frog. Solid Sr. Chew. Vol. 15, pp. 173-255. 1984 007%6786/R4 $o.tn~ + .x1
Printed in Great Britain. All rights reserved. Copyright 0 I984 Pergemon Press Ltd
FROM BONDS TO BANDS AND
MOLECULES TO SOLIDS
Jeremy K. Burdett”
Department of Chemistry, The University of Chicago, Chicago,
Illinois 60637, U.S.A.
CONTENTS
I. INTRODUCTION
1.1 Scope
1.2 The Molecular Orbital Approach
174
174
174
2. ENERGY LEVELS OF MOLECULES 177
2.1 Linear TI Systems 177
2.2 Cyclic Systems 184
2.3 Jahn-Teller Instabilities 193
3. ELECTRONIC STRUCTURE OF SIMPLE SOLIDS
3.1 Energy Bands
3.2 The Peierls Instability
3.3 Building up niore Complex Systems
4. MORE ORBITALS AND MOP& DIMENSIONS
4.1 Variations on the One-Dimensional Problem
4.2 Three Dimensions
196
196
201
210
229
229
244
5. CONCLUSIONS 253
REFERENCES 254
*Camille and Henry Dreyfus Teacher-Scholar
173
174 J.K. Burdett
I. INTRODUCTION
1.1. Scope
Solids, of course, are just infinite molecules. However, understandingconcerning their
geometricaland electronic structurehas lagged considerablybehind the dramatic progress made
in the molecular area over the past twenty years. With the recent availabilityof fast and
cheap computationand a gradual enlightenmentby physicists and chemists alike concerning each
other's viewpoint in this field, the time is surely ripe for progress to be made. Our major
goal in this article will be to show the striking similaritiesbetween the electronic structure
of molecules and solids and to suggest that there is profit to be gained by extending
ideas developed for molecules to the realm of the solid state. We will rely very heavily on
the linear combination of atomic orbitals approach of the chemist or the tight-binding
approach of the solid-statephysicist. These are two models identical in all but name. We
will also make extensive use of symmetry arguments, in the form of group theoreticaltechniques
and will use perturbation theory to access results of interest. For the reader who is
unfamiliarwith these methods reference to the books by Cotton' and by Heilbronner and Bock'
is strongly suggested. The reader who feels comfortable with such concepts can jump to
Section III. We will focus almost exclusivelyon very simple systems, for it is here that
the workings of the theory is most transparentand the analogies between molecules and solids
easiest to appreciate. We make no apologies for spending approximatelya third of this article
in developing orbital ideas for molecules. Many of the orbital tricks we will use in the
rest of the article have their foundationhere.
1.2. The Molecular Orbital Approach
The SchrEdingerequation can be solved exactly for the case of one-electronatoms, e.g.,
H,He',Li2+. For the case of many-electron atoms approximate (but in many cases very good)
solutionsmay be obtained numerically. In many-electronmolecules one way in which approximate
solutionsmay be obtained is via the linear combination of atomic orbital (LCAO)molecular
orbital method. We will describe one version of this which will enable the generation
of one-electronenergy levels of molecules and (eventually)solids. We refer the reader
elsewhere for more complete accounts concerning the generation of more sophisticatedorbital
models.3y4
Let us take the valence orbitals {'pi}of the atoms which make up the molecule and
write a LCAO molecular orbital wavefunctionwhich we hope will suffice to describe the
energy levels of the molecule.
$ = &Vi
i
Now $J is an eigenstate of some one-electronHamiltonian H, i.e.,
1~9 = EJ,.
This leads to an expression for the energy of the state described by this wavefunction:
(1)
(2)
(3)
where the integrationoccurs over all space. Substitutionof equation (I) into equation (3)
leads to
Z~CiCj(~iIHI~j}
E = ;;
ij
(4)
From Bonds to Bands and Molecules to Solids 175
This expression contains three terms of interest. (Q~]Q~) = S.. is the overlap integral
iJ
between atomic orbitals on different atoms. It is always <I. If our atomic basis set {'_oi}
is normalized then ('pi/'pj) = 1 for the case i=j. ('P~IH\Q~) = Hii represents the energy
of an electron in orbital 1~~. It is called the Coulomb integral and is often approximated
by the ionization energy of an electron from that orbital. (9ilHi~j) = H.. is an integral
iJ
which represents the interaction between orbitals (0; and Q.. It is called the resonance
J
integral by molecular chemists and the hopping or transfer integral by physicists. It is
often estimated using the Wolfsberg-Helmholz relationship
H =
ij
$KS;~(H~~+H..) ,
JJ
(5)
where K is a constant, usually set equal to 1.75. The energies of the molecular orbitals
are obtained from equation (4) in the following way. The Variation Theorem states that all
approximate wavefunctions of a system will give energies that are never lower than the true
ground-state energy of the system. So minimization of equation (4) with respect to all the
coefficients c c. will lead to the best estimate of the orbital energy using the expansion
i' J
of equation (1). If there are n atomic orbitals in {vi} then there will be n equations
demanded by the minimization expressions 8E/aci=0. They have the particularly simple
form of equations (6).
(H1l-E)cJ+(Hl2-S12E)C2 + .... + (Hln-SlnE)Cn = o
41
-S21E)c1+(H22-E)c2 + .... + (H2n-S2nE)~n = o
(6)
(Hnl
-SnlE)cl (Hn2-Sn2E)~2 + . . . . + (H -E)c = 0 .
nn n
(7)
These equations will be consistent only if the secular determinant of equation (7) is equal
to zero.
H
11
-E H
12-S12E
.*.... H
ln-SlnE
H
-S21E
H
22
-E H
21
......
2nVS2nE
. . , = 0.
. . .
. .
H
nl -SnlE
H
n2-Sn2E
...... H -E
nn
Given the values of the Hii, S.. and H..,
=J iJ
equation (7) may be solved to give n values of
the energy - the energy levels of the molecule. Each value of E may in turn be used with
the collection of equations (6) to give the values of the orbital coefficients, the c. of
1
equation (1).
An example will illustrate the approach. Assemble an Hz molecule from two hydrogen
atoms which carry singly occupied 1s valence orbitals. The secular determinant is
H1l-E
H
12-S12E
0. (8)
H12-S12E H11-E
Since the two hydrogen Is orbitals are equivalent we have put H11=H22. (Also
H12=H21 by symmetry.) Solution of this determinant gives
(9)
(10)
(Hll-E) = ‘(H12-S12E)
H 11 +H12
E=
1?S12 -
176 J.K. Burdett
The result is shown pictorially in 1.- Notice that the higher energy orbital is destabilized
0-a I-I,,- HP
(relative to a free hydrogen AO) more than the lower orbital is
binding) energy of H2 on this model is simply
*[Hi&H12 _H11] = 2(U;2;;19*11) .
12 12
!.
stabilized. The bond (or
With these values of the energies we can now go back to equation (6) and evaluate the
coefficients. With a little bit of algebra it is easy to find for $, that c1/c2=1 and for
6, that cl/c2 =-1. The extra piece of information we need to fix the values of cl and c2
is the normalization condition
This leads simply to
I$*$dr=I . (12)
JI,=
I
X/2(1+s12) (Cp1+'p2)
(13)
+, =
1
d2(1_-s12)
bl -‘p, 1
In +, the orbitals are mixed in phase - the bonding orbital, and in $, they are mixed out
of phase -the antibonding orbital. Notice that because of the sign before S12 in the denominators
the coefficients are larger in $,. 1 shows a useful pictorial representation of these
functions where the sign of the mixing coefficient (ci of equation (1)) is indicated by the
presence or absence of shading. Note that the absolute signs have no meaning but the relative
phases are important.
There is another useful way to generate the molecular orbitals of H2, and indeed more
complex systems, which we will use below, and that is by taking advantage of the synrmetry of
the problem. H2 belongs to the Dooh point group. Using standard techniques' we can show
that the two Is orbitals transform as U ++au+.
g
A useful group theoretical result enables
us to construct symmetry adapted linear combinations of orbitals with these transformation
properties. For a point group G with syrmnetry elements gEG, and a basis set of orbitals
IV& a wavefunction which transforms as the kth irreducible representation is simply given
byq(k)
= 1 X,(g) g'p,. (14)
gFIG
Here x,(g) is the character of the kth irreducible representation. (pm is any member of
the set {Vi} . For the H2 problem it is a simple matter to use equation (14) and a character
table for the Dmh group to give
From Bonds to Bands and Molecules to Solids 177
qJ(o
e
‘) a (PI+ (0
‘2
beg+1 a q--v2 .
(15)
Normalization of these functions gives the same expressions as those in equation (13). The
energies of $, and $b may then be obtained by substitution of these normalized functions
back into equation (3). The reader should check to ensure that the expressions that result
are identical to those of equation (IO).
The hydrogen molecule of course is a very simple example but the numerical generation
of the energy levels of complex molecules, by using ionization energies as estimates of the
H
ij
and evaluating the S..
rJ
by integration of wavefunctions, has proven to be an extremely
valuable way to study molecular electronic structure.
5
2. ENERGY LEVELS OF MOLECULES
2.1 Linear 71 Systems
There is a simplification we can make to the approach in Section 1.B which will be
extremely useful to us in the generation of orbital diagrams of molecules and solids. In the
Hickel approximation, 2,s initially developed for the TI levels of conjugated organic molecules,
all H.. values for the carbon pi AO's are put equal to ~1, all H.. put equal to
H if the ato: are bonded together (zero otherwise) and all the Sij
rJ
(i+j) in equations (6)
and (7) put equal to zero. The two orbital problem of H2 described above is then isomorphic
with the IT orbital problem of ethylene 2. The secular determinant of equation (8) then-
becomes
o--E 8
= 0, (16)
e o--E
which has roots 3 E=afH. Notice that in this model the bonding orbital is stabilized tothe
same extent that the antibonding orbital is destabilized (8). The TI bond energy of ethy.
lene is then 28. Since the overlap integrals have been dropped in the Hiickel approximation
JPSSC 15:3-C
178 J.K. Burdett
the wavefunctions for the bonding and antibonding combinationsare particularly simple.
IJ,= 2+ (Vl+cp,)
(17)
6, = 2-g ('PI--V*).
All conjugated organic systems are open to an analogous treatment. The secular determinant
for ally1 4, the three orbital problem, is just
Notice that the HI3 and Hgl entries are equal to zero since atoms 1 and 3 are not bonded
to each other. Solution of the secular determinant gives the results shown in 6. Oncea+
J2p
again symmetry arguments will enable the same result to be achieved in a neat fashion. We do
not need in fact to make use of all the symmetry elements of the point groups to which the
molecule belongs. (In 4 we drew the carbon skeleton as a straight line when in fact it is
bent.) The important symmetry properties of the functionswe need are those associatedwith
the mirror plane of 5. All functions need either to be symmetric or antisymmetricwith res--
pect to reflexion in this plane. The character table for a point group which contains
(in addition to the identity operation) this reflexion operation is given in Table 1.
Table 1
Using atomic orbitals of 4 as a basis for a representa--
tion it is easy to show that they transform as 2A + B.
E o
?=
Use of equation (14) with 'pm=~l or (p3, two symmetry
equivalent orbrtals, gives after normalization
A I 1
a species
B I -1
5, = 2-!!(CP,.+V,)
b species 5, = 2-$ (cp,---cp,).
(19)
Use of equation (14) with '~,=~$gives the other a species function
a species 5, = ‘p, . (20)
We may now set up a secular determinant for this problem using not 'pl-!$ as a basis
but the functions 5,
-5, of equations (19) and (7.0)
From Bonds to Bands and Molecules to Solids 179
H
11
-E H
12
H
13
H
21 H22-E
H
23 =
0. (21)
H31 H32 I’33
-E
The elements Hii of equation (21) are easily evaluated using the explicit form of the
'i
Hll = {~-~(IP~+'P~)IH~~-~('P~+~~))= a
H22 = (2-B(~P1-~3)~H~2-t(~I-~3))= a (22)
H33 = ('P21HI$) = a
One of the rules of group theory is that orbitals of different symmetry do not interact with
each other, but orbitals of the same symmetry can. (Under some conditions the interaction
energy may however be zero.) We can show this by evaluating
HI3 =(2-'(~l+'P3)~H~2-'('P1-'P3))= $(a-o) = 0 (23)
and
HI2 = (2+ (cp,+'P~)~HI'P~)=J;i8 . (24)
The use of symmetry here has then reduced the 3~ 3 determinant of equation (21) to one (equation
(25)) which contains a 2X 2 block plus a IX 1 block along the diagonal. The
a-E 4i3 0
fig a-E 0 =o, (25)
0 0 o-l?
simplificationthat this has produced is the reduction of a cubic equation in equation (21)
to a quadratic in equation (25). Solution of the 'a block' of equation (25) gives
E=o*&B. (26)
By analogy with the ethylene problem described above the lower energy level $I corresponds
to the bonding, in phase combinationof 6, and 5, _7 and the higher energy level JI,
to the antibonding,out-of-phase combination 8.- The third level $, corresponds to Es. It
L
180 J.K. Burdett
is of the wrong symmetry to interact with any other orbital. Since also H.. =0 between all
13
nonbonded atoms, this orbital remains nonbonding. The use of symmetry considerations,while
perhaps not obvious in this example actually results in general in a considerableeconomy of
effort in many orbital problems as we will see below.
There is a simple expression6 for the energy levels (equation (27)) and orbital
coefficients (equation (28)) of a one-dimensionalchain containing n 71 orbitals. For the
jth level
jx
Ej = of 28cos -n+l
j =1,2,3 , *..n (27)
and for the coefficienton the rth center.
2.
'jr
rjx
= n+l srn n+l o (28)
The number of nodes in the wavefunction increases by one as the energy increases, This is
shown pictorially in Fig. 1 where we show the level structure of the first few members in
the series. Note that the orbital energies are syrmnetricallylocated about E=a 0 This
means that for odd membered chains there is a central, exactly nonbonding, orbital with this
energy.
Fig. 1. Energy levels of the first few linear polyenes
One interestingpoint concerning the energy levels of allyl, is the charge distribution
in molecules of this type as a function of the electronic configuration. Recall that the
square of the wavefunction, $*JI, g'Ives the probability distributionof electrons in that
orbital. So for an arbitrary wavefunction $ written in terms of an LCAO using two atomic
orbitals X, and X, (equation (29)) the electronic charge
VJ= c1x1 + czx2 9 (29)
in orbital X, is NC; where N is the number of electrons in this orbital, (This simple
expressionneeds to be modified' if we go beyond the Hiickelapproximationand include over-
From Bonds to Bands and Molecules to Solids 181
lap, but it will suffice for our purposes here). Correspondingly the electronic charge in
X2 is NC
2
2"
The total number of electrons is obviously equal to N(cf+cE)=N since the wavefunction
is normalized. Applying this technique to the ally1 problem of 6_, with two
electrons in JI, (allyl+) the charge distribution is shown in 2. Note that the central atom
carries more negative charge than the 'ligands' or end atoms. For the allyl- ion itself
with two more electrons (which enter $2) the charge distribution is now different, Here the
A_
0 ’
1-I
-; t -1
ally1 + OIlyI-
IOlargest
electron density is located on the terminal atoms, These results are more conventionally
shown in a slightly different way by calculating the atomic charges on each center
10, All this entails is subtraction from the results of 9 the number of electrons assigned- to
each orbital before interaction, In our case this is just one electron per orbital.
It is instructive to take the details of the ally1 orbitals and see how the situation
is affected energetically by the replacement of an atom by one with a different electronegativity.
We will simulate this by allowing this new atom to have a different c1 value
from the old. There are several ways to obtain the orbital energies of this new species. One
could, for example, solve the secular determinant of equation (18) which has been suitably
modified. The route we will take employs the techniques of perturbation theory.
2,5,5
This
allows us to take the energy levels of a symmetrical molecule (such as allyl) for which it is
easy to obtain the orbital energies and wavefunctions, and derive the corresponding details
for a related molecule which differs in some way. In the present case the perturbation which
we apply is just a change in c1 on one of the atoms. Later, in the next section we will
assemble the orbital diagrams of 'complex'molecules from those of simpler fragments. In this
case the perturbation then consists of making the linkages between the two fragments; i.e.,
Hn,, between two atomic orbitals increases from 0 to the value it takes in the molecule.
