James R. Henderson and Jonathan Tennyson- Very highly excited vibrational states of LiCN using a...

8/3/2019 James R. Henderson and Jonathan Tennyson- Very highly excited vibrational states of LiCN using a discrete variable

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MOLECULAR PHYSICS,1990 , VOL. 69 , No . 4 , 639-6 48

V e r y h i g h l y e x c i t e d v i b r a ti o n a l s t a t e s o f L i C Nus ing a d i scre te var iable representa t ionb y J A M E S R. H E N D E R S O N a nd J O N A T H A N T E N N Y S O N

D e p a r t m e n t o f P h y si c s a n d A s t r o n o m y , U n i v er s it y C o ll eg e L o n d o n ,G o w e r S tr ee t, L o n d o n W C 1 E 6 B T, E n g l a n d

( R e c e i v e d 3 N o v e m b e r 1 9 8 9 ; a c c e p t e d 2 8 N o v e m b e r 1 9 8 9 )Calcula t ions are presented for the lowest 900 v ibra t ional ( J = 0) s ta tes o fthe LiCN f loppy sys tem for a two dimensional potent ia l energy sur face ( rcNfrozen). Most of these states l ie well above the barrier separat ing the two l ineari somers of the m olecule and the po in t w here the c lass ica l dynam ics of thesystem beco me s chaotic. Analysis of the w avefunctions of individual states inthe h igh energy region shows th a t whi le m ost hav e an i r regular nodal s t ruc ture ,a s ignif icant num ber of s ta tes appea r regular -~correspo nding to so lu t ions ofs t anda rd , 'm od e loca l i zed ' hami lton ians . M ot ions co r r e spond ing in ze ro -o rde rto Li -CN and Li -NC normal modes as wel l as f ree ro tor s ta tes are ident i f ied .Th e distr ibutio n of level spacings is also studied a nd yields results in goo dagreem ent wi th those o bta ined by analys ing nodal s truc tures .

1. I n t r o d u c t i o nA d v a n c e s i n b o t h e x p e r i m e n t a n d t h e o r y h a v e l ed t o c o n s i d e ra b l e i n t e re s t in t h e

h i g h l y in g r o - v i b r a t i o n a l b o u n d s t a t e s o f m o l e c u l e s . T h e s e s t a t e s a r e o f i n t e r e s tb e c a u s e o f t h e ir u n u s u a l d y n a m i c s , c o m p a r e d t o m o r e f a m i l ia r st a te s w i t h o n l y a f e wq u a n t a o f v i b r a t i o n a l e n e r g y , a n d b e c a u s e o f t h e i r a s s o c i a t io n w i t h c h e m i c a l re a c -t i o n s s u c h a s is o m e r i z a t i o n .

T h e L i C N m o l e c u l e in p a r t ic u l a r h a s b e e n t h e f o cu s o f m a n y s tu d i es [1-83 of i t sb o u n d s t a t e n u c l e a r m o t i o n d y n a m i c s . A ll e x c e p t t h e fi rs t o f t h e s e s t u d i e s e m p l o y e dt h e t w o - d i m e n s i o n a l , C N b o n d l e n g t h f ro z e n , S C F p o t e n t i a l e n e rg y s u r fa c e o f E s s er se t a l . [ 9 ] , s e e f i g u r e 1 . T h i s s u r f a c e h a s f e a t u r e s w h i c h l e a d t o i n t e r e s t i n g d y n a m i c sw h i c h i s t h o u g h t t o b e t y p i c a l o f m a n y t r i a t o m i c s y s t e m s [5 -1 b u t a r e e n c o u n t e r e df o r L i C N a t l o w e r e n e r g i e s t h a n o t h e r i s o m e r i z i n g s y s t e m s . T h e s u r f a c e p r e d i c t s al in e a r L i N C a b s o l u t e m i n i m u m , i n a g r e e m e n t w i t h e x p e r i m e n t [1 0-1 , a n d a m e t a -s ta b le l in e a r L i C N m i n i m u m a t 2 28 1 c m - 1 a b o v e L i N C . T h e m i n i m a a r e s e p a r a te db y a b a r r i e r o f 3 45 5 c m - 1 f r o m t h e a b s o l u t e m i n i m u m a t L i N C . C l a ss i ca l c al c u -l a ti o n s [ 4 ] s h o w a n o n s e t o f c h a o s a p p r o x i m a t e l y h a l f w a y t o t hi s b a rr i er .

