Chenuca l Physics 88 (1984) 375-389 375 _ Nor th-Hol land , A m s t e r d a m

13VO-EXCITON SPECTRA O F H O AND HBr CRYSTALS

Franco BOGANI

D,part:mento d: Fts:¢a, Unt~erslta d, Firenze and Umta G N S M dt Ftren=¢, Largo E Fermf 2, 50125 Florence, Italy

and

Roberto G I U A and Vincenzo SCHETTINO

D:Fartmwnto d: Chtmwa. Laboratorto dl Spettroscopta Molecolare, Umvers:ta d: F:renze. ma G Cappom 9. 50121 Florence, itai~

Recel,.ed 27 December 1983

The mechan i sms for double excttattons of the internal v~bratton m crysta lhne HCi and HBr are dxscussed The renormahzed one- and two-exoton Green funcaons of interest are calculated a s suming that renorrnahzat ton of impor tance ts only due to mtramolecular cubte and quart lc po tenua ls It ts shown that the mechan t sms of tx~o-exciton absorpt ion are different m the HCI and HBr crystal In HCI the contr ibut ions o f non-local abso rp tmn involving induced dtpoles and of local absorpt ion mvolwng mechanmal molecular anharmomcaty are comparable In HBr the local absorpt ion gives a p redominan t con tnbu t ton

I. Introduction

Vibrational double excitations m molecular crystals were first reported by Ron and Hornig [1] for the HCI crystal and have since been the object of considerable interest. In fact, the structure of the two-exciton bands in the infrared and Raman spectra bears an intimate connecaon w~th the one- and two-exc~ton densllaes of states and is a significant probe of intra- and rater-molecular couplings. The subject has recently been reviewed with reference to the experimental and theoretical aspects [2,3]. A model for the interpretation of two-exciton band shapes and intensities tn the infrared spectra of molecular crystals has been d~scussed by one of us (FB) [4,5]. The model assumes that the mtermolecular couphng is harmomc and expresses the absorptton coefficient m terms of appropriate Green funcnons renormalmed by the intramolecular anharmonicity, viewed as a local impurity. The theory is sufficiently general to allow the discussion of the essential features of vibrational double excitations in terms of resonances and biexciton bound states split off the cont inuum by the local anharmonicity. In the particular case where the dispersion of one of the excitons can be neglected, closed expresstons for the renormahzed Green funcuons of interest can be obtained. Under these circumstances the theory has been applied to several molecular crystals including CO 2 [5], N 2 0 [5], O C S [5] and SF 6 [6] leading to a satisfactory explanation of the band shape of the two-exciton cont inuum and of its intensity relative to the bound state, whenever present in the spectrum.

A quahtatlve discussion of the general trends occurring in the spectra of vibratlonal double excttations has been reported again in recent papers [7-9] starting from a model (semicircular) density of states and the discussion has been extended to consider the overtone infrared spectrum of the HCI crystal. However, the discussion of this particular system does not appear to have much progressed beyond qualitative considerations, despite the fairly large amount of informat ion available. This is due to the rudimental form

0301-0104/84/$03.00 © Elsevaer Science Publishers B.V. (North-Holland Physics Publishing Division)

3 7 6 F Bogant et al / T~o-erc l ton spectra o f HC! and HBr costaLv

o f the dens i ty of s ta tes used a n d to neglect o f i m p o r t a n t m e c h a n i s m s o f d o u b l e exc i t auon . The re fo re , fea tures o f the ove r tone H C I s p e c t r u m like the d o u b l e t s t ruc tu re o f the b o u n d state, its re la t ive pos i t ion a n d In tens i ty with respect to the c o n t i n u u m and the b a n d s h a p e o f the c o n t i n u u m itself r ema in u n e x p l a i n e d so

far. T h e p u r p o s e o f the presen t p a p e r ~s to r epor t a q u a n t i t a t i v e t r e a t m e n t o f the two-exc i ton a b s o r p t i o n

s p e c t r u m in the HCI and H B r c rys ta l s Recent ly , an a p p r o p r i a t e m t e r m o l e c u l a r po ten t i a l for the in te rna l vabrat ion m crys ta l l ine HCi has been e l a b o r a t e d and the one- and two-exc l ton d e n s m e s o f s ta tes have b e c o m e avmlah le [10]. T h e s a m e type o f i n t e rmolecu la r po t en tml can be used for the H B r crystal . S ta r t ing wi th th~s basic in fo rma t ion , ~t will be s h o w n m the p resen t p a p e r tha t a c losed s o l u u o n for the G r e e n f u n c u o n s o f interest for the o v e r t o n e m HCI and H B r can be o b t a i n e d and f r o m these the two-exc i ton b a n d shapes wdi be ca lcula ted using mo lecu l a r p a r a m e t e r s d e t e r r m n e d e x p e r i m e n t a l l y o r ad jus ted for best fit to expe r imen t s . A sa t i s fac tory exp lana t ion o f mos t o f the fea tures o f the spec t r a has been o b t a i n e d and the paral le l s tudy of the HCI and H B r crys ta l has a l lowed s o m e in teres t ing o b s e r v a t i o n s on the m e c h a m s m s respons ib le for two-exc t ton a b s o r p t i o n in these sys t ems

2. T h e o D

In the present sect ion the theory o f Bogam [4,5] o f the a b s o r p t i o n coeff ic ient in the two-exc i ton region o f m o l e c u m r crysta ls x~di be br ief ly revaewed T h e theory will then be e~ tended for app l i ca t i on to the o v e r t o n e r egmn of the hydrogen hahde crystals .

2 1. Co , s ia l h a m t l t o m a n

T h e v ibra t ,ona l excl ton sys t em can be desc r ibed m t e rms of crys ta l n o r m a l c o o r d i n a t e s Qj (~l t ) o r o f the - / ,

c o r r e s p o n d i n g c rea t ion A~+(~.lt ) and a n n l h d a u o n Aj (,~ I t ) o p e r a t o r s In the fo l lowing k wdl indica te the w a v e v e c t o r in the first B n l i oum zone. . / the v ib ra t iona l exct ton cons ide red a n d et the exc l ton b ranch . It is a s s u m e d that no r m x m g of d i f ferent m t r a m o l e c u l a r m o d e s occurs m the crystal . F o r the p resen t p u r p o s e s it has p roved c o n x e m e n t to x~ork In a local r ep re sen ta t i on by def in ing the o p e r a t o r s

(?;(~)=~2hN! ~b(~lk~)[O_.(~.)-,Q,(~.)/,.,(k~)] exp[-,~-x(l~)]. (1) / , e ,

Q;(~) = ( % ~/-'~lb*(.It.,~)[Q,(~.) +,Q~(~.)/%(~:~)] e~p[,k-x(t.)] (2) ~, 2 h N ] ~,,

where , . , , (ka) and % are the crys ta l and f ree -molecu le f requencms, respecuve ly , ) b ( / ~ l k a ) the crys ta l e igenvectors , N the n u m b e r of unit cells, x ( l t t ) the vec to r pos i t ion o f cell i a n d site /.t. In the l irmt of v a m s t u n g in te rmolecu la r coupl ing, the o p e r a t o r s (1) and (2) r ep resen t c rea t ion and a n m h l l a t t o n o p e r a t o r s for m o d e j at site i/t.

