Valence Bond Theory and Superconductivity - ACS Publications

Chapter 2 Valence B o n d T h e o r y a n d Superconductivity 1,2

2

Richard P. Messmer and Robert B. Murphy 1

General Electric Corporate Research and Development, Schenectady, NY 12301 Department of Physics, University of Pennsylvania, Philadelphia, PA 19104

Downloaded by UNIV OF PITTSBURGH on May 3, 2015 | http://pubs.acs.org Publication Date: August 28, 1987 | doi: 10.1021/bk-1987-0351.ch002

2

Two necessary criteria for a theory of superconductivity are a phase coherence of the wave function and an attractive electron-electron interaction. We review how the BCS theory achieves these criteria and then show how a valence bond wave function can also meet these conditions. The energy scale of the latter approach has a larger range in principle than that possible via the electron-phonon interac tion.

The recent discovery of ceramic high-7 superconductors has forced a re-examination of the basic concepts and physical assumptions employed in current theoretical approaches. In re examining basic concepts, it is well to remember that the true N-electron wave function may be expanded in terms of components each of which is made up of Ν single particle functions and that this expansion can be made in (at least) two different ways: c

Φ = Σ c„ Φ„ ν

(1)

{molecular orbitals /Block orbitals /delocalized basis) Φ = Σ d Φ* ν (valence bond orbitals/localized basis) u

(2)

The former expansion is the one typically assumed both in molecular and solid state work. The ease with which the single particle basis can be obtained in this case is certainly a significant advantage. Furthermore, it might be argued that either approach is equivalent in the end, and hence it makes sense to choose the mathematically more straightforward approach. In fact, one always considers only a small fraction of the terms in either expansion and the more relevant question is which is more rapidly convergent and/or more physically

0097-6156/87/0351-0013S06.00/0 © 1987 American Chemical Society

In Chemistry of High-Temperature Superconductors; Nelson, D., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

14

CHEMISTRY OF HIGH-TEMPERATURE

SUPERCONDUCTORS

m o t i v a t e d . T h e c o n c e p t s d e r i v e d b y t h e t w o a p p r o a c h e s m a y b e q u i t e different. i l l u s t r a t e d r e c e n t l y i n a s e r i e s o f m o l e c u l e s a n d c l u s t e r s , (1-5)

T h i s has b e e n

for w h i c h calculations u s i n g

E q u a t i o n (2) w e r e c a r r i e d o u t . F o r m e t a l c l u s t e r s it w a s f o u n d (1,2)

that e l e c t r o n s b e c a m e

l o c a l i z e d i n t o i n t e r s t i t i a l r e g i o n s . A n d f o r d o u b l e (3), t r i p l e (4) a n d c o n j u g a t e d b o n d s

(C H ) 6

6

(5), it h a s b e e n f o u n d t h a t b e n t - b o n d s m a d e u p o f e s s e n t i a l l y t e t r a h e d r a l h y b r i d s d e s c r i b e t h e b o n d i n g . T h e s e c o n c l u s i o n s a r e q u i t e different t h a n t h o s e b a s e d o n M O

theory.

A k e y f a c t o r i n s u p e r c o n d u c t i v i t y is t h e p r e s e n c e o f a n e n e r g y g a p w h i c h s e p a r a t e s

the

g r o u n d state ( s u p e r c o n d u c t i n g state) f r o m t h e c o n t i n u o u s s i n g l e p a r t i c l e s p e c t r u m c h a r a c t e r i s t i c o f t h e n o r m a l state o f a m e t a l . L e t u s w r i t e t h e m a n y e l e c t r o n w a v e f u n c t i o n as

Φ = Σα V


without specifying the representation. t h e f u l l H a m i l t a n i a n is H = H

Q

Ι

Φ

/

(3)

ν

A s s u m e Φ„ are solutions of Η

( Η Φ „ = EjbJ)

and

ο

+ U . T h e q u e s t i o n is w h a t f o r m s d o U a n d Φ n e e d t o h a v e i n

o r d e r t o p r o d u c e a s i g n i f i c a n t e n e r g y l o w e r i n g o f t h e g r o u n d state w i t h r e s p e c t t o t h e T h a t is, h o w c a n a n energy gap be produced?

W

I α* I

= EE„

+

Σ

(4)

U^ala

u

A s i g n i f i c a n t l o w e r i n g i n e n e r g y c a n b e a c h i e v e d , as is w e l l k n o w n (6), i f t h e E

v

e q u a l a n d the

are a l l nearly equal.

Ε„Ί

T h e t o t a l e n e r g y is

I n this case, i f t h e Ε

υ

~ E

0

are a l l nearly

a n d the

~ -V

t

the

e n e r g y is

W

= E

0

- V

w i t h the lowest energy obtained w h e n all the a

Σ

α

μ

(5)

a

v

are the same. If there are m terms contribut

v

i n g i n t h e e x p a n s i o n o f t h e w a v e f u n c t i o n , t h e r e s u l t is

W

= E

0

(6)

- mV

w i t h the energy gap

Δ

=

(E

0

-W)

= V

Σα*

μ

a

= mV

u

(7)

H e n c e to p r o d u c e a gap, one has to devise a physically m e a n i n g f u l wave function that has p h a s e c o h e r e n c e a n d e q u a l a m p l i t u d e s f o r t h e Φ^ a n d a p o t e n t i a l U w h i c h is a t t r a c t i v e (i.e., leads to matrix elements

= -V).

I n t h e next s e c t i o n , w e r e v i e w h o w B a r d e e n , C o o p e r a n d

S c h r i e f f e r (7) ( B C S ) m e t t h i s c h a l l e n g e . R e v i e w of the B C S T h e o r y of Superconductivity H e r e w e p r e s e n t a b r i e f r e v i e w o f t h e e s s e n t i a l aspects o f t h e B C S t h e o r y o f s u p e r c o n d u c t i v i t y w h i c h w a s i n t r o d u c e d i n 1957 a n d is s t i l l t h e m o s t s u c c e s s f u l a n d c o m p l e t e t h e o r y o f s u p e r c o n d u c t i v i t y w h i c h exists. T h e p u r p o s e o f this s e c t i o n is to stress the u n i q u e f e a t u r e s o f t h e s u p e r conducting wave function w h i c h should be preserved

i n any new theory w h i c h involves

a

different m e c h a n i s m f o r o b t a i n i n g t h e g r o u n d state w a v e f u n c t i o n . I n the B C S theory, the n o r m a l (non-superconducting) B l o c h s i n g l e p a r t i c l e eigenstates (8)

state o f t h e m e t a l is d e s c r i b e d b y

|k> l a b e l l e d b y a w a v e v e c t o r k,


2.

