(4) 8mr 9r which i n c l u d e s t h e a d i a b a t i c e n e r g i e s e ( r ) , a c e n t r i f u g a l term and t h e d i a g o n a l a d i a b a t i c c o r r e c t i o n . The l a t t e r can be 8mr approximated v i a a f i n i t e d i f f e r e n c e scheme (22,28) and f i n a l l y the e f f e c t i v e p o t e n t i a l s i m p l i f i e s t o n
n
2
2
m
n
2
n
n
U (r)
— j (1 - < ^ ( r + Ar,) (5) 8mr m(Ar) Hence, i n t h e DIVAH t h e o r y t h e r e a c t i v e s c a t t e r i n g p r o c e s s i s reduced t o an e l a s t i c s c a t t e r i n g problem on t h e e f f e c t i v e DIVAH p o t e n t i a l U ( r ) . The o n e - d i m e n s i o n a l r a d i a l e q u a t i o n can now be s o l v e d f o r the bound as w e l l as t h e m e t a s t a b l e s t a t e s whose l i f e t i m e T may be determined by T = 2 h d n (E) / dE. Here n ( E ) I t h e phase s h i f t . Both bound and m e t a s t a b l e s t a t e s a r e r e l a t e d t o resonance e n e r g i e s o f t h e c o r r e s p o n d i n g r e a c t i o n p r o c e s s . Two c a t e g o r i e s o f resonances a r e e x p e c t e d : shape-type resonances as m e t a s t a b l e s t a t e s on u" (r) o c c u r r i n g i n P ( E ) r e a c t i o n p r o b a b i l i t i e s and Feshbach-type resonances - due io t h e i n t e r a c t i o n o f t h e one-dimens i o n a l s t a t e s w i t h t h e c o n t i n u a o f lower l y i n g a n g u l a r s t a t e s . n
=
e ( ) r
R
n
s
n
n
n
Truhlar; Resonances ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
356
RESONANCES
The
s i g n a t u r e o f these resonances
and
off-diagonal (P
n
n
may
appear i n d i a g o n a l (P
i ( E ) ) reaction probabilities.
n
(E))
'
In F i g . 1 the e f f e c t i v e p o t e n t i a l s U ( r ) f o r the f o u r l o w e s t a n g u l a r s t a t e s o f the system (gerade (g) and ungerade (u) symmetry) a r e compared t o the c o r r e s p o n d i n g a d i a b a t i c e i g e n e n e r g y c u r v e s e ( )« F o r the e x c i t e d a n g u l a r s t a t e s we f i n d w e l l s i n t h e a d i a b a t i c p o t e n t i a l s c o n f i r m i n g the i d e a a d i a b a t i c t r a p p i n g . However, the d i a g o n a l a d i a b a t i c c o r r e c t i o n which r e f l e c t s the d y n a m i c a l c o u p l i n g t o the o t h e r a n g u l a r s t a t e s and which i s always r e p u l s i v e i n n a t u r e ( 2 7 ) , modulates the shape o f the e f f e c t i v e p o t e n t i a l q u i t e a b i t . Only when t a k i n g i n t o account both terms and c a l c u l a t i n g the bound and m e t a s t a b l e s t a t e s on t h e e f f e c t i v e p o t e n t i a l u" (r) does one o b t a i n e x c e l l e n t agreement w i t h t h e c o r r e s p o n d i n g resonance e n e r g i e s i n the P ,(E) r e a c t i o n p r o b a b i l i t y c u r v e s . A comparison w i t h a c c u r a t e c o m p u t a t i o n s i s p r o v i d e d i n T a b l e I . Furthermore the i n t e r p l a y o f both o f t h e s e c o n t r i b u t i o n s t o the e f f e c t i v e p o t e n t i a l i s r e s p o n s i b l e f o r the change o f the resonance energy l o c a t i o n s r e l a t i v e t o the c o r r e s p o n d i n g t h r e s h o l d e n e r g i e s as one goes from n=l t o h i g h e r quantum numbers. As can be seen from F i g . 1, f o r n=l,2 t h e a d i a b a t i c c o r r e c t i o n s h i f t s the r e s o n a n c e s over the t h r e s h o l d energy. With i n c r e a s i n g quantum number n t h i s e f f e c t i s f i n a l l y overcome by a more and more bonding e i g e n e n e r g y c u r v e ( r ) t u r n i n g the o n e - d i m e n s i o n a l m e t a s t a b l e s t a t e s i n t o one-dimen s i o n a l bound s t a t e s w i t h r e s p e c t t o t h e i r t h r e s h o l d energy. r
n
n
£
Table I.
Resonance e n e r g i e s f o r the c o l l i n e a r H+H^
reaction:
a
comparison o f o n e - d i m e n s i o n a l model p r e d i c t i o n s i n c o m p a r i son t o e x a c t quantum m e c h a n i c a l r e s u l t s on the P o r t e r Karplus II s u r f a c e (49).
Resonance e n e r g i e s
(eV)
a) E
n
(eV)
0 1 2 3
-4.474 -3.953 -3.464 -3.007
From e x a c t quantum-mechanical
DIVAH
Simple
c a l c u l a t i o n o f P^(9,48)k
model
spherical
0
0
"
hypermodel
(14)
-3.870 -3.443 -3.022
-3.872 -3.446 -3.028
-3.930
a) E
^
P^
= t h r e s h o l d energy f o r v i b r a t i o n a l c i a t i o n i n t o t h r e e f r e e atoms. o
s t a t e n, r e l a t i v e
i s the g r o u n d - s t a t e - t o - g r o u n d - s t a t e
reaction
to
disso
probability.