T h e crys ta l h a m d t o m a n for the v l b r a t m n a l exc l tons can be d iv ided into a h a r m o m c , H o, and an a n h a r m o m c , H , , par t . T h e h a r m o m c ha r ruhon ian inc ludes mt r a - a n d in t e r -mo lecu la r c o n t n b u u o n s . In the local r e p r e s e n t a t m n the h a r m o n i c ha rml ton ian can be wr i t ten as

+ I - - I ! I I " ~ I - - -o = E E,o,e, (,)e, E (,)e, (3)

where

¢,, ('~ ~:)= N-' Y'.wT(~ I ka)P-,(,~l k ),,~(~'1 k~) exp[ik-x(l/t, ! ' /~')] k a

(4)

F Bogant et a l / Two-p tc t ton spectra o f H e ! and H B r co'stals 377

a n d

G ( , - I k ) = , , ( k ~ ) - - , . ( 5 )

The crystal frequencies coj(ka) are ob ta ined as eigenvalues of the dynanuca l matr ix

D~(~'I~:) = u) + Z 0J:'(~ ~ ')exp[ik.x(l#. l~ ')] (6)

and the possible forms of the in termolecular coupl ing coefhc ien ts ,~t2>t~ C) have been discussed m detml m r ] ~..u.H. -"

ref. [1]. For the purpose o f the present paper it will be suff icient to consider a n h a r m o m c terms within a single

molecule. The a n h a r m o m c hami l t oman includes cubic and qua ruc terms H~ = H ° ~ + Ht4L in the local representat ion the cubic h a n u i t o m a n ~s g~ven by

H ' 3 ' = ) -~ Y~. v j . j ~ , Q ~ ( ~ ) Q f , ( ~ ) Q ~ ( ~ ) . (7) //x Ja ->Jw

For the r enormahza t lon of the two-e,cctton Green funct ions discussed in the fol lowing tt has been found convenient [4] to work with an effective quar t ic hanu i ton ian

H ( 4 ' = E E x,..,e=(;) e ; ('.) e ; (') e ; ( 8 )

lit Jz ~Jt

which also includes second-order cont r ibut ions of the cubic terms (7) Add i t iona l a n h a r m o m c terms not included m (7) and (8) and cont r ibu t ing to the r enormahza t ton of the single molecule energy are accounted for by using an effective free-molecule frequency.

2 2 Green funct:ons

As discussed in ref. [41 the vabratlonal two-exo ton spec t rum of molecular crystals ,s g~ven by the fol lowing Green funcuons

G( . , ; [ ~ ' ~ : ) = - - , O ( t ) ~ + ', ~ O + a ,+ .... 0 - 1 " I O ~ O - e ,+m4 "" ,~, •, <<Qj, ( . , I t . ~_j..~,._ -I t ) ; . . j , . . , , _ . . . j _ . . , , . 10)>>. (9) Ii1~

O (/.t dx3) -- i0 (t))-". + t' . = ( (Q~, (~, I t ) , Q~(~' 1 0 ) ) ) , ( 10 )

G ( m ]#!#2#3) = - l O ( t ) ~_, L : O + ( '' I t ~ o + ( t ' + " J t ) , Q~(~; [0 ) ) ) . (11) xx~--Jl x,tl I I - / r.-j2x/t 2 IiIj

The Green funct ions (9)-(11) are connected to each o ther and the basle quan t i ty is the two-exciton Green funct ion (9) The e quauons for these Green funct ions canno t be solved analyt ical ly m general and on ly In some par t icular case.s can a closed so luuon be ob tmned . The vibrat ional e x o t o n sys tem is character ized by relatively tugh f r equenoes and it is thus a reasonable a p p r o x i m a u o n to consider the system at zero temperature . A fur ther s i m p h f y m g assumpt ion is that the mtermolecu la r coupl ing coefhctents depend only on the in termolecular separauon. With in these approxzmatmns the ha rmonic two-excl ton Green funct ion is given by

g(~',~] [ ~' ~-'.) = Y~ Y'~ e x p [ , k - x ( m . # . , roman) ] exp{ - ik . [ x ( # , ) -- x ( # 3 ) ] }

wj,(~, I k~)wj*(/~ 31 k a ) ~ ( # , I k f l )wj . (~4 I k f l ) x : - - ( 1 2 )

,~ - ,~,. ( k , ~ ) - , ~ , : ( t , ~ )

3"78 F Bogant et a l / T~o-e~oton spectra of HCI and HBr costa&

wtth poles at ~0 = % , ( k a ) + % , ( k f l ) . The two-exct ton Green funct ion (9) ~s renormal ized by an effective quarDc a n h a r m o m c potentml o f the form (8) w~th the on ly a n h a r m o n i c terms of interest w~thm a single molecule. The system then closely resembles, f rom a formal point o f view, a crystal lattice with a poin t defect [11] and tt can be shown that the Dyson equa t ion for the a n h a r m o m c Green funct ion ~s

G = g + ( X / N )PgG.

wi th a formal so lu t ion

O -- [E - ( X / N )Pg] - ' g .

