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7/26/2019 Cyclopropane - Propene Isomerism
1/7
:5' ~ ~, .'
LS V I R
3 O c tober 1997
C h e m i c a l P h y s i c s L e t te r s 2 77 ( 1 9 9 7 ) 2 5 7 - 2 6 3
C H E M I C L
P H Y S I C S
L E T T E R S
t h e o r e t i c a l s t u d y o n t h e i s o m e r i z a t i o n o f c y c l o p r o p a n e t o
p r o p e n e w i t h a b i n i t i o a n d D F T m e t h o d s
K a n g - N i a n F a n a , Z h e n - H u a L i a W e n - N i n g W a n g a H s in g - H u a H u a n g
W e i H u a n g b
a Department of Chemistry Fudan University Shanghai 200433 China
b Departm ent of Chemistry. National U niversity of Singapore Singapore 1 19260
b
Rec e ived 4 A pr i l 1997; in f ina l f o rm 31 J u ly 1997
Ab stra ct
The isomerizat ion of cyclopropane to propene had b een s tudied by a b ini tio pos t-H F and DFT methods. S ingle-point
energy calculat ions a t UMP4, UCCSD(T), UQ CISD (T) and UB ecke3LY P levels were carded out on the UM P2/ 6-3 1G * *
fully optimized structures. The reaction heat and activation energies for the whole isomerization reaction and the structural
isomerization o f the C2-sy mm etry trimethylene intermediate were evaluated. The correlation m ethods em ploye d release an
accuracy orde r o f UBecke3LYP > UQ CIS D( T)- - (U CC S D (T)> UMP 4 > UMP 2. UBecke3LYP i s found to be the bes t
method to reproduce th e experimental results. The spin contamination prob lem can be solved by using projected MP
energies. 1997 Elsevier Science B.V.
1 I n t r o d u c t i o n
I s o m e r i z a t i o n o f c y c l o p r o p a n e t o p r o p e n e w a s
f i rs t s tu d i e d b y C h a m b e r s a n d K i s t i a k o w s k y i n 1 9 3 4
[ 1 ] . S i n c e t h e n t h e s t e r e o m u t a t i o n a n d t h e w h o l e
s t ru c t ur a l i s o m e r i z a t i o n o f c y c l o p r o p a n e ( S c h e m e 1 )
h a s b e c o m e t h e s u b j e c t s o f d o z e n s o f e x p e r i m e n t a l
a n d t h e o r e t i c a l s t u d i e s . M a n y e x p e r i m e n t a l [ 2 - 4 ] a n d
t h e o r e t i c a l [ 5 - 7 ] s t u d i e s c o n c e n t r a t e d m a i n l y o n t h e
t h e r m a l s t e r e o m u t a t i o n s o f n o n - o r i s o t o p i c a l l y l a -
b e l e d c y c l o p r o p a n e , a n d s i n g l e t t r i m e t h y l e n e s a s t h e
i n t e r m e d i a t e s o f t h e s t e r e o m u t a t i o n w e r e w e l l e s t a b -
l i s h e d . O t h e r e x p e r i m e n t s m a i n l y f o c u s e d o n t h e
i s o m e r i z a ti o n o f c y c l o p r o p a n e t o p r o p e n e a n d t h e
k i n e ti c b e h a v i o r s o f t h e w h o l e i s o m e r i z a ti o n r e a c t i o n
h a d b e e n w e l l s t u d i e d b y v a r i o u s m e t h o d s [ 1 , 8 - 1 0 ] .
U n f o r t u n a t e l y t h e t h e o r e t i c a l s t u d i e s o n t h e w h o l e
s t r u c t u r a l i s o m e r i z a t i o n o f c y c l o p r o p a n e t o p r o p e n e
w e r e f a r l e s s s a t i s f a c t o r y . I n 1 9 7 6 , J u g K a r l p e r -
f o r m e d a n I N D O ( S I N D O ) s t u d y o n t h e w h o l e i s o -
m e r i z a t i o n r e a c t i o n [ 1 1 ] . T h e a c t i v a t i o n e n e r g y o b -
t a i n e d f r o m t h e i r s t u d i e s w a s a b o u t 2 5 l e s s t h a n t h e
e x p e r i m e n t a l v a l u e . I n 1 9 8 8 V a l k o , S i m o n a n d V a n
C u o n g s t u d i e d t h e s t r u c t u r a l i s o m e r i z a t i o n o f
t r i m e t h y l e n e s w i t h a n e x t e n d e d b o n d - e n e r g y a n d
b o n d - o r d e r m e t h o d ( E B E B O ) [ 1 2 ] . T h e a c t i v a t i o n
e n e r g y t h e y o b t a i n e d w a s 3 8 . 9 k J / m o l . R e c e n t l y ,
D o u b l e d a y s t u d ie d t h e l i fe t i m e o f t h e t r i m e h t y l e n e s
w i t h R R K M t h e o r y [ 13 ]. T h e a c t iv a t i o n e n e r g y o f t h e
s t r u c t u r a l i s o m e r i z a t i o n o f t r i m e t h y l e n e t o p r o p e n e
h e o b t a i n e d w a s 2 8 . 9 k J / m o l a n d w a s b e l i e v e d t o b e
a l it t le l a r g e r th a n t h e v a l u e p r e d i c t e d b y W a a g e a n d
Rabinovi tch in 1973 [14] .
A l t h o u g h t h e g e o m e t r i c a l i s o m e r i z a ti o n o f c y c l o -
p r o p a n e w e r e w e l l s t u d i e d b y a f o r e m e n t i o n e d t h e o -
r e t i c a l m e t h o d s , a n d t h e s t r u c t u r a l i s o m e r i z a t i o n o f
0 0 0 9 - 2 6 1 4 / 9 7 / $ 1 7 . 0 0 1 9 97 E l s e v i e r S c i e n c e B . V . A l l r i g h t s r e s e r v e d .
