Direct Ab Initio Dynamics Methods for Calculating Thermal Rates of

Chapter 6

Downloaded by OHIO STATE UNIV LIBRARIES on March 16, 2013 | http://pubs.acs.org Publication Date: May 5, 1996 | doi: 10.1021/bk-1996-0629.ch006

Direct Ab Initio Dynamics Methods for Calculating Thermal Rates of Polyatomic Reactions Thanh N. Truong, Wendell T. Duncan, and Robert L. Bell Department of Chemistry, University of Utah, Salt Lake City, UT 84112

W e present a d i r e c t ab initio d y n a m i c s m e t h o d o l o g y for c a l c u l a t i n g t h e r m a l rate constants f r o m d e n s i t y f u n c t i o n a l t h e o r y ( D F T ) . D y n a m i c a l t h e o r y is b a s e d o n a v a r i a t i o n a l t r a n s i t i o n state theory plus m u l t i - d i m e n s i o n a l semi-classical tunneling a p p r o x i m a t i o n s . P o t e n t i a l e n e r g y surface i n f o r m a t i o n is c a l c u l a t e d f r o m a c o m b i n e d D F T / a b initio M o l e c u l a r O r b i t a l t h e o r y a p p r o a c h . W e also present a p p l i c a t i o n s of this m e t h o d to p r e d i c t i n g d e t a i l e d d y n a m i c s of a h y d r o g e n a b s t r a c t i o n r e a c t i o n a n d p r o t o n transfer in a m o d e l b i o l o g i c a l s y s t e m to i l l u s t r a t e its v e r s i l i t y , accuracy a n d prospects for m o l e c u l a r m o d e l i n g of reactive d y n a m i c s of p o l y a t o m i c c h e m i c a l reactions.

I. I N T R O D U C T I O N T h e p r e d i c t i o n of r e a c t i o n rates f r o m first p r i n c i p l e s a l l o w s o n e to m a k e d i r e c t c o m p a r i s o n s b e t w e e n t h e o r y a n d e x p e r i m e n t a n d hence to d e d u c e r e a c t i o n m e c h a n i s m s o n the m o l e c u l a r l e v e l . F o r t h i s r e a s o n , it has b e e n a m a j o r g o a l of theoretical c h e m i s t r y . H o w e v e r , it also has b e e n a c h a l l e n g e p a r t i c u l a r l y for p o l y a t o m i c reactions for the f o l l o w i n g reasons. T h e c o n v e n t i o n a l a p p r o a c h of reactive d y n a m i c a l c a l c u l a t i o n s u s i n g e i t h e r the f u l l q u a n t a l d y n a m i c s , classical or s e m i c l a s s i c a l trajectory m e t h o d , or v a r i a t i o n a l t r a n s i t i o n state t h e o r y ( V T S T ) r e q u i r e s the a v a i l a b i l i t y of a n accurate a n a l y t i c a l p o t e n t i a l e n e r g y f u n c t i o n ( P E F ) . " D e v e l o p i n g s u c h a p o t e n t i a l e n e r g y f u n c t i o n is not a t r i v i a l task a n d is a m a j o r obstacle for the 1

3

0097-6156/96/0629-0085$15.00/0 © 19% American Chemical Society In Chemical Applications of Density-Functional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.


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C H E M I C A L APPLICATIONS O F DENSITY-FUNCTIONAL T H E O R Y

d y n a m i c a l s t u d y of a n e w r e a c t i o n despite the steady i m p r o v e m e n t i n c o m p u t e r speed. T h i s is because; i) the e x p l i c i t f u n c t i o n a l f o r m for a p o t e n t i a l e n e r g y f u n c t i o n is s o m e w h a t a r b i t r a r y a n d m o s t l y d e p e n d s o n the i n v e s t i g a t o r ' s i n t u i t i o n , ii) f i t t i n g this f u n c t i o n a l f o r m to a set of ab initio e n e r g y p o i n t s a n d a n y a v a i l a b l e e x p e r i m e n t a l d a t a is t e d i o u s a n d yet does n o t guarantee c o n v e r g e n c e or correct g l o b a l t o p o l o g y , i i i ) the n u m b e r of e n e r g y p o i n t s n e e d e d g r o w s g e o m e t r i c a l l y w i t h the n u m b e r of g e o m e t r i c a l i n t e r n a l c o o r d i n a t e s . A s the s y s t e m size increases, this task b e c o m e s m u c h m o r e c o m p l e x i f it c a n s t i l l be a c c o m p l i s h e d at a l l . T h u s , d e v e l o p m e n t s of n e w m e t h o d o l o g i e s for s t u d y i n g d y n a m i c s , k i n e t i c s a n d m e c h a n i s m s of l a r g e p o l y a t o m i c reactions are of great interest. D i r e c t d y n a m i c s m e t h o d s , " i n c l u d i n g those b e i n g d e v e l o p e d i n o u r l a b ' ' " offer a v i a b l e alternative for s t u d y i n g c h e m i c a l reactions of c o m p l e x systems. I n the direct d y n a m i c s a p p r o a c h , a l l r e q u i r e d energies a n d forces for e a c h g e o m e t r y that is i m p o r t a n t for e v a l u a t i n g d y n a m i c a l p r o p e r t i e s are o b t a i n e d d i r e c t l y f r o m electronic s t r u c t u r e c a l c u l a t i o n s rather t h a n f r o m e m p i r i c a l a n a l y t i c a l force fields. O u r e a r l i e r c o n t r i b u t i o n s to t h i s area i n c l u d e the d e v e l o p m e n t of t w o different m e t h o d o l o g i e s for c a l c u l a t i n g t h e r m a l rate constants a n d related p r o p e r t i e s . O n e a p p r o a c h is to e s t i m a t e t h e r m a l rate constants a n d t u n n e l i n g c o n t r i b u t i o n s u s i n g the i n t e r p o l a t e d V a r i a t i o n a l T r a n s i t i o n State T h e o r y ( V T S T ) w h i c h has p r o v e n u s e f u l w h e n a v a i l a b l e accurate ab initio e l e c t r o n i c s t r u c t u r e i n f o r m a t i o n is l i m i t e d . T h e other a p p r o a c h is to use a s e m i e m p i r i c a l m o l e c u l a r o r b i t a l H a m i l t o n i a n at the N e g l e c t of D i a t o m i c D i f f e r e n t i a l O v e r l a p ( N D D O ) l e v e l as a f i t t i n g f u n c t i o n i n w h i c h p a r a m e t e r s h a v e b e e n r e a d j u s t e d to a c c u r a t e l y represent a c t i v a t i o n b a r r i e r s . F u l l V T S T calculations w i t h multidimensional semiclassical tunneling a p p r o x i m a t i o n s t h e n c a n be c a r r i e d o u t u s i n g this N D D O H a m i l t o n i a n w i t h specific r e a c t i o n parameters. B o t h of these a p p r o a c h e s h a v e b e e n s u c c e s s f u l l y a p p l i e d to v a r i o u s c h e m i c a l r e a c t i o n s . ' - ' ' ' ' ' ' H o w e v e r , m a n y d i f f i c u l t i e s persist. F o r instance, i n the f o r m e r i n t e r p o l a t e d V T S T a p p r o a c h , it is d i f f i c u l t to correlate v i b r a t i o n a l m o d e s i n the t r a n s i t i o n state r e g i o n to reactant a n d p r o d u c t a s y m p t o t e s w h e n m o d e crossings o c c u r as they often d o . I n the later, it m a y p r o v e to be d i f f i c u l t to adjust the o r i g i n a l N D D O p a r a m e t e r s to accurately d e s c r i b e the t r a n s i t i o n state r e g i o n if the o r i g i n a l N D D O p o t e n t i a l e n e r g y surface differs s i g n i f i c a n t l y f r o m the reference accurate ab initio surface. Recent development i n combining both approaches ' has s o m e p r o m i s e . 3

