Origin of the minimum in the total energy curve of diamond using the extended Hückel theory

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Origin of the minimum in the total energy curve of diamond using the extended Hückel theory

M. Lannoo

To cite this version:

M. Lannoo. Origin of the minimum in the total energy curve of diamond using the extended Hückel the- ory. Journal de Physique, 1972, 33 (11-12), pp.1105-1113. �10.1051/jphys:019720033011-120110500�.

�jpa-00207337�

ORIGIN OF THE MINIMUM IN THE TOTAL ENERGY CURVE

OF DIAMOND USING THE EXTENDED HÜCKEL THEORY

M. LANNOO

Laboratoire de

Physique

^des

^Solides (*), ISEN, 3,

^rue

François-Baës, 59,

^Lille

(Reçu

^le

2 juin 1972,

^{révisé le}

7 juillet 1972)

Résumé. ²⁰¹⁴

L’objet

^{de ce travail est}

l’application

de la méthode de Hückel étendue au cas du diamant. On montre d’abord dans des cas

simples

^{comment les}

intégrales

^de recouvrement donnent lieu à un terme

répulsif

^dans

1’énergie

totale. Des formes

approximatives

^{de cette}

énergie

^sont

données en vue de leur

application

^{à des}

systèmes plus complexes.

^{Dans le diamant} ceci conduit à

une forme

analytique simple

^de

l’énergie

^{totale montrant} que le minimum est dû

principalement (dans

^la méthode de Hückel

étendue)

^{au terme}

répulsif

^créé par le recouvrement entre des orbitales liantes

adjacentes.

Abstract. ²⁰¹⁴ This work is concerned with the

application

^{of EHT}

(extended

^Hückel

Theory)

to a diamond

crystal.

^{It is first shown in}

simple

^{cases how}

overlap integrals

^can

give

^{rise to a}

repulsive

part in the total _energy.

Approximate

^{forms of this}

repulsive

energy ^are

given

^{in view of}

their use in more

complex

systems.

Application

^to diamond is

made, giving

^a

simple analytic

^form

for the total _energy, which shows that the minimum is

mainly

^due

(in EHT)

^{to the}

repulsive

energy

provided by

^the

overlap

^between

adjacent bonding

^orbitals.

Classification Physics ^Abstracts

17.10

Introduction. ^- The extended Hückel

Theory (EHT)

^{has been}

applied

^{with success to}

hydrocarbons by

^Hoffmann

[1] in determining equilibrium

^distances

and also in

calculating

^{barriers to} internal rotation in these

systems.

In view of this success the method has been

applied recently

^{in solid state}

physics mainly by

^Messmer

and Watkins

[2]

^to

analogous problems,

^{i. e. defect}

centers in covalent

systems.

The results seem

quite

accurate

especially

^for

nitrogen

ⁱⁿ diamond where

quantitative agreement

^{was obtained.}

However EHT has been criticized in its

principle

and also in some of its

applications [3]

^{which gave}

unphysical conclusions,

for instance in the case of

an interstitial carbon atom in

graphite.

Let us first recall the

principles

of EHT. It is essen-

tially

^{a LCAO} method where one takes into account all resonance and

overlap integrals.

However to avoid

complicated

calculations one uses

semi-empirical

values determined in the follow-

ing

way :

- the intraatomic terms

Hii

^{are taken}

equal

^to

the

corresponding

ionization

potentials ;

- the

overlap integrals Su

^are determined with

simple

Slater forms for the atomic orbitals i

and j ;

- the resonance

integrals Hij

^are obtained from the

overlap integral by

where K is a constant whose better value was found to be 1.75 for carbon.

From this one then solves the electronic

problem, computes

the total energy of the

system

^{as the sum} of the one-electron

energies

and then minimizes it to find the stable

configuration

^{of the}

system.

If we assume that the

semi-empirical

^terms

correctly

simulate the Hartree-Fock matrix elements the most serious error

(especially

ⁱⁿ

determining

^{the stable confi-}

gurations)

lies in the way ^one determines the total energy. The

binding

energy obtained in this way ^can be valid

only

^{in cases where the nuclear}

repulsion

^is

cancelled either

by

half the difference between mole- cular and atomic ^core

energies

^or

by

the difference between molecular and atomic electron

repulsions.