A l t h o u g h t h e r e h a s b e e n a g r a d u a l i m p r o v e m e n t in th e m e t h o d s u s e d t o t r e a tt h e v i b r a t io n a l s t a t e s o f L i C N , a l l th e w o r k s c i te d a b o v e c o n c e n t r a t e d o n l y o n t h el o w e s t 13 1 s ta t e s o f t h e s y s t e m . F e w o f t h e s e s ta t e s li e a b o v e t h e b a r r i e r t o i s o m e r -i z a t io n o f t h e s y s t e m ; th e b e h a v i o u r o f t h e s y s t e m w e ll a b o v e th i s b a r r i e r t h u sr e m a i n s a n o p e n q u e s t i o n . I t is t h i s q u e s t i o n t h a t w e a d d r e s s h e r e .

2 . C a l c u l a t i o n sC o n v e n i e n t c o o r d i n a t e s t o r e p r e s e n t L i C N a r e r c N , h e r e f i x ed a t 2 . 1 86 a o , R , t h e

d i s t a n c e o f L i t o t h e C N c e n t r e o f m a s s , a n d 0 , t h e a n g l e b e t w e e n rcN a n d R . 0 r u n sf r o m 0 ~ f o r l in e a r L i C N t o 1 8 0 ~ f o r l i n e a r L i N C . B a c i c a n d L i g h t ( B L ) [ 7 ] w e r e t h e

0026-8976/90 $3.00 9 1990 Taylor & Francis Ltd


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6 40 J . R . H e n d e r s o n a n d J . T e n n y s o n7

6

5

i k I i i I i I

0 60 120 180Figure 1 . Co ntou r p lo t o f the poten t ia l energy surface for LiN C/L iCN [9] . Th e ver t ica l ax isgives values of R / a o . 0 = 0 ~ c o r r e sponds to the L iCN min imum a nd 0 = 180~ to

LiNC . Th e con tours a re g iven a t in te rva ls of 240 cm - 1 .f ir st t o e m p l o y a d i s c r e te v a r i a b l e r e p r e s e n t a t i o n ( D V R ) o f t h e 0 c o o r d i n a t e f o rL i C N .

I n f i n it e b a si s s e t m e t h o d s , t h e m o t i o n i n t h e 0 c o o r d i n a t e i s g e n e ra l l y c a r r i e d b yL e g e n d r e p o l y n o m i a l s [ 2 , 6 ] a n d i n t e g r a t i o n i n t h is c o o r d i n a t e p e r f o r m e d a n a l y t i -c a l ly t o g i v e a n e f f e ct i ve r a d i a l h a m i l t o n i a n [ 1 2 ]. I n t h e i r w o r k , B L a p p l i e d a0 - c o o r d i n a t e t ra n s f o r m a t i o n t o t h is e f fe c ti v e h a m i l t o n i a n b a s e d o n N ~ p o i n t G a u s s -L e g e n d r e q u a d r a t u r e . T h i s r e p r e s e n t a t i o n , w h i c h i s n o w d i s c r e t i z e d i n t o N ~ a n g l e so r ' r a y s ' , h a s f o r m a l e q u i v a l e n c e s w i t h t h e u n t r a n s f o r m e d p r o b l e m . H o w e v e r ,w i t h i n a D V R , i t i s p o s s i b l e t o d i a g o n a l i z e a o n e - d i m e n s i o n a l r a d i a l h a m i l t o n i a n f o re a c h d i s c r e t e a n g le . B L d i d t h i s u s in g a n o n - o r t h o g o n a l b a s is o f d i s tr i b u t e d s p h e r -i ca l g a u s s i a n f u n c t i o n s f o r t h e r a d i a l c o o r d i n a t e .

B L t h e n u s e d t h e l o w e s t L s o l u t i o n s o f t h e o n e - d i m e n s i o n a l p r o b l e m s t o e x p a n dt h e fu ll , t w o - d i m e n s i o n a l p r o b l e m . L , th e n u m b e r o f s o l u t i o n s u s e d t o e x p a n d t h ef i n al p r o b l e m , c a n b e f i x e d i n a d v a n c e o r b y s e l e c t in g a ll t h o s e s o l u t i o n s l y i n g b e l o ws o m e c u t - o f f e n e r g y , E R Ay . E i t h e r w a y , B L f o u n d t h a t t h is p r o c e d u r e g a v e m a n ym o r e v i b r a t i o n a l l e v el s f o r le ss c o m p u t a t i o n a l e f f o r t t h a n s i m i l ar c a l c u l a t i o n s t h a tu s e d a b a s is s e t e x p a n s i o n f o r t h e a n g u l a r c o o r d i n a t e .

