Metal Bonding and Interactions in High Temperature Systems

tial curves and wavefunctions for the eight lowest-. -lying electronic ... mechanical investigations of eight low-lying states o_f each of. L i 2 and ...
0 downloads 0 Views 1MB Size
1 Electronic Structure and Spectra of Light Alkali Diatomic Molecules and Their Molecular Cations D. D. KONOWALOW and M. E. ROSENKRANTZ

Downloaded by 80.82.77.83 on May 9, 2017 | http://pubs.acs.org Publication Date: March 8, 1982 | doi: 10.1021/bk-1982-0179.ch001

State University of New York, Binghamton, Department of Chemistry, Binghamton, NY 13901

A l l - e l e c t r o n a b - i n i t i o c a l c u l a t i o n s of the potential curves and wavefunctions f o r the eight lowest- l y i n g e l e c t r o n i c s t a t e s of Li and of Na and the s i x l o w e s t - l y i n g e l e c t r o n i c s t a t e s o f Li + and Na + are used to p r e d i c t the s p e c t r a l features of a v a r i e t y of t r a n s i t i o n s . We s c a l e these r e s u l t s to p r e d i c t q u a l i t a t i v e l y s e v e r a l aspects o f the s t r u c ture and spectra of K , K +, Rb , Rb +, Cs and Cs +. 2

2

2

2

2

2

2

2

2

2

Recently we have reported ab i n i t i o a l l - e l e c t r o n quantummechanical i n v e s t i g a t i o n s of eight low-lying s t a t e s o_f each o f L i and N a (JL-6) and of s i x low-lying s t a t e s of L i (]_>§) • The computation of the p o t e n t i a l energy curves f o r the low-lying s t a t e s of N a at l a r g e separations (15^R^30 bohr) w i l l be reported s h o r t l y (9). Here, we r e p o r t , f o r the f i r s t time, ab i n i t i o computations of the s i x l o w e s t - l y i n g e l e c t r o n i c s t a t e s of Na2 . These comput a t i o n s u t i l i z e the b a s i s set developed to describe the low-lying states of the n e u t r a l N a molecules (6) and u t i l i z e i n t e g r a l s which have been computed p r e v i o u s l y (6,9). The molecular energies computed at the s i n g l e - c o n f i g u r a t i o n s e l f - c o n s i s t e n t f i e l d (SC-SCF) l e v e l are l i s t e d i n Table I. These SC-SCF comput a t i o n s should provide r e l a t i v e l y r e l i a b l e p o t e n t i a l curves f o r what are e f f e c t i v e l y one-electron systems. We do not attempt to describe the e l e c t r o n c o r r e l a t i o n a s s o c i a t e d with the core e l e c t r o n motions nor that a s s o c i a t e d with the p o l a r i z a t i o n of the core e l e c t r o n s by the s i n g l e valence e l e c t r o n . Thus, while d i s persion e f f e c t s are not w e l l described, the f i r s t order i o n induced d i p o l e i n t e r a c t i o n and the major e l e c t r o s t a t i c i n t e r a c t i o n s of the valence e l e c t r o n are probably reasonably w e l l desc r i b e d at the SC-SCF l e v e l . Note i n Table I I , where we l i s t molecular constants f o r Na? , that the 1 E s t a t e i s bound. I t s 1 °E"' counterpart i n the n e u t r a l molecule i s p r e d i c t e d to be s t r i c t l y r e p u l s i v e a t the SC-SCF l e v e l . The -Ci+R ion-induced d i p o l e i n t e r a c t i o n accounts f o r the d i f f e r e n c e . 2

2

2

2

+

2

+

2

+

t

-4

0097-6156/82/0179-0003$05.00/0 ©

1982 A m e r i c a n Chemical Society

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

4

METAL BONDING AND INTERACTIONS

TABLE I

2

2

bohr

Downloaded by 80.82.77.83 on May 9, 2017 | http://pubs.acs.org Publication Date: March 8, 1982 | doi: 10.1021/bk-1982-0179.ch001

a + P o t e n t i a l energy curves f o r Na obtained from e l e c t r o n , ab ini£io computations 1 E+ g

4.00 4.25 4.50 4.75 5.00 5.50 6.00 6.20 6.50 6.80 7.00 7.20 7.50 8.00 8.50 9.00 10.00 11.00 12.00 13.00 15.00 18.00 21.00 24.00 27.00 30.00

.506383 .520333 .531860 .541350 .549070 .560082 .566501 .568090 .569656 .570418 .570568 .570477 .569974 .568396 .566201 .563637 .558144 .552856 .548282 .544730 .539781 .536632 .535612 .535264 .535122 .535053