( I 3 )

(14)

where 1= ts the umt matrix, g the h a r m o m c Green func tmn (12). X the a n h a r m o m c i t y cons tan t and P the projec t ion ope ra to r def ined as

PC',:. I ~' ~]) = 8o.,, 8,,,..,,,fi,,,,,.8,<,~,8,,,~ . ( 1 5 )

"1 he r enormahzed Green func tmn has. bemde the poles o f g. addi t ional poles at

deilE - ( X / N )Og I = O. (16)

A case of par t icular interest ~s when one of the v~brational exc~tons, say]~, has a negligible dispersLon. ThLs, for instance ~s the case m dtpole app rox ima t ion for the c o m b i n a t i o n o f an infrared- and a R a m a n - a c u v e m o d e in cen t rosymmet tac molecules The tv, o-exc~ton Green funct ion then s tmphfies into

a(:;:; I ~', ~;) = 8~,~ ?(:;:~ i"'" ., .;) (/7)

and the solut ion (14) becomes

"" g("* J~*~'l ( X / N ) . g ( o . l ~ , * o ~ , ) / f ( c o ) . a (,,,5, l ~ ,~,) = . .... ?,,. ,,;, + '" , ' ) .q( .... l+, 0 8 )

with

f ( w ) = I - N---~ g ( o l ~ , , ) . (19)

where s ~s the number of molecules m the umt cell The b e h a v m u r of the a n h a r m o m c Green funct ion ts de te rmined by the resonant d e n o m m a t o r ] ( ~ o ) which, f rom (12) and (17) and using the proper t tes o f the elgenvectors can be writ ten explu.~tly as

i (i xf~ ,,(_-)d: ) . . . . . . . + ~., X,, (,, , - % , ) , x y, ,o, ( k ~ ) - , ~ , , , - , o , , - . t ' ( , , , ) = 1 ,-~s~,, , o - w , , - _ (20)

~ h e r e n ( z ) is the densi ty of states for the dispersed exc i ton jz and the suffix :~ indicates the pnncJpa l par t o f the integral. For sufficiently large ~alues o f the a n h a r m o m c t t y cons tan t X, the real par t of the func tmn f(~o) can have zeros reside as well as outs ide the two-excl ton con t inuum. T h e new poles of the G r e e n funct ion outs ide the c o n t i n u u m co r re spond to b o u n d pairs of exci ta t ions The r eno rmahzed densi ty of s tates v~thm the t-vo-exc~ton c o n t i n u u m ~s gtven by

p ( w ) = ( ' ~ N ) - ] I m G , , , = n ( w - ,,,,, ) / I f ( a , ) I-" (21)

W h e n a b o u n d state occurs at f requency a~ B the Green func t ion for the b o u n d state Is given by

(i ) G B = N X-" , , ( : ) d - 1 (22) "~" (,o _ % _ _ . ) z ,o-,on+_lC"

F Bogam et a l l T~o-exctton spectra of HCi and HBr crystals 379

T h e ca lcula t ion o f the G r e e n func t ions (10) an d ( l l ) is p e r f o r m e d with a s imilar p rocedure . The i r r e n o r m a l i z a u o n by the th i rd -o rde r h a m i h o n i a n (7) gives rise to coup led equa t ions that in ma t r ix no ta t ion and indica t ing by G ~'-~ and G °~ the G r e e n func t ions (10) an d (11), respect ively, can be wri t ten as

G (3) = G ( K / N ) G (2), G(2) = g(2) + g ( 2 ) V ( f ( / N ) G ° ) , (23)

where g(:) is the h a r m o n i c one -exc i ton G r e e n func t ion (I0) , V the cub ,c force cons t an t an d K the matr ix

K(mFIF_, l b t3 ) = 8,,,08t,,t,~t,_,~, J. (24)

T h e func t ion G gwen by (14) comple t e ly de t e rmines the o th e r func t ions G (3) and G (z) which, therefore , have poles m the two-exc i ton region.

2. 3 Green func t ions f o r over tones

T h e exphcl t express ion (18) for the t w o - e x o t o n G r e e n func t ion discussed m sect ion 2.2 was der ived w~th the a s sumpt ion than one o f the exet tons has a neg l i~b le d~spersion. This result, therefore , Is not valid for an over tone . T o ex tend the pre~aous t r ea tmen t we wri te the Dyson equa t ion m mat r ix fo rm for the o v e r tone o f a non-degene ra t e vabration as

G = g ( E + J) + 2( X/N )gPG, (25)

where

J(',',:". I~,', ~'-.) = 8,,,: . . . . . 8~,,~,A.~, ' (26)

and all o the r symbols have the same mean ing as m the p reced ing sections. Eq. (25) di f fers f rom the c o r r e s p o n d i n g equa t , on (13) as a consequence o f the fact that ,n the case o f an o v e r t o n e the two exci ta t ions c a n n o t be dmt lngmshed. A formal so lu t ,on o f the type (14) can be wtatten for the o v e r to n e G r e e n func t ion However . for the ca lcu la t ion o f the abso rp t ion coeff ic ient m the o v e r to n e reg,on on ly the G r e e n func t ion G ( ° 1~. ~.)___-= G ( F F ' ) are ac tual ly needed. :.'t ,s there fore poss, ble to limit o u r a t t en t ion to the reduced G r e e n func t ,on G

= i (GK, (27)

with K def, ned in (24) Cons tde rmg the p roper t , es o f K

Ki( = P, JK = K, (28)

the fol iowmg equation for the reduced Green function can be obtmned from (25)

= 2[E - 2( X/N ) l (gK] - ' i ( g K . (29)

F o r the v ibra t iona l ove r tone m the HCI and H B r crystals, o f interest m the presen t paper , on ly two-d imens iona l matr ices are invo lved in (29) and the revers ion can be easily pe r fo rmed . Wi th

a =g(°0]~ ~) and e = g(°ol~. ~,), (30)

~t ~s found

( a + c a - - e ) 17(° I~. ~ --- G ( F F ) = 2 1 -- ( 2 X / N ) ( a + c) + 1 -- ( 2 X / N ) ( a -- c ) " (31)

This equa t ion can be rea r ranged in to a conven ien t fo rm for numer ica l compu ta t i ons . Using the o r t h o n o r -

380 F Bogam et al / T~o- e.rctton spectra o f HC! and HBr cos ta ls

m a h t y o f the e igenvec tors it ts found tha t

1 1 1 a + c = - ~ E g ( l t l ~ ' ) = - ~ E 2 w ( k a ) "

E. Iw Ot l k~ ) lZ lw( ~ l kB ) l 2 l 1 a - - " = ~ E g ( ~ ) - ~ E g ( m " ) = ,7___,

~ " ,~,,,~ ,,, - , , , ( k , , ) - , ~ ( k ~ a )

It is con , .en icnt to def ine the foliox~ang densi t ies o f s ta tes

, t ( ~ ) = ~-~N ,~__, B (w - 2w ( k ~ ) ) -

~ i k / ~ ) l -~ ) a ( ~ - ,~ (~,- ,~) - ~ ( ~ , - / ~ ) ) ,

1 1

A ~ 0 J - -

(32)

(33)

(34)

(35)

T h e i m a g m a r y par t o f (31) is then given by

p( ta) = lm G = N'zr[ tz(t~) I F ( . , ) 1 2 + [2 . - rXt , ( ,~ ) ] z

~ (~_~)_ :,1_(~,) /

+ I-" }-" ] " I f - (,,-,) + (",,~x [ 2 v ( ~ ) - , , (~ ) ] (36)

where

f ' - ( ~ o ) = l -- 2 , k ' f t t ( z ) d _ : f - - ( ~ a ) = I - , x f .[2o(:)__--__u_(z)]d: (37) J....~ 6 0 - - - " J ~ t,,O--,.7.

