2351
J . Phys. Chem. 1986, 90, 2351-2357 In order to examine the possible effects of non-Condon scattering, we have made a hypothetical calculation using the parameters of Stallard et aL2' who found that their resonance Raman data on cytochrome c was best fit by adjusting the ratio, R, of the non-Condon to Condon scattering a t the amplitude level to the value of -0.13. We have repeated the model calculation given above using this hypothetical value in the amplitude to see what the effect of such a non-Condon term would be. Figure 3 shows the results of the perturbation of this highly structured excitation profile when this value is used. As may be seen in Figure 3, the effect is quite large. It appears that the method developed here is suitable for the calculation of resonance Raman intensities for anharmonic vibrations involving both Condon and non-Condon sources of scattering. Acknowledgment. This work was supported by Grant PCM8021 197-03 from the National Science Foundation and Grant GM15547-17 from the National Institutes of Health.
(lJV)(VJO)=
(I(V)(V(O)
[
=A A2u - 21:F2] 2112 2"v!
( 0 ) ~=) (A2"/2"v!) ~
(OIV
-
('44)
Equation A4 may now be placed in eq 7 to obtain ffl0
= (011v1)2S(w - vIQJ - i
i u120
(Olv - 1)2S(w- vlQl)] (A5)
VI21
Now in the sums over u1 which start with 1 we can replace the dummy index v1 with vi = v1 1 to obtain
+
ffl0
=
(Ollul)S(w - vlQJ - i
i U,>O
Appendix Mathematical Equivalence of the Transform Methods of BIazej and Peticolas and Tonks and Page for Harmonic Oscillation Wave Functions and Nonthermally Excited Vibrations. To show the mathematical equivalence of two approaches, we note that the sum over the Franck-Condon factor products (ll~ul)(vl~O must l ) be replaced by sums over (Ollu1)2and (Ol(vl - 1 ) 2 . To do this we use the following well-known relations for the harmonic oscillator
A @OIv)2-
u1>0
(Ollvl)S(w - u lQl - 0Jl (A61
Note that in the above equations the sums are unbounded since the harmonic oscillator has no upper limit on its quantum number. Following Tonks and Page we define the quantity @(w) to be equal to @(w)
= C ( O l l v l ) T ( w- u l Q d "I
+ ic(Ollvl)S(w- ulQl)
(A71
"I
then the resonant Raman scattering tensor has a particularly simple form
(All ('42)
Note that (Olv - 1 ) = 0 if u = 0, and the homogeneous factor exp(-A?/2) has been omitted. If eq A2 and A3 are placed in eq A l , one obtains eq A4
This is the formula used by Tonks and Page.3 Note that AI, the shift factor in the excited state, appears directly in this equation as a result of the harmonic oscillator approximation. Although the Tonks-Page method has been extended to be applicable to thermally excited vibrations, there does not appear to be any convenient way to extend the Blazej-Peticolas formalism, including anharmonicities, to include effects due to thermally excited vibrations.
Intramolecular Electron-Transfer Contributions to Bonding in the Ground and Excited States of Metal-Ligand Complexes: A Simplified Group Function Approach Pieter E. Schipper School of Chemistry, University of Sydney, N.S.W . 2006, Australia (Received: December 11, 1985)
A simplified group function (SGF) approach is applied to a description of the ground and excited states of metal-ligand systems. By making successive approximations in the electron interchange symmetry, a simple model for metal-ligand bonding is derived leading to expressions similar to (but by no means identical with) those based on simple molecular orbital (SMO) and angular overlap model (AOM) approaches. However, the physically drastic approximations inherent in SMO approaches are replaced by less severe and well-defined approximations in the electron interchange symmetry. In the perturbative limit, the various chemical contributions to the bonding and excitation energies emerge both naturally and additively and provide a clear rationale for why parametrization schemes for excitation energies have little relevance to a discussion of the ground-state bonding.
Introduction The current conceptual basis of metal-ligand bonding is rooted in either simple molecular orbital (SMO) models or ligand field (LF) Both may be used to justify the increasingly (1) Lever, A. B. P. Inorganic Electronic Spectroscopy; Elsevier: Amsterdam, 1984; 2nd ed. (2) Gerloch, M. Magnetism and Ligand-Field Analysis; Cambridge University Press: London, 1983.
0022-3654/86/2090-2351$01.50/0
popular angular overlap model (AOM)." Quantitative analyses based on such approaches rely heavily on empirical parametrization. The theoretical foundations for such semiempirical methods have been discussed on a predominantly formalistic level in terms of rigorously defined effective H a m i l t o n i a n ~ . ~ - ~ ~ ~ (3) Burdett, J. K. Molecular Shapes; Wiley: New York, 1980. (4) Jmrgensen, C. K. Modern Aspects of Ligand Field Theory; NorthHolland, Amsterdam, 1971.