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3 7

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1 7

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2 4

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4 1

I n this c h a p t e r , w e w i l l focus o n l y o n o u r recent c o n t r i b u t i o n s to the d e v e l o p m e n t of d i r e c t ab initio d y n a m i c s m e t h o d s i n w h i c h n o e x p e r i m e n t a l d a t a other t h a n p h y s i c a l constants w e r e u s e d for c a l c u l a t i n g t h e r m a l rate constants of gas-phase p o l y a t o m i c reactions. T h e d y n a m i c a l

In Chemical Applications of Density-Functional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.


6. TRUONG ET AL.

Direct Ab Initio Dynamics Methods

87

m e t h o d is b a s e d o n f u l l V T S T t h e o r y p l u s m u l t i - d i m e n s i o n a l s e m i c l a s s i c a l t u n n e l i n g corrections. T h e m a i n difference w i t h o u r p r e v i o u s w o r k , h o w e v e r , is i n the w a y the p o t e n t i a l e n e r g y surface i n f o r m a t i o n is o b t a i n e d . I n o u r n e w a p p r o a c h , d e s i r e d quantities are o b t a i n e d d i r e c t l y f r o m ab initio e l e c t r o n i c s t r u c t u r e c a l c u l a t i o n s , t h u s n o f i t t i n g is i n v o l v e d . F o r q u a n t i t a t i v e p r e d i c t i o n s of k i n e t i c p r o p e r t i e s , the p o t e n t i a l e n e r g y surface m u s t be a d e q u a t e l y accurate. If s u c h i n f o r m a t i o n is to be c a l c u l a t e d f r o m a s u f f i c i e n t l y accurate l e v e l of ab initio m o l e c u l a r o r b i t a l t h e o r y , the c o m p u t a t i o n a l d e m a n d c a n be s u b s t a n t i a l . I n this case, these m e t h o d s are o n l y u s e f u l for s m a l l systems, a n d t h u s they stop short of o u r g o a l . T o a l l e v i a t e this d i f f i c u l t y , w e h a v e i n t r o d u c e d t w o n e w m e t h o d o l o g i e s w h i c h c a n be u s e d i n c o m b i n a t i o n . O n e is a f o c u s i n g t e c h n i q u e o r a n a d a p t i v e g r i d m e t h o d i n w h i c h m o r e c o m p u t a t i o n a l resources are spent o n r e g i o n s that are m o s t s e n s i t i v e to the d y n a m i c s a n d less resources e l s e w h e r e . T h i s a l l o w s o n e to o b t a i n a n o p t i m a l a c c u r a c y w i t h a m i n i m u m c o m p u t a t i o n a l cost at a g i v e n l e v e l of theory. T h e other is the use of a c o m p u t a t i o n a l l y less d e m a n d i n g electronic s t r u c t u r e m e t h o d , d e n s i t y f u n c t i o n a l t h e o r y ( D F T ) , for the c o m p u t a t i o n a l l y m o s t e x p e n s i v e step i n o b t a i n i n g the p o t e n t i a l e n e r g y i n f o r m a t i o n r e q u i r e d for rate c a l c u l a t i o n s . T h e later raises a n i n t e r e s t i n g a n d i m p o r t a n t q u e s t i o n . W o u l d D F T m e t h o d s be s u f f i c i e n t l y accurate for this p u r p o s e ? R a p i d developments i n new functionals have significantly i m p r o v e d the a c c u r a c y of D F T m e t h o d s i n the past few years. P r e v i o u s l y , m o s t a p p l i c a t i o n s of D F T w e r e for p r e d i c t i n g p r o p e r t i e s of stable equilibrium structures. " R e c e n t l y , m o r e s t u d i e s ' o n the a c c u r a c y of D F T m e t h o d s for t r a n s i t i o n state p r o p e r t i e s h a v e b e e n r e p o r t e d . A g e n e r a l c o n c l u s i o n is that for t r a n s i t i o n state p r o p e r t i e s the n o n - l o c a l D F T a n d the h y b r i d D F T m e t h o d s i n w h i c h a p o r t i o n of H a r t r e e - F o c k e x c h a n g e is i n c l u d e d y i e l d results of c o m p a r a b l e accuracy to the s e c o n d - o r d e r M o l l e r Plesset ( M P 2 ) m e t h o d b u t at a m u c h cheaper c o m p u t a t i o n a l cost, p a r t i c u l a r y for large systems. T h u s , the c o m p u t a t i o n a l a d v a n t a g e of D F T w o u l d a l l o w a p p l i c a t i o n of the direct ab initio d y n a m i c s m e t h o d to s t u d y i n g reactions i n v o l v i n g l a r g e r p o l y a t o m i c m o l e c u l e s . 4 4

4 6

4 7

5 3

T o i l l u s t r a t e the a p p l i c a b i l i t y , accuracy a n d v e r s a t i l i t y of this d i r e c t ab initio d y n a m i c s a p p r o a c h , w e present t w o different a p p l i c a t i o n s . O n e is the h y d r o g e n a b s t r a c t i o n C H 4 + H C H 3 + H 2 reaction. T h i s r e a c t i o n has s e r v e d as a p r o t o t y p e r e a c t i o n i n v o l v i n g p o l y a t o m i c m o l e c u l e s a n d has p l a y e d a n i m p o r t a n t role i n the t h e o r e t i c a l a n d e x p e r i m e n t a l d e v e l o p m e n t s of c h e m i c a l k i n e t i c s . I n a d d i t i o n , it has a n i n t r i n s i c i m p o r t a n c e to c o m b u s t i o n k i n e t i c s a n d is of f u n d a m e n t a l interest to o r g a n i c r e a c t i o n m e c h a n i s m s . F o r this r e a s o n , a m p l e e x p e r i m e n t a l rate d a t a is a v a i l a b l e for c o m p a r i s o n . A l s o this r e a c t i o n is s m a l l e n o u g h so that


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accurate ab initio M O c a l c u l a t i o n s c a n also be p e r f o r m e d to test the a c c u r a c y of D F T m e t h o d s .

