02cr*’ excited states of nicu - chemistrychem.utah.edu/_documents/faculty/morse/57.pdf ·...

Liganci-field theory applied to diatomic transition metals. Results for the &A&d states of Ni2, the &Nid$a states of NiCu, and the &Ni(3F)dL\02cr*’ excited states of NiCu

Eileen M. Spain and Michael D. Morse Department of Chemistry, University of Utah, Salt Lake City, Utah 84112

(Received 14 February 1992; accepted 18 June 1992)

A ligand-field theory has been developed for transition-metal diatomics having electronic configurations of dy,~!~~“, $,&r?, and &(3J’)dhoddr’. The theory treats each atom as a point charge and includes spin-orbit interactions. No contributions due to d-orbital chemical bonding are included. Since the d orbitals are quite small compared to the bond lengths in these molecules, the only inputs to the theory are the ligand charges (Z, and Z,), the radial expectation values (r”,} nd, ( ?B}nd, (rf;) no, and (P;),~, the atomic spin-orbit parameters & and cfi, and the bond length, R. Calculations employing no adjustable parameters (setting ZA,B z + 1.0, and using radial expectation values and spin-orbit parameters from atomic tables) provide essentially quantitative agreement with ab initio results on the $Nid$d manifold of states in NiCu, and on the &2‘&? manifold of states in Ni2. This demonstrates that the ligand-field model has some validity for metal molecules containing nickel, primarily because of the compact nature of the 3d orbitals in this element. Similar calculations of the d9,d9d manifold of states in Pt, and the &i&&d manifold of states in NiPt are presented for comparison to future ab initio or experimental measurements, although the possibility of d-orbital contributions to the bonding in these species makes the ligand-field model less favorable in these examples. The dhi(3F)dga”dr1 excited electronic states of NiCu, which are well known from resonant two-photon ionization spectroscopy, are also investigated in the ligand- field model. As a final example, the G!;,( 3F)do*’ excited electronic states of NiH are also examined using the same treatment as that employed for the dii(“F)d&\,d?drt excited manifold of NiCu.

I. INTRODUCTION

Achieving a detailed understanding of the electronic structure and chemical bonding in the diatomic transition- metal molecules, particularly in the 3d series, presents a formidable challenge to theoretical chemistry.’ A major problem in this regard is the correlation of electronic mo- tions in systems with partially filled d orbitals. In addition, the existence of low-lying excited states of the component atoms forces the theoretician to properly account for mixing of the various accessible separated atom asymptotes, which introduces further difficulties. Finally, proper treatment of the exchange interaction is needed to obtain quantitative results. Indeed, in the 3d series of transition-metal diatomics the ground state of the system often results from a delicate balance between exchange effects (which favor high-spin configurations) and chemical bonding (which fa- vors low-spin configurations). As a result, minor errors in the treatment of electron correlation, exchange, or in the consideration of excited separated atom limits, can lead to results which are both quantitatively and qualitatively in- correct. Finally, and perhaps most troubling, even when such theoretical attempts are successful, one is often left with a complicated multiconfigurational wave function which may obscure the underlying physical effects which control the electronic structure of t.he molecule.

Despite the intrinsic difficulties in describing the eleo- tronie structure and chemical bonding in the transition- metal molecules, theoretical chemistry has provided great

insight into t.he nature of the bonding in these species. For example, Bauschlicher and Langhoff have systematically elucidated the chemical bonding and electronic structure of ligated, unsaturated transition-metal molecules and 3d transition-metal dimers using ab irzitio techniques employing high-level electron correlation and configuration- interaction methods.2 Balasubramanian has worked exten- sively with the 4d and 5d series transition-metal dimers by incorporating relativistic effects in his ab initio calcula- tions3 The local-spin-density method has also been used to study the 3d and 4d homonuclear transition-metal dimers.4 All of these efforts are important and worthwhile, yielding qualitative and quantitative results that have provided insight into transition-metal chemical bonding. However, the computations are colossal and the results are often difficult to interpret in terms of simple, physically intuitive concepts. This situation could result because the bonding in these species is so complicated that simple concepts are insufficient; alternatively, we may have simply not yet found the simple concepts which can organize our thinking about these species.

Within the past decade, Field and others have developed a conceptually simple, computationally manageable theoretical approach5 to the study of the rare-earth oxides6 and halides,7 the transition-metal hydrides,* and the cal- cium halides9-” Kn this approach ligand-field theory has been revived as a method to compute the electronic states of metal-containing diatomic molecules which possess

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4642 E. M. Spain and M. D. Morse: Ligand-field theory applied to diatomic transition metals

compact nonbonding orbitals that are electrostatistically perturbed by a single fixed ligand. Unlike ab initio methods, small basis sets and simple, physically intuitive, molecular Hamiltonians are sufficient for the analysis. Fur- thermore, the computations are trivial compared to ab initio methods, since nothing more complicated than a matrix diagonalization routine combined with a program to evaluate angular momentum coupling coefficients is required. The application of this ligand-field method to the electronic states of the diatomic lanthanide oxides .and halides has met with wonderful success. For example, Schall, Dulick, and Field successfully employed a zero-free parameter, multiconfigurational ligand-field approach to predict the experimentally observed low-lying electronic states of LaF.’ In this model, LaF is considered as a free atomic metal ion, La+, perturbed by a nonpolarizable point F- ligand. Schall, Dulick, and Field, found that this zero-free parameter calculation was in agreement with the observed energy levels of LaF to within a few hundred wave numbers (cm-‘), provided several excited configurations of the La+ ion were included. In the present work we extend these ideas to the diatomic transition-metal molecules, and demonstrate that simple ligand-field treatments of NiCu and Niz provide predicted energy levels which differ only slightly from the results of previous ab initio calculations.

Of course, ligand-field theory provides wave functions in addition to energy levels, and a more critical test of its validity is the extent to which properties calculated from the wave functions are in agreement with experimental measurements. Thus an important confirmation of the ligand-field treatment of the lanthanide oxides was obtained when the Zeeman and hyperfine parameters of CeO and PrO were accurately calculated.6 A correspondingly important test of the ligand-field calculation presented here is the prediction of the heterogeneous perturbation experimentally found between the B[11.9]2.5 (primarily 2As,2) and c[11.9]1.5 (primarily 2113,2) excited states of NiCu, described in the preceding paper.12

The success of ligand-field theory depends on its application to molecules in which the component atoms possess physically compact valence orbitals. For example, the successful application of ligand-field theory to the lanthanide oxides CeO and PQ6 resulted from the extremely contracted f orbitals of the rare-earth metal atom. Like the 4f orbitals of the lanthanides, the 3d orbitals of nickel are very tightly contracted and spatially inaccessible, in con- trast to the outer 4s electron, which is readily available for chemical bonding. Field and co-workers have successfully applied ligand-field theory to the {Ni+ 3ds “D] supermul- tiplet of NiH, I3914 where they compare ab initio predictions to the NiH deperturbed molecular constants rather than erroneously comparing them to effective molecular constants. They found that the ab initio calculations were much better than originally thought, providing a synergism among ab initio theory, ligand-field theory, and experiment. Based on this success, it is clear that nickel possesses the necessary characteristics for effective application of the ligand-field theory.

In this article, we present theoretical results using a

ligand-field plus spin-orbit model for the low-lying electronic states of NiCu (d&d&d manifold) and Ni2 (&.&? manifold), where we can make comparisons to ab initio results. In addition, we apply the model to the excited electronic states of NiCu [d&i(3F)d&0,a20*’ manifold], where we can make a direct comparison to experimental results.i2 The application of this simple physical concept based on atomic parameters to successfully predict the electronic properties of molecular systems is both sat- isfying and useful. In Sec. II, the theoretical methodology for application of the ligand-field plus spin-orbit model is described. The theoretical results are presented, inter- preted, and compared to experiment and ab initio results in Sec. III. A summary of the work is then provided in Sec. IV.

II. THE LIGAND-FIELD MODEL FOR LATE TRANSITION-METAL DIATOMIC MOLECULES

Consider a diatomic molecule composed of two d”s” cations separated by a internuclear distance, r,, with physically contracted (closed or open) d subshells, surrounded and chemically bound by a (relatively) diffuse S(T bonding orbital containing the two electrons which leave the system uncharged. Provided the d orbitals are sufficiently compact, no chemical bonding will occur between them, and the energetics of the system will be determined by simple electrostatic and spin-orbit effects. In essence, the known electronic states of the two M+ ions may be considered to provide a set of Hund’s case (a) basis functions for the molecule, and the Hamiltonian matrix resulting from the electrostatic and spin-orbit interactions in the system may be evaluated to provide molecular energies and eigenfunctions. Hence, the term atomic-ion-in-molecule is used to describe the ligand-field model.15

A. ihe d@‘> states of a heteronuclear transition- metal molecule

The most straightforward application of the ideas described above is to the case of a &Adgd manifold of states, as presumably occurs in NiCu, NiAg, and NiAu. This manifold contains only 10 states, which are described in Hund’s case (a) as 2Xc+, 211, and 2A. However, these are split by spin-orbit interactions into one a= 5/2, two fi =3/2, and two fl=l/2 states, all of which are doubly degenerate. To the extent that the 6r electrons occupy orbitals which are diffuse and physically removed from the &A atom, or are roughly of spherical symmetry about this atom, their electrostatic effect on the 8 core remains in- dependent of the arrangement of the ds electrons, and does not influence the splittings between the states of the &d”g manifold. Accordingly, we treat the d” atom as pfotiding a nonpolarizable + 1 point charge, which exerts an electrostatic influence on the ds configuration of atom A. For the purpose of evaluating the splitting of the states arising from the &,d2c? manifold, we ignore the neutral- izing effect of the 2 electrons. In addition, the spin-orbit energy of the open G? subshell must be included, since the electrostatic energy in the nickel-containing systems is comparable to the spin-orbit energies.

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E. M. Spain and M. D. Morse: Ligand-field theory applied to diatomic transition metals 4643

TABLE I. Basis wave functions for a heteronuclear &;o’ molecule.

Molecular tern1 Coniigurdtion’ Basis wave functionb

:p -t du 211i/J

21 Idm4 dr

“rl?;?J *:I I d,,P)

d?r ?A Q.2

A,1 I GP) dS

‘A,-. <,l49)

dS A I Id,,4

The configuration is represented in terms of the d hole on atom A. “The b&s wave function is an antisymmetrized product of spin orbitals, equivalently+written as a Skater determinant. In this case, one hole is present so A, = 1. The tni quantum number is given as 0. 1, or 2 as a subscript.

We begin by forming a basis set in Hund”s case (a), with electronic states constructed in the form 1 A,S,X,Q). For simplicity, this set of basis functions is obtained by treating the 4; atom as possessing a single d hole rather than a 4d configuration of electrons. The five basis wave functions are listed in Table I, for the general case of any heteronuclear molecule with a @4dAod manifold of states.

