three subspaces: inactive, active, and exter-nal orbitals. The inactive orbitals are assumed to be doubly occupied in allconfiguration functions (CFs) used to build the wave function. The inactiveorbitals thus constitute a Hartree–Fock ‘‘sea’’ in which the active orbitalsmove. The remaining electrons occupy a set of predetermined active orbitals.The external orbitals are assumed to be unoccupied in all configurations.Once the assignment of electrons to active orbitals is done, the wave functionis fully defined within the set of active orbitals. All CFs with a given spaceand spin symmetry are included in the multiconfigurational wave function.This concept of CAS was introduced by B. O. Roos in the 1980s.10,11 Ascheme of how the orbitals can be subdivided is presented in Figure 1.

The choice of the correct active space for a specific application is not tri-vial, and many times one has to make several ‘‘experiments.’’ It is difficult toderive any general rules because every chemical system poses its own problems.The rule of thumb is that all orbitals intervening in the chemical process must beincluded. For example, in a chemical reaction where a bond is formed/broken,all orbitals involved in the bond formation/breaking must be included in the

Figure 1 Orbital spaces for CAS wave functions.


active space. If, on the other hand, several electronic states are considered, themolecular orbitals from/to which the electronic excitation occurs have to beincluded in the active space. There is also a tight connection with the choiceof atomic orbital (AO) basis, which must be extensive enough to be able todescribe the occupied molecular orbitals (MOs) properly. Moreover, the sizeof the active space is limited, being in most software packages around 15 forthe case where the number of orbitals and electrons are equal. This is themost severe limitation of the CASSCF method and makes it sometimes difficultor even impossible to perform a specific study. In this chapter, we shall exem-plify how active orbitals are chosen for compounds that pose special difficultiesin this respect because of the large number of valence orbitals that may contri-bute to actinide chemical bonds (5f, 6d, 7s, and possibly 7p). We shall also illus-trate one case, the Cu2O2 models, in which all affordable active spaces do notdescribe the system in a satisfactory way.

An extension to the CASSCF method exists that has not been used muchbut may become more applicable in the future: The restricted active space(RAS) SCF method12,13 where the active subspace is divided into three regions:RAS1, RAS2, and RAS3. The orbitals in RAS1 are doubly occupied, but a lim-ited number of holes are allowed. Arbitrary occupation numbers are allowedin RAS2. A limited number of electrons is allowed to occupy the orbitals inRAS3. Many different types of RAS wave functions can be constructed. Leav-ing RAS1 fully occupied and RAS3 empty, one obtains the CAS wave function.If there are no orbitals in RAS2, a wave function that includes all single,double, etc. excitations out of a closed shell reference function (the SDTQetc.-CI wave function) is obtained.

The interesting feature of the RAS wave function is that it can workwith larger active spaces than CAS, without exploding the CI expansion. Itthus has the potential to perform multiconfigurational calculations that can-not today be performed with the CASSCF method. The problem with aRASSCF wave function is how to add the effects of dynamic electron correla-tion. For CASSCF, wave function second-order perturbation theory(CASPT2, see below) can be used to accomplish this, but this is not yet pos-sible for RASSCF wave functions. Recent developments in our research groupand in the Lund group of Roos indicate, however, that this may become pos-sible in the near future through the development of a RASPT2 method, thusextending the applicability of the multiconfigurational methods to newclasses of problems that cannot be treated today. This work is currently inprogress.

Multiconfigurational Second-Order PerturbationTheory, CASPT2

If the active space has been adequately chosen, the CASSCF wave func-tion will include the most important CFs in the full CI wave function. In this


way we include all near-degenerate configurations, which describe static cor-relation effects, as for example, in a bond breaking process. The CASSCF wavefunction will then be qualitatively correct for the entire chemical process stu-died, which can be an energy surface for a chemical reaction, a photochemicalprocess, etc. The energies that emerge are, however, not very accurate. Weneed to include the part of the CF space that describes the remaining (dynamic)correlation effects. This requirement is as necessary in the multiconfigura-tional approach as it would be if we started from the HF single determinantapproximation.

How can dynamic electron correlation be included? In a single configura-tion approach, the obvious choices are preferably CC methods, or if the systemis too large, second order perturbation theory (MP2), which is already accu-rate. A practical multiconfigurational CC theory does not exist yet. A methodthat has been used with great success since the 1980s is Multi-Reference CI(MRCI), where the most important of the CFs of the CAS wave functionare used as reference configurations in a CI expansion that includes all CFsthat can be generated by single and double replacements of the orbitals inthe reference CFs.14 The method is still used with some success because ofrecent technological developments.15 It becomes time consuming for systemswith many electrons, however, and has also the disadvantage of lacking size-extensivity, even if this latter problem can be corrected for, approximately.

Another way to treat dynamic correlation effects is to use perturbationtheory. Such an approach has the virtue of being size-extensive and ought tobe computationally more efficient than the MRCI approach. Møller–Plessetsecond-order perturbation theory (MP2) has been used for a long time totreat electron correlation for ground states, where the reference functionis a single determinant. It is known to give accurate results for structural,energetic, and other properties of closed-shell molecules. Could such anapproach also work for a multiconfigurational reference function likeCASSCF? This approach was suggested soon after the introduction of theCASSCF method,16 but technical difficulties delayed a full implementationuntil the late 1980s.17,18 Today it is the most widely used method to com-pute dynamic correlation effects for multiconfigurational (CASSCF) wavefunctions. The principle is simple: One first computes the second-orderenergy with a CASSCF wave function as the zeroth-order approximation.That said, we point out that there are some problems to be solved that donot occur in single determinant MP2. One needs to define a zeroth-orderHamiltonian with the CASSCF function as an eigenfunction. It should pre-ferably be a one-electron Hamiltonian in order to avoid a too complicatedformalism. One then needs to define an interacting space of configurations.These configurations are given as

EpqErsjCASSCFi ½1�


Equation [1] is an internally contracted configuration space, doubly excitedwith respect to the CAS reference function j0i ¼ jCASSCFi; one or two ofthe four indices p;q; r; s must be outside the active space. The functions ofEq. [1] are linear combinations of CFs and span the entire configuration spacethat interacts with the reference function. Labeling the compound index pqrsas m or n, we can write the first-order equation as

X

m

½Hð0Þmn � E0Smn�Cn ¼ �V0m ½2�

Here, Hð0Þmn are matrix elements of a zeroth-order Hamiltonian, which is chosen

as a one-electron operator in the spirit of MP2. Smn is an overlap matrix: Theexcited CFs are not in general orthogonal to each other. Finally, V0m representsthe interaction between the excited function and the CAS reference function.The difference between Eq. [2] and ordinary MP2 is the more complicatedstructure of the matrix elements of the zeroth-order Hamiltonian; in MP2 itis a simple sum of orbital energies. Here H

ð0Þmn is a complex expression involving

matrix elements of a generalized Fock operator F combined with up to fourth-order density matrices of the CAS wave function. Additional details are givenin the original papers by Andersson and coworkers.17,18 We here mention onlythe basic principles. The zeroth-order Hamiltonian is written as a sum of pro-jections of F onto the reference function j0i

H0 ¼ P0FP0 þ PSDFPSD þ PXFPX ½3�

where P0 projects onto the reference function, PSD projects onto the interact-ing configurations space (1), and PX projects onto the remaining configurationspace that does not interact with j0i. F has been chosen as the generalized Fockoperator:

F ¼X

p;q

fpqEpq ½4�

with

fpq ¼ hpq þX

r;s

Drs ðpqjrsÞ � 1

2ðprjqsÞ

� �½5�

With such a formulation, fpp ¼ �IPp (Ionization Potential) when the orbital pis doubly occupied and fpp ¼ �EAp (Electron Affinity) when the orbital isempty. The value of fpp will be somewhere between these two extremes foractive orbitals. Thus, for orbitals with occupation number one,fpp ¼ � 1

2 ðIPp þ EApÞ. This formulation is somewhat unbalanced and will


favor systems with open shells, leading, for example, to low binding energies,as shown in the paper by Andersson and Roos.19 The problem is that onewould like to separate the energy connected with excitation out of an orbitalfrom that of excitation into the orbital.