B L s s u c c e s s h a s l e a d o t h e r w o r k e r s , i n c l u d i n g o u r s e l v e s [ 1 1 ] , t o a p p l y t h e D V Rm e t h o d t o t h e p r o b l e m o f r o - v i b r a t i o n a l s t a t e s o f m o l e c u l a r s ys t em s . T h e o n l ym a j o r d i f f e r en c e b e t w e e n o u r a p p r o a c h a n d B L s is t h a t f o r e a c h a n g l e w e u se t h es a m e s e t o f o r t h o g o n a l p o l y n o m i a l s t o c a r r y t h e r a d i a l m o t i o n .

I n t h i s w o r k w e r e p o r t c a l c u l a t i o n s o n L i C N u s i n g th e s u r f ac e o f E s s e rs e t a l . ,s ee f i g u re 1 , a n d o u r D V R p r o g r a m E l i ] . T h e s e c a l c u l a ti o n s g iv e r e s ul t s c o n v e r g e dw i t h r e s p e c t t o b a s i s s e t t r u n c a t i o n e r r o r s f o r t h e l o w e s t 9 0 0 v i b r a t i o n a l ( i . e . J = 0 )s t a te s o f t h e s y s t em . T h i s l a r g e e x t e n s i o n o f t h e n u m b e r o f s t a te s m e a n s t h a t i n f o r -


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V i b r a ti o n a ll y e x c i te d s t a t e s o f L i C N 641marion is available for the first time on the behaviour of the system well above thebarrier to isomerization.

Final calculations were performed using a DVR grid in 0 based on N, = 80Gauss -Legen dre quadra ture points. Nn = 56 previously optimized Morseoscillator-like functions [4] were used to diagonalize the one-dimensional hamilto-nians for each of the angular quad ratu re points or ' ra ys ' I-7, 11]. The final hamilto-nian matr ix was diagonalized using the L = 1870 lowest of the 4480 solutions of theone-dimensional hamiltonians. This selection criterion is equivalent to choosing allsolutions of the one-dimensional problem with energy less than ERAY=15231 cm -1. This basis converged the lowest 700 vibra tiona l states of LiCN towithin 0.5 cm -1 ; the next 150 states were converged to 2- 5 cm -1 ; the highest statespresented here are only converged to about 10cm -1. Table 1 demons trates con-vergence of a selection of levels with respect to changing the parameters of thecalculation.

In order to analyse the results of this caculation we have plotted the wavefunc-tions of the lowest 900 states of the system. This method of analysing the system wasused by Tennyson and Farantos [4, 5] to study the lowest 80 vibrational states ofthe system. They were able to characterize (a) regular states localized about theLiNC minimum, (b) regular states localized about the LiCN minimum, (c) irregularstates localized about the LiNC minimum, (d) irregular delocalized states and poss-ibly (e) two states which has free rotor-like character. In their work the distinctionbetween regular and irregular states was made according to whether approximatequantum numbers could be assigned on the basis of the observed nodal structure.The majority of our states are irregular in structure and delocalised. Howeverinspection of our wavefunctions revealed regular states corresponding to normalTable 1. Convergence of the LiCN band origins as a function of parameters used in thecalculations. N n gives the number of Morse oscillator-like function used for the Rcoordinate. N, gives the number of discrete points used in the angular coordinate, 0.ERAY gives cut-off energy for solutions of the one-dimensional radial problem in cm- 1relative to the LiNC minimum of the potential yielding a final hamiltonian matrix ofdimension L. All band origins are given in cm-I relative to the LiNC ground state at512.4cm-1. Comparison of levels below level 500 show them all to be converged to

within 0.1 cm- 1 by the calculations presented here.NR 56 56 46 51 56 56 56 56N, 70 80 90 90 90 80 80 80ERAy/Cm -1 13866 13866 14042 13950 13866 12613 13866 15231L 1455 1661 1870 1870 1870 1470 1661 1870Level

