Theoretical Study of the Electronic Structure of the Transition Metal Dimers Sc2,Cr, Mo and Ni BY CAROLWOOD,MARK DORAN AND IANH. HILLIER Chemistry Depart men t University of Manchester Manchester Ml 3 9PL AND MARTYN F. GUEST S.R.C. Daresbury Laboratory Daresbury Warrington WA4 4AD Received 20th July 1979 MCSCF calculations of the potential energy curves of the ground states of the transition metal species Sc, Cr, Mo2and Niz are described. The ground state of Scz is calculated to be ’C; corre-sponding to the orbital occupancy o~(s)a~(d)~~(d)o~(s).In Ni the metal-metal bonding is due to the doubly occupied o,(s) orbital a number of very closely spaced states arising from the weakly interacting pair of nickel d9configurations. In Cr and Mo a major feature of the bonding is the reduction in the order of the “ sextuple ” bond due to correlation effects particularly in Cr,.Bind-ing energies of all four diatomic species are calculated and compared with available experimental data. Transition metals play an important role in chemisorption and catalysis and there is growing interest both experimental and theoretical in the electronic structure of small clusters of these elements. The isolation in argon matrices of transition metal diatom species and higher aggregates has allowed spectroscopic studies to be made of these species.l-s In recent years many calculations on the electronic structure of such clusters have been reported. Thus aggregates of the metals Ni Ag Au Pd Cr Mo and Cu have been studied using semi-empirical molecular orbital methods9-15 and the scattered-wave (SW) Xcc method.l69” These techniques have also been em- ployed to investigate the interaction between gases and models of metal surfaces.’s-22 In this paper we consider the calculation of the electronic structure of the transition metal dimers Sc, Cr2 Mo and Ni by ab initio molecular orbital techniques.Al-though calculations on larger aggregates of transition metals will probably be re- stricted to more approximate methods particularly by pseudo-potential techniques we consider that more accurate calculations on the metal dimers are necessary both to obtain a satisfactory description of their particular mode of bonding and to provide “ bench-marks ” against which the more approximate calculations may be judged.This set of metal dimers was chosen to provide a representative sample of these species and because their dissociation energies have been determined e~perirnentally.~~-~~ Whilst the present work was in progress an investigation of the reaction Ni + H2+ Ni2H2 which included calculations on Ni, using ab initio methods with an effective core potential was rep~rted,~’ together with GVB calculations of Ni,28 which also used an effective core potential. TRANSITION METAL DIMERS COMPUTATIONAL DETAILS BASIS SETS The calculations were carried out using atomic bases of Gaussian type functions. For the first row transition metal atoms the (12s6p4d) basis of Roos et al.29was con- tracted to (5s2pld) and augmented with an additional s p and d function with the following exponents 0.45 0.26 and 0.15 for Ni; 0.24 0.13 and 0.08 for Sc; 0.4 0.4 and 0.2 for Cr.These additional primitive Gaussian functions were added following the conclusions of ROOS~~ concerning the inadequacy of the (12s6p4d) basis in the valence region. Hay3' has shown that the addition of an extra d function to this basis provides an improved description of the 4s13dnf1configuration relative to the 4s23dnconfiguration of the atom. The relative energies of the 4s23dnand 4s13dnS1 configurations of these atoms calculated with this basis are compared with the experimental separations 31 and with those resulting from near Hartree-Fock calcula-tion~~~ in table 1. The basis used in the calculations on Mo was the (27sl3p9d) basis of Huzinaga3' contracted to (8s4p3d).TABLE 1.