Very recently, a modified zeroth-order Hamiltonian has been suggestedby Ghigo and coworkers20 to accomplish this, which removes the systematicerror and considerably improves both dissociation and excitation energies.Equation [5] can be written approximately as an interpolation between thetwo extreme cases

Fpp ¼ �1

2ðDppðIPÞp þ ð2�DppÞðEAÞpÞ ½6�

where Dpp is the diagonal element of the one-particle density matrix for orbitalp. The formula is correct for Dpp ¼ 0 and 2 and for a singly occupied openshell. Assume now that when exciting into an active orbital, one wants itsenergy to be replaced by �EA. This is achieved by adding a shift to Eq[6].

sðEAÞp ¼ 1

2DppððIPÞp � ðEAÞpÞ ½7�

Contrarily, if one excites out of this orbital, its energy has to be replaced by�IP. The corresponding shift is

sðIPÞp ¼ � 1

2ð2�DppÞððIPÞp � ðEAÞpÞ ½8�

The definitions of ðIPÞp and ðEAÞp are not straightforward. Therefore,ðIPÞp � ðEAÞp was replaced with an average shift parameter E. The two shiftsare then

sðEAÞp ¼ 1

2DppE ½9�

sðIPÞp ¼� 1

2ð2�DppÞE ½10�

A large number of tests showed that a value of 0.25 for e was optimal. Themean error in the dissociation energies for 49 diatomic molecules was reducedfrom 0.2 eV to 0.1 eV. Using an average e was particularly impressive for triplybonded molecules: The average error for N2; P2, and As2 was reduced from0.45 eV to less than 0.15 eV. Similar absolute improvements were obtainedfor excitation and ionization energies.20

Perturbation theory like MP2 or CASPT2 should be used only when theperturbation is small. Orbitals that give rise to large coefficients for the


states in Eq. [1] should be included in the active space. Large coefficients inthe first-order wave function are the result of small zeroth-order energy dif-ferences between the CAS reference state and one or more of theexcited functions. We call these functions intruder states. In cases wherethe interaction term V0m is small, one can remove the intruder using a levelshift technique that does not affect the contributions for the other states.21–23

The reference (zeroth-order) function in the CASPT2 method is a prede-termined CASSCF wave function. The coefficients in the CAS function are thusfixed and are not affected by the perturbation operator. This choice of thereference function often works well when the other solutions to the CASHamiltonian are well separated in energy, but there may be a problem whentwo or more electronic states of the same symmetry are close in energy. Suchsituations are common for excited states. One can then expect the dynamiccorrelation to also affect the reference function. This problem can be handledby extending the perturbation treatment to include electronic states that areclose in energy. This extension, called the Multi-State CASPT2 method, hasbeen implemented by Finley and coworkers.24 We will briefly summarize themain aspects of the Multi-State CASPT2 method.

Assume several CASSCF wave functions, �i; i ¼ 1;N, obtained in astate average calculation. The corresponding (single state) CASPT2 functionsare wi; i ¼ 1;N. The functions �i þ wi are used as basis functions in a‘‘variational’’ calculation where all terms higher than second order areneglected. The corresponding effective Hamiltonian has the elements:

ðHeff Þij ¼ dijEi þ h�ijHjwji ½11�

where Ei is the CASSCF energy for state i. This Hamiltonian is not symmetric,and in practice, a symmetrized matrix is used, which may cause problems ifthe non-Hermiticity is large, so it is then advisable to extend the active space.One can expect this extension of the CASPT2 method to be particularlyimportant for metal compounds, where the density of states is often high.

Treatment of Relativity

Nonrelativistic quantum chemistry has been discussed so far. But transitionmetal (starting already from the first row) and actinide compounds cannot bestudied theoretically without a detailed account of relativity. Thus, the multicon-figurational method needs to be extended to the relativistic regime. Can this bedone with enough accuracy for chemical applications without using the four-component Dirac theory? Much work has also been done in recent years todevelop a reliable and computationally efficient four-component quantum chem-istry.25,26 Nowadays it can be combined, for example, with the CC approach forelectron correlation. The problem is that an extension to multiconfigurational


wave functions is difficult and would, if pursued, lead to lengthy and complexcalculations, which allow only applications to small molecules. It is possible,however, to transform the four-component Dirac operator to a two-componentform where one simultaneously analyzes the magnitude of the different termsand keeps only the most important of these terms. The most widely used trans-formation of this type leads to the second order Douglas–Kroll–Hess Hamilto-nian.27,28 The DKH Hamiltonian can be divided into a scalar part and a spin-orbit coupling part. The scalar part includes the mass-velocity term and modifiesthe potential close to the nucleus such that the relativistic weak singularity of theorbital is removed. The effect on energies is similar to that of the Darwin term,but the resulting operator is variationally stable. This part of the relativistic cor-rections can easily be included in a nonrelativistic treatment. Usually, only con-tributions to the one-electron Hamiltonian are included. For lighter atoms, thescalar relativistic effects will be dominant and calculations on, say, first row tran-sition metal compounds, can safely be performed by adding only this term to theone-electron Hamiltonian that is used in nonrelativistic quantum chemical meth-ods. The scalar DKH Hamiltonian has been implemented recently into theCASSCF/CASPT2 version of the multiconfigurational approach by Roos andMalmqvist.29

The scalar terms are only one part of the DKH Hamiltonian. There isalso a true two-component term that, as the dominant part, has the spin-orbitinteraction. This is a two-electron operator and as such is therefore difficult toimplement for molecular systems. However, in 1996, an effective one-electronFock-type spin-orbit Hamiltonian was suggested by Hess and coworkers30

that simplifies significantly the algorithm for the subsequent calculation ofspin-orbit matrix elements. Two-electron terms are treated as screening cor-rections of the dominant one-electron terms, at least for heavy elements.Theatomic mean field integrals (AMFI) method is used, which, based on theshort-range behavior of the spin-orbit interaction, avoids the calculation ofmulti-center one- and two-electron spin-orbit integrals and thus reduces theintegral evaluation to individual atoms, taking advantage of full spherical sym-metry. The approach reduces the computational effort drastically but leads toa negligible loss of accuracy compared with, e.g., basis set or correlationlimitations as shown by Christiansen et al.31

The treatment of the spin-orbit part of the DKH Hamiltonian within theAMFI scheme is based on the assumption that the strongest effects of SOC arisefrom the interaction of electronic states that are close in energy. For these states,independent CASSCF/CASPT2 calculations are performed. The resultingCASSCF wave functions are then used as basis functions for the calculation ofthe spin-orbit coupling. The diagonal elements of the spin-orbit Hamiltoniancan be modified to account for dynamic correlation effects on the energy by,for example, replacing the CASSCF energies with CASPT2 energies. To beable to use the above procedure, one needs to compute matrix elements betweencomplex CASSCF wave functions, which is not trivial because the orbitals of two


different CASSCF wave functions are usually not orthogonal. A method to dealwith this problem was developed by Malmqvist in the late 1980s.32,33

The method has become known as the CASSCF State Interaction (CASSI)method and is also effective for long CAS-CI expansions and was recentlyextended to handle the integrals of the spin-orbit Hamiltonian.34

Is the method outlined above accurate enough for heavy element quan-tum chemistry? Several studies have been performed on atoms and molecules,showing that the approach is capable of describing relativistic effects in mole-cules containing most atoms of the periodic system with good accuracy, withexception of the fifth-row elements Tl-At. Here the method gives larger errorsthan for any other atoms in the periodic system.35 Studies on actinide atomsand molecules show, however, that the method works well for the f-elements.Several examples will be given below.

Relativistic AO Basis Sets

It is not possible to use normal AO basis sets in relativistic calculations:The relativistic contraction of the inner shells makes it necessary to design newbasis sets to account for this effect. Specially designed basis sets have thereforebeen constructed using the DKH Hamiltonian. These basis sets are of theatomic natural orbital (ANO) type and are constructed such that semi-coreelectrons can also be correlated. They have been given the name ANO-RCC(relativistic with core correlation) and cover all atoms of the PeriodicTable.36–38 They have been used in most applications presented in this review.ANO-RCC are all-electron basis sets. Deep core orbitals are described by aminimal basis set and are kept frozen in the wave function calculations. Theextra cost compared with using effective core potentials (ECPs) is thereforelimited. ECPs, however, have been used in some studies, and more detailswill be given in connection with the specific application. The ANO-RCC basissets can be downloaded from the home page of the MOLCAS quantum chem-istry software (http://www.teokem.lu.se/molcas).