0 1986 American Chemical Society
2352
The Journal of Physical Chemistry, Vol. 90, No. 11, 1986
The work of this paper approaches the problem from a different direction. Instead of manipulating the Hamiltonian, the rigorous group function (GF) m e t h ~ d ~is- progressively ~ simplified by making explicit approximations in the electron interchange symmetry, i.e. in the form of the wave functions. By retaining the full Hamiltonian, electron-electron repulsion is consistently retained, the approximations in the interchange symmetry leading, at most, to a neglect of some two-electron exchange integrals. The essential feature of the G F method is the chemically inspired division of the molecule into weakly exchanging chromophoric units (groups) each with a well-defined quota of electrons. The states of the molecular system are developed in terms of the group product functions which are, in principle, already optimized for intragroup exchange. The G F products based on the “ground state” partitioning of electrons to specific groups (which we shall refer to as predominantly excitonic (PE) configurations) may be antisymmetrized with respect to all intergroup electron interchanges. G F products corresponding to (single) electron transfer (ET) configurations may be defined by removing one electron from one group and assigning it to another. Restriction to the PE configurations leads to the crystal field (plus metal and ligand polarization corrections) description of the d-d states and the in-ligand excitations. Similarly, the ET configurations provide a description of the charge-transfer states. The central thesis of this paper is, however, that the so-called “overlap” contributions to the description of the ground state and the d-d excitation properties arise predominantly from the mixing of ET configurations with the PE states. This mixing coupled with a perturbative analysis leads to a particularly explicit resolution of the energetics of the ground and excited states into additive contributions corresponding directly to chemically based concepts such as a-bonding, a-back-bonding, and so on, as well as to a resolution into additive ligand contributions. The final expressions for the excitation energies reduce, for sufficiently simplified forms of the GF‘s, to forms similar to (but not identical with) those of the AOM but have the added advantage of having rigorously defined limits of application. The analysis of the ground-state energetics leads to bonding contributions which are quite distinct from the contributions to the d-d excitation energies and thus serves to stress that parametrization schemes based on the latter cannot be directly used to discuss bonding. Above all, the results suggest that the G F approach is ideally suited to the development of conceptual models for chemical systems without the dramatic loss in quantitative power characteristic of, for example, SMO models. In fact, we shall argue that the SMO procedures may be more readily justified in terms of simplified group function (SGF) methods (which they in fact parallel relatively closely) with a simple reinterpretation of the basis functions and the dominant integrals, than in terms of SCF/CI expansions. The SGF Method Consider two weakly interacting groups A (with N A electrons) and B (with NB electrons). In the limit of no interaction between the groups, the wave functions of the AB system have the simple product form7 I @ a b ( N ~ 3 N=~KAABIQX(NA) )) )I@(NB))
(1)
which we shall refer to as predominantly excitonic (PE) configurations. I Q i ( N A )is) s state function of the isolated A system, which is antisymmetrized with respect to all internal A-system electron interchanges, with a similar definition for I‘kk(NB)).AAB is the intergroup antisymmetrizer for all electron interchanges (Le., interchanges between A and B) such that NA and NB are 1 (i.e., separately unchanged, and K is a normalizer. If A,, (5) Freed, K. F. J . Chem. Phys. 1974, 60, 1765. (6) Freed, K. F. Chem. Phys. 1974, 3, 463. (7) McWeeny, R.; Sutcliffe, B. T. Methods of Molecular Quantum Mechanics; Academic Press: London, 1969. (8) McWeeny, R.; Klessinger, M. J . Chem. Phys. 1965, 42, 3343. (9) McWeeny, R.; Cook, D.;Hollis, P. Mol. Phys. 1967. 13. 553.
Schipper intergroup interchange is neglected), the PE configurations reduce rigorously to pure excitonic configurations. A proper description of the interacting AB system requires the consideration also of configurations in which the electrons are redistributed between A and B in all possible ways: Le., NA’ electrons in A, NE’ electrons in B, subject to the constraint that NA‘ + NB’ = NA NB. We shall only consider the cases where only one electron is effectively transferred, Le., to the configurations (@“’(NA- 1,NB 1)) = K’AA+B-l*i+(NA - l))l*K-(NB + 1 ) ) (2)
+ +
+
I@a“b”(NA 1,NB -1)) = K”AA-B+(*r-(NA
+ l))l@i(NB- I ) )
(3)
which we shall refer to as the BA electron transfer (BAET) and ABET configurations respectively. The states of the interacting AB system may therefore be written in the linear variational form7 I@’) = CCibl@ab(NA,NB)) (PE COnfignS) a,b
xC:,b,l@a’b’(NA - 1,NB a’,b’
+ 1))
x ci,,b+Da”b”(NA+ l , N B- 1))
-t
(BAET configns) (ABET configns)
a”,b”
(4) Each type of basis ( e g , PE configurations) is internally orthonormal when the strong orthogonality criterion introduced by Parr et al. is used.IO Functions representing different types of configurations are not, in general, orthogonal. However, for small overlaps (which is usually ensured by the manner in which the molecule is partitioned into groups in the first place), we may exploit the Lowdin orthogonalization procedure.” This leads to only a trivial notational generalization. For our purposes, we shall simply assume such orthogonality and note that this assumption may be removed by interpreting our final expressions as rigorously applying to a Lowdin-orthogonalized basis. The matrix elements for such configurations have been full derived by M ~ W e e n y . ’ ~ . ’ ~ However, as we shall introduce simplifications of the configuration functions themselves, and not in the form of the Hamiltonian, the evaluation of the matrix elements will be deferred to a later section. The antisymmerizer AAB of the PE configuration leads to what may be loosely referred to as purely “covalent” contributions to the AB interaction energy. Gerloch2 has briefly discussed the G F method as applicable to metal-ligand systems in which A and B may be identified with the metal and ligand chromophoric systems. He implicitly assumes that such “covalent” contributions dominate the metal-ligand bonding, with the ET configurations leading to a description of the charge-transfer states and playing only a minor secondary role in the bonding. In the SGF approach, we effectively reverse this assumption, by attempting to show that AAB, AA+B-, AA-B+ may be totally neglected, and that the metal-ligand bonding arises predominantly from the mixing of the ET configurations with the PE states. The role of metal-ligand overlap in such a model enters exclusively through the PEIET configuration mixing terms. This effective reversal of priorities may be rationalized in the following way. Any interactions between the A and the B groups arising from AAB lead predominantly to two-electron exchange integrals (in MO terminology). These may be interpreted as the mutual electrostatic repulsion of two small overlap densities, and such integrals are known to be small. (Similar arguments apply to the effect of the other antisymmetrizers.) In any case, such interactions lead to no large effective electron “drift” between A and B, as NA and NB are unchanged. However, the mixing of the ET configurations into a PE configuration (e.g., the PE ”ground” state) leads directly to both one- and two-electron in(10) 1106. (1 1) (12) ( I 3)
Parr, R. G.;Ellison, F. 0.;Lykos, P. G . J . Chem. Phys. 1956, 24, Lowdin, P. 0. J . Chem. Phys. 1951, 18, 365. McWeeny, R. Proc. R . Sor. London, Ser. A 1959, 253, 242. McWeeny, R. Rei.. Mod. Phys. 1960, 32, 335.