The Hamiltonian for the dy,dFt? system is constructed following methods provided by Dulickn6 Since only one valence hole exists in the @ !dFc? system, there is no need to include electron-electron repulsions or exchange energy in the Wamiltonian. The total Hamiltonian is then simply

fi(q4dfd) =kdf +eLF, (2.1)

where 6 represents the internal Hamiltonian for the 4d ‘hLF ion, and Pi is the ligand-field Hamiltonian, which provides an electrostatic perJurbation to the @d ion. In the present case of a 4f ion, H.d contains only spin-orbit interactions,

g*, &p? (2.2)

@ ” is a one-electron operator of the form

@L c ~(r;,l& (2.3) i and the II”’ matrix in our basis set is readily evaluated to be

gp=

‘z ;j2 ‘)IIli2 ‘II,,, ‘Aji2 2A5i;

0 -fj’“j/~ 0 0 0

. -6’““si/2 c/T 0 0 0

0 0 --c/2 -f 0

0 0 -5 ; 0

0 0 0 0 -g

, (2.4)

where j is the spin-orbit parameter for the g1 ion. This is obtained by converting the spin-orbit operator [Eq. (2.3)] into an operator acting on holes, rather than electrons, which may be done quite generally by reversing the sign of the spin-orbit coupling parameter Cc) wherever it appears. I6

The ligand-field Hamiltonian is given as

fiLF= - x. ZLe2/riL, i

(2.5)

where the summation is over all of the electrons of the gd ion, Z, is the ligand charge (taken as + 1 in the present calculation), and rig is the distance between the ith electron and the point charge of the ligand. Alternatively, we may express this electrostatic interaction in terms of the repulsion of the single d hole on the 8 atom by the positive charge of the ligand by writing the ligand-field Hamil- tonian as

ELF=z,e2/r,, (2.6)

where rL now gives the distance between the hole and the ligand. With the standard definition of the B$ radial integrals, given as

Bk-zLe2 0- s om R&(I/“,/J”,+‘)?dr, (2.7)

where &(r) is the radial wavefunction of the hole, and rc and r, are the lesser and greater of r (the distance of the hole from the nucleus of the $ atom) and R (the distance between the 8 atom and the d” ligand), respectively, the one-hole matrix elements may be evaluated to give

(3d,,,c 1 eLF I3d,) =S,,,t c B;( - 1 k jrn$ ,” -a)

2 k 2

xo 0 0’ ( ) (2.X)

where (i 6 -‘,j and (i 6 i) are 3-j symbols which liiit the contributing terms in the sum to k=O, 2, and 4.

The Bt radial integrals may be broken up into the region from 0 to R and from R to CO, giving

B,k = ZLe2

+ R’ rlwkdr nl (2.9)

It is sufficient for practical purposes to consider the second integral to be negligible, and to extend the range of the first integral to infinity, essentially assuming that the electrons of the &A system never extend out to the location of the ligand, allowing the Bt integrals to be written as

B,k = Z,e” (‘

co Ri,rke2dr 1

/Rk+1=ZLe2(#‘)/Rk+‘. 0

(2.10)

The (#) moments required for the evaluation of Eq. (2.10) have been computed by Desclaux using numerical relativistic Dirac-Fock methods for all of the elements,” and these values are used in Sec. III where specific molecules (such as NiCu) are considered. In our final IiLF matrix we omit the constant Bg term, since this shifts all of the levels from a given configuration by the same amount. When both regions of Eq. (2.9) (r<R and r>R) make significant contributions, however, the fl$’ term can have a



major impact on the relative energetics of different configurations, leading for example to a stabilization of dns’ (or ~5’) configurations relative to the d”+’ (or f”+‘) configurations in the transition-metal halides (or lanthanide ha-

lides).5 Since only a single configuration is considered in this manuscript, however, the constant PO term may be omitted. The final HLF matrix is then given after explicit evaluation of the 3-j symbols as

2/7B$?-2/7& 0 0 0 0

0 1/7B&4/21& 0 0 0

HLF= 0 0 1/7B&4/2@ 0 0

0 0 0

0 0 0. .

The ligand-field Hamiltonian is diagonal in the Hund’s case (a) basis of Table I, but the spin-orbit Hamiltonian mixes case (a) basis functions with the same value of a. The total Hamiltonian, which is the sum of the spin-orbit and ligand-field Hamiltonian matrices, therefore block di- agonalizes into one 1 X 1 and two 2 X 2 submatrices, which may be readily solved to obtain eigenvalues and eigenvectors. The block diagonalized total Hamiltonian matrices for the !Fl= l/2, 3/2, and 5/2 blocks, are given in Table II for the d’,dro” states of a heteronuclear diatomic like NiCu, NiAg, NiAu, or PtCu, or a @As& molecule like NiAl, NiGa, or NiIn, or a d9Ad molecule like NiH. Simple examination of the matrix elements shows that as long as 2, is taken as positive, the ground term is predicted to be fi = 5/2, which is a pure Hund’s case (a) 2A5,2 state. This has been experimentally determined to be the ground state for NiCu,” NiAu,19 PtCu,” NiQ2’ and NiH.13

TABLE II. Hamiltonian matrix for a heteronuclear d9,dpd molecule.a

Cl=; block (2x2): 22+ l/Z 2b2

%+$B;:

- k4 J

i-k=; block (2x2):

2n3/2 ‘A,,2

‘fA is the spin-orbit parameter for the & configuration of atom A, .and & and & are defined in Bq. (2.10).

(2.11)

-2/7B;+ l/21& 0

0 -2/7B;+ l/21&

I

B. The de,&‘2 states of a homonuclear transition- metal molecule

In going from a d’,dFc? manifold of states in a heteronuclear molecule (such as NiCu) to the corresponding d9d”g manifold in a homonuclear molecule (such as C&s), the basis wave functions are doubled into g/u pairs due to inversion symmetry. The case (a) basis functions for this situation are listed in Table III. Although one might expect this to increase the complexity of the situation, it is straightforward to extend the treatment of the preceding subsection, since the total Hamiltonian may now be block diagonalized according to fi and g/u symmetry. The Hamiltonian must be generalized slightly to allow for spin-orbit interactions on each center, and for each ion core to be perturbed by the electrostatic field of the other. In the end, however, the Hamiltonian breaks up into blocks which are identical to those obtained in the heteronuclear case, with no distinction between the g and u blocks for a given Q value. As a result, each calculated

TABLE III. Basis wave functions for a homonuclear d9,d$‘d molecule.

Molecular term Coniigurationb Basis wave function’

2x+ 2x5

d% zlC I hoa) i- I dma)3/v2

2r;1,zg

duu 411 I d,oa) - 14&3/t/z d%

ZII A,,{ I &,P) -I- 1 d,,P)3/fl

1,zu d?r, ZIl

A^,{ Id,&) - I &,B)3/fl

3/2g dT8

2n3,2u

A,,{ I d,d + I4&3/fi

d?r,

2A3/zg

~dIdAc-4--ldd,~a)3/~

d%?

2A3,2u

A,1C Id,&> f I W9LW db

2Avzg

A,,{ Id,i9) - I&&)3/~

%

2A5,2u

TIC I dA24 + I &dI/ti df4 A,CldA2a)--ld&)3/fl

“Note that the 8,dpc? configuration in a homonuclear molecule only occurs in ions, with a net charge of f 1, h3, *5, etc.

bathe configuration is represented in terms of the orbital occupied by the d hole.

‘The%& wave function is described as a linear combination of terms with the hole localized on atom A or on atom B, and with m, given as a subscript.


level is predicted to be quadruply degenerate, with a degeneracy of 2 coming from the =+=a degeneracy, and another factor of 2 coming from the g/u degeneracy. As in the heteronuclear case discussed above, the lowest electronic state deriving from this manifold is predicted to be “A sjzg or ‘AsjzU, which are degenerate in this model. Any splitting of the d orbitals into bonding and antibonding orbitals, however slight, will of course lift the g/u degeneracy, leading to a 2As/2U predicted ground state. Owing to the extremely poor overlap between the 8, and 8s atomic orbitals, however, the lifting of the g/u degeneracy is expected to be very minor, particularly in the 3d series.

The Cu$ cation belongs in this category. Although the ground term of this cation is undoubtedly d.\“d~ocr~, 2E:, the first manifold of states above the ground state derive from the gddy$ configuration, and the predictions of this model are relevant to them. All that is required to apply this model to the #Ad:4 states of Cu$ is an estimate of the equilibrium bond length and values of (&Jcu, (r$d)cu, and the appropriate spin-orbit parameter [&,( 3d)] for the Cu’ ion in its 8s’ state.

C. The d9,&& states of a hetero- or homonuclear transition-metal molecule

In the case of a #A&d manifold in a heteronuclear molecule, one must explicitly consider a system with two holes, properly account for the different spin-orbit interactions on the two unlike atoms, and include the electrostatic perturbation of each ion core on the other. In per- forming this calculation, probably the most difficult aspect is that of selecting physically meaningful Hund’s case (a) basis wave functions in a manner which places one hole on each atom. The basis set that we have used for both the homonuclear and hcteronuclear 4dd9$c?’ calculations is available from the Physics Auxiliary Publication Service” (PAPS) of the American Institute of Physics or from one of the authors (M.D.M.).

111 forming the set of Hund’s case (a) basis functions for the d9&(r” states of either like or unlike atoms (“D +-“09, we have chosen to work with functions that correspond to SS, 83i; &r, rrr, na, or oo configurations of holes, with one hole on each atom. This allows us to analyze the final eigenfunctions in terms of their SS, &T, etc. character quite readily. For like atoms, the (‘D+ “0) separated atom limit generates the following Hund’s case (a) states:22-‘4 ‘~.~(3), ‘Z,(2), ‘I~IJ29, ‘B,(2), ‘A,(2), ‘AU, &IQ, ‘QU, ‘l-&n “Z;(J), 3XJ2>, “I&(2), 311,(29, 3A,(29, 3A, 3+U1 3S, and “Ir Of these case (a) states, the SS configuration of hol,es generates “Iu, ‘18, 3Xzs 3X;g. ‘Xl, and ‘2; terms; the fin configuration generates 3QU, 3Qp ‘QU, rap 311U, “II, ‘II,, and ‘II, terms; the 6a configuration generates 3A “A ‘A 222 p w ates ‘Alp, ‘A#,

and ‘Ag terms; the TV configuration gener- 3P’ “8’ ‘&“, and ‘22 terms; the ~TCT con-

figuration genera% ‘I!:, 3 II, ‘II,, and ‘II, terms; and the CTU configuration generates ‘Hz and ‘Xl terms. From these Hund”s case (a) terms, case (c) states arise which may be labeled by R, g/‘/u, and for R=O by +/- as O;(7), O;(2), O:(2), O;-(7), l,(7), l,(9), 2,(69, 2,(69, 3,(3), 3,(S), 4J29, q,(2), and 5,. For unlike atoms, the

g/u symmetry designations are dropped and the block diagonalization of the final Hamiltonian matrix is not so complete, but the number of basis states and their case (a) basis wave functions remain the same. Wave functions for the &A&2 configuration are generated in Appendix A and are available from the Physics Auxiliary Publication Ser- vice2t of the American Institute of Physics or from one of the authors (M. D. M. 9.