THE MUTIPLE METAL–METAL BOND IN Re2Cl2�8AND RELATED SYSTEMS

In 1965 Cotton and Harris examined the crystal structure of K2½Re2Cl8��2H2O39 and reported a surprisingly short Re-Re distance of 2.24 A. This wasthe first reported example of a multiple bond between two metal atoms andthe Re2Cl2�8 ion (Figure 2) has since become the prototype for this family ofcomplexes.

Cotton analyzed the bonding using simple MO theory and concludedthat a quadruple Re–Re bond was formed.39,40 Two parallel ReCl4 units areconnected by the Re–Re bond. The dx2�y2 ; px; py, and s orbitals of the valence

The Mutiple Metal–Metal Bond in Re2Cl2�8 and Related Systems 259

shell of each Re atom form the s bonds to each Cl atom. The remaining dz2

and pz orbitals with s symmetry relative to the Re–Re axis, the dxz and dyz

with p symmetry, and the dxy with d symmetry form the quadruple Re–Rebond. The system thus contains one s bond, two p bonds, and one d bondbetween the Re atoms. Because there are eight electrons (Re3þ is d4) to occupythese MOs, the ground state configuration will be s2p4d2. The presence of thed bond explains the eclipsed conformation of the ion. In a staggered conforma-tion, the overlap of the d atomic orbitals is zero and the d bond disappears.The visible spectrum was also reported in these early studies. The notion ofa quadruple bond is based on the inherent assumption that four bonding orbi-tals are doubly occupied. Today we know that this is not the case for weakinter-metallic bonds. The true bond order depends on the relation betweenthe occupation of the bonding and antibonding orbitals, respectively. Such adescription is, however, only possible if a quantum chemical model is used thatgoes beyond the single configuration Hartree–Fock model. The CASSCF mod-el has been used to study this system, and it has been demonstrated that thetrue bond order between the two Re atoms is closer to three than to four.

Because the Re2Cl�28 ion is such an important entity in inorganic chem-

istry, we decided to study its structure and electronic spectrum using multicon-figurational quantum chemistry.41 Scalar relativistic effects and spin-orbitcoupling were included in this study. The geometry of Re2Cl�2

8 was obtainedat the CASPT2 level of theory, and several excited states were calculated atthis geometry. The calculations were performed using the active space formedby 12 active electrons in 12 active orbitals (12/12) (reported in Figure 3).It comprises one 5ds, two 5dp, and one 5dd Re–Re bonding orbitals andthe corresponding antibonding orbitals, and two Re-Cl d bonding orbitalsand the corresponding two antibonding orbitals. They are nicely paired suchthat the sum of the occupation numbers for the Re–Re bonding and antibond-ing orbitals of a given type is almost exactly two. The two bonding Re–Cl

Figure 2 The structure of Re2Cl2�8 .


orbitals are located mainly on Cl as expected, whereas the antibonding orbi-tals have large contributions from 3dx2�y2 . The occupation is low, and theseorbitals are thus almost empty and may be used as acceptor orbitals for elec-tronic transitions.

The strongest bond between the two Re atoms is the s bond, with anoccupation number of Zb ¼ 1:92 of the bonding and Za ¼ 0:08 of the anti-bonding natural orbital. We could estimate the effective bond order asðZb � ZaÞ=ðZb þ ZaÞ, and for the s bond, we obtain the value 0.92. The cor-responding value for the p bond is 1.74. The d pair gives an effective bondorder of only 0.54. Adding up these numbers results in a total effectivebond order of 3.20 for Re2Cl�2

8 . The main reduction of the bond orderfrom 4.0 to 3.2 is thus from the d bond. Note that the calculation of naturalorbital occupation numbers that substantially deviate from zero and two isindicative of the need for a CASSCF description of Re2Cl�2

8 .Vertical excitation energies and oscillator strengths have been deter-

mined at the CASPT2 level with and without the inclusion of spin-orbit cou-pling. Although we refer the interested reader to the original manuscript41 forthe details of the calculations, we describe here only the most significant fea-tures of the spectrum. The most relevant transitions are reported in Table 1.

The lowest band detected experimentally occurs at 1.82 eV (14,700 cm�1) with an oscillator strength of 0.023. It has been assigned to thed! d�ð1A1g !1 A2uÞ transition. Our 12/12 calculation predicts an excitationenergy of 2.03 eV at the CASPT2 level with an oscillator strength equal to0.004. Calculations with enlarged active spaces were also performed, for exam-ple, using 16 electrons in 14 orbitals (16/14). These calculations predict a

Figure 3 The molecular orbitals describing the bonds in Re2Cl2�8 .


CASPT2 excitation energy that varies between 1.68 and 1.74 eV, and the oscil-lator strength that varies between 0.007 and 0.092, which shows that the oscil-lator strength is very sensitive to the active space. The low energy of thistransition is a result of the weak d bond, which places the d� orbital at lowenergy.

In the region of weak absorption between 1.98 and 3.10 eV, (16,000–25,000 cm�1), the first peak occurs at 2.19 eV (17,675 cm�1) and has beenassigned to a d! p�ð11A1g ! 21A1gÞ transition located mostly on Re. We pre-dicted it to be at 2.29 eV, and it is a forbidden transition. Two bands have thenbeen assigned to charge transfer (CT) states. They occur at 3.35 eV and3.48 eV, respectively. It was suggested that they correspond to two A2u spin-orbit components of two close-lying 3Eu states.43 We have not studied the tri-plet ligand to metal charge transfer (LMCT) states, but our first singlet CTstate was predicted at 3.56 eV, corresponding to a Clð3pÞ ! d�ð1A1g ! 1EuÞLMCT transition. Thus, it seems natural to assign the upper of the two bandsto this transition. The peak at 3.35 eV has been assigned to a metal localizedtransition.

A ðd; pÞ ! ðd�; p�Þ ð1A1g ! 1A1gÞ transition is predicted at 3.91 eV and ap! p�ð1A1g ! 1B1uÞ transition at 4.00 eV. No corresponding experimental

Table 1 Spin-Free Excitation Energies in Re2Cl8 (in eV) Calculated at the CASSCF(CAS) and CASPT2 (PT2) Level.

State �E(CAS) �E(PT2) Expta Q(Re)

d! d�, 1A2u 3.08 2.03(0.0037) 1.82(0.023) 1.03d! p�, 1A1g 2.90 2.29(f) 2.19(weak) 1.03p! d�, 1Eg 3.41 2.70(f) 2.60 1.04d! p�, p! d�, 1Eg 3.87 3.10(f) 2.93(very weak) 1.04d! s�, 1B1u 4.47 3.10(f) 1.00d! dx2�y2 , 1A2g 3.96 3.37(f) 1.11ðd; pÞ ! ðd�Þ2, 1Eu 4.20 3.38(0.29E-03) 3.35 1.04Clð3pÞ ! d�LMCT, 1Eu 6.37 3.56(0.60E-04) 3.48 0.84d! dx2�y2 , 1A1u 4.24 3.59(f) 1.13p! p�, 1A1u 5.02 3.76(f) 1.04ðd; pÞ ! ðd�Þ2, 1Eu 4.81 3.80(0.92E-04) 1.04ðd; pÞ ! ðd�p�Þ, 1A1g 5.01 3.91(f) 1.05p! p�, 1B1u 5.17 4.00(f) 1.05Clð3pÞ ! d�;LMCT, 1Eu 6.54 4.08(0.08) 3.83(intense) 0.88s! d�;p! p�, 1B1u 6.01 4.13(f) 1.05p! dx2�y2 , 1Eu 6.15 4.17(0.009) 1.08ðd; pÞ ! ðd�p�Þ, 1B2g 5.66 4.30(f) 1.04dp! d�s�, 1Eu 6.79 4.40(1.0E-04) 4.42(complex) 1.03s! s�; p! p�, 1A2u 6.66 4.56(0.015) 4.86(intense) 1.04

a From Ref. 42Notes: Oscillator strengths are given within parentheses. Q(Re) gives the Mulliken charge on

one Re atom.


bands could be found. An intense CT state is found in the experimental spec-trum at 3.83 eV, and it is assigned to the Clð3pÞ ! d�ð1A1g ! 1EuÞ transitionthat we predict at 4.08 eV with an oscillator strength of 0.08. Togler et al.43

have suggested that the complex band found at 4.42 eV should be a mixture oftwo LMCT transitions. We find no evidence of this mixture in the calculations,but a weak 1Eu state is found at 4.40 eV and there are other symmetry forbid-den transitions nearby. An intense band is found at 4.86 eV with a tentativeassignment p! p�ð1A1g ! 1A2uÞ. We agree with this assignment and computethe state to occur at 4.56 eV with an oscillator strength of 0.015.