The perturbed energy may be written in the following fashion for the ith energy level
E; = Ei(o)+Ei(1)+Ei(2) + . . . . (30)
Where E;(l) and Eit2) are the first and second order energy corrections to Ei
(0)
the unperturbed
energy (lEi(l)I > lEi(2)1) . If the perturbation is the change in the values of some of
the Coulomb and interaction integrals, 6H
uv
then these energy corrections become
E
i
(') = "(~i~H~$i)
and
E (2) =
‘- I'('iIHI~j) I2
i i E (o)_-E (0)
j i j
where
(31)
(32)
(33)
The sum in equation (32) is over all other energy levels, j, of the problem.
182 J.K. Burdett
These expressionsmay readily be applied to the ally1 problem and in fact the use of
this example nicely illustrates the workings of the mathematics of equations (31) to (33).
The first order correction is easy to see. The perturbationwe consider here is just the
changing of c1 on one orbital to CI+ 6a. So Ei(1) Cc2i16a where cil is the coefficient in
orbital i of the atomic orbital (say 'pl of 4) where the perturbationoccurs._ In Fig. 2 we
show the first order energy changes that result by applying this prescription for the two
cases of terminal and central atom substitution,by using the values of the coefficients (the
ci) for 6.
The second order correction is a little more tricky. Consider first the basic case of
two arbitrary orbitals X, and Xp. Then for i = 1
(2) k+l’lb~~2)1’El =
E1
(O)_E (0)
2
(34)
and for i=2
(2)
E2 =
I+‘, b+J’,)I2
E2
(O)_E (0)
I+JJl ii:, I2
=E
(0)_E (0) * (34)
1 2
The result of such a perturbation is to push the lower energy orbital out of X, and X, to
lower energy still, and the higher energy orbital to yet higher energy 11. In the present-
+/IE
!!
Elx’\_
perturbation
case
the second energy correction for orbital i is
2 c?
E1
(2) = ‘il J
1(6d2
E (o)_E (0)
i j
Using the values for the Ei and cil from 6 the second order energy shifts are shown inFig.
2.
Table 2
Substitutionposition
Energy Change
two electron four electron
(35)
central atom -l&Xl +(&#/4& -]&cl\+(6a)2/4fiB
terminal atom --;lScll+5(8~()~/16fiB -;/&al +5(&~)~/16ti2
The total energy shifts for the two electron (allyl+) and four electron (allyl-) cases
are obtained by weighting the energy changes of Fig. 2 with the number of electrons in each
orbital. The results are given in Table 2 above, and show that for substitutionby a more
From Bonds to Bands and Molecules to Solids 183
a)centralatomsubstitution
Firstorder Second order
energy shift energy ahift
b) terminal atom substitution
Second order
energy shift
First order
energy shift
5s:
32AE
Fig. 2. Perturbation treatment of central versus terminal substitution
in a linear three orbital problem. (AE =dk3<0) .
electronegativeatom (i.e., 6a<O) the maximum stabilizationfor the two electron case is for
central atom substitution. For the four electron case the maximum stabilizationoccurs for
end atom substitution. Although the extension of this result to more complex orbital problems
is not obvious, such substitutionpatterns are indeed
12 shows some examples. Here the twelve electron speciesGo-O-Go
N-N-O
I- I-I- I-I--Bractually
observed in'real' systems.
Ga20, where the two perpendicular
TI orbitals corresponging to 9, of 6 are full but those correspondingto q3 are empty,
shows central atom substitution. However the sixteen electron system N20, where both sets
of TI orbitals corresponding to Q1 and $, are full, shows terminal substitutionby the
electronegativeatom. In the polyhalide ions, where the 71manifold is full but the orbitals
corresponding to Q1 and Q3 of the u set are full, the more electronegativeatom is always
located in a terminal position.
The correlation of these results with the charge distributionof 10 is important.Notice
that the more electronegativeatom always occupies the site which carries the highest
184 J.K. Burdett
negative charge in the unsubstitutedparent.
9,lO
This is a result which will be useful to
us later on.
2.2 Cyclic Systems
Let us see how to generate the level structuresof cyclic systems. We shall do this in
three different ways, each of which will be illustrative in its own right. First we will
assemble the cyclobutadienelevels from those of two ethylene molecules 13. 14 shows a mirror--
2 3
--_____
II L4
14
plane present in both sides of 13 which will be useful in dividing the problem into two sym--
metry blocks, one containing functions symmetricwith respect to this plane and one containing
functions antisymmetricwith respect to this plane. The two symmetric functions are shown in
15_* They are just the in-phase bonding orbitals of the ethylene molecule of 2 with energies
of E=a+8. Now the interactionintegral between E1 and c2 is
(5,bk,) = (2-‘(‘pl +‘p2)/H12-’(‘p3+94))
= $(911H194) + ;((~&19~)
= 8,
and so the secular determinant for the synrmetricblock of orbitals is
%1-E %2
I I
(a+B)-E B
= = 0
H21 H22-E B (a+8)-E
This has roots E=a,a+28. The new wavefunctionsare just
$, = 2-'(5,-*c,) = $ (9,+9,+9,+9,) for E = a+26
and
$2 = 2-+52) = $ (9,+9,-9,-9,) for E = a,
(36)
(37)
derived in an exactly analogous way to the ethylene problem of 3. The two antisymmetricfunctions
are shown in 16.- They are the antibonding ethylene combinationswith E=a-8.
Solution of the secular determinant for these levels leads in a similar way to E=a-28 and
E=a. Their respective wavefunctions are $, = 2-i
vJ4= 2-9(&+5J =~(91-92+93-9J.
(5,-c,) =~(c~,-9~-9~+94) and
The complete assembly process is shown in Fig. 3.
One prominent feature of this diagram is the double orbital degeneracy in the middle of the
level stack. It will figure in several discussions later on, Since these levels are degenerate
we have a choice as to how their wavefunctions are written, Although the form of q2
and Q3 of Fig. 3 is perfectly fine, so is a linear combinationof them, 17.- These descrip-
From Bonds to Bands and Molecules to Solids 185
D II
cQ
0
=+B
d
Fig. 3. Assembly of the pv orbitals of cyclobutadiene
tions better emphasize perhaps the nonbonding nature of this pair located at E=c~,
the
energy of an isolated p orbital.
!I
0
%-J; E-J- l-7 = L--J-J- o-o
186
In a second route
18.- The point group of
J.K. Burdett
we will assemble the orbital diagram from four separate p orbitals,
the cyclobutadiene molecule is D4h but for our purposes it is sufficient
to use the group C,,, the cyclic group of order 4. Its character table is shown in
Table 3
c4 E C4 c2 Ck3
A
B
E
I
I 1 1 1
1 --I 1 -1
1 i -I -i
1 -i -I i
Table 3, Using the four px orbitals as a basis for
a representation it is easy to show that they transform
as a+b+e, i.e., each irreducible representation
of cq is contained just once in the reducible representation.
This is a general result which has implications
later in application of these ideas to solids.
For a cyclic (CH), molecule there will be n?T molecular
orbitals, one belonging to each irreducible representation
of the group C,. Using the character table for
C,, and the projection operator of equation (14), it
is easy to write down a normalized wavefunction of a symmetry
$(a) =+(cp, +Q2+'P3+'P4).
By substitution into equation (3) its energy is simply
(38)
Similarly for the wavefunctions of e symmetry
q(e) = i (‘PI + i(P2 + ‘P3 + i(P4 )
$‘(e) = + (cp, + icp, +‘P3 + iv,)
(40)
where in the normalization procedure we have made sure to use I$*$d-r = I. (i.e., used the
complex conjugate JI*) . These two functions may be converted into two new (real) ones using
the license allowed us for degenerate wavefunctions
qN(e) = 2-t($(e) + Q'(e)) = 2~'(rpl- 'p3)
JIN'(e)= 2-' (q(e) - Q'(e)) = 2+((p2- cp,)
(41)
q"(e) and q"'(e) are identical to those of 17.- They are nonbonding orbitals and substitution
into equation (3) gives E=a. Use of the characters for the b representation of Table
3 leads to
Q(b) =;(cp, -'p, +(P3 -'p4) (42)
and an energy of E = a- 26 as found above.
The general result6 for a cyclic system with n atoms is a very simple one. The energy
of the jth orbital is given by
2jx
Ej = a+ 28 cos --n- (43)
where j runs from 0, +I, +2 . . . (*n/2 for n even) or ((n-l)/2 for n odd). The simple
form of equation (43) leads to a useful mnemonic (a Frost circle) for remembering the energy
levels of these molecules. Inscribe in a circle of radius 28 an n-vertex polygon such that
one vertex lies at the bottom. The points at which the two figures touch define the Hiickel
From Bonds to Bands and Molecules to Solids 187
19
energy levels of the molecule as in 19.- The coefficient of the pth atomic orbital determines
the form of the wavefunction as
2aij(p-I)
~j = 2 Cjp(Pp -' f e n=n- .(pp
p=l p=I
(44)
It is an expression which is very similar to that of equation (14). Indeed rewriting it as
in equation (45) highlights the similarity.
n 2aij(p-I)
qj = n-' 1 e n CP--1 *
n Ql
(45)
p=l
The exponential is simply the character xj (Cz-1) of h y 1t e c c ic group of order n as the
reader may readily check for the case of n=4 shown in Table 3. The prefactor of n-f is a
(Hi;ckel)normalizationconstant.
This complex form of the wavefunction is very useful and will be especially so in our
discussions later on solids. A linear combinationof the wavefunctions of a pair of degenerate
orbitals produces two new orbitals which are equivalent in every respect. Using this
fact the functions of equation (44) for the case of degenerate species may be rewritten in a
somewhat nicer form by making use of the trigonometricidentity e
ix
= cosx + isinx (equation
(46)) .
$‘j =+($j+$_j)= in-$ i cos
2xj (p-1) ,(r)
n 'P
p=l
lLy=i($j-$_j) =tn-$ ft_sin 2nj~p-1)_,.pp.
(46)
p=l
It is interesting to see how the wavefunction of equation (44) leads to the energies of equa-
20tion
(43). Substitution of equation (44) into equation (3) gives an expression (equation
(47)) for the energy. This represents the sum of all neighbor interactions (20)
(47)Ej
= cr+B
c
(c.
lp'j(p-1) + 'jp'j(p+l) ),
-2aij(p-I) Laij(p-I) 2Sij -2rrij
E:
-1 n
e -1
--
n n
an e e -l-e n 1
188
=a+8 ic .2 COS /%.I
P
\ * /
27rj
= cr+2Bcos~ .
J.K. Burdett
(48)
Figure 4 shows the energy levels of the first few cyclic molecules. Note how the number of
nodes increases as the level stack is climbed.
00
00
=0”
2
= e;
Fig.4. Energy levels of the first few cyclic polyenes
As a third route to cyclobutadienewe will assemble the diagram in an approximateway
from the levels of ally1 plus an isolated atom 21 using perturbation theory. Again we will-
lol
%8
-b2u
88
- O2"
v
72-a ;
make use of symmetry arguments by classifying the levels as either
with respect to the mirror plane in 2, the same symmetry element
ing the levels of ally1 itself. 22 shows the interactiondiagramsymmetric
or antisymmetric
we used before in generatwhere
we show symmetry(S)
or antisymmetry(A) with respect to the mirror plane of 21.- The middle level of ally1 (q3)
finds no symmetrymatch with an orbital of the single atom, and so remains unchanged in energy
and unchanged in character. As we described earlier the interactionenergy between two orbi-
From Bonds to Bands and Molecules to Solids 189
tals X1 and X2 with energies E1 and Ep respectively is simply given by second order perturbation
theory as
AE12 =
16(x,lHlx,)t
%
(0)-E (0) *
2
With reference to 22 Q1 is depressed in energy by-
1+J,1Ek+)?
(o+fi8)- Co))
the numerator can be evaluated as
which means that
AEC2) = -!- = 0 707 8
fi8 *
(49)
(50)
(51)
and the perturbed energy of Q1 is o+2.12 8. 'P, is destabilizedby the same amount,0.7078.
Now $, and $, may couple in exactly the same way. It is a simple matter to show that the
second order energy numerator is the same as before which means '3,+is depressed in energy by
0.707 S and Q2 raised in energy by 0.7076. The overall result is that 'p,,remains unchanged
in energy but $1 ends up at o+2.128,23. Perturbation theory also tells us about the newa
-2.128
t
ati/-
E
a-
-_
=. - 23a
l1.4lp
-1 /
a +2.128
wavefunctions. With respect to a generalized pair
in a bonding way
of orbitals X, and X,, X2 mixes into X,
190 J.K. Burdett
x;=x1 +
6(Xl IHIX2)
El
(0) _E (0) x2
2
(52)
and X1 mixes into X2 in an antibondingway
x: = x2 -
6(Xl b-h,)
E1
(0) -E(O) Xl *
2
So the new wavefunctionsbecome
(53)
(54)
The new wavefunctionsdescribing the level at E=o have just the forms we found before by
other routes. This is not the case for the new wavefunctions $I and $h. A part of the
problem lies in a mixing, which we have ignored, between two orbitals on the same fragment
(orthogonalbefore the perturbation)under the influence of the perturbation, If x, and x3
are two levels on the same fragment they mix as a result of interactionwith x2 of another
fragment in the followingway
Ix,
-x1
+
80 9: looks like
$: =& + 'cp +$P,+
ti2
+1bIX2) 0,1HIX,)
-
(El
(0) _-E (0)
3 >(El
(O)_E (0)
2 >
(55)
and
= 0.625~p~+O.582(p~+O.625(~~+0.582 'p, (56)
$; = 0.625~~~-0.582'~~ +0.625'03 -0.582~4. (57)
These new functions, though improved, are still not exactly what we found by exact solution
of this problem above, and indeed they should not be. Perturbation theory did not give exact
values for the energies of the top and bottom orbitals of cyclobutadiene,and correspondingly
the wavefunctionswill be approximate too. We could have performed the necessary numerology
by solving two secular determinants,one for the symmetric levels and one for the antisymmetric
ones. The results are of course those of Fig. 3, E=c.x(antisymmetricblock)and E=a,
CX+~ ,a-B (symmetricblock). However the perturbation treatment is a very useful approach
for understandingwhere these levels have come from. In Fig. 5 we show a similar result for
the assembly 24 of the levels of pentalene,- from those of cyclopentadieneand allyl, two
building blocks whose orbitals we have derived already.
result is not perfect.
0+)=03
Again the agreementwith the 'exact'
24
From Bonds to Bands and Molecules to Solids 191
-A-2.066(2.000)
E
I
-1.616
A,S=
-- 1.617(1.814)
s
S
-l.4400.414)
-1.414
S
2.000
'-------Yh 2.424t2.343)
0 > co
Fig. 5. Perturbation treatment for the assembly of the ~II levels of pentalene.
Exact values shown in parentheses. (The energies are in units of 6
relative to cl=O.) The labels A and S refer to the parity of the
wavefunction with respect to the mirror symmetry of the problem
The atomic charges in cyclopentadienyl will of course all be equal from the symmetry of
the molecule, but pentalene has three symmetry inequivalent sites. Either by working out the
form of the wavefunctions of the molecule by using the ideas of perturbation theory, or more
realistically, looking up the coefficients in the compendium of Streitweiser, Brauman and
Coulson7 we may calculate the x-charge distribution in pentalene and pentalene-. They are
shown in 25. Notice that on formation of the negative ion the extra electron density hasappeared
only on the atoms at the I position. Making use of our observation above concerning
the site preferences in substituted molecules, 24 leads to the prediction of electropositiveatoms
in the 1 position and the electronegative atoms elsewhere. This is just what is found
in the molecule B,+N,Hg 6.