MESSMER

15

Valence Bond Theory and Superconductivity

A N D MURPHY

I k>

h

= e

I k>

k

w h e r e h is the single electron H a m i l t o n i a n a n d e

k

(8)

i s t h e s i n g l e p a r t i c l e e n e r g y o f state

|k>.

T h e w a v e f u n c t i o n o f t h e m e t a l i s d e s c r i b e d b y t h e o c c u p a t i o n o f s i n g l e p a r t i c l e states |k> a n d i n t h e g r o u n d state a l l s i n g l e p a r t i c l e l e v e l s a r e f i l l e d u p t o t h e F e r m i e n e r g y Ep w i t h states above E

b e i n g unoccupied. T h i s B l o c h m o d e l does not include correlations between elec

F

trons d u e to C o u l o m b forces n o r the interaction o f the electrons w i t h the lattice vibrations (phonons).

I n t h e s u p e r c o n d u c t i n g state t h e p h o n o n i n t e r a c t i o n i s a c c o u n t e d f o r , b u t t h e e l e c

t r o n i c c o r r e l a t i o n i s n e g l e c t e d e x c e p t f o r effects p r o d u c e d b y t h e e l e c t r o n - p h o n o n

coupling.

T h e a r g u m e n t f o r i g n o r i n g o t h e r e l e c t r o n i c c o r r e l a t i o n effects i s t h a t t h e y a r e c h a r a c t e r i s t i c o f b o t h the n o r m a l a n d superconducting phases a n d therefore cannot b e responsible for p r o d u c i n g a gap.


T h e B C S s u p e r c o n d u c t i n g state i s c h a r a c t e r i z e d b y a n a t t r a c t i v e p o t e n t i a l b e t w e e n e l e c trons arising f r o m a n electron-phonon interaction. W e w i l l n o w review h o w this interaction is derived (9). T h e direct e l e c t r o n - p h o n o n interaction is a result o f the potential b e t w e e n t h e electrons a n d the nuclear charges w h e n the nuclei vibrate about their e q u i l i b r i u m positions. T h e e l e c t r o n - i o n (e-i) i n t e r a c t i o n i s e x p a n d e d i n t e r m s o f d i s p l a c e m e n t s Q/ o f t h e n u c l e a r c o o r d i n a t e s f r o m t h e i r e q u i l i b r i u m p o s i t i o n s Ry,

V (r -R -Q ) e4

r

1

7

j

= V (r -R )

j

= Υ

e4

Λ

electron

1

+ Q 'VV (r -Rj)

j

j

coordinate;

(r

V

e4

e4

7

+

1

-R )

=

;

e4

(9)

—τ

—

ι -Ϊ ' ; I R

γ

B y making a Fourier transformation of V

···

a n d u s i n g a p l a n e w a v e b a s i s |k> f o r t h e

electrons, w e c a n consider the electron-ion interaction i n the usual scattering terms i n w h i c h we consider the amplitude M

q

t o t r a n s f e r m o m e n t u m q t o t h e e l e c t r o n state |k> " s c a t t e r i n g "

it i n t o state |k + q > . T h e c o r r e s p o n d i n g d i a g r a m f o r this p r o c e s s i n F i g u r e 1. F r o h l i c h (10) s h o w e d h o w s e c o n d o r d e r p e r t u r b a t i o n t h e o r y c o u l d b e a p p l i e d t o d e r i v e a n effective i n t e r a c t i o n b e t w e e n e l e c t r o n s f r o m t h e d i r e c t e l e c t r o n - i o n i n t e r a c t i o n s . T h e p h y s i c a l i d e a i s that as o n e e l e c t r o n scatters f r o m a n u c l e a r c e n t e r it d i s t o r t s t h e l a t t i c e , t h i s d i s t o r t i o n i s felt b y a n o t h e r e l e c t r o n , a n d t h u s t h e e l e c t r o n s e x p e r i e n c e a n i n d i r e c t i n t e r a c t i o n . T h e r e s u l t i s t h a t w e c a n t h i n k o f t h e e l e c t r o n s as e x c h a n g i n g p h o n o n m o m e n t u m q i n a n e l e c t r o n e l e c t r o n s c a t t e r i n g p r o c e s s s h o w n i n F i g u r e 2. T h e effective p o t e n t i a l o f i n t e r a c t i o n b e t w e e n the electrons f o r a scattering involving a change i n m o m e n t u m q is (11),

| M | q

h ~ Τ

V

fac " Sk

where M

q

and e

k + q

2

x2 x2 + q) - C M , ) 2

(

1 0

)

2

is the amplitude for the electron to directly absorb a p h o n o n o f m o m e n t u m q; e

k

a r e t h e s i n g l e p a r t i c l e e n e r g i e s b e f o r e a n d after s c a t t e r i n g , a n d Ji w i s t h e e n e r g y o f q

t h e p h o n o n . T h e i m p o r t a n t p o i n t i s that f o r |e - e k

k + q

|
t

n

e

potential is attractive a n d

it i s t h i s a t t r a c t i v e i n t e r a c t i o n w h i c h l e a d s t o t h e s u p e r c o n d u c t i n g state. C o u n t e r b a l a n c i n g this attractive i n t e r a c t i o n is the repulsive C o u l o m b interaction w h i c h i n m o m e n t u m space is

creating the potential for the scattering of a n electron f r o m

| k > to | k > i n the C o u l o m b x

2

field of another electron.


CHEMISTRY O F HIGH-TEMPERATURE

16

SUPERCONDUCTORS

T h e object o f t h e B C S t h e o r y w a s t o m a x i m i z e t h i s attractive e l e c t r o n - p h o n o n ( e - p h ) i n t e r a c t i o n s i n c e i t c o u l d p o s s i b l y l e a d t o a p h y s i c a l l y different state, l o w e r i n e n e r g y t h a n t h e n o r m a l m e t a l . T o m a x i m i z e the interaction B C S considered only electron-phonon interactions for w h i c h V

ei>h

i s n e g a t i v e w h i c h r e q u i r e s s c a t t e r i n g o f e l e c t r o n s f r o m states |k> t o |k' >

s u c h t h a t \s - e ' k

\ < # w where # w i s some average p h o n o n frequency o f the o r d e r o f a c

k

c

I n o t h e r w o r d s o n l y |k> states w i t h i n a r e g i o n ± ) i ω o f t h e F e r m i e n e r g y

D e b y e frequency.

€

can contribute t o t h e scattering process.

