Download Molecular Orbital, Valence Bond, and Ligand Field

A comparison of theories
Andrew D. Liehr
Bell Telephone Laboratories, Inc.
Murray Hill, NewJersey
Molecular Orbital, Valence
Bond, and Ligand Field
Before jumping into the intricate details
of the various theories of valency, let us first pause a
hit and try to obtain some historical insight into their
course of development, their successes, and their
failures. Only in this manner shall we be able to
assess with any assurance the future progress of the
theory of the chemical bond.
Soon after the announcement of the Schrodinger
equation for electronic motions, there were proposed
and utilized three approximate means of formulating
solutions of this equation as it applied to molecular
problems: (1) the valence bond technique of Heitler,
London, Slater, and Pauling; (2) the molecular orbital
technique of Hund, Bloch, Mulliken, Lennard-Jones,
and Hiickel; and (3) the crystal field technique of
Bethe, Kramers, and Van Vleck. Each of these
techniques had its limitations, its strong points, and
its weak points. And the years 1930-45 saw a struggle
among the three for pre-eminence in the minds of
chemists and physicists, even though Van Vleck had
shown in 1935 that they were absolutely equivalent
when carried to completion (2,3). These years witnessed the adoption of the valence bond and molecular
orbital methods by the organic chemist, the valence
bond method by the inorganic chemist, the molecular
orbital method by the molecular and solid-state
chemist and physicist, and the crystal field method by
the magneto-physicist. Each of these adoptions had
the same driving reason: most chemists and physicists
of this era were primarily interested in the ground
electronic states of chemical systems.
With the end of the war, scientific interest began to
swing toward a concern over the excited electronic
states of molecules. This precipitated a rapid fall
from favor of the valence bond theory, and a consequent
rise in esteem of the molecular orbital and crystal field
theories. Thus in all fields of chemistry except inorganic, valence bond ideas concerning the nature of
excited molecnlar states made their exit in the years
1945-55. But with the accelerated interest in inorganic spectroscopy by inorganic chemists commencing around 1955, even this last stronghold of
valence bond concepts has begun to fall. Indeed, in
the particular area of inorganic chemistry called
coordination chemistry, valence bond theory has
suffered its most grevious blows; even its descriptions
of the ground electronic charge distributions have
proved false in many important cases, especially with
regard to the much vaunted, but completely specious
--
Presented at the Symposium an Ligand Field Theory, 140th
Meeting of the ACS, Chicago, September, 1961.
magnetic criterion. [A most recent example of such
a blow is the demonstration by McGarvey (4) that
the odd electron in square planar CU+=complexes is
in a 3d-like orbital and not a 4p-like orbital.] At the
present time the valence bond theory of coordination
compounds is being systematically supplanted by a
molecular orbital-crystal field amalgamation, which
has been aptly dubbed ligand field theory. [Crystal
field theory was originally an ionic theory of chemical
bonding until modified by Van Vleck. I n its modified
form it was a highly simplified molecular orbital theory
of the nd, (n l)s, (n l)p, and nf electrons in which
orbital energies were of the molecular orbital type, but
electron-electron electrostatic repulsion energies and
spin-orbit energies were of the atomic type. Ligand
field theory is a molecular orbital theory of the nd,
(n
l)s, (n
l)p, and nf electrons in which both the
individual orbital energies and the electron-electron
repulsion and spin-orbit energies
are of a molecular
type. I
Why has this particular pattern of historical dynamics evolved? The answer to this question is simple:
the valence bond theory, although it is by far and away
the superior outlook for ground electronic states,
becomes hopelessly complex as a description of excited
electronic states. The jungle of ionic valence bond
struebures nverruns all attempts to describe electronic
excitations by this technique. Molecular orbital and
crystal field theory on the other hand suffer from the
complementary deficiency: they provide adequate
pictures of the excited electronic states, hut not of
the ground electronic state of most molecules (they usually introduce too much ionicity into the ground electronic state). Therefore as long as ground electronic
state properties were the vogue, valence bond theory
shone, but when the properties of electronically excited
states became the style its gleam was dulled.