T h e s e c o n d e x a m p l e is the p r o t o n transfer i n

f o r m a m i d i n e - w a t e r c o m p l e x w h i c h has i m p o r t a n t i m p l i c a t i o n s i n b i o l o g i c a l processes. D u e to the l i m i t t e d space, w e c a n o n l y focus o n the a c c u r a c y of the m e t h o d o l o g y

a n d not so m u c h o n the c h e m i s t r y of these

reactions. W e refer readers to o u r o r i g i n a l p a p e r s ' ' 5

3 4

3 5

for s u c h d i s c u s s i o n .

II. T H E O R Y


A . Variational transition state theory V a r i a t i o n a l t r a n s i t i o n state t h e o r y a n d m u l t i d i m e n s i o n a l semiclassical tunneling methods have been described i n detail elsewhere. ' 5 4

6 0

I n this chapter, w e o n l y c a p t u r e the essence of the t h e o r y

a n d the a p p r o x i m a t i o n s i n v o l v e d i n the a p p l i c a t i o n s p r e s e n t e d here. V T S T is b a s e d o n the i d e a that b y v a r y i n g the d i v i d i n g surface a l o n g the m i n i m u m e n e r g y p a t h ( M E P ) to m i n i m i z e the rate, one c a n m i n i m i z e the e r r o r d u e to " r e c r o s s i n g " trajectories.

T h e M E P is d e f i n e d as the steepest

descent p a t h f r o m the s a d d l e p o i n t to b o t h the reactant a n d p r o d u c t d i r e c t i o n s i n the m a s s - w e i g h t e d c a r t e s i a n c o o r d i n a t e s y s t e m .

The reaction

c o o r d i n a t e s is t h e n d e f i n e d as the distance a l o n g the M E P w i t h the o r i g i n l o c a t e d at the s a d d l e p o i n t a n d is p o s i t i v e o n the p r o d u c t s i d e a n d n e g a t i v e o n the reactant side. F o r a c a n o n i c a l e n s e m b l e at a g i v e n t e m p e r a t u r e T , the c a n o n i c a l v a r i a t i o n a l t h e o r y ( C V T ) rate constant for a b i m o l e c u l a r r e a c t i o n is g i v e n b y k (T) cvr

= m i n k (T,s) s

(1)

GT

where

k (j,s) GT

= ^ . Q ^ ^ fih Φ*(Γ) G

T

T

e

- p w )

( 2 )

I n these e q u a t i o n s , k ( T , s ) is the g e n e r a l i z e d t r a n s i t i o n state t h e o r y rate G T

constant at the d i v i d i n g surface w h i c h intersects the M E P at s a n d is o r t h o g o n a l to the M E P at the i n t e r s e c t i o n p o i n t , σ is the s y m m e t r y factor a c c o u n t i n g for the p o s s i b i l i t y of m o r e t h a n one s y m m e t r y - r e l a t e d r e a c t i o n p a t h a n d c a n be c a l c u l a t e d as the r a t i o of the p r o d u c t of the reactant r o t a t i o n a l s y m m e t r y n u m b e r s to the t r a n s i t i o n state one.

F o r e x a m p l e , the

r o t a t i o n a l s y m m e t r y n u m b e r s for C H 4 (Ta), C H 3 (D3h), H 2 (Dooh) a n d the H 3 C . H . . H g e n e r a l i z e d t r a n s i t i o n state ( C 3 ) are 12, 6, 2 a n d 3, r e s p e c t i v e l y . V

C o n s e q u e n t l y , σ e q u a l s 4 for b o t h the f o r w a r d a n d reverse d i r e c t i o n s of the


6. TRUONG ET AL.

89


C H 4 + H C H 3 + H 2 reaction, β is (koT)' w h e r e k is B o l t z m a n ' s constant a n d h is P l a n c k ' s constant. Φ ( Τ ) is the reactant p a r t i t i o n f u n c t i o n (per u n i t v o l u m e for b i m o l e c u l a r reactions). V ( s ) is the c l a s s i c a l p o t e n t i a l e n e r g y (also c a l l e d the B o r n - O p p e n h e i m e r p o t e n t i a l ) a l o n g the M E P w i t h its z e r o of e n e r g y at the reactants, a n d Q ( T , s ) is the i n t e r n a l p a r t i t i o n f u n c t i o n of the g e n e r a l i z e d t r a n s i t i o n state at s w i t h the l o c a l z e r o of e n e r g y at V ( s ) . Both Φ (Τ) and Q ( T , s ) partition f u n c t i o n s are a p p r o x i m a t e d as p r o d u c t s of electronic, v i b r a t i o n a l a n d r o t a t i o n a l p a r t i t i o n f u n c t i o n s . F o r the e l e c t r o n i c p a r t i t i o n f u n c t i o n , the g e n e r a l i z e d t r a n s i t i o n state electronic e x c i t a t i o n energies a n d degeneracies are a s s u m e d to be the same as at the t r a n s i t i o n state. F o r r o t a t i o n s , since the r o t a t i o n a l e n e r g y levels are g e n e r a l l y closely s p a c e d , r e p l a c i n g the q u a n t a l r o t a t i o n a l p a r t i t i o n f u n c t i o n s b y the classical ones y i e l d s v e r y little loss i n a c c u r a c y . F o r v i b r a t i o n s , i n the present s t u d y , the p a r t i t i o n f u n c t i o n s are c a l c u l a t e d q u a n t u m m e c h a n i c a l l y w i t h i n the f r a m e w o r k of the h a r m o n i c a p p r o x i m a t i o n . N o t e that a n h a r m o n i c i t y s o m e t i m e s c a n h a v e a noticeable effect o n r e a c t i o n rates p a r t i c u l a r l y at h i g h e r t e m p e r a t u r e s . H o w e v e r , it is not i n c l u d e d i n the a p p l i c a t i o n s p r e s e n t e d below. 1

B

Κ

M E P

G T

Κ


M E P

G T

T h e c a n o n i c a l v a r i a t i o n a l t r a n s i t i o n state t h e o r y d e s c r i b e d a b o v e y i e l d s the h y b r i d (i.e. classical r e a c t i o n p a t h m o t i o n w i t h v i b r a t i o n a l degrees of f r e e d o m q u a n t i z e d ) rate constants. F u r t h e r m o r e , if the g e n e r a l i z e d t r a n s i t i o n state is located at the s a d d l e p o i n t (s=0), eq. (2) r e d u c e s to c o n v e n t i o n a l t r a n s i t i o n state t h e o r y . T o i n c l u d e q u a n t a l effects for m o t i o n a l o n g the r e a c t i o n c o o r d i n a t e , w e m u l t i p l y the C V T rate constants b y a t r a n s m i s s i o n coefficient, κ (Τ). T h u s , the f i n a l q u a n t i z e d rate constant is k(T)

= κ

CVT/G

(T)k

CVT

(T)

(3)