1

2

Z

+

u

— —

.421169



.439966 .455180 .467831



.478515



.487618



.495409 .502080 .507782 .512625 .520194 .525482 .529089 .531489 .534025 .535150 .535265 .535187 .535106 .535049

2

n

2

U

n

2

2 Z+ g

g

.463534

.397796



.327497 .344265 .358824 .371545 .382675 .400960 .415139 .419921 .426332



.407381



.435330

.428153



.442172 .451944 .458789



.466749



.468842 .470112 .470787 .471025 .470718 .469832 .468730 .467600 .465611 .463675 .462796 .462480 .462391 .462380

all-

— — — .356177 .372738 .386396







.415482 .422352 .428196 .433164 .441043 .446786 .450986 .454058 .457954 .460670 .461714 .462113 .462272 .462343

.442688 .448781 .453869 .458115 .464645 .469099 .471975 .473527 .474092 .471013 .467378 .465051 .463872 .463304

2

2 E

— — — — .321218 .337683



.352273



.365310 .376990 .387441 .396754 .404994 .418660 .429165 .437208 .443377 .451794 .458482 .461434 .462587 .462934 .462966

-1 Energies l i s t e d i n h a r t r e e atomic u n i t s e^/a = 219474.6 cm A l l energies have a p r e f i x of -323. Thus, tRe energy of the 1 E s t a t e at R = 4 bohr i s -323.506383 e / a . 2

+

+

u

2

Q

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

1.

KONOWALOW AND ROSENKRANTZ

TABLE I I

Light

Alkali

Diatomic

Molecules

5

Molecular constants f o r s e v e r a l e l e c t r o n i c s t a t e s of Na +

2

State

2

1

l+ g

2

1 I

2

Downloaded by 80.82.77.83 on May 9, 2017 | http://pubs.acs.org Publication Date: March 8, 1982 | doi: 10.1021/bk-1982-0179.ch001

Data Source a b c

+

R (bohr) e

7.02 6.69 6.24

D e

(cm

-i

l

«e. cm

a) xe cm

116.0 126

0.43

1 1

)

7796 8230 7905

e

1

u

a c,d

20.3 25

a b,d c

9.14 (9.0) 9.8

1900 (2170) 1860

49.9

0.33

U

a b, d c, d

14.37 (12) (16)

2399 (2850) (3030)

39.1

0.16

n

9.68

48.5 43.9

0.45

l

2 Z+ g

^Present work. Reference (10). ^Reference (11). Values i n p a r e n t h e s i s have been estimated c r u d e l y by us from t a b u l a t e d data i n the r e f e r e n c e c i t e d . +

At present, the l o w - l y i n g s t a t e s of N a are b e t t e r chara c t e r i z e d computationally than e x p e r i m e n t a l l y , although m u l t i photon i o n i z a t i o n experiments may change that p i c t u r e e v e n t u a l l y . We f i n d reasonably c l o s e agreement between the r e s u l t s of our a l l - e l e c t r o n computations, pseudopotential (10), and model potent i a l (11) computations. The l a t t e r two kinds of computations may give more accurate r e s u l t s than our ab i n i t i o computations s i n c e they may account f o r at l e a s t c e r t a i n core p o l a r i z a t i o n e f f e c t s . The a d i a b a t i c p o t e n t i a l energy curves f o r these e l e c t r o n i c s t a t e s c a l c u l a t e d i n the Born-Oppenheimer approximation, are given i n F i g u r e 1. Since we have d i s c u s s e d the choice of b a s i s funct i o n s and the choice of c o n f i g u r a t i o n s f o r these m u l t i c o n f i g u r a t i o n s e l f - c o n s i s t e n t f i e l d (MCSCF) computations (12) p r e v i o u s l y (1-9), we s h a l l not explore these questions i n any d e t a i l here. S u f f i c e i t to say that the b a s i s set f o r L i d e s c r i b e s the lowest S and P s t a t e s of the L i atom at e s s e n t i a l l y the Hartree-Fock l e v e l of accuracy, and i n c l u d e s a set of c r u d e l y optimized d f u n c t i o n s to accommodate molecular p o l a r i z a t i o n e f f e c t s . The b a s i s s e t we employed f o r c a l c u l a t i o n s i n v o l v i n g Na i s somewhat l e s s w e l l optimized than i s the L i b a s i s ; i n p a r t i c u l a r , sa molecular o r b i t a l s are not as w e l l d e s c r i b e d f o r Na (relatively speaking) as they are f o r L i . 2

2

2

2

2

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

METAL BONDING AND INTERACTIONS

Downloaded by 80.82.77.83 on May 9, 2017 | http://pubs.acs.org Publication Date: March 8, 1982 | doi: 10.1021/bk-1982-0179.ch001

6

R Figure 1.