500 9425.4 94 25 .4 94 25 .4 94 25. 4 94 25. 4 94 25 .6 9425.4 9425.3550 9941.2 994 1.2 994 1.2 9941 .2 994 1.2 9941 .3 994 1.2 9941-2600 1044 2- 1 10442.1 10442.1 10442.1 10442.1 10443.2 10442.1 10441.9650 10 91 2. 5 10912.4 10912.7 10912.4 10912.4 10914-4 10912-4 10912-2700 11 38 7. 8 11387.8 11390.9 11389.4 11387.8 11390.9 11387.8 11387.2750 11 81 6- 3 11814.6 11829.9 11815-0 11814.6 11840.2 11814.6 11813.2800 12 25 0- 9 12250-6 12265.4 12258.6 12250.1 12322.6 12250.6 12247.4820 12 43 2. 8 12432.5 12453.0 12432.3 12432.5 12541.7 12432-5 12430-3840 12 59 6. 7 12597.6 12623.7 12602.3 12597.5 12732.6 12597.6 12592.2860 12 76 6. 7 12765.6 12796-7 12773.8 12765.3 12978.6 12765.6 12758.5880 129 38 .0 12982.0 12978.2 12940.8 12927.7 13205.5 12928.0 12916.3900 13 10 9. 7 13108.7 13138-1 13113.4 13108.7 13452.8 13108.7 13086.5


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642 J .R . Hende rson and J. Ten nyso nmodes of LiNC, normal modes of LiCN and free rotor states. These correspond toclasses (a), (b) and (e) above. Details of these states are given in tables 2, 3 and 4respectively. We t abulat e a complete set of nor mal mo de states as no ta bul ati on ofthese appears t o have bee n given previously. As we have onl y consi dered J = 0, onlyeven qu ant a of bend are o bta ine d for the quasi-linear states and only ~ free-rotorstates.

Above the barrier there are very few spatially localized irregular states, such asclass (c) above. Figure s 2 to 5 show typical wav efun cti ons for the systems of type (a),(b), (d) and (e) respectively. These contour plots were obtained using the same

Table 2. Assignments to LiNC ' no rmal mod e' states. States are assigned by inspection ofthe wavefunction (see figure 2) quanta of Li-NC stretch, vs, and bend, vb. States forwhich the nodal structures are greatly distorted are denoted by a ?.Assignments AssignmentsLevel Frequency/ Level Frequency/No. cm - 1 / ) s / 3 b N O . cm- i D s / ) b

1 0'0 0 0 51 2954'8 4 02 247.2 0 2 53 2995'5 3 10?3 469'0 0 4 54 3025.0 0 36?4 665'6 0 6 57 3112-8 3 12?5 754'4 1 0 59 3188'8 4 2?6 836"8 0 8 68 3382"0 4 47 982"3 0 10 75 3565'5 4 6?8 998-1 1 2 81 3667"1 5 09 1111'7 0 12 91 3898'5 5 210 1212"8 1 4 93 3921"5 3 14?11 1245'3 0 14 100 4083'5 5 412 1390'3 0 16 117 4368'9 6 013 1397.7 1 6 128 4597'6 6 214 1498'4 2 0 139 4775'9 6 415 1545.4 0 18 144 4874'8 6 816 1550.4 1 8 156 5060'1 7 017 1671'5 1 10 169 5286.3 7 218 1708'4 0 20 180 5455"0 7 419 1738"7 2 2 199 5740.6 8 020 1786'6 1 12? 213 5964"4 8 221 1874-8 0 22? 245 6410"6 9 022 1919"2 1 14? 262 6632"2 9 223 1946'4 2 4 272 6780'3 9 424 2036'8 0 24? 294 7070"0 10 026 2118.1 2 6 347 7718"6 11 028 2231'8 3 0 365 7935"9 11 229 2244.9 0 26? 400 8356"6 12 030 2246'8 2 8? 457 8984'2 13 032 2340.4 2 10? 515 9601'4 14 034 2418'2 0 28? 536 9814'7 14 235 2452"3 2 12? 575 10208'0 15 036 2469"0 3 2 598 10418.8 15 237 2531"3 0 30? 638 10804'4 16 041 2669"1 3 4 701 11390 17 044 2761"8 0 32? 725 11597 17 245 2825"4 3 6 767 11965 18 048 2918'3 3 8? 832 12531 19 049 2918"4 0 34? 900 13086 20 0


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Vibra tiona ll y exc i t ed s ta tes o f L iC N 643Table 3. Assignments to LiCN 'no rma l mod e' states. States are assigned by inspection ofthe wavefunction (see figure 3) quanta of Li-CN stretch, vs, and bend, vb. States forwhich the nodal structures are greatly distorted are denoted by a ?. Frequencies are

relative to the Li CN (0, 0) state which lies 2286.6 c m- 1 above the LiNC (0, 0) state.Assignments AssignmentsLevel Frequency / Level Frequency/No. cm - 1 vs vb No. cm - 1 Us / )b