B. M u l t i d i m e n s i o n a l Semiclassical T u n n e l i n g Methods T o calculate the t r a n s m i s s i o n coefficient, w e first a p p r o x i m a t e the effective p o t e n t i a l for t u n n e l i n g to be the v i b r a t i o n a l l y a d i a b a t i c g r o u n d state p o t e n t i a l c u r v e d e f i n e d b y (4) w h e r e €? (s)

denotes the z e r o - p o i n t e n e r g y i n v i b r a t i o n a l m o d e s

t

t r a n v e r s e to the M E P . T h e g r o u n d state t r a n s m i s s i o n coefficient, K

C V T / G ( x ^ i t h e n a p p r o x i m a t e d as the ratio of the t h e r m a l l y a v e r a g e d s


90


m u l t i d i m e n t i o n a l semiclassical ground-state transmission probability, P ( E ) , for r e a c t i o n i n the g r o u n d state to the t h e r m a l l y a v e r a g e d classical G

t r a n s m i s s i o n p r o b a b i l i t y for o n e - d i m e n s i o n a l scattering b y the g r o u n d state effective p o t e n t i a l V f O ) .

'

2 4

5 6

"

state for t e m p e r a t u r e Τ as s {T), E*(T),

G

Gvr

f

d e n o t e d as

is the q u a s i c l a s s i c a l g r o u n d - s t a t e t h r e s h o l d e n e r g y , t h e n the

t r a n s m i s s i o n coefficient κ ^


If w e d e n o t e the C V T t r a n s i t i o n

6 5

the v a l u e o f V [s (T)]

Gvr

ν τ

/ ( T ) c a n be expressed as G

\P (E)e

dE

G

K (T)

= ^

CVr,G

-

AbT

.

— \e

(5)

dE

/kbT

E.(T) T h e s e m i c l a s s i c a l t r a n s m i s s i o n p r o b a b i l i t y P ( E ) accounts for b o t h n o n c l a s s i c a l r e f l e c t i o n at energies a b o v e the q u a s i c l a s s i c a l t h r e s h o l d a n d also n o n c l a s s i c a l t r a n s m i s s i o n , i.e., t u n n e l i n g , at energies b e l o w that t h r e s h o l d . H o w e v e r , the B o l t z m a n n factor i n E q . (5) m a k e s t u n n e l i n g the far m o r e i m p o r t a n t c o n t r i b u t i o n . G

S e v e r a l a p p r o x i m a t i o n s for the s e m i c l a s s i c a l t r a n s m i s s i o n p r o b a b i l i t y P ( E ) are a v a i l a b l e , h o w e v e r , o n l y t w o , n a m e l y , the z e r o G

curvature

5 8

a n d the c e n t r i f u g a l - d o m i n a n t s m a l l - c u r v a t u r e s e m i c l a s s i c a l

adiabatic ground-state p r e s e n t e d here.

6 5

a p p r o x i m a t i o n s u s e d i n the present s t u d y are

F o r c o n v e n i e n c e , w e l a b e l t h e m as Z C T a n d S C T for the

z e r o - c u r v a t u r e t u n n e l i n g a n d s m a l l - c u r v a t u r e t u n n e l i n g cases, respectively.

Since the Z C T a p p r o x i m a t i o n is a s p e c i a l case o f the S C T

a p p r o x i m a t i o n , w e present o n l y the f o r m a l i s m for the S C T b e l o w . T h e S C T u s e d here is a g e n e r a l i z a t i o n o f the M a r c u s - C o l t r i n a p p r o x i m a t i o n i n w h i c h the t u n n e l i n g p a t h is d i s t o r t e d f r o m the M E P o u t to the c o n c a v e - s i d e v i b r a t i o n a l t u r n i n g p o i n t i n the d i r e c t i o n of the i n t e r n a l c e n t r i f u g a l force.

T h i s p h e n o m e n o n is c o m m o n l y refered to as

"corner c u t t i n g " . Instead o f d e f i n i n g the t u n n e l i n g p a t h e x p l i c i t l y , the c e n t r i f u g a l effect is i n c l u d e d b y r e p l a c i n g the r e d u c e d m a s s b y a n effective r e d u c e d m a s s , M ff(s), w h i c h is u s e d to evaluate i m a g i n a r y a c t i o n i n t e g r a l s e

a n d thereby t u n n e l i n g p r o b a b i l i t i e s . N o t e that i n the m a s s - w e i g h t e d c a r t e s i a n c o o r d i n a t e s y s t e m , the r e d u c e d mass μ is set e q u a l to 1 a m u . T h e g r o u n d - s t a t e t r a n s m i s s i o n p r o b a b i l i t y at e n e r g y Ε is


6. T R U O N G E T A L .

P (E)=

91


(6)

G

{\ + e-

}

2(KE)

w h e r e Θ(Ε) is the i m a g i n a r y a c t i o n i n t e g r a l e v a l u a t e d a l o n g the t u n n e l i n g path,


Θ(Ε)

= *?£

^ (s)\E-VÏ(s)\

ds

ejr

(7)

a n d w h e r e the i n t e g r a t i o n l i m i t s , s\ a n d s , are the r e a c t i o n - c o o r d i n a t e turning points defined by r

V [ {E)]=V [s {E)} G

a

G

Sl

a

= E

r

(8)

N o t e that the Z C T results c a n be o b t a i n e d b y setting μ ff(s) e q u a l to μ i n E q . e

(7).

T h e effect o f the r e a c t i o n - p a t h c u r v a t u r e is i n c l u d e d i n the effective

r e d u c e d mass

\XQH{S)

w h i c h is g i v e n b y

= μ χ min

μ ^ 0 )

< e

xp{-2«(5)-[«( )] +(%) } S

2

2

(9)

1 where â{s)

=

(10)

\K{s)t{s)\

T h e m a g n i t u d e o f the r e a c t i o n - p a t h c u r v a t u r e κ (s) is g i v e n b y Î3N-7 K(s)

=

\ /2 ]

2

Σ Κω]

(11)

V. m=l w h e r e the s u m m a t i o n is o v e r a l l 3 N - 7 g e n e r a l i z e d n o r m a l m o d e s a n d K (s) i s the r e a c t i o n - p a t h c u r v a t u r e c o m p o n e n t a l o n g m o d e m g i v e n m

by

6 6

K (s) m

and where L

=

m

-L F|vv|

(12)

T

m

2

is the t r a n s p o s e of the g e n e r a l i z e d n o r m a l m o d e

e i g e n v e c t o r of m o d e m , F is the force constant m a t r i x ( H e s s i a n m a t r i x ) ,


92


V V is the g r a d i e n t . F i n a l l y , t(s)

is the m a x i m u m c o n c a v e - s i d e v i b r a t i o n a l

d i s p l a c e m e n t a l o n g the c u r v a t u r e d i r e c t i o n for w h i c h there is n o t u n n e l i n g i n the v i b r a t i o n a l c o o r d i n a t e s . a p p r o x i m a t i o n , t(s)


t(s)

where w

=

m

f

\

W i t h i n the h a r m o n i c

is g i v e n b y

\ Σ [ * Μ ] *

2

Μ

13

is the g e n e r a l i z e d v i b r a t i o n a l frequency of m o d e m .