The spectrum of Re2Cl�28 was recomputed with the inclusion of spin-

orbit coupling, leading to no change of the qualitative features of the spec-trum. There is a small shift in the energies and intensities, but we do not seeany new states with intensities appreciably different from zero. We may, how-ever, have lost some information because we have not studied the LMCT tri-plet states and the corresponding effects of spin-orbit splitting.

Four compounds containing metal–metal quadruple bonds, the½M2ðCH3Þ8�

2n� ions where M ¼ Cr; Mo; W; Re and n ¼ 4; 4; 4; 2, respec-tively, have also been studied theoretically46 using the same CASPT2 methodemployed in the Re2Cl2�8 case. The molecular structure of the ground state ofthese compounds has been determined, and the energy of the d! d� transitionhas been calculated and compared with previous experimental measurements.The high negative charges on the Cr, Mo, and W complexes lead to difficultiesin the successful modeling of the ground-state structures, which is a problemthat has been addressed by the explicit inclusion of four Liþ ions in these cal-culations. The ground-state geometries of the complexes and d! d� transitionare in excellent agreement with experiment for Re, but only satisfactory agree-ment for Mo, Cr, and W.

The primary goal of this study44 was to provide a theoretical understand-ing of the apparently linear relationship between metal–metal bond length andd! d� excitation energy for the octamethyldimetallates of Re, Cr, Mo, andW. As we demonstrated, these seemingly simple anionic systems represent asurprising challenge to modern electronic structure methods, largely becauseof the difficulty in modeling systems (without electronegative ligands) thathave large negative charges. Nevertheless, by using the CASPT2 methodwith Liþ counterions, one can model the ground-state geometries of thesecomplexes in a satisfactory way. This multiconfigurational approach, whichis critical for the calculation of excited-state energies of the complexes, doesa fairly good job of modeling trends in the d! d� excitation energy withthe metal–metal bond length, although the accuracy is such that we are notyet able to explain fully the linear relationship discovered by Sattelbergerand Fackler.45 Progress on these systems will require better ways to accommo-date the highly negative charges, which are in general difficult to describe,because of the intrinsic problem of the localization of the negative charges.These efforts are ongoing.


THE Cr–Cr MULTIPLE BOND

The chromium atom has a ground state with six unpaired electrons(3d54s, 7S). Forming a bond between two Cr atoms could, in principle, resultin an hextuple bond, so it is not surprising that the chromium dimer hasbecome a challenging test for various theoretical approaches to chemicalbonding. Almost all existing quantum chemical methods have been used.The results are widely varying in quality (see Ref. 46 for references to someof these studies). It was not until Roos applied the CASSCF/CASPT2 approachto Cr2 that a consistent picture of the bonding was achieved.46 This studyresulted in a bond energy (D0) of 1.65 eV, a bond distance of 1.66 A, andan oe value of 413 cm�1 (experimental values are 1:53� 0:06 eV,47

1.68 A,48 and 452 cm�1, respectively49).Do the two chromium atoms form a hextuple bond? The calculations by

Roos46 gave the following occupations of the bonding and antibonding orbi-tals: 4ssg 1:90, 4ssu 0:10, 3dsg 1:77, 3dsu 0:23, 3dpu 3:62, 3dpg 0:38,3ddg 3:16, and 3ddu 0:84, yielding a total effective bond order of 4.46. Thed bond is weak and could be considered as intermediate between a ‘‘true’’chemical bond and four antiferromagnetically coupled electrons. The chro-mium dimer could thus also be described as a quadruply bonded systemwith the d electrons localized on the separate atoms and coupled in a wayto give a total spin of zero.

The difficulty in forming all six bonds arises mainly from the large dif-ference in size between the 3d and 4s orbitals. When the Cr–Cr distance is suchthat the 3d orbitals reach an effective bonding distance, the 4s orbitals arealready far up on the repulsive part of their potential curve, a behavior thatexplains why the bond energy is so small despite the high bond order. The dif-ference in orbital size decreases for heavier atoms. The 5s orbital of Mo ismore similar in size to the Mo 4d orbital. Even more pronounced is the effectfor W, where the relativistic contraction of the 6s orbital and the correspond-ing expansion of the 5d orbital makes them very similar in size. The result is amuch stronger bond for W2 with a bond energy above 5 eV and an effectivebond order of 5.19.50 The tungsten dimer can thus be described as a nearlytruly hextuply bonded system. The occupation numbers of the bonding orbi-tals are never smaller than 1.8, which is a value that is the highest bond orderamong any dimer of the Periodic Table.

Nguyen et al. synthesized a dichromium compound with the generalstructure ArCrCrAr, where Cr is in the þ1 oxidation state.51 This is the firstexample of a compound with Cr in that oxidation state. A bond distance of1.83 A was determined for the Cr–Cr bond, and it was concluded that a quin-tuple bond was formed. CASSCF/CASPT2 calculations on the model com-pound PhCrCrPh (Ph¼ phenyl) subsequently confirmed this picture.52 Thenatural orbital occupation numbers (NOONs) were found to be 3dsg 1:79,3dsu 0:21, 3dpu 3:54, 3dpg 0:46, 3ddg 3:19, and 3ddu 0:81, which were very


similar to the chromium dimer with again a weak d bond. The total effectivebond order is 3.52, so the bond is intermediate between a triple bond with fourantiferromagnetically coupled d electrons and a true quintuple bond. Thebond energy was estimated to be about 3.3 eV, which is twice as much asfor the chromium dimer. The reason for this large bond energy is the absenceof the 4s electron in the Cr(I) ion.

Dichromium(II) compounds have been known for a long time. In parti-cular, the tetracarboxylates have been studied extensively since the first syn-thetic work of Cotton.53 The Cr–Cr bond length varies extensivelydepending on the donating power of the bridging ligands and the existenceof additional axial ligands. The shortest bond length 1.966 A was found forCr2ðO2CCH3Þ4 in a gas phase measurement.54 A CASSCF calculation atthis bond distance yields the following natural orbital occupation numbers:3ds1:68, 3ds� 0:32, 3dp3:10, 3dp� 0:90, 3dd 1:21, and 3dd� 0:79, giving aneffective bond order of only 1.99. Note that, as in other examples discussed inthis chapter, the calculation of NOONs for the antibonding orbitals signifi-cantly greater than zero (which they would be in a HF or DFT calculation)indicates the need for a CASSCF description. This is far from a quadruplebond, thus explaining the great variability in bond length depending onthe nature of the ligands. Another feature of these compounds is theirtemperature-dependent paramagnetism, explained by the existence of low-lying triplet excited states, which arises from a shift of the weakly coupledd electron spin.55 A general picture of the Cr–Cr multiple bond emergesfrom these studies. Not unexpectedly, fully developed bonds are formed bythe 3ds and 3dp orbitals, whereas the 3dd orbitals are only weakly coupled.The notion of a hextuple bond in the Cr2 system, a quintuple bond inArCrCrAr, and a quadruple bond in the Cr(II)–Cr(II) complexes is thereforean exaggeration. The situation is different for the corresponding compoundscontaining the heavier atoms Mo and W, where more fully developed multiplebonds can be expected in all three cases.50

Cu2O2 THEORETICAL MODELS

An accurate description of the relative energetics of alternative bis(m-oxo) and m� Z2 : Z2 peroxo isomers of Cu2O2 cores supported by 0, 2, 4,and 6 ammonia ligands (Figure 4) is remarkably challenging for a wide varietyof theoretical models, primarily because of the difficulty of maintaining abalanced description of rapidly changing dynamical and nondynamical elec-tron correlation effects and the varying degree of biradical character alongthe isomerization coordinate.

The isomerization process interconverting the three isomers depicted inFigure 4, with and without ammonia ligands, has been studied recently,54,58

using the completely renormalized coupled cluster level of theory, including

Cu2O2 Theoretical Models 265

triple excitations, various density functional levels of theory, and the CASSCF/CASPT2 method. The completely renormalized coupled cluster level of theoryincluding triple excitations and the pure density functional levels of theory,agree quantitatively with one another and also agree qualitatively with experi-mental results for Cu2O2 cores supported by analogous but larger ligands. TheCASPT2 approach, by contrast, significantly overestimates the stability ofbis(m-oxo) isomers.

