Metal Bonding and Interactions in High Temperature Systems

16 Ionic-Covalent Interactions in Alkali Hydrides Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on October 7, 2015 | http://pubs.acs.org Publication Date: March 8, 1982 | doi: 10.1021/bk-1982-0179.ch016

SZE-CHENG

YANG

University of Rhode Island, Department of Chemistry, Kingston, R I 02881 WILLIAM C. S T W A L L E Y University of Iowa, Iowa Laser Facility and Departments of Chemistry and Physics, Iowa City, I A 52242

The recently available spectroscopic data and the RKR potentials of the a l k a l i hydrides allow us to de termine the "experimental" values of the parameters relevant to the transition probability of the charge transfer processes. In the Landau-Zener model these parameters are the energy gap between the A Σ and X Σ adiabatic potentials at the avoided crossing d i s tance and the coupling matrix elements. In this paper the coupling matrix elements are evaluated in a two -state ionic-covalent interaction model. The systematic trends found in the a l k a l i hydride series for their X Σ potentials are presented. This leads to a simple model for the ionic potentials. 1

1

+

1

+

+

C r a w f o r d a n d J o r g e n s e n (1_) i n t h e i r e a r l y s p e c t r o s c o p i c mea s u r e m e n t s o f t h e L i H A - X t r a n s i t i o n s d i s c o v e r e d an anomaly i n t h e s p e c t r o s c o p i c c o n s t a n t s , B « and A G ' + ^ o f t h e A ! * s t a t e . Both B i a n d A G « i ^ f i r s t i n c r e a s e as a f u n c t i o n o f t h e v ' t o a m a x i mum b e f o r e t h e y r e v e r t t o t h e u s u a l m o n o t o n i c d e c r e a s i n g b e h a v i o r at higher v ' . Mulliken l a t e r offered a q u a l i t a t i v e explanation i n t e r m s o f t h e a v o i d e d c r o s s i n g o f t h e i o n i c and c o v a l e n t p o t e n t i a l c u r v e s ( 2 J . The c o n c e p t has been v e r y u s e f u l f o r u n d e r standing not only the o p t i c a l spectra o f a l k a l i hydrides but also a l a r g e number o f i m p o r t a n t p r o c e s s e s : charge t r a n s f e r r e a c t i o n s , e l e c t r o n i c t o v i b r a t i o n a l (E •> V) e n e r g y t r a n s f e r , c h e m i - i o n i z a t i o n , t h e h a r p o o n i n g mechanism o f t h e c h e m i c a l r e a c t i o n s , p r e d i s s o c i a t i o n , i o n p a i r f o r m a t i o n and i o n - i o n r e c o m b i n a t i o n . The o p t i c a l s p e c t r a o f a l k a l i h a l i d e s p r o v i d e a n o t h e r exam p l e o f t h e e f f e c t o f i o n i c - c o v a l e n t c u r v e c r o s s i n g on s p e c t r a . The q u a l i t a t i v e f e a t u r e s a r e e x p l a i n e d n i c e l y by t h e t h e o r y o f n o n - a d i a b a t i c t r a n s i t i o n s due t o B e r r y ( 3 ) . U n f o r t u n a t e l y t h e s p e c t r a c a n n o t be r e s o l v e d and i d e n t i f i e d " t o y i e l d q u a n t i t a t i v e data relevant t o the i o n i c - c o v a l e n t i n t e r a c t i o n . For a l k a l i h a l i d e s t h e s e i n t e r a c t i o n s a r e more amenable t o a t o m i c beam s c a t t e r i n g experiments ( 4 ) . In c o n t r a s t t h e o p t i c a l s p e c t r a o f 1

v

v

v

V

+

0097-6156/82/0179-0241 $05.00/0 © 1982 A m e r i c a n Chemical Society

In Metal Bonding and Interactions in High Temperature Systems; Gole, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1982.

242

METAL BONDING AND INTERACTIONS

a l k a l i h y d r i d e s a r e s i m p l e and a s s i g n a b l e . The a c c u r a c y o f t h e s p e c t r o s c o p i c measurements and t h e abundance o f d e t a i l s a l l o w us t o e x t r a c t i n f o r m a t i o n r e l e v a n t t o the dynamics o f the charge t r a n s f e r process. New s p e c t r o s c o p i c measurements (5_-l_l_) o f t h e p a s t few y e a r s a l l o w the c o n s t r u c t i o n of the X ! s t a t e RKR p o t e n t i a l s (5.-12) very c l o s e to the avoided c r o s s i n g r e g i o n . S i n c e the e l e c t r o n i c c h a r a c t e r o f t h e a l k a l i h y d r i d e s c h a n g e s as t h e i n t e r n u c l e a r d i s t a n c e i s changed a d i a b a t i c a l l y a c r o s s t h e a v o i d e d c r o s s i n g r e g i o n , t h e p o t e n t i a l c u r v e s a r e e x p e c t e d t o bend s h a r p l y i n t h a t region. The shape and t h e e n e r g y gap o f t h e c u r v e s c o n t a i n q u a n t i t a t i v e i n f o r m a t i o n about the i o n i c - c o v a l e n t i n t e r a c t i o n . In t h i s s t u d y we f i r s t e x a m i n e t h e s y s t e m a t i c s t h a t e x i s t i n t h e RKR p o t e n t i a l s o f t h e X ! * s t a t e t o e s t a b l i s h t h a t t h e p o t e n t i a l curves are s t r o n g l y i o n i c f o r R « R (R i s the d i s t a n c e of the pseudocrossing p o i n t ) . N e x t we e v a l u a t e an e s s e n t i a l l y experimental value of the parameters r e l e v a n t to the cross s e c t i o n f o r the charge t r a n s f e r process i n the Landau-Zener a p p r o x i mation. We a l s o c o n s t r u c t a model i o n i c p o t e n t i a l w h i c h can be used t o d e s c r i b e t h e c h a r g e t r a n s f e r p r o c e s s i n t h e i o n i c r e g i o n .

Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on October 7, 2015 | http://pubs.acs.org Publication Date: March 8, 1982 | doi: 10.1021/bk-1982-0179.ch016

1

1

c

Systematics

o f the X ! *

RKR

1

c

Potentials

T a b l e I shows some o f t h e p a r a m e t e r s

o f t h e RKR

potentials.

T a b l e I. Parameters o b t a i n e d from the o p t i c a l s p e c t r a the a l k a l i hydrides. A ^

X ^

1

1

IP-EA (cm- ) 1

LiH NaH KH RbH CsH

37403.3 35366.0 28926.8 27606.1 25323.7

D

(cm- )

R

1

e

20286.6 15900 14500 14240 14500

± ± + ± ±

of

0.5 500 500 500 500

e

D

(A)

1.5957 1.8872 2.2424 2.3673 2.4938

R (A)

(cm" ) 1

e

8681.6 10137 8732 8598 7286

± ± ± ± ±

e

0.5 500 500 500 500

2.5839 3.1934 3.7629 3.8642 3.9823

The p a r a m e t e r s g i v e n i n T a b l e I a r e b a s e d on t h e f o l l o w i n g : IP = i o n i z a t i o n p o t e n t i a l o f t h e a l k a l i atom (14); EA = e l e c t r o n a f f i n i t y o f t h e h y d r o g e n atom ( 1 5 ) ; D = b o n d i n g e n e r g y o f t h e p o t e n t i a l w e l l ( 1 6 ) ; R = i n t e r n u c l e a r d i s t a n c e a t t h e minimum o f the p o t e n t i a l curve. When we p l o t t h e e n e r g y o f t h e X s t a t e p o t e n t i a l minimum r e l a t i v e t o t h e e n e r g y o f t h e i o n p a i r s as a f u n c t i o n o f R g , t h e i n v e r s e o f t h e i n t e r n u c l e a r d i s t a n c e a t t h e minimum, we g e t a s t r a i g h t l i n e c o r r e l a t i o n among t h e a l k a l i h y d r i d e s ( s e e F i g u r e 1 ) : e

e

l

V(R ) e

where u n i t s

are

= -(IP in cm"

- EA - D ) e

1

= - ^ 8

-

and i\ f o r V and R . e

7709 Figure

2

expresses


16.

YANG AND STWALLEY

lonic-Covalent

i

Interactions

i

243

i

o


•EA)

-4

t—*

R b H ^ H

-5

-

NaHiV

•i ii N^LiH

-6

>

1

04

1

05

06 l/R ( A"') e

Figure 1. Correlation between the energy of the potential minima relative to the ion pair dissociation limit and the reciprocal of the equilibrium bond distance for the X*X state alkali hydrides.

Figure 2. Energy of the potential minima relative to the free ions vs. the equilibrium bond distance. The curve V(R) — — (e /R) is the Coulombic point charge potential. 2