Truhlar; Resonances ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
ROMELT AND POLLAK
Vibrationally Bonded Molecules
357
F i g u r e 1. A d i a b a t i c e i g e n e n e r g i e s e ( r ) (dashed l i n e s ) and e f f e c t i v e p o t e n t i a l s U ( r ) ( f u l l l i n e s ) f o r t h e c o l l i n e a r H+H r e a c t i o n (22) on t h e P o r t e r - K a r p l u s II s u r f a c e (49) f o r n = 0 , l , 2 , 3 . The e n e r g i e s a r e r e l a t i v e t o t h e d i s s o c i a t i o n energy i n t o t h r e e f r e e atoms. 2
Truhlar; Resonances ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
358
RESONANCES
The F+h^ system (on the Muckerman V surface (50)) i s taken as representative of the class of asymmetric reactions. In asymmetric reactions, the asymmetric nature of the electronic potential energy surface may introduce a new feature i n form of avoided crossings (cf. F i g . 2) i n the adiabatic eigenenergy surfaces. In t h i s case the adiabatic diagonal correction includes two d i f f e r e n t types of contributions: the f i r s t coming from the near degeneracy of two adiabatic energies e ( r ) and e i ( ) *- the region of the avoided crossing, and the second r e s u l t i n g from the potential coup l i n g of the angular states as discussed for the system. It can be shown (17,27) that i f the crossing i s strongly l o c a l i z e d , as i t i s i n the F+l-L system, a potential spike occurs due to the curve crossing e f f e c t (51) (cf. F i g . 2). But as long as i t does not disturb the energetic structure of the bound states i n the adiabatic wells, the model i s s t i l l s a t i s f a c t o r y (cf. Table II) for comparison to accurate quantal c a l c u l a t i o n s . r
R
Table I I .
a
n +
Resonance and threshold energies for the c o l l i n e a r F+h^ FH+H reaction on the Muckermann V potential energy sur face (50). Resonance Energies (eV) a
E > (eV)
Exact quantum-mechanical calculation (15,17)
F+H (v=0)
-4.480
FH(v=3)+H
-4.463
-4.465
FH(v=4)+H
-4.035
-4.052
F+H (v=l)
-3.966
2
2
DIVAH model
-4.460 -4.064
-3.95-0.01
b)
-3.959
E = threshold energy for the corresponding v i b r a t i o n a l state, r e l a t i v e to d i s s o c i a t i o n into three free atoms. ^
Estimated from F i g . 3 of Ref.
(17).
The adiabatic correction i s not always necessarily large. Consider the C1+HC1 •> C1H+C1 exchange reaction on the extended LEPS surface of Ref. (28). The coupling of the v i b r a t i o n a l states i s small, the adiabatic correction minor and the reaction proceeds nearly a d i a b a t i c a l l y . This statement can be e a s i l y v e r i f i e d from the reaction p r o b a b i l i t i e s displayed i n F i g . 3. The off-diagonal p r o b a b i l i t i e s are small compared to the diagonal ones. Also the signature of resonance states i n the n-th adiabatic surface U (r) R R 0 i s most noticeable i n the P^ (E) or P ^ ( E ) reaction p r o b a b i l i t i e s but dies o f f very rapidly wi?h increasing An. It has been shown that t h i s a d i a b a t i c i t y of the reaction depends primarily on the masses of the atoms involved (15,16,28,52). The adiabatic approxi mation becomes better with decreasing skew angle, i . e . for C1+HC1 the e f f e c t i v e potentials U (r) and the adiabatic energies e (r) +
n
n
Truhlar; Resonances ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
-4.2- 1 M
1—
I
1
F +H
E/eV -4.3-
359
Vibrationally Bonded Molecules
ROMELT AND POLLAK
2
1
1
1 i\ A
-4.4-
iil \\ V 1
II
\A
\
1
\
W \
5
\
4
v
• •
/
-4.51 \ / / 1 \J 1
1
t
/
/
-
-4.61
1
4
5
1—~"
6
1 —
7 r/j
F i g u r e 2. A d i a b a t i c e i g e n e n e r g i e s e ( r ) (dashed l i n e s ) and e f f e c t i v e p o t e n t i a l s U ( r ) ( f u l l l i n e s ) o f t h e c o l l i n e a r F+H + FH+H r e a c t i o n f o r F + H ^ v r O ) (^ e U. ) and FH(v=3) + H (= e U^) ( 2 7 ) . The e n e r g i e s a r e r e l a t i v e So t h e d i s s o c i a t i o n energy i n t o t h r e e f r e e atoms. 2
y
Truhlar; Resonances ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
360
RESONANCES
1.5 E/eV Figure 3. Reaction probability versus t o t a l energy for the c o l l i n e a r C1+HC1 •> C1H+C1 reaction. The t o t a l energy i s measured from the minimum of the asymptotic HC1 potential well. The arrows and numbers on the abscissae indicate the vibrational energies and associated quantum numbers of the isolated HC1 molecule (28).
Truhlar; Resonances ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
19.