In the case of the d9,&& configuration, the molecular Hamiltonian contains portions describing the i$emal skmcture of the two 8 ions, which we may label as Hcd and HB, along with portions describing the electrostatic influence of the &a ion on the electrons (or hole) of the &A ion, and vice versa, giving

(2.12)

In the present case, we consider the internal Hamiltonian for each ion to consist only of the spin-orbit Hamiltonian for the electrons on that atom. Thus, for example,

2,4= C Lih9rii,&, i


where the summation is over all of the electrons on atom A, &(ris4) describes the radially dependent spin-orbit parameter for atom A, and the orbital angular momentum T& is measured about the nucleus of atom A. Again, this operator may be converted into an operator acting on the holes by reversing the sign cf the spin-orbit parameter, &;a( nd) .16 The expres$on for,HB follows analogously. Matrix elements of H.d and HB in the Hund’s case (a) basis are readily evaluated in terms of atomic spin-orbit parameters cd( nd) and c,(nd) using the methods described by Lefebvre-Brion and Field.‘s

The gectrostatic portion of the Hamiltonian, given by H?;F + eF, is evaluated in a similar manner as for the dg,di’c? system described above. The electrostatic perturbation of the 4A ion by the c& ion, for example, leads to

i?>F= - C Zs2/riB , i

(2.14)

where the summation is over all of the electrons of the &A ion, Z, is the ligand charge of atom 3 (taken as + 1 in the present calculation), and ris is the distance between the ith electron and the point charge of ligand B. The expression for fiy follows similarly. The correspondence between expression (2.14) and Eq. (2.5) allows the derivations to proceed similarly, giving simple one-electron matrix elements analogous to those given in Eqs. (2.8) and (2.10):

G4,~ I &“I nd,d

=S,,,, c B,“( - lY5 k

where

B,“=Zf12(#)A,,d/Rk+? (2.16)

Of course, entirely analogous expressions obtained for the one-electron matrix elements of @jF, and combinations of


L cl z- ? ii+ 3

$

s

if 3

i :: $ -L

?i

4 h h hh *.*.nw* h *,*.*-*~*.*~*,*.*~ i~~~~zSZ6?~=11MMMMMM Yhhhhhhhhhhhn hcoc907hh~o-JcaoYh~c9A~~~A~ 499=l994499=l9~~~~~=J~ ~vvvvvvvvv ---cnbcnca~cx -0e _____ -__-- 4 c? w m cp 09 hl9 0 op m vvvvv-

’ I’,

+,~~~~~~“~~~~~- ~~~.z”~“y~~~Y~~ -*nhcrJ~~nhhhhhh 33 3

q=)~~oYo?c.7h~~h v-v-eJhceoYhho2 v WEVIEL=E, - z- ,- c- ,- ,- ,- ~)~)~~~~~‘>~)~)~> _-- w - w “-,-,,-,-,-+-y qM+&*~*n UaE:M+Za~~=l~~M q-n, ~,,~wdsL-!n+ 9 3 =l,-q Y,-55Fg:922z:&: q --EVvOp E ‘Nvzv’,,“,- =- ,- ,- II =v - II ;; p ll Il v v 11 yy I ll ll I vy I II

w+” I +J%L~~“Ps’~ k I

y-PI&” ivy+ A + pJ+ NW,+

$$‘>‘h ;sp hl 1 P

05; t P

$L y E3 :;

\ $I. g

!2$ Y

Fw - 23 23 Irr I

F t \ P

2

~1 I I

a I! t Lz B

$ 2 X \o

F z ii; 2

,I z g, s: “7

2 ii ? z 9 2 it

E 6 e. B E 2. ?+ B B D 8 1 k

! 3 8 5

5 L%?F awl3 ;gp-, c hlf2 &; * &‘&I 2’

- 3 3w -sg a Qr! g-e ~~~ ii;, r g-S

xa g-3 b e 3 3’ z adj 8 &w gr“c $2 g g$g la 3-g * iG gg.g opww

3. 3' 3 $3 is m 3.i 3' T;,cW 1 L 0

m T +F 4. 5 ; ? 5 I . . c: fn 2 9 z h 5 1 w z b 8 cz fu, ;. e 5 g: s i? G iiT

Downloaded 03 Apr 2001 to 128.110.196.147. Redistribution subject to AIP copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

h .h 8% A c- * .^b ..-. .- s, 6 * .h - - - - t..* .a a,, cu 14 r., w CI G88888888”eae h F. n A ,T n n .--, h r= CI z. h cn cn cn h cn cm cn cz cc cx w: 0, WI 3 14 4 3 9 q =4 4 4 3 =I 9 vvvvwvvvvwvIv

c-3 m “r ‘s, 0 ir, m 6% @a 3-r * y. 6-a ------------_

~j~)=,=,~,~,~j~j~~~~~j~j~ ---------____ - w w ir - w IL.. - mu, Is,, s., - c*

487D@DDQi38+iD~@ c m. - n n -. h n h P-5 ,-. h e... scnmbncnq mco~,cr,cncxDn 5~z9335;~8~.33

=, r- c*tO*v 5, c ““,C, ~v*%.o”y*~*

II II 11 If II II I/ II II II II jl II ytg? s4 I lr’i r~og&&;l& SJ, -..9 B \ \, b-l *, +“” bd g \ I -x-P .A\. I p ,, \ WC. \Z’ f -~*.!A--% I z $ P z

5 i

FL

I

+. 4, \ ‘. y.

;c; b 3 z F w 5 8 ?? 22 Y bk 3 -4 T 6 2 2 e. P ” ; w 8 VI 2:: VI $ T 3 i: 5 b F

& +

*--*-hh~*~*******h*********** ,d s* w - - w ,w x,2 w w Ld aI w SW u w u u - - . - - - -

=DDDDDDDzzz~~~~=~~~JDDDDDDD h-~------~.nnnhnhh~,hFhhhhhhhh q1J~~~~~w.cncno?o,cr,cn~~~~q$~~~yYo? 9 =I =I Q Q Q Q q q =I =I 4 4 g q q q q q q 9 q g 1 =I 9 q ------~~L~vvvvvv YYYYYYYYYY-- L = = z-z.-5’E~ = = = = = E oi 9 +T w *i * * og 0 * eR

u-0 + $

t -u-r 2 P $

>

i?- R 3 i?i 12. I2 ” x” m @ x” 0, B zr 3 2 0 iz 2.



TABLE IV. (Continued. )

a=4 block (4 x 4 for heteronuclear; 2 x 2 CB 2 x 2 for homonuclear)

a=5 block (1x1)

(3r(c%),lfi~3r(L%),)=-2&$+/7+~+/21-~+ where c* = cd * & and & = (ZB(~)~.,~~ZA(~>~,,~)~~/R~+’

D. The &3F)d~&91 states of a heteronuclear transition-metal molecule

As a final example of the application of ligand-field theory to diatomic transition-metal molecules, we consider the d>( 3F)d~~a*’ states of a heteronuclear molecule. This corresponds to the excited states of NiCu which were experimentally studied in the preceding paper.12 Again, we choose to work in a Hund’s case (a) basis, corresponding to the 4Q, 4A, 4H, 42-, 2<p, 2A, 2H, and 28- states which derive from the d~(3F>d~~a*’ configuration, with basis wave functions written as Slater determinants in terms of holes rather than electrons. These wave functions are generated in Appendix B and are available from the Physics Auxiliary Publication Service2i (PAPS) of the American Institute of Physics or from one of the authors (M.D.M.).

The Hamiltonian for the di( 3F)dFgdic1 system is constructed as

(2.17)

where sA again represents the internal Hamiltonian for the di ( 3F) ion, e describes the electrostatic perturbatign of the di ( 3F) ion by the electric field of the dp ion, and HA++ describes the exchange interaction between the 3F ion and the unpaired dr electron. As before, the internal Hamil- tonian for the d:(3F) ion is taken to consist of the spin- orbit contribution only, given as

.&?A= 2 CA(rj,.4)L*~i9 i

(2.18)

where 6 is the spin-orbit parameter for the di ( 3F) ion, and the sum runs over all electrons. Again, this is quite generally converted into a summation over holes by reversing the sign of the spin-orbit parameter, &16

The ligand-field Hamiltonian is again given as

ii;== - c z&r,, i

(2.19)

where the summation is over all of the electrons of the di ion, 2, is the charge of the dp ion, and r/B is the distance

between the ith electron and the point charge of the ligand. Evaluation of the one-electron (or one-hole) matrix elements of giF proceeds exactly as in the previous cases, again yielding expressions (2.15) and (2.16) for the one- hole matrix elements (ndA,, I@ 1 nd,& . The complete evaluation of the matrix elements of eF in the Hund’s case (a) basis then leads to combinations of the one- electron matrix elements, as are given in the overall Hamil- tonian matrices of Table V for the di(3F)dFc?dlc1 system.

Finally, the exchange between the 3F core and the & electron is a significant factor in this molecule. For example, the corresponding interaction in the 3d8(3F)4s’, 2p4F states of the Ni+ ion leads to a splitting of the 2F and 4F states of about 4800 cm-‘. In a molecule like NiCu this should be reduced since the di: electron is not so localized on the d~i center, but it is nevertheless an important interaction which must be included in modeling the system. In the related treatment of the lanthanide oxides, for example, the analogous exchange parameter between the 4f and 6s electrons drops from 300 cm-’ in a free Ce2+ ion to 103 * 12 cm-’ in the ligand-field fit for the CeO molecule.5 In the d>( 3F)dyc?fi’ system considered here this exchange effect is modeled in HA-.,+ by adding an exchange energy, G2, to the diagonal matrix elements of the doublet terms and subtracting 2G2 from the diagonal matrix elements of the quartet terms, as given by Condon and Shortley26 for the analogous d2s’ configuration. Table V presents the final block-diagonalized Hamiltonian matrices for the di( 3F)d~~& system. In all cases, the use of Slater determinants to form the Hund’s case (a) basis functions facilitated the evaluation of the matrix elements. These were analytically evaluated using the Slater-Condon rules26 of one- and two-electron operators. The spin-orbit portions of these matrices were verified by setting the ligand-field portion to zero, diagonalizing the. matrices, and making certain that the degeneracy pattern for 4F and 2F terms was correctly obtained.

J. Chem. Phys., Vol. 97, No. 7, 1 October 1992


E. M. Spain and M. D. Morse: Ligand-field theory applied to diatomic transition metals

T.4BL.E V. Nonzero Hamiltonian matrix elements for a heteronuclear ~4(3F)d~~dc molecule.

C!= Ii’2 block (6x6):

i4A,:,j~14A,:1)-5=2G*c~~3 - (4A1,i;!lH14i11,1?)=-~~~6

(*rI,i21~14z,,j=-~/;/J6

i4A1,Jz?j2nlnj=g psi12 (2n,(* 1 HI 2z;;*) = -2g/t;3

~411,,l~~Jnr.,2j-3~~35~~~21i-~,6-2~2 (41;l i?lA14~,/2)=4B~35--2B~-2G2

j'II,,~~~*II,_;Ij=~/~~ (4z,/,~~~4n -&=g (2~:1121~;II '~~.2)=4B~35-22B4d7+G2

ii=3/'2 bIock (6x6)

(4~3~21~14~,3,-2j~-~~7-~~7$3~/2-2G2 {4Q3;2 j fit 4A3iz) = -c/v?

(“A32 @l”lT,,) = - &6

i4'ST2 1 fi\ 2A3f2j = &'2 ("Az,.z Iii1 ?rJ,qj = - /lOj/3

(4A3,.:1~.j4A.~iz)=~~3+~/3-2G? (4n3:211314n,z)=3~~35--B4d21-5/6-2G~

E;4A3,:2( 81 ‘A& =,;4;/3 (4II;,, I @I *nJJ =Qii

(4A~,:~ / 21411s;t,> = -- @y3 1411,,,1 H142,j = -g

(4A3f2j $1 21'13,2) - &'6 ('II,/,l~l:~,,,)=3B~35-~~21-j/3+G2 (*Il3,*1 HI 2,) = --g/v2

(zA,i2 j HI ‘A& =&j/L; -+ 2j;/3 + Gz (4n,21ij14z,)=4~~35-2~~7-2Gz

X1=5/2 block (5x5)

14G, Ifi1 4’DssrE) = - 327 - B$7 -+- iJ/‘2 - 2G: iJa$* 1 I;il’@ & = j/v2

(4A~,~~~~4As,~)=~~3-~/3-2G2

+Dvr 161 ‘A,;J = - C&j?? (4As,2 (81 *A,) =vT5/3

(4~ip,,~&~ijLA.,1) =sr, /12 (‘A,,, ItI 411s;d = -5,j%

(2~Snl~l”8~~~~=-~~7-~~7-t-j+GL (‘A,,~~1-1~*A,,, j=BQd’3-2j/3+G2

(“P,;,I fil ‘A& = -c/g &jz@14n,/2)= -tJ5/12

P’r$.* 1 El 2A& = - &p/3 {‘ni,,:lN141T,;2)=3R~35-~~21-~/2-2G2

Cl=7/2 block (3x3)

R-9/2 block (1x1)

<409;;1 1 Z?l 4cb& = - &7- $J7 - 35/2 - 2G,, where II,, is defined in Eq. (2.10).