This is

certainly

^{not the}

general

^case

though

^{this was}

shown to be almost

justified

^near the minimum in a

few molecules

[4].

^{However such an}

assumption

^can-

not be

expected

^{to be true for all} internuclear distances.

What we intend to do here is to

analyze

^the

origin

of the minimum in the total energy of a diamond

crystal

^treated

by

the extended Hückel method. We shall be able to show that the amount of

repulsive

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019720033011-120110500

1106

energy necessary ^to

produce

^a rise in the curve is

mainly provided (in

^this

method) by

^the

overlap

between

adjacent bonding orbitals,

^{situation which}

is not encountered for instance in diatomic molecules where the EHT fails.

To do this we first examine the influence of an over-

lap integral

^{on the}

properties

^{of a cubic s band.}

After that similar considerations are

applied

^{to a}

diamond

crystal

^and

finally

^the minimum of energy is discussed.

I. Influence of

overlap

^{for a s band. - The sim-}

plest application

^{of the}

tight-binding

^{method to}

crystals

^{is the case of a s band}

only including

^reso-

nance

integrals Â

^{between nearest}

neighbours.

^{In this}

case one can also include without trouble the

overlap integral S

^{between nearest}

neighbours

^{and see how}

does the mean energy of the band behave as a func- tion of S. To the exact value will be

compared approxi-

mate evaluations which will be useful for diamond where ^an exact calculation is still

possible

^numeri-

cally

^{but where a}

simple analytical

form for the total energy will

give

^{much more}

insight

^{into the}

problem.

1. EXACT CALCULATION. ^- We shall treat the one

dimensional case and the three dimensional case for

a

simple

cubic lattice.

For the linear chain the energy is

Eo being

the atomic energy.

From this one determines

easily

^{the mean} energy per ^atom

One sees that S acts as a

repulsive term,

^{the mean} energy

being higher

than when S ⁼ 0

(if - Eo

^{S is}

negative).

E

can be

computed exactly

^{in the same} way for the cubic lattice where

The

computed

^{result for S}

1/6

^is

given

^on

figure

^1.

2. SERIES EXPANSION. ^- For more

complicated systems

it will be

interesting

^{to have an}

approxi-

mate form for E. For this we shall

try

^{to relate the}

properties

^{of the}

system including overlap

^{to those}

FIG. 1. ^- Comparison ^{of the exact and the} linearized _expres- sion for (E -

Eo)/(2(Â

^{- Eo}

S».

a) Linear chain.

b) Simple cubic lattice.

when S ⁼ 0. For the

system

^without

overlap

^{one has}

to solve

which

gives

fk being

^{a well} defined function of the resonance inte-

gral.

^When

including overlap

^{the new}

equation

^to

solve is

whose solution can be written

This

equation

^{can be solved}

by

successive iterations.

Here we shall

only expand

the solution to first order

in S. For this we choose the same

starting

^{value a}

for all

E(k).

^To first order we have

where

is a solution where the effect of the

overlap

^{would be}

contained in an effective resonance

integral A -

^aS.

From

this

^{we can deduce the new}

barycenter

^{of the}

band E as a function of the two first moments 03BC1, 92 of the band without

overlap

which here reduces to

3. ACCURACY OF THE FIRST-ORDER APPROXIMA- TION. ^- For a S band with nearest

neighbours

^inter-

action,

^{one can write}

where n ⁼ 2 for the linear chain

n ⁼ 6 for the cubic

system.

From this we obtain

We have now to choose a as the best

starting point

for all

E(k).

A reasonable

starting

^{value is}

naturally

the

barycenter

^{of the}

system

^without

overlap

^which

is

Eo

^{in our case. From this}

Comparaison

^{of this} linearized

expression

^{and the}

exact value is made in

figure 1, a)

^and

b)

^{from S = 0}

to

8max

which is the maximum

possible

^{value of S}

for which denominators in

E(k)

^{never vanish.}

There are common features in the two cases.