P I I S 0 0 0 9 - 2 6 1 4 ( 9 7 ) 0 0 9 0 5 - 6
7/26/2019 Cyclopropane - Propene Isomerism
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258
K.-N. Fan et al. / Chemical Physics Letters 277 1997) 257-263
R ~
//C~ opemuing Stmctm-al
C C ~ - C H 2 C H 2 C H z ~ C I -I ~C H C H 3
e y e l o p r o p a n e R 0 ) tr im e t h y le n e p r o p e n e P )
Scheme 1.
t r i m e t h y l e n e b i r a d ic a l t o p r o p e n e w a s a l s o w e l l s t u d -
i e d b y D o u b l e d a y [ 1 3 ] , t h e w h o l e s t r u c t u r a l i s o m e r -
i z a ri o n o f c y l o p r o p a n e t o p r o p e n e h a s n o t b e e n s t ud -
i ed by any ab in i t io me thods . S ing le t b i rad ica l s a re
i m p o r t a n t i n t e r m e d i a t e s i n v o l v e d i n m o s t t h e r m a l
d e c o m p o s i t i o n r e a c t i o n s , w h e r e e l e c t r o n c o r r e l a t i o n
e n e r g y p l a y s a v e r y i m p o r t a n t r o l e. S i n c e m o s t c o r r e -
l a t i o n m e t h o d s a r e v e r y t i m e c o n s u m i n g , w e h a v e t o
b a l a n c e b e t w e e n a c c u r a c y a n d t i m e t o f i n d a n a p p r o -
pr i a t e too l in dea l ing wi th b i rad ica l s . Addi t iona l ly
H - s h i f t in g r e a c t i o n i s a ls o a v e r y c o m m o n r e a c t i o n i n
o r g a n i c c h e m i s t r y . P r o b l e m s e n c o u n t e r e d h e r e a r e
p r o t o t y p i c a l a n d t h e y p r o v i d e a n i n t e r e s ti n g a n d c h a l -
l e n g i n g t e s t g r o u n d f o r q u a n t u m c h e m i c a l r e s e a r c h e s.
2 . C a l c u l a t i o n m e t h o d o l o g y
A l l c a l c u l a t i o n s w e r e c a r r i e d o u t b y u s i n g t h e
G a u s s ia n 9 2 / D F T p a c k a g e o f a b in i ti o pr o g r a m s
[15] w i th the 6-3 1G * * bas i s s e t . Ge om et ry op t imiza -
d o n s w e r e p e r f o r m e d a t t h e U M P 2 l e v e l w i t h a l l
f o r c e c o n s t a n t s c a l c u l a te d b y u s i n g a n a l y t i c a l m e t h o d .
A l t h o u g h i t i s w e l l k n o w n t h a t M P 2 i s s u p e r i o r t o
I - IF , he re UI- IF me thod was a l so engaged for op t i -
m i z a t i o n t o b e c o m p a r e d w i t h t h e U M P 2 r e s u l t s . A l l
t r a n si t io n s t a te s t r u c tu r e s w e r e f u r t h e r c o n f i r m e d b y
i n t ri n s ic r e a c t i o n c o o r d i n a t e a R C ) c a l c u la t io n s . F o u r
t y p e s o f c o r r e l a ti o n m e t h o d s , U M P 4 , U C C S D T ) ,
U Q C I S D T ) a n d U B e c k e 3 L Y P w e r e u se d f o r s in g l e
p o i n t e n e r g y c a l c u l a t i o n o n t h e U M P 2 o p t i m i z e d
s tructures .
A n a l y t i c a l l y e v a l u a t e d f r e q u e n c i e s a t t h e U M P 2
l e v e l w e r e s c a l e d b y a f a c t o r o f 0 . 9 3 [ 1 6] . I n o r d e r t o
o b t a i n t h e a c t i v a t i o n e n e r g y , t h e r m a l e n e r g i e s o f
s p e c i e s w e r e c a l c u l a t e d a d o p t i n g i d e a l g a s a p p r o x i -
m a r i o n o f s t a ti s ti c a l t h e r m o d y n a m i c s . C a r e f u l a n a l y -
s e s w e r e p e r f o r m e d o n t h e c a l c u l a t e d f r e q u e n c i e s o f
t h o s e s p e c i e s a n d t h e l o w e r f r e q u e n c i e s r e l a t e d t o
i n t r a m o l e c u l a r r o t a ti o n s w e r e t r e a t e d a s f r e e r o t a t io n s
i n t h e c a l c u l a ti o n o f t h e t h e r m a l e n e r g i e s a b o v e r o o m
t e m p e r a t u r e .
3 . R e s u l t s a n d d i s c u s s i o n
3.1. Geometries
Four t rans i t ion s t ruc tures and two in t e rmedia t e s
w e r e l o c a t e d o n t h e p o t e n t i a l e n e r g y s u r f a c e P E S ) .
T h e c o m p l e t e d i a g r a m o f t h e w h o l e r e a c t io n i s i l lu s -
t r a te d i n S c h e m e 2 . I R C c a l c u l a ti o n s s h o w t h a t R 1 i s
the in t e rmedia t e tha t undergoes s t ruc tura l i somer iza -
t i o n to p r o p e n e P ) . T h e s t e r e o m u t a t i o n o f R 1 a n d
R2 demons t ra t ed he re i s i den t i ca l w i th tha t i n Refs .