T o simplify matters further a n average constant

a t t r a c t i v e p o t e n t i a l - V w a s d e f i n e d as - V =

ei>h

c

p h o n o n p o t e n t i a l a n d t h e C o u l o m b p o t e n t i a l o v e r t h e k r e g i o n d e f i n e d f o r attractive

V ,h. ei

T h e next c r u c i a l step i n t h e t h e o r y w a s the d e v e l o p m e n t o f a s u p e r c o n d u c t i n g w a v e f u n c t i o n that b o t h o p t i m i z e s t h e attractive p h o n o n i n t e r a c t i o n a n d m i n i m i z e s t h e C o u l o m b r e p u l s i o n . T h i s m e a n s s p e c i f y i n g a p a r t i c u l a r o c c u p a t i o n o f |k> states. C o n s i d e r i n g t h e m a t r i x e l e m e n t s o f t h e a t t r a c t i v e p o t e n t i a l , B C S s h o w e d that t h e o p t i m a l w a v e f u n c t i o n i n v o l v e d a n o c c u


p a t i o n o f k states s u c h that i f state

| kcr> w i t h m o m e n t u m k a n d s p i n σ i s o c c u p i e d t h e n t h e

state w i t h o p p o s i t e m o m e n t u m a n d s p i n ,

| -k

is also occupied. T h i s condition is referred

t o as e l e c t r o n p a i r i n g a n d is r e s p o n s i b l e f o r t h e c o h e r e n c e o f the w a v e f u n c t i o n . T h e scatter ing from V h ei>

o n l y o c c u r s f r o m t h e p a i r state |k,-k> t o a n o t h e r p a i r state | k ' , - k ' > w h e r e k '

= k + q as s h o w n i n F i g u r e 2 a n d a g a i n k a n d k ' a r e i n t h e r e s t r i c t e d r e g i o n n e a r t h e F e r m i l e v e l . T h e p a i r e d w a v e f u n c t i o n is w r i t t e n as,

*5cs=n(v

+ ii rt)|0>

k

(12)

k

k where u

k

i s t h e a m p l i t u d e t o h a v e p a i r state |k,-k> o c c u p i e d , w i s t h e p r o b a b i l i t y t o o c c u p y k

p a i r state |k,-k>, v

k

i s t h e a m p l i t u d e t o h a v e p a i r state |k,-k> u n o c c u p i e d , b

k

= cb

c\ ^, i s k

t h e p a i r c r e a t i o n o p e r a t o r i n s e c o n d q u a n t i z e d f o r m , a n d |0> is t h e v a c u u m . I n t h e g r o u n d state at 0 Κ o f t h e n o r m a l m e t a l w , t h e p r o b a b i l i t y t o o c c u p y p a i r state k

|k,-k>, i s u n i t y u p t o t h e F e r m i e n e r g y after w h i c h i t i s z e r o . I n the s u p e r c o n d u c t i n g state u

k

differs f r o m t h e F e r m i d i s t r i b u t i o n b y t h e e x c i t a t i o n o f s o m e p a i r states a b o v e t h e F e r m i l e v e l . T h e s e p a i r s i n t e r a c t via t h e a t t r a c t i v e p o t e n t i a l w h i c h m o r e t h a n c o m p e n s a t e s f o r t h e e x c i t a t i o n e n e r g y a b o v e t h e F e r m i l e v e l . T h u s a r o u n d e d F e r m i d i s t r i b u t i o n is o b t a i n e d ( F i g u r e 3 ) . T h e H a m i l t o n i a n w h i c h d e s c r i b e s these i n t e r a c t i o n s is s i m p l y ,

H cs B

= Σ e k, t o | k ' , - k ' > f o r k , k ' i n t h e r e g i o n n e a r t h e F e r m i l e v e l . T h e e n e r g y is s i m p l y

E

BCS

T h e coefficients u

k

and v

= 2 Σ u e k 2

k

2

k

= 1/2 ( 1 -e /E \ k

It i s a s s u m e d that t h e m a t r i x e l e m e n t F

Kk

u

k

v

k

u> v> k

k

(14)

a r e d e t e r m i n e d v a r i a t i o n a l l y g i v i n g (see F i g u r e 3 ) ,

k

u

o n l y i n t h e v i c i n i t y o f e ).

- Σ V , k,k'

k

k

v

= 1/2 ( 1 + e /E )

2

k

k

k

(15)

~ V (i.e., a r e i n d e p e n d e n t o f k a n d a r e n o n - z e r o

T h i s leads t o a gap parameter

Δ

= V Σ u> v< k

k


(16)

2.

MESSMER

A N D MURPHY


17

lk + q>


F i g u r e 1. E l e c t r o n o f m o m e n t u m k b e i n g s c a t t e r e d b y p h o n o n i n t o state w i t h m o m e n t u m k +

lk -q> 2

F i g u r e 2 . T w o e l e c t r o n s w i t h m o m e n t a k^ a n d 1^ e x c h a n g i n g a p h o n o n o f m o m e n t u m q i n a n electron-electron scattering process.

ι - — Normal Metal ! Distribution

2

uk

Ε

For Superconductor k

Fermi

F i g u r e 3. O c c u p a n c y o f k-states f o r t h e n o r m a l m e t a l a n d i n t h e s u p e r c o n d u c t i n g state.


CHEMISTRY OF HIGH-TEMPERATURESUPERCONDUCTORS

18

w h i c h m a y b e c o m p a r e d t o E q u a t i o n (7) a n d a n e w q u a s i p a r t i c l e e n e r g y g i v e n b y E

=

k

y/el+A

2

(12). T h a t a set o f n o n z e r o u

k

h a s b e e n o b t a i n e d f o r k>k

F

means that a n e w m a n y - e l e c t r o n

state h a s f o r m e d w i t h a l o w e r e n e r g y t h a n t h e n o r m a l state. T h e k e y f e a t u r e o f t h e n e w state is i t s s t a b i l i t y t o s i n g l e p a r t i c l e e x c i t a t i o n s .

I n the n o r m a l metal, single particle excitations

above the F e r m i level c a n b e m a d e w i t h vanishingly s m a l l energies.