Approxi-

mate values are lower. Examination of the curves

shows that in both cases the linear

approximation

^is

valid within ten per ^cent of the exact value for S up to 0.4

8max.

We must notice that one can

expand (7)

^to

higher

orders and find the first correction to

(13)

^{to be of}

order

S’

and of the same

sign

^{as the} first order term.

One can also find the

general

^{term of the}

expansion

but this _goes

beyond

^{the aim of this work.}

II. Influence of

overlap

^{for diamond. - We shall}

now

apply

similar considerations ^to diamond

type crystals, starting

from sp3

hybrid

^{orbitals and includ-}

ing

the interactions

step by step

^{in order to}

analyze clearly

^{their influence on the mean energy of the}

valence band.

1. DEFINITION OF THE PARAMETERS. ^- In the

tight- binding

method the most

important

^{term in the}

study

of the electronic structure of diamond is the reso- nance

integral p

^{between nearest}

neighbour’s

sp3

hybrid

^orbitals

pointing

towards each other

(Fig. 2).

If one

only

takes this

integral

^{and the}

corresponding overlap

^{S into}

^account,

^one obtains flat

degenerate

valence and conduction bands centered around

energies

FIG. 2. ^- Schematic representation ^{of the} sp 3 hybrids ^on adjacent ^bonds.

for the valence band and

for the conduction

band,

^the

origin

^{of the} energy

being

^{taken at the mean atomic} energy

Em

^{for the}

hybri-

dized sp3 state. The

eigenfunctions

^are

simply bonding

and

antibonding

orbitals built ^on

pairs

^{of nearest}

neighbours

^as in diatomic molecules.

Starting

^{from this situation one can then}

analyze

the influence of the

remaining part

^{of the Hamilto-} nian which can

conveniently

^{be written}

HB mixing only bonding

^states,

HA antibonding

states,

HBA introducing

the interaction between bon-

ding

^and

antibonding

^states.

Here we shall

study

^{the main}

part

^{of these Hamil-} tonians which are

given by

^the interaction

L1 B

^between

adjacent bonds, dA

^between

adjacent

^{antibonds and}

also

d BA

^between

adjacent

bonds and antibonds

(Fig. 2).

At the end the effect of the

corresponding overlap

1108

integrals SB, SA

^and

SBA

^{will be}

analyzed

^{to first order}

in the same way ^as in Section 1.

2. INFLUENCE OF

Hg

^AND

HBA

WITHOUT OVERLAP. -

The effect of

HB

^on the valence

(pure bonding)

states is

simply

^{to widen the}

degenerate

^flat

^band,

^the

barycenter remaining

^{fixed at}

Em

+ YB-

This is not true for

HBA

^which

couples

^the

bonding

and

antibonding

^{states. As these are} located far

apart

we shall take this term into account

by perturbation theory.

^{To second order one can write the}

displace-

ment

c5B

^{of the}

barycenter

of the valence band :

where the summation is made over the pure

bonding

valence band

states v

> and _pure

antibonding

conduction band

states ( c

> ^whose

energies

^are

Ev

^and

E,,. (N being

the number of

bonding states.)

One must notice here that

HBA gives

^no third order terms so that

(16)

^is

quite

^a

good approximation.

Now,

^{in order to evaluate}

ÔB explicitely

^we

expand Ev - Ec

^with

respect

^{to its} average value f.1B - MA’

This

gives (Appendix A)

This is an alternate

expansion

in powers of

, ..., ,

3. INTRODUCTION OF OVERLAP INTO THE CALCULA- TION. ^- We shall now introduce

overlap integrals SB, SA

^and

SBA

^between

adjacent

^orbitals.

They

^cor-

respond to L1 B, L1 A

^{and L1} BA. For this we use the

approxi-

mate

expression

for E which has been derived in Sec- tion I.