[6 ,13] . But the t rans i t ion s t ruc ture of the 1 ,3 H-sh i f t
o f t r i m e t h y l e n e t o f o r m e t h y l c a r b e n e o p t i m i z e d b y
D o u b l e d a y w a s n o t o b t a i n e d b y u s . A s w i ll b e s h o w n
l a te r , T S 2 , R 2 a n d T S 3 m a k e n o c o n t r i b u t i o n t o t h e
r e a c t i o n r a t e c o n s t a n t o f t h e w h o l e r e a c t i o n o r t h e
s t ruc tura l i somer iza t ion of R1 and wi l l no t be d i s -
c u s s e d h e r e . G e o m e t r i e s o f c y c l o p r o p a n e , T S 1 , R 1 ,
T S 4 a n d p r o p e n e o p t i m i z e d a t t h e U M P 2 a n d U H F
leve l s a re shown in F ig . 1 .
U H F , U M P 2 a n d C A S S C F o p t i m i z e d s t r u c t u r e s
a re ve ry s imi l a r [13] . I t i s s een f rom F ig . 1 tha t CH
b o n d s o p t i m i z e d a t th e U H F l e v e l a r e a l w a y s s h o r t e r
t h a n t h o s e o p t i m i z e d a t t h e U M P 2 l e v e l . B u t t h i s
s i m p l e r e l a t i o n s h i p d o e s n o t a l w a y s h o l d f o r C C
b o n d s . A s s h o w n i n F i g . 1 , C C b o n d s i n c y c l o -
p r o p a n e , C 2 C 3 b o n d i n T S 4 a n d C i C 2 b o n d i n
p r o p e n e o p t i m i z e d a t t h e U H F l e v e l a r e a l l s h o r t e r
t h a n t h o s e a t t h e U M P 2 l e v e l . B u t o t h e r C C b o n d s
opt imized a t t he UI- IF l eve l a re longer . Pu lay e t a l .
[ 1 7 ] h a d p r o p o s e d a s e t o f e m p i r i c a l c o r r e c t io n p a -
r a m e t e r s f o r t h e t h e o r e t i c a l H F b o n d l e n g t h s . C o r -
r e c ti n g U H F o p t i m i z e d C H o r C C b o n d l e n g t h w i th
t h e s e p a r a m e t e r s , w e i m m e d i a t e l y g e t t h e b o n d
l e n g t h s w h i c h i s v e r y c l o s e t o o u r U M P 2 r e s u l t s .
T h e r e f o r e , d e s p i t e th e l a c k o f e x p e r i m e n t a l g e o m e t r y
d a t a o f t h e s h o r t - l iv i n g s p e c ie s , t h e U M P 2 o p t i m i z e d
s t ruc tures should be ve ry c lose to the rea l s i tua t ion .
T h i s i s t h e r e a s o n w h y w e u s e U M P 2 a s t h e o p t i -
m i z a t i o n m e t h o d .
TS4 CL)
R 0 . R I C D - P C s )
T S 2 C s ~ / T S3 C , )
~ l t 2 C s )
Scheme 2.
7/26/2019 Cyclopropane - Propene Isomerism
3/7
K.-N. Fan et al. / Chemical Physics Letters 277 1997) 257-263
259
1 0 7 6 T 1 .0 75 ~ 1 . 0 9 0 C C C 1 12 4 0
1 1 2 . 6 0
1 511w r 5o
1 . 0 7 8 ~ 1 . 4 9 4
1 . 4 9 7 U H F / 6 - 3 1 G * * / I - [ O 3 C 5 0.8 9 / H b C C C - 13 2 .6 9
1 . 5 0 2 U M P 2 / 6 - 3 1 G * * 4 5.5 5 - 14 0 .4 3
C y c l o p r o p a n e R 0 ) I ~ h ) T S 1 C 2 )
1 . 0 9 2 ~ L C C C 1 1 2. 84 H 4C 2 1 . 2 3 9 _ _ H 4
~ . 0 9 8 ~ . .1 3 . 1 1 . 0 9 8 1 1 3 . I6 _ 1 . 1 8 6 ~ 1 .4 35
I - l B l ~ a m a l b0 7 ~ ' ~ a ~ [ ~ ' 4 5 0 3 ' ~ ~ ~ 1 . 4 9 7
/ ~ C U C 3 5.8 9 / H b ( Z I2 - 1 6 0 .8 9 1 . 0 7 9 1 A S 1 ~ 1 . 0 7 6
36.90 -15~85
RI C2)
1.076 I -~ 1 .087
1 . 0 7 8 ' ~ C 1 I - ~ . U 2 ~ /
1 . 0 8 2 - - 1 . 4 9 8 ~ i5 - -
C I C 2 1 .3 1 8 L C C C 1 2 5 .3 0
1 . 3 3 7 1 2 4 . 4 5
Propene C s )
C 1 C 2 1 .4 1 3 / C C C 1 2 5 .7 4
1 . 4 3 0 1 2 6 . 3 6
I-I3C2 1.08 0
1 0 8 5
H 5 C 3 1 . 0 8 4
1 . 0 8 9
TS4 CI)
Fig. 1. Geometriesand impo rtant geom etryparameters for the reactant (R0), produ ct (P), one intermediate R) and two transition structures
(TS1 and T S4 respectively).
3.2. Energy profiles
T h e c a l c u l a t e d t o t a l e n e r g i e s , z e r o - p o i n t v i b r a -
t i o n a l e n e r g i e s a n d r e l a t i v e e n e r g i e s t o R 0 f o r a l l
s p e c i e s w i t h s i x d i f f e r e n t m e t h o d s a r e l i s te d i n T a b l e
1 . T h e P E S o f t h e t f i m e t h y l e n e i s v e r y f l a t [ 6, 1 3] .