I n the superconducting

state t h e r e i s n o t o n l y a n e n e r g y l o w e r i n g o f t h e w h o l e s y s t e m r e l a t i v e t o t h e n o r m a l m e t a l b u t there is also a finite energy gap to single particle-like excitations o f the order o f the gap parameter Δ . T h i s g a p to single particle excitations is responsible for most o f the physically observable Δ/kT)

p r o p e r t i e s o f a s u p e r c o n d u c t o r i n c l u d i n g a s p e c i f i c h e a t w h i c h g r o w s as e x p ( -

a n d perfect diamagnetism to b e discussed below. W h e n a magnetic field is applied to a superconductor, a current is i n d u c e d i n the

m a t e r i a l w h i c h c r e a t e s a n e w f i e l d o p p o s e d t o t h e a p p l i e d f i e l d s u c h that Β = 0 i n t h e m a t e r i a l


( M e i s s n e r effect). T h e q u a n t u m m e c h a n i c a l c u r r e n t d e n s i t y j i n t h e m a t e r i a l i s g i v e n b y ,

j = -

2wc

ΓΦ * V Φ - ( ν Φ ) * Φ 1 - — ΑΦ * Φ me

(17)

w h e r e A is t h e vector potential d e f i n e d b y Β = V χ A . T h e first t e r m is the p a r a m a g n e t i c c o n tribution to the current a n d the second is the diamagnetic term. Φ is the wave function i n the p r e s e n c e o f t h e f i e l d A w h i c h differs f r o m t h e g r o u n d state s u p e r c o n d u c t i n g f u n c t i o n Φ

0

to

first o r d e r i n A by,

+

Σ

< n

A

n

w h e r e |n> a r e e x c i t e d states a b o v e Φ . E 0

j (E

nfO

- E

n

0

' -

'

P

0

>

l » >

(18)

E) 0

i s at least as l a r g e as t h e g a p p a r a m e t e r

Δ

w h i c h m a k e s t h e p e r t u r b a t i o n t o Φ r e l a t i v e l y s m a l l ; t h u s w e c a n u s e Φ = Φο i n t h e c a l c u l a t i o n o f j . I n t h e g r o u n d state Φο, t h e p a r a m a g n e t i c t e r m i n t h e c u r r e n t i s z e r o , h e n c e t h e o n l y r e s p o n s e o f t h e s y s t e m i s p u r e l y d i a m a g n e t i c , j = ~(e /mc) 2

A

= A « e x p ( - V (4ime

2

Α Φ*Φ. Since V

2

A = -(47rn/c) j ,

· z ) , w h i c h s h o w s , i n t h e o n e - d i m e n s i o n a l case, t h e e x p o n e n t i a l

/mc ) 2

d r o p o r t h e f i e l d as ζ i n c r e a s e s i n t o t h e m a t e r i a l f r o m t h e s u r f a c e . T h i s s h o w s t h a t d i a m a g n e t i s m i s a d i r e c t c o n s e q u e n c e o f t h e s t a b i l i t y o f t h e s u p e r c o n d u c t i n g state t o s i n g l e p a r t i c l e l i k e excitations. F i n i t e t e m p e r a t u r e effects a r e e a s i l y i n c l u d e d i n t h e t h e o r y b y m o d i f y i n g a l l t h e m a t r i x elements b y i n c l u d i n g F e r m i - D i r a c factors, l/(exp(-/?£ ) + 1). V a r i a t i o n a l l y m i n i m i z i n g t h e k

f r e e e n e r g y G = U - T S s h o w s t h e g a p f u n c t i o n d e c r e a s i n g as a f u n c t i o n o f t e m p e r a t u r e u n t i l i t r e a c h e s z e r o at t h e c r i t i c a l t e m p e r a t u r e T

Ci

T h e expression for T

c

i s , kT

c

at w h i c h t h e t r a n s i t i o n t o t h e n o r m a l state o c c u r s .

~ Ι/ω e x p ( - l / N ( O ) V ) , f o r N ( 0 ) V < 1 . €

N ( 0 ) is the density o f

states o f t h e n o r m a l m e t a l at t h e F e r m i l e v e l . T o summarize, the B C S theory o f superconductivity provides a n energy gap a n d a physi c a l m e c h a n i s m t o a c h i e v e : (1) p h a s e c o h e r e n c e i n t h e w a v e f u n c t i o n a n d (2) a n a t t r a c t i v e i n t e r a c t i o n . B o t h o f t h e s e a r e r e q u i r e d t o p r o d u c e a g a p as s e e n i n t h e d i s c u s s i o n o f t h e I n t r o d u c t i o n . T h e B C S m e c h a n i s m o f p a i r i n g electrons w i t h k a n d - k m o m e n t a is t h e k e y t o a c c o m p l i s h i n g these goals. W h y , then, are there so m a n y discussions o f n e w mechanisms to explain the high superconductors?

T

c

F i r s t , there is a l i m i t a t i o n to the m a g n i t u d e o f V i f it arises f r o m the

e l e c t r o n - p h o n o n i n t e r a c t i o n s ; s e c o n d , t h e d e n s i t y o f states f o r t h e s e o x i d e m a t e r i a l s i s q u i t e low. T a k e n together, these observations m a k e it difficult to u n d e r s t a n d a transition t e m p e r a t u r e o f 90°K. R e c e n t l y , t h e b a n d s t r u c t u r e o f t h e t e t r a g o n a l La Cu0 2

4

compound has been

c a l c u l a t e d ( 1 3 , 1 4 ) . T h e r e s u l t s i n d i c a t e a p a r t i a l l y f i l l e d b a n d at t h e F e r m i l e v e l w h i c h c a n b e w e l l d e s c r i b e d i n a t i g h t b i n d i n g m o d e l as C u d ^ x

f o r m the four C u - O bonds i n the plane.

a n d Ο ρ orbitals i n the plane hybridizing to

T h e F e r m i surface o f the undistorted tetragonal


2.

MESSMER AND MURPHY

lattice i n 2 - D is just a square.

19

Valence Bond Theory and Superconductivity T h e lattice vibrations associated w i t h a transition f r o m

the

t e t r a g o n a l p h a s e t o a n o r t h o r h o m b i c s t r u c t u r e h a v e w a v e v e c t o r s that e x a c t l y s p a n t h e F e r m i s q u a r e a n d h e n c e c a n c a u s e a P e i e r l s d i s t o r t i o n (15) t o t h e o r t h o r h o m b i c p h a s e a n d o p e n s a g a p at t h e F e r m i l e v e l , c r e a t i n g a s e m i c o n d u c t o r .

T h e B a substituents are believed to change

the F e r m i level thus changing the shape of the F e r m i surface a n d destroying the Peierls insta bility, b r i n g i n g the symmetric structure b a c k without the semiconducting b a n d gap.

Unfor

t u n a t e l y , n e w e x p e r i m e n t s (16) s e e m t o i n d i c a t e t h a t t h e l a t t i c e d i s t o r t i o n c a n o c c u r i n t h e s u p e r c o n d u c t i n g state. C a l c u l a t i o n s of the p h o n o n m o d e s a n d B C S electron p h o n o n c o u p l i n g constants suggest t h a t a h i g h T m o d e (w )

rather than a strong electron phonon coupling parameter.

c

lated was 4 0 K , l i m i t e d b y the highest C u - O frequency. 1 8

(17)

is the result of the h i g h frequency of a C u - O ( l ) in-plane bond-stretching

c

Ο i n t h i s s y s t e m (18) s h o w s n o c h a n g e i n T

c

T h e highest T

c

Unfortunately replacement o f

1 6

calcu Ο with

casting doubts o n this m e c h a n i s m .