-CALCULATED AND EXPERIMENTAL RELATIVE ENERGIES (a.U.) OF THE STATES OF ATOMIC Sc Cr AND Ni numerical this work Hartree-Fock experimental 0.0 0.0 0.0 0.008 0.047 0.001 (-0.001)" 2D(4s23d')"{ 4F(4s13d2) 0.0 0.02 0.0 0.037 0.0 0.052 7S(4S13d5) 0.0 0.0 0.0 cr{ SD(4s23d4) 0.059 0.047 0.035 a On averaging the J components of each state. COMPUTATIONAL METHODS In this work we are concerned with the determination of potential energy curves for these diatomic species so that single determinantal wavefunctions which do not dissociate correctly will be inadequate for this purpose. Multiconfigurational wave-functions which include at least those configurations leading to correct dissociation were calculated by standard configuration interaction (CI) methods,33 and in many cases the orbitals used to construct such expansions were optimized by the multi- configuration self-consistent field (MCSCF) method previously described.34 In this way a compact wavefunction of high accuracy can be obtained.A more restrictive MCSCF method the antisymmetric product of strongly orthogonal geminals (APSG) method,35 was used to obtain correlated wavefunctions for the closed-shell configura- tions of Cr and Mo2. We now discuss the calculations carried out on the individual diatomic molecules. C. WOOD M. DORAN I. H. HILLIER AND M. F. GUEST 161 COMPUTATIONAL RESULTS GROUND STATE OF SC To determine the ground state of Sc several states with different spin multi- plicities were examined.The procedure adopted was to perform MCSCF calculations for a number of selected states with S = 0 1 and 2 corresponding to the orbital occupancies shown in table 2. The molecular orbitals la and la,* may be regarded as the 4s bonding and antibonding orbitals the ten remaining valence MOs (2o, ln, ln, 16, la, 2a,) having mainly d character. In atomic Sc the 4s orbital is lower in energy than the 3dorbitals and this suggests that both the la and la MOs may be occupied in the ground state of Sc,. The selected orbital occupancies were obtained by occupying the 4s bonding orbital la and then distributing the remaining valence electrons amongst the lo, 2a, In and 16 MOs thus creating configurations with the maximum number of bonding orbitals occupied.The MCSCF calculations performed on these states were each to the same level of approximation and included all the important correlating excitations for each of them being double and quadruple excitations to the unoccupied internal orbitals. In all the MCSCF and CI calculations for Sc the core orbitals were kept fully occupied and in the MCSCF calculations only internal excitations were considered. The MCSCF wavefunction for each state has only one dominant configuration in the bonding region (typically z80%). The MCSCF energies are presented in table 2. A minimum in the MCSCF TABLE 2.-SCF AND MCSCF ENERGIES (a.u.) OF SELECTED STATES OF Sc electronic state configuration SCF energy MCSCF energy 'x; 10,210,220,2 -1516.477 -1516.578 1x; 10;1n4 -1516.478 -1516.590 3m7 10;2a,210:17T -1516.570 3xc 10;20;1n; -1516.437 -1516.573 5x; 10,22o30;1n; -1516.533 -1516.619 5As 10;2a; 1nf16; -1516.572 energy of the 5C; state occurs at 2.57 A and the energies of all the states are given at this bond length.If the ground state of Sc is a singlet or a triplet it must formally dissociate to 2D(~2d1) + ,D(s2d1). The lowest dissociation products from a quintet state of Sc are ,D + 4F(~1d2), and from a septet 4F+ 4F. From these considerations the ground state of Sc is unlikely to be a septet state since a relatively large binding energy would be required to make it lower in energy than states which were bound with respect to a lower dissociation limit.For this reason MCSCF calculations were not performed on any septet states of Sc,. The MCSCF energies of the singlet and triplet states in table 2 are all higher in energy than the separate atoms ,D + ,D,the lowest being the lZ; (loilzi). To improve the description of the lZ; state relative to the ,D + ,D limit a CI calculation was performed using the MCSCF vectors as basis MOs. The CI expansion included all double single and some