-0.20
co-0.17
0.19
-0.20
8
26There
are several results which have their basis in graph theory which are worth mentioning
here. The reader may have noticed that the level structures of almost all of the systems
we have studied (with the exception of cyclopentadiene and pentalene) contain a mirror symmetry
about E=cx. i.e., all the levels occur in pairs at E = cz+ x8. If there are an odd
number of atoms then there is at least one level at E=o. (The level structure of pentalene
(Fig. 5) shows no such sysxaetry.) This is a general feature of orbital situations which,when
all centers are labeled either with a star or left unstarred, no two starred or no two
unstarred centers are adjacent. Such molecules are described as alternant hydrocarbons6 or,
in the jargon of graph theory, as bipartite systems. 27 and 28 show some examples of alter-- nant
systems. For molecules of this type the levels are always symmetrically placed about
E=o. 29 shows some nonalternant hydrocarbons where two starred atoms are adjacent. This-
192 J.K. Burdett
napthalene
azulene
will always be the case for molecules containing odd membered rings. For these systems there
is no such symmetry. Of particular interest to us is the variation in the energy difference
between structuresas a function of the number of PTI electrons (Npx). 30 shows the energy
differencebetween napthalene and the molecules in 28 and 29.- - Ploticethat the curve representing
the energy difference between the two alternant systems is, like their individual
orbital structures, symmetricalabout the half-way point, but the energy difference between
an alternant and nonalternantmolecule shows no such symmetry at all. (Of course for real
systems the only points of 30 which have any chemical meaning is the region around IO elec--
fulvalene
trons (five pairs) where each carbon atom contributes just one pa electron. At this point
napthalene,with two six membered rings is more stable than any of the other molecules.
Another result, coming from graph theory,
11
is that equation (58) holds for regular
graphs, i.e., those systems where the coordinationnumber (v) is the same for all atoms:
From equation (48) therefore for all annulenes where each atom is two coordinate
+ c4B2 cos2 - = 28*.
2aj
n
j
(59)
This is a well known trigonometricidentity. Equation (59) applies to all systems, irrespective
of their alternant or nonalternant character.
A further observation of Guttman and Trinajsticl* is that for neutral carbon compounds,
where each carbon atom contributes one PIT electron, loops of length (4m+2) stabilize the
structure but loops of length 4m destabilize the structure Cm= 1,2 ...). That this is the
From Bonds
case is quite clear from 30 wherestable
for five 7~ electron pairs.
around this half-filled point.
to Bands and Molecules to Solids
the structure with two six rings (napthalene) is most
The molecule 23 is unstable, compared to naphtalene-
193
2.3 Jahn-Teller Instabilities
An extremely useful theorem of Jahn and Teller
4,13
allows us to predict some of the
conditions under which a symmetric structure will lie at a local energy maximum with respect
to particular distortions away from that structure. We will not discuss the operational
details14 here but the basic philosophy is easy to follow. If there is an asymmetric occupation
of a degenerate set of levels at a particular geometry then the energy will be lowered
by a distortion which removes the degeneracy as shown in 31 for the case where the twoI
i
-%E
ic 31‘St
%
C
distortion
electrons have their spins paired. For the triplet situation where the electron spins are
parallel and the Tauli principle demands that the two electrons lie in separate orbitals, then
on distortion one electron goes up in energy while the other goes down and there is no resultrng
stabilization. The cyclobutadiene molecule is an interesting example of this situation.
With four prr electrons the electronic configuration is (a)2 (cz)~ and singlet and triplet
states are possible, The former being Jahn-Teller unstable, and the latter Jahn-Teller stable
at the square geometry. Although the Jahn-Teller approach does not tell us in detail how the
molecule will distort, 32 shows how the energy levels will change on 'dimerization'. This-
u-2/9
-\-
E
I <IQ, _______---
la Eli
EU-p
E a+/3
II II
32dimer
structure lies somewhere along the pathway to fragmentation into two double bonded units
(also shown in 32) and is just the reverse of the assembly process we used above to derivethe
cyclobutadiene levels in the first place. Numerically of course the x energy is equal
JPSSC 15:3-D
194 J.K. Burdett
to 4(a+28) at both the left- and right-hand sides (where two bonds are completelybroken)
of 27. In viewing the distortion however we can assume that two of the linkages shorten
(Ig,I>IBI) while the other two lengthen ([B,\<I@[) . The reader can readily verify using
the first order perturbation
that the new TI energies are
expression of equation
those given in 33. If(31)
along with the wavefunction of Fig. 3
8,+f3,N 28 then the 71 stabilizationon
E
B
cl
a,
II@2 33
distortion is just 2(B,-B,) . The experimentalevidence for singlet butadiene points to a
nonsquare planar molecule, but whether it is a rectangle or rhombus is still open to question.15
We can ask how to stabilize a singlet cyclobutadienemolecule against such a distortion
by mimicking in some way the opening up of the energy gap between HOMO and LUMO which
occurs on geometricaldistortion. One way this can be done16 is by substitutionof the hydrogen
atoms by electron withdrawing and electron donating groups. These will respectively
increase and decrease the value of (~11of the atom to which they are attached. Two disubstituted
possibilitiesarise, 34 and 35.- - Let us use perturbation theory to see which one will
,\ ,\
X Y
‘. ,
,nI’\
, \ 34
Y X
be best. All we need to do is to calculate the energy shifts of the four orbitals of Fig. 3
as a result of increasing c1(by da) on two atoms and decreasing CI(by 6cr)on the other two.
The problem is no more complex than that of the substitutedally1 system of Fig. 2 but we can
use symmetry considerationsto help us out. We will choose the form of the e species wavex
; Y
El 35x
i Y
functions to match the point symmetry of the substituted systems; the orbitals of 17 for 34- and
those of Fig. 3 for 25, respectively.
First let us look at the alternating substitutionpattern of 34. The resulting firstand
second order changes are shown in 36.- Notice there is no first order shift for JI, or
@, since
E
1
(‘I = [(;)2+(+)2]6 a+ [(y+($)2](-6a) = 0. (60)
Also note that the splitting between $2 and Q, is exactly 26~~. Both $, and $,+ are symmetric
with respect to reflection in the two mirror planes 34 one of which passes throughthe
two X atoms and the other through the two Y atoms. Q2 and Q3 are antisymmetricwith
From Bonds to Bands and Molecules to Solids
first second
order order
b
36-
second
order
inclusion
of a/3
respect to one of these operations. So @, and 9, interact energetically in second order
but $, and @, remain unchanged. The interaction energy is
[(;+a+(-; -$(-&a)]2 (sa)2
=-.
(a+2B) -_(a-2!3) 48
(61)
Second, for the pattern of 35 all the first order energy corrections are identicallyzero,
since they are all of the form of equation (60).
C-Q,
of Fig. 3 may be classified
according to their parity under the mirror plane of 35. In second order the symmetric func--
tions Q1 and $, will interact with each other. Similarly the antisymmetric functions Q2
and q, will interact with each other. In both cases the interaction energy is
[(t+ $)So +(-;-$)(-6c()]2 (&a)2
=-.
(o+28)-a
(62)
28
The overall result is shown in 37. With a total of four TT electrons, the stabilizationenergy
of 34 is 2&a+ (6a)2/28 and that of 35 is 2(6a)2/8. Assuming that IE(l)]> ]EC2)1,- then
for this electronic configuration the pattern 34 is preferred. Indeed all 'push-pull'cyclobutadienes
that have been made are of this type 38. I7 The molecule B2N2Rq, where R isa
substituent, has this pattern too.
For the hypothetical case of two n electrons or of six n electrons then the results
of 36 and 37 suggest that the XXYY pattern of 35 should be preferred. 39 shows schematic-- - - ally
an energy difference plot as a function of the number of II electron pairs (P,).
F)C2%
o=c
n
0
N(CB'$JR
(C,H&NVP
AE
38XYXY
A
1 \
f \
I
\
I---,I’ 2 ‘\* n: -4P
\ I /.
\ ’ \ /
\.’ ‘.I
,
I XXYY
196 J.K. Burdett
The picture here is a simplifiedone. According to the Wolfsberg-Helmholz formula
(equation (5)), changing the value of Hii 1a so leads to a change in Hij(B). This consideration
will not affect the energies for the substitutionpattern 34 since all the linkagesinvolve
unlike atoms and here the change in g is given by
68=(&a-&X)=0 . (63)
For the two linkages in 30 which involve like atoms then 6Ra 6a and 68~~--6clrespectively,Using
a proportionalityconstant of q in these last two expressions leads to the changes
indicated at the right-hand side of 37.- The only changes occur in contributions to the
second order energy. The result is a change in the form of the plot of 39. If these secondorder
correctionsare small then an asymmetry 40 develops.- If these effects are large then
there is a reversal of the form 41 of the more stable isomer for three pairs of II electrons,-
.
AE I
XYXY p\
I ‘.
AE
The molecule S2N2, with this electronic configurationis an example of this situation. It
has the pattern 34. We shall see a similar series of plots in the solid state later.Another
way to stabilize the square geometry is to add two more electrons to cyclobutadiene.
The molecules S,+
+2
, Seq+2 and
they have a square geometry.18
3. ELECTRONIC
Te,++'have this electronic configuration,and indeed
STRUCTURESOF SIMPLE SOLIDS
3.1 Energy Bands
In this section we
the extended solid state
simplicity,and consider
tackle the orbital problem for the case of that infinite molecule,
array. First we look at the one-dimensionalsystem with its obvious
the situation presented by an infinite chain of carbon atoms each
carrying one px orbital, i.e., polyacetylene, (CH),. (As we will see, very similar results
apply to other systems with one 'frontier'orbital per atom or structuralunit). In all our
deliberationsin this article we will consider only the case of crystallinematerials, i.e.,
those with a regularly repeating motif. The unit cell of the infinite chain is of length a
42. From the results described in Section 2.A we know qualitativelywhat the orbitals of-
From Bonds to Bands and Molecules to Solids
this infinite system looks like. Equation (27) tells us immediately there is an infinite set
of orbitals, the lowest energy of which is at E = a + 25 and is bonding between all adjacent
pairs of atoms. The highest energy orbital is at E =a--25 and is antibonding between all
197
adjacent atom pairs. Between them lies a continuum of orbitals 43 which we call an energyband
with a width of 4151. Just as in the finite linear molecule case of Fig. I, the number
of nodes increases as the energy increases. Right in the middle of the stack at E=cl there
is a nonbonding orbital.
One easy way to describe the energy levels of such an infinite system is to impose
Born-von Karm>n boundary conditions on the problem, In practice this entails imperceptibly
bending the one-dimensional chain of atoms into a loop, as shown schematically in 44. Of
-
0
b--d
. - - : : -
0 I 2 3 4 5"' I
p 05”
2
3
4 5"'
44course
the number, N, of atoms (orbitals) in this loop is huge. So the bending of the chain
is 'imperceptible' only as far as the atoms of the chain are concerned. Equation (43) then
tells us that there will be N energy levels whose energies are given by equation (64)
where j takes all integral values
sion since N/2 is extremely large
ing a new index k such that
Here a is the unit cell length of
E.
J
= a+Z@cos(Zjn/N) (64)
from 0, ?I, ?r2 ... ?N/Z. This is a very unwieldy expresbut
it may be rewritten in a much neater fashion by definE(k)
= a+28coska. (65)
42 and k=Zjx/Na, called the wavevector takes values- .^
from 0 continuously to *r/a. Figure 6 shows the transition" from the finite to the infinite
case, Recall that for an n membered ring j in equation (43) took values from 0, fl,
+ 2 through (n-l)/2 for an odd-membered ring, So for the five-membered ring the extremal
value of (j1 is j,,,= 2. For the fifteen-membered ring the corresponding value of j,,, is 7
198 J.K. Burdett
a. b.
”\ tT"-" /
E E=a+@coa ka
\i/
or
il 2*j
= a+ 2~~0s~
(N ia very large)
a+2P
-u/a u/a kor
(!I$.!) jFig.
6. The transition from the finite to infinite case pn energy levels
of (a) cyclopentadiene,(b) 15-annulene,and (c) an infinite loop
or one-dimensionalsolid as a function of the j or ir index.
What happens if values of Ijl larger than jm,, are used? The reader may readily show that
the energy levels already derived with j<jmax are generated: i.e., use of j>jmax gives
redundant information. Similarly in equation (65), use of values of Ikl>n/a also leads to
no new information. In the crystalline state the levels lying between -n/a<a<n/a are
called the (first) Brillouin zone. k=O is the zone center, k = ?a/a is the zone edge and
the variation of the energy with k is the dispersion of the band. Figure 6c is just then a
'smoothedout' version of Fig. 6b with a continuum of levels (in the limit N-*m) rather than
a discrete set.
The wavefunctionsof this infinite unit are also easily written down by the substitution
of k=2ja/Na in equation(45) as
qj = N-4 f e
2nij(p-1)
n GP-1
n 'pl
p=l
N (66)
,,,(k) = N-s 1 eik(p-l)a ‘pp
p='l
Group theory allows us to write down a similar expression for the orbitals of this infinite,
periodicallyrepeating chain, An expression analogous to equation(14)for the molecular
case is given in equation (67) for the translationgroup T. This is an infinite group made
up of all translations tET. There are correspondinglyan infinite number of k values
vJ(&)= 1 x,(t) w, (67)
tET
= Ce
ikR,
CD&-$) 3 (68)
tET
From Bonds to Bands and Molecules to Solids 199
but the characters take on a simple exponential form. Here Rt is the distance along the
chain which the translation operation t moves 'p, and cp,(r-_R,)is an orbital translationally
equivalent to (PI, i.e., tcp,(r)=Qt(r-$t). If we write Rt= (p--l)a then it is obvious
that equations (66) and (68) are identical except for a normalization constant. This was to
be expected since via the construction of 44 the infinite translation group and the cyclicgroup
of infinite order are isomorphous. In other words k, the wavevector, in equations
(66) and (68) are the same. In three dimensional systems the vector nature of k becomes
apparent and, as we will see later, we need to replace the exponential in this equation with
a vector dot product exp(ik*R ) where I& = 1 (pi-l)fii,- -t
a sum over the primitive lattice
vectors a.. Just as the vector R
-1 -t
(with Dimensions of length) maps out a direct space
(x,y,s coordinates) so k (with dimensions of (length)-') ma s out a reciprocal space.p The
symmetry adapted functions $(k) are called Bloch functions, $($I = I:c
p kp "p'
As in the case
of the cyclic system, all the levels turn up in pairs (positive and negative k values). TO
understand the orbital structure of the (one-dimensional) solid we only need one set of
values; we choose the positive set, the right-hand side of Fig. 6c.
The energy levels themselves can also be derived by using an approach identical to the
one in Section 2.2 and shown in 20. The energy of the level of the infinite chain describedby
a given value of k is given by multiplying by N, the number of orbitals in the chain,
the energy contribution from the interaction of a single orbital with its neighbors
E(k) = Nx[(Nmtexp (-ikpa)(Pp/HIN-f(exp(-ik(p-l)a)~pp_l+
exp (ikpa)cpp+exp (ik(p+l)a)vp+,)]
= Nx[o+B(exp(ika)+ exp(-ika))]
(69)
= a+28coska .
The top of the band occurs at k=rr/a where coska =-I and E=o-28. The bottom of the band
is found at k=O where cdska =I and E = a+28. The middle of the band occurs at k=n/2a
where coska =0 and E=a. At k=O the phase factor linking an orbital with its neighbor
is, from equation (68), equal to +I and so the Bloch function looks like 45. This is bond--
ing between all adjacent atom pairs. At k=r/a the phase factor is -I and the Bloch function
looks like 46. The number of states with a given energy n(E) is usefully shown in adensity
of states plot (Fig. 7b). n(E) turns out to be proportional to (aE/ak)-1 for the
solid state continuum of levels, " a function which the reader will see has the shape shown.
We will describe below how the density of states is constructed in general. For comparison
the density of states plot for the discrete molecular benzene case is shown in 47. Fillingup
the levels of the energy band with with electrons each will give a total of 2N electrons
for the N atom chain (N is large!), Normally when describing band occupancy we refer to
the number of electrons per unit cell, and so this TI band for the infinite chain may contain
a maximum of 2 electrons,
The collection of pn orbitals we have just studied would describe the electronic 71
band of polyacetylene 48 in a geometry where all the C-C distances are equal. Since thereis
one electron contributed per carbon pn orbital the 71 band is exactly half full. We
show this by the crosshatching of the occupied states in Fig. 7b, Throughout this article
we will use the simple representation of this situation shown in 49. Note that such a system-
200 J.K. Burdett
a) b)
IE a
n/20 T r/a n(E)
k- kF
Fig. 7. (a) E versus k (dispersion) curve for a one-dimensions1 chain of p" orbitals;
(b) density of states plot, n(E). The energy levels are filled up (indicated
by shading) to the Fermi level EF, corresponding to kF. For polyacetylene the
band would be half full.