T h e f i r s t c o n c l u s i o n t o b e d r a w n f r o m t h i s w o r k is that c a l c u l a t i o n s b a s e d o n t h e B C S m o d e l cannot presently explain the 7 y s o f order 90 Κ given the p h o n o n frequencies a n d b a n d structures calculated.

T h i s p o i n t s o u t that p e r h a p s a m e c h a n i s m o t h e r t h a n a n

electron-

p h o n o n m e c h a n i s m is involved, w h i c h depends o n parameters w i t h larger energy scales. possibility is a n electronic m e c h a n i s m , m a n y of w h i c h have b e e n p r o p o s e d .

A

W e w i l l discuss

o n e o f t h e s e i n t h e next s e c t i o n . T h e s e c o n d p o i n t i s that t h e o x i d e e l e c t r o n i c s t r u c t u r e is d i s c u s s e d i n a r e a l s p a c e l o c a l bonding framework

although traditionally b a n d structure/k-space methods have b e e n used.

T h e s i m p l e picture of the k , - k p a i r i n g i n B C S seems to b e lost i n these c o m p l i c a t e d b a n d struc t u r e c a l c u l a t i o n s . It s h o u l d b e n o t e d a l s o that t h e r e h a v e b e e n s e r i o u s d i f f i c u l t i e s w i t h a p p l y i n g t r a d i t i o n a l b a n d s t r u c t u r e concepts oxides.

f o r p r e d i c t i n g t h e normal

associated

behavior of metallic

A s a typical example, simple b a n d filling models for M n O cannot avoid attaining a

p a r t i a l l y f i l l e d b a n d i n d i c a t i n g t h a t it s h o u l d b e a c o n d u c t o r ( 1 9 , 2 0 ) ; h o w e v e r , M n O is o n e o f the best insulators k n o w n i n nature. S u c h difficulties l e d to t h ë u s T o f the H u b b a r d m o d e l

(21)

o f m e t a l s , w h i c h i n c l u d e s s t r o n g a t o m i c - l i k e c o r r e l a t i o n effects i n a m o d e l H a m i l t o n i a n f o r m t o r a t i o n a l i z e s o m e o f these b a n d p r o b l e m s . M o r e o f t h e h i s t o r y o f t h e i m p o r t a n c e o f c o r r e l a t i o n effects i n t h e o x i d e s c a n b e f o u n d i n R e f e r e n c e s 19,20,22. T h e p o i n t i s that c o r r e l a t i o n effects h a v e b e e n k n o w n t o b e v e r y i m p o r t a n t f o r p r e d i c t i n g t h e p r o p e r t i e s o f o x i d e s , a n d t h i s s h o u l d b e kept i n m i n d w h e n using the results or concepts of m e a n f i e l d theories w h i c h neglect t h e s e v e r y i m p o r t a n t effects. Valence B o n d s and Superconductivity O n e o f t h e a u t h o r s (22), h a s p r o p o s e d a m o d e l o f s u p e r c o n d u c t i v i t y w h i c h is b a s e d o n a h i g h l y c o r r e l a t e d d e s c r i p t i o n o f m e t a l s , c l o s e r t o t h e W i g n e r l a t t i c e l i m i t o f e l e c t r o n l o c a l i z a t i o n (23) t h a n to the free electron l i m i t usually assumed. F o r s m a l l m e t a l clusters of L i atoms a n d G o d d a r d (1,2) h a v e s h o w n u s i n g c o r r e l a t e d ab-initio

McAdon

m e t h o d s that e l e c t r o n p a i r s l o c a l i z e

i n t e t r a h e d r a l i n t e r s t i c e s o f t h e l a t t i c e . It a l s o h a s b e e n s h o w n (24) that o n e c a n o b t a i n a v e r y g o o d a p p r o x i m a t i o n to the experimental charge density of B e using this description. F u r t h e r m o r e , s t u d i e s o n m o l e c u l a r systems (3-5)

have s h o w n a n energetic preference i n sp valence

a t o m s f o r b e n t b o n d s f o r m e d f r o m tetrahedral based o n M O theory.

hybrids

r a t h e r t h a n t h e t r a d i t i o n a l σ,π

bonds

T h e u s e o f t e t r a h e d r a l h y b r i d s f o r B e m e t a l suggests that t h e e l e c t r o n

p a i r s c a n l o c a l i z e i n t o t e t r a h e d r a i n t w o s e p a r a t e ways, as s h o w n i n F i g u r e 4. T h e f u l l y s y m m e t r i c wave function is a coherent superposition of the two structures. H o w e v e r , each of these s t r u c t u r e s is o n l y a s h o r t h a n d w a y o f d e s c r i b i n g a l a r g e n u m b e r o f v a l e n c e b o n d s t r u c t u r e s w i t h alternative hybrids f o r m i n g bonds. B u t , what does this have to d o w i t h superconductivity? L e t us r e t u r n to the discussion i n the I n t r o d u c t i o n a n d r e c a l l that to o b t a i n a theory o f s u p e r c o n d u c t i v i t y it is n e c e s s a r y (see E q u a t i o n s 4-7) t o h a v e a n a t t r a c t i v e p o t e n t i a l , - V , a n d t o have the a

u

o f E q u a t i o n (3) b e e q u a l . I n t h e B C S t h e o r y , E q u a t i o n (1) w a s u s e d as t h e b a s i s

o f t h e m o d e l . H e r e , w e u s e E q u a t i o n (2) as o u r s t a r t i n g p o i n t . T h e q u e s t i o n , as it w a s a b o v e i n t h e d i s c u s s i o n o f B C S t h e o r y , is h o w d o w e o b t a i n a p h y s i c a l l y m e a n i n g f u l w a v e f u n c t i o n a n d p o t e n t i a l t o satisfy t h e s e c r i t e r i a ?


CHEMISTRY OF HIGH-TEMPERATURE

20

SUPERCONDUCTORS

R e t u r n i n g , t h e n , t o t h e e x p a n s i o n o f E q u a t i o n (2), w e n o t e t h a t t h e t e r m s

represent

different v a l e n c e b o n d s t r u c t u r e s . W h y s h o u l d t h e y a l l h a v e t h e s a m e a m p l i t u d e a n d p h a s e ? T h i s situation is very s i m i l a r to the p r o b l e m o f d e t e r m i n i n g the "resonance energy" o f b e n z e n o i d m o l e c u l e s (25,26,27).