ROMELT AND
POLLAK
361
Vibrationally Bonded Molecules
almost coincide. From Table I I I , which includes a comparison to accurate quantal calculations for the same potential energy surface, i t can be seen that the predictions are exact to within 10 eV (28). Most of these very narrow resonances ( c f . F i g . 3) were f i r s t pre dicted by the DIVAH model and have subsequently been v e r i f i e d by two-dimensional quantal calculations. The resonance structure strongly resembles a spectrum of t r a n s i t i o n complex. It i s a very appealing idea to invert t h i s 'spectral' information to obtain the v i b r a t i o n a l structure of the t r a n s i t i o n complex. With regard to the width of resonances, i t i s well known that the widths of Feshbach resonances are related to the coupling potential des c r i b i n g the interaction of the bound or metastable state with the continuum of a lower lying angular state. As discussed above t h i s coupling decreases with the skew angle of the system. Consequently, the Feshbach-type resonances become sharper and sharper with de creasing skew angle. Table I I I . Approximate and exact resonance energies for the c o l l i n e a r C1+HC1 exchange reaction. The exact resonance energy i s for the t r a n s i t i o n indicated.
n
a )
Parity 7
0 0 1 1 1 2 2 2 2 3 3 3 3 3
g u g g u g g g u g g g g u
b )
E(eV) n
c )
E , (eV) v' «- v
d )
v' + v
Type of resonance 7 r
0. 58835 0. 61991
0.58848 0.61960
0 1
0 1
Feshbach Shape
0. 84459 0. 88312 0. 91576
0.84492 0.88359 0.91609
1 1 1
1 1 1
Feshbach Feshbach Feshbach
1. 12229 1. 16072 1. 19597 1. 22551 1. 23193
1.12272 1.16158 1.19691 1.22592 1.23194
2 2 2 2 3
2 2 2 2 3
Feshbach Feshbach Feshbach F eshbach Shape
Quantum number of the e f f e c t i v e v i b r a t i o n a l l y adiabatic p o t e n t i a l . y
k Parity (u or g) of the e f f e c t i v e v i b r a t i o n a l l y adiabatic p o t e n t i a l . c) Approximate one-dimensional resonance energy obtained from the solution of equation (3). d
)
r
4-
Exact resonance energy. The most exciting result of these investigations, however, was the r e a l i z a t i o n that i f DIVAH theory predicted a bound state below the zero point energy of reactants and products, then t h i s state would represent a truly bound molecule since a non-adiaba-
Truhlar; Resonances ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
RESONANCES
362
t i c t r a n s i t i o n to lower lying vibrational states i s not possible (30). In fact, the bound state computations of Meyer showed that c o l l i n e a r IHI does have four t r u l y bound states on a minimum-free potential energy surface (31). Furthermore, the exact c o l l i n e a r energies agreed excellently with the DIVAH predictions (cf. Table IV), and therefore vibrational bonding can be regarded as an extreme case of resonances i n a 'direct' scattering process. We had thus predicted the existence of a new type of chemical bond which i s formed because of an e f f e c t i v e v i b r a t i o n a l l y adiabatic well - hence the name v i b r a t i o n a l bonding. The mass combination i n the IHI system (heavy-light-heavy) resembles very much the molecule. The elec tron i s located mainly between the two protons, attracting both of them and thus overcoming the repulsion between the two protons and binding the system. The fast electronic motion provides an e f f e c t i v e (adiabatic) potential for the slowly moving nuclei. Ana logously, in v i b r a t i o n a l bonding the bond i s a result of the fast nuclear motion of a l i g h t - p a r t i c l e group of nuclei which provides an e f f e c t i v e binding potential for the rest of the system. a) Table IV.
Bound-state energy levels (kJ/mol) of c o l l i n e a r Exact solution of mathematical-twodimensional problem
E
n
Og E, ig E E^ 3g 0 2 g
Vibrationally adiabatic approximation
-299.175
-299.173
-297.846
-297.844
-296.609 b) -295.512 J
IHI
-296.605 -295.506
Surface A of Ref. (8,10). The levels l i e between the saddle point at -303.78 kJ/mol and the HI zero point energy at -294.75 kJ/mol, r e l a t i v e to dissociation into three free atoms. ^ Converged within ~ 0.005 kJ/mol. c) Note that the excellent agreement can only be obtained after insertion of the diagonal correction. The magnitude of t h i s term i s < 0.01 kJ/mol. The potential energy surfaces (LEPS A from Refs. (8,10)), the e f f e c t i v e potentials U (n=0; gerade and ungerade symmetry) and the bound states of the c o l l i n e a r IHI and IDI are shown in F i g . 4. As argued for the C1+HC1 system the reaction i s very adia batic, and thus the e f f e c t i v e potentials and the adiabatic eigenenergies e (r) coincide within 10" eV. It should be noticed that isotopic suGstitution of the hydrogen atom by i t s heavier isotope deuterium tends to destabilize the vibrational bonded molecule. This inverted isotope effect i s mainly due to a smaller variation of the IDI zero point energies, indicating that the heavier deuterium
Truhlar; Resonances ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
ROMELT AND POLLAK
Vibrationally Bonded Molecules
1.0 I 26
19
i
1
28
'
i
1
i
30
21
i
'
363
r.
