Inorganic Chemistry

177 Pages·1981·3.011 MB·English

by  Malcolm Gerloch, John H. Harding, R. Guy Woolley (auth.)

Table of Contents
The Context and Application of Ligand Field Theory
M. Gerloch, J. H. Harding, G. Wooley ................
Structure and Bonding of Transition Metal-Sulfur Dioxide Complexes
R.R. Ryan, G.J. Kubas, D. C. Moody, P. G. Eller .......... 74
Non-Commensurate (Misfit) Layer Structures
E. Makovicky, B. G. Hyde ....................... 101
Author-Index Volumes 1-46 ....................... 171
The Context dna Application of
Ligand Field Theory
Malcolm Gerloch ,1 John H. Harding I and R. Guy Woolley 2
1 University Chemical Laboratories, Lensfield Road, Cambridge CB 2 1 EW, England
2 Cavendish Laboratory, Madingley Road, Cambridge CB 30HE, England
A formal development of ligand field theory from the elements of quantum chemistry is traced to
provide an explicit context for its procedures and practice. The nature of the assumptions and
limitations of ligand field theory in general is described and the significance of interelectron repul-
sion and spin-orbit coupling parameters within its application is discussed. Finally, a detailed justifi-
cation for the angular overlap model is presented, the parameters of which are shown to behave in
most respects as first postulated ad hoc. The discussion includes an interpretation of the phenome-
non of "d-s" mixing, recently invoked in studies of planar-co-ordinated complexes.
1 Introduction ..................................... 2
2 Effective Hamiltonian Theory ............................ 9
3 Group Product Functions and the Basis Orbitals ................... 01
4 The Primitive Ligand Field Parameterization ..................... 31
5 Parameter Renormalization ............................. 61
6 The Angular Overlap Model ............................. 32
6.1 The AOM as a Parameterisation Scheme ....................... 32
6.2 The Ligand Field Hamiltonian ............................ 27
6.3 The Physico-Chemical Interpretation ......................... 30
6.3.1 The Static Contribution ................................ 30
6.3.2 The Dynamic Contribution .............................. 38
6.3.3 The Potential in Empty Cells ............................. 39
7 Summary ....................................... 14
8 References ...................................... 54
2 .M Gerloch et .la
1 Introduction
After a history of some 50 years, ligand field theory remains a vital procedure in the study
and interpretation of the spectral and magnetic properties of transition metal and lan-
thanide complexes. Yet the nature of the many parameters spawned by ligand field
models often appears unclear since the relationship between the theory and underlying
fundamental principles is seldom apparent. Our purpose in this review is to investigate
this relationship and so provide a viewpoint from which ligand field theory may be seen
to emerge naturally from the main body of quantum chemistry. In so doing, we hope to
define the basic assumptions of the theory more clearly than hitherto, to interpret the
parameters employed, and to discover the bounds of applicability of the model. Our
attention throughout is confined to the spectral and magnetic properties of single-centre
systems.
Interpretation of the phenomenology of the subject has been enormously successful
even though the chronology of achievement has reflected the late arrival of sufficiently
sensitive spectrophotometers: historically, magnetism has provided the path by which
ligand field theory was established. One assumption above all gave rise, via the early
exploitation of the techniques of group theory, to the various qualitative generalities
which established the success of ligand field theory. It is that the electronic levels which
determine the spectral and magnetic properties of metal complexes derive from a simple
atomic-like d n configuration in the transition metal block, or ff in the lanthanides. It is
interesting to speculate whether such an assumption would have been entertained so
readily had the subject begun some years later. However, predictions of the spin and
orbital degeneracy of the ground states of complexes classified according to their
geometry and d configuration, as evidenced by their bulk paramagnetism and anisotropy,
were totally successful. In later years, the identification of broad bands in d-d spectra as
spin-allowed transitions confirmed electronic spectroscopy as the other major experi-
mental technique by which the qualitative (group theoretical) predictions of ligand field
theory could be verified. Ballhausen )1 has recently reviewed these pioneering develop-
ments within an historical context, identifying 1956 as roughly the year in which the
subject came of age.