I
1- I-
E
‘+ 47-
E
-cI2
n(E)
48-
0 Fl____!I
49 50- is
metallic. In 50 we show another electronic possibility where all the levels are filledwith
unpaired electrons. A solid with this electronic arrangement would be a magnetic insulator'.
The interplay between the stabilities of these two distributions will be discussed later
The top of the occupied stack of levels is called the Fermi level with an energy EF. The
corresponding k point (Fig. 7a) is labelled kF.
This LCAO type of approach applied to solids is called the tight-binding model by
solid state physicists. It will be clear that it is really no different to the LCAO scheme
for molecules. In Section 1.2 we showed that the levels of a molecule were accessible by
solution of the determinantal equation
H
ij -SijE ="
Here the H.. and SIJ
1J
were the interaction and overlap integrals between a basis set of
From Bonds to Bands and Molecules to Solids
atomic orbitals {Al}. In the solid state the analogous equation is
Hij(k)-Sij(k)E =0,
201
where now the H.. and S
iJ
ij integrals are those between the Bloch functions (equation (68))
constructed for each atomic orbital contained in the unit cell. Whereas for the molecule
equation (70) is solved just once to generate the energy levels, equation (71) needs to be
solved (in principle) at an infinite number of k points. Sometimes we are lucky and areable
to write down an analytic expression (as in equation (69)for example) for E(k) but more
usually we need to solve equation (71) numerically for a mesh of k points which will repre--
sent the behaviour of the whole energy band when averaged. We will use the method later on
to estimate the energies of structural alternatives. This 'special points' method
21,22
is in
general a very useful way of replacing a complex integration over k space with a set ofindividual
calculations. For a solid state system in general the density of states may be
obtained by using the technique in Fig. 8. Solution of equation (71) at a large number of kpoints
will generate an even larger number of energy levels which may be arranged in order of
increasing energy. The number of such levels contained within a small energy increment can
be evaluated and used to produce a histogram for n(E). A little cosmetic smoothing produces
the desired result. Generation of a dispersion diagram (behavior of E with respect to k)
proceeds in the same manner. Smooth curves are drawn between the sets of points representing
the energies at different k values.
E
n(E) n(E)
Fig.8. Schematic showing the method of generating
the energy density of states n(E)
In section 2 we extensively used the Hiickel approximation by putting S..= 6.. in equa-
1-J
tion (7). In much of this section too we will do the same by writing Sij'k': 6.. since it
iJ
will allow algebraic access to several results of interest. Later we will relax this restric,
tion when we look at more complex systems.
3.2. The Peierls Instability23
In the previous section we chose a repeat unit for the calculation which contained a
single orbital. If we choose a two-atom repeat unit as in 51 how do the results change?-
202 J.K. Burdett
Any observable will have the same calculatedvalue, of course, but the E(k) diagram will be
different since there are now two pn orbitals per cell. We need to solve equation (71)
(with S..(k)= 6..
equationl'(66).
1J
in the H%kel approximation)where the Bloch basis orbitals are given by
With reference to 51 we may write the Bloch function asql(k)
= n-3 (...(pleSika' +'P2+'P3e
ika'
...I (72)
q2(k) = nmt (...(pemika’12 + v5 e
ika'/2
+ 'PBe
3ika'/2
4 1 (73)
Now to set up the 2X 2 secular determinantwe need to calculate the H..
=J
and
HI1 = ($l(k)lHl$l(k))= nef l n-$ l (no) = a (74)
Hz2 =($2(k)IHIJ12(k))= I-I-~*II-'*(no) =a . (75)
Both of these elements are independentof k since there is no interaction (in the Hiickelapproximation)between
(for example) 'pI in one cell and (p2 or 'pg in adjacent cells. H12,
however does contain b and is dependent on k.
HI2 =($l(k)lH1q2(k)) =n -i *n-tn(eika"2 + ewika"')S
= 2$coska'/2. (76)
The secular determinant is then
a-E 2Bcoska'/2
=0 , (77)
2bcoska'/2 u--E
with roots E=af2bcoska'/2, a result shown in 52_. Remembering that a in 39 is half thea'
of 51 the relationshipbetween Fig. 7 and 52 is clear to see. The E(k) diagram of the- a+2P
-
52
two orbital cell is just that of the one orbital cell with the levels folded back along
k =r/2a 53. Now there are two orbitals for each value of k.El0
*/a’
k
53-
k
From Bonds to Bands and Molecules to Solids 203
Another way to derive this result is to take as a basis the II and ?l* levels of the
diatomic unit in the unit cell, (+1 and $, respectively) and to set up a secular determinant
as before. From 54 it is easy to see that HI1 = (a+ g)+ 24.23 8(,ika'+ .-ika' 1 = ta+~) +
B
coska', and from 55 that Hz*= (c~-8)+2-~(-2~')H(e
ika'
fe -ika') = (cl-_B)-Hcoska'.
HI2 from 56 is 2-?T-2-$)Eeika'+ 2-H(+ 2-9)Heika'=-iH sinka'. Hz1 from 51 is
(2-3)(2-q)Teika' + (,-5)(-,-B) E e-ika' = ii3sink.a'. The secular determinant becomes
(a+B)+Bcoska'-F - ib sinka'
= 0.
iE sinka' (a-8)-8coska' -E
Note that this is Hermitian, i.e., HI2 =H&. Equation (78) has roots
(781
E = c1 + d/'(2+2coska')
= c1 + 2Hcoska'/2 (79)
which is the same result as before. Notice that both at the zone edge and the zone center
HI2 =O. So the form of the wavefunctions are easily written down 58. They are just simple-
0
k
*/a
-
58
combinations of the II or TI* functions, in phase at k=O and out of phase at k=r/a'. We
could then regard the one-dimensional energy band of 52 as being made up of 71 and TI* bandsas
in 59 which touch at one point in the Brillouin zone 59. There is a qualification con-- cerning
such a simple viewpoint. Because of the zero off-diagonal entry at k=O, ala' in
equation (78) there is no mixing between the IT and Al* functions at these points. However
204 J.K. Burdett
a-2P
a
a-2fl B
\_/**\_/7r
a-P
a+P
59the
two functions can and will mix together at other k points so that a description in terms
of two separate bands (one IT, the other r*)is not a rigorous one.
At k= a/a' there is an orbital degeneracy. 60 shows one way of writing the wavefunc--
tions at this point, which we have just employed in 58. However a linear combinationofthem,
as in 61 is just as good and emphasizes their nonbonding character. Recall, we had asimilar
choice for cyclobutadiene.
60Polyacetylene
itself however does not have the regular structure shown in 48 but is asemiconductorz4and
exhibits the bond alternation shown in 62_* The energy bands for this
arrangementare readily derived with reference to 63. Now, in addition to different valuesx0’
(I- da’
63
r a
of Rt in equation (68) there are two different interaction integrals, 8, and 8, to be
included. The secular determinant is easily derived by suitablemodification of HI2 (equation
(76)) and becomes
a--E
81e
ikxa'+8 .-ik(l-x)a'
2
=O.
8,e
-ikxa'
+B2e
ik(l-x)a'
a-e
Solution of this determinant leads to
(80)
E = o*(8,2+822+281 82coska')' . (81)
Note that explicit dependence on x has disappeared. (8, and 8, will, of course, be x
dependent). At k= 0, E = o*(B, +8,) and at k=n/a'E = a*(B1- 8,). The E(k) diagram
From Bonds to Bands and Molecules to Solids 205
which results now looks like 64.- For the case where 161]> ]f32] the corresponding density
of states is also shown in 65.- Let us estimate the energy of the distorted structure of 62compared
to that of 43.- We should integrate the function E(k) to get the best answer but
k
64-
65
n(E)
for our purposes it will be sufficient to represent the energy as being the average of that
at the zone edge and that at the zone center. For the symmetric structure 43E=
2~;[(~.+2B)+(a)]=Z(a+S) . (32)
For the distorted structure 62E
= 2*$[(o+B1+82)+(o+B1-B2)]
= 2(a+S1). (83)
If we assume that 28=B,+B, then the stabilization energy on distortion is B,-B, for a
unit cell containing two atoms (lB,]> 1(3,]). This result is an extremely important one.
With one electron per orbital the PII band of 58 is half full,- there is no HOMO-LUMO gap
and there is a degeneracy at the zone edge. On distortion 48+-z, which results in alowering
of the orbital energy, a HOMO-LUMO gap (a band gap, E
g
in the language of the
solid state) is opened up, and the degeneracy is removed. The situation is strongly reminiscent
of that of singlet cyclobutadiene of Section 2.3. There the symmetrical structure
distorted to a dimer structure. Here the atoms of the chain have dimerized in a similar way.
This distortion is then the solid state analog of the Jahn-Teller distortion. It is called
a Peierls distortion. As we will see throughout this article there are many similarities
between the two. The distortion energy in both the molecular and solid state analogs is the
same, (f31-B,)/2 per atom. (In some ways this result is artificial since we have averaged
the zone-center and zone-edge energies, when integration of equation (81) should have been
performed. This approximation will, however, serve our purpose.) 66 shows the analogy in-
206 J.K. Burdett
a -28
/
s--
a pictorial fashion where the energy levels, and the energetic locations of the energy bands
are those we have derived here and in Section 2.2. Table 4 shows some examples of Peierls
distorted systems some of which are described in more detail later in this article. Notice
that the system does not always distort to convert the metallic half-filled band into an
insulator. Also increasing the temperature (e.g., in the V02 example) is often a way to
reverse the effect, Application of pressure is also effective in this regard. How can we
stabilize a system such as that of Fig. 7 against the Peierls distortion? In the case of
cyclobutadiene two extra electrons per four atom unit remove the Jahn-Teller instability.
In the present case we need to fill the entire band with electrons. The result is the structure
of fibrous sulfur and of elemental selenium and tellurium, where there are chains of
atoms with equal distances between them.l* The chain however, is now nonplanar, and a spiral
structure is found.
Table 4. Peierls Distortion in Linear Chains
2)
3)
4)
5)
6)
Polyacetylene
(bond alternation)
Doped Polyacetylene
(no alternation?)
NbI4 chain NbI4 under pressure
pairing up of metal atoms metal atoms equidistant
VOp chain (rutile structure) VO2
pairing up of metal atoms
at higher temperatures
metal atoms equidistant
Elemental hydrogen
H-H dimers
High pressure-metallic
behavior
Presumably... H-H-H-H...
chains
BaVS3 [VS3 chain]
metallic at room
temperature
(TTF)B~
(TTF):2 dimers
Metal-insulator transition on
cooling, but a magnetic
insulator 50 rather
than diamagnetic 49-
(TTF)B~~_~
metallic conduction in chain
(cf. polyacetylene)
Another route, suggested by the observation of the B2N2RI, molecule and the discussion
concerning the relative stabilities of the substitution patterns 34 and 35 of cyclobutadiene- is
via substitution of the chain atoms in a similar way. To generate the levels of the
... XY... solid we need to take those of the two atom cell and apply a perturbation. We will
increase the c1 value of X by 6~ and decrease that of Y by 6a. The resulting energy level
From Bonds to Bands and Molecules to Solids 207
t
E
a ___- _____ _-_______ ____---___-.
____^______ ____ _ ___--___
(8a)‘l
k WQ 4p
lo I.
I
8a________-___--____ _- 67
6Q _/“O’k
7rja
shifts are shown in 67. At the zone edge the energy correction is in first order and is
n.n-t - -6a - &a for the X located level and n-n -a *n-$(-&o) = -61 for the Y located level
At the zone center the energy correction occurs in second order and is (&o)'/48. The wavefunctions
at the zone edge remain unchanged as a result of the perturbation but at the zone
center they mix together. From equation (52) the new wavefunction for the energy level at
E - u+28+ (aa)'/48 is given by 68 and describes a function weighted more heavily on themore
electronegative (Y) atom. Similarly the higher energy orbital at k=O becomes 69 andis
predominantly X located. Perhaps the most important result is the opening up of a band
68gap
at the zone edge. Substituted polyacetylenes or their analogs 70 however are eitherunknown
or poorly characterized but we see no reason why the molecular reasoning should not
prevail here too.
70Having
shown how ...XY... substitution opens up a band gap we need also to examine, as
we did in the molecular case, the relative merits of . ..XYXY... and . ..XXYY... substitution.
To probe this we need to generate the energy levels of a four orbital cell since we
shall be interested in comparing . ..XXYY... and . ..XYXY... alternatives. By making use
of the folding back trick of 53 this is quite simple to do and is shown in 71. Just as the- -
208 J.K. Burdett
levels of the two atom cell could simply be obtained by using the n and Al* orbitals of
ethylene as a basis, so the nodal properties of the levels of the four atom cell can be
obtained from those of butadiene (of Fig. I). Half of the perturbationproblem has been done
for us already since the form of the functions of cyclobutadieneare identical to the levels
of 71 at k=O, The energy shifts at this point for the ...XXYY... and ...XYXY...substitutionsare
then given by 36 and 37 respectively, In fact the energy shifts at k=O- for
the ...XYXY... pattern are just the sum of those shown
cell at k=O plus that at k=a/a.
is simply that of ...XY... folded
the zone edge for the four atom cell
k, =/a
in 67 for the simple two atomThis
comes about because the diagram for ...XYXY...
back along k=nr/2a. This leaves the energy shifts at
of 71. For the ...XYXY... pattern we will make the.
.approximationthat the mean of the shifts at k=O and k=n/a of the ...XY... problem of
67 will suffice. For the ...XXYY...- pattern it is easy to see that the levels split apart
in first order by 26a and that there is no second order correction. The energy shifts for
the two substitutionpatterns are shown in 72.- Approximating the band energy as before by
28a
28a
72-
From Bonds to Bands and Molecules to Solids 209
averaging the zone-center and zone edge values of a filled level the stabilization energies
of the two alternatives as a function of band filling are given in Table 5 where the more
stable structure is indicated with an arrow. Notice the symmetry associated with this Table,
the entries for the quarter filled and quarter empty bands are identical.
Table 5. Stabilization Energies of Structural Alternatives
Band filling . . .XYXY... . ..XXYY...
0 0 0
114 (60)/2+@a)2/Sg 6c(+(6c02/2B +-
112 2 t (W2 /B
314 (W/2+ (W*/gg 6cl+(6a)2/2B c
I 0 0
The relative stabilities of the two possibilities then vary with band filling as in 73where
we show the results obtained by numerical solution of the problem but understandable
using our discussion. At the l/4 and 314 filled band the . ..XXYY... structure is more
stable but at the l/2 filled band the . ..XYXY... structure is more stable in exact analogy
with the case of the substituted cyclobutadienes 39 of Section 2.3.- Again our treatment
here has been virtually the simplest possible. The problem can be reworked by including the
variations induced in the interaction integrals (B) by the changes in ~1, in an appropriately
similar way to the cyclobutadiene problem of 37. The result is very similar.25 The anti--
bonding part of the band receives an extra destabilization for the . ..XXYY... substitution
in an analogous way to that shown at the right-hand side of
AE
37 for the molecular case.-
73-
74JPSSC
15:3-E
210 J.K. Burdett
Taking this into account leads to the two possibilities related to the molecular pictures of
Go, 41 shown in 74 and 75 depending upon the size of the effect. We shall see an example- - of
74 later but the case of 75 is found for (SN), polymer. This material has three n- electrons
per IT band and is found as the -S-N-S-N- isomer and not as -S-S-N-N-. A
band structure calculation25 using the observed geometry of the polymer confirms the state of
affairs in 75. has the alternating arrangement too. Most- Recall that the molecule S2N2
organic donor-acceptor complexes, based on the stacking up of planar molecules crystallize in
the . ..XYXY... arrangement.26 Although we do not discuss these systems in any detail here,
the basic electronic arrangement is one which at its simplest involves the interaction of a
doubly occupied donor level with an empty acceptor level on the adjacent molecule. This
corresponds to a one electron pair per two orbitals problem, and the alternating arrangement
of donor and acceptor is understandable from our discussion above. However, the system
NBP*TCNQFb and Ni(tfd)2.PTZ crystallize in the . ..XXYY... structure. Both of these
species correspond electronically to one electron pair per four orbitals.25 The observation
of this isomer for these two cases is in nice agreement with the theory. A sample of some of
the structures found for various band fillings is given in Table 6.
Table 6. . ..XXYY... versus . ..XYXY...