The relative energies of m� Z1 : Z1 (trans end-on) and m� Z2 : Z2 (side-on) peroxo isomers (Figure 4) of Cu2O2 fragments supported by 0, 2, 4, and 6ammonia ligands have also been computed with various density functional,CC, and multiconfigurational protocols. Substantial disagreement existsamong the different levels of theory for most cases, although completelyrenormalized CC methods seem to offer the most reliable predictions. Thesignificant biradical character of the end-on peroxo isomer is problematicfor the density functionals, whereas the demands on active space size andthe need to account for interactions between different states in second-orderperturbation theory prove to be challenging for the multireference treatments.For the details of the study, the reader should refer to the original papers.56,57

We focus here on the CASSCF/CASPT2 calculations and try to understandwhy in the current case the method has not been able to produce satisfactoryresults. As stated, the method depends on the active space. What are the rele-vant molecular orbitals that need to be included to have an adequate descrip-tion of Cu2O2? A balanced active space would include the molecular orbitalsgenerated as a linear combination of the Cu 3d and O 2p atomic orbitals. Inprevious work, the importance of including a second d shell in the activespace for systems was also discussed, where the d shell is more than halffilled58,59 (the double-shell effect). In total this would add up to 28 activeelectrons in 26 active orbitals. Such an active space is currently too large tobe treated with the CASSCF/CASPT2 method. Several attempts have beenmade to truncate the 28/26 active space to smaller and affordable activespaces, but with little success. The Cu2O2 problem represents a case in whichthe CASSCF/CASPT2 method, currently, still fails. The relative energies (kcalmol�1) of the triligated bis(m-oxo) and m� Z2 : Z2 (side-on) peroxo isomersare reported in Table 2.

Figure 4 Some isomers of two supported copper(I) atoms and O2.


Although CC and DFT (in agreement with experiment) predict the per-oxo structure to be more stable than the bis(m-oxo structure by about 10 kcalmol�1, CASPT2 always overestimates the stability of the bis(m-oxo) isomer, byabout 30 kcal mol�1, independent of the active space used. We believe thatsuch a result is from the inadequacy of the active spaces used. Similar resultsare obtained for the isomerization reaction interconverting the trans end-onm� Z1 : Z1 and the side-on m� Z2 : Z2 isomers. Extending the CASSCF/CASPT2 approach to RASSCF/RASPT2, so as to handle larger active spaces,up to 28 electrons in 26 orbitals, seems to give promising results.60

SPECTROSCOPY OF TRIATOMIC MOLECULESCONTAINING ONE URANIUM ATOM

The chemistry of uranium interacting with atmospheric components, likecarbon, nitrogen, and oxygen, poses a formidable challenge to both experi-mentalists and theoreticians. Few spectroscopic observations for actinidecompounds are suitable for direct comparison with properties calculated forisolated molecules (ideally, gas phase data are required for such comparisons).It has been found that even data for molecules isolated in cryogenic rare gasmatrixes, a medium that is usually considered to be minimally perturbing, can

Table 2 Relative Energies (kcal mol�1) of the bis(m-oxo)Isomer of Cu2O2 with Respect to the m� Z2 : Z2 PeroxoIsomer with Various Methods.

Method � E

CCSD(T) 6.3CR-CCSD(T) 4.3CR-CCSD(T)L/BS2 13.1CR-CCSD(T)La 10.1CASSCF(8,8) 17.9CASSCF(16,14) 29.8CASSCF(14,15) 22.5CASPT2(8,8) 12.1CASPT2(16,14) 17.2CASPT2(14,15) 16.6BS-BLYP 8.4BS-B3LYP 26.8BS-mPWPW91 9.1BS-TPSS 7.9

Notes: CCSD(T): coupled cluster method. BLYP, B3LYP,mPWPW91, and TPSS: Various density functional theory-basedmethods.

BS means broken symmetry DFT. See Refs. 56 and 59 for adescription of the details of the calculations.

Spectroscopy of Triatomic Molecules 267

be influenced by the host. Calculations on isolated molecules are thus of greathelp to understand the interpretation of such experimental measurements.

We have studied several triatomic compounds of general formula XUY,where X;Y ¼ C;N;O, and U is the uranium atom in the formal oxidation state4þ, 5þ, or 6þ. We have determined the vibrational frequencies for the electro-nic ground state of NUN, NUOþ, NUO, OUO2þ, and OUOþ61 and have com-pared them with the experimental measurements performed by Zhou andcoworkers.62 The CASSCF/CASPT2 method has proven to be able to reproduceexperimental results with satisfactory agreement for all these systems.

The electronic ground state and excited states of OUO were studiedextensively.63–65 The ground state was found to be a (5ff)(7s), 3�2u state.The lowest state of gerade symmetry, 3H4g, corresponding to the electronicconfiguration (5f)2 was found to be 3300 cm�1 above the ground state. Thecomputed energy levels and oscillator strength were used for the assignmentof the experimental spectrum,66,67 in energy ranges up to 32,000 cm�1 abovethe ground state.

The reaction between a uranium atom and a nitrogen molecule N2 lead-ing to the formation of the triatomic molecule NUN was investigated.68 Thesystem proceeds from a neutral uranium atom in its (5f)3(6d)(7s)2, 5L groundstate to the linear molecule NUN, which has a 1�þg ground state and a formalU(VI) oxidation state. The effect of spin-orbit coupling was estimated at crucialpoints along the reaction coordinate. The system proceeds from a quintet statefor UþN2, via a triplet transition state to the final closed shell molecule. Aneventual energy barrier for the insertion reaction is caused primarily by thespin-orbit coupling energy. The lowest electronic states of the CUO moleculewere also studied.69 The ground state of linear CUO was predicted to be a�2 (a � state with the total angular momentum � equal to two). The calculatedenergy separation between the �þ0 and the �2 states is�0.36 eV at the geometryof the �þ0 state [(C–U)¼ 1.77 A and (U–O)¼ 1.80 A], and �0.55 eV at the geo-metry of the �2 state [(C–U)¼ 1.87 A and (U–O)¼ 1.82 A]. These results indi-cate that the �2 state is the ground state of free CUO. Such a prediction doesnot confirm the experimental results,70 supported also by some DFT calcula-tions. According to the results of Andrews and co-workers, the ground stateof the CUO molecule shifts from a closed shell ground state to a triplet groundstate, when going from a Ne matrix (analogous to free CUO) to an Ar matrix.Other groups are also working on the topic,71 which remains under debate.

For the systems here described, a multiconfigurational treatment isneeded, especially in the case of OUO, where the ground state is not a closedshell and several electronic states are lying close in energy to the ground state.In general the ground state and low-lying excited states of these systems aredescribed in a satisfactory way in comparison with experiment with theCASSCF/CASPT2 approach, whereas the high-lying excited states are in lessaccurate agreement with experiment, because it becomes difficult to includeall relevant orbitals in the active space.


ACTINIDE CHEMISTRY IN SOLUTION

The elucidation of actinide chemistry in solution is important for under-standing actinide separation and for predicting actinide transport in the envir-onment, particularly with respect to the safety of nuclear waste disposal.72,73

The uranyl UO2þ2 ion, for example, has received considerable interest because

of its importance for environmental issues and its role as a computationalbenchmark system for higher actinides. Direct structural information on thecoordination of uranyl in aqueous solution has been obtained mainly byextended X-ray absorption fine structure (EXAFS) measurements,74–76

whereas X-ray scattering studies of uranium and actinide solutions are morerare.77 Various ab initio studies of uranyl and related molecules, with a polar-izable continuum model to mimic the solvent environment and/or a number ofexplicit water molecules, have been performed.78–82 We have performed astructural investigation of the carbonate system of dioxouranyl (VI) and (V),½UO2ðCO3Þ3�

4� and ½UO2ðCO3Þ3�5� in water.83 This study showed that only

minor geometrical rearrangements occur upon the one-electron reduction of½UO2ðCO3Þ3�

4� to ½UO2ðCO3Þ3�5�, which supports the reversibility of this

reduction.We have also studied the coordination of the monocarbonate, bicarbo-

nate, and tricarbonate complexes of neptunyl in water, by using both explicitwater molecules and a continuum solvent model.84 The monocarbonate com-plex was shown to have a pentacoordinated structure, with three water mole-cules in the first coordination shell, and the bicarbonate complex has ahexacoordinated structure, with two water molecules in the first coordinationshell. Overall good agreement with experimental results was obtained.