In the 25 years or so since then, most work concerned with ligand field theory,
especially of d block complexes, has been performed by the chemist rather than by the
physicist in the capacity of user rather than phenomenologist. Of course, to some extent
the theory has been continuously checked in that, as calculations have become increas-
ingly sophisticated and experiments more detailed, the ability of ligand field models to
reproduce experimental data quantitatively has been demonstrated repeatedly. The cen-
tral feature of so many latter-day studies in this area, however, is the emphasis upon the
system rather than on the phenomena. Usually the chemist wishes to use ligand field
theory, in conjunction with optical spectroscopy and paramagnetism, as a tool with which
to probe the nature of the bonding in specific molecules. Until recently his realization of
this ambition has fallen well below the expectation which derived from the early, largely
group-theoretical, achievements in the prediction of coordination number, stereochemis-
try or even something of the thermodynamics of heats of hydration, for example. In
solving the problems of less-than-cubic molecular symmetry and spin-orbit coupling as
revealed in optical term splittings, polarization ratios, or anisotropy of paramagnetic
The Context and Application of Ligand Field Theory 3
susceptibilities and e.s.r, g values, sight of the ultimate purpose of the effort has often
been lost. The quantities deduced from these specialized studies frequently reflected
little of obvious chemical interest and have not been understood by the more general
chemist. In consequence, interest in ligand field theory has decreased markedly in the
past 51 years. Perhaps this has been a necessary phase of development. There is, how-
ever, ample evidence to show that the early optimism was justified and that, in its latest
form, ligand field theory genuinely does offer means by which the bonding in transition
metal complexes may be revealed in some detail and in a way not easily accessible by
other means.
The central feature of this approach stems from a recognition of the commonly
powerful chemical concept of the functional group. Earlier ligand field models in which
tetragonal distortions of an octahedron, for example, were parameterized by the quan-
tities Dq, Ds, Dt failed explicitly to recognize obvious local features of the electronic
potential. Such parameters, while correctly representing those electronic features not
determined by group theory, are global and are inevitably unable to provide any com-
mentary upon individual, local interactions between each ligand and the central metal
atom. Equally problematic is the increasing lack of utility in purely symmetry-based
parameters in molecules with even lower symmetry than axial, including the most com-
mon circumstances of all; namely, molecules with no symmetry whatever. It has recently
been argued )2 that magnetic properties are especially sensitive to detailed molecular
structure and that idealizations of molecular geometry to artificially high symmetry are
generally wholly unacceptable. By contrast, it appears that the angular overlap model
(AOM) of Sch/iffer and J0rgensen )7-3 may be exploited to overcome most of these
difficulties.
In common with the early electrostatic crystal field theory, the angular overlap model
considers the ligands around a central metal ion as setting up an effective potential and
seeks to account for the non-spherical part of that potential. Detailed descriptions of the
underlying assumptions of the model have been confused, however, by an implicit desire
to blend the interpretation of the model parameters with a sort of divorce from the
historical origins of the approach. The essence of the AOM as it si actually used in
computation is to consider the whole of the non-spherical ligand-fleld potential as divided
up into non-overlapping regions of space, each region associated with a separate ligand or
donor group in such a way that the ligand potential is diagonal with respect to the metal d
(or f) orbitals when quantized in a coordinate frame reflecting the local metal-ligand
pseudosymmetry. This basic assumption distinguishes the AOM from the point-charge or
radially-directed-dipole crystal field models except when the local metal-ligand symmetry
is ,~ooC i.e. for so-called linear ligators.
By way of example, we might consider the interaction of a quinoline ligand whose
donor nitrogen lone pair is directed exactly toward the metal. The local pseudo-sym-
metry is ~2C and we may label the metal-nitrogen axis as Z, with Cartesian coordinates X
and Y lying in, and perpendicular to, the quinoline plane respectively. The central
assumption of the AOM is that the d orbitai energy matrix is diagonal within this coordi-
nate frame and we write
( )dlniuqVid = dqei(quin) (1-1)
or in a fuller, more transparent notation,
4 .M Gerloch et .la
(dy~Vldy~) = e~y, (dvVdv) = e~xy,
( dxLy2lVldx2_y2 ) = 2y_2xde (l-a)
The next step is to express this diagonal energy matrix relative to some global molecular
frame and sum contributions from each metal-ligand interaction. The resulting energy
matrix is a function of the e parameters of all ligands and is no longer diagonal. The
transformation from local to global coordinates is simply a rotation of the angular parts of
the central metal functions. Subsequently, using the usual techniques of vector coupling,
the many-electron states of a given complex may be constructed from the general orbital
energy matrix and from these, spectral and magnetic properties may be calculated.