(BOHR)

R

(BOHR)

Potential curves for low-lying states of Li , Li , Na , and Na from the all-electron calculations reported in Refs. 1-9. 2

+

2

2

+

2

obtained

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

1.

KONOWALOW AND ROSENKRANTZ

Light

Alkali

Diatomic

7

Molecules

Since the research which produced the p o t e n t i a l curves i n Figure 1 has been (or w i l l be) reported over a span of some s i x years, i t seems a p p r o p r i a t e now to make some very general observ a t i o n s about them. F i r s t we note that these p o t e n t i a l s are expected to be h i g h l y accurate at l a r g e i n t e r n u c l e a r s e p a r a t i o n s say, f o r R>15 bohr. Because of l i m i t a t i o n s on the kinds of e l e c t r o n i c c o n f i g u r a t i o n s which we could t r e a t , (see Ref. (1) f o r a d i s c u s s i o n ) , the IT s t a t e p o t e n t i a l curves, f o r the n e u t r a l molec u l e s are somewhat l e s s accurate than t h e S s t a t e curves, e s p e c i a l l y at i n t e r n u c l e a r separations of R < 15 bohr. Our best r e s u l t s f o r £ s t a t e s of L i and a l l s t a t e s of L i are probably i n e r r o r by 2% or l e s s . The corresponding e r r o r s f o r Na and N a are probably about three times l a r g e r . We obtained these e r r o r estimates by s c a l i n g our p o t e n t i a l curves so that they reproduced i n the l e a s t squares sense the experimental energy l e v e l spacing of the X Z+ and A !* s t a t e s . Becaule they are i s o v a l e n t molecules, the s i m i l a r i t y of the p o t e n t i a l curves f o r comparable s t a t e s w i t h i n the L i and the Na systems i s , of course, no s u r p r i s e . As expected, the p o t e n t i a l curves f o r the Na system are s h i f t e d to the r i g h t of the c o r r e s ponding curves i n the L i system and each of the Na curves i s " f a t t e r " and shallower than i t s L i counterpart. Consequently, the bound s t a t e s i n the Na system are expected to have more bound v i b r a t i o n a l l e v e l s which are more c l o s e l y spaced than t h e i r L i counterparts. These expectations are borne out f o r s t a t e s such as X Zg, A !* and B T T which are w e l l c h a r a c t e r i z e d experimentall y and f o r these and other s t a t e s which are so f a r best c h a r a c t e r i z e d by computations such as our own. Note that the lowest £ curve crosses the lowest T I curve for both L i and L i but not f o r N a and Na . T h i s can lead to strong p r e d i s s o c i a t i o n of the T I curves i n the L i system but not i n the Na system. The d i f f e r e n c e i n behavior apparently has three main causes: the f i r s t , the 3s-3p l e v e l s e p a r a t i o n i s some 2050 cm l a r g e r i n Na than the 2s-2p s e p a r a t i o n i n L i . I f we were to t r a n s l a t e r i g i d l y upwards the L i 2s + 2p asymptote together with a l l i t s curves, by some 2050 cm" , we would f i n d the s J II curve c r o s s i n g to occur at about 4.1 bohr i n s t e a d of the a c t u a l 4.6 bohr, and the lowest s e v e r a l v i b r a t i o n a l l e v e l s of the T I s t a t e would be r e l a t i v e l y f r e e of p r e d i s s o c i a t i o n . Secondly, the I I s t a t e i s about 30% deeper i n L i than i n Na . Thus, the IT curve comes much c l o s e r to the lowest ns+ns asymptote (and consequently c l o s e r to the E * curve) i n L i 2 than i n Na2. Thirdly, s i n c e Na and N a are s u b s t a n t i a l l y more p o l a r i z a b l e than L i and L i , the r e p u l s i v e E curves i n the Na system are " s o f t e r " (more a t t r a c t i v e ) than i n the L i system. While these q u a l i t a t i v e arguments have concentrated on the presence or absence of the E IT curve c r o s s i n g i n the n e u t r a l systems, the arguments extend i n an obvious manner to the E J - f t p r e d i s s o c i a t i o n (or not) i n L i (and N a ) . +

2

2

+

Downloaded by 80.82.77.83 on May 9, 2017 | http://pubs.acs.org Publication Date: March 8, 1982 | doi: 10.1021/bk-1982-0179.ch001