In

(9)

^{one has to}

replace

pi

by

The evaluation of _,u2 first

requires

^the calculation of the contribution _/12B due to the interaction between

bonding

^{orbitals. When one} introduces the interac- tion between

bonding

^and

antibonding

^{orbitals one}

finds

approximately (Appendix B)

Introducing

^these

expressions

ⁱⁿ

(9)

^{allows us to find}

Taking

^{for ce} the value

Em

+ YB ^one

finally

^obtains

Note that here

bB

^and /12B ^are functions of ^a obtained

by replacing Hij ^by Hij - asij.

^{One then takes the}

derivatives with

respect

^{to ce and}

finally gives

^{to a the}

value

Em

+ /1B.

With the same

assumptions

^{as before one finds}

the detailed

formula, neglecting

^{for the moment}

dc5 B/ drx

where the

4§

^{are defined}

by

This is the

simplest

^{formula one can derive. It is}

not

exactly

^accurate but it contains the essential terms and will be useful to

demonstrate

the

origin

of the minimum in the extended Hückel

approxima-

tion. In the

following

^{section we shall discuss its}

validity.

III. Numerical results and discussion. ^- We shall first use our

approximate expression (22)

^{for the}

mean energy of the valence band. After that we

discuss its _accuracy and

validity

^and

finally the origin

of the minimum in the extended Hückel

approxima-

tion.

1. NUMERICAL RESULTS. ^-

There

^are three contri- butions to the mean

energy E

ⁱⁿ

(22) :

^the

bonding

energy ,uB ^as in diatomic

molecules,

^the

following

term

due to the interaction between

bonding

^{and anti-}

bonding

^{states, the last term -}

6 4 h SB corresponding

to

overlap

effects and

acting

^{as a}

repulsive quantity.

For numerical calculations in EHT one

usually

^takes

Slater forms for the atomic 2 s and 2 p orbitals

To determine the Hamiltonian matrix elements one

takes :

- For the intraatomic terms :

which are the atomic ionization

potentials.

- For the interatomic terms :

Then the whole calculation reduces to the determi- nation of the

overlap integrals

which are two center

integrals.

These can

easily

be calculated

using spheroidal

coordinates and

expressed

^{in terms} of tabulated inte-

grals [5]. They

^all

depend

^{one one}

parameter

R

being

^{the nearest}

neighbour’s

interatomic distance.

Here tabulated values

[5]

^{were used for v 7.}

For

greater v

^we calculated a few

points

^{and then}

interpolated.

^{This was done}

by

^{hand and was suffit-}

cient for our purpose but future work will

require

more

precise

evaluation.

The results for _li, and also _MA as a function of v are

given

^on

figure

^{3. One can see} that there is ^no

FIG. 3. ^- Curves of _PB and _PA vs v. The dashed part ^has simply ^been interpolated ^{from v = 2 to v = 0.}

minimum in the

interesting region,

^{i. e. near 4.75}

which is the value of v for the

equilibrium

^distance

1.54

A

in diamond

(c j2

^{= 1.625 a.}

u.).

^{One can}

however

point

^{that a} minimum exists near v ⁼ 2 but it has no influence on the success of

EHT.

We have next studied the

repulsive

^{term in}

^E.

^The

corresponding

curves are

given

^on

figure

^{4. This}

term -

6 Ja SB

^rises

quite strongly

for v smaller than 5 and it is at the

origin

^{of the minimum. To see}

clearly

^what

happens

^we

^give

^{on the same}

figure

- 6

dB SB,

/lB and their sum.

FIG. 4. ^- Curve showing ^different parts ^{of the} mean energy ^{vs v.}

(1) - 6 dB SB.

(2) ^{The best} analytic ^form (26) ^{for the} repulsive ^term.

(3) YB-

(4) ^{Sum of - 6} dB SB and PB.

The

following

^term

ô, gives

^a

gain

in energy increas-

ing

^{with v and}

having significant

^values

only

^{for v grea-}

ter than 4.25. On

figure

5 the upper and lower ^curves

represent

two extreme values for this term. As

they

are

appreciably

^{different we} shall derive a more accu-

rate value before

including

^{it in the total} energy. It is

FIG. 5. ^- t5B ^vs v,

- upper ^{curve :}

- dashed curve :

- lower curve :

1110

TABLE 1

Values which are not

given

^{in the table were not} necessary for our

study.

important

^to notice that

b B

^can

only

influence the exact value of the minimum for values of v

greater

than 4.5.