Electronic
Situation 114
Band Filling
If? 314
1;11;;(,I;;;;;;Tz1iifi;;!g1::.“::N)x
aThese examples are discussed in Section 4.2.
3.3 Building up more Complex Systems
The results achieved for the linear chain may be readily extended to produce the energy
bands of more complex systems. Many of the systems we shall discuss are as yet unknown in
the laboratory. The structure of the ladder, 76 is easily generated by linking two chainstogether.
The secular determinant is a trivial one. If the Bloch orbitals on the two chains
are IJJ~and q2 then
a+28coska-E B
0 (84)
8 a+28coska-E
with roots E= af8+28coska. The result is shown in 77. Another way to generate thisresult
is to use as a basis the II and 71* levels of the ladder unit cell 78 in which casethe
secular determinant is
From Bonds to Bands and Molecules to Solids 211
(a+ 8) +28coska 0
I 0 (a=8)+28coska
78=o
(85)
with the same roots as before. Note that in this new formulation there is no mixing between
@I and jl, _of 78 since they have different parity with respect to reflection in the mirror
plane which bisects the chain. The two bands of 77 may then be regarded as TI and E* bandsof
the ladder.
n(E)
79We
can use the form of the density of states of Fig. 7b to construct the density of
states for the ladder orbitals as in 79. A third way of generating these energy levels isto
'polymerize' cyclobutadiene 80. Here the symmetry of the basis orbitals with respect to-
Q
212 J.K. Burdett
reflection in the mirror plane bisecting the ladder is very useful in simplifying the problem.
81 shows the wavefunctionsof Fig.- 3 which are symmetricwith respect to reflection, and 82I
I I I_-
q
_-
2
--
2 2 2
t, %
0
82
F
r I._- I
2 2 2
a a-2/3
shows the functions which are antisymmetric. For the symmetric block we need to solve the
secular determinant
Now
HI1 (k) -E
Hgl (W
H1 3 W
=o (86)
H33(W -E
HII =(a+28)+28(i*f)eika+ 28($*i)e-ika= (a+28)+8coska (87)
and in general
HII = (a+ n8) + 2ycoska (88)
where y is the interaction integral between one basis orbital and its neighbor in the next
cell evaluated in terms of 8 and the products of orbital coefficients,and n gives the
energy of the orbital of the isolated unit. Equation (86) then becomes
(cr+28)+ coska-E i8sinka
= 0
-iBsinka (cr)-8coska-E
(89)
with roots E=(a-8)* 28cos2(ka/2). The antisymmetricblock can be similarly constructed
and the roots are found to be E=(ct+8)+28cos2(ka/2). The energy bands then look like 83.-
a-3p
t
E
a
83a
+3p
From Bonds to Bands and Molecules to Solids 213
These are just like those of 77 but folded at k=T/2a. The cos2(ka/2) dependence on khas
arisen in exactly the same fashion as that for the single stranded chain when the repeat
unit was doubled in 52.A
similar variety of approaches allows construction of the level pattern of polyacene
where only half of the linkages are made between the two chains 84. We will generate theband
structure of this system first by 'polymerizing' butadiene as shown in 85.- The energy
lcLI<l- Qgca a
85
levels of the four orbital chain are shown in Fig. 1. To simplify our problem we shall divide
them into functions symmetric 86 and antisymmetric 87 with respect to the mirror plane- a+
1.62/3 a- 0.628
a-1.62/3 a+ 0.628
lying perpendicular to the plane of the polymer. For the symmetric block the secular determinant
entries are
HI1 = (a+l.62B) + (2X 2X 0.372X 0.601) Bcoska
H33 = (a-0.62B) - (2X 2X 0.373X 0.601) Bcoska
H13 = [-2X (0.372)']eika+[2X (0.601)2]e-ika
(90)
which leads to the messy expression for the energies of
E=
(Za+B)* 6~(2.24+1.76coska)2+4(0.6-0.4 cos 2ka)
2
(91)
At k=O, E=c1+2.556, a-I.556 and at k=ti/a, E=o,a+B. For the antisymmetric block the
arithmetic is similar and we find at k=O, E=a-2.558, a+1.55$ and at k=xr/a, E=a, a-6*
Qualitatively the band structure looks like 88 with a degeneracy at the zone edge at E=a.Using
these bands we may construct a density of states for the x levels shown in 89. Figure
-
214 J.K. Burdett
a-26.
a-P-
a-
a+2P-
9 shows a band structure for polyacene using the extended Hiickel approach. The broad features
E
88-
/
89
n(E) n(E)
of the TI bands we have generated here are retained but the degeneracy at the zone edge has
been converted into a crossing at smaller k. This occurs as a result of the inclusion of
overlap across the rings, ignored in our treatment. How this occurs is difficult to see
using our approach here. It becomes quite transparent however if the band structure is
assembled via the process in 90. This we will do in a qualitative fashion. The left-handside
of 90 has a band structure which is simply two superimposed diagrams 58. This is shown--
Fig. 9. (a)
(b)
(a) (b)
Band structure for polyacene using the Extended Hiickel approach.
Band structure for a distorted polyacene using the Extended
Hiickel method. (Adapted from Ref. 28.)
From Bonds to Bands and Molecules to Solids
-o+o-o+o- -o+o-Ot_Q-
215
-o~o-o_co-- -0j-d,_cd- 9o
by the dashed lines of 91. As a first approximation at k=O we will just construct bondingand
antibonding pairs from the levels at the bottom and top of the band. (These will not be
the final energies of the bands since the orbitals we have generated are not orthogonal and
there will be a second order energy correction.) At k=m/a, making similar combinations of
..
.- -
S
the orbitals of 58 leads to one bonding, one antibonding and two nonbonding orbitals. Thetwo
nonbonding orbitals are degenerate in 91 only because overlap across the six-memberedring
is neglected. If this is included then they will split apart in energy as in Fig. 9.
The bonding partner (symmetric (S) with respect to the mirror plane lying parallel to, and
bisecting the chain) obviously then drops to lower energy. Since this must correlate with a
symmetric function at the left-hand side of 91 it therefore goes up in energy with decreasingk
and so crosses the other orbital with E>a at a little less than k=n/a. Figure 9b shows
the result of a calculation for polyacene where each of the carbon chains has undergone a
pairing distortion. As in the polyacetylene example itself, a band gap opens up at the zone
edge. We will discuss later the problem of Peierls type distortions in polyacene and related
systems containing more polyacetylene chains.
The energy bands of these chains have been easy to derive since we have gained a considerable
simplification by making use of the mirror symmetry of the problem. A much more
difficult problem arises in the generation of the bands of the unknown species polyazulene 92where
there is no such symmetry. The level structure can be generated as in 93 by usingperturbation
theory in a way analogous to the derivation of the pentalene levels of Fig. 5.
216 J.K. Burdett
Although instructive,this is extremely tedious. 94 shows the TI bands generated by numericalsolution
of the 8X 8 secular determinant. Note that in contrast to the polyacene problem,
the levels do not lie symmetricallyabout E=cr, a direct result of the presence of odd membered
rings. The polyazulenenet is not bipartite and so its level structure at all k lackssuch
symmetry.
94
lool 96The
level structure of poly-[8]-annulene,95, (for lack of a better label) may be derived
in a way analogous to that of the ladder above, by starting off with the orbitals of the eightatom
ring, 96, as a basis. This exercise is left to the reader. The energy bands are shown
in 97.-
97-
From Bonds to Bands and Molecules to Solids 217
Of particular interest is the energy difference between these three structures as a
function of band filling. This is shown in 98.
NPT
is the number of pi electrons per atom.Notice
that the energy difference curve for poly-[8]-annulene is symmetric with respect to
reflection about this point (Npn= I) as demanded by the bipartite nature of both networks.
+
AE
90The
curve for polyazulene on the other hand lacks such symmetry. It has its maximum stability
just after the half-filled point at the band filling of about 0.7. As known for years by
physical organic chemists and implied by the result"
.
of Gutmann and Trinajstic for molecules
discussed earlier, the most stable structure at the half-filled point is the one with six
rings. Notice too that the poly-[a]-annulene structure (with a-rings and 4-rings) is the
least stable alternative at the half-filled point, a result analogous to that found in 3D forthe
8-4 molecular case. Also the stability of polyazulene (5-rings and 7-rings) relative to
polyacene reaches a maximum in approximately the same place in 98 as does the 5-7 relative to-
6-6 molecular structures in 98_.
Not all of the systems we shall describe are one-dimensional ones. Most 'real' systems
are three dimensional in extent. For the latter case we may write the translation group as a
simple product group involving translations along the three lattice vectors,
2.1. In this case the exponential in the Bloch sum of equation (68) becomes
e
i(k b .e,~,)ei(k2b_2.112_a2)ei(k3b_3.R3~3) =ei(k*z)1-l
where b:a. = 2n6..
-1 -, iJ
and the reciprocal lattice vectors b-i are defined by
ai of Section
(92)
(93)
Just as the direct lattice vectors define this lattice, so these reciprocal lattice vectors
define a reciprocal lattice. A simple example will illustrate its construction. In 99 weshow
a primitive orthorhombic lattice. The direct lattice vectors _ai are given by
,.
= bi;
h
-al= ax; 22 %I
=cz , (94)
,. h A
where x, 41 and z represent unit vectors along x, y and z directions. The hi are then
simply
b-1 = (2Irr/a)z; b-z = (2a/a) 2 ; b3 = (2nla) 4, (95)
218 J.K. Burdett
a
and the reciprocal lattice as in 100. Now the first Brillouin zone is defined as the volumeenclosed
by the set of planes which bisect perpendicularly all the lines drawn from one
lattice point to all others in the reciprocal lattice. In practice only a small number of
close points are needed. 101 shows the constructionfor the reciprocal lattice of 100.--
2*
T
b, b,
k
4
100
101-
From Bonds to Bands and Molecules to Solids 219
Various points which lie on the faces, edges or vertices of the Brillouin zone are usually
given symmetry labels. 102 shows the conventional choice for our example. In units of 2x/a
the values of kI, k, and k3 are
r;o,o,o K;O,3, Z;O,O,l
y;-$,O,O T;--g,O,$ lJ;O,?!,$ (96)
g;-$,t,O KG-_?,+,:
Notice that the one-dimensional example which we have exclusively described until now is the
special case of 101, with b2=b3=0. We have chosen for our example, a particularly simple
zone. Other lattices give rise to zones which correspond to more complex polyhedra.20,2g
For the primitive hexagonal lattice, which we will use shortly, the situation is a little
more complex. If the primitive direct lattice vectors are as in 103 then the reciprocal
lattice vectors, by the construction of equations (93) become those of 104. The first
Brillouin zone, also has hexagonal symmetry and is shown in 105. Notice that the point M
is just ($,0,0)2i~/a but K is (f,i,ll)2n/a by simple geometry.
First we tackle the problem of the square net, 106 which has a set of primitive lattice
vectors as, bf and ci where we may visualize c as being very large and a=b (01' 100)...--
The two-dimensional zone we need to consider is therefore the one given in 107.
106-
107-
220 J.K. Burdett
The energy band of the square net is easy to derive since it is a one-orbital problem.
By analogy with the one-dimensional chain of Section 3.1 the E(k) dependence is written as
E(k) = a + ZBcos(k*a )+ LBcos(k*a )
- -1 - -2 (97)
which therefore has a maximum energy of a-48 and a minimum energy of 1x+48. There is a
pictorial problem is showing the dispersion of the energy in two dimensions but what we can
do is trace the energetic behavior along lines joining syrmaetry points of the Brillouin zone.
a-46
a
\
__ ___
/
_____ ______.
a+4p L
M r X M
a-4P
E
a
a-4P
n(E)
108-
109
Using this technique the energetic dispersion is shown in 108. Its density of states isshown
in 109 and indicates a maximum at the half-filled band. The band structure of thisnet
may also be generated by linking together one-dimensional chains. The first step in
this process is shown in 76, 77, where we constructed the energy bands of the ladder. Clearly
the energy bands of a connected set of n chains will have an energy
Ej(k)= cx+ 28cos(jv/n+l) + 2Bcoska ,
by using equation (27) (j = 0,1,2,3 . . . n). Equations (97) and (98)
infinite collection of chains. Notice that the density of states of
some of the features of that of the square net.
dependence of
(98)
become identical for an
the ladder already has
From Bonds to Bands and Molecules to Solids 221
It is not much more difficult given these results to generate the TI bands of the 4a2
net of 110, especially if it is redrawn to emphasize its construction as a square net with
cyclobutadiene at each node. There are now four energy bands, each with a & dependence
given by an equation (99) of exactly the same form as that of equation (97) but
E(k) = (a+ng)+ 2y1cos(~*_al)+ 2y2cos&*a2) (99)
where yi (i= 1,2) now represents the interaction integral along the ai directions between
unit cells associated with the cyclobutadiene orbitals 111-114. It is easy to see that--
3
a+28
n=2
A
a
n= 0
a
n=O
-3
a-2/3
n= -2
yi = B/4 for _lJ and 114. y,=B/2 and yp=O for 113 while yI=O, and y,=g/2 for _l&.The
resulting band structure looks like 115. Notice that at (4,4)2lr/a, the level pattern-
a-3P
M td r X M
is identical with that of the isolated cyclobutadiene system since from equation (99) at this
point cos(k.a.)= 0.
- -1
It is difficult to understand the density of states plots for two- and
three-dimensional systems in the same way we used for one-dimensional ones. The plot for 110
however, is shown in 116 and, if we restrict ourselves to looking at that section of the zonebetween
r and M, we might expect something resembling 117 which is not too different.-
222 J.K. Burdett
n(E)
a-3/3
E
a
Q -30
117-
n(E)
If we were to estimate the energy of this system by choosing the & point in the middle
of a quadrant of the Brillouin zone 107 at ($,4)2n/a then the level structure is identicalto
that of cyclobutadiene. It is therefore very interesting to find that the only known
examples of isolated nets of this sort with a half filled IT band are for M
II
B2C2 where the
squares contain alternating boron and carbon atoms 3o just as in the molecular case, and for
exactly the same reasons we have described exhaustively before.
A slightly more adventurous derivation is of the r band of graphite. The unit cell we
will use is shown in 118 with the primitive lattice vectors needed in equations (92),(93)-
a2
/-a,
118
shown. Here we
functions using
not orthogonal.
minant becomes
will need to take into account the vector nature of k and evaluate Blochthe
phase factor exp(ik*R ) a little more carefully, since ~1 and a2 are- -t
Using as a basis the two pr orbitals of the cell in 118 the secular deter--
From Bonds to Bands and Molecules to Solids 223
o-E
ik * (fgl-$5,) ik*(4al+La
+e +e 3 2)
=o (100)
a-E
with roots E = cl&A 1 H where
A = [3+2cos(~l+_a2)*k+2cos i2*k +2cos a_I *&)I * (101)
Clearly again we cannot show the E(k) dependence in two dimensions in an analogous way to
the one-dimensional case but we will depict the energy changes along line in the Brillouin
zone 102.- At the three symmetry points T,M and K the energy is given by
r E=o+3H; M E=a+B; K E =a+OE . (102)
c/
K
Fig. 10. Band structure for graphite using the Extended
Hiickel approach. (Adapted from Ref. 28.)
119 shows the graphite IT band structure using these results.- Figure 10 shows a band structure
for graphite using an extended Hiickel method which includes the u bands too. Notice
M K I- M
that the levels at E>u are destabilized more than the levels for which E are stabilized.
This has an explanation identical to that discussed in Section 1.2. Inclusion of overlap
destroys the symmetry associated with the orbitals or bands about the E=a level. Notice
224 J.K. Burdett
however that the degeneracy at E=o at the point K is maintained even when overlap is
included. With one electron per PIT orbital in graphite, this band is half filled and IcF
lies at the point K. Graphite is thus a zero-gap semiconductor,one where valence and
conductionbands just touch.
= o”L7 u*0
e
0 120Exactly
the same results are found if, instead of using the individual pa orbitals as a
basis, we use the TI and x* functions of the two orbital cell 12.0. The secular determinantis
then
(o,+B)+Ecosk*a +Bcosk*al-E
-2 -_ iE(sink*al+sink'a2)--
=o
-ig(sink*al + sink*a2) (a-_)-Bcosk*a -gcosk.a--E- -2 - -1
with roots E = cr?A'g as before (equation (101)).