It zs seen that the Imag ina ry par t o f the G r e e n fuflct ton c o n t a m s two d i f fe ren t r e sonan t d e n o m i n a t o r s , f f - ( to ) a n d f - ( o a ) Both can have zeros ou t s ide the c o n t i n u u m c o r r e s p o n d m g to d i f fe ren t b o u n d s ta tes at ,.,~ and ~ a - T h e G r e e n funct ion for the b o u n d s ta tes is g iven by

- " 4 X z (~o~ - : ) - w - w ~ -r-t, (w a _ : ) - w - ~ +__l~

T h e G r e e n funct ions G ~-'~ and G °) are sull re la ted t h rough coup led equa t ions to the two-exc t ton G r e e n funct ion and can thus be o b t a i n e d f rom the latter.

2 4 T ~ o - e x c t t o n a b s o r p t i o n coe fJ tczen t

T h e electric s u s c e p u b d t t y of a crys ta l ,s p r o p o r t i o n a l to the Four i e r t r a n s f o r m o f the c rys ta l d ipole m o m e n t c o r r e l a t i o n f u n c t i o n

x(,o)= 3~t j-~ _ d t ( g ( t ) ' M ( O ) ) e x p [ l ( o ~ - , e ) t ] , (39)

where V is the vo lume o f the c rys ta l and ( ) deno te s the t he rma l average . T h e a b s o r p t i o n coeff ic ient a ( w ) ,s re la ted to the m m g i n a r y par t o f X(tO) and. a s s u n u n g tha t the re f rac t ive index n ts c o n s t a n t in the region o f weak two-exc l ton a b s o r p t i o n bands , ts ~ v e n by

4 ~,~ 1 / ' ~ ~ ( t..o ) = - - l m j _ d t ( / ~ t ( t ) - M ( O ) ) exp [ l ( t a -- ze) t ] (40)

3 n c h V

F Bogam et al_ / Two-exct ton spectra o f HCI and HBr crystals 3-81

By express ing the crysta l d ipole m o m e n t as a su m o f molecu la r m o m e n t s a n d e x p a n d i n g in the molecu la r n o r m a l coo rd ina t e s Q ( t ) it is poss ib le to recognize v a n o u s co n t r i b u t i o n s to the two-exc i ton a b s o r p t i o n coeff ic ient . In fact, in the expans ton o f the d ipo le m o m e n t it is poss ible to d is t inguish " l o c a l " c o n t r i b u - t ions, involving s u m m a t i o n o f single molecu le quant i t tes , f rom non- loca l terms. Fo~ a crysta l o f cha tomic molecules the crysta l d ipole m o m e n t can be wr i t t en as

_ I" I I" M = M, + Me.+ M._ = E d ' " ( t ~ ) a ( D + Z d " - ' ( ~ ) O z ( ~ ) + Z E d ( - " ( t , . . . ) 0 ( . ) 0 ( : ) . /~t I~ It* / ' i

(4z)

where

a , ' , ( . ) = aM('.) a , : , ( . ) = ° ' M ( D a,-',(" ";) = O-'M (42)

oo( o(e(,)) , , 0Q(~)aQ(~. )

T h e f i rs t -order te rm Mt and the second te rm M e are local con t r i bu t ions m the sense de f ined above . T h e non- local t e rm M, with expans ion J('~t~ r coeff ic ients a - ,~ has c o u n t e r p a r t m _ ~,.) no the Isolated molecule . If a (~) ~s the d ipole po la r i zab th ty at site I/~, these expans ion coefficzents can be o b t a i n e d f rom the express ion of the crystal induced d ipole m o m e n t

a s

' " " M C : ) P = E e(~) = E E a ( J T ( ~ , / ) tt, //~ t ' i

(43)

__ aMC;) a,~C') ,. aMC) d (a V) -- DQ('.) igQ(~;) ~)Q(t;) i )Q(t )

where T(t~ r / ) is the usual t ransfer mat r ix for d i p o l e - d i p o l e mterac t ton . It zs c o n v e m e n t to discuss separa te ly the local and non- loca l cont rabut tons to the two-exc t ton abso rp t ion

coefftczent. F o r the local con t r t hu t lon we have

a:(~o) = a ' ' ( a , ) + a 2 -'(~o) + a ' 2(~0) = 5 n c hi,, lm -3,1, ( 0 ) ) e x p [ , ( w - i t ) t ]

By express,ng the crystal dipole moment in the ereauon and destruction operators it ~s easdy seen that the local abso rp t ,on coeff , c ,ent ,s s~mply related to the G r e e n func t ions de f ined in the p reced ing sect ions. Def in ing an ef fec t ,ve s econd -o rde r m o m e n t

b':' = ½(h / . . , ) a ' : , + ( h/,. , ) : ( v /2 ,o )a , ' , (46)

and cons ,der ing the relat ions a m o n g the var ious G r e e n func t ions discussed in sect ion 2.2, tt can be shown that the local a b so r p t i on coeff ic ient is g~ven in te rms o f the renormal ized denszty o f s tates (36) by

, / ( ,o ) = ~ ( ~ , ,o / ,~ ) ( 1 / h v )( D('-' ) 20 ( ,,, ) . (47) Two-exc i ton a bso rp t t on th rough local m o m e n t s occurs by essentzally the same m e c h a m s m respons ib le

for the a p p e a r a n c e o f c o m b i n a t i o n and ove r tone bands in the in f ra red spec t ra o f isolated molecules . By the ef fec t o f in t r amolecu la r (electrical and mechanica l ) a n h a r m o n i c i t y the a b s o r p t i o n occurs wi th in a single molecule . T h e mter rno lecu la r interactaon spreads the exc i ta t ton t h r o u g h o u t - t h e crysta l chang ing the s t ruc tu re o f the a b s o r p t i o n band .