From this

simple

calculation one

already

^{can say}

that the existence of the minimum in the extended Hückel method is due

mainly

^{to the} inclusion of the

overlap.

^{However as shown}

by figure

^{4 the curve is}

very flat for v

greater

than 4.5 and the exact

localiza-

tion will

certainly require

^{a more accurate form for}

E.

It is what ^we intend to do in the

following

^{where we}

check the

validity

^of

(22).

2. VALIDITY OF THE APPROXIMATIONS FOR E. - We shall

firstly analyze

^the

validity

^of

(22) always taking

into account

only

interactions between

adjacent

^orbi-

tals.

Second-order

perturbation theory

^is

certainly quite good

^for

ôB

^{because even for v}

equal

^to 6.50 the ratio of this term to /lB - MA is less than 0.1

(Table I)

and there are no third order terms.

Another

point

^to be considered is the

expansion

of the second-order term

b B

itself due to bandwidth effects. Detailed considerations

(Appendix A)

^lead

to a better

approximation

^for

bB given by

where

ÔB.

^{and x are defined}

by

The

corresponding

^{curve is}

compared

^on

figure

⁵

to the two extreme values.

Table 1 shows that the correction

d5B/det

^{is not}

very

serious,

^{of order a} few per ^cent.

One ^can also

question

^the

validity

^of the first order

FIG. 6. ^- Total mean energy E ⁼ /-lB - 6 A SB + (Ô.B) E ^{vs v}

expansion (9)

^which

gives

^{here - 6}

Je SB.

^{From a}

determination of a few

higher

^{order terms it is}

possible

to derive a

quite

^accurate

expression

^{for this}

repulsive

terms which turns out to be

It

gives exactly

the first three terms in the

expansion

and it

diverges

^at

SB equal

^{to 0.5 as it should be in}

this model

owing

^to the existence of a

doubly dege-

nerate flat band at the

top

^{of the} valence band.

Compa-

rason of this almost exact result and the linear term is made on

figure

^{4. The linear term} overestimates the

repulsion

^{for v} in the range 3.5 ^to 6

by

^{at most ten}

per ^cent. It

gives

^{the exact value at 3.5.}

Another fundamental

point

^{concerns the influence}

of more distant resonance and

overlap integrals.

We have found that

c5B

^is

only

^increased

by

^a few per cent i. e. to a

negligible

^{extent. This is not so obvious}

for the

repulsive

^{term. However a}

preliminary

^calcu-

lation done for v

equal

^{to 4.75}

gives

for the corrected

repulsive

^{term a} value of 1.47 which lies between the two curves of

figure

^{4. The details of the more refined}

calculations will be the

subject

^{of a}

subsequent work, together

^{with an extension} of the formulae used here.

Figure

6 shows the total _energy

given by

^{the linear}

term -

6 Ja SB

and the corrected value for

c5B (25).

We believe that it is a fair

approximation

^{to the}

exact curve.

3. DISCUSSION OF THE RESULTS. ^- It is necessary to recall the

origin

^{of the}

important parts

in the energy.

The main

part

^near the minimum is

provided by

^the

bonding

energy YB ^as it should be. Over ^a

large

range of interatomic distances _YB behaves

exactly

^{as it}

would when

applying

^{the EHT}

procedure

^{to the}

H2 molecule,

^{i. e. it is a}

uniformly increasing

^function

of v.

Only

^at small distances

(less

than half the

equi-

librium

value)

^does ,uB

present

^a minimum. One ^can then _say that near the

equilibrium

^{distance values of}

YB determined

by

^{EHT fail to}

predict

^{a minimum}

exactly

^{as in the case} of diatomic molecules. From this fact one can conclude

that,

ⁱⁿ

diamond,

^EHT

certainly

^{fails to} describe the

repulsive part

^{of the} energy ^at small distances

exactly

^as it does for the

H2

case.