2

l

x

2

1

1

1

U

U

u

+

+

2

2

2

2

U

-1

1

3

3

3

3

3

U

2

2

U

3

+

+

u

3

U

3

U

2

2

u

+

2

+

2

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

U

U

METAL BONDING AND INTERACTIONS

8

Scaled P o t e n t i a l Curves f o r K Rb2, C s and t h e i r Molecular Cations Note that a l l the a l k a l i atoms have ns S and np P as t h e i r two l o w e s t - l y i n g atomic terms, ( c l e a r l y n i s the p r i n c i p a l quantum number). T h i s suggests that we should be able t o s c a l e our p o t e n t i a l curves f o r the l i g h t a l k a l i s i n order t o estimate, however c r u d e l y , the p o s i t i o n s and shapes of the corresponding s t a t e s of K , Rb , C s and t h e i r molecular c a t i o n s . Let us o u t l i n e a very simple s c a l i n g scheme. We d e f i n e s c a l e d i n t e r n u c l e a r s e p a r a t i o n ( R ) and energy ( E ) v a r i a b l e s : 2 >

2

2

2

2

2

2

g

Downloaded by 80.82.77.83 on May 9, 2017 | http://pubs.acs.org Publication Date: March 8, 1982 | doi: 10.1021/bk-1982-0179.ch001

R

= R

s

= ( R / JUp R ,

C

X

P

W

=

g

e

°

(

R

C

)

C

=

( D

1

e

D

}

e

e C ( r C )

>

where p and e are s c a l i n g f a c t o r s f o r i n t e r n u c l e a r s e p a r a t i o n and the d i s s o c i a t i o n energy, R and D are the corresponding values of the c a l c u l a t e d b i n d i n g energy curve E ( R ) which i s to be s c a l e d and E ( R ) i s the r e s u l t i n g s c a l e d b i n d i n g energy curve. I t would be t y p i c a l to take R and D* to be experimental values i f t h i s data were a v a i l a b l e . T h i s i s the s o r t of s c a l i n g that i s c a r r i e d out i n a p p l y i n g the Law of Corresponding States (13). We t e s t e d t h i s scheme f i r s t by s c a l i n g our L i and L i pot e n t i a l s so that we estimate the corresponding p o t e n t i a l s f o r Na and N a . Instead of experimental values of R* and we used our values c a l c u l a t e d f o r the X !*" s t a t e o f N a to o b t a i n p and e. Thus we use only the L i and L i curves and the R and D values for the X s t a t e o f N a to estimate the e n t i r e p o t e n t i a l curve f o r each of e i g h t e l e c t r o n i c s t a t e s of N a and s i x e l e c t r o n i c s t a t e s of N a . Except f o r the E g and I I s t a t e s the s c a l i n g worked w e l l as i s evident from F i g u r e 2. Let us now t u r n to the e s t i m a t i o n of the p o t e n t i a l curves f o r l o w - l y i n g n e u t r a l and c a t i o n i c diatomic molecules f o r the heavy a l k a l i s . For each molecule we take R* and t o be the experimental values (14) f o r the corresponding X !^ s t a t e . We a l s o a d j u s t the s e p a r a t i o n of the asymptotes to correspond t o the a p p r o p r i a t e experimental resonance t r a n s i t i o n (ns-np) and ns S i o n i z a t i o n energies. (We have ignored the s p i n - o r b i t s p l i t t i n g (15) of the P s t a t e of the heavy a l k a l i s as we had f o r L i and Na. The s i n g l e S + P asymptote was made to correspond to the degeneracy-weighted energy of the P s t a t e . C l e a r l y t h i s approximat i o n becomes p r o g r e s s i v e l y more s e r i o u s i n K2, Rb2 and CS2 where the comparable s p l i t t i n g s are about 58 cm" , 237 c m and 554 cm" , r e s p e c t i v e l y . ) The s c a l e d p o t e n t i a l curves a r e shown i n F i g u r e s 2, 3 and 4. I t i s c l e a r that the s c a l e d curves which are based on our L i system and those based on our Na system agree reasonably w e l l with each other except f o r the curves f o r the Z g and I I s t a t e s . (This discrepancy was expected i n view of our comparison of the L i and Na systems.) In view of our e a r l i e r remarks, i t appears that the p r e d i c t i o n s based on L i may be the C

c

C

g

C

s

e

+

2

2

+

2

1

2

+

2

2

e

e

2

2

+

3

3

2

U

1

2

2

2

2

1

3

-1

3

U

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

1

Downloaded by 80.82.77.83 on May 9, 2017 | http://pubs.acs.org Publication Date: March 8, 1982 | doi: 10.1021/bk-1982-0179.ch001

KONOWALOW AND ROSENKRANTZ

Light

R

Alkali

Diatomic

Molecules

(BOHR)

Figure 2. A comparison of the potential curves for Na and Na * computed ab initio with those obtained by scaling the corresponding curves of Li and Li * Identity of curves is shown in Fig. 1. 2

2

2

.