A s d e s c r i b e d a b o v e , i n o r d e r to carry o u t f u l l V T S T c a l c u l a t i o n s a n d m u l t i - d i m e n s i o n a l s e m i c l a s s i c a l z e r o - or s m a l l - c u r v a t u r e t u n n e l i n g corrections, geometries, energies, g r a d i e n t s a n d H e s s i a n s are n e e d e d at the s t a t i o n a r y p o i n t s a n d a l o n g the M E P . H e s s i a n c a l c u l a t i o n s a l o n g the M E P are c o m p u t a t i o n a l l y the m o s t e x p e n s i v e step. B e l o w w e describe t w o d i f f e r e n t m e t h o d o l o g i e s for m i n i m i z i n g the c o m p u t a t i o n a l cost of this step. C. Focusing technique T h e f o c u s i n g t e c h n i q u e w a s d e v e l o p e d to assure the c o n v e r g e n c e of the c a l c u l a t e d rate constants w i t h a m i n i m a l n u m b e r of H e s s i a n s r e q u i r e d . T h i s m e t h o d i n v o l v e s t w o separate steps. F i r s t , a p r e l i m i n a r y rate c a l c u l a t i o n w i t h a coarse H e s s i a n g r i d is c a r r i e d o u t to estimate r e g i o n s c o n t a i n i n g the t e m p e r a t u r e - d e p e n d e n t c a n o n i c a l t r a n s i t i o n states o r those h a v i n g l a r g e c u r v a t u r e w h e r e the "corner c u t t i n g " effect w o u l d be large. A finer H e s s i a n g r i d is t h e n u s e d for these r e g i o n s to i m p r o v e the a c c u r a c y of the c a l c u l a t e d C V T rate constants a n d the S C T t u n n e l i n g p r o b a b i l i t y . T h e t e c h n i q u e w i l l be i l l u s t r a t e d i n m o r e d e t a i l b e l o w .

D. E m p l o y i n g D F T methods N o n - l o c a l D F T m e t h o d s s u c h as the c o m b i n a t i o n of Becke's H a l f a n d - H a l f ( B H & H ) o r three p a r a m e t e r (B3) h y b r i d e x c h a n g e w i t h L e e Y a n g - P a r r ( L Y P ) c o r r e l a t i o n f u n c t i o n a l s can be u s e d to calculate the geometries a n d H e s s i a n s a l o n g the M E P . T h e h y b r i d B H & H f u n c t i o n a l as i m p l e m e n t e d i n the G 9 2 / D F T p r o g r a m consists of 5 0 % H a r t r e e - F o c k a n d 5 0 % Slater exchange c o n t r i b u t i o n . T h e D F T energies, h o w e v e r , are not a l w a y s s u f f i c i e n t l y accurate. I n this case, to o b t a i n m o r e a accurate p o t e n t i a l e n e r g y a l o n g the M E P , o n e c a n either p e r f o r m a series of s i n g l e p o i n t c a l c u l a t i o n s at a m o r e accurate l e v e l of t h e o r y w i t h a l a r g e r basis set 6 7

68

6 9

7 0




93

for selected p o i n t s a l o n g the M E P o r scale VMEP(S) b y a constant factor to


m a t c h a m o r e accurately c a l c u l a t e d classical b a r r i e r a n d r e a c t i o n e n e r g y . F r o m o u r e x p e r i e n c e , w e h a v e f o u n d that the B 3 L Y P m e t h o d is s l i g h t l y m o r e accurate t h a n the B H & H L Y P m e t h o d for c a l c u l a t i n g e q u i l i b r i u m s t r u c t u r e s . F o r f i n d i n g t r a n s i t i o n state s t r u c t u r e s , the B H & H L Y P m e t h o d has s h o w n to be m o r e reliable for o p e n - s h e l l s y t e m s . F o r c l o s e d - s h e l l s y s t e m s , o u r p r e l i m i n a r y results i n d i c a t e that b o t h B 3 L Y P a n d B H & H L Y P are adequate. I n practice, w e often c a r r y o u t accurate ab initio M O c a l c u l a t i o n s at the s t a t i o n a r y p o i n t s to check the a c c u r a c y of D F T m e t h o d s p r i o r to their use i n rate c a l c u l a t i o n s . III. A P P L I C A T I O N S A . C H + H C H + H 4

3

2

reaction

W e u s e d this r e a c t i o n as a test case w h e r e the r e a c t i o n v a l l e y (geometries, energies, g r a d i e n t s a n d H e s s i a n s a l o n g the M E P ) w a s c a l c u l a t e d w i t h b o t h the Q u a d r a t i c C o n f i g u r a t i o n I n t e r a c t i o n i n c l u d i n g a l l S i n g l e a n d D o u b l e excitations ( Q C I S D ) a n d the h y b r i d B H & H L Y P m e t h o d s . B o t h m e t h o d s use the same 6-311G(d,p) basis set. I n p a r t i c u l a r , w e h a v e p e r f o r m e d a n e w b e n c h m a r k c o n v e r g e d direct ab initio d y n a m i c s rate c a l c u l a t i o n s for this reaction. I n the n e w c a l c u l a t i o n s , the r e a c t i o n p a t h w a s c a l u c a t e d at the Q C I S D / 6 - 3 1 1 G ( d , p ) w i t h the step size of 0.01 a m u / b o h r w h i c h is a n o r d e r of m a g n i t u d e s m a l l e r t h a n i n o u r p r e v i o u s study. H e s s i a n g r i d s are also m u c h finer. F u r t h e r m o r e , i n s t e a d of s c a l i n g the p o t e n t i a l e n e r g y a l o n g the M E P as i n the p r e v i o u s s t u d y , s i n g l e p o i n t P M P 4 / 6 - 3 1 1 + G ( 2 d f , 2 p d ) c a l c u l a t i o n s w e r e p e r f o r m e d at the H e s s i a n g r i d s . 1

2

3 4

R e a c t i o n energies a n d b a r r i e r heights are l i s t e d i n T a b l e 1. T h e B H & H - L Y P classical barriers w e r e f o u n d to be too l o w b y 2.7 a n d 0.6 k c a l / m o l for the f o r w a r d a n d reverse reactions, r e s p e c t i v e l y , as c o m p a r e d to the C C S D ( T ) / c c - p V Q Z results. S i n g l e - p o i n t s p i n projected f o u r t h - o r d e r M o l l e r - P l e s s e t p e r t u r b a t i o n theory ( P M P 4 ) c a l c u l a t i o n s w i t h the l a r g e r 6311+G(2df,2pd) basis set at the B H & H - L Y P / 6 - 3 1 1 G ( d , p ) geometries b r i n g the differences i n the classical barriers to less t h a n 0.7 k c a l / m o l a n d also y i e l d the r e a c t i o n e n t h a l p y at 0 Κ to be -0.36 k c a l / m o l as c o m p a r e d to the C C S D ( T ) / c c - p V Q Z v a l u e of -0.3 k c a l / m o l a n d the e x p e r i m e n t a l v a l u e f r o m J A N A F t a b l e s of -0.02 k c a l / m o l . 74