Observed experimental features are then reproduced by variation of the e parameters of
each ligand, together with parameters for interelectron repulsion, spin-orbit coupling
and, where appropriate, orbital reduction factors. Interpretation of the best fit e param-
eters finally centres around the association of eo with a bonding between metal and
ligand, ~,e with z~ bonding etc. Such interpretations should be seen as ex post facto and not
as intrinsic to the technical procedures and assumptions of the AOM itself.
The chemical significance of the e parameters is clear cut within the original formula-
tion of the AOM. Although similar frameworks were described by McClure )s and by
Yamatera )01,9 with reference specifically to orthoaxial chromophores, the AOM proper
began with papers of Jcrgensen, Pappalardo and Schmidtke )3 and by Schfiffer and
J0rgensen .)5'4 It was derived from the Wolfsberg-Helmholz formulation )11 of the
Hfickel molecular orbital model. A metal-ligand bond was regarded as strongly
heteropolar and the shift in energy of a given metal d orbital was taken to be proportional
to the square of the appropriate metal-ligand overlap integral. This viewpoint automati-
cally yields the notion of a locally diagonal d-orbital energy matrix in view of the sym-
metry classification of the overlapping orbitals. Further, with heteropolar bonding and a
presumption of negligible ligand-ligand overlap, the additivity of different ligand con-
tribution follows immediately. The technical matter of transforming local effects from
different ligands into a common global frame is exactly as outlined above. The immediate
appeal of this formulation of the AOM is the significance it lends the e parameters. As
discussed elsewhere )zl ze is related to the radial part of the overlap integral S~L(2) and
the diagonal metal and ligand energies HM, HE (e.g. VSIE's or VOIP's) by
(1-3)
ez \HM__ HL
Further, ligand donors have H M > H L and hence e parameters greater than zero; con-
versely, negative e values are associated with ligand acceptors. Ligands like NH 3 are not
expected to enter into s~-bonding with the metal and so we should find e~(NH3) = 0.
It is worth mentioning straightaway that the e values found by best fit procedures to
spectral and magnetic data do indeed tend to support this qualitative view. It has been
found that for ammonia and tertiary amines one can consistently associate vanishing ~e
values for these ligands in various complexes .)31 In M(pyridine)4(NCS)2; M=Co(II),
Fe(II), the ~e parameters defined in the planes of the pyridine groups are zero and take
small (positive) values (ca. 100 cm -1) perpendicular to these groups .)41 By contrast, the
The Context and Application of Ligand Field Theory 5
formally sp 2 nitrogen donor atom of imine groups, while again being associated with zero
~e parameters in the plane of the -I~=C moiety, have been characterized '51 )61 by signifi-
cant positive ~,e values (ca. 1000 cm -1) parallel to the direction of the unhybridized N
p-orbital. Halogen ligands show variations in oe and ~e values which seem to reflect, at
least qualitatively, variations in halogen diffuseness and in bond lengths associated with
tetrahedral or octahedral coordination, for example .)71 Phosphine groups in M(PPh3)2X2;
M=Co II, Nin; X=C1, Br )si or in Ni(PPh3)X3-; X=Br, )911 display large negative r,e
parameters as would be expected for such rJ acids: similar results have been observed )°2
in low-spin planar-coordinated, cobalt(II) complexes L2Co(PEt2Ph)2; L=mesityl,
a-methylnaphthyl, ,51C6C C6F .5 Details of these and other studies are reviewed in .)2 The
apparently successful association of e parameter values with known chemical characteris-
tics of functional groups cannot be taken to support the proportionality in (1-3), how-
ever, not least because of defects in the Wolfsberg-Helmholz model from which that
relationship was effectively derived.
Some years ago the Wolfsberg-Helmholz method enjoyed some popularity as a means
of calculating energy levels in transition metal complexes. Its quantitative success was
very limited, however, and in common with many other semi-empirical m.o. methods has
been the subject of considerable criticism for its theoretical inconsistencies .)12 It is inter-
esting, therefore, that Schfiffer subsequently suggested '6 )7 a "perturbation" formalism for
the AOM which does not refer to the Wolfsberg-Helmholz scheme. He sets out three
assumptions for his perturbation model:
I That the ligand field potential acts as a first-order perturbation on a d (or f) orbital
basis, in the sense that admixtures of ligand functions into the basis need not be
considered;
II that the perturbation is diagonal within the local metal-ligand coordinate frame; and
III that the perturbations from different ligands are additive.
