HARMONIC AND ANHARMONIC REPRESENTATION OF LIBRATIONAL MOTIONS IN MOLECULAR CRYSTALS

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HARMONIC AND ANHARMONIC

REPRESENTATION OF LIBRATIONAL MOTIONS IN MOLECULAR CRYSTALS

B. Kuchta, T. Luty

To cite this version:

B. Kuchta, T. Luty. HARMONIC AND ANHARMONIC REPRESENTATION OF LIBRATIONAL MOTIONS IN MOLECULAR CRYSTALS. Journal de Physique Colloques, 1981, 42 (C6), pp.C6-587- C6-589. �10.1051/jphyscol:19816171�. �jpa-00221250�

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JOURNAL DE PHYSIQUE

CoZZoque C6, suppZ6ment au nO1 2, Tome 42, dbcembre 1981

HARMONIC AND ANHARMONIC R E P R E S E N T A T I O N O F L I B R A T I O N A L MOTIONS I N MOLECULAR CRYSTALS

B . Kuchta and T. Luty

Institute of Organic and Physical Chemistry, Technical University, 50-370 Wroc Em, Po Zand

Absfracf

.-

The o r i e n t a t i o n a l c r y s t a l p o t e n t i a l , represented a s a sur9ace i n n-dimensional space spanned on L i b r a t i o n a l normal coordina-bes, has been c a l c u l a t e d f o r oC -N

,

ôC-0 ândÎ)^-0

c r y s t a l s and compared with i t s harmonic $epresgn%ation.

Signi-

f i c a n t d i f f erencies i n symmetries of t h e s u r f a c e s show t h a t molecular Librations a r e highly anharmonic and t h a t any proper d e s c r i p t i o n of t h e e x c i t a t i o n s i n molecular c r y s t a l s has t o t a k e i n t o account t h e f u l l symmetry of an exact o r i e n t a t i o n a l c r y s t a l p o t e n t i a l o

1. Introduction.- The harmonic approximation which i s commonly used i n t h e l a t t i c e dynamics c a l c u l a t i o n s of molecular c r y s t a l s i s belie- ved t o be well s a t i s f i e d f o r t r a n s l a t i o n a l degrees of freedom but not f o r r o t a t i o n a l ones. R e s u l t s of c a l c u l a t i o n s / l i b r a t i o n a l phonon frequencies/ f o r diatomic molecular c r y s t a l s , i n p a r t i c u l a r f o r so- l i d n i t r o g e n

[ll,

support t h e conclusion. I n t h e present paper we s h a l l d i s c u s s t h e problem of a d e s c r i p t i o n of l i b r a t i o n a l e x c i t a t i - ons a t t h e c e n t e r of 3 r i l l o u i n zone i n c r y s t a l s , where a r o t a t i o n a l p a r t of t h e c r y s t a l p o t e n t i a l can be separatedr For t h e discussion we choose nitrogen and oxygen c r y s t a l s f o r t h e follo@ing reasons:

t h e ab i n i t i o intermolecular N2-N2 p o t e n t i a l i s known 123, and t h e c r y s t a l s t r u c t u r e s of oxygen phases /one molecule i n a u n i t c e l l / simplify a r e p r e s e n t a t i o n of t h e r o t a t i o n a l p o t e n t i a l .

2. R o t a t i o n a l potentials.- I n c r y s t a l s where molecules occupy c e n t e r of i n v e r s i o n s i t e s , t h e r o t a t i o n a l c r y s t a l p o t e n t i a l f o r zero wave v e c t o r can be separated from t r a n s l a t i o n a l p a r t and w r i t t e n a s an

expansion with r e s p e c t t o a complete s e t of r o t a t i o n a l coordinates.

It i s convenient, f o r o u r purpose, t o expancl t h e p o t e n t i a l i n t e r n s of normal coordinates of l i b r a t i o n a l e x c i t a t i o n s and t o represent a s a s u r f a c e spanned on t h e coordinates, The l i b r a t i o n a l normal co- o r d i n a t e s a r e t h e eigenvectors of t h e harmonic dynamical matrix corresponding t o t h e p a r t i c u l a r normal mode / Ag, Bg f o r monoclinic

d -02, E f o r hexagonal

P

-02 and Eg

,

^Tg

,

T f o r cubic ^oC^-N2/.

g g

A q u a l i t a t i v e d i f f e r e n c e between harmonic and anharmonic represen-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19816171