(103)
N’B’N’B’,:Q ,A
‘“:
gl
B\N/B,N/B
The secular determinant of equation (100) may easily be rewritten for the case of BN
121 by using two different values, oN and ctB.- The energy levels may be evaluated by expansion
of the secular determinant in the usual way, as
E = ;(c~B+~)f$&~-cr$~ + 4A2 . (104)
The most striking result of this substitutionis the removal of the degeneracy at the point
K. Here the energies become E=clN and E=oB. 122 shows schematicallythe result predicted.-
122M
K r M
With two IT electrons per pair only the lower band is filled. BN is thus an insulator and,
in contrast to the metallic sheen of graphite,BN is a plain white solid. An extended Hiickel
band structure is shown in Fig. 11. Graphite and BN have half filled pi bands and it is
interesting to see that the observed structure of BN is one where the boron and nitrogen
atoms alternate in two dimensions. Recall that for one-dimensionalchains, with a half-filled
From Bonds to Bands and Molecules to Solids 225
band the structure . ..XYXY... was favored over the alternative . ..XXYY... . Analogous
arguments for the BN system favor 121 over, for example, 123.
-:
3
2 -IC
W
-1:
I
\
/
E,-
f
Fig. 11. Band structure for boron nitride (in the graphite structure)
using the Extended Hiickel approach. (Adapted from Ref. 28.)
The energy levels in graphite are filled up to the nonbonding level at E'o. Occupation
of deeper lying levels contributes to carbon-carbon bonding. Occupation of higher lying
levels has a destabilizing effect and should give rise to an increase in the carbon-carbon dis.
tances. Figure 12 shows how this distance increases with the concentration of intercalated
1.42c11xmCK,
Fig. 12. Experimental values of the CC intraplanar distance as a function
of the extent of intercalation of potassium. (Adapted from ref. 52.)
JPSSC 15:3-F
226 J.K. Burdett
donor atom, in nice agreement with this prediction. (But see Ref. 31 for a further analysis
of this problem.)
A tough problem to iackle using the quantitative method we have adopted above, is the
generation of the band structure of the net shown in 124 which contains 5 and 7 rings. It is
not bipartite and lacks a lot of the symmetry we have found useful before. Its bands need to
be generated numerically. The net is actually found for the nonmetal sheets of ScI$_C2 where
124the
metal ions lie between such nets. If the scandium atoms are considered to contribute
their three valence electrons to the sheets then the unit (B2C2)3- has an average of 5/4
electrons per atom, i.e., the IT band is 518 full. 125 shows another related net, that ofthe
boron atoms in Y2LnB6. 126 shows the energy difference between the graphite, 482 andScB2C2
nets as a function of band filling. (NpT is the number of PIT electrons per atom).
126As
in the case of the one-dimensional examples of 98 (and indeed in the molecular species of-
30) the six-ring net is the most stable at the half-filled point, (i.e., as observed forgraphite)
the net containing 5 and 7 rings is most stable just after this point (i.e., as
observed for ScB2C2) 32 and the482 net is stable at the very beginning and very end of the
filling curve. Notice that the nonbipartite nature of the net 124 shows up in the asymmetryof
its energy difference curve with the bipartite graphite net. Beyond the scope of this
article is discussion concerning the underlying physics of the plots of 98 and 126. In- brief
however we can show
33
that the energy differences between two structures of the type
From Bonds to Bands and Molecules to Solids 227
energetically depicted in 126 (where each atom has the same coordinate number) is simply
given by equation (105)
E(X) =I [Nr(l)-f$-(2)] f(r,x) 3 (105)
r
where N,(l) and N,(2) are the numbers of r-rings in each of the two structures and x(0+1)
is the extent of band filling. The functions f(r,x) are universal expressions which describe
the energetic contributions from an r-ring. Equation (105) also holds approximately for the
one-dimensional cases of 98 (here not all the centers are three coordinate) and also moreapproximately
in the molecular case of 30. The result of Gutman andlier
is simply the special molecular case of x=0.5, and emphasizes
and instability of 4- and 8-rings at the half-filled point.
In the ScBzC2 net of 124 there is clearly a preference for theto
occupy specific sites in the net. A calculation on an all carbon
Trinajstlc'* noted earthe
stability of 6-rings
carbon and boron atoms
net with the geometry of
124 shows indeed that the carbon atoms in ScB2C2 occupy sites of higher negative charge in
the unsubstituted parent. Band structure calculations*' which include both u and 71 bonding
manifolds of orbitals show a stabilization of about 20 kcal/mole for the observed structure
compared to the one where the boron and carbon atoms have been exchanged.
In Section 2.2 we discussed at some length the Peierls distortion on one-dimensional
polyacetylene and showed how the distortion energy was :(B,-S,)/2 per atom. It is of some
considerable interest to calculate the distortion energy in polyacene 84 and graphite 118- since
both of these systems have a degenerate pair of orbitals at the Fermi level. For
polyacene we consider the distortion in 127 which retains the mirror symmetry bisecting the
&&jJ+yJJ-u !27
I I
polymer. So the labels S and A used in 91 to describe the parity of the bands with res--
pect to this plane are still good labels to use during and after the distortion. Immediately
a striking difference between the distortion in polyacene and polyacetylene is apparent. In
polyacene on distortion, the energy changes associated with the levels will occur via a
mixing of the two antisymmetric bands, and via a mixing of the two symmetric bands, i.e.,
the energy change will occur in second order. In polyacetylene the splitting apart of the
levels at the zone edge occurred in first order. We may readily calculate the energy shifts
for the S and A pairs of 91 by solving the relevant secular determinant. The off-diagonalelement
linking either the two S bands or the two A bands at the zone edge is just
(gl-8,)/2 and so for the antisymmetric block
(o-8)-E 2(8,-a,)
=o (106)
$03,~8,) a-E
which has roots E = (a-8/2)~(82f(81-82)2)5/2. Expanding this in terms of (8,-8,)/B
leads to the two new antisymmetric levels at the zone edge
E=C1+ (8,-S,)*
48
; E = a__B - @l-$* . (107)
Similar expressions apply to the synrmetric block and lead 128 to a band gap of (Bl-B,)*/28for
the polyacene distortion 127. Compare this with a value for polyacetylene of (B,-8,)/2.
228 J.K. Burdett
128Let
us build up the levels of a three stranded polymer to see how it will distort. We will
just derive the level structure at the zone edge via the process 129 for the undistortedspecies.
This is easy to do and is shown schematicallyin 130. We will leave it to thereader
to show that during the distortion 131 q5 pushes $, down in energy by exactly the
same amount as +, pushes Q4 up in energy.-JII, therefore remains unchanged in energy on
distortionup to second order. Analogously q3 remains unchanged in energy too in second
order. It can be shown that the splitting apart at the zone edge of VI, and $,+ occurs in
From Bonds to Bands and Molecules to Solids 229
1:0 0 0 .’ .. ”
?!??Im..0 0 0 I_-- ,. .;’ E
third order in the energy and that the band gap is (13,-B,)‘/4Bz. In general34 for polymers
of this type the energy gap goes as (Sl-S,)6n-1 where n is the number of strands and
6 = (8,-B,)/W 1. So as n increases the stabilization energy drops off sharply. 132 shows
the results of some numerical extended Hiickel calculations 25 on systems of this type. The
,
AE 0 acetylene
i polyocene
.\T
\
%.
132
I 2 3 4 5 tcphite
number of strands
ordinate represents the relative energy differences between the undistorted parent and a
slightly distorted version. (The numerical scale of the ordinate will depend on the size of
the distortion.) The rapid decrease in stabilization energy on distortion as n increases is
apparent. Opposing such a stabilization associated with the n levels is a small destabilization
on distortion for the o levels (the so-called elastic forces of the solid state
physicist) which eventually outweighs the 71 distortion energy. For graphite the distortion
is energetically unfavorable.
4. MORE ORBITALS AND MORE DIMENSIONS
4.1 Variations on the One-Dimensional Problem
Our discussion so far has centered on the band structures of systems built up from
single atomic px levels. The results are transferable in many cases to several other
systems. Algebraically the one-dimensional results apply to a chain of atoms bearing s or
da2 orbitals as in 133 and 134. Both of these problems are characterized by an interaction
integral g of the same type as that
linear chain of hydrogen atoms 135.in
42. The simplest problem corresponding to 133 is a- This
should undergo a Peierls distortion since it has
but one electron per orbital, in an exactly analogous fashion to polyacetylene. The result:
ing dimers 136 would be in accord with traditional ideas which we have concerning bondingin
this molecule. On application of high pressure (w2Mbar) the process can be reversed and
136 -+ 135. The physical properties of 135 (or rather its three-dimensional analog) are
interesting. 49 shows that such a structure should be metallic and indeed at these high
-
230 J.K. Burdett
-H-H-H-H-H-H- E
-H H-H H-H H- 136
pressures metallic conduction has been observed.35 This particular result is of interest to
geoahysicists, since the 'atmosuheres' of several ulanets have been sueaested to be made from
such material.
An example of 134 is found in the salts of the ion Pt(CN),2-. This species is square
planar and, in the solid state, the planes stack, 36 one above the next as in 137-* The
valence orbitals of such a molecular unit are shown in 138 along with the orbital occupancy-
-
E
1
138expected
for a low spin d* (Pt(II)) configuration. The z2 orbitals of each unit interact
with the next in the fashion shown in 134 and a band, filled with two electrons, results 139./I
/I----I
_/Z2
\I/II 139- _/ZZ
\lll I 140K2[Pt(CN),+l
is
of 3.48A. The
of the chain to
\I \u
a white solid and an insulator (5X 10-7Q-1cm-1) with equal Pt-Pt
salt can be cocrystallized with elemental bromine which results in
give nonstoichiometric material K2(Pt(CN)4)Br6. 3H2Q. The chain
x--9 1
formulated as Pt(CN),+" - and, as a result electron density is removed from the zL
distances
oxidation
ion is now
band 140.This
material is metallic and, since the electron density has been removed from the very top
of the band where (43)maximum antibonding interactions are found, it has a significantly shor--
ter Pt-Pt distance (2.88A for 6=0.3).
From Bonds to Bands and Molecules to Solids 231
There are many other materials which are built up using exactly the same principles. For
example, the d* complexes of the glyoximate and dioximate ligands 141 and Ni, Pd and Ptalso
lead to stacks of planar units.37 On cocrystallization with halogen, conducting
y+i-- -$I 7 ‘;’
H, O-..“...?
i
AC= r;\ ,N=C/
% HC’
I I
RHc\N / \ d,“R
+bNyM ‘N=C,C*C’H
&_+___~
rlr
&..“...t,
Ii
141Mtgly),
R=H
M(dpg), R=Ph
MPC
materials are produced. Similar features are found in phthalocyaninnes MPc 142 and porphyrins.
Often the crystal structure of the doped, conducting material shows regularly stacked
143 and that of the pure material, slipped stacks of planar units 144.
-M-
144-M-
-M-M-
-MConceptually
very similar to these Pt(CN),, (often abbreviated as TCP) systems are a
whole series of 'organic
typified by the stacking
metals' which contain no metal atoms at all. Such species again are
up of planar molecules such as those in 145.3s In the absence of-
tetrathiatetracene
perylene TTT
Se-Se
c@@Q
Se-Se
S
Q=aJS
145
tetraselenotetracene
TSeT
tetrathiofulvalene
TTF
232 J.K. Burdett
dopants the crystals are not conductors of electricity. However just as in the TCP case,
cocrystallizationwith halogen leads to conductingmaterials. Some examples are
(perylene)0'4(I: .212)o 4 5-SO(Rcm)-l
(TTF)Br0.7 300-500
(TTT& I; 2700
(TSeT)C10 5 2100 .
A particularly interestingseries is shown in 146.- The parent material is an insulator but
insulator -4OO(Rcm)-’ ClO-6 KLcmP’ insulator
the nonstoichiometricallydoped material ((TTF)Br0_7)is a metal. With a half filled band as
in (TTF)22Br- a Peierls distortion leads to dimerizationand the observation of (TTF)i+
pairs in the crystal. Finally, if the band is emptied completely, individual ions (TTF)2+
are found. Not all of the structures of these species end up as neatly stacked planar molecules.
In (TTT)21; for example the stacks are slipped somewhat as in 144.There
is an obvious difference between the energy bands derived from 42, 133 and 134- _
and that associatedwith a single pa orbital at each center 147. Here since the positive
lobe of one orbital overlaps with the negative lobe of the ps orbital carried by an adjacent
atom in the chain, the interaction integral between them is positive rather than being negative.
The only difference this makes to our discussion above is that maximum bonding between
the atoms of the chain is now found at the zone edge, and maximum antibonding at the zone
center, 148.-
148-
k
We are now in a position to qualitativelyoutline the o band structure of a linear
atomic chain containing s and p orbitals on each atom. We need to solve the relevant secular
determinantwhich will include three different values of S, one for PO-po, one for
s-s and one for PO-S interactions. This is
os+2Sss coska-E
--2igspsinka
ZiBsp sinka
=o.
ap-28
PP
coska-E
(108)
From Bonds to Bands and Molecules to Solids 233
We will not write down the (messy) expression for the energy levels but just note that at
k=O and k=r/a the off-diagonal element is identically zero, i.e., there is no s-p mixing
at either of these points. Hsp is at a maximum in fact for k=r/2a. 149 shows a qualita--
tive diagram for such a system constructed by making use of this result and the fact that
IasI>Iopl. The dashed lines show the dispersion in the absence of sp mixing. Here we have
k n/a
-
0 k wa
149assumed
that the s-p separation is large compared to the values of H for the s and p bands
150 shows a more realistic case where the unmixed s and p bands (dashed lines) cross inenergy.
As Ssp increases then the mixing in the middle of the zone gives rise to the energy
dependence shown by the solid lines. Note that because of the different k-dependence of the
p and s orbital energies the 's' band, while purely s-s bonding at the zone center, is
purely p-p bonding at the zone edge. In addition to the presence of po orbitals on the
chain atoms there will be two pr orbitals (p, and p,). These will be degenerate at all k
(i.e., the IT label is still a good one for the chain) and their behavior will be just like
that found for polyacetylene. There is no overlap between these orbitals and any of the
orbitals of o type. A composite picture for a one-dimensional chain of atoms is shown in
151_.
234 J.K. Burdett
Let us now ask the following question. We known that the molecule ferrocene (CSHS)2Fe
and bicyclobutadiene nickel (C,H,),Ni are stable molecules with geometries shown in 152. With
these electron configurations, the eighteen electron rule is satisfied and there is a significant
energy gap between HOMO and LUMO. For what metal will a similar energy gap be produced
in the infinite species of 153 and 154 ? Starting with the case of 154 we need to-- _
consider the five metal d orbitals 155 and the four cyclobutadiene orbitals of Fig. 3. We
have used 156 the alternative form of the degenerate pair of orbitals Q2 and $, shown in
17-. Symmetry arguments are useful here in simplifying this problem. Just as in the chain
of sp atoms noted above, we could separately treat the orbitals of u and TI type, so here
we can classify both the metal d and cyclobutadiene orbitals in terms of their u, TI and 6
symmetry. Let us start off with the orbitals of o symmetry. The secular determinant is
ad-E 2i$sinka/2
= 0. (109)
-2iSsinkaf2 ol-E
Here ~1~ is the energy of the cyclobutadiene orbital +I (=a+26 in the notation of Sections
II and III), and cxd is the d orbital Coulomb energy. Note that neither diagonal entry contains
any k dependence since we have ignored M-M and cyclobutadiene-cyclobutadiene inter-
From Bonds to Bands and Molecules to Solids 235
actions. B in equation (108) is the interaction integral between I), and z2.
It is probably
quite small since the geometry is such that the cyclobutadiene orbitals lie close to the
conical node in 2z 157 which occurs at 8=54.73". The u band structure then has a smalldispersion
and probably looks like 158. The secular determinant for the 6 block will be
158-
236 J.K. Burdett
very similar to equation (108) and 6 will again be small since the overlap between $, and
one of the 6 components is not very favorable. Only in the case of the 71 type interaction
is the interaction integral significant 159.- The secular determinant is given by
ad-E
12bcoska/2
In section 3.2 we looked at a problem of
bands. The bottom of the upper one lies
(See 67 and the associated discussion.)these
results and is shown in 160.-
2$coska/2
= 0. (110)
op- E
this type using perturbation theory. There are two
at c4
d
and the top of the lower one lies at op.