4649

II. RESULTS AND DISCUSSION

A * The &&? states of NlCu

Using the basis functions of Table I and the Hamil- tonian matrices of Table II, a ligand-field calculation on the &Nidgio” manifold of NiCu has been performed. In an initial calculation the parameters were fixed at the values {CNi,3&) =603 cm- 1,27 and Zc-= + 1, and the values of (r?,i),,=O.3466 A’ and (r~i)3d=O.3204 A4 were obtained by averaging the corresponding 3dJ12 and 3dTi2 values obtained in numerical Dirac-Fock calculations on the d8s2 configuration of atomic nickel by Desclaux.” Although the radial expectation values would be expected to change slightly upon moving to the 8.~~ configuration, these values are suitable for testing the va1idit.y of the ligand-field model. The results of this calculation are given in Table VI, where comparison is made to the only ab initio calculation in existence on the NiCu molecule.28

Shim’s ab initio calculation on the NiCu molecule uti- lized a Hartree-Fock procedure, followed by configuration

interaction allowing full reorganization within the 3d and 4s subshells. The configuration-interaction calculation was performed separately on the ‘A, *II, and 22+ states deriving from the &Nidgd configuration, using orbitals optimized in the Hartree-Fock calculation for each of these states. Following the configuration-interaction calculation, spin-orbit coupling was introduced to give the final set of states and their parentage in terms of the Hund’s case (a) ‘A, “II, and 22’ basis functions. The molecule was found to possess a 2A5,2 g round state, with ~,=2.41 A, D,= 1.54 eV, and a,=347 cm-‘. These values differ con- siderably from the experimental values (r,=2.235 A, D, =2.05i=O.10 eV, and a,=273 cmW1),12 indicating that Shim’s theoretical treatment is not quantitatively accurate; nevertheless, it is not clear that errors calculated in these properties necessarily imply errors in the calculated splittings between the states which make up the dgNid&a’ manifold.

Table VI demonstrates a nearly quantitatiue agreement



TABLE VI. Comparison of ab initio and ligand-field theory for the &d&g states of NiCu.

State

2A5/2 SIl=3/2 n=1/2 0,=3/2 n= l/2

Relative energy (cm- ’ ) Composition

Ab initio” Ligand fieldb Ligand field’ Ab initio” Ligand fieldb Ligand fieldC

0 0 -40.5(40.5)_ 1QW2A, 100% 2A5,2 40%2rI+60%2A 31%*II+69%‘A

100% ‘A, 719 798.5 73016( - 11.6) 35% ‘n + 65%‘A

1602 1775.6 1626.9( -53.7) 28% ‘X+ .< 72% ‘l=l 33% ‘E+ + 67% ‘IT- 35% ‘2+ + 65% ‘II 1952 2098.4 2005.7( -24.9) 60% ‘Il + 40% *A 69% ‘l-l + 31% ‘A 65% ‘ll + 35% ‘A 3239 3346.2 3189.3(49.7) 72% ‘IX+ + 28%~‘II 67% *2+ + 33% ‘Il 65% ‘2+ + 35% ‘II

‘Reference 28. bCalculated holding Zc-= + 1.0, 5,,=603.0 cm-‘, (?),,=0.3466 AZ, (r4),,=0.3204 i4, and R=2.233 A. Obtained by allowing Z,,, &, and the term energy to vary, in a least-squares fit to the ab initz’o results. The difference between the ab initio and the fitted ligand-field results is given in parentheses. Fitted parameters are Z,,= +0.935; 5,i=607.4 cm-‘.

between the results obtained in the present ligand-field calculation (with no adjustable parameters) and this ab initio investigation. In addition, both calculations are in agreement with experiment in predicting a 2A5,2 ground state. To test the sensitivity of the model to the specific values of the parameters, the values of {(Ni,3&) and Zcu were allowed to vary, and a least-squares fit to the energies obtained by Shim was also performed. This yielded the optimized values of Z,,=+O.935 and &Ni,3&>=607.4 cm-‘, very close to the ligand-field model parameters of Zcu= + 1 and g=603 cm-‘, further supporting the ligand- field model for this system.

Although we would prefer to measure the validity of this calculation against experimental data for NiCu, the only experimental result for the d9Nid~~~ manifold is the identification of the ground state as 2A5,2. However, there are experimental results for the low-lying (d9Ni~‘) elec; tronic states of NiH,t3 a diatomic molecule that is directly analogous to NiCu. These states have recently been treated by a ligand-field method which is similar to that presented here.14 In this calculation, the observed levels were fitted to a ligand-field plus spin-orbit Hamiltonian which included all of the effects given above. In addition, however, configurational mixing between the d9,iO2, 2E+ state and the dgo’, 22+ state was considered, and this was found to stabilize the d9,iO2, 22+ state and dilute its spin-orbit coupling to the &iO2, 211r,2 state. Although one would also expect this effect to be important in NiCu as well, this cannot be demonstrated until further experimental data are available on the low-lying d9,id~~a2 states of the molecule.

B. The ~-&de&~ states of NiZ and Ptp

Tables VII and VIII present the results of the ligand- field calculation on Ni2, along with a comparison to the ab initio results of Shim,2gp30 which are essentially identical to those obtained in’ two other ab initio calculations- on Ni 31p32 The essentially quantitative agreement between the ligznd field and ab initio calculations is superb, again with no adjustable parameters. Perhaps most remarkable is the agreement between the two calculations.in the composition of the final wave functions. Although the agreement is not perfect, it is clear that the ligand-field treatment repro- duces most of the physical interactions which govern the splittings and mixing of the wave functions in this system.

A nonlinear least-squares fit in which cNi( d9) and ZNi were varied, and the resulting energies fitted to Shim’s ab initio results is presented in Tables VII and VIII as well. The converged values of cNi(d9) =623.9 cm-’ and ZNi = +0.953 which were obtained in the least-squares fit are close to the values ~~i( d9) = 603 cm-’ and ZNi = + 1 .O, lending further support to the ligand-field model. More- over, the splitting pattern among the &$o$ set of states has been derived from a physical model based on atomic parameters which does not require huge basis sets and treatment of electron correlation for its application.

Again, we would prefer to test the ligand-field model for Ni2 by comparison to experiment, rather than to ab initio theory. Unfortunately, our experimental knowledge of the d9Ad9dg manifold of states in Ni, is rather limited. A previous gas-phase experimental study by Morse et al. 33 established the ground state as possessing Q”=4, with a bond length of ro=2.200*0.007 A, based on the rotational resolution of a single vibronic band of jet-cooled Ni2 near 11430 cm-‘. Subsequently, Spain and Morse34 rotationally resolved a second 58Ni2 band at 11 523.015 cm-’ (this time with proper calibration based on the I2 atlas),35 and again found the transition to originate from an a” =4 electronic state, with a bond length of ro=2.199 *to.005 A. These experimental results, which assign the ground state as possessing Q” =4, differ from the predictions of both ab initio and ligand-field theory. At this time we do not fully understand this discrepancy between experiment and theory. Perhaps the Ni2 molecules are not sufficiently cooled in the supersonic expansion, thereby allowing the lowest Sz =4g state, predicted to be about 800 cm-’ above the ground electronic state, to remain populated. This state might well be resistant to cooling, since the only states predicted to lie below it in energy are O;, 0: , and 5,. Deactivation of an fi =4g state to. a 0, or Og’ state would require a reversal of the orbital angular momentum of one of the 6 holes in the d9 core of one of the nickel atoms. This h/z = 4 transition might well have a very low collision cross section. Likewise, deactivation of an fi=4, state to the fi=5, state might be improbable because the ds cores are well shielded from the outside world by the c$ framework, making the conversion from a “g” coupling of the cores to a “u” coupling improbable as well. Of course, it would be ridiculous to think that the ground


TABLE VII. Comparison of ah initio and ligand-field results for the gerade $,&& states of Ni,.

Energy (cm-‘) Combination in % of the case (a) basis functions’

Ligand Ligand State 4 4b initio’ field’ field’ ‘r#s) 3xp-(-Iss) “qss) %@rj l’I$(sa) 311,(&i-) “n,gm SA,&‘W lA,@d ‘A,J 4 “S;i m) ‘x&z i mj 3IIs( mj ‘I-I& m.7) lx; ((T(7)

0; 0

-5 828 1, 865 0: 1675 -38 1731 2s 1746 1, 2195 48 2241 1, 2575 % 2632 0: 2974 3, 3005 0; 3055 2s 3225 3, 3235 0; 3694 1, 3849 3 3919 1, 4163 2s 4229

0: 4329 CR’ 5036 1, 5356 9 5436 1, 5547 28 5604 0; 6740

0 816 816 1633 1850 1850 2139 2139 2666

30 4S(SO) 837 76i71) 837 71(71) 1645 27i25)

52(50)

25i25j

19(21)

O(O)

4i4) oioj

OiO)

WI O(O)

OiO) 5i4) 4i4)

OiO) OiO)

24(16) OiO) WH WJ) OiO)

18i20) 18(11) 16(20)

27i21) 17(21)

415) 9(9) OiO)

26(34) oioj WN oioj

21(23) 59(48j

W5) 58(48) li%

7i24) WI 24(25) 16(22) X0)

20(23j

23(25) 27(22j

20(22) 6ilO) 24(22j

7(5) S(5)

4Oi43) 211) 4i22) 14( 10) 32(20)

l(O) 4i13) 30(50)

loO( 100) 49(24) 55(48) 56(43j

O(O)

QiO)

Wl

17(1Oj

li0) 53(43)

29(47)

24i29j 14(15) 39(42)

31(34j 52(68) 32i35j

15(15) O(O) O(O)

oioj

S(5)

3ill)

17( 10)

65(50)

lO(24j

m is: 4? 5 2 a 5 P ic r! 3 6 9 CL -* CD h 9 8 2 P, z m P 0” El: 2 2. 0 4 0) 7. % z

23(34) O(O)

76(71)

W48)

33i5w

oioj 44iW

46i32j 23(16)

39i35) oioj 1802 2162 2162 2610 2610 2970 2970 2970 3420 3420 3575 3935 3935 4228 4228 4295 5193 5193 5193 5553 5552 6810

29(29) 24(29j

OiO) 24(24) OiO)

17(17) 62(5Oj

loo( loo) 48(32)

7i16) 1iO)

1ilO) OiO)

25(11) OiO)

43(42) O(O) O(O) MO) 4i5) oioj O(O)

18(24j 31(22) 17(11)

2955 2956 2956 3437 3437 3699 3989 3989 4254 4254 4278 5287 5287 5287 5576 5576 6874

20(21) L 0 z ? z 3

$

3 c r

i? 0 g 2 ic

5iOI

26(34) 49(68j

OUN OiO) O( 10) 35i9)

20(5j 24(11j 19(48j

704)

ww

52(22) OiO)

4i4) O(O) O(O) O(O) l(O)

OiO) 4i5) 13(20) 6i 10) 9i9)

OiO)

“db initio results are from Ref. 30. bLigand-tield treatment with ZNi= 1.0, c=603 cm-‘, (&3,=0.3466 A2, (r4),,=0.3204 A4, and R=2.20 A. “Ligand-field energies resulting from a least-squares fit to the ab initio energy levels, yielding ZNi= +0.953; c=623.92 cm-‘. ‘The % contribution is given for the ab initio result, followed by the percent contribution found from the ligand-field treatment with ZNi= + 1.0, f=603 cm-’ in parentheses.

5 P T


ii E

m s d 5 2 : a 6 z co c % i3 23 i% 9 zi 2 al H (D a 0” a 2 3 5 =r 9 -. % Z’ 2 Ei

TABLE VIII. Comparison of ab initio and ligand-field results for the ungerude d9*d9& states of Nis.