244


t h i s c o r r e l a t i o n i n a n o t h e r manner. We can see t h a t t h e V ( R ) curve runs p a r a l l e l t o the coulombic p o t e n t i a l , V - ) (R) = -! /R. T h i s s i m p l y r e f l e c t s the f a c t t h a t the ground e l e c t r o n i c s t a t e i s p r i m a r i l y i o n i c , i . e . M H". The f a c t t h a t V ( R ) p o i n t s s t a y s m o o t h l y on a c u r v e above V Q I ( R ) s u g g e s t s t h a t t h e r e p u l s i v e exchange i n t e r a c t i o n a l s o f o l l o w s the t r e n d s i n t h i s s e r i e s . I t i s i n t e r e s t i n g to note t h a t the s y s t e m a t i c t r e n d i s o b t a i n e d by c o n s i d e r i n g t h e i o n i c c h a r a c t e r a l o n e even t h o u g h t h e a d i a b a t i c curves d i s s o c i a t e i n t o n e u t r a l atoms. T h e r e i s a n o t h e r s t r i k i n g s i m i l a r i t y among t h e a l k a l i h y d r i d e systems. F i g u r e 3 shows a r e d u c e d p l o t o f t h e X s t a t e RKR t u r n i n g p o i n t s (5^ - V 3 ) . The minima o f t h e p o t e n t i a l w e l l s a r e t a k e n as t h e z e r o e s o f e n e r g y and t h e i n t e r n u c l e a r d i s t a n c e R i s s c a l e d by t h e e q u i l i b r i u m bond d i s t a n c e R o f t h e a l k a l i h y d r i d e . The p o t e n t i a l s f a l l i n t o a l m o s t t h e same r e d u c e d f o r m by t h i s simple s c a l i n g . Note t h a t t h e v e r t i c a l a x i s i s n o t s c a l e d . The d i s s o c i a t i o n l i m i t s o f t h e m o l e c u l e s a r e i n d i c a t e d i n t h e figure. I t i s r e m a r k a b l e t h a t a common r e d u c e d p o t e n t i a l f o r a l l t h e a l k a l i h y d r i d e s i s m a i n t a i n e d up t o an e n e r g y v e r y c l o s e to t h e i r s i g n i f i c a n t l y d i f f e r e n t d i s s o c i a t i o n l i m i t s . However i f we t e m p o r a r i l y i g n o r e t h e a v o i d e d c r o s s i n g and p r e t e n d t h a t t h e m o l e c u l e s d i s s o c i a t e i n t o t h e i r i o n p a i r s , t h e n t h e common r e d u c e d f o r m i s n o t so s u r p r i s i n g b e c a u s e t h e y r e p r e s e n t l e s s t h a n t h e b o t t o m 30% o f t h e i o n i c p o t e n t i a l w e l l . The i m p l i c a t i o n i s t h a t the molecules are dominantly i o n i c a t a l l i n t e r n u c l e a r d i s t a n c e s s h o r t e r than those i n the avoided c r o s s i n g r e gion. M o r e o v e r , t h e e n e r g y o f t h e X !" " p o t e n t i a l c u r v e i n t h e avoided c r o s s i n g region i s q u i t e c l o s e to the a d i a b a t i c d i s s o c i a t i o n l i m i t o f the m o l e c u l e s . The r e d u c e d p o t e n t i a l c u r v e s a r e e x p e c t e d t o d e v i a t e f r o m e a c h o t h e r e v e n t u a l l y upon e n t e r i n g the avoided c r o s s i n g r e g i o n . The c u r v e s w i l l t h e n bend s h a r p l y t o t h e i r d i f f e r e n t d i s s o c i a t i o n l i m i t s . The RKR p o t e n t i a l o f NaH (8) i s an e x a m p l e w h i c h a l r e a d y shows some d e v i a t i o n f r o m t h e common r e d u c e d f o r m a t t h e h i g h e r v i b r a t i o n a l l e v e l s w i t h o u t e r t u r n i n g p o i n t s i n the avoided c r o s s i n g r e g i o n . However, i f t h e a v o i d e d c r o s s i n g r e g i o n i s i g n o r e d , t h e X !" " p o t e n t i a l s o f a l l t h e a l k a l i h y d r i d e s a p p e a r t o be w e l l r e p r e s e n t e d by a s i n g l e f u n c t i o n o f R/R o n l y . E

2

C O U

+

E

U


G

E

1

1

1

1

E

Coupling Matrix

E l e m e n t and t h e E n e r g y o f t h e C r o s s i n g

Point

In t h e above we have a r g u e d t h a t t h e p o t e n t i a l c u r v e i n the a v o i d e d c r o s s i n g r e g i o n c o n t a i n s i n f o r m a t i o n about the charge t r a n s f e r process: , M + H -> M + H" We c o n s i d e r t h e L a n d a u - Z e n e r model (V7) f o r c a l c u l a t i n g t h e charge t r a n s f e r c r o s s s e c t i o n f o r high c o l l i s i o n e n e r g i e s : Q

Z L

(E)

= 47rg R

2 q

(1 - H . . ( R ) / E ) G ( X ) C


16.

YANG AND STWALLEY

G(X)

lonic-Covalent

=

exp(-Ax)

2, T » x i c

(hv

Interactions

[1 - e x p ( - A x ) ]

dH..(R) •n ' dR

x" dx 3

dH (R) cc ' dR R=R

x

[1 -

245

v

r

H..(R )/E]^ C

where QZL i s t h e t o t a l c r o s s s e c t i o n f o r c h a r g e t r a n s f e r r e a c t i o n , E i s t h e c o l l i s i o n e n e r g y , H-j-j(R) and H ( R ) a r e t h e i o n i c and c o v a l e n t c u r v e s , R i s t h e c r o s s i n g d i s t a n c e , g i s a symmetry f a c t o r , v i s t h e r e l a t i v e v e l o c i t y o f t h e c o l l i s i o n and T - j i s the c o u p l i n g m a t r i x element. In t h e t w o - s t a t e c o n f i g u r a t i o n a p p r o x i m a t i o n (18> - 20)


C C

c

c

T

ic

=

I W

"