quadruple excitations from the determinant la,21zi to * The orbital numbering in this paper does not include the core orbitals. TRANSITION METAL DIMERS the first twenty unoccupied orbitals. However this only lowered the energy of the state by 0.014 a.u. still leaving it higher in energy than the 2D + 2Ddissociation limit.On the basis of the MCSCF calculations it is proposed that the ground state of Sc is 5X~ (la,22a~ln303. The MCSCF curve (fig. 1) has a minimum at 2.57 A and is bound by 29 kJ mol-I with respect to the dissociation limit 2D+ 4F. -0.44 -0.48 ? -0.52 t -0.56 >r E" K d a -0.60 0 c 0 c -0.64 I1 I I 1 I 2.1 2.5 2.9 3.3 3.7 4.1 R /B FIG.1.-Potential energy curves of 5C; ground state of Sc2. An improved description of the 'Z; state was obtained by using the MCSCF vectors as basis MOs for a larger CI calculation. The CI expansion which ensured correct dissociation behaviour included all configurations which could be generated from the internal set of orbitals with the general valence description s2d4or s3d3and all other single and double excitations from la~2a~la~ln~ to the unoccupied orbitals (30,.. . 16, 2ng. . . 26,). The CI curve for the ground state shown in fig. 1 has a depth of 109 kJ mol-1 with respect to 2D+ 4Fatoms and 53 kJ mol-I with respect to two 2Datoms to be compared to the experimental value for the dissociation energy of 124 & 20 kJ m01-I.~~ From the CI curve the equilibrium bond length is 2.6 A. The results of this present work are not in agreement with the findings of previous workers. Cooper et a1.l3performed Extended Hiickel calculations and proposed a (la,21n:) ground state and an equilibrium bond length of 2.2 A. Busby et aL7 also used this method for their calculations which they performed at a bond length of 2.3 A.They proposed a 5A (la~2a~In~16~) ground state. Both the 'Z and the 5As states have been shown to be higher in energy in the bonding region than the proposed ground state (table 2). GROUND STATE OF Ni2 The ground state of the Ni atom is 3Fc~rre~p~nding to a configuration (Ar)4s23d* while the first excited state is 3Dcorresponding to a configuration (Ar)4s13d9. How-ever on averaging the J components of each state (corresponding approximately to ignoring spin-orbit coupling) the ground state is 3D(~1d9) while 3F(s2d*)is only 0.03 163 C. WOOD M. DORAN I. H. HILLIER AND M. F. GUEST eV higher. In both these states the partially filled 3d shell is more tightly bound than the 4s electrons This indicates that the interactions of two Ni atoms will be domin- ated by the 4s electrons on each centre.In this case two s2dsatoms will interact in a repulsive manner as they approach one another but two s1d9atoms can interact in a bonding manner. An s2d8atom coupled with an s1d9atom will lead to a total of three electrons in the 4s shell and would not be expected to bond as strongly as in the s'd -s1d9 case. The basic picture for the lower Nizstates will thus be of a single 4s bond analogous to the alkali metals or H,. The determination of the ground state of Ni2 therefore reduces to a problem of finding the most favourable coupling between the 3d electrons on each centre. A strong candidate for the ground state of Ni is 'Xi (I ail n,41n,41(Ti16:2a,2)* and indeed has been calculated to be the ground state using the SCF-Xcr-SW rnethod.20p22 However using semi-empirical MO method^'^^'^ the ground state is found to be 3Z; (1aila,21 ?Ti1 ?T;1 8; 16ff203.The present work was begun by performing SCF calculations on these two states for R(Ni-Ni) = 2.3 .