32 — x / i
'
23-x/A
F i g u r e 4. V i b r a t i o n a l bonding i n the c o l l i n e a r IHI and IDI systems. The LEPS A p o t e n t i a l energy s u r f a c e (8,10) i n masss c a l e d r a d i a l c o o r d i n a t e s £x = (m /m ) r , y = r a r e shown i n the l o w e r p a n e l s . Contours a r e a t - 5 6 8 . 4 , - 2 9 4 . 7 and 304.0 kJ/mol f o r IDI. The p o s i t i o n o f the s a d d l e p o i n t i s i n d i c a t e d by £. The a d i a b a t i c gerade and ungerade s u r f a c e s as w e l l as the s t a t i c minimum p o t e n t i a l energy a r e shown i n t h e upper p a n e l s as a f u n c t i o n o f the r a d i a l c o o r d i n a t e r. Bound (resonance) s t a t e s a r e g i v e n as s o l i d (dashed) l i n e s . E n e r g i e s a r e r e l a t i v e t o d i s s o c i a t i o n i n t o t h r e e f r e e atoms. A l s o shown i s the r e l a t i v e I-I internuclear distance r . A
B C
B C
A
B C
T T
Truhlar; Resonances ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
B C
RESONANCES
364
i s less e f f e c t i v e i n overcoming the repulsive forces. In a l l cases the binding energy i s of the order of 1-10 kJ/mol, thus being roughly comparable to the binding energy of van der Waals molecules (38,39). We have seen that the DIVAH model i s i d e a l l y suited for l i g h t atom transfer systems. Here, the r a d i a l motion corresponds to the slow heavy atom motion while the angular motion corresponds to the asymmetric fast l i g h t atom movement. Thus there i s a good separation of time scales and an adiabatic approximation i n r a d i a l coordinates i s appropriate. Resonances of course appear also i n heavy atom transfer systems for which, at least at low t o t a l energies one expects an adiabatic reaction path hamiltonian approach to be quite reasonable as shown by Truhlar and coworkers (18,53,54). Aquilanti and coworkers (55,56) have studied the a p p l i c a b i l i t y of adiabatic r a d i a l coordinate theories for heavy atom transfer reactions. Schor and Manz (57) also find that DIVAH model i s successful for heavy atom transfer systems. S i m i l a r l y , i t should be stressed that v i b r a t i o n a l bonding i s not l i m i t e d to l i g h t atom transfer systems. Atabek and Lefebvre (35) have given an example for v i b r a t i o n a l bonding i n non-reactive systems. Pollak (36) and Meyer (37) have studied vibrational bonding in heavy atom transfer reactions. Resonant Periodic Orbits In p r i n c i p l e , adiabatic potential energy curves may also be con structed s e m i c l a s s i c a l l y . I f r i s the r a d i a l (hyperspherical) coor dinate and cp the (hyperspherical) angle, then for each fixed r one may evaluate an action i n t e g r a l for the angular motion. Quan t i z i n g t h i s action would then y i e l d the semiclassical estimate for the adiabatic p o t e n t i a l . I f the adiabatic approximation were exact, then a c l a s s i c a l trajectory i n i t i a t e d at a barrier or well of the n-th adiabatic surface (with no r a d i a l momentum but with angular action corresponding to the n-th v i b r a t i o n a l state) would stay at the well or barrier forever: i t would be a periodic o r b i t (44,58). We have shown that t h i s statement may be turned around: a periodic o r b i t i s an adiabatic b a r r i e r or well of some adiabatic potential energy surface. Consider then an adiabatic well i n the hyperspherical coordinate system. C l a s s i c a l l y , the motion of the periodic o r b i t at the well would be an o s c i l l a t i o n from a point on the inner equipotential curve i n the reactant channel to a point on the same equipotential curve i n the product channel. This i s q u a l i t a t i v e l y the motion of what are termed "resonant periodic o r b i t s " (RPO's). For example the RPO's of the IHI system are given i n F i g . 5. Thus, finding adiabatic wells i n the r a d i a l coordinate system corresponds to finding RPO's and quantizing t h e i r action. Note that i n F i g . 5 we have also plotted a l l the periodic o r b i t dividing surfaces (PODS) of the system, except for the symmetric stretch. By d e f i n i t i o n , a PODS i s a periodic o r b i t that s t a r t s and ends on d i f f e r e n t equipotentials. Thus the symmetric stretch PODS would be an adiabatic well for an adiabatic surface i n reaction path coordinates. However, the PODS i n the entrance and exit channels shown i n F i g . 5 may be considered as adiabatic barrieres i n either the r a d i a l or reaction path coordinate systems. Here, the b a r r i e r i n r a d i a l coordinates, has quantally a tunneling path between the entrance and exit channels.
Truhlar; Resonances ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
19.
ROMELT AND
Vibrationally Bonded Molecules
POLLAK
365
Thus the PODS are the c l a s s i c a l l y allowed part of the angular motion. The tunneling does cause though a more serious d i f f i c u l t y . The RPO's move across a double well potential so that to obtain a good semiclassical estimate of the quantal energies one would have to incorporate tunneling below the barrier and r e f l e c t i o n contributions above the barrier (56). This d i f f i c u l t y can be c i r cumvented by using a p ^ i x i t i v e semiclassical approach (45,46). Consider f i r s t the case of a symmetric exchange reaction. For each fixed value of r, the angular potential i s a symmetric double well (or, at small r, a single well p o t e n t i a l ) . The quantal states w i l l be either gerade or ungerade with respect to r e f l e c t i o n about the symmmetric stretch. As r goes to i n f i n i t y , the b a r r i e r separating the two wells becomes very large so that for states well below the three-body d i s s o c i a t i o n energy, U^(r)=U (r). For large r the doublet n and n+1 v i b r a t i o n a l states are well approximated semic l a s s i c a l l y by an (n+l/2)h action condition i n either well or o v e r a l l by a (2n+l)h action condition over the double well. As r decreases, U