31 0.0 0 0 253 4229.9 5 640 323.0 0 2 260 4267.7 6 247 610.1 0 4 281 4608.8 7 052 689.1 1 0 284 4638-2 6 4?58 853.7 0 6 302 4875.6 6 664 1016.2 1 2? 309 4963.3 7 276 1306.3 1 4 330 5228.8 8 080 1368.0 2 0 335 5271.9 7 488 1550.6 1 6 353 5516.6 7 696 1700-0 2 2 360 5585.2 8 2111 1992.2 2 4 381 5836-9 9 0113 2055.7 3 0 413 6199.8 9 2?123 2234.2 2 6 433 6435.5 10 0131 2372.4 3 2 468 6802.8 10 2148 2666.8 3 4 489 7026.3 11 0151 2695-6 4 0 525 7398.8 11 2163 2912.3 3 6 545 7606.3 12 0171 3034-7 4 2 603 8177-4 13 0

192 3345.5 5 0 663 8739.4 14 0206 3573-4 4 6 723 9205 15 0215 3687-7 5 2 785 9842 16 0234 3978.5 6 0 854 10421 17 0236 3991.2 5 4

Table 4. Assignments to LiC N 'free ro to r' states. States are assigned by inspection of thewavefunction (see figure 5) quanta of Li- CN stretch, vs, and free rotation (or bend), m.States for which the nodal structures are greatly distorted are denoted by a ?. Fre-quencies are relative to the LiNC (0, 0) state.Assignments AssignmentsLevel Frequency/ Level Frequency/No. cm - 1 vs m No. cm - 1 vs m

65 3311"3 0 24 366 7938.3 0 53?71 3474-8 0 26 375 8055"2 2 48?74 3552"1 0 27? 382 8141"9 0 5482 3671"8 0 28 399 8349'5 0 55?87 3819'4 0 29? 409 8454.9 1 52?94 3968-4 0 30? 459 8989'7 0 58110 4275'1 0 33? 499 9418"6 0 60145 4896"6 1 31? 519 9634-0 0 61?150 4971"9 0 37? 563 10075.7 0 63160 5139-7 1 33? 586 10299'5 0 64?181 5468"6 0 40 609 10520.5 0 65191 5623"3 0 41? 682 11197-5 0 66?237 6295'2 0 45? 732 11652 0 68?333 7539"8 0 51 783 12108 0 70?349 7737"1 0 52 810 12337 0 73


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6 4 4 J . R . H e n d e r s o n a n d J . T e n n y s o n

R 5

37

R 5

STATE 93 i

i :

i 0 . .9

STATE575

!c,,

STATE 365

I

! STATE900

0 C) 180 0 ~) 180Figure 2. Con tour p lots of 4 typical LiNC norm al mo de states, Solid (dashed) contoursenclose regions where the w avefunction has positive (negative) amplitude. C ont ou rsare draw n at 4, 8, 16, 32 and 64 per cent of the ma ximu m ampli tude of the wavefunc-t ion. The outer da shed contou rs represent the classical turning point of the potent ial

for the associated eigenvalue.

Figure 3 .

R 5

37

STATE111L

i ! '!

STATE 330

6 C,a 0 ~

STATE 413 STATE 854

~o . . , , . . o {....... ' o

. . . . . . . . . .

0 ~) 180 0 ~) 180Co nt o ur p l o t s o f 4 t yp ica l L i CN nor m a l mo de s ta te s. C on t o u r s a s in f igu r e 2 .


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Vibrationally excited states o f L iC N 645

Figure 4.

R

R

7 STATE 200

I g . . . . . . . .i . I=i i

STATE 401

~' o ~ g~ o'd ~ a 4~-~ ,~

~ o ]

!

5

3

0 @ 180 0 ~ ) 1 8 0Co nt ou r plots of 4 typical unassignable states. C ont our s as in figure 2.

coor d ina te range as the p lo t o f the po ten t ia l , f igu re 1, and a l so have a con tourden ot ing th e class ical turn ing p oint for the s ta te in quest ion . This a l lows judg em entsto be ma de ab ou t how ind iv idua l wavefunc t ions re f lec t the shape o f the under ly ingpo ten t ia l func t io n and the degree to wh ich these wavefunc t ions sample the ava i lab lecoord ina te space .

7 i STATE82

R 5

37

R 5

Figure 5.

I

STATE 160 ]I

:'~I # '~ % . 7

STATE 382 STATE 810

~ ' ~ = . ]O Q ' : I : 0 : : i

James R. Henderson and Jonathan Tennyson- Very highly excited vibrational states of LiCN using a...

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