94

CHEMICAL APPLICATIONS OF DENSITY-FUNCTIONAL T H E O R Y

T a b l e 1: H e a t o f r e a c t i o n a n d b a r r i e r h e i g h t s ( k c a l / m o l ) for the C H 4 + H 3

C H 3 4-H2 r e a c t i o n

Level B H & H L Y P / 6-31 l G ( d , d p )

ΔΕ

Δ// °

1.4

-1.9

0

11.2 (13.0)

12.6

a

(ell.l)* PMP4/6-311+G(2df,2pd)


//BH&HLYP/6-311G(d,p) QCISD/6-311G(d,p) PMP4/6-311+G(2df,2pd) //QCISD/6-311G(d,p) CCSD(T)/cc-pVQZ //CCSD(T)/cc-VQZ J3

C

Expt.

a

b

c

b

2.9

-0.4

14.6 (13.1)

11.7 (13.5)

2.5

-0.7

16.3 (14.8)

13.8 (15.5)

3.6

0.3

14.6 (13.0)

10.9 (12.7)

3.5

-0.3

15.3 (13.1)

11.8 (13.3)

2.8

-0.02

12.9 (11.8)

10.1 (11.9)

2.6

d

d

-1.3 -0.02e d

(13.3 ± 0 . 5 )

d

(14.6 ± 0 . 4 )

d

Zero-point energy corrected barriers are given in the parentheses. F r o m Ref. 71. From Ref. 72.

d

From Ref. 73.

e

From Ref. 74.

S i m i l a r results w e r e o b t a i n e d i f the Q C I S D geometries w e r e u s e d i n the P M P 4 c a l c u l a t i o n s except the c a l c u l a t e d r e a c t i o n e n t h a l p y is s l i g h t l y p o s i t i v e . U s i n g the B H & H - L Y P z e r o - p o i n t e n e r g y c o r r e c t i o n , the P M P 4 / / B H & H - L Y P z e r o - p o i n t e n e r g y corrected b a r r i e r s for b o t h f o r w a r d a n d reverse reactions are w i t h i n 0.2 k c a l / m o l o f the C C S D ( T ) / c c - p V Q Z v a l u e s a n d are also i n g o o d a g r e e m e n t w i t h e x p e r i m e n t a l d a t a .

7 3


6. TRUONG ET AL.

2.2


M I 11 j 11 11 j 1 1 1 1 ; I M I ( f ι 1 1 1 1 1 1 1 1 1 1 0

95

The geometries along the m i n i m u m e n e r g y p a t h c a l c u l a t e d b y b o t h the B H & H L Y P a n d Q C I S D levels are s h o w n i n F i g u r e 1. W e f o u n d that the B H & H - L Y P m e t h o d y i e l d s the active C H a n d H - H b o n d lengths a n d H - C - H angle as f u n c t i o n s o f the r e a c t i o n coordinate i n excellent a g r e e m e n t w i t h the Q C I S D results. M o r e s p e c i f i c a l l y , Figure 1 shows an u n n o t i c e a b l e difference i n the active b o n d lengths a n d a difference o f less t h a n 1 degree i n the H - C - H angle b e t w e e n the t w o m e t h o d s . a

a


a

0.6 * ' " * ' ' ' ' ''' '' -1.5 - 1 -0.5 0 0.5 1

1

1

1

s (amu

1/2

1

1

1

1

' ' ' ' ' ' go 1 1.5

1

1

1

bohr)

Figure 1: BH&HLYP (solid lines) and QCISD (dased lines) geometrical parameters along the MEP of the H + CH —>H +CH reaction

4

2

a

3

vs the reaction coordinate s.

16

Vc-DFT Vc-QCISD — VC-PMP4//QCISD -I - VC-PMP4//DFT ' ' •

-1.5

-1.0

-0.5

0.0

s (amu

1/2

1

•

1

• •

0.5

1

•

1

"

1.0

• •

1.5

bohr)

Figure 2: Classical potential energy curves { PMP4// QCISD (solid), PMP4//BH&HLYP (dashed), scaled BH&HLYP (dashed-dotted), scaled QCISD (dotted)}.

The g o o d agreement between P M P 4 / / Q C I S D and P M P 4 / / B H & H L Y P potential c u r v e s as s h o w n i n F i g . 2 indicates that the a c c u r a c y o f the p o t e n t i a l e n e r g y a l o n g the M E P c a n be i m p r o v e d b y carrying out single point PMP4/6-311+G(2df,2pd) c a l c u l a t i o n s at selected p o i n t s a l o n g the D F T M E P . W h e n such single point calculations o n the H e s s i a n g r i d s are computationally expensive, it is p o s s i b l e to scale the p o t e n t i a l e n e r g y a l o n g the M E P b y a constant factor to adjust the b a r r i e r h e i g h t t o m o r e accurate c a l c u l a t i o n s . A s s h o w n i n F i g . 2, w e s c a l e d the B H & H L Y P a n d Q C I S D p o t e n t i a l c u r v e s to best r e p r o d u c e b o t h the f o r w a r d


96


4~*5000 111 ι ι ; 11 11 ι ι ι 111 ι ι ι 1111 ι ι 11 I I I

Έ ο

:

:

a n d reverse classical b a r r i e r s c a l c u l a t e d at the C C S D ( T ) / c c p V Q Z level of theory by K r a k a et a l .

7 1

N o t e that b o t h

the scaled D F T a n d Q C I S D classical p o t e n t i a l c u r v e s h a v e a b o u t the s a m e w i d t h c o m p a r e d t o the


P M P 4 / / Q C I S D curves, t h o u g h the b a r r i e r h e i g h t s differ b y a b o u t 1 k c a l / m o l . I n g e n e r a l , s c a l i n g the r e a c t i o n p r o f i l e b y a constant to o b t a i n m o r e accurate b a r r i e r h e i g h t s does not guarantee to s (amu

1/2

i m p r o v e the shape as w e l l as

bohr)

Figure 3: BH&HLYP (solid lines) and QCISD (dashed lines) harmonic frequencies along the reaction coordinate s.

the a s y m t o t i c regions o f the reaction profile.