In attempting to draw this perturbation approach and the overlap-oriented model
together, Sch~iffer asserts that the first of these assumptions (I) is equivalent to the idea
that the metal orbital energy shift is proportional to the square of the relevant overlap
integral. He comments, too, that the overlap criterion of the historically earlier model
"serves to clarify" assumption II: this is obviously correct, for assumption II of the
perturbation model is a strong assumption otherwise. One may be left with the impres-
sion that the "equivalence" of the perturbation and overlap approaches implies that the
AOM relies upon energy shifts being proportional to the squares of overlap integrals,
even if the Wolfsberg-Helmholz foundation is discarded. Such is not the case, of course,
for the only equivalence required (and which is trivially true) is that the angular parts of
the overlap integrals transform identically to the angular parts of the basis (d) wavefunc-
tion. Indeed the name of the AOM follows from this. The Wolfsberg-Helmholz model
carries a much stronger assumption of simple proportionality between resonance and
overlap integrals which is irrelevant for the AOM. In principle, the energy shifts of the
metal orbitals need not be simple functions of overlap integrals at all: such would be the
case in a purely ionic crystal field situation, for example. The formalism and procedures
of the AOM would remain intact, however. So the question of the relationships between
the e parameters and overlap or resonance integrals, configuration interaction and so on,
though of central importance to the chemical significance of the e parameters, does not
bear on the technical procedures undergone in ligand field analyses within the localized
potential model we call the AOM.
6 .M Gerloch et .la
While the chemical interpretation of the e parameters is a matter of real concern to
us, there are also several other difficulties which are, however, more apparent than real.
Consider the question of the calculation of magnetic properties in transition metal com-
plexes - paramagnetic susceptibilities and e.s.r, g values. In contrast to the study of
eigenvalues for optical transition energies, these require descriptions of the wavefunc-
tions after the perturbation by the ligand field, interelectron repulsion and spin-orbit
coupling effects. In susceptibility calculations it is customary to use Stevens' orbital
reduction factor k in the magnetic moment operator
/~ -== ( k ~l + 2 s~) ; a = x, y , z (1-4)
supposedly because the wavefunctions are no longer characterized by an integral l value.
In terms of the original introduction z2/of the k factor, the formation of molecular orbitals
from metal and ligand functions reduces the expectation values of .~l Orgel has
described )32 the k factor as an electron delocalization factor, though the relationship
between its numerical value and "covalency" has been shown )42 to be complex. Often,
the e.s.r, spectroscopist prefers to fit his g value data without the use of the effective
operator (1-4), parameterizing his model instead by explicit molecular orbital mixing
coefficients. It is not unusual to see ligand orbital mixing coefficients as large as 0.3 or
more from such analyses.
The question apparently arises, therefore, whether the "impurity" of the d orbital
basis, evidenced by either of these two approaches, invalidates the use of the transforma-
tion matrices in the AOM (or other ligand field models) which are functions of the
angular parts of pure d (or f) orbitals. We recall Sch~iffer's first assumption above, in
which he explicitly ignores admixtures of ligand functions into the metal orbital basis. If
the answer to this question is that we uniformly regard such admixtures as negligible,
then we must point again to the large magnitude of the mixing coefficients often estab-
lished by e.s.r, spectroscopists. ,.
A similar line of thinking focusses attention on the interelectron repulsion parameters
and the nephelauxetic effect. Jcrgensen in particular has been much concerned '21 )52 with
the desire to recognize and establish differential orbital expansion by the introduction of
the nephelauxetic parameters/333,/335,/355. Consistently lower values for/333 associated
with the eg orbitals of octahedral configurations relative to/355 associated with tzg, have
been taken as evidence for the greater nephelauxetic effect of a orbitals relative to n. In
turn, these effects, which derive from experimental spectral features, are considered to
indicate differing radial properties amongst the d orbital basis. Again, if this is so, we
might ask if it is satisfactory to use the simple, "pure", rotation matrices in the remainder
of the ligand field calculation.
We are questioning the use of a pure d(f) orbital basis in ligand field calculations,
therefore; something which must be answered from the outset, for all our ligand field
procedures, including those of the AOM, depend on that basis. The problem is more
apparent than real, however. As a matter of formal manipulation we may always express
the ligand field, projected onto some convenient basis. By suitably transforming the
Hamiltonian and all other relevant operators (cf. Eq. 1-4), we occasion no approximation
whatever by formulating the ligand field problem within a pure d (or f) orbital basis. It is
a matter for discussion as to whether such a projection is useful, that is whether it yields,
say, a parameterization scheme of suitable transparency, but we may always work in this
The Context and Application of Ligand Field Theory 7
way. As to the problem of the m.o. mixing coefficients referred to earlier, or the "aniso-
tropic nephelauxetic parameters"; again there is no formalistic difficulty here for those
quantities refer to different models with different bases. They are further examples of the
common mistake of comparing different quantities. The chemical conclusions drawn
from any one of these models may well have some validity but the connections between
the different approaches are not obvious.