With this historical perspective behind us, let us
now see how to extend the ligand field technique to
encompass all inorganic compounds. In two frequently
overlooked papers by Kimball (5) in 1940 and by
Eisenstein (6) in 1956, there are tabulated the symmetry classifications of the primary atom and ligand
atom orbitals for most geometries of interest. With
these classifications written down once and for all,
it is a simple matter to construct a molecular orbital
energy level diagram, and with this diagram to predict
the number and classification of the excited electronic
states. We shall demonstrate this fact explicitly
for a few exemplary systems.
In Figure 1 we show the orbitals which are primarily
involved in the bonding of a linear MX2 transition
+
+
+
+
Volume 39, Number 3, March 1962
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135
metal compound (e.g., CuC12). We have written down,
from Kimball's and Eisenstein's tables, the correct symmetry designations of the primary atom and ligand
atom orbitals prior to compound formation; that is,
for symmetrically disposed reactants at infinity. Then
of the ligand orbitals and the a.+* and T.* antihonding
orbitals are composed mostly of the primary atom
nd+ and nd,,,
orbitals, respectively. Hence, for
CuCI2 we expect the ground electronic state to be a
2Z,+ state [as there is one unpaired electron in the
cg+* orbitarcapital letters denote the over-all state
electronic distribution], a state with zero orbital angular momentum along the molecular axis, and the first
two excited states to be the so-called nd excitations,
TI, and 2Ax,which arise from the one electron jumps
s.*-te,+* and .
8
c.+*, each. This is what is actually observed (8).
As a second example, we present in Figure 2 the
energy levels of a square planar complex such as
-
. .
.
Figure 1. Molecular orbitals for a linear triotomic transition metal compound ( D d . Note that the <-bond strudure is closely approximated b y
the valence bond hybrids r'p'dl, and nots'p' or pld'alone.
recalling that the symmetry quantum numbers,'
c.+, c.+, s,,6,, etc., are exact quantum numbers at all
internuclear distances, we allow the reactants to approach one another in a symmetrical fashion to produce the h a 1 molecule, and we combine only those
primary atom and ligand atom molecular orbitals to
form the complete over-all bonding, antibonding,
and nonbonding molecular orbitals, which have the
self-same symmetry designations (i.e., symmetry quantum numbers). We can, of course, obtain only so
many complete bonding and antibonding molecular
orbitals of a given symmetry type as there are primary
atom and ligand atom orbitals of this same symmetry
classification initially present. The ordering of the
resultant molecular orbitals of the product molecule
is completely based on qualitative concepts: the more
the primary atom and ligand orbitals are directed
toward one another, the deeper the consequent bonding
orbitals will lie and the higher the consequent antibonding orbitals. will lie. For example, the cs+*
antibonding orbital lies higher than the s.* antibonding
orbital, as the primary atom c.+ orbital, nds2,is directed more strongly toward the ligand cg+ orbital than
the primary atom s, orbital, nd,,,,,, is toward the
ligand T. orbital (the molecular axis is the z axis).
Moreover, as the ligands have a greater affinity for
their electrons than does the primary atom, the c.+
and s,bonding molecular orbitals are composed mostly
'For the linear molecule the Greek letter symbols ( G , r , 6, p
etc., replace the stomic designations a, p, d, j,etc. Just as these
Latter symbols denoted orbital angular moments of 0,1,2, 3, ete.,
respectively, in the atom, the former now indicate the magnitude
of the component of orbital angular moment 0, 1, 2, 3, etc., along
the internuclear axis (thez axis) of the moleoule.
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Journal of Chemical Education
P e I M A W ATOM
ORBITALS
Figure 2.