Q
U
the doublet s p l i t s , IP < U , as a result of tunneling and abovebarrier r e f l e c t i o n . However, one may define an average eigenvalue U-(r) as n U-(r) = | (U^(r) + U V ) )
(6)
and U-(r) does have a well defined primitive semiclassical analog: The action over the double well i s quantized by a (2n+l)h action condition. In other words, the resonant o r b i t with (2n+l)h action w i l l give the adiabatic well energy and location of the U-(r) adia batic potential energy curve. For example, i n F i g . 6 we compare the energies and locations of quantized RPO's of the IHI system with the quantal (averaged) adiabatic surfaces. A s i m i l a r method may also be devised for asymmetric reactions. Instead of considering the general case, we w i l l deal with a s p e c i f i c system - the c o l l i n e a r FHH reaction; generalization to any asymmetric system i s straightforward. As r-*-°° , one finds for the angular coordinate an asymmetric double well p o t e n t i a l . Since the barrier separating the two wells i s very large, the quantal states are well l o c a l i z e d i n either the entrance or exit channels; hence, we l a b e l these quantal states by an entrance channel index j or an exit channel index k. U. and are well approximated by (j+l/2)h or (k+l/2)h action conditions in the entrance or exit channel wells, respectively. I f for a certain pair of indexes U. -U^ then, just as i n the symmetric case, the two states w i l l mix as r decreases, because of tunneling or above-barrier r e f l e c t i o n . In the FHH system, the lowest resonance occurs on an adiabatic potential energy curve which may be correlated with the asymptotic H (j=0) and HF(k=3) v i b r a t i o n a l states (cf. F i g . 2). Using the primitive semiclassical prescription, we may approximate t h i s well by looking for an RPO with {(j+l/2)h + (k+l/2)h} = 4h action. S i m i l a r l y , the next c o l l i n e a r resonance occurs on a well that mixes the asymptotic H«(j=l) and HF(k=4) v i b r a t i o n a l states and so may be approximated B y an RPO with 6h action. Thus, i n general, the adiabatic well responsible for a resonance may be approximated by an RPO with (j+k+l)h action, where j and k are determined by the c o r r e l a t i o n of the well with 2
Truhlar; Resonances ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
RESONANCES
366
xla.u.) Figure 5. RPO's of the c o l l i n e a r IHI system on the LEPS A surface of Ref. (8,10). Equipotential contours are at 0.1, 0.5 and 0.9 eV r e l a t i v e to the bottom of the asymptotic HI well. The mass scaled coordinates x,y are defined i n F i g . 4. The RPO's are the heavy l i n e s denoting t r a j e c t o r i e s that have their two turning points on the same equipotential l i n e s . Reproduced with permission from Ref. 45.
r(A)
Figure 6. Average adiabatic surfaces for the IHI system. The crosses denote the energies and locations of the quantized o r b i t s . The s o l i d l i n e s show U-(r) determined from quantal computations. The coordinate on the right hand side denotes the asymptotic vibrational energies of HI. Reproduced with permission from Ref. 45.
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19.
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POLLAK
Vibrationally Bonded Molecules
367
a s y m p t o t i c r e a c t a n t and p r o d u c t s t a t e s o f n e a r l y the same energy. Given the a d i a b a t i c w e l l s , one may use them t o study the p r o p e r t i e s o f resonances formed because o f them. As we have shown elsewhere ( 6 0 ) , g i v e n a p e r i o d i c o r b i t one can e v a l u a t e the a d i a b a t i c frequency a) f o r motion p e r p e n d i c u l a r t o the w e l l r b a r r i e r formed by t h a t o r b i t . We have found ( 4 4 ) , i n a l l systems we a n a l y z e d , t h a t the a d i a b a t i c , T - q u e n c i e s o f the RPO's are r e a l , c o n f i r m i n g t h a t RPO's a c t u a l l y are a d i a b a t i c w e l l s . One o f the main advantages o f d e a l i n g w i t h p e r i o d i c o r b i t s i s t h a t they a r e u n i q u e l y determined by the e l e c t r o n i c p o t e n t i a l energy s u r f a c e and the masses o f the system. T h i s i s not t r u e f o r the q u a n t a l a d i a b a t i c s u r f a c e s . As shown above, the q u a n t a l a d i a b a t i c a p p r o x i m a t i o n became q u a n t i t a t i v e l y r e l i a b l e o n l y w i t h the advent o f DIVAH t h e o r y - t h a t i s one must i n c o r p o r a t e d i a g o n a l a d i a b a t i c c o r r e c t i o n s w i t h i n the a d i a b a t i c a p p r o x i m a t i o n . By s t u d y i n g the p r o p e r t i e s o f RPO's we have shown t h a t t h e a d i a b a t i c d i a g o n a l c o r r e c t i o n s are b a s i c a l l y c u r v a t u r e c o r r e c t i o n s t h a t a r e a u t o m a t i c a l l y i n c o r p o r a t e d i n the RPO's ( 4 5 ) . In a d d i t i o n t o the a d i a b a t i c frequency co^ f o r the motion p e r p e n d i c u l a r t o the RPO one can a l s o determine the s t a b i l i t y f r e quency 0 ) £ or what i s known as a c h a r a c t e r i s t i c e i g e n v a l u e (61) f o r the motion p e r p e n d i c u l a r t o the RPO, by l i n e a r i s a t i o n o f the e q u a t i o n s o f motion about the p e r i o d i c o r b i t . P o l l a k has shown (60) t h a t the a d i a b a t i c frequency i s j u s t the f i r s t - o r d e r Magnus a p p r o x i m a t i o n t o the s t a b i l i t y f r e q u e n c y . T h i s i m p l i e s t h a t when the a d i a b a t i c assumption i s good then u) ^ For example, i n the H-j system, t h e frequency ft o f the RPO i s ~ 2000 cm" w