382 F Bogam et al / T~o-exc~ton spectra of HCI and HBr cr)stals

In recent discussions o f the two-exci ton bands m the HC! crystal [8 -10] on ly the abso rp t ion due to local m o m e n t s has been considered. However , the non- loca l s econd-o rde r m o m e n t s al low the appea rance o f two-exct ton absorp t ton bands even m the absence o f molecular anha rmontc i ty . The second-o rde r non- loca l m o m e n t in fact couples d~reetly the g round state to the two-exct ton states. Tins mechan i sm of two-exci ton absorp t ion has been discussed in detatl by n o w s and Sche t tmo [12]. The most impor t an t term in the absorp t ion coefftctent due to the non- local m o m e n t s is given by [5,12]

3 n c 4 ~ , , # - ,~

and ha~ the form of a weighted denst ty of states. The molecular a n h a r m o m c t t y in t roduces addt t tonal con t r ibu t ions to the non-local abso rp t ton as well as interference terms between the local and non- local m e c h a m s m s These addt t tonai terms have been d,scussed by Bogam [5] and can be neglected in m a n y cases.

3. The oser tone infrared spectrum of HCI and H B r cr). stals

The vibrat ional exc~ton in the o r t h o r h o m b m HCI crystal has been discussed m a recent pape r [10] usmg an internlolecular potentml including interact~ons be tween d y n a m i c d~poles a n d quadrupo les The parameters of the electrostat tc potentml are collected m table 1 and were taken f rom expertments or f rom q u a n t u m - m e c h a m c a l calculat tons wtth only a minor ad jus tment o f the d y n a m i c quadrupo le for best fit to the expermlen- tal Daxj, dov spht tmg. All the mare features o f the infrared spec t rum (Dax j ' dov and T O - L O spht- tings, band shapes and relattve mtensttles) were sa t is factordy explained by the model potent ia l that was then used for the calculat ton of the one- and two-exc~ton densmes o f states The latter extends over a regaon of --- 120 c m - ~ and exinb~ts a central peak w~th some stde structure. These general fea- tures compare sa t t s fac tonly wHh the shape of the b road structure of the infrared spec t rum m the over tone regton [1.13-17]

In the present work the same model potent ia l employed for the HCI crystal has been used to calculate the vibrat ional e,tctton ene-gtes m the H B r crystal. The molecular pa ramete r s used are repor ted m table 1 and the results of the calcula- ttons, including the exct ton dispersto~ curves, the one-exc~ton and two-exci ton densit tes o f states, are shown m ftg. 1 The general behav lour ~s sarmlar to that found in HCI and, m part tcular , the w~dth of

the two-exct ton densi ty of states ts m good agree- ment wtth the width o f the b road infrared over- tone absorp t ion band [15-17] . It ts wor th no t ing that the dtspers ton o f the excl ton frequencies m the (,~00) and (00~) direct tons ts large. Since these are direct ions pe rpend icu la r to the hydrogen- b o n d e d H X chains, tt can be conc luded that the m t e r a c u o n s a m o n g different chains are not negh- gtble and. m this context , it would not be ap- p ropr ia te to cons tder the hydrogen hahde crystals as quas l -one-d tmens tona l arrays. As a mat te r o f fact m the HCI case tt has a l ready been noted [6] that the one-exc t ton denst ty o f states calcula ted with the present model ts qui te different f rom that ob ta ined m the one-d tmens tona l case [18]

A s u m m a r y of the exper imenta l mformat~on avat lable on the mf ra red over tone spectra of HCi

T a b l e 1 M o l e c u l a r p a r a m e L e r s u s e d m the c a l c u l a u o n o f t h e v t b r a t t o n a l

e x o t o n ene rg , c~ ,n o r t h o r h o m b t c H C I a n d H B r

H C I H B r

d~po le m o m e n t s

M o ( d e b y e ) 1 07 0 788

~,lt z ( e su ) 177 4 140 1

q u a d r u p o l e m o m e n t O t ( d e b ) e ) 4 8 5 6

p o l , t n z a b ~ h a e s ,",~ ( A a ) 3 i 3 4 22

a~ I ( c su ) 0 456 × 10 - 4 0 542 :,< 10 - 4

a f t ( e su ) -- 1 402 × 1 0 - 4 -- l 671 × 1 0 - 4

F. Bogam et aL / T~o-exctton spectra of HCI and HBr crystals 383

20

10

0

- tO

- 2 0

- 3 0

C19

. > ~ ° ~

k --~

f

f

L 1

"-i

I

- | 30 20 10 0 -10 -20 -30 -40 -50 -60 cat

F ig 1 E n e r g e n c s of the v l b r a n o n a i exc t ton m the t t B r c rys ta l RJght d m g r a m exc l ton d i spe r s ion curves m the 1~00 ( - - - - - - ) , 0 ~ 0 ( ~ ) and 00~j ( - - ) d t r e c t t o n s a n d one -exc t ton de ns t t y of s t a t e s Left d t a g r a m two-exc t ton den~t ty of s t a t e s

and H B r c rys ta l s is c o n t a i n e d m tab le 2 a n d the m o l e c u l a r c o n s t a n t s used m the ca lcu la t ions de- sc r ibed m the fo l lowing are r epo r t ed in tab le 3 and c o m p a r e d wtth the c o r r e s p o n d m g g a s - p h a s e r a l - lies.

T a b l e 2

T h e two-exc t ton mf ra r ed s p e c t r u m of HCI a nd H Br c rys ta l s

HCi H Br

A ( c m - t) =~ 120 80

i n / ! C b~ 1 10 ltw / l f *0 0 002 0 008

=) A ts the s e p a r a t i o n of the b o u n d s t a t e f rom the c o n t i n u u m

e d g e b) IB and I C are the m t e n s m e s of the b o u n d s t a t e s a n d of the

c o n t i n u u m , respec t tve ly c~ it,, ' a n d If a re the , n t ens iue s m the two-exc t ton a nd m the

f u n d a m e n t a l regton, respec t tve ly

As has been d iscussed m the p reced ing sect ions , two-exc t ton in f ra red a b s o r p t i o n can ar ise t h r o u g h local o r non- loca l m o m e n t s . In the local m e c h a - m s m the a b s o r p t i o n coeff ic ient for the H C I a n d H B r ove r tones ts p r o p o r t t o n a l to the r e n o r m a h z e d dens t ty of s ta tes (36) a n d ts thus g o v e r n e d b y the r e sonan t d e n o m i n a t o r s f+(~0) a n d f - (~0 ) . A c c o r d - m g to (37), the b e h a v t o u r o f these func t ions wtll d e p e n d on the va lue o f the a n h a r m o m c t t y c o n s t a n t