H

ii

( R

c>

S

l

where H j and S a r e t h e o f f d i a g o n a l H a m i l t o n i a n m a t r i x e l e m e n t and t h e o v e r l a p i n t e g r a l between t h e i o n i c and c o v a l e n t r e p r e sentations, respectively. Our i n t e n t i o n i s t o use as much o f t h e e x p e r i m e n t a l RKR p o t e n t i a l s a s p o s s i b l e t o e v a l u a t e t h e r e l e v a n t q u a n t i t i e s T j and Hii(Rr). F i g u r e 4 shows t h e known RKR t u r n i n g p o i n t s f o r b o t h t h e X and A s t a t e s o f t h e a l k a l i h y d r i d e s . The i n t e r n u c l e a r d i s t a n c e i s s c a l e d by t h e e q u i l i b r i u m bond d i s t a n c e i n t h e X Z state. I t c a n be s e e n t h a t t h e A s t a t e p o t e n t i a l s ( 6 , 7 , TO., 1_2> 13) a r e known a c c u r a t e l y i n t h e a v o i d e d c r o s s i n g r e g i o n . The X s t a t e RKR p o t e n t i a l s a r e known t o i n t e r n u c l e a r d i s t a n c e s v e r y c l o s e t o t h e c r o s s i n g r e g i o n , b u t n o t i n t o i t . However, w i t h a v e r y s h o r t e x t r a p o l a t i o n t h e X ^ " " RKR p o t e n t i a l c a n be e x t e n d e d t o t h e c r o s s i n g distance. We a d o p t e d t h e f o l l o w i n g p r o c e d u r e t o e x t r a p o l a t e t h e X s t a t e p o t e n t i a l c u r v e s o f NaH, KH, RbH and CsH b a s e d on t h e a v a i l a b l e t h e o r e t i c a l X state potential curves. The e n e r g i e s o f t h e t h e o r e t i c a l p o t e n t i a l c u r v e s (21_- 23) a r e s c a l e d t o make them s m o o t h l y f i t w i t h t h e o u t e r m o s t o f t h e e x i s t i n g RKR t u r n i n g p o i n t s t o e x t r a p o l a t e t h e RKR p o t e n t i a l s i n t o t h e c r o s s i n g r e gion. The e x t r a p o l a t i o n o f t h e NaH p o t e n t i a l i s b a s e d on t h e t h e o r e t i c a l c u r v e s o f O l s o n and L i u ( 2 1 ) . The e x t r a p o l a t i o n s o f KH and RbH a r e b a s e d on t h e t h e o r e t i c a l c u r v e s o f S t e v e n s and H i s k e s ( 2 2 ) . The t h e o r e t i c a l c u r v e o f S t a l l cop and L a s k o w s k i (23) i s s c a l e d t o e x t r a p o l a t e t h e RKR c u r v e s f o r C s H . The u n c e r t a i n t y i s e s t i m a t e d t o be ^ 3 0 0 c m " b a s e d on t h e d e v i a t i o n o f t h e t h e o r e t i c a l c u r v e s f r o m t h e known p o r t i o n s o f t h e RKR c u r v e s . L i H i s unique because the d i s s o c i a t i o n energy i s a c c u r a t e l y d e t e r m i n e d ( 1 6 ) . The h y b r i d p o t e n t i a l f o r L i H ( 2 4 ) i s b e l i e v e d t o be h i g h l y r e l i a b l e . T h i s i s f u r t h e r s u p p o r t e d by t h e h i g h q u a l i t y ab i n i t i o c a l c u l a t i o n s on L i H by P a r t r i d g e and L a n g h o f f (25) and by Docken and H i n z e ( 2 6 ) . The f o r m e r a g r e e s w i t h t h e hybrid potential within a f r a c t i o n of a percent. We s i m p l y use the r e p o r t e d h y b r i d p o t e n t i a l (24) f o r L i H i n t h e a v o i d e d c

c

X

+

1

1


246


1

1

1

Li + H - - - —

-

Na + H Cs+H-^

A©

o A o


S

0

-

x'z

1 1

i°

AO*

/

o

\ -

0.5

-

Rb+H'

/

\

\ \

/

/ *

l

1.0

1

1.5

2.0

R"(-R/R ) e

Figure 3. Potential energy curves as a function of reduced distances (R/R ) for the X*V state of the alkali hydrides. Key: A , LiH; O, NaH; A, KH; X , RbH; and m, CsH. e

Figure 4. RKR turning points of the X*T and A !,* states of the alkali hydrides plotted vs. the reduced internuclear distance R/R . Key: see Fig. 3. 1

e


16.

YANG AND STWALLEY

louic-Covaient

Interactions

247

crossing region. We d e t e r m i n e t h e c r o s s i n g d i s t a n c e , R , and t h e a d i a b a t i c e n e r g y g a p , A V ( R ) , a t t h e c r o s s i n g d i s t a n c e as follows: F i r s t we d e f i n e t h e c r o s s i n g d i s t a n c e a s t h e i n t e r n u c l e a r d i s t a n c e where t h e e n e r g y gap between t h e A and X s t a t e p o t e n t i a l s i s a minimum. I t i s a l s o a t t h i s p o i n t where t h e two p o t e n t i a l c u r v e s have t h e same s l o p e , i . e . c

C

dV (R)

dV (R)

dR

dR

x


A

The

R=R.