The Journal of Physical Chemistry, Vol. 90, No. 11, 1986 2353
Bonding in Metal-Ligand Complexes teraction terms which are directly responsible for describing the shift of a fraction of an electron between A and B, that fraction (and the direction of transfer) being directly determined by the weighting of each relevant E T configuration. Such contributions may be referred to as dative in that they arise from the mixing of the ionic configurations (relative to the initial P E partitioning), and we shall give a more accurate interpretation later which highlights their dependence on metal-ligand overlap. A full justification of these arguments requires a more detailed development of the model, which we turn to now. Applications to Metal-Ligand Bonding In the analysis that follows, we shall explicitly consider the first transition series, but the partitioning scheme is readily generalized to other metal complexes, including those of both d- and f-block elements. The inner-metal chromophore M is defined as described by the 3d" free-ion configurations built from only the five 3d orbitals and an effective core potential from the lower shells. The atomic orbitals (4s, 4p, ...) of higher energy, which may be considered as vacant in the P E configurations defined later, are considered, if occupied (as in the ET configurations) as constituting an outer-metal chromophore m. Finally, the ligand system is taken to constitute a single chromophoric ligand system L, containing all the electrons initially associated with the free ligands. W e define the P E configuration functions as those in which the NM metal electrons are wholly constrained to the inner-metal chromophore, and the N L ligand electrons to the ligand chromophoric system, i.e,, have the form
I~P)
= IM")IL")
(5)
where a indexes the d" states, and Y the ligand states. The outer-metal states are unoccupied in these configurations, and thus there is no m-configuration function to consider. Note that antisymmetrization with respect to M-L interchanges is rigorously neglected in the above definition, but intragroup antisymmetrization is implied. In order to develop the form of the dominant ET configurations, we consider first the inner-metal states arising from removal or addition of a single electron. For a one-electron removal, the M+ chromophore is represented by the states corresponding to the d"' configuration, viz. IM?')
= la+)
(6)
where a+ is an index running over the various states of the M+ chromophore. Similarly, for a one-electron addition, the states corresponding to the 3d"+' configuration may be written IMF) = la-)
(7)
We shall not define these states in detail at this stage, except to state that, in order to obtain a tractable analysis, we assume that the same set of basis orbitals (e.g. the 3d free-ion basis) is used in the definition of all three manifolds la),la+), and la-). The ligand ionic chromophores L+, L- may be similarly described by the states Finally, the participation of the outer-metal orbitals proceeds through the definition of a one-electron m- chromophore with states [ m y ) = Iw-)
(9)
where Iw-) is a singly occupied outer metal state, i.e., runs over (4s, 4p, ...). The E T configurations of interest to this work involving only one-electron transfers may then be defined as the nonexchanging product states Ia+,w--,v) Ja,w--,v+)
IM"+)lrn_")lL")
(MmET)
(10)
= IM")lmY)LY$)
(LmET)
(11)
)a-,v+)
I
lMY)lLl;+)
(LMET)
(13)
Thus four types of ET configurations emerge: the intrametal (MmET), the ligand-outer-metal (LMET), the inner-metal-ligand (MLET), and the ligand-inner-metal (LMET) configurations. The totality of the PE, E T configurations of eq 5 and 10-13 serves as a basis for the variational determination of the states of the metal-ligand system with respect to the "exact" Hamiltonian of the NM+NL- electron system. This Hamiltonian has the explicit form
where k ( ~is) the kinetic energy operator of electron K , u ~ ( Kthe ) potential energy operator describing the interaction of electron K with the totality of effective core charges on both the metal and ligand systems, u(K,K') the electron-electron repulsion of electrons K,K', and V, the totality of core-core interactions. Spin-orbit coupling terms may be formally absorbed into a redefinition of V, if required, but we shall not consider them explicitly here. It is important to note that we shall define all the matrix elements with respect to this total Hamiltonian at all stages of our analysis, all approximations being defined explicitly with reference to the electron interchange symmetry of the basis configurations, and with respect to the functions in terms of which the configurations are defined. As each configuration is separately an (NM+NL) electron function, there is never any need to neglect, say, electron-electron repuslion, which is a major drawback of one-electron S M O procedures. Approximations in interchange symmetry lead to a neglect of certain (well-defined) two-electron integrals, but always retain all coulombic and exchange electron repulsion terms within and coulombic repulsion terms between the separate chromophoric systems thus generated. The variational determination of the metal-ligand states 107) with respect to the Hamiltonian of eq 14 proceeds through the determination of the coefficients CY in the linear expansion