T a b l e 3 M o l e c u l a r p a r a m e t e r s of HCI a n d H B r m the gas p h a s e a n d m

the crystal

HCI H B r

gas crystal gas crystal

X ( c m -~ ) - 5 2 - 7 5 - -40 - -60 D 2 / D = 0 1 0 03 0 075 0 0B5

384 F Bogant et al. / T .o-excl ton spectra o f t tCI and HBr c~. stals

7

//

/ 200 //" 300

/

50 1oO 150 200 250 300 cm

Fig 2 Behaxlour of the f u n c t i o n s / ~ ( w ) ( f u l l hne) a n d / - ( ~ ) (do t t ed hne) m HCI for Y = - - 7 5 cm - I In the inser t the

o r d i n a t e scale has b e r n e x p a n d e d 1000 l imes

X A typical plot o f the func t ions for the HCI crys ta l ts s h o ~ n m fig 2 It can be seen that for large values of X bo th func t ions have zeros out - s ide the t~o-exc~ton c o n t i n u u m at f requencies ¢0~

a n d w ~ , c o r r e s p o n d i n g to two-exc i ton b o u n d states . F r o m the def in i t ion o f the r e sonan t d e n o m i n a t o r s it ~s ev iden t tha t the a p p e a r a n c e o f two d i f fe ren t b o u n d s ta tes is a chrect c o n s e q u e n c e o f the d o u b l e o c c u p a n c y o f the uni t cell. A c c o r d - m g to (38) the f requenc ies o f the b o u n d s ta tes d e p e n d on the densi t ies o f s ta tes (34) a n d (35) F o r the H C I crys ta l these a re s h o w n in fig. 3 where ~t can be seen that the dens i ty o f s ta tes v ( a 0 has a p r o n o u n c e d p e a k and that the q u a n t i t y 2 v ( o ~ ) - u(o~) a p p e a r i n g m (38) will be even m o r e sha rp ly peaked . A s u m m a r y o f the e n e r g e a c s a n d in tens i ty d J s t n b u t t o n in the two-exc i ton region acco rd ing to the local m e c h a n i s m is shown pic tor ia l ly for the H C I and H B r crys ta ls in fig. 4 as a func t ion o f the va lue o f the a n h a r m o n i c l t y cons tan t . T h e essentmi fea tures o f the s p e c t r u m are as follows. T h e sep- a_ration o f the b o u n d s ta tes f r o m the c o n t i n u u m mcrea se s wzth increas ing a n h a r m o m c i t y . T h e s h a r p b o u n d - s t a t e p e a k s a p p r o a c h each o the r at large a n h a r m o m c l t l e s a n d have a p p r o m m a t e l y equal hetght m the region o f interest , ca r ry ing m o s t o f the t ~ o - e x c i t o n mtens t ty . F o r smal le r a n h a r m o m c - ities the zeros o f the r e sonan t d e n o r m n a t o r s fall

r 1

r

I

7

r- L

- r ~ r ' ~ r ~ L

~'o 40 6b 8b

F,g. 3 The dens t tms of s ta tes v ( ~ } ( - - - - - - ) and u ( ~ ) (.

L

i ~ n

I I

~6o ~o

) In the HCI cr2,~ta!

lliO cm'

F. Bogam et a L / T~o-exctton spectra o f HCI and HBr crystals 385

%

,%

5O

100

15( cn~t HCI HBr

20 40 60 80 cm 20 40 60 80 cm j

100 %

50

50

100

150 cm I

X, a n h a r m o n l c l t y

Fig 4 Energettcs and intensity d is t r ibut ion in the over tone regmn of HCI and HBr crystals as a funct ion of the anhar - momct ty cons tan t X Lower d iagrams poszuon o f the bound states relative to the edge of the c o n t i n u u m (dot ted hne) U p p e r

d m g r a m s relative intensit ies of the b o u n d states and of the

c o n t i n u u m

w~thm the c o n t i n u u m and , m this case, an en- h a n c e m e n t o f the in tens i ty at the pos t t ion o f the r e sonances occurs wi th s t rong a l t e ra t ion o f the b a n d prof i le wi th respec t to the h a r m o n i c dens i ty o f s tates . T h e genera l t rend is i l lus t ra ted tn f~g. 5 s h o w m g the r e n o r m a h z e d dens t ty o f s ta tes in H C i for d i f fe ren t va lues o f the a n h a r m o m c l t y cons t an t .

As can be seen f r o m fig. 4 and tab le 2, the g a s - p h a s e va lue o f the a n h a r m o n i c i t y c o n s t a n t m H C i ( - 52 c m - l ) gives rise to b o u n d s ta tes fa lhng m u c h c loser to the c o n t i n u u m than o b s e r v e d ex- pe r imen ta l ly . A g o o d fit with e x p e r i m e n t s is o b - t a ined wi th X = - 75 c m - 1 a va lue m close agree- m e n t wi th tha t o b t m n e d b y Sha la and C a h d l [19] f r o m s tudies o f the in f r a red spec t r a o f lSOtopically mixed H C I / D C I crystals . Such a subs t an t i a l m - c rease o f the a n h a r m o n i c i t y c o n s t a n t ~s qu i te t o m -

p rehens ib l e as a c o n s e q u e n c e o f h y d r o g e n b o n d i n g in the crysta l . Howeve r , wi th this va lue o f X the in tensRy o f the b o u n d s ta tes re la t ive to the con - t m u u m is 30 : 1, m u c h larger than the e x p e r i m e n t a l va lue o f t ab le 2. I t is thus imposs tb le to reconci le the r e l auve p e a k pos i t ions a n d intensi t ies in the two-exc i ton s p e c t r u m o f HCi . T lus incons i s t ency has been no t ed in recent p a p e r s [7,8].