MOLECULAR ORBlTAL5
OF T H E COMPOUND
LlCAND MOLECULAR
ORBITALS
Molecular orbitdr for o square planar transition metal complex
I D d Observe that the o-bond structure is well approximated b y the
volence bond hybridsr'p'ff, ond notr'p2d'orpzd2alone.
PtC14-2.2 The placement of the bonding and antibonding orbitals is again accomplished by qualitative
principles and may be subject to reordering. The
location of the a@* orbital, which is primarily nd,,
% T h emolecular symbols corresponding to the atomic s, p, d , f,
etc., designations for non-linear compounds are a, b, e , and 1
(an icosahedral molecule has the additional symbols y. and h,
which are not to be confused with tho analogous atomic terms).
The molecular symbol a corresponds to the atomic s (and the
linear molecule designation a)-it means that the molecular wave
function does not change sign under a rotation of 2 r l n about the
molecular n-fold rotational axis of symmetry (e.g., the four-fold z
axis of PtCL-2): the symbol b means that it does. (In a very
direct sense this is equivalent to saying that the molecular a type
orbitals have a component of angular momentum along the n-fold
rotational axis of symmetry (the e axis) whose magnitude is a
multiple of n (e.g., 0 or n), and the b type a companmt which is
an odd multiple of n12.) The symhals e and 1 mean that the
molecular wave function is degenerate (just as the atomic states
p, d, f, eto., am degenerate), with two-fold and three-fold d o
generacy, respectively (the icosahedral g and h symbols indicate
four-fold and five-fold degeneracy). Such degenerate sets of
in character, is especially vague as i t depends quite
strongly upon axial perturbations (the z-axis is the
four-fold axis) which are always present in solid or
solution. With our assignment [similar to that of
Fenske, Martin, and Ruedenberg ( l o ) ] we find the
ground electronic state to be 'Al., and the first three
excited nd states to be 'I?,., lA2., and 'E.. These
transitions should be strongly enhanced due to vibronic (vibrational-electronic) intensity theft from the
nearby en* and bl,* n-antibonding orbitals. The
'E, electronic state should exhibit a Jahn-Teller energy
surface similar to that of Figure 3. The nuclear
Figure 4. The nuclear dirplocements dlowed a rquors p l m m complex.
Intensity enhancement is provided only by the odd (01 coordinoter, and JohnTellw forces only b y the wen (gl81, ond Bag madinmter
I
Figure 3. The JohmTeller energy
surface characterirtic of a square
planar system ( i 11. Nuclear motions
on this surface ore essentially one
dimensional.
displacements which cause these Jahn-Teller motions
and intensity enhancement are given in Figure 4.3
Figures 5-8 depict the molecular energy level d i a
grams (in the absence of spin orbit coupling) for the
trigonal bipyramid (PFk), the tetrahedron (SiFp, VCL,
and MnOa-2), the octahedron (SF, RepB, and TiFe-a),
and the cube (Ti+S:CaFd.
Discussion of the result-,
ant excited states is similar to that given previously.
Note especially the number of Jahn-Teller states
expected for these systems. Several such Jahn-Teller
states have been recently observed for SiF4 (9).
By now the procedures utilized to construct molecular
orbitil energy level diagrams are apparent, and so my
purpose in presenting this lecture has been accomplished.
I sincerely hope that what I have said today will be
of aid to those of vou who are concerned with the
excited states of inorganic systems, and that it will
stimulate a fervent interest in the science of inorganic
spectroscopy. I eagerly look forward to the day when
the idem of Mnlliken (1) and Van Vleck (2, 3) which
were here outlined will be universally accepted.
functions transform one function into the other under certain of
the rotations and reflections permitted the overall molecular
symmetry, and are identified in practioe in this way. This
criterion for degeneracy is entirely equivalent to saying that such
states have components of angular momentum (better yet, "permutational momentum") about some d o l d rotational axis of
symmetry of the molecule which are not multiples of nI2. The
subscripts "g" and "u" indicate that the wave functions are even
or odd, respectively, under inversion in the center of symmetry;
and the subscripts 1 and 2, etc., that they are even or odd under
reflection in some given plane of symmetry.