I@?)= c Y ( a , ~ ) l a , ~ ) (PE)
(15)
+ cY(a+,w-,v)Ia+,w-,v) (MmET) + c~(a,w--,v+)Ia,w-,~+) (LmET) + cY(a-,v+)la-,v+) (LMET)
(17)
+ cY(a+,v-)la+,v-)
(19)
(16)
(MLET)
Summations over all indices (except y) are implied. The strong orthogonality assumption ensures orthonormality of configurations of a given type but configurations of different types may have some overlap. We may, however, assume orthogonality of all configurations and include the effect of overlap later by simply interpreting all results as referring to the Lowdin-orthogonalized basis. To effect the variational analysis, the various matrix elements of the Hamiltonian must be determined. W e desist from their evaluation at this stage (as we shall simplify our configuration functions further) and simply develop the notation P(a+,w-,v) =
(a+,W-,YIHJa+,W-,Y),
H(a,v;a+,w-,v) = (a,vlHla+,w-,Y)
The products are explicitly ordered (electron 1, ..., electron NM + NL)in both bras and kets, although the intragroup order is immaterial because of intragroup antisymmetrization. The matrix elements fall into three classes. The diagonal terms represent the total energy of the particular configuration, comprising the energies of the individual groups in their indexed states together with their mutual electrostatic interactions (including electron-electron repulsion). For the "ground" state (0,O)P E configuration, for example, E'(0,O) describes the crystal field limit. The off-diagonal terms between configurations of the same type may be considered as polarization "corrections" and correspond to the inductive and dispersive interactions between the groups. (Although the latter terms are usually reserved for intermolecular forces in which the
2354
The Journal of Physical Chemistry, Vol. 90, No. 11, 19616
additional simplification of a multipolar form for the coulombic interaction operator is used, they are used here in the more general sense where the integrals are calculated “exactly”; Le., without the additional multipole approximation.) The third class of matrix elements, and perhaps the most significant in our analysis, is the set of elements between configurations of different types, particularly those between the PE and E T configurations. These integrals will be referred to as transfer integrals and we will attempt to show that they play a role similar to the “resonance’! integrals of SMO procedures. Only certain of the transfer integrals will have a significant value, as we shall see later. To illustrate most clearly the role of the transfer integrals, we consider the particular case when the diagonal elements of the PE manifold are energetically remote from those of the ET manifolds, Le., when the transfer integrals are small compared to the separation of the P E j E T manifolds. This may be expected whenever the d-d excitations are well-separated from the charge-transfer excitations in the absorption spectrum. In this case, we shall consider only those PE states corresponding to d-d excitations, with the ligands in the ground state. (The other PE states will describe the in-ligand excitations at much higher energy, whereas the predominantly E T solutions will yield the chargetransfer manifold.) We may then use the perturbative limit of the variational problem. (It is emphasized that whenever these approximations are not valid, such as in complexes with d-d transitions having a large amount of charge transfer character, we need simply return to the full variational problem. Thus although we shall restrict consideration in the remainder of the paper to the perturbative limit, the simplifications in the matrix elements apply with equal validity to their appearance in the variational expressions.) In addition, for notational simplicity, we take the inner-metal states as preadapted to the crystal field limit, i.e., prediagonalized in the v = 0 P E basis. In this case, the metal-ligand states of predominantly d-d character may be conveniently developed from perturbation theory in the following form: I@(M”Lo))
Ia,O)
(CF)
(20)
- H(a,O;a’,v)[E?(a’,u) - E0(a,O)]-’Ia’,u)(Poln)
(21)
- H(a,O;a+,w-,v) x [E?(a+,w-,u) - Eo(a,O)]-’la+,o-,u)(MmET) (22) - H(a,O;a’,w-,v+) x [E?(a’,w-,u+) - ~ ( a , O ) ] - l ( a ’ , w - , u + )
(LmET) (23)
- H ( a , O ; a - , ~ f )X - E“(~r,o)]-’lc~-,~+) (LMET) (24)
[Eo(.-,.+) - H(a,O;a+,v-)
x
[EO(a+,p)- E?(c~,O)]-’la+,v-)
(MLET) (25)
with implicit summations over all a’, u, a+, a-, u+, u--, and w--. The perturbative form resolves the wave function into separable and additive contributions, each of which carries its own characteristic chemical interpretation as a well-defined contribution to the energy and thereby parallels our chemically intuitive goal of classifying the various contributions to the bonding. The corresponding expression for the energy (which in this formalism follows trivially from the form of the perturbed wave function) is E(M”Lo)
N
Eo(a,O)
(CF)
- (H(a,0;a’,u)J2[Eo(a’,u) - EO(a,O)]-’
(26)
(Poln)
(27)
- IH(cr,O;a+,o-,~)l*[E~(a+,w-,u) - E?(a,O)]-’