In the analysis following we discuss the parameterization of interelectron repulsion,
the ligand field and the AOM, spin-orbit coupling, and of an externally imposed magne-
tic field. Our discussion attempts to put all these quantities on the same footing and so
provide the experimentalist with more confidence in the models he uses. Some problems
of interpretation remain and we do not attempt to disguise these: we hope the study may
stimulate further enquiry but always with the goals of chemical utility in view.
It is conventional that the ligand field problem for systems with Nd > 1 d electrons
requires the diagonalization of an effective Hamiltonian operator composed for the
electronic kinetic energy T, and both one-electron "ligand field" terms, and two-electron
Coulomb interactions .)62
dN dN
W = T + Y VLF(ri) + Y 1/(rq) (1-5)
i i<j
W acts in the finite-dimensional function space composed of Nd-electron Slater deter-
minants built up from spin-orbitals with pure I = 2 angular transformation properties (in
all there are ~NCOI such functions). Only FLV carries the point symmetry of the complex,
while the interelectron repulsion operator leads to reduced values of the Raeah B, C or
Condon-Shortley f~ parameters familiar from atomic spectroscopy. The technical part of
this review is concerned with the relationship between this effective Hamiltonian and the
full n-electron Schr6dinger equation for a transition metal complex.
The outline of the review is as follows: in the next section (Sect. 2) we introduce the
basic ideas of effective Hamiltonian theory based on the use of projection operators. The
effective Hamiltonian (1-5) for the ligand field problem is constructed in several steps:
first by analogy with n-electron theory we use the group product function method of
Lykos and Parr )7z to define a set of n-electron wavefunctions {eNd} which define a
subspace O ° of the full n-particle Hilbert space in which we can give a detailed analysis of
the Schr6dinger equation for the full molecular Hamiltonian H (Sect. 3 and 4). This
subspace consists of fully antisymmetrized product wavefunctions composed of a fixed
ground state wavefunction, WLo, for all the electrons in the molecule other than the Nd
electrons which are placed in states, {l'IIMrn} , constructed out of pure d-orbitals on the
metal atom. The use of such d-orbitals is always possible provided that either all other
orbitals used in the construction of ~Lo are orthogonal to the d-orbitals (as in Sect. 3) or
that the Hamiltonian H is augmented by "pseudopotential" terms that take the place of
the orthogonality constraints which these orbitals otherwise would have to satisfy. We
call the ligand field theory derived from this analysis the "primitive parameterizafion" of
the ligand field since we recognize that the restriction of our basis functions to the
subspace 5 ¢ of the full Hilbert place implies that we can only recover the exact electronic
energy levels of H if we are prepared to construct )82 a modified Hamiltonian operator
to be used in the Schr6dinger equation for the eigenfunctions .}~v~q{ A corollary of this
statement is that if we require matrix elements of operators other than the Hamiltonian
8 M. Gerloch et al.
in the basis }ev~q{ we shall also have to modify the functional form of these operators. If
an eigenfunction of the true Hamiltonian, /2/ is ,k~ff and the eigenfunction of the effective
Hamiltonian ~ belonging to the same eigenvalue is q~, the relationship between a true
operator /3 and the equivalent effective operator ~ is completely determined by the
equation, (Sect. 5),
('I',~lZ~l'I'k) ('/'~%+/'~9
(1-6)
once the subspace b ° has been specified (Freed29)). The most obvious application of this
equation is, of course, with/3 = /q. In magnetic studies, for example, we need matrix
elements of the total orbital angular momentum operator, and its conventional modifica-
tion with Stevens' orbital reduction factor can be interpreted theoretically with the aid of
equation (1-6) (see Sect. 5). Equation (1-6) also implies that the Coulomb operator
Z jir/1 in Eq. (1-5) must be modified, and this has repercussions in the interpretation of
i<j
interelectron repulsion parameters.