I n t h a t case, o f a l l t h e p o s s i b l e v a l e n c e b o n d s t r u c t u r e s w h i c h

m i g h t c o n t r i b u t e , only t h e K e k u l e ' s t r u c t u r e s a r e u s e d . F o r l a r g e b e n z e n o i d systems t h i s i s o n l y a s m a l l fraction o f the total n u m b e r o f structures. F u r t h e r m o r e , it is a s s u m e d that they a l l enter w i t h e q u a l e x p a n s i o n coefficients

(i.e., e q u a l a m p l i t u d e a n d p h a s e ) .

I n addition, the

m a t r i x e l e m e n t s w h i c h c o n v e r t o n e s t r u c t u r e i n t o a n o t h e r a r e set e q u a l t o a c o m m o n v a l u e , determined empirically. T h u s , the energy lowering associated w i t h "resonance" i n b e n z e n o i d molecules has a mathematical structure which maps onto the discussion i n the Introduction. H o w e v e r , there are s o m e important

differences.

A n e c e s s a r y , b u t n o t sufficient, c o n d i t i o n f o r p r o d u c i n g a s u p e r c o n d u c t i n g g r o u n d state is t h a t t h e n u m b e r o f a v a i l a b l e o r b i t a l s e x c e e d s t h e n u m b e r o f e l e c t r o n s .

I n order to describe


a s u p e r c o n d u c t i n g m e t a l , t h e l o w e r i n g o f t h e g r o u n d state p r o d u c e s a g a p w i t h r e s p e c t t o a c o n t i n u u m o f single particle excitations. F r o m a valence b o n d viewpoint such a c o n t i n u u m is easily achieved w h e n the n u m b e r o f orbitals (hybrids) is significantly larger t h a n the n u m b e r o f e l e c t r o n s (e.g., t h e r a t i o v

6

f °{ °^ ^ Φ of electrons ta

s

m J

a

v

b e i n t h e r a n g e ~ 1.1 - 4.0). F o r t h e o x i d e m a t e r i ' B

als it i s t h e r a t i o o f o r b i t a l s t o h o l e s w h i c h i s i m p o r t a n t . However,

for the benzenoid

molecules

t h i s r a t i o i s u n i t y a n d t h e g r o u n d state i s a n i n s u l a t o r ; t h e l o w e s t l y i n g e x c i t e d states w i l l h a v e s i g n i f i c a n t e x c i t o n i c a n d p o l a r o n i c effects.

T h u s , i n spite o f the coherent

superposition of

a l t e r n a t i v e b o n d i n g s t r u c t u r e s ("spatial r e s o n a n c e " ) i n these m o l e c u l e s , w h i c h i s o n e o f t h e necessary conditions for superconductivity, there are insufficient orbitals for the n u m b e r o f electrons, therefore failing another of the requirements. For

superconducting metals, the n o n - K e k u l e '

valence

bond

structures (analogs o f

D e w a r a n d "long-bond" structures) provide the basis for constructing the c o n t i n u u m o f single p a r t i c l e e x c i t a t i o n s . R a i s i n g t h e t e m p e r a t u r e i n this m o d e l w i l l e v e n t u a l l y d e s t r o y t h e s t a b i l i t y gained f r o m the resonance because the entropie t e r m , -ST, i n the free energy w i l l increase w h e n the e l e c t r o n pairs take o n the m a n y other possible configurations that are available t o t h e m t h a n just those w h i c h m a x i m i z e the resonance energy. A m o r e quantitative discussion w i l l b e given elsewhere (28). I n order t o discuss the n e w superconductors w i t h this m o d e l , it must b e s h o w n that such a c o h e r e n t s p a t i a l r e s o n a n c e c a n o c c u r i n these m a t e r i a l s as a c o n s e q u e n c e o f t h e i r l o c a l b o n d i n g . A s a f i r s t step, w e h a v e c a l c u l a t e d t h e e l e c t r o n i c s t r u c t u r e o f t h e S F ^ m o l e c u l e t o g a i n some insight into the b o n d i n g occurring i n octahedral complexes w h i c h are a n important part of the environment i n the n e w superconductors. Bonding i n a Octahedral Environment T h e molecule sulfur hexafluoride ( S F , ) has recently challenged b o t h molecular

spectroscopy

w i t h i t s u n e x p e c t e d r o t a t i o n a l s p e c t r a (29) a n d e l e c t r o n i c s t r u c t u r e t h e o r i e s w i t h n o v e l c o r r e l a t i o n effects ( 3 0 , 3 1 , 5 ) .

T h e electronic structure must explain the molecule's h i g h stability,

octahedral symmetry, and, most importantly, provide a simple picture o f the b o n d i n g . A t first glance, the traditional c h e m i c a l models d o not appear to b e appropriate because sulfur seem ingly f o r m s six b o n d s t o fluorines, yet the sulfur s p 2

4

v a l e n c e c o n f i g u r a t i o n a l l o w s f o r at m o s t

two covalent bonds. I n t h i s s e c t i o n w e r e p o r t o n s o m e p r e l i m i n a r y r e s u l t s w h i c h suggest a n o v e l i n t e r p r e t a t i o n o f t h e e l e c t r o n i c s t r u c t u r e o f S F ^ b y c o n s i d e r i n g t h e w a v e f u n c t i o n as a c o h e r e n t s u p e r p o sition o f l o w symmetry generalized valence b o n d structures involving ionic b o n d i n g a n d little sulfur d-orbital character.

T h i s coherent superposition provides a significant fraction o f the

correlation energy b y including b o t h intra- a n d inter-pair correlation while retaining a local picture of the bonding. A t the lowest level, S F ^ is described b y the molecular orbital H a r t r e e - F o c k wave func t i o n w i t h d o u b l y o c c u p i e d o r b i t a l s fa w h i c h a r e d e t e r m i n e d self c o n s i s t e n t l y i n t h e m e a n f i e l d of the other pairs,


2.