(MmET) (28)
- IH(a,0;a’,w-,v+)J2[@(a’,w-,u+) - E‘(a,O)]-I
(LmET) (29)
- (H((U,O;CU-,V+)~~[EO(~-,U+) - Eo(a,0)]-’ (LMET) (30) - IH(~,O;~+,~-)~’[EO(~+,V-) - E O ( ~ , O ) ] - ’ (MLET) (31)
Schipper with implicit summations over all indices except a. The contributions to the energy of a particular perturbed inner-metal excited state (or the ground state if a = 0) may then be interpreted in the following general way. The first term corresponds to the crystal field limit. The contribution of eq 27 describes inductive and dispersive interactions. Strictly speaking, the MmET contribution of eq 28, though formalistically introduced in terms of PE/ET mixing, may be absorbed into the preceding polarization term as the m manifold still derives from the metal orbitals and the In+)Iw-)state may be seen as a metal excitation. The remaining contributions are, however, of unequivocal PEjET mixing origin. Equation 29 describes the ligand to outer-metal E T contribution to the energy and encapsulates the role of ligand-(4s, 4p, ...) metal ”overlap”. This can be of either u- or *-donating character. Equation 30 describes the analogous ligand-inner-metal (ligand4 electron) ET contributions arising from electron-donating ligands. Equation 3 1 represents the innermetal-ligand ET contributions and may be expected to be important for, say, *-acceptor ligands (Le., “back-bonding”). The transfer integrals and the energy denominators determine directly the weighting with which the ET configurations contribute to the wave function of the perturbed 3d“ manifold. For example, the LMET configurations appear with a weighting wLMET
= ~H((U,O;CU-,V+)~~[EO(CY-,V+) - E?(CI,O)]-~ (32)
leading to an effective transfer of a fraction wLMET of an electron from the ligand to the metal (3d) system. The chemical goal of describing the redistribution on bonding is therefore realized in terms of well-defined, individual, and additive contributions. The role of the ET configurations in stabilizing the ground state arises not because of the intrinsic ionic contributions to the energies of the ET configurations themselves. They are usually of higher energy than the ground-state PE structures (for this determines our original group partitioning), which manifests experimentally in the higher transition energies of the charge-transfer states. The stabilization arises from the PEjET mixing terms, which may be considered as interaction terms describing “exchange” transition densities interacting with the cores (one-electron terms) and with each other (two-electron terms). The transfer integrals are therefore characteristic of the change in going from the PE to the ET configuration, not of the properties of individual configurations. It is for this reason that their contribution to the bonding is not strictly ionic (or covalent) in the conventional sense but corresponds most closely to conventional dative bonding contributions. This interpretation may be considered the “electron-transfer excitation” analogue of conventional exciton coupling. The latter describes the inductive and dispersive stabilization between two separated atoms or groups, in which higher energy excitonic states are mixed into the ground state. The stabilization in that case arises from ”exciton”-transfer integrals in which transition moments (again characteristic of the change from the ground to the excited states) of the individual atoms are coupled through photon exchange. The above expressions encapsulate the general features of the SGF model applied to metal-ligand states. In the remainder of the paper, we consider further simplifications in the configuration functions in order to highlight the practical and conceptual utility of the approach. It is interesting to note that further simplification proceeds in a manner consistent with our basic SGF philosophy, Le., by successively subdividing each group into smaller groups and thereby neglecting further interchange symmetry at each stage. A Localized Ligand Analysis We introduce now a simplification of the ligand states in order to obtain additive contributions from individual ligands, based on the chemically reasonably assumption that direct ligand-ligand overlap is usually negligible. This may be formally incorporated into our analysis by a further subdivision of the ligand GF‘s into nonexchanging products of the individual ligand states. Each individual ligand (denoted 1) is considered as having, in isolation, doubly occupied valence orbitals (Ii), where i indexes the type of orbital (Le., whether it is of u- or a-symmetry with
The Journal of Physical Chemistry, Vol. 90, No. 11, 1986 2355
Bonding in Metal-Ligand Complexes respect to the local metal-ligand axis). For ease of notation, we consider a t most, one of each type (per ligand), generalization proceeding through a trivial augmentation of notation. Similarly, the lowest-lying excited ligand states of that symmetry are denoted by ll*. The ligand ground state is developed in terms of the two-electron individual ligand GF’s
MI,))=
(33)
11,)FI)
with the bar denoting spin B in the usual way. The state !Lo) may then be approximated as the group product ILO) =
rIlg(1,))
(34)
I,!