It is

possible, however, to a d d a small number of single point c a l c u l a t i o n s as d i s c u s s e d a b o v e a n d to i n t e r p o l a t e the energy corrections a l o n g the M E P . S u c h a p r o c e d u r e is n o w b e i n g tested i n o u r l a b . Generalized frequencies c a l c u l a t e d at the B H & H - L Y P / 6 - 3 1 1 G ( d , p ) level v e r s u s the r e a c t i o n

coordinate

are p l o t t e d i n F i g u r e 3 a l o n g w i t h the p r e v i o u s Q C I S D / 6 3 1 1 G ( d , p ) results. N o t e that excellent agreement

was

f o u n d b e t w e e n the B H & H - 9 η!• •• .ι. , . . ι . . . .ι. . . . ι . . . ,ι. . . . 0.5 1 1.5 2 2 . 5 3 3.5 1000/Τ (Κ) Figure 4: Arrhenius plot for the forward H + C H ^ reaction. Symbols are experimental data. Lines are the C V T / S C T results {PMP4//QCISD

(solid),

P M P 4 / / B H & H L Y P (dashed), scaled B H & H L Y P (dotted), scaled QCISD (dashed-dotted)}.

L Y P a n d Q C I S D results, t h o u g h the f o r m e r are s l i g h t l y larger b y a b o u t 3%. The A h r r e n i u s plots of the c a l c u l a t e d a n d experimental forward and reverse rate constants are


6. TRUONG ET AL.

97

s h o w n i n F i g u r e s 4 a n d 5, r e s p e c t i v e l y . N o t e that E our C V T / S C T - 1 3 P M P 4 / / B H & H - L Y P results \ -14for the f o r w a r d rate 3 O - 1 5 -constants are i n excellent Q) O a g r e e m e n t w i t h the E - 1 6 ; \ J e x p e r i m e n t a l d a t a for the ^ - 1 7 =temperature range from î 300-1500 Κ w i t h the largest 1 8 d e v i a t i o n factor o f 1.5 j o 3 -1 9 r c o m p a r e d t o the recent • Ζ recommended • 2 0 H.j.11 11 1111 1 1 1 1 11 Γι r 0.5 1 1.5 2 2.5 3.5 experimental values. 1000/T (K) Similar results were Figure 5: Arrhenius plot of the C H + H reaction. f o u n d for the 3 2 PMP4//QCISD Symbols are experimental data. Lines are the C V T / S C T results {PMP4//QCISD (solid), calculations. T h e PMP4//BH&HLYP(dashed), scaled B H & H L Y P d e v i a t i o n factor is s l i g h t l y (dotted), scaled Q C I S D (dashed-dotted)}. larger for the reverse rate constants f r o m b o t h the P M P 4 / / Q C I S D and P M P 4 / / B H & H L Y P c a l c u l a t i o n s , p a r t i c u l a r l y r a n g i n g f r o m 2.5-3.0 for the t e m p e r a t u r e r a n g e f r o m 300-1500 K . Rate constants c a l c u l a t e d f r o m the s c a l e d B H & H L Y P a n d Q C I S D p o t e n t i a l are also i n r e a s o n a b l y g o o d a g r e e m e n t w i t h the e x p e r i m e n t a l d a t a . - 1 2



J I M

I 1 1 1 1 I 1 |1ï 1I I 1τ 1

ι

111 ι

ι

1 1 1 1.

7 5

T h e a b o v e results s h o w that w e c a n use c o m p u t a t i o n a l l y less d e m a n d i n g D F T m e t h o d s t o p r o v i d e geometries a n d H e s s i a n s a l o n g the M E P . D F T energies m a y not be sufficient for rate c a l c u l a t i o n s , h o w e v e r , p o t e n t i a l energies a l o n g the M E P c a n be i m p r o v e d b y s c a l i n g b y a factor t o r e p r o d u c e classical f o r w a r d a n d / o r reverse b a r r i e r s f r o m m o r e accurate c a l c u l a t i o n s o r p e r f o r m i n g a series o f s i n g l e p o i n t c a l c u l a t i o n s at a m o r e accurate ab initio M O l e v e l .


98


Β. Proton transfer i n formamidine-water complex Η

Η

I H

Η.


,H 4

H

H

W e u s e d p r o t o n transfer i n the f o r m a m i d i n e - w a t e r c o m p l e x as a b a s i c m o d e l f o r s t u d y i n g p r o t o n transfer i n h y d r o g e n b o n d e d s y s t e m s , t h o u g h f o r m a m i d i n e a n d its a m i d i n e class also h a v e t h e i r o w n b i o l o g i c a l and pharmaceutical importance. P r e v i o u s s t u d i e s ' f o u n d that a d d i n g a w a t e r m o l e c u l e to b r i d g e the p r o t o n d o n o r a n d acceptor sites stabilizes the t r a n s i t i o n state. T h i s l o w e r s the b a r r i e r b y 27 k c a l / m o l , a n d as a consequence, s i g n i f i c a n t l y enhances the transfer rate. D u e to the s m a l l p r o t o n m a s s , w e also f o u n d that the t u n n e l i n g c o n t r i b u t i o n i s s i g n i f i c a n t , p a r t i c u l a r l y at l o w t e m p e r a t u r e s , b y c o m p a r i n g the C V T a n d C V T / S C T r e s u l t s . F u r t h e r m o r e , the "corner c u t t i n g " effect i n c l u d e d i n the C V T / S C T c a l c u l a t i o n s o n the m u l t i d i m e n s i o n a l surface g r e a t l y e n h a n c e s the tunneling probability for p r o t o n transfer i n the formamidine-water complex (see F i g u r e 6). T h i s i s i l l u s t r a t e d b y the large • 10 increase i n the C V T / S C T rate constants w h e n c o m p a r e d t o the C V T / Z C T rate. N o t e that -20 i n Z C T calculations, tunneling is restricted to b e a l o n g the -30 « • ' ' * • ' • ' * • • • • I M I I 1 I I I I I I 1 2 3 4 5 6 7 M E P . The reaction valley i n 10007T(K) this case w a s c a l c u l a t e d at the M P 2 level. The potential Figure 6: Arrhenius plot of calculated CVT, CVT/ZCT and CVT/SCT thermal rate constants. energy a l o n g the M E P w a s (Adapted from Ref. 5). further scaled b y a factor o f 1.123 to m a t c h the 5

7 6




99

C C S D ( T ) / / M P 2 classical b a r r i e r heights. I n a l l ab initio M O a n d D F T c a l c u l a t i o n s for this s y s t e m , the 6-31G(d,p) basis set w a s u s e d .