Energy (cm-‘) Combination in % of the case (a) basis functionsd

Ligand Ligand State Ab initioa fieldb fieldC ‘ru(6S) 3L:(SS) ‘H;(SS) 3@“(sn) ‘@ “(&r) 311u(6n) ‘k(sa) 3A,(W ‘A,(Sd ‘A.,(m) 32:(rv) ‘H;(m) 3n,(m) ‘II,(~) 3x:(m,)

O(O)

O(O) O(O)

w-3

O(O) 12CJ) 6(5) O(O) O(O)

56(43) 34(22) 53(22) 22( 11) OWN O(W O(O)

Loo( 100) 6(13) 4(7)

35(48) 16(24) 37(43) lS(22)

O(O)

O(O) O(O)

O(O)

O(O)

O(O)

13(10) 13(10)

O(O)

O(O) #rn WV

39(43) 40(43)

wa 48(47) 47(47)

O(O)

3(4)

19(21)

15(23)

32(25)

23(22)

7(5)

48(50) 52(50)

68(71) 28(25) 25(25)

32(29)

1W)

21(,21)

O(O) O(O)

O(O)

OjO) 3(4)

om O(0)

W) O(O) O(O)

20(21)

P(O)

3(4)

O(O)

O(O)

O(O) O(O) 0; 5” 4” 1” 0; 3” 2” 3” 4” 1” 2” 1” 0+ 0: 3, 2, 3, 0; 1, 2, 1, 2, 1, 0; 3, Of 0; 1, 2” 1" 0; 1,

12 0 58 0 835 817 853 816 1670 1633 1674 1633 1741 1850 1763 1850 2111 2139 2151 2139 2565 2666 2621 2666 2968 2956 3087 2955 3114 2955 3241 3437 3264 3437 3621 3699 3661 3699 3904 3989 3964 3989 4142 4254 4194 4254 4469 4278 4491 4278 4889 5287 5396 5287 5415 5287 5467 5576 5524 5576 6727 6874 6754 6874

30 30 lOO( 100) 837 66(71) 837 1645 1645 42(50) 1802 1802 8(O) 2162 34(29) 2162 2610 2610 2970 2970 2970 43(42) 3420 3420 W) 3575 3575 3935 3935 4228 4228

4295 4295 6(9) 5193 5193 5193 5552 5553 6810 6810

34(29) O(O)

O(O)

6(5)

22(22)

31(11)

O(10)

WY

18(24)

20(22)

16( 15) 41(42)

Wl) 58(68)

45(34)

32(35) 0(O)

24(24) lOo( 100) 25(17)

9(8) 41(32)

24( 16) O(O) O(O) O(O)

19( 10) O(P)

4(11) 33(42)

17(21) O(O) O(O) O(O) O(O) 6(5) f-w) O(O)

16( 15) O(O) O(O) O(O) 3(4)

O(O)

WO)

14(21)

15(23) O(O)

%W

W9) 28(25)

32(22) O(O)

41(22) g(5) O(O)

42(21) O(O)

ll(34) 66(71)

45(48)

5(O) 20( 16) 34(32)

36(35)

21(24)

O(O) O(O) 15(11) 16(11) 12(20) 32(22) O(O)

I- CI z 3 3 Y f $

s $ “4

ii a E G is

l(O) 31(42) 29(34) 30( 34) O(O) 59(68) l(O)

O(O) 30(47) O(5) O(5) 9(4f4 l(9) O(O)

29(24) 25(24) 38(9j 54(48) O(O)

O(O)

16(S) O(O)

16( 16)

O(O)

18( 10)

4(11)

37(20)

7(22)

62(50) 15(21)

W)

5(5)’

O(O)

O(O) 54(43) 7(10) 6(10) 41(22) 13(20) O(O)

ll(9)

WJ) 15(20)

“Ab initio results are from Ref. 30. bLigand-field treatment with ZNi= + 1.0, 5‘=603 cm-l, (?),=0.3466 A*, (r4),=0.3204 A4, and R=2.20 A. ‘Ligand-field energies resulting from a least-squares fit, to the ab initio energy levels, yielding ZNi= +0.953; [=623.92 cm-‘. dThe % contribution is given for the ab initio result, followed by the result for the ligand-field calculation with ZNi= + 1.0, 5=603 cm-’ in parentheses.

^.

. .

I:, I



state of Ni2 would not be populated in the jet-cooled molecular beam, so a more thorough investigation of Ni, should reveal the expected Q ‘02 , 0;) and 5, states which both ab initio and ligand-field theory predict to be nearly degenerate? and are the strongest candidates for the ground state. Stimulated emission pumping experiments probably offer the greatest hope for untangling the gAd”dg manifold of states in Niz, and this work will be critical for testing the ligand-field model of the electronic structure in this molecule.

Unlike nickel, the ground term of atomic palladium corresponds to the 4d”5s” configuration, and the ground state of Pd, probably derives from the singly promoted 4d’05so+4$5,~1 separated atom limit. Both theory36 and experiment.“7 are in agreement that the ground state of Pdz is best described as d~~~~~~~ti~‘d~~~d~~‘s~~, ‘Zc. The doubly promoted ~c~#fli states of Pdz fall at higher energies and have not yet been experimentally investigated. For this reason they will not be discussed further here.

The ground state of Pt2 derives from two 8s’ atoms, so the ligsnd-field model for the @d&&g manifold is applicable to this case. Unlike Ni,, however, Pt, shows significant d-orbital contributions to its bonding. Strong evidence for this statement comes from the optical spectrum of Ptz, which has been investigated in a jet-cooled molecular beam by the resonant two-photon ionization method.38 By ob- serving the onset of predissociation in a very congested vibronic spectrum, Taylor et al. were able to determine the bond strength of diatomic platinum to be 3.14 eV.3s This is 0.85 eV greater than the corresponding bond strength in .4uzJi9~” where splitting of the d orbitals into bonding and antibonding orbitals occurs, but no net d-orbital bonding results, because the d subshells are filled. The possibility of d bonding is not contained in the ligand-field theory as described in Sets. II C and II D, so one would not expect the ligand-field model to correctly predict the energies of the ~f~d&~ states of Ptz. Nevertheless, we have calculated the expected pattern of states assuming that the amount of d-orbital bonding is negligible, and the results are compared to m ab Xtio investigation”’ in Table IX. As in the case of Ni,, the lowest levels correspond to a degenerate set of 5,, 0:’ and 0; states, deriving primarily from the S,S, configurat.ion of holes.

In the most complete ab initio study of PtZ currently available, Balasubramanian predicts an “‘0: [32% “q-es;), 30% tqcs;,, 17% lq(s;), 15% “lx; (s;)] ground state for Ptz, with a low-lying Q=5, [93% ‘I”,(&&)] state only 614 cm-’ higher in energy.“’ Both of these states derive from a 6’ configuration of holes, which runs counter to the expectation that the d orbitals in Ptl are strongly split into bonding and antibonding orbitals. In this strongly split scenario, one would expect the most favorable chemical bonding for a dof, configuration of holes, which would give a net d-electron bond order of one, due to the unbalanced dd pair of electrons. The emergence of the fi=O;+ and a= 5, states (arising from SS configurations of holes) as the lowest lying states implies rather little d-orbital contributions to the chemical bonding in this calculation on Pt,. Despite the quantitative differences,

TABLE IX. Ligand field and ab initio results for Pt,.

Ligand field* Ab initiob

States’ Energy (cm-‘) State Energy (cm-‘)

5,,0:,0; 4,4,,1,1, 3,WgJu 3,,0,‘,0, 2,2,?1,L l”,O,+,O, 4&V4”lkl” 3,3,,o,;,o;,o;i-,o, ?&&A, 2,2,,1,1, 2;2,,18,1, l,l,,o,+,o;,o:,o; 3,,0;,0; &w,~” l&,0,

0.0 1 897.5 3 644.0 3 795.0 5 541.5 7 288.0

10 751.4 12 648.9 13 571.3 14 395.4 15 468.8 17 215.3 21 502.8 24 322.7 27 142.6

O,‘G,W 0 5”(3r”>~$“) 614 4”(3@“,6,7Q 1074 0; (+rff) 3064 2,(3GJ”@ 3 214 l,cn,Pn) 3 556 2,(3A,o;6,) 5 838 ‘J3qJ$) 5 935 0; (3rI”,w,) 7500 0: ( 3rr,,cr”~J 7 501 4J’rsYs6) 7 886 3J’A,u,S,I 8 164 1J3Z,s&Q 8 180 4,(3r,,s,Q 8 259 l&.( ‘z; ,6’) 8 282 o;(3z:,s,s,) 8 755 L(I~“,Od 9 556 0; G,,nn) 9 765 #+A,) 10 810 2JA.,uN 11286 3,~3@,,s,~.J 15 234 3,,(3r,,6,6J 15 806

‘Ligand -field calculation employing Z,,= + 1.0, &=4052.8 cm‘ ‘, R =2.471 A, (?),d=0.9211 A2, and (r&)= 1.7306 A4.

“Complete-active-space self-consistent-field first order configuration interaction results from Ref. 41. States which consist of greater than 70% of a single configuration are identified by the full term symbol and orbital designation of the d holes in the leading configuration. When spin-orbit or configurational mixing are more important, spin and orbital g/u dea- ignations are omitted.

‘States are designated by their 0, +/--, and g/u quantum numbers. In the ligand-field calculation many states with different quantum numbers are nevertheless degenerate. These would be split in a more realistic calculation.

this is similar to the results of the ligand-field treatment described here. A major distinction, however, arises because the molecular-orbital plus configuration-interaction approach41 allows both covalent 448&g and ionic ( dh”dvg and didfgg) contributions, while the current implementa- tion of ligand-field theory is similar to simple valence-bond theory in that only d9,d9&* configurations are considered. In light of the demonstrated contribution of the d electrons to the bonding in PtL, however, it seems likely that neither calculation will be correct; this cannot be stated with cer- tainty without a great deal of further experimental work on Pt,, however.

C. The c&&? states of NiPt

Diatomic NiPt belongs to the category of a heteronuclear d9,&& molecule, which has been treated by the ligand-field plus spin-orbit method in Sec. II C. In addition, NiPt has been spectroscopically investigated in the gas phase and found to possess a ground electronic state with 0=0.42 It is not straightforward to determine if the ligand-field model is applicable to NiPt. It is true that an a=0 state is one of the lowest electronic states coming from a &A#@* manifold of states in the ligand-field plus spin-orbit model, arising from spin-paired 6 holes on the


4.654 E. M. Spain and M. D. Morse: Ligand-field theory applied to diatomic transition metals

TABLE X. Ligand-field predicted states of NiPt.’

Energy Energy Energy Stateb (cm-‘) State (cm-‘) State (cm-‘)

5,0+,0- 0 -~ 291 4444 3,0+,0- 11 833 1 814 3,0+,0- 4 602 291 13 104 4 1066 3,2 4 908 3,0+,0- 13 152 4 1115 291 5 176 3,2 14 136

3,2 2 131 291 6 283 221 15 126 1 2 133 1,0+,0- 6 890 291 15 447 1 2 469 2,1 7015 1,0+,0- 16 893

3,0+,0- 3 283 1,0+,0- 8 475 291 17 012 4 3 799 4 10 455 1,0+,0- 18 478

32 4 194 1 11020

Valculation performed with Zm=Zr.,= +l.O, &=603.0 cm-t, & ~4052.8 cm-‘, R=2.205 A, (?(Ni))sd=0.3466 A’, (r4(Ni))sd =0.3204 A4, (?(Pt))sd=0.9211 A’, and (r4(Pt))sd=1.7306 A4.

bStates are designated by their Q quantum numbers. In many cases several states of different a are degenerate.

Ni and Pt atoms. On the other hand, an f’L=O term would be generated from the dddr4dcY4dS*4d?r*4sc? (‘2+, Cl =O+ ) molecular configuration as well, and this would be expected for the ground state if the d orbitals were strongly split into bonding and antibonding orbitals.

Since the available experimental data for the splitting pattern of the &d9& manifold of states in NiPt is limited to a knowledge of the ground state, we must rely on comparisons of bond strengths and bond lengths to evaluate the d-orbital contributions to the chemical bonding in this molecule. As in Pt,, the onset of predissociation in a very dense set of vibronic energy levels has allowed the bond strength of NiPt to be determined as D0 (NiPt) =2.798 eV.42 This may be compared to the analogous closed d subshell coinage metal diatomic, CuAu, which possesses a bond strength of DO (CuAu) =2.344 eV.43 The greater bond strength of NiPt implies significant but not over- whelming contributions to the NiPt chemical bond by the d orbitals of the component atoms. This is further substan- tiated by the reduced bonds length in NiPt (rs=2.2078 *to.0023 A),42 as compared to CuAu (rc=2.3302*0.0006 A)?3 On this basis the ligand-field plus spin-orbit model presented here, which omits metal-ligand orbital overlap, is probably not applicable to the NiPt diatomic molecule, since it appears that the d orbitals are split to some degree into bonding and antibonding pairs. Nevertheless, Table X presents the results of the ligand-field plus spin-orbit calculation for comparison to future ab initio or experimental results.