where the product spans only the occupied group states, and antisymmetrization between the different g(l,) (i.e., for different i or 1 or both) is neglected. The possible excited states of the neutral and charged (relative to 11O)) ligands may then be written in the form lL(ll-4J*)) = ~ L o ) ~ l l ) - ~ = ~ llll-lJ*) J*)
The transfer integrals may also be simplified. This simplification is somewhat more subtle and warrants some discussion. Whenever the tranfer integral can be written in the form (X,YIHIX,Z) where 1X,Y) = JX)IY),I X J ) = IX)lZ) than the value of the transfer integral will be dominated by the interaction of the YZ exchange (“overlap”) density with the totality of core charges and the static electronic density of the X configuration (group). As the latter electronic density is generally diffuse, and the interaction with the core charges will be dominated by contributions from the closest cores, it is possible to write (with Hxx an “effective” Hamiltonian incorporating the effects of the static field due to the group X) (X,YIHlX,Z) = (YIffXXlZ)
(YlHlZ)
With the above simplifications, the perturbation expansion for the predominantly d - d states reduces to the form ~(M*Lo)N )
la,o)
(CF)
(35)
- H(a,l,;a,l*J)[Z(l,~lJ*)]-’~a,l,~lJ*) (Poln)
fl,*-)
(36)
- H(a;a+,w-)[Z(a+,a) E(w)]-’Ia+,w-,O)
IL+O+(lJ) = lLo)ll,)-1 = Ill+)
(37)
lL-O-(l,*)) = ILO)F,*) _=
The neutral basis of eq 35 is that leading to the usual exciton analysis of the in-ligand state^.'^,'^ The charged basis of eq 36 results from adding an electron to a vacant ligand state, whereas eq 37 describes the total removal of an electron. The individual ligand basis effectively neglects exchange integrals between the group functions, allowing the charged basis to be defined in terms of removal or addition of an electron of arbitrarily defined spin. We shall conventionally consider electrons removed from the ligand system to be of a spin, whereas those added to the ligand system (Le., removed from the inner-metal chromophore) to be of B spin. Equations 34-37 therefore fully define our simplified ligand basis, which will in turn significantly reduce the number of E T and PE configurations. Some physical assumptions regarding the energy denominators may now also be introduced, principally to highlight the nature of the approximations that lead to maximum transferability in comparing different metal-ligand systems. Such assumptions are additional to those inherent in the further group resolution of the configuration functions and may be relaxed if deemed inappropriate at any stage in the analysis by remembering that the energy denominators are rigorously defined in terms of the configuration energies (E‘) of the states involved in that particular perturbation term. For example, @(a-,v+) describes the energy of the unperturbed la-,v+) configuration, comprising that of the isolated inner-metal “anion” and ligand “cation”, plus their mutual electrostatic energy. Similarly, @(cr,O) describes that of the unperturbed Ia,O) configuration. We shall approximate the energy denominator arising from the mixing of these configurations as EO(a-,l,+)
- EO(a,O) N E(a-,a) + Z(1,)
(38) where E(a-,a) is a (usually negative) electron-affinity-type term describing the difference in the energy of the relevant d”’], d” configurations, both for simplicity calculated in the ground-state ligand crystal field, and Z(1,) is the free-ligand ionization potential (with the static perturbation of the metal conveniently neglected). Z(a+,a) defines the analogous inner-metal ionization energy (i.e., the difference in the “cationic” and “neutral” inner metal configurations in the appropriate states. I( l,+lJ*) is the free-ligand transition energy, and E ( w ) is the one-electron energy of the free-outer-metal state. The latter can of course only be realistically defined by including the electrostatic contributions from (at least) the inner-metal chromophore. Finally, E( l,*) is the isolated-ligand electron affinity. With these approximations, some degree of parametrization may be introduced, with the limitations succinctly defined by the nature of the approximations discussed above. (14) Mason, S. F. Inorg. Chim. Acta Rev. 1968, 2, 89. (15) Bosnich, B. J . A m . Chem. Soc. 1968, 90, 627.
(39)
+
(40) (41) (MmET) (42)
+ E(w)]-’(a,w-,l,+) (LmET) (43) - H(~,l,+;a-)[Z(I,) + E(c~-,~~)]-’la-,l,+) (LMET) (44) - H(C~;O+,I~*-)[Z(C~+,CY) + E(l,*)]-’Jc~+,l,*-) (MLET) - H(ll+;w-)[Z(ll)
(45) with implicit summations over the a-, a+, w--, I,, and lJ*. The corresponding expression for the energy follows trivially by inspection. Equations 40-45 and corresponding energies are fully partitioned into additive ligand contributions and represent the simplest form for these quantities that can be obtained by simplifying the ligand system alone. The neglect of direct ligand-ligand exchange is generally consistent with the relatively large separation of the ligands. Thus the above simplifications (apart from perhaps an oversimplication of some of the energy denominators, which is readily reversed) should not drastically affect the quantitative applicability of the model. I t is also for this reason that the inner-metal states have been retained in their unsimplified form, as the role of exchange within the d manifold is seldom negligible. However, we shall now turn to a further simplification of the inner-metal manifold in the spirit of reducing the S G F approach to its simplest form, which can be compared with SMO and AOM approaches. The analysis of the following section may therefore be considered as quantitatively oversimplistic, but as a conceptual basis for the more quantitative approach, it should serve well. In leading to results commensurate with those of S M O procedures, it also serves to highlight the exact nature of the approximations implicit in such treatments. A Localized Metal Analysis A one-electron SMO analysis effectively determins the optimal one-electron states and sets up the various configurations in an aufbau-type manner. Spin plays only a secondary role in limiting the MO occupancy, and spin degeneracies remain unresolved. Thus exchange effects are to some degree ignored, as is electron-electron repulsion. The roughly parallel treatment in the SGF approach is not quite as restrictive, except that the spin resolution and exchange will be ignored, but in a somewhat different manner. In a way, the resulting analysis gives some insight as to why the SMO approach works, as the final treatment leads to expressions mimicking those of the SMO procedures, but with a significant reinterpretation of the basis states. The analysis starts with the set of 10 spin orbitals {ld,(s,,))] where s, denotes spin a or B. The unpaired electrons may be taken to be of spin j3, as is any electron removed from M in the definition of ET configurations. By neglecting exchange between the d electrons, we are simply subdividing the inner-metal chromophore
2356
The Journal of Physical Chemistry, Vol. 90, No. I I I986
Schipper
~
to the extreme situation, where each spin orbital is a separate group, and neglecting (as we have done consistently throughout this work) intergroup exchange. The ground state of the d” configuration then has the product form IMo) =
n Id&))
(46)
K,S#
(WC)
where the product spans only the occupied orbitals. The individual d, functions may be expressed as linear combinations of the real d orbitals adapted to the metal-ligand symmetry to facilitate the analysis. The excited d” configurations based on a one-electron excitation d,* (with the asterisk denoting that the relevant orbital from d, is either singly occupied or unoccupied in the ground state) may be written in the form
-
IF+P’*)
= IMo)ld,(s,) )-‘ld,,t*(s,,,))
(47)
Two types of charged configurations may be defined. For an electron transfer from or to the ground-state configuration, we have
but+)= lM0)ld>(s,))-’
(48)
IF*-)
(49)
= IMo)ld,*(s,))
where the dagger implies that the relevant orbital is either singly or doubly occupied in the ground state. The second type of charged configuration describes an electron transfer from or to an excited d“ configuration Ip+p’*); viz.
