31 Molecular Vibration States: CH2 Asymmetric Stretch CHRISTOPHER A. PARR and JAMES L. ROOKSTOOL
Downloaded by CORNELL UNIV on October 6, 2016 | http://pubs.acs.org Publication Date: June 1, 1977 | doi: 10.1021/bk-1977-0056.ch031
The University of Texas at Dallas, Box 688, Richardson, TX 75080
S t a t e - t o - s t a t e chemistry can be modelled t h e o r e t i c a l l y given molecular p o t e n t i a l - e n e r g i e s and dynamics c a l c u l a t i o n s . Proper s i m u l a t i o n and a n a l y s i s r e q u i r e s that i n i t i a l and f i n a l molecular s t a t e s be quantized modes o f motion. Even f o r c l a s s i c a l o r q u a s i c l a s s i c a l dynamics, wherein molecular reactants and/or products are not n e c e s s a r i l y q u a n t i z e d , the c r e a t i o n o f reactants and the a n a l y s i s o f products r e q u i r e s a proper decomposition o f molecular energies i n t o v i b r a t i o n a l / r o t a t i o n a l components. In t h i s paper, we ignore the c o n t r i b u t i o n o f r o t a t i o n t o the molecular motion problem. We concentrate i n s t e a d on the establishment o f the c l a s s i c a l v i b r a t i o n s t a t e s o f a r e a l i s t i c model o f ground s t a t e methylene B C H . We seek the natural v i b r a t i o n modes i n t o which the v i b r a t i o n a l Hamiltonian i s "most n e a r l y " separable; we wish to r e duce the (3N-6)-dimensional problem t o 3N-6 quasi-one-dimensional problems. Without such a r e d u c t i o n , exact s t a t e - t o - s t a t e c a l c u l a t i o n s are not p o s s i b l e . We report here the s u c c e s s f u l i s o l a t i o n of the natural asymmetric s t r e t c h mode a t energies from near zero p o i n t t o near a t o m i z a t i o n . The natural v i b r a t i o n modes are a s y m p t o t i c a l l y e q u i v a l e n t t o the normal v i b r a t i o n modes (1_) i n the l i m i t o f i n f i n i t e s i m a l v i b r a amplitude. The dynamics o f normal v i b r a t i o n are c h a r a c t e r i z e d by i n v a r i a n t p e r i o d i c i t y , i n v a r i a n t energy, and a one-dimensional t r a j e c t o r y i n coordinate space. These p r o p e r t i e s are destroyed i n f i n i t e v i b r a t i o n amplitude dynamics (2_). The same p r o p e r t i e s may be r e t a i n e d by the natural v i b r a t i o n modes through the ergodic l i m i t ( 3 , 4J beyond which v i b r a t i o n dynamics need not be " r e g u l a r " and " q u a s i - p e r i o d i c " . We use c l a s s i c a l t r a j e c t o r y methods {2) t o i d e n t i f y the natural v i b r a t i o n modes, h e r e i n a f t e r c a l l e d r e g u l a r modes. An i t e r a t i v e search s t r a t e g y i s developed wherein the i n i t i a l c o n d i t i o n s o f a t r a j e c t o r y are perturbed u n t i l the r e s u l t ant t r a j e c t o r y i s a c l o s e d , s e l f - t r a c i n g , one-dimensional curve i n coordinate space. The s o l u t i o n i s a pure r e g u l a r mode, and i t s energy conservation and p e r i o d i c i t y requirements are s a t i s f i e d a f o r t i o r i . The f a m i l y o f such s o l u t i o n s , each member a t a d i f f erent energy, complete the c h a c t e r i z a t i o n o f the r e g u l a r mode. Asymmetric s t r e t c h v i b r a t i o n s f o r a methylene model p o t e n t i a l 3
1
2
250
Brooks and Hayes; State-to-State Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 1977.
Downloaded by CORNELL UNIV on October 6, 2016 | http://pubs.acs.org Publication Date: June 1, 1977 | doi: 10.1021/bk-1977-0056.ch031
31.
PARR AND ROOKSTOOL
251
Molecular Vibration States
Figure 1. Potential-energy contours and selected asymmetric vibration trajectories for Bt methylene. Insert shows relevant geometry parameters and defines the axes.'Numberedcontours are curves of constant potential energy. The energies themselves (in kcal mol' relative to zero at equilibrium) are the squares of the identifying numbers. While the contours are thus unevenly spaced in energy, quadratic potentials would yield such contours evenly spaced in the co-ordinates. The observed uneven spacing in coordinates reflects the degree of anharmonicity in the potential. (This choice of contouring reduces the severe crowding observed in most potential maps.) Lettered curves represent regular asymmetric vibrations. Parameters for those vibrations are given in Table I. 3
1
(5) are shown as l e t t e r e d curves i n F i g u r e 1. The search s t r a t egy c o n s i s t e d of v a r i a t i o n of asymmetric k i n e t i c energy and i n i t i a l bond angle at a given symmetric s t r e t c h c o o r d i n a t e . This s t r a t e g y was s u c c e s s f u l because symmetric and bend c o n t r i b u t i o n s to r e g u l a r asymmetric s t r e t c h are at extrema f o r C ? geometries. These c o n t r i b u t i o n s do not destroy the p u r i t y o f the r e g u l a r asymmetric mode because they occur at the asymmetric s t r e t c h f r e quency. That symmetric motions c o n t r i b u t e to "asymmetric s t r e t c h " v
Brooks and Hayes; State-to-State Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 1977.