It
i s s e e n f r o m T a b l e 1 t h a t T S 1 a n d R 1 a r e v e r y c l o s e
i n e n e r g y a n d t h e r e l a ti v e e n e r g y o f T S 1 o r R 1 t o R 0
e v a l u a t e d a t t h e I - I F l e v e l i s m u c h l e ss t h a n t h a t
c a l c u l a t e d b y o t h e r c o r r e l a t i o n m e t h o d s . T h i s c a n b e
e x p l a i n e d b y t h e f o l l o w i n g f a c t s. S i n c e t w o e l e c t r o n s
b e t w e e n t w o b o n d i n g c a r b o n a t o m s i n R 0 a r e d i -
v i d e d a n d l o c a t e d o n t w o r e m o t e c a r b o n a t o m s i n
T S 1 o r R 1 , t h a t p a r t o f c o r r e l a t i o n e n e r g y i n v o l v i n g
t h e s e t w o e l e c tr o n s in R 0 a n d T S 1 o r R 1 m a y b e l o n g
t o t w o d i f f e r e n t t y p e s . I n R 0 i t is d y n a m i c c o r r e l a -
t ion , whi l e in TS1 and R1 i t i s ma in ly s t a t i c . Not i c -
i n g t h at t h e S C F w a v e f u n c t i o n s o f R 0 a r e R H F a n d
c o r r e c t n o d y n a m i c c o r r e l a t i o n w h i l e t h o s e o f T S 1
a n d R 1 a r e r e a l U H F a n d c o r r e c t s ta t ic c o r r e l a t io n t o
s o m e e x t e n t , s o c o m p a r e d w i t h c o r r e l a t i o n m e t h o d s
w h i c h c o n s i d e r b o t h d y n a m i c i n R 0 a n d s t a t i c c o r r e -
l a t io n i n T S 1 o r R 1 , H F m e t h o d g i v e s l o w e r r e la t i v e
e n e r g y d i f f e r e n c e f o r T S 1 o r R 1 t o R 0 . S i n c e T S 1
a n d R 1 a r e v e r y c l o s e i n e n e r g y , e l e c t r o n i c c o r r e l a -
t i o n a n d Z P E c o r r e c t i o n s m u s t h a v e a s u b s t a n t i a l
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2 6 0 K . - N . F a n e t a l . / C h e m i c a l P h y s i c s L e t te r s 2 7 7 1 9 9 7 ) 2 5 7 - 2 6 3
T a b l e 1
T o t a l e n e r g i e s i n h a r tr e e ) , z e r o - p o i n t v i b r a t io n a l e n e r g i e s Z P E ) i n h a r tr e e a n d s c a l e d b y a fa c t o r o f 0 .9 3 ) , r e l a ti v e e n e r g i e s i n k J / m o l )
a n d S 2 ) f o r a l l s p e c i e s
S p e c i e s c y c i o p r o p a n e R 0 ) T S I R I T S 4 p r o p e n e P )
M e t h o d s D 3 h ) C 2 ) C 2 ) C I ) C s )
U H F - 1 1 7 . 0 6 8 9 4 2 - 1 1 7 . 0 0 8 6 8 0 - 1 1 7 . 0 0 8 9 9 9 - 1 1 6 . 9 5 8 1 7 4 - 1 1 7 . 0 8 1 0 2 7
0 . 0 1 3 6 . 6 1 5 8 . 2 ) a 1 3 7 . 0 1 5 7 . 4 ) 2 6 6 . 8 2 9 0 . 8 ) - 3 7 . 5 - 3 1 . 7 )
U M P 2 - 1 1 7 . 4 9 7 1 4 2 - 1 1 7 . 3 8 8 3 2 6 - 1 1 7 . 3 8 8 4 1 9 - 1 1 7 . 3 5 5 1 1 5 - 1 1 7 . 5 0 3 9 1 4
0 . 0 2 6 4 . 1 2 8 5 . 7 ) 2 6 5 . 1 2 8 5 . 4 ) 3 4 8 . 9 3 7 2 . 9 ) - 2 3 . 5 - 1 7 . 8)
U M P 4 - 1 1 7 . 5 4 5 9 4 5 - 1 1 7 . 4 4 2 4 7 0 - 1 1 7 . 4 4 2 4 8 0 - 1 1 7 . 4 0 9 3 2 2 - 1 1 7 . 5 5 6 4 8 4
0 . 0 2 5 0 . 1 2 7 1 . 7 ) 2 5 1 . 3 2 7 1 . 6 ) 3 3 4 . 7 3 5 8 . 7 ) - 3 3 . 4 - 2 7 . 7 )
U C C S D T ) - 1 1 7 . 5 4 7 1 0 2 - 1 1 7 . 4 4 5 6 8 0 - 1 1 7 . 4 4 5 5 7 5 - 1 1 7 . 4 2 2 9 8 2 - 1 1 7 . 5 5 8 4 6 0
0 . 0 2 4 4 . 7 2 6 6 . 3 ) 2 4 6 . 2 2 6 6 . 6 ) 3 0 1 . 9 3 2 5 . 9 ) - 3 5 . 6 - 2 9 . 8 )
U Q C I S D T ) - I 1 7 . 5 4 7 2 7 5 - 1 1 7 . 4 4 5 8 8 9 - 1 1 7 . 4 4 5 7 7 4 - 1 1 7 . 4 2 3 0 1 2 - 1 1 7 . 5 5 8 7 1 0
0 . 0 2 4 4 . 6 2 6 6 . 2 ) 2 4 6 .1 2 6 6 . 5 ) 3 0 2 . 2 3 2 6 . 2 ) - 3 5 . 8 - 3 0 . 0 )
U B e c k e 3 L Y P - 1 1 7 . 9 0 4 2 0 8 - 1 1 7 . 8 0 7 7 1 1 - 1 1 7 . 8 0 7 2 3 0 - 1 1 7 . 7 9 2 1 2 0 - 1 1 7 . 9 1 6 4 2 7
0 . 0 2 3 1 . 7 2 5 3 . 4 ) 2 3 4 . 2 2 5 4 . 6 ) 2 7 0 . 3 2 9 4 . 3 ) - 3 7 . 8 - 3 2 .1 )
Z P E 0 . 0 7 8 3 7 9 0 . 0 7 0 1 5 0 0 . 0 7 0 6 2 4 0 . 0 6 9 2 3 2 0 . 0 7 6 1 8 8
S 2 ) b 0 . 0 1 . 0 1 3 0 . 1 7 2 ) d 1 . 0 1 8 0 . 1 7 2 ) 0 . 8 5 8 0 . 2 0 8 ) 0 . 0
S 2 ) c 0 . 0 0 . 9 6 0 0 . 0 4 6 ) 0 . 9 8 7 0 . 0 4 9 ) 0 . 1 5 3 0 . 0 0 1 ) 0 . 0
a V a l u e s i n p a r e n t h e s e s a r e n o t c o r r e c t e d b y Z P E .