.

.

i

.

.

.

2

i

15 27 R (BOHR)

3

Figure 3. Estimated potential energy curves for K and K obtained by scaling computed potentials for Li and Na . Identity of curves is shown in Fig. 1. 2

2

2

2

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

Downloaded by 80.82.77.83 on May 9, 2017 | http://pubs.acs.org Publication Date: March 8, 1982 | doi: 10.1021/bk-1982-0179.ch001

METAL BONDING AND INTERACTIONS

Figure 4. Estimated potential energy curves for Rb and Rb obtained by scaling computed potential for Li and Na>. Identity of curves is shown in Fig. 1. 2

2

2

Figure 5. Estimated potential energy curves for Cs and Cs obtained by scaling computed potentials for Lij and Na,. Identity of curves is shown in Fig. 1. 2

2

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

1.

KONOWALOW AND ROSENKRANTZ

Light

Alkali

Diatomic

Molecules

11

more r e l i a b l e . There i s no question that these estimates f o r the p o t e n t i a l s f o r K2 (and Rb2 and CS2) are easy to o b t a i n . The r e a l question i s to assess the r e l i a b i l i t y of p r e d i c t i o n s based on these curves. F i r s t note that the E * s t a t e of K2 i s p r e d i c t e d not to p r e d i s s o c i a t e the IT s t a t e a t a l l s t r o n g l y s i n c e the curves don't cross at low e n e r g i e s . T h i s question i s important, f o r example, i n terms of understanding p o s s i b l e A*X and B-*X l a s e r emission i n K2, and i n terms of understanding the use of K as a working f l u i d i n a s o l a r powered engine (16). C u r i o u s l y enough, these t r i p l e t curves a r e p r e d i c t e d not to cross no matter whether we base our p r e d i c t i o n on L i 2 (where these curves do cross) or on Na (where they don't c r o s s ) . E v i d e n t l y the shrinkage of the two curves toward t h e i r r e s p e c t i v e asymptotes i n the course of s c a l ing the b i n d i n g energy curves i s s u f f i c i e n t to overcome the s h i f t of the ns + ns - ns + np asymptotes c l o s e r together i n the systems f o r the heavy a l k a l i s than they l i e i n e i t h e r the L i or Na s y s tems . Recently, Bhaskar and coworkers (17) have a t t r i b u t e d an IR a b s o r p t i o n band between 1.1 and 1.6 y t o the E * E ^ transit i o n i n K2. C l e a r l y , our s c a l e d curves suggest that such a band system could e x i s t with reasonable Franck-Condon f a c t o r s i n that wavelength r e g i o n . ( I t would, i n f a c t , be p o s s i b l e to o b t a i n s c a l e d t r a n s i t i o n d i p o l e moment f u n c t i o n s , (see below), c a l c u l a t e Franck-Condon f a c t o r s based on them and our s c a l e d p o t e n t i a l curves, and make more d e t a i l e d q u a n t i t a t i v e estimates of the E+ «- E+ band system i n K .) Before t u r n i n g from our crude c o n s i d e r a t i o n s on the heavy a l k a l i s , l e t us make a f i n a l p r e d i c t i o n . That i s that there s h a l l be observed i n K2 vapor an a b s o r p t i o n f e a t u r e which corresponds to the II E+ a b s o r p t i o n which l i e s somewhat to the blue of the atomic resonance l i n e v = 13000 cm" ; we p r e d i c t i t s peak to occur a t 710nm. [We have learned a t t h i s symposium (18) that a f e a t u r e i n the K a b s o r p t i o n spectrum a t 720 nm has been a t t r i b u ted to t h i s t r a n s i t i o n . ] T h i s f e a t u r e corresponds t o the absorpt i o n i d e n t i f i e d by Koch, Stwalley, and C o l l i n s (19) i n L i 2 , and by Woerdman and deGroot (18) i n Na2. The f e a t u r e i n L i was found at 588 nm, w h i l e our c a l c u l a t i o n s p r e d i c t e d i t to be a t around 595 nm. We estimate the TI -«- E"*" a b s o r p t i o n peak to l i e a t 546 nm i n Na2 while In Rb2 such a f e a t u r e w i l l occur a t about 727 nm and i n CS2 the corresponding feature w i l l occur a t about 800 nm, according to rough estimates we've obtained from our s c a l e d p o t e n t i a l curves. 3

3

U

2

Downloaded by 80.82.77.83 on May 9, 2017 | http://pubs.acs.org Publication Date: March 8, 1982 | doi: 10.1021/bk-1982-0179.ch001