A Jahn-Teller molecule is one which exists in a degenerate (or
nearly degenerate) electronic state. Such a molecule, acoording
Roy. SOC.,161A, 220
to the theorem of Jahu and Teller (PTOC.
(1937)), may experience eoulombic forces which tend to destroy
this degeneracy by distortion of the nuclear framework. The
J
~ a h n - ~ e l l theorem
er
and its consequences has been given by the
author elsewhere. See LIEHR,A. D., Revs. Mod. Phys., 32, 436
(1960); "Annual Reviews of Physical Chemistry," Val.. 13,
Annual Reviews, Ine., Palo Alto, California, 1962; Progr. Inorg.
Chem., 3, 281 (1961); and P r o y . Inonorg. Chem., 4, 000 (1962).
C. J., Adv. Chem. Phys., 4, 000
In addition, see BALLHAUSEN,
(1961). A similarly graphic-type discussion of the closely related intensity problem has also been given elsewhere. See
LIEHR,A. D., Adu. Chem. Phys. 4, 000 (1961); BALLHAUSEN,
C. J., Progr. Inorg. Chem., 2,251 (1960).
Figure 5. Molecular orbitals for a trigand bypyramid geometry (DIJ.
See that the <-bond rtrvcture is nicely approximated b y the valence bond
hybridsr'padJ, and not r1p3d'or r'pWalone.
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137
Literature Cited
(1) MULLIKEN,
R. S., Phys. Rar. 40, 55 (1932).
(2) VANVLECK,J. H., J. Chem. Phys. 3 , 803 (1935).
Rev. Mod. Phys. 7 ,
(3) VANVLECK,J. H., AND A., SHERXAN,
-lfi7
-. f l-D-R--6 ,) .
\
McG~~viw
B., R., J. Phys. Chem. 60, 71 (1956).
KIMBALL,
G. E., J. Chem. Phys. 8 , 188 (1940).
J. C., J. C h . Phys. 25, 142 (1956).
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COULSON,
C. A,, "Valence," Clarendon Press, Oxford, 1952.
HOUGEN,
J. T.. LEROI,G. E., AND JAMES,T . C., J. Chem.
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(4)
(5)
(6)
(7)
(8)
+BONDS
PrtlMAW ATOM
01181TAL5
MOLECULAR ORBITALS
OF THE COMPOUND
".80NDI
(9) HEXTER,R. M., private communication, February 1961.
K., p"(10) FENSKE,R. F., MARTIN,D. S., AND RUEDENBERG,
"ate eammuni<:i~tion,May 1961, also 140th Meeting of
ACS, September, 1960.
(11) LIEHR,A. D., XVIIIth International Congress of Pure and
Applied Chemistry, Montreal, Canada, August 6-12,
1961. 'This reference pertains to the general theory of
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Recent Qualitative Discurrions
NYHOLM,
R. S., Biochem. Sac. Symposia, No. IS, 1 (1958);
Record Chem. Progr., 19, 44 (1958); La Ricerea Seiatijea,
Suppl., p. 3 (1958).
KIMBALL,
G. E., AND LOEBL,E. M., J. CHEM.EDUC.,36, 233
(1959).
PLIR3
LICI\ND MOLECULAR
ORBiTAL5
Figure 6(0)
Figure 61bl. Molecular orbitols far a tetrahedral arrangement ITd)
about a cectrml nr np nd type atom ore shown in Figure 6i.I.
The same ir
1 ) s in
i l p type .tom.
rhown in Figure 6(b) for the nd in
Perceive
thot the .-bond rtrvctvre is neatly opproximated by the valence bond hybrid. r'#da, and not +pa or r'dJ alone.