The band structure is then simply derived from
160-
metal
atom
bands of
153
q
To fill all the bonding and nonbonding levels we need a total of twelve electrons per
unit, a metal atom from the iron group is required. Such a system however, would have a
zero band gap because of the touching of the (xz , yz) band and the z2 and (x2-y',xy)
bands. In fact a numerical calculation,which models the cyclobutadienylring properly (the
carbon atoms do carry s orbitals) results in a pushing up of the (xy,yz) band at the zone
edge and an opening up of a band gap of about 1.5eV. The polymetalcyclopentadienylsystem
153 presents a similar case.- Analogous arguments lead to the prediction of a manganese group
metal for this polymer.
Another problem which may be tackled along similar lines is the generation of the band
structure of the hypothetical MB,, system shown in 161.- The chain is composed of edge!-I
H H
z -l-l. ‘I’+. 1_.H. ’ H
X lHjh;\H/X\H.
@ MC 162
H
AH -
From Bonds to Bands and Molecules to Solids 237
sharing MHs octahedra. First we have to decide on the basis set to use for the problem. As
we have seen in Section 3 the same result may be reached in a variety of different ways. Perhaps
the easiest route for this example is to first of all set up the valence orbitals of the
butterfly MH,, unit, 162. With HMH angles of 90" and 18C" this is very easy and is shown in
163. z2- is destabilized (by 2.5eo using the angular overlap mode14) more than xy (by l.Se,)
l-l
I/H E
b-l
t
H
163and
the three other d orbitals remain nonbonding. We know enough about energy bands by now
to be able to write down an approximate band structure. This is shown in 164.-
t
22
E
XY
X2
YZ
MH4 xp-Yi
fragment
energies
)
CX2
__
(Y
II-__ _._
x2-y2
164
The energetic behavior of the levels with respect to k is easily understood. Recall that
whether maximum bonding or antibonding is found at the zone center just depends on whether
the overlap integral of one basis orbital is positive or negative respectively with its part-
238 J.K. Burdett
ner in the next unit cell. (The pi orbitals of the polyacetylene chain is an example of the
former, the pu orbitals of 147 and 148 an example of the latter). The dispersion of the
z* and xy bands is set by the sign and magnitude of the M-H interactions between cells.
From 163 the overlap is negative for z*- but positive (and larger in magnitude) for xy.
Notice the correspondingly larger dispersion of the xy band with the opposite dispersion
behavior to that of z*. If the energetics were dominated by metal-metal interaction, since
the z2 overlap is positive but the x2-y2 overlap negative, the dispersion of the two bands
would be reversed. The xz,yz and x*-y* orbitals have no hydrogen orbital character and
their dispersion is set by the positive overlap for x*-y* and yz and the negative overlap
for xz.. We show a tiny dispersion for yz since it is unfavorably situated for good M-M
interaction compared to the o and TI interactions of x*-y2 and xz respectively. The band
structure ends up as a two above three pattern typical of octahedral coordination. In 164we
show the Fermi level for a d' metal and inquire how the system might distort to lower its
energy. The situation is not quite as simple as the one-dimensional Peierls instability of
polyacetylene or elemental hydrogen, since there are three accessible energy bands to worry
I I I I‘..M_‘~‘..M_‘.‘..M__. “.M__*
0,v \/ \I \ 165
I I I
about.
However a pairing distortion 165 to 166 does stabilize the system considerably as--
shown in 167 and a semiconductor is generated. This orbital problem is very similarj' to
clX*-y2
I=
167-
Ix2
cltl --Ef
x2-y2
166 165
that of NbX,, (X = halogen) which forms chains of the type in 166. NbI,, under pressurebecomes
metallic. Just as in the case of elemental hydrogen (135, 136) the Peierls distortionhere
may be reversed by the application of pressure.
Note that in the distortion shown in 166, and indeed in all of the distortions of the
Peierls type we have studied, the bandwidths of each of the components on distortion together
are less than the undistorted width. This arises as a simple consequence of the fact that the
overlap integrals decrease as the interatomic distance increases. The corresponding interaction
matrix elements then become smaller on distortion.
In fact although the band structure of NbC14 looks very similar to that depicted in 164
the picture which emerges from a quantitative calculation3g is a little more complex. Our
From Bonds to Bands and Molecules to Solids 239
depiction of the dispersion behavior of the lower set of levels in 164 is perfectly fine foran
MHq chain, but with TI bearing ligands the situation is more involved. The basis set of
orbitals now needs to include the effects of TI interactions. With donor ligands the 'd'
orbitals are metal ligand antibonding and may be written as in 168-170. The overlap integral--
xz 169 -49x2-y’&Q* 170
of 168 with its neighbor in the next cell contains a negligible metal-metal component (ignoredbefore)
and a negative d--pr overlap. Maximum bonding is then assured at k=n/a. 169 is a
little problematical. Metal-Metal overlap is negative but d--pn overlap is positive. Calculations
on this system show the two to virtually cancel and a flat band results. Similarly,
the overlap integral of 170 with its neighbor is the sum of goodlaps.
It then receives its maximum bonding at k=O. 171 showsto
that of 164 but with a subtle difference in orbital labeling.
IE
x2-
X2
YZ
positive metal-ligand overthe
new result - very similar
0
k--
n/a
A system where there are three bridging atoms occurs 172 in the chains of MX3
stoichiometry that occur in BaMS3 (M = 'Jr,V, Ta) for example and also in a series of ternary
chlorides AMC13 (A = Cs ,Rb ,N(CH3)4; M = V ,Cr ,Mn ,Fe ,Co ,N~,CU).~' The orbitals we
will use for the MSg unit, assuming SMS angles of 90", are shown in 173_* We have shown the
situation for an MH3 unit for simplicity. There are two M-H o antibonding orbitals (desta-
173C&I
- a, (z2)
bilized by 1.5eo using the angular overlap model) which form a degenerate pair and three
nonbonding d orbitals at much lower energy. (Only one component of each e pair is shown.)
The aI, .z2 orbital is nicely set up for good metal-metal interaction across the shared face
of the coordination octahedron of 172. Figure 13 shows the calculated band structure for
this system. It has similar features to that of the NbCl, species we have just described.
240 J.K. Burdett
2e
PI2e Ill2e
E
la la
HI
le
le le on
la
Fig.13. The d-block bands of a face-sharingoctahedral MX3n- chain.
0 ?r/a
k
The energy variation with wavevector left and a block
diagram right. (Adapted from Ref. 40.)
174 shows how the bands split apart as a result of a pairing distortion 175 of the type des-- cribed
extensively in this article. 2, in fact is the result of a numerical calculationfor
za
VS3 chain. The shaded area indicates the bands which are doubly occupied. If the distor-
t
E
E,--
w
distortion
‘a,
le
174tion
proceeds far enough then a Peierls insulator results. However BaVS3 is a metal. Just
as in the molecular case where dynamic Jahn-Teller processes allow an undistorted structure
to be observed, so too with its solid state equivalent. Although the theory of the distortion
indicateswhen it will occur, it is difficult to predict the size of the effect in general,
especially if, as in the present case, several bands overlap to complicate the picture. (See
also Table 4).
A slightly more complex problem lies in understandingthe electronic structure of the
mixed valence PtII(L)4. ptIV (L)4x;+ (X = halide) chains3' 177 which, with L=NH3 orNEtH*,
are present in Wolfram's red salt and Reihlen's green salt. These are obvious distortions
of the synnuetric'PtIII'structure176. Let us start with the unit cell of 176 shown-
From Bonds to Bands and Molecules to Solids 241
I,’ I,# II’ I,I
-PPt-x-pt-x- - Pt X-Pt-X
/I 11 fr /I
176 177in
178.- We saw the energy levels of the square planar ML,, unit in 138. The only orbitalthat
we need to consider is r2 since this is the only one with appreciable overlap with the
X orbitals. On X we need to include valence s and pu orbitals. The secular determinant
4b 178-
a
for this three orbital problem is simply constructed as in equation (111). Obviously we will
not try to solve this in closed form as it stands. First we will study the result obtained
by setting
-2 8
pd
isinka/:! = 0
2Bsd
coska/2
28pd
isinkaJ2
"P = as*
and equate both interaction integrals (8sd=8pd=@). The solutions are easily
written down
E=a
P (=a,)
(112)
E=
(ap+ad) f J(ap-adlz+ 168’
2
and show dramatically that none of the energy bands have any dispersion at all, i.e., there
is no k dependence. Second we put as=ap but allow Bsd#Spd. The roots now become
E = ap (=a,)
(Orp+Crd)? J(C%p-ad)2+16[8~d-cos2(ka/2)(8pd-8sd)21
(113)
E= --.9L
Now, arbitrarily assuming that lBpdl > 18sd[, we may draw out the
expansion of equation (113) as a power series the roots are
and
4(82d
E=a - '
-.os2(ka/2)(8;d-S:d))
P
'p- od
E=apt
4(8;d -cOS2(ka/z)(B~d-B~d))
op - "d
band structure as in 179. By-
(114)
(115)
The result is actually reminiscent of the behavior we have just noted for NbCl,. Interaction
of .2 with the s orbital on X leads to a dispersion with a cos2(ka/2) dependence with
maximum bonding at k=O, but interaction of r2 with the p orbital on X leads to a
sin2(ka/2) dependence with maximum bonding at k=Tifa. The result, if the two interactions
JPSSC 15:3-G
242 J.K. Burdett
I
ad -B -
E
B=&&a,-ad)
179
are equal, is a dispersion-free band as we found above. The form of the wavefunctions at
the top and bottom of the '2" band are shown in 180
state of the platinum in 176 is Pt'II and thus this
and 181 respectively. The oxidationI+@1
!!?!
'z" band is half full of electrons.
Figure 14 shows the result of an extended Hiickel calculation as the symmetric system 176 isdistorted
to 177.- The initially metallic state has become insulating and an energetic
stabilization has occurred. As in the case of NbI,, the conductivity of these salts
increases markedly on the application of pressure.
Let us look in a little more detail at the form of the orbitals since it will give clues
as to how these systems distort. 182 shows the z2- band for a doubled PtL,+X cell. At the
zone edge the obvious choice of wavefunctions, intermediate in character between those at the
top and bottom of the band, has been made. As in all of the degenerate orbital problems we
have looked at SO far,a linear combination of these two orbitals as in 183 and 184 leads to
--
another perfectly good pair. NOW if the system is distorted slightly (176 + 177) then a band-__
gap opens up at the zone edge. 183 goes up in energy since the Pt-X distances decreasearound
this metal atom and the antibonding interactions become stronger. 184 goes down in-
From Bonds to Bands and Molecules to Solids 243
183energy
since the Pt-X distances around this metal atom increases with a concurrent weakening
of the Pt-X antibonding interactions. This is shown in 185 in an exaggerated way. As the
176
185distortion
proceeds the interaction of
halogen orbitals of the bridge becomes
parts of this band become very narrow,
the z2 orbital on the now planar Pt atom with the
small and the bandwidths of both the upper and lower
as in Fig. 14. The lower band (now full) then corresponds
largely to a zz orbital on the square planar Pt atom 186 which is now Pt
II
. Theupper
band (now empty) corresponds to a strongly antibonding orbital on the octahedral Pt
IV
center 187. The result is a classical mixed valence compound, Pt I1Lq *Pt
IV
- L4X2.
(b)
176 177
Fig.14. (a) d-block bands of [Pt(NH3)4 l Pt(NH3),,C12] chains (176 left and
177 right). (b) The widths of the split z2 band as a function
of the distortion 176 + 177. (Adapted from Ref. 40.)
244 J.K. Burdett
4.2 Three Dimensions
In three dimensions things get quite complicatedbut there are places where we may make
some simplificationsto help us out. Firstly we will consider the simple cubic structure of
188.- In many ways we may regard this as a simple sum of three one-dimensionalchain problems,
Ol 03
k
02
= Y
k
X
188
one lying along each Cartesian direction. For a solid composed of s orbitals (cubium) the
energy dependence on k is, from equation (65) for a one atom cell,E(K)
= 2B(cosk,a+coskya+cosk,a) 16)
This leads to the dispersion curve in 189 where F, M, K and X represent the points
(k,,ky,k,) = (O,O,O) Za/a, (llh,i)2~ia, ($,&,0)2Tl/a and (0,0,&)2a/a, respectively
a-6p
a
a-6P
189X
r M K
For three p orbitals located on a single atom another simple result applies. If we
neglect pn-pT interaction 190 between orbitals on adjacent- atoms and only consider u overlap,
191 then the energy dependence on k for the three p orbitals is, from 148 simply- PX
E(k)=a-2gcosba
pY E(k)=a--2Bcoskya
FZ. E(&)=o--2Bcosksa .
A picture similar to that for the s orbital problem of 189--I
(117)
is shown in 192. Just as the-
From Bonds to Bands and Molecu
a-2p
a
a-2k
__
__
[
X I- M K
half-filled one-dimensional chain of Section 3 underwent a
that the simple cubic lattice with three p electrons would similarly be unstable. The structures
of elemental arsenic and black phosphorus may be viewed in this way. Let us assume for
simplicity that two electrons per atom reside in a deep lying valence s orbital. For these
to Solids
192
245
Peierls distortion, so we expect
Group V elements, this leaves three electrons to half occupy the three p orbitals. 193
shows how the energy bands change on a distortion which involves pairing up all the atoms
along the x, y and r. directions. There are several ways in which the pairing up may be
simple
cubic
arsenic
193done.
It may be shown41 that there are a total of 36 different possibilities for a simple
cubic cell containing eight atoms. Two of these correspond to the black phosphorus 194 andarsenic
195 structures, which we show schematically. On application of pressure to black
phosphorus the metallic simple cubic structure is regenerated.42 So, as in several examples
we have seen earlier, the Peierls distortion is reversed by the application of pressure.
For the Group IV elements these ideas lead43 to the prediction of a structure where the
bonds have broken along two Cartesian directions only. The band description is shown in 196We
have artificially half-filled two p orbitals at the simple cubic structure and left the
third
shown
in an
vacant. One way of executing the distortion leads to the (metallic)white tin structure
in 197, structurally related to diamond 198.
Can we stabilize the simple cubic structure against a Peierls distortion by substitution,
exactly analogous way described above for molecules and simpler solids? The answer is
246 J.K. Burdett
As
196-
simple
cubic
P-tin
-- --I-- -
,---__
77
-2
__:;--,--=_ __ _ _---,
] ;-y: ; ]
,--> +-$:;zz:L+~~ 197
i #ci/,I I-,-
,
L_ - d-gd\fl
II;
I
- -L-I_ ,_
--_
_-__ I-_ ,-_ =
=, _
__,_ ,I&:-_-;
From Bonds to Bands and Molecules to Solids 247
yes but in practice there is a twist. 199 shows what happens to the energy bands of thesimple
cubic structure when we make ...XYXY... substitutions in all three perpendicular
chains. Geometrically the result is production of the rocksalt structure. Eventually if the
simple
cubic
rocksalt
199electronegativity
difference is large enough then both the s and p orbitals of X, the least
electronegative atoms rise above the orbitals of Y. Using these ideas we can see that solids
with four electrons per atom (or eight electrons per XY atom pair, the so-called octets),
unstable at the left-hand side of 199 are stabilized with respect to distortion by increasingthe
XY electronegativity difference. It is interesting tonote that all octets with the
NaCl structure have44 the atomic orbital arrangement shown at the right-hand side of the diagram
(i.e., one where both s, p on X lie higher than s, p on Y). Octets with other orbital
patterns, and therefore with the s ,p orbitals on X close to those on Y, invariably have
either the sphalerite or wurtzite structure. Becall that the diamond structure (degenerate
sphalerite) is geometrically close to that of white tin as we described above (187 and 198).- Clearly
we are not in a position to discuss the energetic stability of one geometry out of
white tin, diamond (hexagonal or cubic), Si(III), or graphite over another for these systems
but it is interesting to be able to cast some light on their electronic structure.