R ' s a r e d e t e r m i n e d g r a p h i c a l l y and l i s t e d i n T a b l e I I . c Table II. The e n e r g y ( i n c m " ) o f t h e a d i a b a t i c p o t e n t i a l s at the crossing distance with respect to the ion p a i r d i s s o c i a t i o n l i m i t M + H' 1

w

R (*> c

LiH NaH KH RbH CsH

3.90 4.08 4.70 4.86 5.15

V (R ) X

-29326.3 -27250.3 -23424 -22633.5 -21323.7

C

-38796.3 -36758 -29959 -28814 -26423.7

The c o r r e s p o n d i n g e n e r g i e s o f t h e X and A p o t e n t i a l s a r e a l s o listed in this table. In t h e two s t a t e c o n f i g u r a t i o n m i x i n g a p p r o x i m a t i o n o f G r i c e and H e r s c h b a c h ( 1 8 ) , Adelman and H e r s c h b a c h (19) and J a n e v and R a d u l o v i c ( 2 0 ) , t h e c o u p l i n g m a t r i x e l e m e n t c a n be r e l a t e d t o t h e energy o f a d i a b a t i c curves a t the c r o s s i n g d i s t a n c e : T

H

i c Mr =

R

C

ii< c>-*W R

(V > - W >

1

V c^-r^ ic

+

R

T

The o v e r l a p i n t e g r a l S has been e v a l u a t e d by G r i c e and H e r s c h bach ( 1 8 ) f o r t h e c r o s s i n g d i s t a n c e s ( l i s t e d i n p a r e n t h e s e s , column 2 o f T a b l e I I I ) slightly d i f f e r e n t from our semiTable III. Parameters f o r the i o n i c - c o v a l e n t i n t e r a c t i o n i n the two-state approximation.

R (A) C

LiH NaH KH RbH CsH

3.90 4.08 4.70 4. 8 6 5.15

(3.92) (4.00) (4.66) (4.82) (5.15)

S(R ) C

0.34 0.33 0.30 0.29 0.27

l ic" iJ l (cm )

Hii(Rc) (cm* )

4187.6 4236.2 2973.4 2830.4 2225

-35671.3 -33572.9 -27871.5 -26630 -24562

H

H

s

- 1

1


248


experimental values. S i n c e the d i f f e r e n c e s i n the R's are s m a l l , we use t h e i r v a l u e o f S l i s t e d i n column 3 o f T a b l e I I I to e v a l u a t e t h e semi e m p i r i c a l v a l u e s o f T. and H..(R ) w h i c h a r e l i s t e d i n columns 4 and 5 o f T a b l e I I I . P r e v i o u s l y t h e c o u p l i n g m a t r i x e l e m e n t s were c a l c u l a t e d by J a n e v and R a d u l o v i c (20) e m p l o y i n g t h e a s y m p t o t i c wave f u n c t i o n s i n a two e l e c t r o n a p p r o x i m a t i o n . In T a b l e IV we compare t h e s e 1


Table

IV.

C

Comparison

1 1

of

c

|H. -H..SI ic n ' 1

J a n e v and Radulovic

This

3

R

LiH NaH KH RbH CsH a

W

c( o> a

7.40 7.66 8.86 9.11 9.74

Reference

2970 3065 2415 2359 2065

7.37 7.71 8.87 9.19 9.73

work

iH^-H.-SKcm- ) 1

4188 4236 2973 2830 2225

20.

values w i t h our e s s e n t i a l l y experimental values. There a r e d i f f e r e n c e s i n the magnitudes o f the c o u p l i n g m a t r i x e l e m e n t s , e s p e c i a l l y f o r the l i g h t e r a l k a l i h y d r i d e s . Note t h a t b o t h t r e a t ments a r e i n t h e framework o f a t w o - s t a t e m o d e l , b u t we f e e l o u r e m p i r i c a l l y b a s e d v a l u e s s h o u l d be p r e f e r r e d f o r c a l c u l a t i o n o f QLZ(E).

.

A s e r m e m p i r i c a l c o r r e l a t i o n among c o u p l i n g m a t r i x e l e m e n t s and c r o s s i n g d i s t a n c e s f o r many m o l e c u l e s was e x a m i n e d by O l s o n , S m i t h and B a u e r ( 2 7 j and O l s o n ( 2 8 ) . The c o u p l i n g m a t r i x e l e ments c a l c u l a t e d f r o m t h e i r s e m i e m p i r i c a l f o r m u l a a g r e e w i t h our values to w i t h i n a f a c t o r of two, which i s not s u r p r i s i n g f o r s u c h an a p p r o x i m a t e c o r r e l a t i o n . The M o d e l i n g

o f an I o n i c P o t e n t i a l

The s y s t e m a t i c s f o u n d i n t h e a l k a l i h y d r i d e X * £ potentials s u g g e s t t h a t i t m i g h t be p o s s i b l e t o model a s i m p l e i o n i c p o t e n t i a l t o r e f l e c t the b e h a v i o r o f the whole s e r i e s o f a l k a l i hy drides. Here we c o n s t r u c t a " p r a c t i c a l " d i a b a t i c c u r v e w h i c h r e f l e c t s t h e p h y s i c a l p r o p e r t i e s , e . g . t h e d i p o l e moment f u n c t i o n . . We e x p e c t o u r d i a b a t i c c u r v e t o f o l l o w t h e i o n i c p a r t o f t h e X ! a d i a b a t i c p o t e n t i a l and t o b e g i n t o d e v i a t e i n t h e a v o i d e d c r o s sing region. We have c h o s e n t o model t h e d i a b a t i c i o n i c c u r v e ( H ^ - t R ) ) as a t r u n c a t e d R i t t n e r p o t e n t i a l w i t h a t u r n i n g - o f f f u n c t i o n f o r the p o l a r i z a t i o n term: +

1


16.