b U H F , U M P 2 , U M P 4 , U C C S D T ) a n d U Q C I S D T ) a ll h a v e th e sa m e S C F r e f e r e n c e w a v e
c U B e c k e 3 L Y P .
o V a l u e s i n p a r e n t h e s e s a r e t h o s e a f t e r h i g h s p i n c o n t a m i n a t i o n i s a n n i h i l a t e d .
f u n c t i o n s a n d s o h a v e t h e s a m e S 2 )
i n f l u e n c e o n t h e i r r e l a t i v e p o s i t i o n s . T a b l e 1 s h o w s
t h a t a t h i g h e r c o r r e l a t i o n l e v e l s s u c h a s U C C S D T ) ,
U Q C I S D T ) a n d U B e c k e 3 L Y P , T S 1 e v e n l ie s
s l i g h t l y b e l o w R I . A f t e r Z P E c o r r e c t i o n T S1 i s
s l igh t ly lower than R1 a t a l l s ix l eve l s .
3.2 .1 . Reac t ion hea t
T h e w h o l e r e a c t i o n i s a n i s o g y r i c p r o c e s s s o e v e n
t h e s i m p l e s t m o d e l S C F ) w h i c h c o n s i d e r s n o c o r r e -
l a t i o n c a n r e p r o d u c e t h e e x p e r i m e n t a l v a l u e c o r r e c t l y
[ 1 8 ] . T h e r e a c t i o n h e a t p r e d i c t e d a t t h e U H F l e v e l i s
3 6 . 8 k J / m o l a n d f a l l s i n t h e a v e r a g e e r r o r r a n g e
o f e x p e r im e n t s . A m o n g t h o s e c o rr e l at i o n m e t h o d s
t h e b e s t o n e i s U B e c k e 3 L Y P , a n d t h e w o r s t i s M P 2 .
U B e c k e 3 L Y P p re d i ct s a v a lu e o f - 3 7 . 2 k J / m o l .
T h e a c c u r a c y o r d e r o f t h e s e s i x m e t h o d s i s
U B e c k e 3 L Y P = U H F > U Q C I S D T ) > U C C S D T )
> U M P 4 > U M P 2 f r o m t h e b e s t t o t h e w o r s t) .
H o w e v e r a l l m e t h o d s u n d e r e s t i m a t e t h e e n e r g y o u t -
c o m e o f t h i s r e a c t i o n .
3.2 .2 . Ac t i va t ion en ergy fo r the s t ruc tura l i somer i za-
t ion o f t rime thy lene R1 to propene E , , s ))
A c c o r d i n g t o t h e c o n v e n t i o n a l t r a n s i t i o n s t a t e t h e -
o r y t h e a c t i v a t i o n e n e r g y o f a e l e m e n t a r y r e a c t i o n i s
e q u a l t o t h e e n e r g y b a r r i e r o f t h e r e a c t io n p l u s R T
[ 19 ]. A c t i v a t i o n e n e r g i e s i n T a b l e 2 a r e c a l c u l a t e d a t
8 0 0 K , s i nc e m o s t e x p e r i m e n t s w e r e p e r f o r m e d n e a r
t h i s t e m p e r a t u r e . I n c o n t r a s t t o t h e c a l c u l a t i o n o f t h e
r e a c t i o n h e a t , t h e a c t i v a t i o n e n e r g y p r e d i c t e d a t U H F
l e v e l i s 1 3 0 . 7 k J / m o l a n d i s t o o h i g h . U M P 2 a n d
U M P 4 h a v e th e sa m e ac c u r ac y - - 8 4. 5 k J / m o l ) a n d
t h is al s o h o l d s f o r U C C S D T ) a n d U Q C I S D T ) =
5 6 . 8 k J / m o l ) , b u t b o t h p a i rs o v e r e s t i m a t e t h e a c ti v a -
t i o n e n e r g y n e e d e d . U B e c k e 3 L Y P m e t h o d p r e d i c t s
a n a c t i v a t i o n e n e r g y o f 3 6 . 9 k J / m o l , e x c e l l e n t l y
r e p r o d u c i n g t h e v a l u e o f 3 9 . 7 k J / m o l t h a t B e n s o n
e s t i m a t e d i n 1 9 6 1 [2 0 ]. B u t t h i s v a l u e i s m u c h h i g h e r
t h a n t h a t o f W a a g e a n d R a b i n o v i t c h [ 14 ]. W a a g e a n d
R a b i n o v i t c h d e d u c e d a v a l u e o f 3 .7 k c a l / m o l f o r t h e
c r i t i ca l ene r gy d i f f e r en ce A E0* o f the kg to kp see
Ref . [ 14] f o r d e f in i t ion) . I f we take A E0* as
A E T S 4 - T S 1 ) I n f a ct , t h e se t w o i t e m s a r e n o t
e q u a l , s i n c e i n o r d e r t o g e t A E T S 4 - T S 1 ) f r o m a n
e x p e r i m e n t w e m u s t e m p l o y s o m e a s s u m p t i o n s r e -
g a r d i n g k i n e t i c i s o t o p e e f f e c t s a n d a r e a s o n a b l e
m e c h a n i s m a b o u t t h e i s o m e r i z a t i o n r e a c t i o n [ 6 ] . ) a n d
t a k e t h e e n e r g y d i f f e r e n c e b e t w e e n T S 1 a n d R 1 t o b e
1 k c a l / m o l [ 6 ,7 ,1 3 ] a n d w e w i ll d e d u c e a v a l u e o f
a b o u t 4 .7 k c a l / m o l 1 9 .6 k J / m o l ) + R T f o r t h e
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K. -N. F an e t a l . / Che m ic a l P hy s ic s Le t te r s 277 1997) 257- 26 3
261
T a b l e 2
A c t i v a t i o n e n e r g i e s o f s t ru c t u r a l i s o m e r i z a t i o n E a s ) ) a n d t h e w h o l e H - s h i f t i n g r e a ct i o n E a w ) ) a n d t h e e n t h a l p y c h a n g e o f t h e w h o l e
r e ac t io n at 2 9 8 . 1 5 K k J / m o l )
E n e rg i es U H F U M P 2 U M P 4 U C C S D T ) U Q C I S D T ) U B e e k e 3 L Y P e x p .