The process of constructing ~ and its matrix elements can be regarded as a "tenor-
realization" of the formal primitive theory described in Sect. 4. At both primitive and
renormalized levels of description we can reduce the n-electron Hamiltonians, H and g~
respectively, to operators acting on only the Na d-electrons, as in Eq. (1-5), by integrat-
ing out all the other electrons with the probability distribution ,oLWo~W and we therefore
have the formal relations
(tlJLolHItlJLo) primitive theory
'gY (Eq. (1--5)) - (1-7)
( ~coIN ~ro) renormalized theory
The reduction of the many-electron matrix elements of ~' to the orbital matrix elements
which are to be parameterized, proceeds as usual using Slater's rules38): in the primitive
parameterization (Sect. 4) we obtain a single set of ligand field parameters from which all
many-electron states can be reconstructed. We show, however, that the reduction of the
true effective Hamiltonian obtained from ~ in Eq. (1-7) is much more complex since
(unlike H) contains 1, 2,... n-electron operators: as a result the orbital matrix element
the "true parameters" in the terminology of Freed )°3 become state dependent so that in
principle every electronic state has its own set of parameters. The usual ligand field
procedures can therefore only be recovered by assuming that we can construct a weighted
configuration average of the true parameters that leads to only modest errors in the
energies of the manifold of the electronic states of d-orbital parentage. This averaging
requirement is an essential assumption on which ligand field theory is founded, and
granted its validity, this review provides a demonstration of how one may proceed by a
consistent and reasonable approximation scheme from the n-electron theory of the full
molecular Hamiltonian for a transition metal complex to an equivalent projected theory
concerned only with the electronic states of metal d-orbital parentage.
The Context and Application of Ligand Field Theory
2 Effective Hamiltonian Theory
It has become clear from recent work that a justification for the parameterized semi-
empirical molecular orbital theories widely used in the electronic theory of molecules
cannot be expected to be obtained from the SCF-Hartree-Fock approximation in molecu-
lar orbital theory .)53-°3 A realistic treatment of electronic correlation, implicit or explicit
through many-body theory, must be incorporated in the formal development of these
models if one wishes to identify with any confidence the parameters which should be
related to experimental data. The distinguishing feature of the ligand-field description is
the recognition of the existence of a set of low-lying excitations in a transition metal
complex which is largely responsible for both the magnetic behaviour and a "band" of
spectral features of weak intensity, and is associated with electronic states dominated by
orbital contributions of metal ion d-electron parentage. The conventional ligand field
theory confines itself to an explicit treatment of just the metal d electrons (cf. Eq. (1-5),
and is not therefore a comprehensive theory of chemical bonding in these metal com-
plexes, although of course the hope is that we shall be able to extract chemically interest-
ing information about metal-ligand bonding from the ligand field parameterization. The
ligand field theory is only concerned with the eigensolutions of an effective "ligand field"
Hamiltonian, ~', Eq. (1-5), spanning a range in energy of a few electron volts above the
ground state: we are thus given the form of the final equations, and our task is twofold,
namely to manipulate the many-body theory of the full n-electron Hamiltonian,/1 into
this form, and to provide a justification for the usual chemical bonding interpretation of
the ligand field parameters.
The reader should note, however, that given what we now know about the treatment
of correlation in many-electron systems '92 ,43 ,)s3 it is perfectly feasible to contemplate a
fresh start for the electronic theory of transition metal complexes, and aim to describe the
chemical bonding, the charge transfer states and the d-d transitions in a uniform, modern
theoretical framework than could either be parameterized using experimental data or
dealt with using the techniques of ab initio quantum chemistry. This admittedly ambitious
programme remains a task for the future but is mentioned here firstly as something to
bear in mind in relation to the somewhat awkward formalism we suggest best corres-
ponds to the conventional ligand field treatment and, secondly, to advertise the theoreti-
cal possibilities in the context of metal complexes.
In the light of these remarks we shall content ourselves in this section with only a brief
resum6 of the effective Hamiltonian theory we require for the ligand field theory, and
defer a more detailed discussion to Sect. 5. We noted above that in ligand field theory we
are only interested in a set of eigenvalues {ek} which lie close to the ground electronic
state, and accordingly we aim to confine attention to the subspace, 5 ,° spanned by the
associated eigenvectors {¢k}, which may be assumed to be a finite orthonormal set. We
can define a projection operator onto this subspace Pa, and a projector onto the ortho-
gonal complement, ~)a,
k tq~) (~,1
i% = Y (2-1a)
i
a~{ = i -/3 (2-1b)