YANG AND STWALLEY

Ionic-Covalent

P

H.^R)

= A exp[-R/p]

(a

+ a )

- \

f(R)

(1)

where f ( R ) = 1 - e x p [ - ( R / Y ) ] . a and a_ a r e t h e i o n p o l a r i z a b i 1 i t i e s ( 3 1 , 3 2 ) , Y t h e d i s t a n c e p a r a m e t e r where t h e c h a r g e o v e r l a p d i m i n u t i o n o f t h e p o l a r i z a t i o n t e r m becomes s i g n i f i c a n t , and A and p a r e p a r a m e t e r s f o r t h e r e p u l s i o n t o be f i t t e d f r o m RKR t u r n i n g p o i n t s . A t l a r g e i n t e r n u c l e a r d i s t a n c e s (R » Y ) , where b o t h t h e e x change and c h a r g e o v e r l a p e f f e c t s a r e n e g l i g i b l e , t h e t u r n i n g - o f f f u n c t i o n f(R) approaches u n i t y . There the p o t e n t i a l i s the usual R" power s e r i e s e x p a n s i o n . In t h e i n t e r m e d i a t e range (R - Y ) , t h e o v e r l a p o f c h a r g e s c a u s e s t h e p o l a r i z a t i o n t e r m t o be r e d u c e d . A t s h o r t i n t e r n u c l e a r d i s t a n c e s (R ^ p ) , t h e r e p u l s i v e exchange i n t e r a c t i o n becomes i m p o r t a n t . We now j u s t i f y t h e i n t r o d u c t i o n o f t h e t u r n i n g - o f f f u n c t i o n f ( R ) . In a s t u d y o f H * , Kreek a n d M e a t h (29) showed t h a t t h e c o e f f i c i e n t s o f t h e u s u a l R" e x p a n s i o n f o r l o n g range f o r c e s a r e r e d u c e d a t an i n t e r n u c l e a r d i s t a n c e o f i n t e r m e d i a t e range due t o c h a r g e o v e r l a p effects. T h i s e f f e c t i s t a k e n as t h e b a s i s f o r t h e i n t r o d u c t i o n of the t u r n i n g - o f f f u n c t i o n f ( R ) . The f u n c t i o n a l f o r m o f f ( R ) was s u g g e s t e d by N u m e r i c h and T r u h l a r ( 3 0 ) . The c h a r g e o v e r l a p e f f e c t i s e x p e c t e d t o be i m p o r t a n t i n t h e a l k a l i h y d r i d e s b e c a u s e o f t h e v e r y d i f f u s e e l e c t r o n d i s t r i b u t i o n i n H" ( t h e p o l a r i z a b i l i t y o f H" i s 3 0 . 5 A (31)!). The p a r a m e t e r Y i s e s s e n t i a l l y f i x e d by t h e r e q u i r e m e n t t h a t the i o n i c curve pass through the c r o s s i n g p o i n t . A t R the e x change r e p u l s i v e t e r m i s s m a l l and 6


2

249

Interactions

+

1

1

3

c

H

1

1

( ^ > « - r - -

!

^ e x p { l

" (7)

6 }

(

2 )

^ c The p a r a m e t e r s A and p a r e d e t e r m i n e d by a n o n l i n e a r l e a s t s q u a r e f i t o f e q u a t i o n (1) t o t h e RKR i n n e r t u r n i n g p o i n t s o f each o f the a l k a l i h y d r i d e s . T h i s f i t t i n g p r o c e d u r e i s j u s t i f i e d by t h e f a c t t h a t t h e m a g n i t u d e o f t h e d i p o l e moment f u n c t i o n s o f L i H ( 2 5 , 26) and NaH (_33) a t t h e s e i n t e r n u c l e a r d i s t a n c e s a r e v e r y c l o s e t o those of opposite p o i n t charges a d i s t a n c e R a p a r t . Based on o u r d e f i n i t i o n f o r t h e " p r a c t i c a l " d i a b a t i c c u r v e s , we r e q u i r e t h a t the i o n i c curve agree w i t h the i n n e r w a l l o f the adiabatic curve. The f i t t e d p a r a m e t e r s A and p a r e l i s t e d i n T a b l e V. T a b l e V. Parameters f o r the i o n i c curve. C

K

a +a_(A ) LiH NaH KH RbH CsH

30.55 30.70 31.36 31.94 32.96

Mem' )

P(A)