The electronic excitations from the
filled nonbonding h irl orbital to the unfilled ontibonding e*(r) arbitd giver
rise to the low lying single excited stater ITt and LTs, while that from fl(*)
is, rl orbital produces the 'A?, 'E, ITlr and
to the unfilled antibonding ta"
'T* states. As the ground electronis state i s 'A1 and the transition dipole
restor, el: transforms a s h only the electronic jumpslA1 -IT2 ore allowed.
TheLAl -'E,'Tr
hops are made vibronically ollowed via intensity theft from
the aIlowedlT~ state by the and n nuclear modes. ThelAz state is strictly
forbidden unless intensity borrowing by the nuclear first harmonics ore considered ithis borrowing is usually inperceptiblel. The stater ' E , ITx, and
IT2 ore, of COUIS~, John-Teller active (1 1).
+
138
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+
Journal o f Chemicd Edvcufion
..
PRIMARY ATOM
OFtB1TALS
<-.
MOLECULIIQ OFlBlTALS
OF T H E COMPOUNO
"...- '.
ON.
LoNC
v.,..,
L I G A N D MOLECULAR
OeBITALJ
Figure 7[b). Molecvlor orbitals for an oetohedrol disposition (Oh)
about 0 central nr np nd type otom ore rhown in Figure 7(4. The some
is rhown in Figure 7ibl for the nd (n
1 lr in
1lp type otom. Discern
that the o-bond structure i s readily opproximated by the valence bond hybrids rLpsd2in agreement with the usual notion. An electronic excitation
from the filled nonbinding h,(d tothe unfilled ontibonding f&) gives rise to
the electronic states A
' I.,
LEm,%u,and
of which only the T
' .I i s electronically accessible from on A
' 1. ground state. The 'E" and ITnu $totes ore vibronimliy dowed, but the A
' ,. date is strictly forbidden [if first hormonic
vibranicinteractionsare discounted). The'E,.'T~,,and
'Tz.stotesallow JohnTeller antics il 1).
+
+
SUTTON,L. E., J. CHEM.EDUC.,37,498 (1960).
LIEHR,A. D., Bell Syrt. l'ech. J . , 39,1617 (1960).
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(1961).
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S u a ~ ~S.,
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R. S., ORGEL,L. E., AND J@ROENSEN,
C. K., in Reports
of the 10th Solvay Conference, Bruxelles, May, 1956.
MOFFITT,W. E., AND BALLHAUSEN,
C. J., i n "Annual Reviews of
Physical Chemistry," Val. 7, Annual Reviews, Inc., P d o
Alto, California, 1956, p. 107.
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Elektroehem.,
,~.
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AND TURNBULL,
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Chemistry," Vol. 1, edited by COTTON,F. A,, Interscience
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Chemistry snd Radiochemistry," Vol. 2, edited by EMELELIS,
H. J., and SHARPE,
A. G., Academic Prcss, New York, 1960,
pp. 1-60.
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C. J., in "Advances in the Chemistry of the Coordination Compounds," edited by KIRSCHNER,
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Books
"Ions of the Transition Elements," Disc. F a d a y Soe., No. 26,
1958.
ORGEL,L. E., "An Introduction to Transition-Metal Chemistry:
Ligand Field Theory," John Wiley & Sons, Inc., New York,
1960.
Figure 8. Molesvlar orbitals for an eight coordinated cubic environs (Oh)
about a central tramition metal atom. Mark thot the c-bond rtruchm is
(to obtoin eight
easily approrirnded by the valence bond hybrids
equivalently directed valence orbitdr o a-bonding f-type orbital must b e
included).
GRIFFITH,J. S., "The Theory of Transition Metal Ions," Csmbridge University Press, London and New York, 1961.
J@RGENSEN,
C. K., "Ab~orptionSpeotra and Chemical Bonding
in Complexes," Pergamon Press, Ltd., London and New York,
1961.
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