This is not

surprising because,

^{even if some cancel-}

lation between the nuclear term and a difference between molecular and atomic electronic terms holds at the

equilibrium distance,

^{there is no reason to be}

so for other distances. In

fact, by analogy

^with

H2 [6]

one

expects

^at small distances an increase in kinetic energy associated to some

compression

^{of the wave}

functions. This could be described

by

^a

mixing

^with excited atomic states or

by taking

^{c in the wave func-}

tions as a variational

parameter.

^{This however cannot} be done with EHT.

This conclusion does not alter an

interesting point

which was stressed before. It is the existence of another

repulsive

^term

mainly provided by

^the

overlap

^bet-

ween

adjacent

^{bonds. The}

problem

^{is to know its}

importance

^{relative to the}

repulsive

^{term which}

ought

to be included in _JlB.

There are two final comments to be made :

- In order that all this work be valid it is neces-

sary ^to have ^no

overlap

^{between the}

bonding

^and

antibonding

bands. We have checked this was the

case with v

ranging

^{from 3 to 6.50}

including integrals

up ^to

adjacent

^{bonds and antibonds.}

- From

figure 6,

^{one can}

easily

^{work out the}

elastic constant

Ci 1

^{+ 2}

C12.

^A crude evaluation

gives

about 12 ^x

1012 dyn/cm2 compared

^{to 13.9 found}

by

^{Messmer and Watkins}

[2]

from their 35 carbon atoms

system.

’

Conclusion. ^- The aim of this work has been to

investigate,

^on the basis of a very

simple model,

^the

origin

^of the minimum in the total energy of diamond when

using

^{the extended Hückel}

theory.

^{The main}

reason for this has been found to be a

strong repulsive

term

arising

^from

overlap

^between

adjacent

^bonds.

We believe that the same

arguments apply

^to

hydro-

carbons where EHT was used with success.

The

advantage

of this model is to express the total energy as a sum of three

separate quantities

^which

are :

- The bond _{energy YB} similar to that in a diatomic

molecule,

for which EHT fails to

give

^the

repulsive part

^{at small distances}

exactly

^{as in}

H2.

- A

repulsive part

^{due to the}

overlap

^between

bonds. This

part

^is essential for the success of EHT.

It will also have an influence in more refined calcula- tions.

- A

stabilizing

^{term due to} the interaction bet-

ween

bonding

^and

antibonding parts.

^{It increases} with interatomic distance.

This

decomposition clearly

shows the limitations of the EHT

procedure especially

^{for YB. It also}

gives

the

origin

^{of its success in some cases.}

Further work will

probably require

^the

following improvements :

-

Application

of Mulliken’s

approximation

^{to the}

potential

matrix elements

only

^{combined with an}

exact calculation of the kinetic _energy terms.

- Introduction of the 1 s states in the calculation.

- A variational calculation for the

screening

^para-

meter in the 2 s, 2 _p orbitals.

We believe that such a

study

^{can be useful in view}

of its

application

^{to the}

study

^of

imperfections

ⁱⁿ

covalent solids where we could obtain an idea of the distortion around the defects.

1112

APPENDIX A

We have to evaluate which can be written

This can be

expanded

Now this

expression

^{can be}

changed easily

^into

where B

> is one of the

bonding

^{states and the}

sum is made overall

antibonding

^states.

Including only

interactions between

adjacent

^bonds

and antibonds

gives

which

finally gives

One can extend such ^a calculation

by noticing

^that

evaluating

^{the different terms in the}

expansion

^cor-

responds exactly

^to the determination of the moments of

operators

^{such as}

(HB - YB)"-HBA... by

^{a walk}

counting technique [8].

^In

doing

^{this care must be}

taken of the

sign

of the matrix elements of

HA

^which

depend

^{on the atom which the two}

adjacent

^bonds

have in commun.