*HF = det fact φ βφ αφ β 1

The

Hartree-Fock

21


MESSMER AND MURPHY

2

]

2

calculations were performed

g e o m e t r y w i t h a S - F b o n d l e n g t h o f 1.564Â. A

(19)

at t h e e x p e r i m e n t a l

(32)

octahedral

standard double zeta basis was used for the

f l u o r i n e a t o m s (33), w h i l e t h e s u l f u r w a s d e s c r i b e d b y a n effective p o t e n t i a l (34) w i t h a v a l e n c e d o u b l e z e t a s-p b a s i s . I n a s e p a r a t e c a l c u l a t i o n a s i n g l e d p o l a r i z a t i o n f u n c t i o n w i t h e x p o n e n t .532 w a s a d d e d t o t h e s u l f u r b a s i s t o assess t h e i m p o r t a n c e o f d f u n c t i o n s . T h e m o s t s t r i k i n g f e a t u r e o f t h e r e s u l t s is that S F ^ i s n o t b o u n d w i t h r e s p e c t t o

the

separated atoms w h e n d functions are not i n c l u d e d i n the sulfur basis while the i n t r o d u c t i o n of t h e d f u n c t i o n o n s u l f u r l o w e r s t h e e n e r g y b y 10.5 e V m a k i n g S F ^ b o u n d b y 5.6 e V . r e s u l t s w e r e o b t a i n e d b y R e e d (30). sulfur has a large

sp d 3

2

c o m p o n e n t f r o m w h i c h six e q u i v a l e n t S - F b o n d s c a n b e m a d e .

F r o m the H a r t r e e - F o c k calculation, the z e r o t h order description of S F ^ is of


Similar

W i t h o u t f u r t h e r e v i d e n c e t h i s r e s u l t w o u l d suggest t h a t fluorine

f o r m i n g partially i o n i c b o n d s to sulfur, w i t h a s m a l l p o p u l a t i o n of the sulfur α function ener getically very important for bonding.

R e e d h a s c o n c l u d e d that t h e d o r b i t a l o c c u p a t i o n i s

s m a l l because the d orbitals are h i g h i n energy. T h e H a r t r e e - F o c k b i n d i n g energy falls far s h o r t o f t h e e x p e r i m e n t a l v a l u e o f 20.1 e V i n d i c a t i n g that t h e r e a r e m a j o r c o r r e l a t i o n effects neglected, w h i c h w e w i l l discuss b e l o w . I n order to gain b o t h a m o r e local a n d a m o r e accurate description of the b o n d i n g i n S F g w e i n t r o d u c e d i n t r a - p a i r c o r r e l a t i o n s via method ( G V B - P P )

(35).

the perfect p a i r i n g generalized

T h e valence electrons are described by generalized

valence

bond

Heitler-London

pairs,

(20)

(ΦαΦι>+Φ*Φα) (οφ-βα) w h e r e the singlet c o u p l e d o v e r l a p p i n g spatial orbitals φ

α>

variationally determined.

φ

ύ

f o r m i n g a l o c a l valence b o n d are

T h e G V B - P P w a v e f u n c t i o n is t h e a n t i s y m m e t r i z e d p r o d u c t o f t h e

pair functions a n d perfect p a i r i n g refers to the orthogonality of the p a i r functions a n d the nature of the spin coupling:

*GVB.PP =

det

M 2 ) &(3) *»(«)

•' '

( ) 21

with

Θ

ΡΡ

= ( « ( 1 X 8 ( 2 ) - /?(1)α(2)) W 3 ) / 9 ( 4 ) - β(3)α(4))

· · ·

(22)

S i x p a i r s o f e l e c t r o n s w e r e d e s c r i b e d as c o r r e l a t e d p a i r s ( E q u a t i o n 20) w h i l e t h e o t h e r p a i r s w e r e t r e a t e d at t h e H a r t r e e - F o c k l e v e l . N o s y m m e t r y r e s t r a i n t s w e r e p l a c e d o n t h e w a v e f u n c t i o n f o r r e a s o n s t h a t a r e e x p l a i n e d b e l o w . T h e b a s i s set i n c l u d e d t h e s u l f u r d f u n c t i o n s . T h e c o r r e l a t i o n e n e r g y o b t a i n e d r e l a t i v e t o H a r t r e e - F o c k i s 3.0 e V i n d i c a t i n g t h e extent

of

intra-pair correlation. T h e s e G V B r e s u l t s suggest t h a t w e m a y t h i n k o f S F ^ f o r m i n g i n a h y p o t h e t i c a l

sequence

i n w h i c h t h e a x i a l f l u o r i n e s f i r s t f o r m l a r g e l y i o n i c b o n d s t o s u l f u r , t h u s p r o m o t i n g a n effective sp

3

v a l e n c e c o n f i g u r a t i o n o f s u l f u r . T h e f o u r s u l f u r e l e c t r o n s a r e left i n a t e t r a h e d r a l o r i e n t a

t i o n o n sulfur available for the bonds w i t h the equatorial fluorines. T h e equatorial bonds have s m a l l ζ components o n the sulfur i n the G V B results.

The

l a c k o f ζ c h a r a c t e r i n t h e o r b i t a l s is a r e s u l t o f t h e h i g h e l e c t r o n e g a t i v i t y o f t h e f l u o r i n e s , c a u s i n g highly p o l a r i z e d b o n d s , the lack of inter-pair correlations i n the perfect p a i r i n g m e t h o d , s t e m m i n g f r o m the orthogonality constraints between the pairs, a n d the neglect of the correla t i o n effects d e s c r i b e d n e x t .



22

CHEMISTRY OFHIGH-TEMPERATURE SUPERCONDUCTORS

F i g u r e 4. S c h e m a t i c representation o f electrons l o c a l i z e d i n tetrahedral interstices o f hep B e m e t a l . T h e e l e c t r o n p a i r s a r e d i s t r i b u t e d a m o n g f o u r h y b r i d s at e a c h site. O n l y a r e p r e s e n t a tive n u m b e r o f electron pairs are s h o w n .

F i g u r e 5. S c h e m a t i c representation o f b o n d i n g i n S F tures.

6

f o r o n e o f t h e spatial

resonance

T h e r e a r e s i x e q u i v a l e n t s t r u c t u r e s , e a c h h a s t w o "ionic" bonds and four

struc

"covalent"

b o n d s . T h e s h a d e d s p h e r e s r e p r e s e n t t h e p o s i t i o n s o f e l e c t r o n pairs; the light connecting lines r e p r e s e n t a p p r o x i m a t e t e t r a h e d r a l h y b r i d o r b i t a l s . T h e d a r k c o n n e c t i n g lines merely show the octahedral geometry.


2.