Ip-w’*,vt+) = IP+F’*)
Id:(s,,)
)-I
(50)
IP+P’*,v*-) = I~+d*)ldg~*(S70) (51) where the vt, v’* refer to the previously defined occupancies but in the excited Ip-+p’*) configuration. If we now use the approximation of eq 39 for the matrix elements, they will reduce (except for the polarization term which we retain in the original form) to simple one-electron integrals. In fact, their form will then be analogous to the definition of the resonance integral in S M O approaches, a point we return to later. Defining = (a(l)lHlb(l)) and assuming our configurations have been preoptimized to the crystal field give the perturbed ground state which has the form I@(MoLo)) N l0,O)
(CF)
(52)
- H(O,~l;O,~J*)[Z(~l-~,*)]~l~O,~l+~J*) (Poln)
- h(d?,w)[Z({’)
+ E(~)]-ll{~+,w-,O)
- h(l,,w)[Z(ll) + E(w)]-’lO,w-,l,+)
(MmET)
(LmET)
- h ( d ~ * J J [ ~ ( l+J E({*)l-11P-311+) (LMET) - h(df,l,*)[Z({’)
+ E(lJ*)]-’I{++,lJ*-)
(MLET)
(53) (54) (55)
(56) (57)
where summations are implied over all l,, lJ*, w, d?, and d,* (the t and * defined relative to the ground-state occupancy). The wave functions for the excited d configurations have the form IQ(M,-,l*Lo))
- H(p+p’*
N
lp+p’*)
,ll;p+p’*
,IJ*) [Z(ll+lJ *) ] -1 (cC-p’* ,l1-4)* )
(58)
(59)
- h(d,t,w)[z(qt) + E(w)l-’l(c~-p’*,v~+),w-,o)
(60)
- h ( l l , ~ ) [ N l+ ) E(w)I-’lpu-~’*,w-,1,+) - h(d,*Jl) + E(v*)I-lI(p+Cc’*,v*-),l,+)
(61)
- h(d,,+J,*)[z(vt) + E(lJ*)l-lt(p+p’*,?t+),lJ*-)
(63)
[ W I )
(62)
with implicit summations over all I,, lJ*, w, qt, and v* (the and * defined relative to the occupancy of the lpr+p’*) configuration)
Again, the expressions for the energy follow directly by inspection. We shall return to a discussion of these results in the next section. This section concludes with the definition of the relationship of our method with those of S M O models. The LmET contribution to the ground state (eq 55 and 61) serves as an example. If we consider just one outer-metal state (e.g., the 4s orbital), the term of eq 5 5 may be expliclty written in the form
where ,tIHand aH denote the relevant Hiickel integrals, then the above correction would arise in an SMO scheme from orbital mixingI6 of the individual ligand orbital with the outer-metal orbital. This formalistic connection arises because we have efficiently partitioned out a single electron group on both the ligand and the metal. Although in our simplfied form of eq 64 we have chosen to ignore electron-electron repulsion terms (through the energy denominator simplification and the approximation of the transfer integrals as one-electron integrals), tbe expressions may be readily used in their fuller form. This formalistic connection is even more interesting if we consider the ll+) states themselves. Again restricting the analysis to the ligand-outer-metal states for convenience, we may develop the one-electron S M O states for the ”cationic” ligand states (which would directly describe the unperturbed charge-transfer states) in the usual form = Ccl,Ili)
+ CAW)
(66)
1,
with the coefficients determined by a variational calculation in the subspace of these one-electron outer-metal and ligand states using the integrals of eq 25. It is interesting to note that the Hiickel solution presumes complete delocalization of the electrons in the relevant MO. We can tackle the same problem in the G F context by writing
where the lu-)lli+) may be considered as basis solutions in which a positively charged hole resides on the relevant ligand, and the ILo) basis solution has the hole in the outer-metal. A separate variational problem in the subspace of these solutions may be readily shown (see, for example, a discussion of the analogous procedure in a discussion of through-bond T , U coupling’’) to reduce, using the integrals of eq 65, to an eigenvalue problem isomorphic with that of the Hiickel calculation of eq 66. We may say, in fact, that the Huckel problem is the limiting case of the simplified GF analysis, providing a rationale for why such S M O procedures are difficult to strictly “derive” from a more sophisticated M O analysis. The only M O aspect of Hiickel theory is the form of the basis of eq 66. In fact, we may say that, strictly speaking, the Hiickel calculation refers to the configurational or “hole” basis of eq 67, which has been rigorously shown to lead to the Hiickel determinantal problem. As the GF scheme delocalizes the ligand “hole” through the outer-metal orbital participation (the ligand-ligand overlap or transfer integrals being negligible) by local one-electron transfers, this isomorphism suggests that the complete delocalization picture inherent in SMO approaches applies to the hole (the “charge excitation”), and not to all the electrons. This is a satisfying result, for it suggests that electrons can be largely identified with particular atoms and bond, and that electron transfer may be considered as occurring through local, nearest-neighbor transfers. A delocalized transfer may therefore be considered as a succession of such local transfers. This is in closer accord with intuitive chemical thinking and (16) Inagaki, S.;Fujimoto, H.; Fukui, K. J . Am. Chem. SOC.1976, 98, 4054. ( 1 7 ) Schipper, P. E. Inr. Rev. Phys. Chem., in press.