To understand the structural and chemical behavior of uranyl and acti-nyls in solution, it is necessary to go beyond a quantum chemical model of theactinyl species in a polarizable continuum medium, by eventually includingseveral explicit water molecules. A dynamic description of these systems isimportant for understanding the effect of the solvent environment on thecharged ions. It is thus necessary to combine quantum chemical results withpotential-based molecular dynamics simulations. Empirical and/or semi-empirical potentials are commonly used in most commercial molecular simu-lation packages (for example, AMBER), and they are generated to reproduceinformation obtained by experiment, or, to some extent, results obtained fromtheoretical modeling. Simulations using these potentials are accurate onlywhen they are performed on systems similar to those for which the potentialparameters were fitted. If one wants to simulate actinide chemistry in solution,this approach is not adequate because there are few experimental data (struc-tural and energetic) available for actinides in solution, especially for actinidesheavier than uranium.

An alternative way to perform a simulation is to generate intermolecularpotentials fully ab initio, from molecular wave functions for the separate

Actinide Chemistry in Solution 269

entities. We have studied the structure and dynamics of the water environmenton a uranyl ion using such an approach (the nonempirical model potential,NEMO, method), which has been developed during the last 15 years.85,86 Ithas been used primarily to study systems like liquid water and water clusters,liquid formaldehyde and acetonitrile, and the solvation of organic moleculesand inorganic ions in water. A recent review article85 by Engkvist containsreferences on specific applications.

The interaction between uranyl and a water molecule has been studiedusing accurate quantum chemical methods.87 The information gained hasbeen used to fit a NEMO potential, which is then used to evaluate otherinteresting structural and dynamical properties of the system. Multiconfi-gurational wave function calculations were performed to generate pairpotentials between uranyl and water. The quantum chemical energies wereused to fit parameters in a polarizable force field with an added charge trans-fer term. Molecular dynamics simulations were then performed for the ura-nyl ion solvated in up to 400 water molecules. The results showed a uranylion with five water molecules coordinated in the equatorial plane. The U–water distance is 2.40 A which is close to the experimental estimates. A sec-ond coordination shell starts at about 4.7 A from the uranium atom.Exchange of waters between the first and second solvation shell is foundto occur through a path intermediate between association and interchange.This study is the first fully ab initio determination of the solvation of theuranyl ion in water.

THE ACTINIDE–ACTINIDE CHEMICAL BOND

After studying single actinide-containing molecules, the next questionthat one tries to answer is if it possible to form bonds between actinide atomsand, if so, what is the nature of these bonds? Experimentally, there is someevidence of such bonds both in the gas phase and in a low-temperature matrix.The uranium diatomic molecule U2 was detected in the gas phase in 1974.88

The dissociation energy was estimated to be 50� 5 kcal=mol. Andrews andco-workers found both U2 and Th2 molecules using matrix isolation spectro-scopy.89 Both molecules were also found in the gas phase using laser vapori-zation of a solid uranium or thorium target.90 Small molecules containing U2

as a central unit were also reported, for example, H2U–UH291 and OUUO.88

Not much was known theoretically about the nature of the chemical bondbetween actinides before the study of U2 by Gagliardi and Roos.6 The samemolecule was studied theoretically in 1990,92 but the methods used werenot advanced enough to allow for a conclusive characterization of thechemical bond.

Is it possible to say something about the bonding pattern of a moleculelike U2 based on qualitative arguments? Before undertaking the study of the


diuranium molecules, some systems containg a transition metal and a uraniumatom were studied, for example, the UAu4 and UAu6 molecules andNUIr.4,93,94 The ground state of the uranium atom is (5f)3(6d)1(7s)2, 5L6

with four unpaired electrons that could in principle form a quadruple bond.The double occupancy of the 7s orbital, however, prevents the unpaired orbi-tals from coming in contact to form the bonds. We find, on the other hand, avalence state with six unpaired electrons only 0.77 eV above the ground level:(5f)3(6d)2(7s)1, 7M6. A hextuple bond could in principle be formed if it isstrong enough to overcome the needed atomic promotion energy of 1.54 eV.There is, however, one more obstacle to bond formation. The 7s and 6d orbi-tals can be expected to overlap more strongly than the 5f orbitals. In particu-lar, the 5ff orbitals, which are occupied in the free atom, will have littleoverlap. Thus, there must be a transfer of electrons from 5f to 6d to form astrong bond. As we shall see, it is this competition between the atomic config-uration and the ideal molecular state that determines the electronic structureof the uranium dimer. To proceed further with the analysis, one needs to per-form explicit calculations, and such calculations were done using a basis set ofANO type, with inclusion of scalar relativistic effects, ANO-RCC, of the size:9s8p6d5f2g1h.6 As pointed out, potentially 13 active orbitals on each atomare involved in the bonding (5f, 6d, 7s). This would yield an active space of26 orbitals with 12 active electrons, an impossible calculation, so the numberof trial calculations were performed using different smaller active spaces. Theresults had one important feature in common: They all showed that a strongtriple bond was formed involving the 7ssg and 6dpu orbitals. The occupationnumbers of these three orbitals were close to two with small occupation of thecorresponding antibonding orbitals. It was therefore decided to leave theseorbitals inactive in the CASSCF wave function and to also remove the anti-bonding counterparts 7ssu and 6dpg. This approximation should work wellnear the equilibrium bond length, but of course it prevents the calculationof full potential curves.

With six electrons and six MOs removed from the active space, one isleft with 6 electrons in 20 orbitals, a calculation that could be performedeasily. Several calculations were thus done with different space and spin sym-metry of the wave function. The resulting ground state was found to be a sep-tet state with all six electrons having parallel spin, and the orbital angularmomentum was high with � ¼ 11. Spin-orbit calculations showed that thespin and orbital angular momenta combined to form an � ¼ 8 state. The finallabel of the ground state is thus 7O8.

The main terms of the multiconfigurational wave function were foundto be

�ðS¼3;�¼11Þ¼0:782ð7ssgÞ2ð6dpuÞ4ð6dsgÞð6ddgÞð5fdgÞð5fpuÞð5ffuÞð5ffgÞþ0:596ð7ssgÞ2ð6dpuÞ4ð6dsgÞð6ddgÞð5fduÞð5fpgÞð5ffuÞð5ffgÞ

The Actinide–Actinide Chemical Bond 271

This wave function reflects nicely the competition between the preferred atom-ic state and the most optimal binding situation. We have assumed that thetriple bond is fully formed. Also two electrons exist in 6d-dominated sigmabonds, 6dsg and 6ddg. The remaining MOs are dominated by 5f. Twoweak bonds (one d and one p) are formed using 5fdg and 5fpu orbitals.Note that there is substantial occupation of the corresponding antibondingorbitals. Finally, the 5ff orbitals remain atomic and do not contribute tothe bonding (equal occupation of the bonding and antibonding combinations).Formally, a quintuple bond is formed, but the large occupation of some anti-bonding orbitals reduce the effective bond order closer to four than five.Because of the weak bonding of the 5f orbitals, the effective bond order inU2 is not five but closer to four. It is interesting to note the occupation ofthe different atomic valence orbitals on each uranium atom. They are 7s,0.94, 6d, 2.59, and 5f, 2.44. Compare that with the population in the lowestatomic level with 7s singly occupied: 7s, 1.00, 6d, 2.00, and 5f, 3.00. We see atransfer of about 0.6 electrons from 5f to 6d, which allows the molecule to usethe better bonding power of the 6d orbitals compared with 5f.

The calculations gave a bond distance of 2.43 A and a bond energy ofabout 35 kcal/mol, including the effects of spin-orbit coupling. An experimen-tal value of 50� 5 kcal=mol was reported in 1974.88

Is it possible that other actinides can also form dimers? We already men-tioned that Th2 has been detected in the gas phase and in a rare gas matrix. Wehave studied this dimer and the dimers of Ac and Pa.95 Some major findingsare reported here.

We present in Table 3 the excitation energies needed to produce avalence state with all orbitals singly occupied. The largest excitation energyis for Ac. The price to pay for forming a triple bond between two Ac atomsis 2.28 eV; for Th, only 1.28 eV is needed, which can then, in principle, form aquadruple bond. Note that in these two cases only 7s and 6d orbitals areinvolved. For Pa, 1.67 eV is needed, which results in the possibility of a quin-tuple bond. The uranium case was already described above where we saw that,despite six unpaired atomic orbitals, only a quintuple bond is formed with aneffective bond order that is closer to four than five.