h i l e (D - 2200 cm"" , t h a t i s u)^ > ft. E v i d e n t l y the a d i a b a t i c a p p r o x i mation cannot be too good, i n f a c t the RPO's are u n s t a b l e , | fc> J ~ 1300 cm" . Note t h a t i n t h i s c a s e , q u a l i t a t i v e l y , the d e v i a t i o n o f 0 ) j . from 0)^ i s p r o v i d i n g i n f o r m a t i o n on the e x t e n t o f n o n - a d i a b a t i c t r a n s i t i o n s . In Ref. (42) we have shown t h a t i f the RPO i s u n s t a b l e then the q u a n t a l resonance w i d t h may be e v a l u a t e d from the i m a g i n a r y s t a b i l i t y f r e q u e n c y . In an A + BC system, as the l i g h t atom B g e t s l i g h t e r and the atoms A,C get h e a v i e r , one e x p e c t s the a d i a b a t i c a p p r o x i m a t i o n about the RPO t o get b e t t e r . T h i s s t a t e ment i s i n complete agreement w i t h the c o r r e s p o n d i n g f i n d i n g s o f the q u a n t a l c o m p u t a t i o n s . In f a c t , f o r the IHI system, a ) ^ « a) ^. a t a l l e n e r g i e s , which i m p l i e s t h a t the RPO i s s t a b l e i n IHI. For l i g h t atom t r a n s f e r systems where the a n t i s y m m e t r i c l i g h t - a t o m motion i s much f a s t e r than the symmetric heavy-atom m o t i o n , the RPO i s s t a b l e a t almost a l l e n e r g i e s below the t h r e e - b o d y d i s s o c i a t i o n l i m i t ( 2 9 , 4 2 ) . C l a s s i c a l l y we f i n d a l a r g e f a m i l y o f bound q u a s i p e r i o d i c o r b i t s a l t h o u g h the p o t e n t i a l energy s u r f a c e employed was minimum f r e e . These o r b i t s were the f i r s t example o f v i b r a t i o n a l bonding. A
0
S
1
A
s
s
s
Three-dimensional
Treatment o f Resonances and V i b r a t i o n a l
Bonding
So f a r we have d e s c r i b e d t h e phenomenon o f resonances and v i b r a t i o n a l bonding f o r c o l l i n e a r systems. Of c o u r s e , the r e a l w o r l d i s t h r e e - d i mensional and hence i t was o f i n t e r e s t t o extend t h e s e c o n c e p t s t o 3D systems. For the s e m i c l a s s i c a l t h e o r y P o l l a k and Wyatt (43,46)
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RESONANCES
have shown that RPO's are useful for determining properties of resonances i n 3D, too. The underlying idea i s an adiabatic periodic reduction method (43^. A general triatomic molecule undergoes rota t i o n a l , bending and stretching motion. Usually the overall rota tion i s much slower than the vibrational motion because of the r e l a t i v e l y large moment of i n e r t i a . Furthermore, i t i s often the case that bending motion i s much slower than the antisymmetric stretching motion. Thus one may use a Born-Oppenheimer-type separa t i o n , i n which one f i r s t freezes the rotational and bending angles, then finds the RPO at each set of fixed angles, and f i n a l l y quantizes i t . Each RPO corresponds automatically to a nearly frozen symmetric stretch motion. Thus we obtain an adiabatic well U ( Y ) dependent on the bend angle y . Averaging the Hamiltonian over the (fast) antisymmetric stretching motion of the RPO provides an e f f e c t i v e Hamiltonian for the bending motion of the adiabatic well with stretch v i b r a t i o n a l quantum number n. The quantized bend states are obtained by semiclassical quantization of the e f f e c t i v e bend Hamiltonian. F i n a l l y one averages the Hamiltonian over the quantized bend motion to obtain the e f f e c t i v e rotational resonance Hamiltonian. To i l l u s t r a t e t h i s method we consider i n some d e t a i l the reso nances of the 3D FHH system on the Muckerman V surface (50). As noted i n the previous section, c o l l i n e a r l y the resonance may be i d e n t i f i e d with an RPO having 4h action. Defining the bend angle y via the Natanson-Smith-Radau (62,63) coordinate system, we f i n d the 4h action RPO at fixed y. This provides the curve E - ( Y ) (shown in F i g . 7) where E^(y ) i s the energy of the y dependent KPO. The kinetic energy^for the bending motion has a y dependent mass l i k e c o e f f i c i e n t B (y) (cf. F i g . 7) which may be found by averaging the t o t a l Hamiltonian over the motion of the y dependent RPO. The e f f e c t i v e Hamiltonian for the bending motion of the resonance i s h ( Y) = B * ( Y ) x
P* + E ( y ) x
(7)
To find the lowest bend l e v e l one may quantize the bend motion s e m i c l a s s i c a l l y . This provides the bend energy l e v e l , shown as a dashed l i n e on the lower right hand side of F i g . 7. F i n a l l y one averages the rotational constant B^(y ) over the bend motion to obtain the rotational constant B^ for o v e r a l l rotational motion. The energy dependence of the resonance on t o t a l angular momentum i s simply "h B^ J(J+1). The dependence of the resonance energy on J has been computed by Redmon and Wyatt (64) using a j conser ving approximation, and by Walker and Hayes (65) using the approxi mate BCRLM methodology. Both of these studies agree well with the prediction of the semiclassical RPO theory as shown i n F i g . 8. We have also used the periodic reduction method to predict with good accuracy the 3D structure of v i b r a t i o n a l l y bonded mole cules. It should be stressed though, that i n p r i n c i p l e i t i s not necessary to use periodic reduction. As shown i n F i g . 9 the RPO's of the IHI system are stable also i n 3D, one can find bound quasiperiodic orbits and quantize them semiclassically d i r e c t l y without resorting to periodic reduction. A similar treatment of the 3D IHI system was given on a quantum mechanical l e v e l (34). For each value of the r a d i a l coordinate r we evaluated within a harmonic approximation the zero point bending
Truhlar; Resonances ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
19.