A H r a - 3 6 . 8 - 2 2 . 8 - 3 2 . 7 - 3 3 . 9 - 3 5 . 1 - 3 7 . 2 - 4 0 . 5 d
E a S b 130 . 7 84 . 7 84 . 3 56 . 6 57 . 0 36 . 9 26 . 3 e
E a w ) 273 . 1 355 . 2 341 . 0 308 . 2 308 . 5 276 . 6 267 . 5 4 - 3 . 6 [ 1 3 ]
276.0 4- 2.1 [9]
2 7 2 - 2 7 6 [ 1 ]
a A E P - R 0 ) a t 2 9 8 .1 5 K . b A E T S 4 - R 1 ) +
R T
a t 80 0 K . c A E T S 4 - R 0 ) +
R T
a t 8 0 0 K . d C a l c u l a te d f r o m t h e h e a t s o f c o m b u s t i o n o f
c y c l o p r o p a n e a n d p r o p e n e f r o m C R C H a n d b o o k o f C h e m i s t r y a n d P h y s i c s 6 9 t h e d i ti o n , 1 9 8 8 - 1 9 8 9 ) . T h i s v a l u e i s d e d u c e d f r o m r e f. [1 4 ]
A E 0* + 1 . 0 k c a l / m o l + R T ) .
a c t i v a ti o n e n e r g y o f R 1 t o p r o p e n e . N e v e r t h e l e s s w e
w o u l d c a l l W a a g e a n d R a b i n o v i t c h v a l u e e x p e r i -
m e n t a l v a l u e f o r c o m p a r i so n .
3.2.3. Activation energ y fo r the whole reaction
(E~(w))
F o r t h e w h o l e i s o m e r i z a t i o n p r o c e s s ( S c h e m e 3 ) ,
e n g a g i n g s t e a d y s t a t e a p p r o x i m a t i o n w e g e t
d[ ] k l
= X
d t
k 4
k _ 1
/ _~ + k 3 + k 2 / / q ) / q
k _ 2 + k 3 + ( k _ 2 / k _ l ) k _ 3 + ( k _ 2 + k _ 3 / k _ l ) k 4
X [ R 0 ] . 1 )
A c c o r d i n g t o c o n v e n t i o n a l t ra n s i ti o n s t a t e t h e o r y , t h e
r a t e c o n s t a n t o f a u n i m o l e c u l a r r e a c t i o n i s g i v e n b y
k B T ( A G * )
k = h e x p ~ - ~ , ( 2 )
w h e r e k n i s B o l tz m a n n c o n s t a n t a n d A G * = G ( T S )
- G ( R e a c t a n t ) [ 19 ]. S o w e h a v e
k 2
k a = ~--~, k _ 3 ,
I - - I
s ince A G ( k 2 ) * - A G ( k l ) * + A G ( k 3 ) * is e q u al t o
A G ( k _ 2 ) * - A G ( k _ i )* + A G ( k - 3 ) * N o t i c in g th a t
e n e r g y h a r r i e r f o r th e s t r u c t u r e i s o m e r i z a t i o n is m u c h
kl k
R O ~ . ~ / 1 - ~ P
S c h e m e 3 .