1081730 1335760 1497650 1558760 1930060

0.3826 0.4030 0.4460 0.4577 0.4530

1

3

+

3.6607 3.7810 4.2320 4.4364 4.9033


250


Figure 5 i l l u s t r a t e s the e s s e n t i a l l y experimental hybrid po t e n t i a l s and t h e f i t t e d i o n i c c u r v e o f L i H . Note t h a t t h e i o n i c curve s t a r t s t o d e v i a t e from the X E p o t e n t i a l i n the avoided c r o s s i n g r e g i o n and r u n s p a r a l l e l t o t h e i o n i c p o r t i o n o f t h e A ! p o t e n t i a l s l i g h t l y o u t s i d e t h e c r o s s i n g r e g i o n and c r o s s e s t h e A !" " c u r v e a t r o u g h l y t h e p o s i t i o n o f t h e s e c o n d c r o s s i n g d i s t a n c e o f t h e A !" " s t a t e . The r e l a t i o n between t h e c o u p l i n g m a t r i x e l e m e n t 1^ and t h e e n e r g y gap R i n t h e t w o - s t a t e i n t e r action approximation i s also indicated there. F i g u r e 6 i l l u s t r a t e s t h e RKR t u r n i n g p o i n t s and t h e i o n i c c u r v e s o f NaH, KH, RbH and CsH. The i o n i c c u r v e s a l l seem t o be s a t i s f a c t o r y i n v i e w o f t h e v e r y s i m p l e model we have e m p l o y e d . Note t h a t i n o u r t w o - s t a t e m o d e l , t h e e n e r g y o f t h e c r o s s i n g p o i n t s a r e above t h e d i s s o c i a t i o n l i m i t o f t h e X !" * p o t e n t i a l . Thus t h e H ( R ) i s r e p u l s i v e a t t h e c r o s s i n g d i s t a n c e i n t h i s t w o - s t a t e model. T h i s may i m p l y e i t h e r t h e t w o - s t a t e a p p r o x i m a t i o n i s not s u f f i c i e n t l y accurate or the i n t e r p r e t a t i o n o f H ( R ) is not simple. In f a c t G a r r e t t et_ al_. ( 4 1 ) and N u m e r i c h and T r u h l a r (30) found t h a t the i n t e r a c t i o n i s n o t w e l l l o c a l i z e d a t t h e c r o s s i n g d i s t a n c e and t h a t t h e i n t e r a c t i o n w i t h a t h i r d s t a t e (although small) i s not n e g l i g i b l e . F o r t h e same r e a s o n s , t h e Landau-Zener approximation f o r the charge t r a n s f e r r e a c t i o n i s n o t e x p e c t e d t o be a c c u r a t e f o r l o w e n e r g y ( t h e r m a l ) c o l l i s i o n s (!!> £ 2 ) ( a l t h o u g h i t i s e x p e c t e d t o be a d e q u a t e f o r medium and high energy c o l l i s i o n s ) . There a r e o t h e r measurable p h y s i c a l p r o p e r t i e s which r e f l e c t t h e change o f e l e c t r o n i c c h a r a c t e r as a f u n c t i o n o f t h e i n t e r nuclear distance R i n the avoided crossing region. Examples c a n be f o u n d i n t h e measurements o f t h e v i b r a t i o n a l l y r e s o l v e d d i p o l e moments o f t h e A ! " s t a t e o f L i H and NaH ( 3 4 ) t o complement e a r l i e r X E r e s u l t s ( 3 5 ) . A l s o , measurements on t h e l i f e t i m e s ( 3 6 , 37) and t h e f l u o r e s c e n c e i n t e n s i t y p a t t e r n s ( 3 8 , 39) p r o v i d e i n f o r m a t i o n a b o u t t h e t r a n s i t i o n moment f u n c t i o n s . Further e x p e r i m e n t a l measurements when c o u p l e d w i t h t h e t h e o r e t i c a l c a l c u l a t i o n s o f t h e d i p o l e moments and t h e t r a n s i t i o n moments ( 2 5 , 26) and t h e r a d i a t i v e l i f e t i m e s ( 4 0 ) w i l l e v e n t u a l l y g i v e a more d e t a i l e d p i c t u r e o f t h e i o n i c - c o v a l e n t i n t e r a c t i o n s , and s h o u l d a l s o more c l e a r l y e s t a b l i s h t h e l i m i t a t i o n s o f a t w o - s t a t e m o d e l . F i n a l l y we n o t e t h a t s i m i l a r s c a l i n g p r o c e d u r e s t o t h a t shown h e r e f o r t h e a l k a l i h y d r i d e s a r e d i s c u s s e d e x t e n s i v e l y f o r t h e a l k a l i d i m e r s i n t h i s volume ( 4 3 ) . X

+

1

1

1

1

1


c

1

1

C C

C C

1

2

4

+

Conclusions New s p e c t r o s c o p i c measurements o f t h e a l k a l i h y d r i d e s have p r o v i d e d i n f o r m a t i o n r e l a t e d t o the dynamical charge t r a n s f e r process f o r these systems. We have e x a m i n e d t h e RKR p o t e n t i a l s d e r i v e d from these s p e c t r a . Several s t r i k i n g r e g u l a r i t i e s f o r t h e X E p o t e n t i a l s a r e p r e s e n t e d a l o n g w i t h an i n t e r p r e t a t i o n b a s e d on a s i m p l e model o f i o n i c p o t e n t i a l s f o r i n t e r n u c l e a r d i s t a n c e s s h o r t e r than the c r o s s i n g d i s t a n c e R . 1

+

c


16.

YANG AND STWALLEY

lonic-Covalent

1 1

1

Interactions

1

1

251

1 1 Li + H" +

3

LiH

2


A'Z

y/

+

.•

••

• /

• i /

."

1

.