In the same way ^we have derived the first correc-

tive terms to

(A. 6)

^{and found the sum of the three}

first terms as

In order to estimate further terms

simply

^{one can}

choose a function

and determine a

and p

^{in such a way that}

expansion of f (x) gives

^{the same} three first terms as

(A. 7).

The result is

One can believe that

f (x)

^will

approximate (A. 3) nicely

^{at least for} small values of x which is the case

here where x is of order 0.2.

A first reason for this is that one can show

(A. 2)

to be a fonction of x alone

(and

^{not of other combi-}

nations of

d B, A’

^{and also one can evaluate the next}

term in the

expansion

^{whose exact value is -}

^{26 x3} compared

^{to -}

^{20 X3} given by f (x).

^{The error is then}

certainly

^{smaller than 6}

^X3

which for x ⁼ 0.2 turns out to be of order 5

%.

APPENDIX B We shall

try

^to derive the

approximate

^formula

(19)

by examining

^{two extreme cases}

corresponding

^to

perturbation theory

^on

degenerate

^{and non}

degenerate

states.

The

problem,

^{for a} fixed value of the wave vector

k,

reduces to the interactions between two 4 x 4 matrices.

If the valence band states can be considered as

degenerate,

^{then one has to}

diagonalize

^{a matrix}

ô

being

^a matrix with

general

^term

I,

the unit matrix.

v and v’ two solutions of the valence band

problem.

In this case the first moment

and,

^{to second order}

Then,

ⁱⁿ the limit where _{/12 B} vanishes

(19)

^{is valid.}

In the

opposite limit,

^at fixed k there are four distinct values

EU

for the valence band levels. Second

order

perturbation theory gives

^{for each of this state}

a shift

ô,,.

In this case :

and for _J12

which to second order in

bUV

^{reduces to}

The last term is a corrective term to

(19)

^{due to}

valence bandwith effects. It can be evaluated

by

^the

methods of

Appendix A, giving

^{for it}

From Table

I,

^this

gives

^{for v = 5.25}

through (2) (dJl2/det)

^{an extra}

gain

in energy of 0.03 eV which

can be

neglected compared

^to

bB

^itself.

It is to be noted that the true situation in interme- diate between these two extremes

for,

ⁱⁿ the valence band of

diamond,

^{there are three near}

degenerate

states and a non

degenerate

^one

lying quite

^lower

in energy. This could

perhaps

^reduce

slightly

^{the cor-}

rective term.

References

[1]

^HOFFMANN

(R.),

^{J. Chem.}

Phys., 1963, 39,

^1397.

[2]

^MESSMER

(R. P.)

and WATKINS

(G. D.), Phys.

^Rev.

Lett., 1970, 25,

^656.

[3]

^{MOORE Jr}

(E. B.)

and CARLSON

(C. M.), Phys.

Rev.,

B, 1971,

4,

6,

^2063.

[4] NEWTON (M. D.),

^BOER

(F. P.)

and LISCOMB

(W. N.),

J. Am. Chem.

Soc., 1969, 88,

^2367.

BOER

(F. P.),

^NEWTON

(M. D.)

and LIPSCOMB

(W. N.),

Proc. Nat. Acad. Sci.

US, 1964, 52,

^890.

GOODISMAN

(J.),

^{J. Am. Chem.}

Soc., 1968, 91,

^6552.

[5]

^KOTANI

(M.),

^ISHIGURO

(E.),

^HIJIKATA

(K.),

^NAKA-

MURA

(T.)

and AMEMIYA

(A.),

^J.

Phys.

Soc.

Jap., 1953, 8,

^463.

[6]

^SLATER

(J. C.),

^«

Quantum Theory

^{of Molecules and}

Solids »,

^Vol.

1, McGraw-Hill,

^1963.

[7]

^NEWTON

(M. D.),

^BOER

(F. P.)

and LIPSCOMB

(W. N.),

J. Am. Chem.

Soc., 1966,

^{88v 2353.}

[8]

CYROT-LACKMANN

(F.),

^{Adv. in}

Phys., 1967, 16, 393 ;

J.

Phys.

^Chem.

Solids, 1968, 29, 1235.