Finally, the analogous NiPd (Ref. 44) and PdPt (Ref. 44) diatomics have also been investigated in the gas phase, but the ground states of these molecules are thought to arise from d~i( ‘F)diic? and dbid&( 3F) 2 state configurations, respectively. This is due to the stable dl’so configuration of atomic palladium. Accordingly, a ligand-field model based on a a$&& manifold of states is inappropri- ate for these molecules, and they will not be considered further. Both molecules would find their counterparts in the d,&,(3F)c? states of CoH, however, and the analysis of

all of these systems in terms of a ligand-field theory will be quite interesting.

D. The dsNi(3F)di\ddr’ states of NiCu

The single example of a manifold of experimentally well-known electronic states in a transition-metal diatomic molecule is the d&i ( 3F) d&\ga*’ manifold of NiCu, which was experimentally investigated in the preceding paper.12 As such, this manifold provides an important test of the ligand-field theory developed in Sec. II E above. Accord- ingly, a calculation was attempted with &,= +l.O, 5 (Ni,3d84?) ~663 cm-1,25 (r2,,),,=0.3466 A2,17 (Y&>3d =O 3204 A4,17 and with the 3dNi-a exchange parameter G2 alldwed to vary. Despite the flexibility afforded by making G2 an adjustable parameter, it was not possible to repro- duce the experimentally observed electronic states of the d~i(3P)d~~02~1 manifold of NiCu with this model. This was troubling, particularly in light of the success of the ligand-field model in comparison with ab initio results for the d9,id~~~ manifold of NiCu and the d9,d9d2, manifold of Ni,.

In the d&(3F)dg”gdi(1 manifold of NiCu, however, one additional electron has been removed from the nickel core, so it might be reasonable for the copper to have an effective charge which is somewhat reduced from Z,, = + 1.0. Accordingly, Z,, was varied along with G2, and a reasonable fit to the experimental data was successfully obtained, as is listed in Table XI. Surprisingly, the calculation converged to values of G2= 5 11.6 cm-’ [substan- tially reduced from G2- - 1600 cm-’ in the free d8(3F)s’ Ni+ ion] and Z,,= - 0.666. A subsequent calculation also allowed g (Ni, 3d84?) to vary, resulting in the fitted parameters of G2=547.3 cm-‘, Z,,=-0.686, and 5 (Ni, 3d842) =592.3 cm-‘. Both fits are listed in Table XI. The emergence of a negative effective charge on the copper atom is surprising and is not presently understood. This suggests an improvement to the ligand-field treatment, in which one quantum mechanically solves for the 28’ system of three electrons in the presence of pseudopotentials describing the d&d& cores, then calculates the electrostatic perturbation to the d&(3F) core including both the point charge of the copper core and the influence of the aah’ electrons. Presumably this calculation would reveal why an effective negative charge resides on the copper atom in the d&(3F)d$!dti1 states of NiCu.

To show how the charge of the copper ligand affects the splitting in the d&i(3F)d&\dti1 manifold of states. Fig. 1 displays the energies of the a =7/2, 5/2, and 3/2 levels as a function of ligand charge, holding the spin-orbit parameter and the exchange energy constant at 663 and 511.6 cm-‘, respectively. Again, it should be noted that this ignores the large configurational reordering effects which can result from variation of ZL.’ The figure shows that the pattern of IR states basically inverts in changing the ligand charge from + 1 to - 1, with the exception of a few avoided curve crossings. From this figure it is clear that we have not missed a minimum with Zo, > 0 in the nonlinear least-squares fit, since the pattern of states is totally wrong for Zcu> 0. A challenge for theorists is to




TABLE XI. Fitkd @and-field results for the d&,(‘F)&$dc manifold of NiCu.

Energy (cm-‘) (Residual given in parentheses) Composition of the wave function (in %)d

State lzspt.a Ligand fieldh Ligand field“ j@ jh “II,, 4n...,,z Qx- %? “A ‘n Jz-

$113.512.5 13 547.33 13 595.X( -48) 13 541.4(6) 4(3) 2(t) O(O) 86(88) 8(8)’ q13.1]1.5 13080.34 13 195.7(-115) 13 121.4(-41) 5(4) O(O) X2) 2(l) 66(68) 25(25)’ O-5 13 159.2 13 087.6 32) W-J) 3U) l(l) 52(53) 40(41) .&3]3.5 unobserved 12 272.97 12 12 232.2(41) 276.4 12 12 302.7(-30) 105.3 76(79) 13(9) 4(3) ls(l2j 5(4) 2(l) l(O) 2(31f 83(88)’

0.5 11 937.5 11 811.8 30(5) 33(3) ll(6) 25(l) O(37) O(49) B[11.9]?.5 11902.11 11 X77.5(25) 11916.1(-141 19(11) l(1) 7(61 4(5) 70(77)’ Ql1.9]1.5 11 903.89 11 770.3( 134) 11 813.0(91) O(O) 13(11) O(0) g(6) 21(23) 58(60)’ 0.5 11 762.5 11 743.4 5(29) 6(35) 6(10) 2(26) 34(O) W W A[11.5]2.5 11 519.51 11558.3( -39) 11477.7(42) 55(65) 25(22) X(6) 9(7) 3(l)f 1.5 unobserved 11 347.2 11 242.9 13(12) 16( 19) 38(39) 25(25) 7(5) l(1)’ 0.5 11 250.9 11 147.8 39(40) l(1) 36(36) 17( 17) 5(4) a21 [IO.8]3.5 10 777.27 10 731.2(4(i) 10 763.8( 13) 71(76) 12( 12) 17( 12)f [10.4]2.5 10414.47 10455.5(-41) 10480.1(-66) 18(17) 25(29) 36(39) l(1) 19(14)’ [11).4]1.511) 10 410.8br 10 363.6’(47) 10 389.8’(21) 4(4) 35(38) O(0) 39(43) 5(4) 16(11)f 0.5 10 320. I 10 343.0 18(19) 23(25) 31(34) 7(8) S(6) 11(8) 4.5 10 105.7 10 173.8 lOO( loo) 3.5 9 751.5 9 804.5 16( 14) 84(U) om’ 2.5 9 649.5 9 702.6 4(4) 47(47) 49(49j O(O) O(O)’ 1.5 9 617.4 9 670.6 l(l) 20(20) 55(55) 24(24) O(O) war 0.5 9 hll.1 9 664.3 5(5) 36(36) 12(12) 47(47) O(O) O(O)

‘Reference 12. ‘ligand-field calculation with & tied at 663 cm-‘, (2),,=0.3466 A’, (r’),,=O.3204 A4, and R=2.339 A; Z,, and Gz were allowed to vary, and sonvergcd to I&,= -0.666 and G,-511.6 cm-‘.

‘L&and-field calculation with (?jX2=0.3466 A’, {r’),d=0.3204 .&‘, and R-2.339 A; Z,, G,, and & were allowed to vary, and converged to &,=-0.686, GL=S47.3 cm-‘, and &,=592.3 cm-‘.

d.The composition of the final wave function is given in % for the fit described in footnote b, and in parentheses for the fit described in footnote c. “Not included in the least-squares tit, because the identification of LIZ- 1.5 is not completely definite. fThis contribution is re.sponsible for the oscillator strength in absorption from the ground X ‘Asi state.

understand why the Ni d” orbitals respond electrostatically to the Cu ligand as if it had a negative charge.

Of course, it is possible that the negative effective charge on the copper ligand is a compensation for some- thing left out of the model, such as perturbations with other d~id~~~a* ’ states, such as the dki( ‘D)d&c?til

NiCu d,,Y’F)dc.‘“r3?g* States

I~igand Charge quarr W=7R ii,& L1’ = s:* tmsgk ,r F ,iL

FE. 1. The energy ordering of the NiCu tiN, ‘(F) &,dti’ states (R’-312, 562, and 7i2 only) as a function of Cu &and charge, with the spin-orbit parameter 5=663 cm- I and the exchange parameter G, = 5 11.6 em- i held constant.

manifold, which lies approximately 12 000 cm ’ higher in energy. Alternatively, configuration interaction with the d&d&\Gy d&:d&fp’, or d”N&a”U*’ manifolds of states may be induced by the presence of the ligand, accounting for the departure from Zcu= + 1.0. If either of these possibil- ities is occurring, however, it is surprising that the exchange parameter G2 makes sense. Further, it is not clear why a variation of the ligand charge alone can account for these effects. In any case, our fit of the energies and 0, values of eight observed electronic states by the variation of just Z,, and G2 [or in the second fit, Z,,, G2, and 5 (Ni, 3d84.?)] seems to be more than chance, and suggests that there is some validity to the effective parameters thereby obtained. To verify whether this is so, we have used the calculated wave functions to make further predictions about t.he d8,i( “F)d&\c?o*’ manifold of states of NiCu.

1. Molecular properties from &and-field eigenvectors

Although the ligand-field fit to the dki(3F)d&\dd’1 manifold of states of NiCu requires a seemingly unphysical negative charge on the copper atom, it nevertheless provides several convincing predictions. Both fits predict the existence of two 0 = 1.5 states, located at approximately 12 200 and 11 300 cm-‘, which were not observed in the spectroscopic work of Spain and Morse. I2 However, absorption oscillator strength from the X 2&,/Z ground state to the a= 1.5 states derives entirely from the ‘II3,2 char-



acter contained within the strongly mixed excited-state wave functions. The two unobserved sZ= 1.5 states consist of only about l%-3% 211s,2 character, whereas the observed a= 1.5 states contain l l%-60% 2113,2 character. Thus the calculation predicts that these two unobserved band systems should have absorption intensities which are approximately one-tenth of that found in the other a= 1.5 band systems. This readily accounts for their absence in the recorded spectra.

Another prediction of the ligand-field calculation is that an Cn= 1.5 state should be found near 10 370 cm-‘. Spain and Morsel2 have detected a state in this energy range, which heterogeneously perturbs the [10.4]2.5 state. It therefore must possess an fi value of either 1.5 or 3.5. Although rotationally resolved studies of this band system were inconclusive, the observation of a strong P branch suggested that the new state possessed Q = 1.5, and consid- erations of the vibrational factor of the perturbation matrix element placed the v=O level of this state at approximately 10 410.9 cm-‘. Although this state was not included in the least-squares fit to the ligand-field model, it was nevertheless predicted to lie within 50 cm-’ of the observed energy of the u=O level.

2. Excifed-state radiative lifetimes The eigenvectors obtained in the ligand-field calcula-

tion should also at least qualitatively account for the radiative lifetimes measured in the resonant two-photon ionization experiments. l2 The measured lifetimes fall into two categories, giving a range of 3.98-8.61 ,us for the A-F states, and lifetimes greater than 40 ps for the low-lying [10.8]3.5, [10.4]2.5, and [10.4]1.5(?) states. Since the ground d9,id~:O,O2 manifold is purely doublet (S= l/2) in character, the radiative decay rate should be roughly pro- portional to the total amount of doublet character in the excited-state wave function, which is given by the sum of the percentage contributions from the doublet states in Ta- ble XI. This gives values of 79%-96% doublet character for the B-F states and only 12%-21% doublet character for the low-lying [10.8]3.5, C10.412.5, and [10.4]1.5(?) states, in rough correspondence with the measured lifetimes. On the other hand, the A[1 1.512.5 state is not accurately represented by the ligand-field calculation, since its total doublet character is only 8%-12%, but its decay rate is comparable to that of the states which are calculated to be 79%-96% doublet in character. Likewise, oscillator strength for the absorption A[1 1.512.5 +X 2A,,2 derives entirely from the 2A5,2 character of the A111.512.5 state, and the ligand-field calculation predicts this to be only l%- 3%. This is far too low to explain the intensity of the A[11.5]2.5+-X2A sj2 system in absorption.