h i g h e r t h a n t h o s e f o r t h e s t e r e o m u t a t i o n o f t h e
t r ime th ylene in t e rmedia t e s [13] a s we l l a s those for
the r ing c lose reac t ion an d k_ 1 , k_ 2 , k3 , k_ 3 a re a l l
o f t h e s a m e m a g n i t u d e o r d e r [ 6 ,1 3 ] , w e h a v e
k - 2 + k 3
- k 4 < < k _ 2 + k 3
k _ l
T h e n f r o m E q s . ( 1 ) a n d ( 2 ) w e g e t
kl k T e x p ( A G * ( T S 4 - R 1 ) )
kw h l e ~ k4 E l = h -
( k n T / h ) e xp { - [ A G * ( T S l - R 0 ) / R r ]
( k n T / h ) e xp { - [ A G * ( T S 1 - R I ) / R T ]
k a T ( A G * ( T S 4 - R 0 ) )
- h e x p - ~ - ~ . ( 3 )
Eq . (3 ) has a s imi l a r fo rm as Eq . (2 ) and can be
w r i t t e n i n A r r h e n i u s f o r m . S o t h e a c t i v a t i o n e n e r g y
f o r t h e w h o l e r e a c t i o n i s e q u a l t o A E ( T S 4 - R 0 ) +
R T. The
ac t iva t ion ene rg ies ca l cu la t ed wi th d i f fe ren t
m e t h o d s a r e l i s t e d i n T a b l e 2 . A t 8 0 0 K ,
U B e c k e 3 L Y P m e t h o d y i e ld s a v al u e o f 2 7 6 .6 k J / m o l
w h i c h a g r e e s v e r y w e l l w i t h t h e e x p e r i m e n t a l v a l u e
o f = 2 7 6 k J / m o l . T h e U H F m e t h o d p r e d ic t s a v al u e
o f 27 3 .1 k J / m o l . U Q C I S D ( T ) a n d U C C S D ( T ) a g a in
h a v e t h e sa m e a c c u r a c y b u t t h e i r v a l u e , - - 3 0 8
kJ /mol , i s s t i l l t oo h igh a s tha t fo r s t ruc tura l i somer-
i z a t i o n . A g a i n w e h a v e a n a c c u r a c y o r d e r o f
U B e c k e 3 L Y P ~ U H F > U Q C I S D ( T ) -- U C C S D ( T )
> U M P 4 > U M P 2 ( f r o m t h e b e s t t o t h e w o r s t ) .
3.3. Com parison o f different methods
F a c t o r s a f f e c t i n g t h e a c c u r a c y o f a n a b i n i t i o
m e t h o d g e n e r a l l y i n c l u d e t h e c o m p l e t e n e s s o f t h e
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262 K. -N. F an e t a l . / Che mic a l P hy s ic s Le t te r s 277 1997) 257 -26 3
b a s i s s e t a n d t h e l e v e l o f t h e c o r r e l a t i o n . S i n c e o u r
s t u d y i n v o l v e s s i n g l e t b i r a d i c a l s a n d e m p l o y s s p i n
u n r e s t r i c t e d m e t h o d s , t h e r e i s a t h i r d f a c t o r w h i c h i s
t h e s p i n c o n t a m i n a t i o n i n t h e r e f e r e n c e w a v e f u n c -
t i o n s o f t h e o p e n s h e l l s p e c i e s . I t s e e m s t h a t 6 - 3 1 G * *
b a s i s s e t i s a d e q u a t e h e r e s i n c e R e f s . [ 7 , 2 1 ] h a v e
s h o w n t h a t b a si s s e ts l ar g e r t h a n 6 - 3 1 G * * d o n o t
g r e a t l y c h a n g e t h e r e l a t i v e e n e r g y o f R 1 o r R 2 t o R 0 .
S o o n l y t w o f a c t o r s , i . e . t h e c o r r e l a t i o n l e v e l a n d t h e
s p i n c o n t a m i n a t i o n a r e w o r k i n g h e r e. A s t h a t p r e v i -
o u s l y m e n t i o n e d , s i n c e t h e w h o l e r e a c t i o n i s a n
i s o g y r i c o n e , t h e H F r e a c t i o n h e a t i s v e r y c l o s e t o t h e
e x p e r i m e n t a l v a l u e . T h e a c c u r a c y o r d e r o f t h e o t h e r
f o u r m e t h o d s o n t h e r e a c t i o n h e a t c a l c u l a t i o n s r e -
f l e e t s t h e i r c o r r e l a t i o n l e v e l . I t i s c l e a r t h a t M P 2 i s
n o t a de q u at e. M P 4 , C C S D ( T ) a n d U Q C I S D ( T ) a r e
b e t t e r t h a n M P 2 b u t i t s e e m s t h a t t h e i r c o n s i d e r a t i o n
o f c o r r e l a t i o n i s s t i l l n o t a d e q u a t e . T h u s B e c k e 3 L Y P
i s t h e m o s t a c c u r a t e m e t h o d h e r e .
T h e c o m p a r i s o n o f t h e c a l c u l a t i o n o f t h e t w o
a c t i v at i o n e n e r g ie s i s s o m e w h a t c o m p l i c a t e d s i n c e i t
i s m i x e d w i t h t h e e f f e c t s o f b o t h c o r r e l a t i o n a n d s p i n
c o n t a m i n a ti o n . C o m p u t e d v a l u e s o f ( S 2 ) a r e a l s o
g i v e n i n T a b l e 1 . F r o m T a b l e 1 w e c a n s e e t h at ( S 2 )
o f t h e t h r e e o p e n s h e l l s p e c i e s is l a r g e a n d t h e s p i n
c o n t a m i n a t i o n h e r e i s s e v e r e . T h e c o n t a m i n a t i o n i n
t h e U B e c k e 3 L Y P w a v e f u n c t i o n s o f th e t h r e e s p e ci e s
i s n o t a s s e v e r e a s o t h e r m e t h o d s , e s p e c i a l l y f o r T S 4 .
L o w e r s p i n c o n t a m i n a t i o n a n d h i g h c o r r e l at i o n l e v el
a r e t h e t w o f a c t o r s t h a t m a k e t h e U B e c k e 3 L Y P
m e t h o d m o r e a c c u r a t e t h a n t h e o t h e r c o r r e l a t i o n
m e t h o d s . I t w o u l d n o t b e s u r p r i s i n g h a d t h e U H F
m e t h o d g i v e n t h e w o r s t r e s u l t s , b e c a u s e i t c o n s i d e r s
c o r r e la t i o n t o s o m e e x t e n t i n i t s re a l U H F w a v e f u n c -
t i o n s o f t h e T S 1 , R 1 a n d T S 4 s p e c i e s b u t c o n s i d e r s
n o n e i n it s R H F w a v e f u n c t i o n s o f t h e R 0 . B u t i t is
s u r p r i s i n g t h a t t h e a c t i v a t i o n e n e r g y f o r t h e w h o l e
c a l c u l a t e d r e a c t i o n i s s o c l o s e t o t h e e x p e r i m e n t a l
v a l u e . T a b l e 3 p r e s e n t s t h e r e s u l t s c a l c u l a t e d f r o m
H F , M P 2 a n d M P 4 e n e r g i e s w i t h s p i n p r o j e c t i o n .