$,twice the covalent radius and using these MOs in CI calcula-tions in which the core orbitals were kept occupied and only excitations amongst the internal orbitals were considered. For the calculation of the singlet states the CI expansion included all such closed shell configurations and all single and double excitations from the orbital occupancy la~ln~1n~ld~l(Tff2a~. The expansion of the triplet states included all single and double excitations from the 3Z; SCF configuration. Thus these two CI expansions include all the configurations with S = 0 or 1 respec-tively which can be produced by doubly occupying the 4s bonding orbital zag and permuting the remaining 18 valence electrons amongst the 3d bonding ar,d anti- bonding MOs.The striking feature of the results of both calculations (tables 3 and 4) is the small energy differences between the various states. This highlights the main problem encountered in trying to determine the ground state of Niz that is the low- lying states are all very close together. These preliminary calculations indicate the triplet states to be lower in energy than TABLE 3.-RESULTS OF THE CI CALCULATION ON THE SINGLET STATES OF Ni AT 2.3 A root state energy/a.u. important configurations 1 IX; -3009.0598 1a;2a,21ntfli n;lS;184 10;20;1 ntf1 nil S;l slf1a," 241ntf1 n;1 S;l s;10," 2 A9 -300!J.0407 la$20:1 ntfl~~;18i1&10~ 1a,220;1ntf 1 n;1 Sil Stf 10 1a;24 n;1 n;1 SilStf 1a,' '44 3 -3009.0398 10,220; 17.~21 nil Sil Stf10; 1a;2a,21 nil n;1 S;ls:l a 1 a,22a.3 nlfl n;1 s;1 s:1a; 4 lrs -3005.0302 10,220,21n:1 nil S$l Sfl a," I 1 a,220;1 n;ll nil S,2l S4,l a 5 'C,+ -3~09.0302j 6 'c -3009.0286 1~,22a:1n:1 n;l Sil S:~CT," 7 'nu -3009.0265 10,22~,21~~1~~16~1Stfl~~ 1a;2a;1 n:n 7riI s;1s:10," * Here the MOs 1o,,1ouy1nuy1ngy16 and 16 refer to the MQs arising from the 3d atomic orbitals and 20, 20 to the 4s orbitals.164 TRANSITION METAL DIMERS TABLE 4.-RESULTS OF THE CI CALCULATION ON THE TRIPLET STATES OF Niz AT 2.3 A root state energy/a.u.important configurations 1 -3009.1133 1 a,22a,21 nil nil 821 Sjlo,” 1 a,22a,2l nil nil s;1 sj10; 2 -3009.1 1 1 9 3 -3009.1 1 1 9 4 -3009.11 17 1 0,2203 np;l n;1s;1 s;1 a,” 1 a,22a,21 nlf1 nil s,21 s91a; 5 -3009.1 115 10,220,21 njl nil s;1 s;1 a,” la,22a,21 nil n;1 s;1 s:1 a,” 6 -3009.1109 7 -3009.1097 8 -3009.1094 J 9 -3009.1077 10,2203 nj1 n;1s; 1 s;1 0,’ 10;2a,21 nil nil s;1 st1 (7 10 -3009.1067 1 42a; 1 n 1 n;1 ail Splla 1a;2a,21 ntfln;ls;ls;lO,Z 10,2203 nlf1 nil sj1 s:1 a,’ 11 -3009.1067 12 -3009.105 1 1 0,224 n 1n;1 8491 s;f 1a,” 13 -3009.1046 } 14 -3009.1014 1 0,220; 1n:1n; 1 s;1 sj1a 15 -3009,0978 1a;2a;1 nj1 n;1 s;1 sjla,‘ 16 -3009.0526 1 0,2203 n:1 nil s;1 sj10,220,‘ singlet states with the same orbital occupancies.However it was subsequently found that this was largely due to the different basis MOs used in the two calculations. To obtain a uniform description of the lowest states produced by these initial calcula- tions a series of fairly limited CI calculations was performed in which only the SCF MOs for the 3X~ state were taken as basis MOs (table 5). Each state was described by a small expansion of configurations which included the essential orbital occupancies (tables 3 and 4) with only their important correlating excitations (i.e. ag2 -+a 7ti -+ n; 6; 36;) as there are only a small number of dominant configurations. The object of these calculations was to provide an adequate description of each state in the bond- ing region so that the ground state might be identified.To improve the description of these states MCSCF calculations were performed with the same set of configurations. The results of these MCSCF calculations (table 5) show that there are a number of singlet and triplet states lying very close together with energies approximately 33 kJ rno1-l below the 3F + 3Fdi~~~ciation limit. These results also show that the favoured orbital occupancies are those in which there are two holes in the 6 sub-shell. The 3ru, resulting six states (Irg,‘C,C IC; 3C;, ”:) are extremely close in energy and it is not possible on the basis of these calculations to be definitive about which is the ground state. From this group of states the lTSwas chosen for a more detailed study.265 C. WOOD M. DORAN I. H. HILLIER AND M. F. GUEST TABLE 5.