ROMELT AND POLLAK
Vibrationally Bonded Molecules
369
•18
0.71 -
2
A B,(Y)
•10 '
E
CD
J
4
1 r~ 0.96 1.0 COS Y F i g u r e 7. A d i a b a t i c a n g l e dependent p o t e n t i a l energy c u r v e s E ( y ) and f o r c e c o n s t a n t s B ( Y ) , B ( Y ) f o r resonances i n 3B FH^ on the Muckerman V (50) p o t e n t i a l energy s u r f a c e . n=l,2 denotes the f i r s t and second v i b r a t i o n a l resonance l e v e l s o f the F H system. The dashed l i n e s denote the q u a n t i z e d bend l e v e l s . Note the bend l e v e l s u b s t r u c t u r e f o r the n=2 c a s e . For f u r t h e r e x p l a n a t i o n see t e x t . Reproduced w i t h p e r m i s s i o n from R e f . 46. — i
1
0.92
2
BCRLM(B=2.45)
IOO
200
300
400
J (J*l) F i g u r e 8: Energy v a r i a t i o n o f resonance e n e r g i e s f o r the 3D F+H^ r e a c t i o n . SCAD denotes the s e m i c l a s s i c a l a d i a b a t i c r e s u l t s based on the a d i a b a t i c r e d u c t i o n o f R P O ' s . QU denotes J c o n s e r v i n g r e s u l t s (46) and BCRLM^are r e s u l t s from Ref. ( 6 § ) . The r o t a t i o n a l c o n s t a n t s B, i n cm" , t h a t emerge from the v a r i o u s t r e a t m e n t s a r e i n d i c a t e d i n p a r e n t h e s i s f o r each c u r v e . Reproduced w i t h p e r m i s s i o n from R e f . 46.
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370
RESONANCES
energy ^( ) = ^- Thus t h e ground s t a t e a d i a b a t i c p o t e n t i a l i n 3D i n c l u d e s a bend l e v e l c o n t r i b u t i o n ; i . e . i n analogy t o t h e c o l l i n e a r approximation e
U
n
D
(
r
l
?
)
=
e
n n (
r
'
+
h
M h
8
)
o og a) b The d i f f e r e n t c o n t r j t u t i o n a r e d i s p l a y e d i n F i g . 10. C l a r y and Connor (32,40) computed t h e e x a c t quantum m e c h a n i c a l , f u l l y c o u p l e d 3D energy a t z e r o t o t a l a n g u l a r momentum J . A l l t h e s e s t u d i e s ( s e m i c l a s s i c a l , approximate quantum m e c h a n i c a l and e x a c t quantum mechan i c a l ) found t h a t f o r t h e LEPS A p o t e n t i a l s u r f a c e , a d d i t i o n o f bending energy t o t h e c o l l i n e a r b o u n d - s t a t e e n e r g i e s l e a v e s o n l y one bound 3D, J=0 IHI s t a t e , w h i l e f o r IDI no t r u l y bound s t a t e e x i s t s i n 3D. These r e s u l t s prove t h a t t h e phenomenon o f v i b r a t i o n a l bonding i s n o t an a r t i f i c a l e f f e c t o f t h e c o l l i n e a r geometry, a l t h o u g h bending seems t o weaken t h e bond c o n s i d e r a b l y . A g a i n , we f i n d an anomalous i s o t o p e e f f e c t : s u b s t i t u t i o n o f hydrogen by deuterium causes d e s t r u c t i o n o f t h e m o l e c u l e i n s t e a d o f s t a b i l i s a t i o n . Because the LEPS A s u r f a c e i s o n l y a crude e s t i m a t e o f t h e t r u e e l e c t r o n i c p o t e n t i a l energy s u r f a c e , f u r t h e r i n v e s t i g a t i o n s have been u n d e r t a k e n u s i n g t h e more r e f i n e d DIM-3C s u r f a c e o f L a s t ( 6 6 ) , and t h e o c c u r r i n g i n t e r p l a y o f t h e van d e r Waals i n t e r a c t i o n and v i b r a t i o n a l bonding has been s t u d i e d ( 3 8 , 3 9 ) . From a l l o f t h e s e i n v e s t i g a t i o n s the f o l l o w i n g r e s u l t s emerged: The d i s s o c i a t i o n energy f o r IHI I+HI ( o r HLH+H+LH i n g e n e r a l ) has a c o n t r i b u t i o n from v i b r a t i o n a l bonding which a p p a r e n t l y does not depend v e r y much on t h e d e t a i l s o f t h e e l e c t r o n i c p o t e n t i a l s u r f a c e , but i s m a i n l y determined by t h e k i n e m a t i c p r o p e r t i e s o f the system. T h i s c o n t r i b u t i o n , due t o v a r i a t i o n s o f t h e a n t i s y m m e t r i c s t r e t c h i n g and t h e bending z e r o p o i n t e n e r g i e s , s t a b i l i z e s IHI by a p p r o x i m a t e l y 6 k J / m o l . Normal mode f r e q u e n c i e s o f t h e a d i a b a t i c m o l e c u l e a r e q u i t e d i f f e r e n t from f r e q u e n c i e s o f t h e a s y m p t o t i c d i a t o m s . The IHI s t r e t c h i n g frequency s h o u l d be s m a l l e r than t h e c o r r e s p o n d i n g v i b r a t i o n a l frequency o f t h e 1^ m o l e c u l e . The bending frequency o f IHI i s l a r g e r than t h e symmetric s t r e t c h i n g f r e q u e n c y , and p r o b a b l y o n l y a few bending l e v e l s a r e bound. The a n t i s y m m e t r i c s t r e t c h o f IHI i s p r e d i c t e d t o possess t h e l a r g e s t f r e q u e n c y . One observes d e s t a b i l i s a t i o n upon d e u t e r a t i o n ( t h e v i b r a t i o n a l bonding c o n t r i b u t i o n d e c r e a s e s t o about 3 k J / m o l ) , and t h i s i s o t o p e e f f e c t s h o u l d be o b s e r v a b l e i n temperature d e p e n d e n t - s p e c t r o s c o p i c s t u d i e s . E x p e r i m e n t a l E v i d e n c e f o r V i b r a t i o n a l Bonded
Molecules