248 J.K. Burdett
From the simple cubic structurewe move to the body-centeredcubic (bee) and facecentered
cubic structures. For these we will derive"' the energy band for a single s orbital
located at each center. The bee structure 200 has a coordinationenvironment around eachcenter
which consists of a cube of eight neighbors plus an octahedron of six neighbors some
14% further away. Since this is a one orbital problem we may immediatelywrite down the
energy dependence on k. For the cubal neighborsE(k)
= cl+2~[cos(~~~~)+cos(~~~p)+cos(~~~g)+cos(~~~24)1 (118)
where
5 = [al+a2+a31; r2 = [aI-a2+a31;
(119)
'3 = [-al+a2+a31 ; rq = [-aI-a2+a31.
A little trigonometryreduces this to
E(k) = a+ 8B[cos(kxa)xcos(kya)xcos(k,a)] (121)
For the octahedralneighbors the problem is identical to that of the simple cubic lattice and
we may write
E(k) = o.+ 2S'[cos(kxa)+cos(kya)+cos(k,a)] . (121)
For the fee structure the local coordination is a cuboctahedronwith therefore 12 neighbors
Q3
J- 201
QI a2 -
fee
201-. Kow we may write down the energy dependence on k for an s band asE(k)
= o+ 28[cos(k.rl)+cos(k.f2)+cos(k.r3)+cos(~~)+cos(k'rg)+cos(k'fls) (122)- --
where
(123)
Again a little trigonometryreduces this to
E(k) = o+ 48[cos(kxa)cos(kya)+cos(~a)cos(k,a)+cos(kya)cos(k,a)] (124)
For illustrativepurposes we will use the special points method, discussed in Section
3.1, to generate a set of representativeenergy levels from these bands. If we want to do
the job properly we would use a larger number of points but here we will use the i points
namely ($,i, f), (i,:,:), (i,:,:) and ($,f By symmetry we need to weight the last
From Bonds to Bands and Molecules to Solids 249
two points three times as heavily as the first two. (In fact the choice of special points
actually depends upon the nature of the lattice. This occurs because the shape of the
Brillouin zones for the simple cubic, body-centered cubic and face-centered cubic lattices
are different. They are shown in Ref. 29. As it turns out, because of our choice of a single
band based on an s orbital, we can use the same set of values for all three structures.) The
levels which result are shown in 202. Notice that the levels for the simple cubic structure
simple
cubic
- 3/28,
fee bee
-2/2P&3J2/3I,
% 2J 2&,*3JLP;
and the bee structure (if either B, or Eb = 0) lie symmetrically disposed about E=a, but
that those for the fee structure show no such symmetry. Similar situations were found in
molecules in Section 3 and the explanation behind these observations is the same as for them.
The simple cubic lattice is bipartite as is the network produced by connecting the central
atom in the bee structure to its first nearest neighbors. The fee lattice (and the hcp
analog too) is not bipartite and so the level structure lacks this mirror symmetry. However
for each of the three structures equation (5S) holds nicely after we divide out the left-hand
side by S. (There are eight sets of levels included by the special points.) For the simple
cubic structure M = 6f3,2and for the fee structures M= 128:. For the cubal neighbors of the
bee structure M = 8Bb2 and for the octahedral neighbors M= 682.
In estimating the relative stabilities of the three structures as a function of band
filling we first need an estimate of the four parameters B,, B,, B, and 8;. These will
vary from system to system. Assuming that all three structures have the same density, then
the interatomic distance is much smaller in the simple cubic structure and so we know that
8,>8,, Eb- Similar distance arguments lead to B,>bb. 203 shows an energy difference
curve between the three structures using the parameters shown (arbitrary units). A very
interesting result is the domination of this picture by the stability of the fee structure
at the quarter filled band and the emergence of the simple cubic structure for the almost
full band. Unfortunately, as is often the case with such ultra-simple examples, there is no
series of structures we can tie in with these results. But the indications both here, and
above, are quite clear. The most stable structure at one band filling may not always be the
lowest energy structure at another filling. Of course molecular chemists are very familiar
with structural changes occurring as a result of a change in electron count. For example,
the geometries of the AF2 molecules are linear for A= Be, Xe (two and five valence electron
pairs, respectively) but nonlinear for A= Si, 0 (three and four valence electron pairs,
respectively).
250 J.K. Burdett
&= 1.41
more stable than
p, = 2p, = I. 12 simple cubic
less stable than
simple cubic
Table 5. The Structures of the Transition Metals
Period N 3 4 5 6 7 8 9 10 II
3d, 4s SC Ti V Cr (Mn) (Fe) (Co) Ni Cu
4d, 5s Y Zr Nb MO Tc RU Rh Pd Ag
5d, 6s (La) Hf Ta W Re OS Ir Pt Au
Structure hcp hcp bee bee hcp hcp fee fee fee
As a final series of examples which show how an extension of the ideas presented in this
article lead to some very dramatic results, we very briefly describe the problem of the
crystal structures of the transition elements. Table 5 shows how the most stable structure
varies across the Periodic Table. Note that, with the exception of the magnetic elements,Mn,
Fe and Co, the structure is determined by the column of the Periodic Table. The sequence
that is found is hcp+ bee+ hcp+ fee as the number of electrons increases, or in the languange
used here, as the d band is filled with electrons. The detailed discussion of this
problem, although fascinating, is beyond the scope of this review. However, in Fig. 15 we
show the results46 of simple Hiickel calculations for the bee, fee and hcp structures.
Fig. 15. Calculated variation in lowest energy crystal structure of the
transition metals with band filling using a Hiickel model.
From Bonds to Bands and Molecules to Solids 251
The computation is more difficult than in the examples we have used before. Here there are
five d orbitals per metal atom instead of the single s or pm orbital. In addition the
overlap integrals between the d orbitals on one atom and those on its neighbors are dependent
upon the d orbitals concerned, and the geometrical location of the central atom with
respect to its neighbors. For example the overlap integral between the two orbitals in 204
is different to that between those
interaction integrals, 8a overlap
in 205_. But, taking this into account, and writing the
204 8 _205
integral leads to the plot of Fig. 15 for the energy
differences between the three structures. Now, the current view46 of the electronic configuration
of the solid metals is one where -I electron resides in an s orbital and the other
valence electrons occupy d orbitals. So the electronic configuration of elemental chromium
is represented as s'd5. Ignoring the effect of the lone s electron on the structure we
can see from Fig. 15 that the bee structure is indeed predicted for chromium with this configuration.
This is the structure actually found (Table 5). In general the agreement
between the observed and calculated structures is quite good. There are some problems with
such a d-orbital-only-model at the right-hand side of the series where the bee rather than
the fee structure is calculated to be more stable.
Iron is found under ambient conditions as a magnetic metal with the bee structure. As
shown in 206 the top group of occupied levels contain unpaired electrons. Under pressure the
hcp structure, predicted in Fig. 15, is found.
207
It is nonmagnetic 207. By knowing how many
unpaired electrons to include in 206 for iron we can recalculate the set of curves of Fig. 15.
at d7 the bee structure is found,46'47 in nice agreement with experiment. The behavior in
206 and 207 is similar to the stereochemical observations associated with high and low spin
molecular complexes. In 208 and 209 we show the relevant orbital patterns and occupancy of
four coordinate d8 systems. One is high spin and distorted tetrahedral and the other low
high spin 208
252 J.K. Burdett
+
low spin 209
spin and square planar. Notice the similarity between the electron occupancy patterns in 206
and 208 and also in 207 and 209_* Associated with the spin change in both cases is a
geometrical change.
Finally we show in Fig. 16 the
derived from the bee structure as a
regarded as arising via the stacking
energy difference curve calculated 4* for two AB alloys
function of band filling. The CsCl structure may be
of square nets in an XYXY sequence and the CuTi structure
as a result of an analogous XXYY stacking. Notice that the shape of the energy difference
curve is very similar to that of 74 and arises for similar reasons.- Figure 16 also
shows the regions where CsCl and CuTi examples are actually found. The agreement is excellent.
CuTi
Fig. 16. Energy difference curve 48 AK between CsCl and CuTi for transition
metal-transition metal alloys as a function of the average number of
d+s electrons per atom (i). In the top part of the diagram the
CsCl structure is more stable and at the bottom the CuTi structure
From Bonds to Bands and Molecules to Solids 253
5. CONCLUSION
In this article the emphasis has lain very much with very simple theoretical ideas,
based largely on the Hiickel simplification of the orbital problem. We have used this approach
because it has provided a vehicle with which to stress the strong underlying symmetry and
connectivity aspects of the level structures we have studied. So we have not just reported
numbers and left the concepts buried in a machine. The cost of such a treatment has of course
been virtually a complete lack of numerology of any type. The most obvious absence from our
discussion of the planar hydrocarbons has been detailed discussion of the role of the o manifold
of orbitals, both in polyacetylene and in the ladders and sheets derived from it. How
do they come into the picture energetically? Do they influence the relative energies of
these structures? In general, the mixing together of s and p or even s, p and d orbitals
requires numerical solution of the electronic problem. The natural extension of the ideas
described in this article is use of the extended Hiickel method, where the determinant of equation
(71) is solved for a basis set containing all the valence orbitals of the unit cell. All
of the band structures we have shown as Figures have been obtained using this method. In
recent years this method, and variation on it, has been a popular one for thecrists of varying
persuasions. Symmetry considerations, however, transcend the calculational method. The
band touching at the point K in the Brillouin zone found in 119 is reproduced in Fig. IO.Degeneracies
appear in the u manifold too at the point r which also have an underlying
symmetry explanation.
We need to go beyond the one-electron model to look at systems with different spin
configurations. The simple ideas presented here will not allow the reader to decide which
out of 206 or 207 will be the more stable arrangement for elemental iron under ambient--
conditions. Another problem to be answered is the prediction of the pressure for the onset
of metallic behavior in any of the one-dimensional Peierls distorted solids we have
discussed. All of these questions require the use of high quality numerical calculations
where the one- and two-electron terms in the energy are properly taken into account. Such
methods are not generally available at present for systems of any complexity. However
pseudopotential-based calculations appear quite promising and recently have been very successfully
applied to the coordination number problem in the octets.4gp50 However the size of the
problem that may be tackled is still quite small as a direct result of computational demands.
There is another problem with such calculations however, a philosophical one. While the
agreement with experiment is spectacular in numerical terms the understanding of why the
results come out the way they do is lacking. They have been described5' as complex ideas
for simple systems and this is one big problem which besets numerical calculations in general
How does one dig out of the numerology concepts and pictures which the nonspecialist can
appreciate? The ideas of symmetry and connectivity which we have stressed in this article,
may be a useful starting point.51
ACKNOWLEDGEMENTS
This research has been supported by the National Science Foundation (NSF DMR8019741)
by the donors of the Petroleum Research Fund administered by the American Chemical Society
and by a grant from the Exxon Foundation. Some of the material in this article has been
assembled from informal discussions within our research group and thanks are due to all of
the participants. Special thanks are due to E. Canadell, T. Hughbanks, G.J. Miller, and
M-H Whangbo who read the manuscript.
1.
2.
3.
F.A. Cotton, Chemical Applications of Group Theory, 2nd Edn, Wiley (1971).
E. Heilbronner and H. Bock, The HfifUMethod and its Application, Wiley (1976).
G.A. Segal, ed., Semi-BnpiricalMethods of Electronic Structure CaZcuZation,Plenum
(1977).
4.
5.
J.K. Burdett, MoZecuZar Shapes, Wiley (1980).
T.A. Albright, J.K. Burdett, and M-H Whangbo, Orbital Interactions in Chemistry,Wiley
(1985).
6.
7.
8.
9.
IO.
II.
A. Streitweiser, Molecular OrbitaZ Theory for Organic Chemists, Wiley (1961).
A. Streitweiser, J.I. Brauman, and C.A. Coulson, Supplemental Tables of Molecular
Orbital CaZcuZations,Pergamon (1965).
R. Hoffmann, Accts. Chem. Res. 5, I (1971).
M.M.L. Chen and R. Hoffmann, J. Amer. C&m. Sot. 98, 1647 (1976).
J.K. Burdett, N.J. Lawrence, and J.J. Turner, Inorg. Cha. 2, 2419 (1934).
For example, A.J. Hoffman in Studies in Graph Theory, Vol. 12, Part II,
D.R. Fulkerson, ed., Math. Assoc. America (1975).
r
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
I. Gutman and N. Trinajstic, Chem. Phys. Lett. 0, 257 (1973).
H.A. Jahn and E. Teller, Proc. Boy. Sot. E, 220 (1937).
J.K. Burdett, Inorg. Chm. 0, 1959 (1981).
J.A. Jafri and M.D. Newton, J. Am. Chem. SOC. 100, 5012 (1978).
R. Hoffmann, Chm. Comun. 240 (1969).
R. Gomper and G. Seybold, Angew. Chem. 80, 804 (1968).
A.F. Wells, Structural Inorganic Chemistry, Oxford University Press (1984).
B.C. Gerstein, J. Chem. Edue. 50, 316 (1973).
N.W. Ashcroft and N.D. Mermin, SoZid State Physics, Saunders (1976).
D.J. Chadi and M.L. Cohen, Phys. Rev. BE, 5747 (1973).
A. Baldareschi, Phys. Rev. B 7_, 5212 (1973).
R.E. Peierls, Quantum Theory of SoZids, Oxford University Press (1955).
For a review see M. Kertesz, Adv. Qua&m Chem. 2, 161 (1982).
J.K. Burdett, unpublished calculations.
See, for example, B.M. Hoffmann, J. Martinsen, L.J. Pace, and J.A. Ibers in Ref. 27,
Vol. III.
J.S. Miller, ed., Extended Linear Chain Compounds, Plenum (1982).
M-H. Whangbo, R. Hoffmann and R.B. Woodward. Proe. R. Soe. e, 23 (1979).
C.J. Bradley and A.P. Cracknell, The MathmatieaZ Theory of Symmetry in SoZids, Oxford
University Press (1972).
30. See, for example, B.G. Hyde and M. O'Keeffe, PhiZ. Trans. R. Soe. 295, 553 (1980).
PI.Kertesz, F. Vonderviszt and R. Hoffmann in IntereaZatedGraphites, M. Dresselhaus,31.
ed., Elsevier (1983).
254 J.K. Burdett
REFERENCES
From Bonds to Bands and Molecules to Solids 255
32. There is no conclusive evidence that the SC atoms are effectively SC(III). In fact
some recent results (E. Canadell) suggest that the B,C, sheet is isoelectronic with
graphite (i.e., the scandium is present as Sc(II).)
33. J.K. Burdett and S. Lee (in press).
34. M. Kertesz and R. Hoffmann, SoZid St. Comn. 47, 97 (1983).
35. P.S. Hawke, T.J. Burgess, D.E. Duerre, J.G. Heubel, R.N. Keeler, H. Klapper, and
W.C. Wallace, Phys. Rev. Lett. 5, 994 (1978). .
36. See, for example, J.M. Williams, A.J. Schultz, A.E. Underhill and K. Carneiro in
Ref. 27, Vol. I.
37. See, for example, L. Alcacer and H. Novais in Ref. 27, Vol. III.
38. See, for example, T.J. Marks and D.W. Kalina in Ref. 27, Vol. I.
39. M-H. Whangbo and M.J. Foshee, Inorg. Che?n. 20, 113 (1981).
40. M-H. Whangbo, M.J. Foshee, and R. Hoffmann, Inorg. Chem. 19, 1723 (1980).
41. J.K. Burdett, T.J. McLarnan, and P. Haaland, J. Cham. Phys. 75, 5764 (1981).
42. J.C. Jamieson, Science, 139, 1291 (1963).
43. J.K. Burdett and S. Lee, J. Am. Chem. Sm. 105, 1079 (1983).
44. J.K. Burdett and S.L. Price, J. Phys. Chem. Solids, 43, 521 (1982).
45. J.R. Reitz, Adv. Sol. State Phys. 1, 2 (1955).
46. D.G. Pettifor, CaZphad, 1, 305 (1977).
47. D.G. Pettifor in ?hysicaZ Metallurgy, 3rd edn. P.W. Kahn and P. Haasen, eds, NorthHolland
(1984).
48. J.K. Burdett and T.J. McLarnan, J. Solid State Chem. 53, 382 (1984).
49. M.T. Yin and M.L. Cohen, Phys. Rev. B 6, 5668 (1982).
50. M.L. Cohen in Structures of Complax SoZids M. O'Keeffe and A. Navrotsky, eds,
Academic Press (1981).
51. J.K. Burdett, J. Phys. Chem. E. 4368 (1983).
52. D.E. Nixon and G.S. Parry, J. Phys. C 2, 1732 (1962).