2

3

3

3

3

2

3

3

1

s p

2

3

3

e

Spectroscopic Considerations R e c a l l that a spectrum corresponding to a one-photon t r a n s i t i o n i s e s s e n t i a l l y a p l o t of i n t e n s i t y (of emission or absorpt i o n ) as a f u n c t i o n of the d i f f e r e n c e i n energy ( u s u a l l y expressed as wavelength or frequency) between the two s t a t e s i n volved i n the t r a n s i t i o n . The d i f f e r e n c e p o t e n t i a l (the

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

METAL BONDING AND INTERACTIONS

12

d i f f e r e n c e between two p o t e n t i a l energy curves) provides rough estimates of t r a n s i t i o n energies. For diatomic molecules, i t i s s t r a i g h t f o r w a r d to s o l v e the Schrodinger equation f o r the v i b r a t i o n a l - r o t a t i o n a l energy eigenvalues and corresponding nuclear motion wavefunctions once the e l e c t r o n i c p o t e n t i a l curve i s known and thus obtain r e f i n e d information about t r a n s i t i o n energies. ( C l e a r l y , we invoke the Born-Oppenheimer approximation here.) I t i s a l s o s t r a i g h t f o r w a r d to c a l c u l a t e the e l e c t r o n i c t r a n s i t i o n d i p o l e moment f u n c t i o n

Downloaded by 80.82.77.83 on May 9, 2017 | http://pubs.acs.org Publication Date: March 8, 1982 | doi: 10.1021/bk-1982-0179.ch001

D(R)

= /** u * e,i e,j

i n terms of the e l e c t r o n i c wavefunctions f o r the s t a t e s i and j involved i n the t r a n s i t i o n , and the d i p o l e moment operator u. I n t e n s i t i e s of a s p e c t r a l t r a n s i t i o n a r e p r o p o r t i o n a l to the square of the i n t e g r a l (20)

R

nk

- '* * n

t D ( R ) ]

*k

where $ denotes the v i b r a t i o n a l - r o t a t i o n a l wavefunction f o r the s t a t e n. Frequently, i n v e s t i g a t o r s approximate the f u n c t i o n D(R) by i t s asymptotic value D(°°) which corresponds t o the t r a n s i t i o n d i p o l e moment o f the r e l e v a n t atomic t r a n s i t i o n . C l e a r l y , t h i s approximation,which we term the atomic approximation, i s worthless f o r molecular t r a n s i t i o n s which are allowed but which correspond a s y m p t o t i c a l l y to forbidden atomic t r a n s i t i o n s . I t i s even i n c o r r e c t to assume that such molecular t r a n s i t i o n s w i l l be weak. We have found t r a n s i t i o n s among e x c i t e d s t a t e s of L i 2 (21) which are forbidden a s y m p t o t i c a l l y y e t which have t r a n s i t i o n d i p o l e values as l a r g e as 24 atomic u n i t s (=61 Debeye)! Even where the t r a n s i t i o n i s allowed a s y m p t o t i c a l l y , the atomic approximation can lead to s u b s t a n t i a l e r r o r i n the p r e d i c t e d i n t e n s i t i e s . I t i s not unusual f o r D(R) to deviate by 20-30% from i t s asymptotic value a t i n t e r n u c l e a r separations where the overlap o f the v i b r a t i o n a l wavefunctions $ and i s appreciable. Thus, i n t e n s i t i e s which are c a l c u l a t e d by using the atomic approximation f o r such t r a n s i t i o n s may be i n e r r o r by 40-70%. A paper i s i n p r e p a r a t i o n (22) which w i l l r e p o r t the D(R) values f o r a l l d i p o l e allowed t r a n s i t i o n s among most of the s t a t e s which are depicted i n Figure 1. n

n

In the f o l l o w i n g paragraphs we give s e l e c t e d examples of the use of our wavefunctions and p o t e n t i a l curves t o p r e d i c t or conf i r m various s p e c t r o s c o p i c features o f the a l k a l i s . We know of plans to observe L i 2 s p e c t r a i n a t l e a s t two l a b o r a t o r i e s (23, 24) so some p r e d i c t i o n s regarding the s p e c t r a appear to be i n order. J u l i e n n e (25) has used our wavefunctions f o r L i 2 to c a l c u l a t e the e l e c t r o n i c t r a n s i t i o n d i p o l e moment f u n c t i o n c o r r e s ponding to the A IT - X Z ^ t r a n s i t i o n and to c a l c u l a t e the matrix element < £ ^ | L - i L y | II > needed to determine the r a t e of p r e d i s s o c i a t i o n of the IT s t a t e by the E s t a t e . Since the +

+

2

2

U

2

2

x

U

2

2

U

U

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

1.