T h e l a r g e "corner c u t t i n g " t u n n e l i n g effect r e q u i r e s m u c h m o r e p o t e n t i a l e n e r g y surface i n f o r m a t i o n t h a n just i n the v i c i n i t y o f the s a d d l e p o i n t . H o w e v e r , d u e t o the size o f this s y s t e m , one w o u l d l i k e t o m i n i m i z e the n u m b e r o f M P 2 H e s s i a n c a l c u l a t i o n s w i t h o u t a s i g n i f i c a n t loss t o the accuracy. T h u s , it is a g o o d e x a m p l e to i l l u s t r a t e the a c c u r a c y o f our focusing technique described below. F o r this d i s c u s s i o n , the M E P w a s c a l c u l a t e d at the M P 2 l e v e l w i t h a m a x i m u m o f 28 H e s s i a n p o i n t s e v e n l y d i s t r i b u t e d b e t w e e n s v a l u e s o f 0 a n d 1.2 a m u / b o h r . D u e to the s y m m e t r y o f the M E P , this is e q u i v a l e n t t o a t o t a l o f 59 H e s s i a n p o i n t s , i n c l u d i n g three s t a t i o n a r y p o i n t s , for the entire M E P . U s i n g the C V T / S C T t h e r m a l rate constants c a l c u l a t e d f r o m these 28 c a l c u l a t e d H e s s i a n p o i n t s as the reference p o i n t , w e h a v e c a l c u l a t e d C V T / S C T rate constants w i t h the n u m b e r o f H e s s i a n p o i n t s less t h a n the f u l l 28, w h i c h w e r e c h o s e n b y the f o c u s i n g t e c h n i q u e , a n d p l o t t e d i n F i g u r e 7 w i t h the percent difference i n the rate constant at 300 Κ v e r s u s the n u m b e r o f H e s s i a n s u s e d . W e f o u n d that a m i n i m u m o f 10 H e s s i a n p o i n t s is r e q u i r e d for the c o n v e r g e n c e o f the rate constant to w i t h i n 10%. F u r t h e r m o r e , u s i n g a l o w - p a s s f i l t e r i n g t e c h n i q u e to r e m o v e n o i s e i n the c a l c u l a t e d effective r e d u c e d m a s s s l i g h t l y i m p r o v e s this c o n v e r g e n c e . N o t e that e v e n w i t h 5 H e s s i a n p o i n t s , the c a l c u l a t e d rate constant at 300 Κ c o n v e r g e s to w i t h i n a factor - - 0 - - not smoothed of t w o i n the 2 8 - p o i n t case. F o r reactions w i t h a s m a l l e r tunneling contribution than this case, a s m a l l e r n u m b e r o f H e s s i a n s m a y b e sufficient. In a d d i t i o n , not i n c l u d i n g the "corner c u t t i n g " effect f r o m the b e n d i n g m o d e s , i.e. vibrational modes w i t h 0 5 10 15 20 25 30 frequency less t h a n 1800 c m " , number of Hessians along M E P only introduces an error of less t h a n 2 0 % . 1

2

1

Figure 7: Convergence of the calculated rate constant at 300 Κ as functions of the number of Hessians along the MEP. (Adapted from Ref. 5)

F i n a l l y , u s i n g the M P 2 rate results for the tautomerization i n the formamidine-water complex


100


C H E M I C A L APPLICATIONS OF DENSITY-FUNCTIONAL T H E O R Y

(Adapted from R e f .

5.)



6. TRUONG ET AL.

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as a reference p o i n t , w e h a v e i n v e s t i g a t e d the a c c u r a c y u s i n g the n o n - l o c a l B H & H - L Y P D F T m e t h o d for c a l c u l a t i n g the p o t e n t i a l e n e r g y i n f o r m a t i o n . I n this case, w e also u s e d the same b a r r i e r s c a l i n g p r o c e d u r e d e s c r i b e d a b o v e . I n a d d i t i o n , w e also i n v e s t i g a t e d a n o t h e r c o m p u t a t i o n a l l y less d e m a n d i n g a p p r o a c h . T h a t is to use H F t h e o r y , b u t i n a d d i t i o n to s c a l i n g the classical b a r r i e r to the C C S D ( T ) / / M P 2 v a l u e , the H F frequencies w e r e also s c a l e d b y a factor of 0.9. I n b o t h H F a n d D F T cases, the M E P ' s w e r e c a l c u l a t e d w i t h the s a m e step s i z e of 0.1 a m u ^ b o h r as u s e d i n the M P 2 c a l c u l a t i o n s . T h e A r r h e n i u s p l o t for the M P 2 , H F a n d D F T C V T a n d C V T / S C T rate constants are also s h o w n i n F i g u r e 8. W e f o u n d the H F C V T rate constants are n o t i c e a b l y s m a l l e r t h a n the M P 2 a n d D F T rates. A s a r e s u l t , the f i n a l H F C V T / S C T rate constants are also s m a l l e r a n d the differences increase as the t e m p e r a t u r e decreases. T h e excellent a g r e e m e n t b e t w e e n the M P 2 a n d D F T rate constants further s u p p o r t s the a b o v e c o n c l u s i o n o n the use of n o n - l o c a l D F T m e t h o d s for d i r e c t ab initio d y n a m i c s c a l c u l a t i o n s . F o r instance at 300 K , the H F rate constant is s m a l l e r t h a n the M P 2 v a l u e b y a factor of 8.0 w h i l e the D F T rate constant is larger b y a factor of 1.6. W e h a v e f u r t h e r d e v e l o p e d these d i r e c t ab initio d y n a m i c s m e t h o d s to calculate v i b r a t i o n a l - s t a t e selected rates of p o l y a t o m i c r e a c t i o n s . The results so far are e n c o u r a g i n g . P a r t i c u l a r l y , it a l l o w s one to correlate features o n the p o t e n t i a l surface to state specific c h e m i s t r y . T h e r m a l a n d v i b r a t i o n a l - s t a t e selected rate constants of p o l y a t o m i c reactions n o w c a n be r o u t i n e l y c a l c u l a t e d f r o m ab initio M O a n d / o r D F T m e t h o d s b y u s i n g o u r T h e R a t e ( T h e o r e t i c a l Rate) p r o g r a m . W e are i n the process of d e v e l o p i n g n e w m e t h o d o l o g i e s for i n c l u d i n g a n h a r m o n i c i t y a n d l a r g e c u r v a t u r e t u n n e l i n g c o n t r i b u t i o n s w i t h i n o u r d i r e c t ab initio d y n a m i c s approach. 1 2 , 3 7

7 7

IV. CONCLUSION T h e d i r e c t ab initio d y n a m i c s m e t h o d w e d e s c r i b e d a b o v e w i t h the use of a n o n - l o c a l D F T m e t h o d to p r o v i d e p o t e n t i a l e n e r g y i n f o r m a t i o n a n d a f o c u s i n g t e c h n i q u e to m i n i m i z e the n u m b e r of H e s s i a n s r e q u i r e d offers a p r o m i s s i n g a l t e r n a t i v e for s t u d y i n g k i n e t i c s , d y n a m i c s a n d m e c h a n i s m s of l a r g e p o l y a t o m i c reactions. W i t h i n the s a m e m e t h o d o l o g y , one c a n f u r t h e r e x t e n d the d y n a m i c a l t h e o r y to treat c h e m i c a l reactions o n c r y s t a l surfaces as w e l l as i n s o l u t i o n s . S u c h steps are n o w b e i n g t a k e n i n our lab. Acknowledgement T h i s w o r k w a s s u p p o r t e d i n part b y the U n i v e r s i t y of U t a h a n d b y the N a t i o n a l Science F o u n d a t i o n t h r o u g h a n N S F Y o u n g I n v e s t i g a t o r A w a r d to T . N . T . .


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