3. Heterogeneous perturbation of the &11.9]2.5 and Cp 1.911.5 states: L-uncoupling and S-uncoupling effects

If the ligand-field model is valid, the ligand-field eigenvectors should also predict the J-dependent perturbation between the B[11.9]2.5 (found to be primarily 2As,2 in the fits of Table XI) and al 1.911.5 (found to be primarily

2Jb2 in the fits of Table XI) states of NiCu. The matrix elements for this rotationally induced heterogeneous perturbation, with selection rule Ati= f 1, are

(A,S,B,R,uI -?+&R21A* l,S,Z,Iczh 1,~‘)

,-B,,,(h,S,B,fi~ C~lA~l,s,z,n~l1)[J(J+l) i

-n(a3= 1)]“2

for the L-uncoupling operator, and

(A,S,8,Q,vj --J^f?/2pR2]A,S,B* l,fi* 1,~‘)

(3.1)

-n(n*l)]“2 (3.2) for the S-uncoupling operator,25 evaluated with Hund’s case (a) basis functions. As described in the preceding paper, the interaction matrix for the heterogeneous coupling of the B[11.9]2.5 and ql1.911.5 states is of the form

i

TO,,(B) +Bd(J+ 1) kB,[J(J+ 1) -?I’” HBc=

kB,,[J(J+1)-$?1’2 To,,(C)+B”J(J+l) i ’ (3.3)

where k is a constant which combines the term-by-term interactions between the component Hund’s case (a) states of the B[l 1.912.5 and 411.911.5 states. With use of the ligand-field eigenvectors, which are linear combinations of Hund’s case (a) basis functions, the off-diagonal coupling constant between the B[11.9]2.5 and q11.911.5 states, induced by the L- and S-uncoupling operators, is calculated to be k=3.36, using either of the two least-squares fits. The corresponding constant determined from experiment is k =2.85.12 Although these are not precisely equal, the ligand-field model nevertheless predicts the coupling between these states with only 18% error.

4. The ffNi (3F)~rr*’ excited sfates of NiH

The & ( 3F)gdi” excited states of NiH are analogous to the & ( 3F) d&\$ti’ excited states of NiCu, particularly if the d&! subshell is taken to remain closed. With this in mind, the ligand-field treatment applied to this manifold in NiCu may be immediately carried over to the corresponding excited states of NiH, where considerable experimental data is available.45 Indeed, the known electronic states of the two molecules fall into the same pattern, indicating that the electronic structure of the two molecules is determined by similar factors. Table XII presents the known electronic states of the d~i( 3F)gdr;1 manifold of NiH, along with the results of a least-squares fit of the experimental data to the ligand-field model described in Sec. II E above.

Table XII presents the composition of the wave function obtained in the fit, along with the fitted energy levels. In these calculations we have not considered the L- and S-uncoupling interactions which occur in the rotating molecule, and which can be quite significant in a light molecule such as a hydride. With further work to include these in-




TABLE ,T;1I. Fitted ligand-field results for the &,( ‘F)dLir manifold of Ni1-I.

state

Energy (cm- ’ ) (Residual given in parentheses)

E.qt. Ligand Gel&’ Ligand field’ % “A

Composition of the wave function (in %I jd

411+il Jll -Ii2 4X 3) LA

c;et 8.412.5 18371 1X389(-18) 18 342(29) 0.5 17 552 17 494 q17.211.5 17 250 17 3661- 116) i7305(-55) ~~17.0]3.5 17 032 17 OOf(25j 17059(--27) liil6.3]1.5 16 27‘) 16 253(26) I6 225(54) 0.5 16 095 16 139 I3[l5.9]2.5 15925 15977(-S2) 14007(-882) 0[16.1]1.5 16091 15911(180) 15 944( 1477) A[l5.5]2.5 15461 15 504(-43) 15 526( -45) 0.5 15 454 15403 1.5 14 953 14 946 3,s 14 801 14 8X0 0.5 14 738 14 735 4.5 14 148 14 271 2.5 14 061 14 121 1.s 14OMl 14047 0.5 13 919 13 985 0.5 13 191 13 306 3.5 13 123 13 230 1.5 13 116 13 230 2.5 13 066 13 179

X2) l(O) l(1) O(O) l(l) 4(3j 2(l) O(O) S(8) 12(10) 18( 17) 17( 17) 5(5) 46(47j

4(4jE 49(48) 48(49) 31(31je 4(4j

5(5j 82(80)

53(h3)

l(l) 3(2) 4(3j O(O) 4(3j 32(31) 41(41)

O(1) lt(14)e 44(47) 46(44)

50(47jC 14( 14j 4ilo) 72(75)

78(81)r 32i32j Q7Y

15(15j 35(34) 33(34)

O(O) O(O) XlY 4(4j

88(90) 2!2)

6(5)’ 7(7j 32(32) 12(12) 2(l) l(l)

lOO(lOOj 7(7j 3(3j

31(32) 41(42) 25(26) g(8) 93(94) 30(30) 56(56)

51(53) 2(2) 18(17j 41(42)

10(S)” 3i3j 6(5je

4(3j 4(3j O(O) O(O)

4(6) l(1) 3(3j

O(OY

O(O) 54(54) 41(41)

16(15j O(O) O(OY

O(OY

=Keference 45. “Ligand-field calculation with &, fixed at 443 cm-’ (r’),d=0.3444 A’, (r4)-,d=0.3204 A4, and R=1.444 A, 2, and Gz were allowed to vary, and c(oncerged to Z,- -0.439 and $=791.94 cm”“‘.

ligand-field calculation with (i),d=O.3466 -G, <r4jjd=0.3204 .&‘, and R= 1.644 A; l&,, Z,, and G, were allowed to vary, and converged to &=614.44 crn~~‘, Z,--0.442, and G+783.41 cm-‘.

%e composition of the tinal wave fun&on is given in percent for the tit described in footnote (bj, and in parentheses for the fit described in footnote ICI.

This contribution is responsible for the oscillator strength in absorption from the ground X ‘AJi2 state.

teractions it would be possible to make predictions of the A-doublings and hctrrogenous perturbations observed in the spectrum of NiH. Nevertheless, the compositions of the wave functions obtained in Table XII compare well to the spectroscopic interpretations of the states of Kadavathu et al. 45

Beginning with the high-energy states, Kadavathu et al. find the G[lS.4]2.5-X1 ‘AVZ excitation to be very weak indeed,“5 particularly for low rotational levels which are reldively un&Ected by heterogeneous perturbations. In emission, the G[18.4]2.5-+X, 1.5 transition is relatively strong. This suggests that the G[18.4]2.5 state is primarily “Ov2 in character, which would lend intensity to emission to the X2 1.5 state, which is primarily 2Ax,2 in character. Indeed, the G[lS.4]2.5 state would only be accessible from the XI ‘AT,? state through its component of 2As,,2 character. In complete accord with these observations, the least- squares fit to the ligand-field Hamiltonian predicts the GtlS.412.5 state to be 93% ‘+)si2 and only 4% ‘Asn in character.

Likewise, Kadavathu et al. suggest that the 1[17.2]1.5 state contains nearly equal admixtures of the ‘113/2 and 2A3t;2 states, with a much lower admixture of “A,, states.“’ The ligand-field calculation predicts this state as 63% 2A_11Z and 31% ‘Iljl’ 2 in character, with very little quartet (S=3/2) character, very much in keeping with this sug- gestion.

Kadavathu et al. also assign the fl17.013.5 state as primarily “QTi2 in character;“5 the ligand-field treatment is in agreement, making this state 94% 2@,,2 in character.

The observed spectroscopic data on the a16.311.5 state of NiH led Kadavathu et al. to suggest that this state is approximately 50% 4A,,2 in character, with admixtures of ‘AJ12 and to a lesser extent 2113/2. The ligand-field treatment is somewhat in accord with this conclusion, giving the composition of the EJ16.311.5 state as 80% 49J/2, 1% 2A 312, and 14% 2113,2.

The B[15.9]2.5 state shows up as the most intense absorption from the ground X, 2A5,2 state and was observed as long ago as 1935.46 It was originally assigned as primarily “A,/, in character, and this seems now to have been confirmed by measurements of the Zeeman effect.47 Again, the ligand-field fit is in agreement, characterizing the B[15.9]2.5 stat.e as 81% 2A5,2.

Reasoning from all of the available experimental data, Kadavathu et al. suggest that the 0[16. l] 1.5 state contains mainly 2113,2 character with some admixtures of ‘A3/2 and 4A3 ‘2 , . The ligand-field lit characterizes this state as having a composition of 47% *IIJ12 and 32% 2Ag/2, with the re- mainder due to quartet states, again in reasonable agreement.

Finally, Kadavathu et al. identify the A[ 15.512.5 state as primarily quartet in character, with 4@p,,,2 as the major contributor. The ligand-field calculation is in total agree-



ment, with the A[15.5]2.5 state having 75% 4@5,2 character.

The general agreement between the experimentally deduced compositions of the wave functions and those obtained in the ligand-field fit is encouraging, and supports the validity of the ligand-field approach. Moreover, this level of agreement suggests that the results presented in Table XII will be useful in locating the remaining states of the d~i( 3F)~2dc1 manifold of NiH. On the other hand, the charge state of the hydrogen atom obtained in the fit, 2, = -0.442, is difficult to rationalize. The deperturbed14 term energies of the ‘A and 211 states of the d9,iO2 manifold of NiH correspond to an effective charge on the H atom of Z,= +0.538, and it is difficult to understand why this parameter would change to Z,= -0.442 in the d~i( 3F)ddr1 manifold, unless the d’ electron is totally localized on the hydrogen atom in the d~i(3F)~2dr1 manifold. In any case, whatever is responsible for this severe change in the effective charge of the ligand is common to both the d~i( 3F)~2dr’ states of NiH and the d~i ( 3F) d~~ddc’ states of NiCu.

In the case of the d~i(3F)d~:0,dd” manifold of NiCu and the dii( 3F) O2dr’ manifold of NiH the ligand-field Hamiltonian has been used as a fitting function to obtain an understanding of these electronic manifolds. The fitted results are robust, in the sense that they predict phenom- ena beyond those used to obtain the fit. Thus, for example, the composition of the wave functions of the dki( 3F) ddrl states in NiH are generally in agreement with that deduced from spectroscopic work. Likewise, the ligand-field model correctly predicts the relative intensities and heterogeneous coupling between states in the d~i(3F)d~:0ua2dr’ manifold of NiCu. Although the model fails in some details, it is clear that it provides a good zeroth-order description of these excited states of both NiH and NiCu. Further refine- ments to include ligand-induced configuration interaction, as was required in the treatment of LaF,7 for example, represent relatively straightforward extensions of this work, and can be incorporated readily.

This mystery of the charge of the ligand will be best understood by examining the electronic structure of the do*’ framework of the NiCu and NiH molecules through ab inifio quantum-mechanical methods. A proper treatment would then allow the three u electrons to move in the potential of a dk( 31;) Ni2+ ion in close proximity to a d& Cu’+ ion. Of course, the effects of the electrons of the cores of these ions would have to be included through pseudopotentials, but one could in principle obtain very reasonable wave functions for the ddr’ portion of the molecule in this fashion. It would then be possible to pursue a proper ligand-field treatment of the splitting among the states of the $Ni( 3F)dk\02dr’ manifold of NiCu and the d~i( ‘F)a2d”’ manifold of NiH by subjecting the d&i( 3F) core to an electrostatic perturbation derived from both the + 1 point charge of the d& core in addition to the field derived from the ddr’ framework. This generalization of the ligand-field treatment would still be far simpler than ab initio quantum chemistry, and would perhaps provide an explanation of the negative effective charge found for the hydrogen and copper ligands.