It is the competition between the needed atomic promotion energy andthe strength of the bond that will determine the electronic structure. In

Table 3 The Energy Needed to Reduce the Occupation Number of the 7sOrbital from Two to One in the Actinide Atoms Ac-U (in eV)a.

Ac: (7s)2(6d)1, 2D3=2 !(7s)1(6d)2, 4F3=2 1.14Th: (7s)2(6d)2, 3F2 !(7s)1(6d)3, 5F1 0.64Pa: (7s)2(6d)1(5f)2, 4K11=2 !(7s)1(6d)2(5f)2, 6L11=2 0.87U: (7s)2(6d)1(5f)3, 5L6 !(7s)1(6d)2(5f)3, 7M7 1.01

aFrom the NIST tables in Ref. 96.


Table 4, we present the results of the calculations, and in Table 5, the popula-tions of the atomic orbitals in the dimer are given. The results illustrate nicelythe trends in the series. A double bond is formed in the actinium dimer invol-ving the 7ssg and the 6dpu orbitals. But the su orbital is also doubly occupied,which would reduce the bond order to one. The Ac2 molecule mixes in the 7porbital character to reduce the antibonding power of the su orbital, whichresults in a unique population of the 7p orbital that we do not see for the otherdi-actinides. The populations are, with this exception, close to that of the freeatom. The calculated bond energy of Ac2 is also small (1.2 eV) and the bondlength large (3.63 A).

Already in the thorium dimer, Th2, we see another pattern. The 7s popu-lation is reduced to close to one. The electron is moved to 6d, and a strongquadruple bond is formed, involving three two-electron bonds and two 6done-electron bonds. We also start to see some population of the 5f orbitalsthat hybridizes with 6d.

The strongest bond is formed between the Pa atoms in Pa2. Here the con-tribution of 6d is maximum, and we see a complete promotion to the atomicstate with five unpaired electrons. A quintuple bond is formed with a shortbond distance and a bond energy close to 5 eV. The bond contains the(7ssg)

2(6dpu)4 triple bond plus a 6dsg two-electron bond and two 6ddg

one-electron bonds. The 5f population is increased to one electron, but we stilldo not see any molecular orbital dominated by this atomic orbital. They are allused but rather in combination with the 6d orbitals.

With the Pa2 dimer, we have reached the maximum bonding poweramong the actinide dimers. In U2 the bond energy decreases and the bondlength increases, which is from the increased stabilization of the 5f orbitalsand the corresponding destabilization of 6d. Large transfer of electrons from

Table 4 The Dominant Electronic Configuration for the Lowest Energy State of theEarly di-Actinides.

Ac2: ð7ssgÞ2ð7s7psuÞ2ð6dpuÞ2, 3��gTh2: ð7ssgÞ2ð6dpuÞ4ð6ddgÞ1ð6dsgÞ1, 3�g

Pa2: ð7ssgÞ2ð6dpuÞ4ð6ddgÞ2ð5f6dsgÞ2, 3��gU2: ð7ssgÞ2ð6dpuÞ4ð6dsgÞ1ð6ddgÞ1ð5fdgÞ1ð5fpuÞ1ð5ffuÞ1ð5ffgÞ1, 7Og

Table 5 Mulliken Populations (Per Atom), Bond Distances, and Bond Energies (D0)for the Early di-Actinides.

7s 7p 6d 5f Re(A) D0(eV)

Ac2: 1.49 0.49 0.96 0.04 3.63 1.2Th2: 0.93 0.01 2.83 0.21 2.76 3.3Pa2: 0.88 0.02 3.01 1.06 2.37 4.0U2: 0.94 0.00 2.59 2.44 2.43 1.2

The Actinide–Actinide Chemical Bond 273

5f to 6d is no longer possible, and the bonds become weaker and more domi-nated by the atomic ground state, even if we still see a complete promotionfrom a (7s)2 to a (7s)1 state. This trend will most certainly continue for theheavier di-actinides, and we can thus, without further calculations, concludethat Pa2 is the most strongly bound dimer with its fully developed quintuplebond having an effective bond order not much smaller than five.

INORGANIC CHEMISTRY OF DIURANIUM

The natural tendency of a uranium atom to be preferentially complexedby a ligand, rather than to form a direct U–U bond, has precluded the isolationof stable uranium species containing direct metal-to-metal bonding. Althoughthe uranium ionic radius is not exceedingly large, the presence of many elec-trons combined with the preference for large coordination numbers with com-mon ligands makes the task of stabilizing the hypothetical U–U bond difficult.The greater stability for the higher oxidation states of uranium would suggestthat if a bond is to be formed between uranium atoms, such species wouldrather bear several ligands on each multivalent U center.

As discussed, the uranium atom has six valence electrons and the U–Ubond in U2 is composed of three normal two-electron bonds, four electronsin different bonding orbitals and two non-bonded electrons leading to a quin-tuple bond between the two uranium atoms. Multiple bonding is also foundbetween transition metal atoms. The Cr, Mo, and W atoms have six valenceelectrons, and a hextuple bond is formed in the corresponding dimers, even ifthe sixth bond is weak. The similarity between these dimers and the uraniumdimer suggests the possibility of an inorganic chemistry based on the latter.Several compounds with the M2 (M¼Cr, Mo, W, Re, etc.) unit are known.Among them are the chlorides, for example, Re2Cl6, Re2Cl2�8 ,39 and thecarboxylates, for example Mo2(O2CCH3)4.97,98 The simplest of them arethe tetraformates, which in the absence of axial ligands have a very shortmetal–metal bond length.99

Recently, calculations have suggested that diuranium compounds shouldbe stable with a multiple U–U bond and short bond distances.100 We have stu-died two chlorides, U2Cl6 and U2Cl2�8 , both with U(III) as the oxidation stateof uranium (see Figure 5), and three different carboxylates (see Figure 6),U2(OCHO)4, U2(OCHO)6, and U2(OCHO)4Cl2. All species have been foundto be bound with a multiply bonded U2 unit.

In the diuranium chlorides, the formal charge of the uranium ion is þ3.Thus, 6 of the 12 valence electrons are available and a triple bond can in prin-ciple be formed. U2Cl6 can have either an eclipsed or a staggered conforma-tion. Preliminary calculations have indicated that the staggered conformationis about 12 kcal/mol lower in energy than the eclipsed form, so we focus ouranalysis on the staggered structure.


The diuranium chloride and diuranium formate calculations were per-formed at the CASSCF/CASPT2 level of theory. The active orbitals were thosethat describe the U–U bond. Enough orbitals were included such that themethod can make the optimal choice between the 5f and 6d orbitals in form-ing the bonding and antibonding orbitals. The number of active electrons waseight for the U4þ

2 unit and six for U6þ2 . A basis set of the atomic natural orbital

type, including scalar relativistic effects, was used.The U–U and U–Cl bond distances and the U–U–Cl angle have been opti-

mized at the CASPT2 level of theory. The ground state of U2Cl6 is a singletstate with the electronic configuration (sg)

2(pu)4. The U–U bond distance is2.43 A, the U–Cl distance 2.46 A, and the U–U–Cl angle 120.0 degrees. At

Figure 5 The structure of U2Cl6.

Figure 6 The structure of U2(OCHO)4.

Inorganic Chemistry of Diuranium 275

the equilibrium bond distance, the lowest triplet lies within 2 kcal/mol of thesinglet ground state. The two states are expected to interact via the spin-orbitcoupling Hamiltonian, which will further lower the energy, but is expected tohave a negligible effect on the geometry of the ground state, because it is asinglet state. The dissociation of U2Cl6 to 2 UCl3 has also been studied.UCl3, unlike U2Cl6, is known experimentally101 and has been the subject ofprevious computational studies.102 Single-point CASPT2 energy calculationshave been performed at the experimental geometry, as reported in Ref. 102,namely a pyramidal structure with a U–Cl bond distance of 2.55 A and aCl–U–Cl angle of 95 degrees. U2Cl6 was found to be about 20 kcal/molmore stable than two UCl3 moieties.