The u l t i m a t e q u e s t i o n r e g a r d i n g v i b r a t i o n a l bonding i s - can i t be observed e x p e r i m e n t a l l y ? At t h i s p o i n t we do n o t have a d e f i n i t e answer, but t h e f o l l o w i n g d i s c u s s i o n s h o u l d c o n v i n c e t h e r e a d e r t h a t i t i s w o r t h w h i l e t o pursue t h e p r o b l e m . The s p e c t r o s c o p i c p r e d i c t i o n s and e s t i m a t e s g i v e n a t t h e end o f t h e p r e v i o u s s e c t i o n show t h a t i n g e n e r a l a v i b r a t i o n a l l y bonded m o l e c u l e has p r o p e r t i e s s t r i k i n g l y d i f f e r e n t from those o f van d e r Waals m o l e c u l e s o r h y d r o gen bonded m o l e c u l e s ( w h i l e h a v i n g about t h e same b i n d i n g e n e r g y ) . J u s t t h e s e p r o p e r t i e s s h o u l d a l l o w an e x p e r i m e n t a l v e r i f i c a t i o n . For example, i n t h e p a s t decade P i m e n t e l (67), Noble ( 6 7 , 6 9 ) , A u l t (69) and o t h e r s (70) have s t u d i e d i n some d e t a i l t h e p r o p e r t i e s
Truhlar; Resonances ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
19.
Vibrationally Bonded Molecules
ROMELT AND POLLAK
Figure 9. A quasiperiodic orbit of the 3D IHI system projected on the c o l l i n e a r plane. Potential contours are given at 0.1 eV i n t e r v a l s , the dashed contour i s at 0.1 eV r e l a t i v e to the bottom of the asymptotic IHI well. The coordinates p , p are massscaled; for their d e f i n i t i o n , see Ref. (33). Reproduced with permission from Ref. 33. Copyright 1983, North Holland.. A
f
-290-
E/kJmol"
1
-300
-310-1 3.25
, 3.50
/
0
r„/&
i 3.75
1-
L0Q
Figure 10. Potential energy for the symmetric IHI and IDI stretching mode U as a sum of the repulsive c o l l i n e a r minimum potential energy 9 . plus the zero-point energy of the a n t i symmetric stretching mode plus that of the degenerate bending mode. The resulting IHI energy E (IHI), as well as the HI ground state energy E (IH) are given as a dashed and dotted l i n e , respectively. ?he position of the saddle point i s indicated by The asymmetric stretch, symmetric stretch and bend l e v e l zero point energies are denoted as 1/2 ti u) , 1/2 ~h u) and "tioj respectively. Reproduced with permission from Ref. 34. North Holland.
Copyright 1983,
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371
RESONANCES
372
of molecules l i k e YHX~(with X,Y = F,CI,Br,I) isolated i n matrices. In Table V we compare the spectroscopically determined normal mode frequencies of IHI and IDI~ as measured by Noble and the theoretical predictions based on our computations on the LEPS A surface. Of course the potential energy surface for IHI i s different from the IHI surface assumed i n our studies. But as noted many properties of such molecules are rather insensitive to the d e t a i l s of the potential energy surface. Not only the isotope effect of the experi mental investigation i s reproduced ( c f . Table V), but even the absolute frequencies are i n reasonable agreement. A l l t h i s suggests that i t i s worthwhile to take a closer experimental look at the IHI (and IDI) systems i n matrices. Table V.
Spectroscopic constants of three-dimensional IHI and IHI".
System
Reference
v (cm" )
IHI" IHI IDI IDI HI DI
Noble (68) Manz et a l . (38) Noble (68) Manz et a l . (38) Herzberg (71) Herzberg (71)
120 137 124 140
1
1
1
v (cm" ) 2
?
280 7
211
v^(cm
)
682 878 470 594 2308 1631
Another p o s s i b i l i t y i s given by gas-phase studies using super sonic beams. V i b r a t i o n a l l y bonded molecules are bound as t i g h t l y as van der Waals molecules; there i s no reason why they should not be observable with the same techniques (72,73). So far the case that has been analyzed i n d e t a i l has one bond breakup and a formation of a single bond; this represents a process i n which b a s i c a l l y only one frequency becomes weaker i n the i n t e r action region. In more complex systems more than one force constant may become weaker and the e f f e c t could be larger. To conclude: At present vibrational bonding provides a challenge to experimentalists - can they verify i t s existence? Acknowledgments
This work i s a result of a close collaboration of the authors with Drs. J. Manz and R. Meyer. We would l i k e to thank them for many stimulating discussions. E.P. would l i k e to thank Prof. W.H. M i l l e r and J.R. i s indebted to Dr. M.S. Child for their kind h o s p i t a l i t y during the writing of t h i s review. We would l i k e to thank Prof. D.G. Truhlar, Dr. S.K. Gray and Dr. E.L. Sibert for their c r i t i c a l comments on t h i s manuscript. The kind support of Prof. S.D. Peyerimhoff i s also g r a t e f u l l y acknowledged. This work was p a r t i a l l y supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Chemical Sciences Division of the U.S. Depart ment (Contract No.: DE-AC03-76SF00098) and by the Fonds der Deutschen Chemischen Industrie.
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19. ROMELT AND POLLAK
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373
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RECEIVED
June 11, 1984
Truhlar; Resonances ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