The magnitude of the ligand charge found for the excited dSNi(3F)d~~ua20*’ states of NiCu and the d~i( 3F)$dr’ states in NiH is surprising and at this point unexplained. It is suggested that a quantum-mechanical treatment of the ddr’ framework in the pseudopotential created by the d&i Ni2+ and d& Cut+ ions, followed by the use of the entire electrostatic perturbation in a ligand-field calculation, would shed light on these anomalous charges. It would also provide a means of evaluating the exchange parameter, G2, which drops significantly in the NiH and NiCu molecules relative to its value in the Nil+ ion. This is reasonable, however, given that the dc electron in the NIH and NiCu molecules is spread over two centers, while it much more localized in the Nil+ ion.

It remains to be seen to what extent the ligand-field

IV. SUMMARY

In this paper ligand-field theory has been applied to the diatomic transition-metal molecules, focusing primarily on the d9,id~~oZ manifold of NiCu, the $Ad9dg manifold of Ni2 and Cu$, and the d~i( 3F)d~~~zdr’ manifold of NiCu. In addition, comparisons to the d9NiaZ and d&i ( 3F) aZd”’ manifolds of NiH have been made. The results of the theory applied to the &Nid&\d manifold of NiCu and the d9,d9& manifold of Ni, are impressive, showing essentially quantitative agreement with ab initio quantum chemistry in a model with no free parameters. Most important is the fact that the ligand-field model provides a conceptual framework for understanding the electronic structure of these complicated molecules which does not require thousands (or even millions) of contributing electronic configurations for its application.

model can be extended to other metal systems. Its success in nickel-containing systems stems from the highly contracted nature of the d orbitals in these molecules. Given the success of the model in calculations on nickel- containing molecules, we are exploring the application to other late transition-metal compounds such as NiCu2, Ni,Cu, Ni3, CoCu, FeCu, CoNi, FeNi, NiAg, NiAu, etc. A comparison of a ligand-field calculation for CoCu and FeCu to experimental results would provide an interesting probe of the degree of d-orbital bonding in the 3d transition-metal-Cu diatomic molecules. It will also be very interesting to apply the ligand-field model to NiCu congeners such as NiAg, NiAu, NiAl, and PtCu, and to make a comparison to experimental results. In this regard we have recently determined the ground electronic states of NiAu,” NiA1,20 and PtCu (Ref. 19) to be 2A=,,2, but the optical spectra are complicated and unlike the spectra of NiCu. It seems likely that the ligand-field model may again succeed for the ground-state manifolds of these molecules, but may not be readily applicable to the excited electronic states.

ACKNOWLEDGMENTS

The authors wish to thank Professor R. W. Field for engaging and fruitful discussions on the application of the ligand-field theory to diatomic molecules. We would like to




thank Chris Gittins (MIT) for introducing us to ligand- field theory applied to diatomics when he visited the Uni- versity of Utah. We appreciate Professor Irene Shim for providing us with detailed ab initio results for Ni,. E. M. S. expresses her gratitude to the University of Utah Chemis- try Department for a scholarship. Finally, we gratefully acknowledge research support from NSF under Grant No. CHE-8912673. Acknowledgment is also made to the Do- nors of the Petroleum Research Fund, administered by the American Physical Society, for partial support of this research.

APPENDIX A: GENERATION OF WAVE FUNCTIONS FOR A dffc$& MANIFOLD OF STATES

The 31’5(u) (S=l, A=4, 2=1, fi=A+Z=5) basis wave function (which is the only 0 = 5 term deriving from a &d9& configuration), was written as a Slater determinant as

13r,,,,)=A^,ls,a(l)~~(2) I. (AlI Here g2 is the antisymmetrization operator, the subscripts A or B label the^atom, and S indicates that ml= +2. Ap- plication of the S- lowering operator to this 31’5(uj wave function then generated the wave functions of the 31’4(u) (X=0) and 31’3(u) ( Z= - 1) terms. The ‘F4(s) wave function was then obtained by forming the linear combination of the two a=4 Slater determinants that is orthogonalJo the 31’4(u) wave function. Following this, application of L- to the ‘I(,) and ‘r(s) wave functions generated the 3@(,) and ‘@ (,, wave functions corresponding to the Srr configuration. From these, the 3@c8j and ‘+(uj wave functions (again belonging to the Srr configuration) were generated by reversal of the appropriate signs. The ‘A(,, and 3AcBJ wave functions belonging to the& configuration were then generated by application of L- to the corresponding ‘a (u) and 3@‘(g) wave functions, and the ‘A(a) and 3Acu) wave functions deriving from the Sa configuration were obtained by reversal of the appropriate signs.

The 3ACUj and ‘A(,, wave functions corresponding to the 7~a- configuration were constructed in analogy to the ‘r (llj and ‘F(s) wave functions of the SS configuration by reducing the ml quantum numbers of the occupied orbitals from 2 to 1, thereby completing the construcgon of the basis wave functions with A>2. Application of L- to these 3A (“) and ‘Acg) states then yielded the 3II(u) and ‘II(s) states deriving from the 7-r~ configuration, and from these the 311(8) and ‘II,,, basis functions were constructed by suitable changes in sign.

The 3&s), 311(u), ‘IIcs,, and ‘II,,, basis functions deriving from the ST configuration were constructed by replacing ml= + 1 with ml= - 1 in the 3@cs,, 3+(,j, ‘Qcgj, and ‘Qp,,) basis functions deriving from the 8rr configuration, respectively. This then completed the construction of all of the Hund’s case (a) basis functions except for the Z states.

The a= 1 component of a 3H state derives from the a( l)a( 2) combination of spins, so the 3X&,,(0 = l(,,) and ‘X&,(fi = l,,)) states deriving from the SS configuration may be simply written as the plus and minus linear

combinations of the 16,(l)a(l) &(2)ar(2) 1 and I&( l>a( 1) aB(2)a(2) I Slater determinants, with the as- signment of the g/u symmetry determined by inspection. Here S indicates ml= -2, while 6 indicates ml= +2. Once these were determined, the 9 =0 components were generated by application of the S- lowering operator. Finally, the ‘X,,(fi = O(u)) and ‘B&,(R = 0;)) states deriving from the SS configuration were obtained by reversing the appropriate signs in the 38&(fi = O&,,) and 32& (a = O- > basis functions, respectively. The corresponding 32&;,%- ‘X;‘g,, and ‘8- (8)’ (uj basis functions deriving from the VT configuration were then obtained by replacing all ml= A2 values with ml= * 1. The last of the Z basis wave functions (deriving from the ou configuration) were then obtained by replacing the ml= f 1 values with ml=0 in the 3Z&, and ‘8&) wave functions arising from the n-r configuration. The annihilation of the wave function when ml= *l was replaced with ml=0 in the rr 32G, and ‘26, basis functions confirmed that the (T(T configuration does not generate these terms, and verified that the procedure described above was correct. The final set of basis wave functions obtained is available from the Physics Aux- iliary Publication Service21 (PAPS) of the American In- stitute of Physics or from one of the authors (M.D.M.).

APPENDIX B: GENERATION OF WAVE FUNCTIONS FOR A d”A(3F)&&c+’ MANIFOLD OF STATES

The bags set wzs derived by applying the lowering operators, L- and S- to the wave function of the 4+9/2

state, which is unambiguously written as an antisymmetrized product of the hole spin orbitals as

14~9,2)=~31~~(l)a(1)n;q(2)a(2)~*(3)a(3) I, (Bl) giving S=3/2,8= 3/2, A= 3, and sZ=9/2. By sequen- tially applying S- to this basis function the 4@7/2, 4Q5,2,

a!d 4@3/2 basis functions were generated. Application of L- to the four components of the 4@ term then generated the corresponding four components of the 4A term, and application of L- once again yielded th,e four components of the 411 term. Finally, application of L- to the 4115,2 and 4K13/2 terms gave the “XG2 and 421/2 terms, respectively. For application of the L- operator, the o* orbital was taken as an s orbital, so it could not be lowered in angular momentum. This method ensured that the basis wave functions correspond to a 3F term of the d: configuration on atom A, and are not contaminated by ‘S, 3P, ‘D, or ‘G couplings of the di electrons.

The S= l/2 terms of the di( 3F)dpgdr1 configuration were slightly more difficult to derive. These were obtained by first writing the wave function of the 21Ys,2 term which results from the di( ‘G)dfc?o*’ configuration, which is uniquely generated as

12r9,2)=Ah311SA(l)a(1)SA(2)P(2)~(3)a:(3) I. WI

The lowering operator, t-, was then applied to generate the 2+7/2 term of the di( ‘G)dk’do*’ configuration. The 2@p,,2 term of the di(3F)dgddc’ configuration was then determined as the linear combination of the three Slater



determinants which give ML=3 and MS= l/2 which is orthogonal to both the 2@7/2 term of the d:( ‘G)dk’c?d”’ configuration and the 4@7/2 term of the d:(3F)dtc?dr1 configuration, as

+C31S,(l)a(1)~~(2)a(2)dr(3)8(3) Il.

(B3) Application of ? to I 2@7,2( 3F>) then generated I 2@5,2( 3F) ), and repetitive application of L^- to these functions generated the remaining 2A, 211, and 22- basis wave functions, which correspond to pure 3F states of the dt ion embedded in the molecule. The 21 basis wave functions obtained by this procedure are available from the Physics Auxiliary Publication Service2’ (PAPS) of the American Institute of Physics or from one of the authors (M.D.M.).

‘C. W. Bauschlicher and S. R. Langhoff, Adv. Chem. Phys. 77, 103 (1990).

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(unpublished). ‘H. Schall, M. Dulick, and R. W. Field, J. Chem. Phys. 87,2898 (1987). *J. A. Gray, M. Li, and R. W. Field 92, 4651 ( 1990). 9S. F. Rice, H. Martin, and R. W. Field, J. Chem. Phys. 82, 5023

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2’Sec AIP document No. PAPS JCPSA-97-4641-3 for 3 pages of tables giving the Hund’s case (a) basis functions for a d9$&? system and a da,(3F)d~$dc’ system. Order by PAPS number and journal reference from American Institute of Physics, Physics Auxiliary Publication Ser- vice, 335 East 45th Street, New York, NY 10017. The price is $1.50 for each microfiche (60 pages) or $5.00 for photocopies of up to 30 pages, and $0.15 for each additional page over 30 pages. Airmail additional. Make checks payable to the American Institute of Physics.

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Diatomic Molecules, 2nd ed. (Van Nostrand Reinhold, New York, 1950).

24R. N. Zare, Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics (Wiley-Interscience, New York, 1988).

25H. Lefebvre-Brion and R. W. Field, Perturbations in the Spectra of Diutomic Molecules (Academic, Orlando, 1986).

26E. U. Condon and G. H. Shortley, The Theory ofAtomic Spectra (Cam- bridge University, Cambridge, 1970), pp. 169-174 and 203.

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browicz, J. Chem. Phys. 73, 2360 (1980). 33M. D. Morse, G. P. Hansen, P. R. R. Langridge-Smith, L.-S. Zheng, M.

E. Geusic, D. L. Michalopoulos, and R. E. Smalley, J. Chem. Phys. 80, 5400 (1984).

34E. M. Spain and M. D. Morse (unpublished). 35S. Gerstenkom and P. Luc, Atlas du Spectre d’Absorption de I4 Molkule

d’Iode (CNRS, Paris, 1978); S. Gerstenkom and P. Luc, Rev. Phys. Appl. 14, 791 (1979).

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(1990). 43G. A. Bishea, J. C. Pinegar, and M. D. Morse, J. Chem. Phys. 95, 5630

(1991). “S. Taylor, E. M. Spain, and M. D. Morse, J. Chem. Phys. 92, 2710

(1990). 45S. Adakkai Kadavathu, R. Scullman, R. W. Field, J. A. Gray, and M.

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