STATE-TO-STATE
252
i s obvious from both the curvature o f the asymmetric t r a j e c t o r j e s and the d e v i a t i o n o f t h e i r midpoints from the e q u i l i b r i u m geomet ry. The bend c o n t r i b u t i o n cannot be seen i n Figure 1 as i t has been p r o j e c t e d o u t ; however, Table I i n d i c a t e s the extent of i t s c o n t r i b u t i o n . As the asymmetric s t r e t c h energy i s i n c r e a s e d , the r e g u l a r motion i n v o l v e s a decreasing bond angle. The o s c i l l a t i o n of t h i s bond angle during the asymmetric s t r e t c h v i b r a t i o n i s min imal (about one degree a t the highest energy s t u d i e d , t r a j e c t o r y f). Because each asymmetric s t r e t c h shown i n Figure 1 occurs at a d i f f e r e n t bond a n g l e , the Figure i s something of a c h e a t ; the numbered p o t e n t i a l - e n e r g y contours were drawn f o r the e q u i l i b r i u m bond a n g l e , θ = 1 2 7 . 7 ° . However, due to the weak bending f o r c e constant (Fee 0.234 mdyn A" r a d " ) , there i s l e s s than a 4 kcal/mol d i f f e r e n c e i n the energy contours between the extremes o f the asymmetric t r a j e c t o r i e s d e p i c t e d . F u r t h e r , the t r a j e c t o r i e s are s a t i s f y i n g l y normal to the contour l e v e l s at t h e i r endp o i n t s ; hence the s p i r i t of comparison between the t r a j e c t o r y and i t s p o t e n t i a l bounds i s preserved. The energy of the asymmetric s t r e t c h v i b r a t i o n i s a good c l a s s i c a l constant of the motion; hence i t i s a s s o c i a t e d with a good quantum number of a quantum mechanical dynamics ( 2 J . The asymmetric s t r e t c h a c t i o n i s the t o t a l c l a s s i c a l a c t i o n i n these t r a j e c t o r i e s ; thus one may o b t a i n JWKB v i b r a t i o n a l quantum numbers from the t r a j e c t o r i e s i n a s t r a i g h t f o r w a r d manner ( 6 ) . These quantum numbers are presented i n Table I. The a t t r i b u t e s of the =
Downloaded by CORNELL UNIV on October 6, 2016 | http://pubs.acs.org Publication Date: June 1, 1977 | doi: 10.1021/bk-1977-0056.ch031
CHEMISTRY
Table I. Label
1
2
Methylene Asymmetric S t r e t c h V
JWKB
V 1
(cm" )
Ε (kJ mol ) -1
(A)
66 m (degrees)
t
η
-0.5
3361
0
1.08192
127.705
0.0
a
0.02
3311
20.54
1.09411
127.05
0.01
b
2.55
3082
117.53
1.15666
124.08
0.07
c
6.07
2780
240.87
1.25000
120.51
0.17
d
12.88
2268
445.39
1.45000
114.87
0.40
e
17.52
1937
555.64
1.60000
111.71
0.61
f
25.89
1249
720.69
2.02000
106.52
1.23
a-f) m) n) t)
Calculated t r a j e c t o r i e s Midpoint geometry f o r asymmetric v i b r a t i o n Normal mode r e s u l t s a t the e q u i l i b r i u m geometry δ θ i s the d i f f e r e n c e i n θ between t u r n i n g and midpoint
Brooks and Hayes; State-to-State Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 1977.
31.
PARR AND ROOKSTOOL
Molecular Vibration States
253
Downloaded by CORNELL UNIV on October 6, 2016 | http://pubs.acs.org Publication Date: June 1, 1977 | doi: 10.1021/bk-1977-0056.ch031
asymmetric s t r e t c h vary so smoothly with JWKB quantum number that one can i n t e r p o l a t e t h e i r values at i n t e g r a l quantum numbers con f i d e n t l y with Lagrange i n t e r p o l a t i o n ( 7 ) . The term values p r e sented i n the t a b l e represent an e a s i l y - o b t a i n e d , f i r s t a p p r o x i mation route to pure asymmetric s t r e t c h s p e c t r a l information from the p o t e n t i a l model. The s u c c e s s f u l e x t r a c t i o n of t h i s pure r e g u l a r v i b r a t i o n mode r a i s e s the p o s s i b i l i t y of the establishment of r e g u l a r v i b r a t i o n modes f o r a r b i t r a r y bound molecular p o t e n t i a l s . These modes and t h e i r combinatorial dynamics w i l l be o f g r e a t u t i l i t y i n the c r e a t i o n and a n a l y s i s of r e p r e s e n t a t i v e v i b r a t i o n s t a t e s i n t h e o r e t i c a l dynamical s t u d i e s .
Literature Cited (1) (2) (3) (4) (5) (6) (7)
Wilson, Ε. B., Decius, J. C., and Cross, P. C., "Molecular Vibrations", McGraw Hill Book Company, Inc., New York, 1955. Parr, C. Α., Kuppermann, Α., and Porter, R. Ν., J. Chem. Phys. (1977) 66 (7) 2914-31 Moser, J., "Stable and Random Motions in Dynamical Systems", Princeton University Press, New Jersey, 1973. Brumer, P. and Duff, J. W., J. Chem. Phys. (1976) 65 (9) 3566-74. Eaker, C. W. and Parr, C. Α., J. Chem. Phys. (1976) 64 (4) 1322-32. Messiah, Α., "Quantum Mechanics", v o l . 1, 239-41, North -Holland Publishing Co., Amsterdam, 1970. Margenau, H. and Murphy, G. M., "The Mathematics of Physics and Chemistry", 2nd e d i t i o n , 470-1, D. Van Nostrand Co., Inc., Princeton, 1956.
Brooks and Hayes; State-to-State Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 1977.