-CI AND MCSCF CALCULATIONS ON SOME OF THE LOW-LYING STATESOF Niz [R(Ni-Ni) = 2.3 A] state dominant configuration CI energy/a.u. MCSCF energy/a.u. -3009.1 180 -3009.1395 -3009.1 181 -3009.1396 -3009.1139 -3009.1380 -3009.1 174 -3009.1384 -3009.1 176 -3009.1 390 -3009.1 152 -3009.1 394 -3009.1 1 1 1 -3009.1268 -3009.1 1 30 -3009.1 347 -3009.1119 -3009.1275 -3009.1 122 -3009.1342 -3009.1 125 -3009.1 324 -3009.101 8 -3009.1264 -3009.1096 - -3009.1094 - -3009.1079 - -3009.108 1 - The MCSCF calculations described above were performed for the ITs state over a range of bond lengths. The MCSCF orbitals were then used in the CI calculations in which all internal configurations of ITs symmetry were included giving the potential 0.22 1.7 2.1 2.5 2.9 3.3 3.7 4.1 4.5 R/B.FIG.2.-Potential energy curves for Ni, (a) SCF curve for 3C; state (la~2a~ln4,1n~16~164,la,2); (b)CI curve for state (la$?a,21n4,1~~16",ls~lo,'). 166 TRANSITION METAL DIMERS curve (fig. 2) and ensuring at least formally correct dissociation. The lowest calcu- lated dissociation limit is ‘F + ’&‘and therefore the state of Ni found to be the ground state should go continuously to that limit. However at a bond length of 4.5A the atoms in the lrqstate of Ni still adopt an s1d9Configuration and the potential curve at this distance lies above the 3F + ’F limit. We might expect therefore a curve cross- ing at a longer bond length if the is the ground State so that the correct dissociation limit is achieved.However a detailed examination of the long range behaviour of the states of Ni would require an improved description of the atomic states. The poten- tial curve produced for the ISqstate gives a dissociation energy of 50 kJ mol-l with respect to ’F + 3F and an equilibrium bond length of 2.36 A. The experimental estimate of the binding energy of Ni is 231 kJ mol-l. GROUND STATES OF Cr AND Mo In the case of Cr and Mo, where 12 valence electrons are to be accommodated in 12 valence MOs it is not practical computationally to decide the ground state elec- tronic configurations. However complexes containing the Cri+ and Mo~+entities are found to be diamagnetic and in such complexes it is considered that the Mi+ electron configuration is ai(d)n:(d)d:(d) corresponding to a formal quadruple metal- metal bond.It is thus likely that the configuration in Cr and Mo corresponds to the “ sextuple bond ” .~(s)a,”(d).~(d)6~(d).At a bond length of 1.9A the SCF calculation of this configuration for Cr yields an energy 1 a.u. above that of the atoms in the 7S(4s13d5) state. A similar situation is found for Mo where at a bond length of 2.0 A the energy of the SCF wavefunction is 0.35 a.u. above that of the isolated atoms. Such a result reflects the lack of left-right correlation in the SCF wavefunction which does not dissociate correctly to ‘S atoms. With a bond of such high multiplicity the error in the SCF wavefunction is considerably greater than that found for Ni and Sc (fig.1 and 2). To ensure correct behaviour of the wavefunction at the dissociation limit requires nearly 1000 bonded functions and represents a calculation that it is not practical for us to carry out at present. For this reason we have used the APSG method which yields a multiconfigurational wavefunction that dissociates to metal atoms not in their lowest energy configuration but to a weighted average of all the atomic states arising from the sldsconfiguration. The corresponding average atomic energy for chromium and molybdenum is calculated to be some 0.1-0.15 a.u. above that of the 7Satomic state. Such an APSG calculation is equivalent to an MCSCF calculation that takes account of the intra-pair correlation energy arising from the 092 -+ at n,2 -+ ni and 6 -+ 6 excitations and from other multiple excitations.The results of the SCF and APSG calculations on Cr and Mo are shown in fig. 3 and 4 respectively. From these calculations including correlation equilibrium bond lengths of 1.9and 2.1 A and binding energies (with respect to the average energies of the s1d5atomic configurations) of 143 and 430 kJ m01-l for Cr and Mo, respectively are found. The formal description of the M-M bond in these two