The Journal of Physical Chemistry, Vo1. 90, No. 11, 1986 2351
Bonding in Metal-Ligand Complexes suggests that the “total electron delocalization” interpretation characteristic of S M O procedures results, in fact, from a spurious interpretation of the basis functions. Another general feature arises from the above analysis. If certain E T configurations interact strongly, it is always possible to prediagonalize the Hamiltonian in the subspace of these configurations and to use the preoptimized solutions as a basis in, for example, a perturbational calculation of the ground state. In a sense, we have implicitly used such a preoptimization for the C F solutions in setting up the contributions of eq 40, 52, and 58 in order to avoid the notational complexity of remixing the Ia,O) basis with other Ia’,O) states. Usually, if the I q O ) basis is already adapted to the symmetry of the ligand field, then the symmetry resolution is sufficient to define a unique CF basis. In the more general case, we may start with any convenient basis (e.g., the configuration functions defined in terms of the real d orbitals) and prediagonalize the la,O) PE configurations in the crystal field. The earlier expressions then refer to the relabeled, prediagonalized manifold.
d-d Excitation Energies The results of eq 52-63 and the corresponding expressions for the energies may be directly used to determine the d-d transition energy. An inner-metal d-d transition in our analysis corresponds to the transition
-
I@(MoLo))
-
@(M,+*,L0))
(68)
i.e., an effective one-electron jump d, d,,*. The polarization terms may be neglected, and the LmET terms rigorously cancel with the energy denominator approximations introduced earlier. In addition, may other terms cancel due to the relevant summations over the d manifold, a point we return to anon. The perturbed transition energy reduces to
Wrc--Cc’*)= ICF(W‘P’*)
(69)
- {Ih(d,!*,m)I’[W*) + E(w)l-’ - Ih(d,,w)I’[Z(p) + E(~)l-’l
going from the initial to the final state. Thus, in general, parametrization schemes based on transition energies cannot be used to extract information on the ground-state energy (Le., the bonding), which is fully described by the energies corresponding to the wave function of eq 52-57. The above form for the transition energy is similar to (but not identical with) that used in the justification of AOM parametrization ~ c h e m e s .This ~ can be made clearer if each transfer integral is approximated by the Wolfsberg-Helmholz formI8 HI,
=
Y 2 ~ ~ l ,+~ H,,) ~ l l
(73)
and with the overlap integrals expanded as products of the radial and angular factors in the usual way.4 If this approximation is used, however, we must also use a rigorously orthogonalized basis using the Lowdin procedure, because quadratic overlap factors will also emerge from the diagonal matrix elements. We prefer to retain the more fundamental transfer integral form, as they have a relatively simple interpretation. Consider, for example, the LM transfer integral h(ll,dp). This has the one-electron contribution (11(1)lUC(1)ld,(l) )
(74)
where vc is the overall core potential. This describes the interaction of the exchange density +,,(l)qd,(1) with the core charges. The integrated exchange density is simply the overlap integral. If the exchange density is remote from the cores, the transfer integral may be approximated by the interaction of the electronic charge X overlap integral with the closest core charges, a rationale for the Wolfsberg-Helmholz form. However, as pointed out elsewhere,” finite transfer integrals may be obtained even for rigorously orthogonal orbitals if the exchange density is asymmetrically distributed with respect to the core charges. The transfer integrals may be interpreted as conductive links, which serve to mix the P E and ET configurations by mechanistically allowing the electron transfer.
Conclusions If has been shown that the SGF method serves as a sound basis for a discussion of metal-ligand bonding. With judicious choice - ~ l ~ ~ ~ , +J E(dM)I-l l ~ 1 -2 I~(d,~*JJI’[~(lJ ~ ~ ~ ~ l ~+ ~ ( d M ~ * ) 1 - 9 of group partitioning and a series of rigorously defined approx(71) imations in electron interchange symmetry, a simple analytic form for the wave functions and energies in the perturbative limit is - {lh(d,t*,l,*)12[Z(p’*) + E(l,*)]-’ - h(d,,1J*)12 X obtained. Although this paper has been principally concerned with [I(cL) + ~(1,*)1-9(72) the energetics of the predominantly d-d manifold, the explicit form of the wave functions serve as a suitable starting point for a with the implicit summations restricted to l,, lJ*, and w . These discussion of d-d intensities, rotatory strengths, and other response contributions arise from crystal field (eq 69),MmET (eq 70), and properties. In addition, it serves as a basis for a unified approach LMET and MLET (eq 71 and 72) perturbations, respectively. to a concurrent treatment of charge transfer and in-ligand states The transition energy is dominated by the different value of the (which in principle emerge as the residual solutions of the full transfer integrals for the relevant d orbitals involved in the variational problem) and encounters no formal difficulties for transition, the energy denominators being less sensitive. strongly mixed d-d/charge-transfer transitions. Studies of these Most of the terms corresponding to contributions to the absolute further applications are in progress. bonding energy cancel in the determination of the transition energy, as we might expect. The transition energy, after all, (18) Wolfsberg, M.; Helmholtz, L.J . Chem. Phys. 1952, 20, 837. monitors only those aspects of the bonding that are changed in (70)