P U I - IF , P M P 2 a n d P M P 4 a l l l o w e r th e t w o a c t i v a t i o n
e n e r g i e s , E ~ (s ) a n d E a ( w ) r e m a r k a b l y . A s f o r th e
r e s u l t s o f th e P U H F , E ~ (s ) i s c l o s e r to t h e e x p e r i m e n -
ta l va lue than Ea(W), bu t Ea (W) i s fa r f rom the
e x p e r i m e n t a l v a l u e . E a (S ) c a l c u l a t e d f r o m t h e P M P 2
a n d P M P 4 e n e r g i e s ar e m u c h c l o s e r t o t h e e x p e r i -
m e n t a l v a l u e t h a n t h a t c a l c u l at e d f r o m u n p r o j e c t e d
M P e n e r g i e s . E a ( w ) c a l c u l a t e d f r o m P M P 2 e n e r g i e s
Table 3
Activation energies for the structural and who le isomerization
reaction obtained from PUH F, PMP2 and PMP4 energies a (in
kJ /m ol )
Activation energy PUHF PMP2 PMP4
ea(s) 83.5 39.7 42.5
Ea(w) 210.7 294.4 283.2
a Spin contamination s annihilated from s + 1 to s +4.
i s g r e a t l y i m p r o v e d b u t i s s t i ll n o t a c c u r a t e e n o u g h .
T h i s a l s o s h o w s t h a t M P 2 m a y n o t b e s u f f i c i e n t f o r
c o r r e l a t i o n c o r r e c t io n . T h e a c c u r a c y o f t h e P M P 4 i s
v e r y h i g h . S o w e c o n c l u d e t h a t t h e s p i n c o n t a m i n a -
t i o n p r o b l e m i n t h e M P w a v e f u n c t i o n s c a n b e s o l v e d
b y s p i n p r o j e c t i o n m e t h o d . T h i s i s i n a g r e e m e n t w i t h
t h e r e s u l t s o f r e f. [ 22 ] a n d [ 2 3] . S i n c e U C C S D ( T )
a n d U Q C I S I X T ) c a l c u l a t i o n s a r e s t i l l p e r f o r m e d o n
t h e s p i n c o n t a m i n a t e d H F w a v e f u n c t i o n s , i t i s e x -
p e c t e d t h at i f w e u s e p r o j e c t e d U C C S D ( T ) a n d U Q -
C I S D ( T ) e n e r g i e s o r M R - Q C I S D ( T ) a n d M R -
C C S D ( T ) , t h e d i s c r e p a n c y b e t w e e n t h e ir c a l c u la t i o n s
a n d t h e e x p e r i m e n t a l r e s u l t s s h o u l d b e h e a l e d . T h i s
i s tr u e f o r E a ( S ) s i nc e D o u b l e d a y ' s C A S - C I S D
m e t h o d g o t a b e t t e r v a l u e f o r i t t h a n u s . H o w e v e r ,
t h i s f u n c t i o n h a s n o t y e t b e e n i n c o r p o r a t e d i n t o
G a u s s i a n a n d w e a r e n o t a b l e to p e r f o r m t h e s e t y p e s
o f c a l c u l a t i o n s a t p r e s e n t .
A l t h o u g h p r o j e c t e d M P 4 e n e r g i e s p r o v i d e v e r y
g o o d r e s u l t s w i t h r e g a r d t o E a ( s ) a n d E ~ ( w ) , a n d s o
w i l l th e p r o j e c te d C C S D ( T ) a n d Q C I S D ( T ) e n e r g i es ,
t h e s e t h r e e m e t h o d s a r e m o r e t i m e c o n s u m i n g t h a n
t h e U B e c k e 3 L Y P m e t h o d . S o w e r e c o m m e n d
U B e c k e 3 L Y P a s t h e f i r s t c h o i c e w h e r e s i n g l e t b i r a d -
i c a l s a r e i n v o l v e d .
4 C o n c l u s i o n s
U B e c k e 3 L Y P i s t h e b e s t m e t h o d w h i c h r e p r o -
d u c e s e x p e r i m e n t a l r e a c t i o n h e a t a n d a c t i v a t i o n e n e r -
g i e s , b o t h f o r t h e s t r u c t u r a l i s o m e r i z a t i o n a n d t h e
w h o l e i s o m e r i z a t i o n r e a c t io n , a c c u r a t e l y . I t c a n w e l l
d e s c r i b e t h e b e h a v i o r s o f t h o s e o p e n s h e l l s p e c i e s
b e c a u s e i t s w a v e f u n c t i o n s c o n t a i n l e s s s p i n c o n t a m i -
n a t i o n . I t i s a l s o t h e l e a s t s o u r c e r e q u i r i n g c o r r e l a t i o n
m e t h o d e m p l o y e d i n t h i s s t u d y . A c t i v a t i o n e n e r g i e s
c a lc u l at e d at U M P 2 , U M P 4 , U C C S D ( T ) a n d U Q -
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K . - N . F a n e t a l. / C h e m i c a l P h y si c s L et t e rs 2 7 7 1 9 9 7 ) 2 5 7 - 2 6 3 263
CISD T) levels are higher t han the corresponding
experimental values. This is mainly because of the
spin contamination in their HF reference wavefunc-
tions. For U MP2, this problem can be p artly over-
come by annihilating the high spin contamination in
its reference wavefunctions. Values derived from
projected MP4 energies agree with the experimental
results very well .
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