MESSMER

AND


MURPHY

I n t e r - p a i r c o r r e l a t i o n s i n t r o d u c e d b y t h e t e t r a h e d r a l sp

3

torial

fluorines

23

orbitals b o n d i n g to the equa

c a n b e s e e n i n F i g u r e 5, w h e r e w e h a v e s h o w n s c h e m a t i c a l l y w h a t t h e e q u a

t o r i a l b o n d s w o u l d l o o k like w i t h m o r e sulfur p

z

character i n the bonds. T h i s figure shows the

inter-pair correlation o c c u r r i n g b y pulling two pairs above the equatorial plane a n d two below, thus r e d u c i n g the P a u l i repulsion relative to having a l l pairs i n the plane. T h e choice of the a x i a l d i r e c t i o n f o r t h e i o n i c c o m p o n e n t s is n o t u n i q u e . T o r e s t o r e t h e s y m m e t r y i n t h i s s c h e m e w e m u s t a l s o i n c l u d e t h e d e g e n e r a t e c o n f i g u r a t i o n i n w h i c h t h i s sp

3

state is r e f l e c t e d a b o u t t h e

equatorial plane a n d consider the four other orientations obtained f r o m the other equivalent " a x i a l " p o s i t i o n s . T h u s w e suggest t h a t t h e S F ^ w a v e f u n c t i o n i s d e s c r i b e d b y a c o h e r e n t s u p e r p o s i t i o n of these six structures,

*SF


6

= 0(Φ!

+ Φ

2

+

+Φ )

(23)

6

S u c h a c o h e r e n t s u p e r p o s i t i o n o f d e g e n e r a t e states l e a d s t o t h e w e l l k n o w n " r e s o n a n c e " e n e r g y d e f i n e d as t h e d i f f e r e n c e i n e n e r g y b e t w e e n t h e e n e r g y o f Φι a l o n e a n d Φ ρ 5

6

(33,5).

This

b o n d i n g scheme for introducing inter-pair correlations beyond the G V B - P P m e t h o d using a superposition of resonance structures emphasizing the atomic hybridization has b e e n s h o w n to b e s u c c e s s f u l f o r b e n z e n e (5). M u c h r e m a i n s to b e d o n e to i m p l e m e n t the ideas outlined here about valence

bond

wave functions, i n order to address the m a n y questions about new h i g h - r materials. H o w e v e r c

t h e fact t h a t it is f o r m u l a t e d i n real space a n d is b a s e d o n c h e m i c a l b o n d s , s h o u l d a l l o w m u c h m o r e d i r e c t c o n t a c t w i t h t h e chemical

aspects t h a n h a s b e e n p r e v i o u s l y p o s s i b l e .

A c k n o w l e d g ment T h i s w o r k was supported i n part b y the Office of N a v a l R e s e a r c h .

Literature Cited [1] McAdon, M.H.; Goddard, III, W.A. Phys. Rev. Lett. 1985, 55 2563. [2] McAdon, M.H.; Goddard, III, W.A.J.Phys. Chem. 1987, 91 2607. [3] Messmer, R.P.; Schultz, P.A.; Tatar, R.C.; Freund, H.-J. Chem. Phys. Lett. 1986, 126 176. [4] Messmer, R.P.; Schultz, P.A. Phys. Rev. Lett. 1986, 57, 2653. [5] Schultz, P.A.; Messmer, R.P. Phys. Rev. Lett. 1987, 58, 2416. [6] Kittel, C. "Introduction to Solid State Physics", Third Edition, p. 624, Willey, 1966, See Reference 11, also. [7] Bardeen, J.; Cooper, L.N.; Schrieffer, J.R. Phys. Rev. 1957, 108, 1175. [8] Bloch, F. Z. Phys. 1928. 52, 555. [9] Mahan, G.D. " Many Particle Physics", Plenum, 1981. [10] Fröhlich, H. Phys. Rev. 1950, 79, 845. [11] Feynman, R.P. "Statistical Mechanics, A Set of Lectures", Ch. 10, Benjamin Press, 1972. [12] Reference [7] and an alternative derivation by Bogoliubov, N.N.; Nuovo cimento, 1958, 7, 794. [13] Mattheiss, L.F. Phys. Rev. Lett. 1987, 58, 1028.


CHEMISTRY OF HIGH-TEMPERATURE SUPERCONDUCTORS

24 [14] [15] [16] [17] [18] [19] [20]

Yu, J.; Freeman, A.J.; Xu, J.H. Phys. Rev. Lett. 1987, 58, 1035. Peierls, R. "Quantum Theory of Solids", Clarendon, 1956. Fleming, R.M.; Batlogg, B.; Cava, R.J.; Rietman, E.A. to be published. Weber, W. Phys. Rev. Lett. 1987, 58, 1371. Batlogg, B;, Cava, R.J.; et al. Phys. Rev. Lett. 1987, 58, 2333. Adler, D. Solid State Physics, 1968, 21. Adler, D.; Feinleib, J. Phys. Rev. Β 1970, 2, 3112.


[21] Hubbard, J. Proc. Royal Soc., A276 238 (1968); ibid. A277 237 (1964); ibid. A281 401 (1964); ibid. A285 542 (1965); ibid. 296 82 (1966); ibid. A296 100 (1966).

[22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]

Messmer, R.P. Solid State Commun. 1987, 63, 405. Wigner, E. Faraday Discussions 1934, 34, 678. Tatar, R.C.; Messmer, R.P. to be submitted for publication. Pauling, L. "The Nature of the Chemical Bond", Third Edition, Chapter 6, Cornell University Press, 1960. Wheland, G.W. "Resonance in Organic Chemistry", Wiley, 1955. For recent work, see for example: Gutman, I.; Cyvin, S.J. Chem. Phys. Lett. 1987, 13 137. Murphy, R.B.; Messmer, R.P. to be published. "Physics Today", July 1984, 17. Reed, A.E.; Weinhold, F.J.Amer. Chem. Soc. 1986, 108, 3587. Hay, P.J. J. Amer. Chem. Soc. 1977, 99, 1003. Herzberg, G. "Molecular Spectra and Molecular Structure", Vol. III, Polyatomic Molecules, Van Nostrand, NY 1966. Dunning, Jr., T.H.; Hay, P.J. "Methods of Electronic Structure Theory" Vol. 3 of Modern Theoretical Chemistry, edited by H.F. Schaefer III, (Plenum, New York, 1977), p. 1. Wadt, W.R.; Hay, Ρ.J.J.Chem. Phys. 1984, 82, 284. Hunt, W.J.; Hay, Ρ.J.; Goddard, III, W.A.J.Chem. Phys. 1972, 57, 738; Goddard, III, W.A.; Dunning, Jr., T.H.; Hunt, W.J.; Hay, P.J. Acc. Chem. Res. 1973, 6, 368; Bobrowicz, F.W.; Goddard, III, W.A. "Methods in Electronic Structure Theory", Vol. 3 of "Modern Theoretical Chemistry", edited by H.F. Schaefer III (Plenum, New York, 1977), p. 79.

[36] Voter, A.F.; Goddard, III, W.A. Chem. Phys. 1982, 57, 253. RECEIVED

July 17, 1987


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