KONOWALOW AND ROSENKRANTZ

Light Alkali

Diatomic

a n c

13

Molecules

p

s p i n - o r b i t s p l i t t i n g between the Pi/2 * 3/2 s t a t e s of L i i s only 0.34 cm (15), the IT s t a t e i s c h a r a c t e r i z e d by Hund's case b c o u p l i n g . Thus the " z j s t a t e only p r e d i s s o c i a t e s the r o t a t i o n a l l e v e l s which give r i s e to P and R branches; the Q branch i s not p r e d i s s o c i a t e d . J u l i e n n e has used the c a l c u l a t e d t r a n s i t i o n d i p o l e moment f u n c t i o n t o o b t a i n the r a d i a t i v e l i f e t i m e s f o r the I I r t t r a n s i t i o n . He f i n d s , f o r example, that the r a d i a t i v e l i f e t i m e of the v =0 Q branch emission i s about 12 ns and increases f a i r l y r a p i d l y with v i b r a t i o n a l l e v e l . He a l s o f i n d s that r a d i a t i o n and p r e d i s s o c i a t i o n r a t e s f o r the P and R t r a n s i t i o n s are comparable for certain r o t a t i o n a l - v i b r a t i o n a l levels. Our curves f o r the molecular ions N a ? . . . C s 9 show that the and IT p o t e n t i a l s do not cross a t an energy where the I I s t a t e i s bound. Thus, the i n t e r e s t i n g p r e d i s s o c i a t i o n p r e d i c t e d to be present i n L i w i l l not be present i n these heavier a l k a lis. The o r i g i n of the d i f f e r e n c e i n p r e d i s s o c i a t i o n behavior can be understood by a simple extension of the arguments we've put forward i n d i s c u s s i n g the I I - E+ i n t e r a c t i o n i n the n e u t r a l molecules. The spectroscopy of the s i n g l e t manifold i n the l i g h t a l k a l i s i s r e l a t i v e l y well-developed i n part s i n c e the molecular ground s t a t e i s a s i n g l e t . We have obtained t r a n s i t i o n d i p o l e moment f u n c t i o n s f o r t r a n s i t i o n s i n the s i n g l e t manifold which are i n reasonable agreement with the l i m i t e d experimental data which i s a v a i l a b l e (26,27). We have not y e t undertaken a l i n e b y - l i n e comparison of c a l c u l a t e d and experimental Franck-Condon f a c t o r s (which are more r e a d i l y a v a i l a b l e than are the D(R)). Although i t i s a s t r a i g h t f o r w a r d task, i t i s a p a r t i c u l a r l y time consuming one. I t has been noted by Wellegehausen (28,29) that one can o b t a i n good l a s e r emission i n the A ^ J - X ^ and B ^ - X ! ] * systems of L i and Na2, only weak l a s e r a c t i o n i n these systems of K but, so f a r , no l a s e r emission has been observed i n Rb and C s . The r e s u l t s presented here, together with unpublished c a l c u l a t i o n s on e x c i t e d s t a t e s of L i (21) suggest that s e l f absorption of any l a s e r emission should become a more probable event the h e a v i e r the a l k a l i dimer. Thus, i t appears that Wellegehausen's observat i o n s might be explained by such s e l f absorptions. By contrast t o the s i n g l e t manifolds of the a l k a l i s , lowl y i n g members of the t r i p l e t manifolds have probably been much b e t t e r c h a r a c t e r i z e d computationally. T h i s i s understandable s i n c e the ground s t a t e of the t r i p l e t manifold, E , i s very weakly bound ( D = 300 cm" = 420 K f o r L i , and D = 180 cm"" = 250 K f o r Na , f o r example). Thus at the temperatures o r d i n a r i l y used to o b t a i n a l k a l i vapor, one expects to f i n d a very small f r a c t i o n of bound E+ molecules. (Thus, i t i s not uncommon f o r s p e c t r o s c o p i s t s to term the s t a t e "the r e p u l s i v e s t a t e " and question only whether i t e f f e c t i v e l y p r e d i s s o c i a t e s the TI or e l s e immediately dismiss i t from f u r t h e r c o n s i d e r a t i o n . They -1

2

2

2

U

f

+

+

Z

Z

Downloaded by 80.82.77.83 on May 9, 2017 | http://pubs.acs.org Publication Date: March 8, 1982 | doi: 10.1021/bk-1982-0179.ch001