A c c u r a t e m e a s u r e m e n t s o f the abso lu t e m f r a r e d intensi t ies o f the o v e r t o n e in the HCi c rys ta l a re not avadab le . F r o m the in f ra red spec t r a o f Brunel [14] and o f Avrd l ie r et al. [15] the in tegra ted two-exc t ton m t e n s i t y can be e sUmated to be 50 c m -2 mrnole - t 0 .e . 0.2% the in tens i ty o f the f u n d a m e n t a l ) . In the local m e c h a m s m a n d accord - lng to (47) this overa l l in tens i ty o f the two-exc i ton s p e c t r u m r eqmres a va lue o f the s e c o n d - o r d e r m o - m e n t 92 = 0 .03D t. T h e ra t io D ~ / D ! in the c rys ta l is thus sma l l e r t han in the gas [20] (see table 2) m a g r e e m e n t wi th the resul ts o f Kathabl a n d Vu [16] showing that the in tens i ty o f the H C I o v e r t o n e does not c h a n g e m u c h f r o m the gas p h a s e to the crys ta l , whale tha t o f the f u n d a m e n t a l is k n o w n to inc rease by o n e o r d e r o f m a g n i t u d e [21]

As has been d iscussed above , the local m e c h a - n i s m and the c rys ta l va lue o f the a n h a r m o n i c l t y c o n s t a n t wou ld p red ic t a two-exc i ton s p e c t r u m wt th the in tens i ty a l m o s t emi r e ly c o n c e n t r a t e d m

the b o u n d states. Th i s ~s not in a g r e e m e n t wRh e x p e r i m e n t s showing tha t the mtens l t y o f the b o u n d s ta tes a n d o f the c o n t i n u u m are c o m p a r a - ble. A n add t t t ona l in tens t ty m the c o n l a n u u m is p red ic t ed b y the non- loca l m e c h a m s m discussed m the p rev ious sect ions . W e have ca lcu la ted the non- local in tens i ty a c c o r d i n g to (48) us ing m the e x p a n s i o n coef f lc ten ts (44) the g a s - p h a s e va lue o f the p o l a r i z a b d i t y d e n v a t w e r epo r t ed m tab le 1. T h e ca lcu la ted non- loca l in tens i ty o f the con - t m u u m is sma l l e r than the o b s e r v e d o n e by a f a c t o r o f three. A r easonab le a g r e e m e n t wi th e x p e r i m e n t s can thus be o b t a i n e d b y a s s u m i n g tha t the p o l a n z a b ~ h t y de r iva t ive m the c rys ta l is l a rger than m the gas phase b y a f ac to r o f 31/2.

Th i s wou ld i m p l y tha t the R a m a n in tens i ty o f the f u n d a m e n t a l in the c rys ta l is three t imes la rger t han in the gas. Such an a s s u m p t i o n seems qui te r e a s o n a b l e cons ide r ing tha t the R a m a n in tens i t ies a re r a the r s ens i twe to the m o l e c u l a r e n v a r o n m e n t

386 F. Boganz et a l / T~o-exczton spectra o f HCI and HBr co,trait

T -

l

X--28 I

J X=-52

X =-75

\ 0 160 -' 260 a00 460 ' ~ "

Ftg 5 The renorrnahzed denslt~ of ~tates p( , - , ) m the HCl cr3,stal for different xalues of the anharmomctty constant

and m hyd rogen -bonded c o m p o u n d s have been reported to mcrease by a factor o f 5 - 6 w~th respect to the gas-phase values [221-

In fig 6 the exper tmental [14] infrared spec t rum of crys ta lhne HCI ts compared with the abso rp t ion coefficients due to the local and non- local mecha- msms usmg the parameters collected m table 3 and discussed above. The model used m the present paper does not include b r o a d e m n g mechan i sms for the bound states However , for the sake o f clari ty in the compar i son with exper iments the calculated spectra were d rawn assuming for the bound states and arbitrary, half width o f 5 c m - I As can be seen f rom fig. 6 the fit to the exper imen- tal band shape ts qui te reasonable. The only major d iscrepancy is concerned w4th the spli t t ing o f the

two bound states that ~s calcula ted (3 cm - l ) to be much smaller than observed (10 cm - l ) and is not resolved m the calculated spectra. The frequencies o f the b o u n d states depend on the densit ies o f

states (34) and (35) shown m fzg. 3. It ts possible that small correc t ions m the denstt ies o f states would improve the calcula ted spli t t ing o f the b o u n d states. In addl tzon it should be noted that in the exper imenta l spectra of Blanchard et al. [13] and Brunel [14] the b o u n d states have m t e n s m e s in the ratso 3"1 wtule they have equal intensi ty m the calcula ted spectra.

The two-exci ton spec t rum o f the H B r crystal can be discussed a long the same lines. The results o f the present calcula t ions are c o m p a r e d w~th experaments [17] in fig. 7 and the pa ramete r s used are collected m table 3. As zn the HCI case the calcula ted spec t rum was d rawn assunung for the b o u n d states an a rb i t ra ry half width o f 5 cm -z. The fol lowing remarks can be made.

(a) As m HCI it is necessary to use an anhar - m o m c i t y cons tan t cons iderab ly larger than in the gas phase and close to the value repor ted by AvriUier et at. [15] for the crystal.

F B ogam et a l / T ~ o - e x c i t o n spectra o f H C I a n d H B r crystals 387

1

O~

OE

04

0 2

b o

:, :,

oE ' i

Fig 6 Overtone infrared spectrum of the ttCI crystal (a) Expenmemal spectrum from ref [14] (b) Calculated spectrum, dotted hne non-local mechamsm, full hne local mechamsm The ordinate scale of the two-exclton continuum m the local rnechamsm has been expanded 50 t~mes An arbitrary half w~dth of 5 cm- ~ has been used for the bound states

075

0,5

0 2 5

o

<~

075

05

025

/

t

I

l I

4700

i

i I

o i

49OO

a

b

C'X'£

(b ) T h e r a t io o f the s e c o n d - to f t r s t - o rde r m o - m e n t is ve ry c lose to the g a s - p h a s e va lue . T h i s is in a g r e e m e n t w i t h o b s e r v a t i o n s t h a t the f u n d a m e n t a l

a n d the o v e r t o n e i n c r e a s e m m t e n s t t y w~th c o n - d e n s a u o n m a l m o s t the s a m e w a y [16].

(c) T h e m a j o r d i f f e r e n c e w i t h the H C I case is

t h a t the e x p e r t m e n t a l i n t e n s i t y r a t i o b e t w e e n the b o u n d s ta tes a n d the c o n t i n u u m is 10 : 1, n o t ve ry

d i f f e r e n t f r o m the v a l u e o f 20 : 1 o b t a i n e d f r o m the

loca l m e c h a m s m . A g a i n the r e s i d u a l i n t e n s i t y in

the c o n t i n u u m is s u p p l i e d b y a b s o r p t i o n d u e to the n o n - l o c a l m e c h a m s m . F o r a g o o d fit to expe r i -

m e n t s i t is n e c e s s a r y to a s s u m e tha t the R a m a n

Fig 7 Overtone mfrared spectrum of the HBr crystal (a) Experimental spectrum from ref [17] (b) Calculated spectrum, dotted hne non-local mechamsm, full hne- local mechamsm The ordinate scale of the tv.o-exclton conlanuum m the local mechamsm has been expanded 8 lames An arbitrary half wzdth of 5 cm-~ has been used for the bound states

i n t e n s i t y i n the crystal is twice as l a rge as i n the gas .