-

x'i

.• / L

+

1

-2

7

i/

/

l R

J1

I

1 I

I

'I

2

3

4

C L

I

I

L

5

6

7

R (A) Figure 5. LiH potential energy curves. The A*T and X*V curves are the hybrid potentials of Ref. 24. Key: , ionic curve of equation 1; • • ionic curve without R' polarization term; and vertical , R. 4

c

I

i

•

NaH

i

'

i >

KH

v . •

: / •/

1

\A y

\ / \

T

'

1 •1

1

RbH

-

v

Jf

\ '

\

;

/ \ /

i

!,

i

,

CsH

v

\

A-

•\/\

^ • :

Figure 6. Potential energy curves for the X*V and A*V states of NaH, KH, RbH, and CsH. Key: • • RKR turning points; , extrapolations of X*X states based on ab initio calculations; , ionic curve of Eq. 1; and vertical , R. c


252


In t h e a v o i d e d c r o s s i n g r e g i o n t h e e x p e r i m e n t a l RKR p o t e n t i a l o f t h e A !.* s t a t e and t h e " e s s e n t i a l l y e x p e r i m e n t a l " p o t e n t i a l of the s t a t e a r e used t o d e t e r m i n e t h e c r o s s i n g d i s t a n c e R and t h e c o u p l i n g m a t r i x e l e m e n t i n t h e two s t a t e approximation. These q u a n t i t i e s a r e r e l e v a n t t o t h e e v a l u a t i o n of the t o t a l charge t r a n s f e r cross s e c t i o n a t high energy ( e . g . i n the Landau-Zener model). In t h i s s t u d y we used t h e s c a l e d t h e o r e t i c a l p o t e n t i a l c u r v e s o f O l s o n and L i u ( 2 1 ) , S t e v e n s , K a r o and H i s k e s ( 2 2 J , and L a s k o w s k i and S t a l l c o p ( 2 3 j t o make a s h o r t e x t r a p o l a t i o n o f t h e X ! RKR p o t e n t i a l s i n t o t h e a v o i d e d c r o s s i n g r e g i o n . Experimen t a l measurements t o o b t a i n s t r i c t l y e x p e r i m e n t a l RKR p o t e n t i a l s f o r t h i s region are i n progress. New o p t i c a l measurements on t h e d i p o l e moments ( 3 4 , 3 5 ) , t h e t r a n s i t i o n moments ( 3 8 , 39) and r a d i a t i v e l i f e t i m e s ( 3 6 , 37) o f t h e a l k a l i h y d r i d e s a r e a l s o becom i n g a v a i l a b l e . T h i s t y p e o f i n f o r m a t i o n w i l l soon p r o v i d e a d d i t i o n a l d e t a i l s about the i o n i c - c o v a l e n t i n t e r a c t i o n s i n these molecules. 1


c

1

Acknowledgements The a u t h o r s w i s h t o t h a n k W. S t e v e n s , A. M. K a r o , J . R. H i s k e s , B. L a s k o w s k i , J . S t a l l c o p , H. P a r t r i d g e and S. R. L a n g h o f f f o r m a k i n g t h e i r work a v a i l a b l e p r i o r t o p u b l i c a t i o n . S. C. Yang w o u l d l i k e t o t h a n k W i l l i a m Meath and D a v i d Freeman f o r t h e i r v a l u a b l e s u g g e s t i o n s and d i s c u s s i o n s . Work a t Rhode I s l a n d was s u p p o r t e d by t h e R e s e a r c h C o r p o r a t i o n ; work a t Iowa was s u p p o r t e d by t h e N a t i o n a l S c i e n c e F o u n d a t i o n and t h e N a t i o n a l A e r o n a u t i c s and S p a c e A d m i n i s t r a t i o n .

Literature Cited 1. 2. 3. 4. 5. 6.

7. 8. 9.

Crawford, F. H . ; Jorgensen, T . , J r . Phys. Rev. 1935, 47, 358; Phys. Rev. 1935, 47, 932. Mulliken, R. S. Phys. Rev. 1936, 50, 1017, 1028. Berry, R. S. J . Chem. Phys. 1957, 27, 1288. Los, J.; Kleyn, A. W. "Alkali Halide Vapors"; Academic Press: New York, 1979; edited by P. Davidovits and D. L. McFadden; Chapter 8, p 275-330. Ennen, G . ; Ottinger, C. Chem. Phys. Lett. 1975, 36, 16. Li, K. C . ; Stwalley, W. C. J. Chem. Phys. 1979, 70, 1736; Orth, F. B . ; Stwalley, W. C. J. Mol. Spectrosc. 1979, 76, 17; Stwalley, W. C . ; Way, K. R.; Velasco, R. J . Chem. Phys. 1974, 60, 3611. Orth, F. B . ; Stwalley, W. C . ; Yang, S. C . ; Hsieh, Y. K. J. Mol. Spectrosc. 1980, 79, 314. Giroud, M . ; Nedelec, O. J . Chem. Phys. 1980, 73, 415; Baltayan, P . ; Jourdan, A . ; Nedelec, O. Phys. Lett. 1975, 36, 16. Cruse, J . A . ; Zare, R. N. J. Chem. Phys. 1974, 60, 1182.


16.

10. 11. 12. 13.


14. 15. 16.

17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37.

YANG AND STWALLEY

lonic-Covalent

Interactions

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RECEIVED September 3,

1981.


Metal Bonding and Interactions in High Temperature Systems

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