U

U

+

2

3

3

U

1

1

2

2

2

2

2

3

U

1

e

1

2

e

2

3

3

U

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

METAL BONDING AND INTERACTIONS

14

have missed the opportunity to i n v e s t i g a t e some f a s c i n a t i n g spectroscopy.) There i s apparently a s u f f i c i e n t amount of E * a v a i l able i n L i , Na, K, and Cs vapors f o r at l e a s t two kinds of t r a n s i t i o n s i n v o l v i n g i t to have been observed (17,1^8,19,30). Bhaskar and coworkers (17) have observed i n potassium vapor an IR absorpt i o n band between 1.1 and 1.6y which they a t t r i b u t e to the E+ -«- E+ t r a n s i t i o n . Zouboulis and coworkers (30) i d e n t i f y a 1.25 - 2.5y band with t h i s t r a n s i t i o n i n CS2. Our s c a l e d potent i a l curves f o r K , f o r example, show that a b s o r p t i o n i n t h i s region could indeed be a t t r i b u t a b l e to that t r i p l e t t r a n s i t i o n . One would expect to observe a band with some s t r u c t u r e as expected f o r a t r a n s i t i o n that i s dominated by a b s o r p t i o n from the continuum to bound l e v e l s of the I * s t a t e . Boundbound t r a n s i t i o n s corresponding to t r a n s i t i o n s from very low v" to rather high v > 40 should be d e t e c t a b l e at wavelengths X < 0.9y. We expect the emission c h a r a c t e r i s t i c s of the E ^ - £ * t r a n s i t i o n i n K to be s i m i l a r to those we have p r e d i c t e d (31) f o r L i and Na . Here, we would expect the peak continuous emission i n t e n s i t y to occur near l . l y . I f the E g s t a t e of K could be populated s u f f i c i e n t l y r a p i d l y , the E * E* transition would comprise a tunable near i n f r a r e d excimer l a s e r with i t s peak about midway between that of the corresponding excimer l a s e r s based on L i X^1.3y and on Na X^0.83y. These estimates f o r K are n e c e s s a r i l y very rough s i n c e they are based on our s c a l e d p o t e n t i a l curves. I t does seem u n l i k e l y that we are i n e r r o r by s u b s t a n t i a l l y more than O.ly, however. A s i m i l a r a n a l y s i s of Rb and C s could be made i n terms of our s c a l e d potentials. 3

3

3

2

Downloaded by 80.82.77.83 on May 9, 2017 | http://pubs.acs.org Publication Date: March 8, 1982 | doi: 10.1021/bk-1982-0179.ch001

3

f

3

3

2

2

2

3

2

3

2

3

2

2

2

3

2

3

The TI Z * t r a n s i t i o n i n the t r i p l e t manifold has app a r e n t l y been observed i n L i (19) and i n Na (17,24). Koch and coworkers (19) observed a bound-free-bound t r a n s i t i o n which they a t t r i b u t e d i n part to n +• Z+ a b s o r p t i o n i n L i . They obtained a peak a b s o r p t i o n at around 588 nm with s u b s t a n t i a l s t r u c t u r e to the red. In F i g u r e 6, we show the p h o t o d i s s o c i a t i o n cross s e c t i o n (32) as a f u n c t i o n of the frequency (and wavelength) of r a d i a t i o n e x c i t i n g the bound-free II E^ transition in L i . We p r e d i c t a peak s l i g h t l y to the blue of 595 nm which agrees n i c e l y with the observed peak at 588 nm. Veza and coworkers (17) c l a i m the f i r s t d i r e c t experimental c o n f i r m a t i o n of the II s t a t e i n Na . They a t t r i b u t e the s a t e l l i t e band i n the B T[„ - X E g band system at 551.5 nm to the maximum i n the TTg d i f f e r e n c e p o t e n t i a l and thus to that t r a n s i t i o n . We estimate, extremely c r u d e l y , that the s a t e l l i t e should l i e at around 546 nm. I t i s c l e a r that ab i n i t i o computations, e s p e c i a l l y of the p o t e n t i a l curves and wavefunctions f o r t r i p l e t s t a t e s has proven to be u s e f u l i n the i n t e r p r e t a t i o n of a v a r i e t y of s p e c t r a l f e a tures i n L i and Na . I t appears that our s e m i - e m p i r i c a l t r e a t ment of the h e a v i e r a l k a l i s may prove u s e f u l u n t i l we are able to t r e a t these systems ab i n i t i o . We have launched a more d e t a i l e d g

2

3

2

3

g

2

3

3

2

3

2

l

2

3

2

2

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

Light Alkali Diatomic

Molecules

Downloaded by 80.82.77.83 on May 9, 2017 | http://pubs.acs.org Publication Date: March 8, 1982 | doi: 10.1021/bk-1982-0179.ch001

KONOWALOW AND ROSENKRANTZ

Figure 6.

The calculated photodissociation cross section for the U