(d ) T w o b o u n d s ta tes a re p r e d i c t e d a l so m the H B r s p e c t r u m w i t h a s e p a r a t i o n o f -- 2 c m - 1. T t u s s p l i t t i n g ha s n o t b e e n r e so lved i n the e x p e r i m e n t a l spec t r a .

388 F Bogant er al / T~o-erct ton spectra o f H C/ and H Br cr)stals

4. Conclusion

The calculat ions pe r fo rmed m the present work using p roper two-excl ton densit ies o f states clearly demons t r a t e the exmtence o f b o u n d states m the over tone region of the hydrogen hahde crystals. Th~s conclusion, a l ready appa ren t m recent discus- sions [7,8]. is now more t ransparen t smce the model adop ted m the present paper has been found capable of accoun t ing for bo th the energet- ~cs and the intensi ty d~stnbut ton m the over tone region of HCI and HBr crystals.

Molecular parameters hke the d~pole moments . anha rn lomc t ty constants , a n h a r m o m c force con- s t a n t s . ~ a o l a n z a b t h t y derivatives, e t c . are faarly well known for the gaseous hydrogen hahdes. Fe~ o f them have been measured for the HCI and HBr crystals as well and found to be quite different than m the gas phase. The o ther parameters needed tn the calculat ions described In the present pape r could not be t ransferred from the gas within the deszred degree of accuracy and were es t imated for best fit ~ t th exper iments The xalues o f the esti- mated parameters appear to be reasonable and the calculat ions pe r fo rmed clarify the baste features o f the tx~o-exc~ton spectra in the HCi and HBr crystals A remarkable point ~s that in HCi the local and non-local m e c h a m s m s give c o m p a r a b l e con t r ibu t ions to the tx~o-exc~ton absorp t ion The mtenslt les arising from local and non-local mo- ments accumula te into the bound states and the c o n t i n u u m respect~vel3,, and the overall spec t rum ss the superpos~t:on of the t~o independent contr i - but ions It ts evident that In the present CdSe the interference terms between the two mechan isms

discussed by Bogam [51 could be o f some mapor- tance These terms have not been taken into account m the present paper Also the HBr spec- t rum results f rom the superposlt~on of two contr i - but ions However, in this case the role of the local m e c h a m s m ~s p r e d o m m a n t and the HBr spec t rum thus closely resembles those of o ther crystals d~s- cussed by Bogam [5]

Flnall) . one may m q m r e whether the tx~o-exct- ton con t inuum intensi ty may par t ly arise f rom over lapping with p h o n o n side-bands. This c anno t be ruled out a praon since p h o n o n s ide-bands are Indeed observed in the fundamenta l [23] and also

in the over tone region [1 ,13-17] o f these crystals. However , there zs var ious evidence against the posszblhty. In par t icu lar it can be men t ioned that s imul taneous t ransi t ions are observed m liquid HCI [16.24] where p h o n o n s ide-bands are obvi- ous ly absent and where the ' .nfrared intensi ty can be ent irely ascr ibed to a m e c h a m s m entirely simi- lar to the non- local mechan i sm discussed in this pape r In addi t ion , in the HCi/DCI mixed crystals a s t rong infrared abso rp t ion has been observed at the sum of the hght and heavy ~sotope frequenczes [14,19] This absorp t ion can only be ascr ibed to a non- local momen t . Finally, tt may be recalled that the over tone mfrared spec t rum has also been mea- sured for pure DCI crystals [13,14] and found to be very similar m shape with the HC! spect rum. In DCI the b o u n d states are observed at 40 c m - I beiox~ the edge o f the two-exct ton c o n t i n u u m [14]. It ~s remarkable to note that using the crystal value o f the a n h a r m o m c s t y cons tan t o f DC! ( - 38 c m - ~ ) [19] the d m g r a m o f fig 4 would predict the b o u n d states 30 c m - I below the band edge This ~s in excellent agreement with exper iments when al- lowance ~s made for the fact that the s t ructure of the fundamenta l band clearly shows that the ~,.~dth o f the tx~o-exclton band in DCi should be some- how smaller than m HCI All this a d d m o n a l mfor- nlat lon is comple te ly consis tent with the present model for the two-excl ton absorp t ion in the hydro - gen hahde crystals.

R e f e r e n c e s

[ll "~ Ron and D F Hormg J Chem Ph3.s 39 (1963) 1129 [2] ~; Cahfano V S c h e t t m o a n d N Neto, La t tzcedynamtcsof

molecular ~.r2,stals (Spnnger Berhn, 1981) [3] M V Eelouso,. m Excttons eds E i Rashba and M D

Sturge (North-Holland Amsterdam 1982) [4] F Bogam. J Phys CI I (1978)1283 [5] F Bogam J Phys CI1 (1978)1297 [6] P R Saint and V Schettmo Chem Ph)s 40 (1979) 413 [7] J Klafter and J Jortner J Chem Phys 77 (1982) 2316 [8] T Holstein R Orbach and S Alexander. Phys Rev I~26

(1982) 4271 [9] J Jortner and S A R~ce, Phys Rev B26 (1982) 4727

[10] V Schettmo and P R Sai,,I. Chem Ph)s 41 (1979) 439 [11] R.J Elhott, m Lattice dynarmes and mtermolecular forces,

ed S Cahfano (Acadenue Press. New York. 1975) [12] D A Do,~s and V Schettmo, J Chem Phys 58 (1973)

5009

F Bogant et a L / T~o-exetton spectra of HCI and HBr crystals -389

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[14] L C. Brunel. Thests, Faculte des Sciences, Umversl te de Lyon. Lyon (1970)

[15] S. Avrdher. S S Mttra and H Vu J Chem Phys 64 (1976) 2202

[16] P. Kathibl and H. Vu, J. Chim Phys 69 (1972) 662 [17] S Avrflher, Thesis, Umverstt6 de Pans VI, Pans (1973) [18] G Zerbt, m Latttce dynamics and intermolecular forces,

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[20] P Geerhngs, D Berckmans and H P Ftgeys, J. Mol Struct 57 (1979) 283

[21] H B Fnedr ich and W B Person, J Chem Phys 39 (1963) 811.

[221 N Abe and M lto, J Raman Spectry 7 (1978) 161 [23] D F Hormg and W E. Osberg, J Chem Phys 23 (1955)

662 [24] J Chesnoy, D Rmard and C Flytzams, Chem Phys 42

(1979) 337 [25] H B F n e d n c h and R E Carlson, J Chem Phys 53 (1970)

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