U2Cl2�8 is the analog of Re2Cl2�8 . The structure for U2Cl2�8 has been opti-mized using an active space formed by 6 active electrons in 13 active orbitals,assuming D4h symmetry. As in the U2Cl6 case, the molecular orbitals are linearcombinations of U 7s, 7p, 6d, and 5f orbitals with Cl 3p orbitals. The groundstate of U2Cl2�8 is a singlet state with an electronic configuration ofð5fsgÞ2ð5fpuÞ4. The molecule possesses a U–U triple bond. The U–U bond dis-tance is 2.44 A, the U–Cl bond distance is 2.59 A, and the U–U–Cl angle is 111.2degrees. U2Cl2�8 is different compared with Re2Cl2�8 in terms of molecular bond-ing, in the sense that the bond in Re2Cl2�8 is formally a quadruple bond, eventhough the dg bond is weak, because Re3þ has four electrons available toform the metal-metal bond. Only a triple bond can form in U2Cl2�8 becauseonly three electrons are available on each U3þ unit.

Based on several experimental reports of compounds in which the ura-nium is bound to a carbon atom, we have considered the possibility that aCUUC core containing two U1þ ions could be incorporated between two steri-cally hindered ligands. We have performed a theoretical study of a hypotheti-cal molecule, namely PhUUPh (Ph¼ phenyl), the uranium analog of thepreviously studied PhCrCrPh compound.102 We have chosen to mimic thebulky terphenyl ligands, which could be potentially promising candidatesfor the stabilization of multiply bonded uranium compounds, using the sim-plest phenyl model. We demonstrate that PhUUPh could be a stable chemicalentity with a singlet ground state. The CASSCF method was used to generatemolecular orbitals and reference functions for subsequent CASPT2 calcula-tions. The structures of two isomers were initially optimized using DFT,namely the bent planar PhUUPh isomer (Isomer A, Figure 7) and the linearisomer (Isomer B, Figure 8). Starting from a trans-bent planar structure, thegeometry optimization for isomer A predicted a rhombic structure (a bis(m-phenyl) structure), belonging to the D2h point group and analogous to thestructure of the experimentally known species U2H2.91 Linear structure Balso belongs to the D2h point group. CASPT2 geometry optimizations for sev-eral electronic states of various spin multiplicities were then performed onselected structural parameters, namely the U–U and U–Ph bond distances,whereas the geometry of the phenyl fragment was kept fixed. The most


relevant CASPT2 structural parameters for the lowest electronic states of theisomers A and B, together with the relative CASPT2 energies, are reported inTable 1. The ground state of PhUUPh is a 1Ag singlet, with a bis(m-phenyl)structure (Figure 1a), and an electronic configuration ðsÞ2ðsÞ2ðpÞ4ðdÞ2, whichcorresponds to a formal U–U quintuple bond. The effective bond orderbetween the two uranium atoms is 3.7.

It is interesting to investigate briefly the difference in the electronic con-figurations of the formal U2þ

2 moiety in PhUUPh and that of the bare meta-stable U2þ

2 cation.6 The ground state of U2þ2 has an electronic configuration

ðsÞ2ðpÞ4ðdgÞ1ðduÞ1ðfgÞ1ðfuÞ

1, which corresponds to a triple bond betweenthe two U atoms and four fully localized electrons. In PhUUPh, the electronicconfiguration is different, because the molecular environment decreases theCoulomb repulsion between the two U1þ centers, thus making the U–Ubond stronger than in U2þ

2 . The corresponding U–U bond distance, 2.29 A,is also slightly shorter than in U2þ

2 (2.30 A). A single bond is present betweenthe U and C atoms.

Inspection of Table 6 shows that the lowest triplet state, 3Ag, is almostdegenerate with the ground state, lying only 0.76 kcal/mol higher in energy.Several triplet and quintet states of various symmetries lie 5–7 kcal/mol abovethe ground state. The lowest electronic states of the linear structure (Figure B)lie about 20 kcal/mol above the ground state of the bis(m-phenyl) structure. Asthe 1Ag ground state and the 3Ag triplet state are very close in energy, they maybe expected to interact via the spin-orbit coupling operator. To evaluate theimpact of such interaction on the electronic states of PhUUPh, the spin-orbitcoupling between several singlet and triplet states was computed at the groundstate (1Ag) geometry. The ordering of the electronic states is not affected by the

Figure 7 The bent planar PhUUPh isomer.

Figure 8 The linear PhUUPh isomer.

Inorganic Chemistry of Diuranium 277

inclusion of spin-orbit coupling. To assess the strength of the U–U bond inPhUUPh, its bonding energy was computed as the difference between theenergy of the latter and those of the two unbound PhU fragments. PhUUPhis lower in energy than two PhU fragments by about 60 kcal/mol, with theinclusion of the basis set superposition error correction. The question thatone would like to answer is how to make PhUUPh and analogous species.PhUUPh could in principle be formed in a matrix, which is analogous to thealready detected diuranium polyhydride species9,10 by laser ablation of ura-nium and co-deposition with biphenyl in an inert matrix. The phenyl ligandmight, however, be too large to be made, and its reactions might be controlled,in a matrix, and so species like CH3UUCH3 for example may be more feasibleto construct.

CONCLUSIONS

Exploring the nature of the chemical bond has been a central issue fortheoretical chemists since the dawn of quantum chemistry 80 years ago. Wenow have a detailed understanding of what the electrons are doing in mole-cules on both a qualitative and a quantitative basis. We also have quantumchemical methods that allow us to compute, with high accuracy, the propertiesof chemical bonds, such as bond energies, charge transfer, and back bonding.In recent years, it has been possible to extend these methods to treat bondinginvolving atoms from the lower part of the Periodic Table.

In this chapter we illustrated how the CASSCF/CASPT2 method can beused to explore the nature of such chemical bonds. Classic cases are the Re–Remultiple bond in Re2Cl2�8 , and the Cr–Cr bond ranging from the quadruplybonded Cr(II)–Cr(II) moiety to the formal hextuple bond between two neutralchromium atoms. The bonding between the 3dd electrons is weak and shouldbe considered as an intermediate between two pairs of antiferromagnetically

Table 6 CASPT2 Optimized Most Significant Structural Parameters (Distances in A,Angles in Degrees) and Relative Energies (kcal/mol) for the Lowest Electronic States ofIsomer A and B of PhUUPh.

Isomer Elec. State R(U-U) R(U-Ph) UPhU PhUPh �E

A 1Ag 2.286 2.315 59.2 120.8 0A 3Ag 2.263 2.325 58.3 121.8 þ0.76A 5B3g 2.537 2.371 64.7 115.3 þ4.97A 5B3u 2.390 2.341 61.4 118.6 þ7.00A 3B3g 2.324 2.368 58.8 121.2 þ7.00A 1B3g 2.349 2.373 59.3 120.7 þ7.14B 3B3g 2.304 2.395 180 þ19.67B 3Ag 2.223 2.430 180 þ22.16B 1B3g 2.255 2.416 180 þ27.62


coupled localized 3d electrons and a true chemical bond. The Cr–Cr case alsoillustrates that no simple relation exists between bond order and bond energy.The energy of the bond in the Cr(I) compound PhCrCrPh is twice as large asthat of the formally Cr(0) compound, Cr2, despite the decreased bond order.On the other hand, the Cr(II)–Cr(II) moiety would hardly be bound at all with-out the help of bridging ligands such as carboxylate ions.

In the study of Cu2O2, the CASSCF/CASPT2 method is unsatisfactory.This and related problems motivate the extension of the CASSCF/CASPT2method to handle larger active spaces.

The chemical bond in systems containing actinide atoms, in particularuranium, was also addressed. A formal quintuple bond was found for the ura-nium diatomic molecule U2 with a unique electronic structure involving sixone-electron bonds with all electrons ferromagnetically coupled, which resultsin a high spin ground state. It was questioned whether the U2 unit could beused as a central moiety in inorganic complexes similar to those explored byCotton et al. for transition metal dimers. Corresponding chlorides and carbox-ylates were found to be stable units with multiply bonded U(III) ions. It mighteven be possible to use the elusive U(I) ion in metal–metal bonding involvingprotective organic aryl ligands in parallel to the recently synthesized ArCrCrArcompound. Many challenges exist, and issues still remain open. The interplaybetween theoreticians and experimentalists will certainly enhance the possibi-lities for further progress in transition metal and actinide chemistry.

ACKNOWLEDGMENTS

A wonderful collaboration and friendship with Bjorn O. Roos over the years has certainlybeen inspiring for the author. All the developers of MOLCAS, whose effort has been essential inorder to study such an exciting chemistry, should also be acknowledged, especially Roland Lindh,Per-Ake Malmqvist, Valera Veryazov, and Per-Olof Widmark. The Swiss National Science Founda-tion, Grant 200021-111645/1, is acknowledged for financial support.

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