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C6-588 JOURNAL DE PHYSIQUE

t a t i o n of t h e l i b r a t i o n a l e x c i t a t i o n s i s c l e a r l y seen from Pigulres 1 and 2. I n t h e c a l c u l a t i o n s of t h e r o t a t i o n a l energy s u r f a c e s we have used t h e "6-exp" atom-atom p o t e n t i a l , optimized t o t h e c r y s % a l s t r u c t u r e s of s o l i d oxygen. The s t r i c k i n g d i f f e r e n c i e s i n t h e sym-

Fig. 1

Fig. 2

Figures 1 and 2. The r o t a t i o n a l energy surfaces f o r oC-0

/Fig. 1/ and (3-0 /Fig. 2; c r y s t a l s i n harmonic /on l e f t ? and anharmonic /og r i g h t / representations.

metry of t h e s u r f a c e s suggest t h a t t h e angular motions of t h e molecules can be q u i t e d i f f e r e n t i n c h a r a c t e r t h a n t h e l i b r a t i o n a l osci- l l a t i o n s around a f i x e d axis. It i s a l s o seen t h a t any quadratic /harmonic/ o r multiquadratic /quasihanonic/ r e p r e s e n t a t i o n of t h e p o t e n t i a l i n c r e a s e s i t s symmetry and cannot properly describes t h e l i b r a t i o n a l e x c i t a t i o n s i n t h e c r y s t a l . The important point i s t o take i n t o account t h e odd powers i n t h e p o t e n t i a l expansion.

3. L i b r a t i o n a l frequencies.- An a p p l i c a t i o n of t h e p e r t u r b a t i o n theory allows one t o t a k e i n t o account an influence of t h e anharmonic terms i n t h e r o t a t i o n a l p o t e n t i a l on l i b r a t i o n a l phonon frequen- c i e s [3]. Kit-bin t h e method, l i b r a t i o n s of t h e molecules a r e t r e a - t e d a s coupled anharmonic o s c i l l a t o r s i n an exact ro-l;ational c r y s t a l p o t e n t i a l . Figure 3 shows cross-sections of t h e i r r e d u c i b l e p a r t 02 t h e r o t a t i o n a l energy i n t h e cubic phase of nitrogen c r y s t a l , cor-

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responding t o T symmetry /coordinates a r e those corresponding t o two g T l i b r a - tions/. The d i f f e r e n c e between harnonic Ct;

/ s o l i d l i n e / and anharmonic /broken l i n e / r e p r e s e n t a t i o n of t h e p o t e n t i a l i s not l a r g e , it g i v e s however a signi- f i c a n t d i f f e r e n c e i n l i b r a t i o n a l f r e - quencies. R e s u l t s of t h e c a l c u l a t i o n s are presented i n t h e f o r t h column of t h e Table. There are seen very i m p o r t a n t

,

q u a n t i t a t i v e d i f f erencies between

I harmonic, quasiharmonic /se lf-consis-

Pig. 3 /see t e x t / t e n t phonon approximation, SCF/ and Table

-

L i b r a t i o n a l phonon frequencies / i n c r 1 / a t ^q= 0 i n

e -N, c r y s t a l .

L

Mode Hamnonic 111 S C P ~ I ~ Anharmonic

133

~ x p e r i m e n t (41

Eg 40.8 39.5 32.0 32.3

Tg 50.7 48.5 36.3 36.3

20. 74.3 70.3 60.5 59.7

anharmonic r e p r e s e n t a t i o n of The l i b r a t i o n a l excitations. Although t h e p e r t u r b a t i o n t h e o r y aoes reproduce t h e l i b r a t i o n a l phonon f r e -

quencies very well, it does not give however a unique i n t e r p r e t a - t i o n of t h e molecular motions. The symmetry of t'ne r e a l o r i e n t a t i o - na3 pot e n t i a l suggests t h a t angular molecular motions can be q u i t e d i f f e r e n t i n c h a r a c t e r than an o s c i l l a t o r and t h a t any proper des- c r i p t i o n of t h e e x c i t a t i o n s i n molecular c r y s t a l s has t o t a k e i n t o accowlC t h e f u l l symmetry of t h e c r y s t a l p o t e n t i a l .

References.

[I] 21. Luty, A . van der Avoird and R.Ivi. Berns, J. Chem. Imys.,

2 ,

5305 (1980), 123 R.kl. Berns and A. van d e r Avoird, J. Chem. Phys.,

2 ,

6107 (1980)

131 ^B. Kuchta and T. Luty, i n preparation,

[43 ^1.K. Xjems and G. Dolling, fhys. Rev., B l l , 1639 (1975).