diatomic species may be discussed by the decomposition of the APSG wavefunctions in terms of their com- ponent configurations (table 6). In the case of Mo the sextuple bond configuration contributes nearly 50% to the multiconfigurational wavefunction whilst for Cr this value is only 19%.The dominant correlating configuration arises from the 8 -+ 6% excitation and reflects the weak interaction between the atomic dxyand dx2-y2 orbitals in these molecules. We may estimate an appropriate M-M bond order (P)in terms of the occupation C. WOOD M. DORAN I. H. HILLIER AND M. F. GUEST -0.5 -0.4 0.3 -n =! 0.2-N a3 0 0.1 -(v + U >r 0-F C -0.1 -d b + 0 -0.2--0.3 -I I 1 I I I 1 FIG.3.-Potential energy curves for Cr2. -I 2 0.7} 1.7 2.0 2.3 2.6 2.9 3.2 3.5 3.8 RJA FIG.4.-Potential energy curves for Mo2. TRANSITION METAL DIMERS TABLE 6.-sUMMARY OF APSG CALCULATIONS ON Cr2(R = 1.9 A) AND Mo~(R= 2.0 A) configuration percentage in wavefunction Crz MO2 numbers (al,a,) of the bonding and corresponding antibonding orbital of each geminal giving P = 2 (aI2-a2”.fJ,%8 ‘This definition of P which merely provides a “ count ” of the number of bonds and provides no indication of the relative strengths of the components of each bond gives a value of 4.6 for the Mo and 3.3 for Cr, in line with the smaller contribution of the sextuple bond configuration to Cr compared with Mo,. DISCUSSION A major aim of the theoretical description of transition metal diatoms is the deter- mination of the ground state configurations which have not generally been determined experimentally. This presents a theoretical difficulty since a number of closely spaced states may arise due to the degeneracy of the atomic d orbitals and their small energy separation from the corresponding atomic s orbitals.The ground state configuration may not correspond to a maximum occupancy of the most bonding molecular orbitals since such an occupancy usually corresponds to a lowering of the spin compared with that of the isolated atoms. The actual configuration adopted will therefore correspond to a balance between the bonding energy and the spin energy. The latter contribu- tion is not considered in Huckel calculations which will predict a ground state corre- sponding to maximum occupancy of the most bonding orbitals. The calculations reported here on Sc and Ni have aimed to obtain an accurate description of these effects by means of quite small expansions of multideterminantal wavefunctions in which the individual orbitals are optimized.In both molecules such a multideterminantal description is necessary to achieve bonding and as expected the calculated binding energies are sensitive to the size of the CI expansion being considerably smaller than the experimental values although the calculated bond lengths are relatively insensitive to the size of the expansion. In the case of Ni, our predicted bond length 2.36 A is shorter than that in the bulk metal (2.49 A) and is to be compared with the value of 2.04 derived from calculations using an effective core-potential. Note that although our calculations predict different ground state configurations from those given by more approximate methods recent density functional calculations36 also yield a T; ground state for Sc and predict a number of closely spaced states for Ni,.In the case of Cr and Mo, we have not carried out the large number of calcula- tions required to determine the ground state configuration of these molecules. How-ever having assumed a ‘C,+ “ sextuple-bond ” configuration we find that a single determinant gives molecular energies with larger deviations from the atom energies C. WOOD M. DORAN I. H. HILLIER AND M. F. GUEST than was found for Sc or Ni, due to the greater formal bond order in Cr and Mo,. The inclusion of correlation effects by the APSG method reduces the bond order to near 3 and 5 for Cr and Moz respectively and yields bond energies of 143 and 430 kJ mol-l. 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