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Implication methods for the determination of quadratic force constants 1971

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IMPLICATION METHODS FOR THE DETERMINATION OF QUADRATIC FORCE CONSTANTS by RAYMOND WINSTON GREEN B.S., Oregon State U n i v e r s i t y , 1963 A THESIS SUBMITTED IN PARTIAL FULFILMENT THE REQUIREMENTS FOR THE DEGREE. OF DOCTOR OF PHILOSOPHY i n the Department of CHEMISTRY We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1971 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Br i t ish Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of Br i t ish Columbia Vancouver 8, Canada i i > A B S T R A C T C u r r e n t l y t h e f o r m u l a t i o n o f a v a l i d f o r c e c o n s t a n t m a t r i x p o s e s t h e l a r g e s t p r o b l e m i n t h e n o r m a l c o o r d i n a t e a n a l y s i s o r t h e m e c h a n i c a l i n t e r p r e t a t i o n o f v i b r a t i o n a l s p e c t r a . U s u a l l y a p r e s e l e c t e d s e t o f t r i a l f o r c e c o n s t a n t s i s i t e r a t i v e l y c o r r e c t e d b y m e a n s o f f i r s t o r d e r p e r t u r b a t i o n t h e o r y a n d t h e p r i n c i p l e o f l e a s t s q u a r e s . T h i s t h e s i s b r e a k s t h a t t r a d i t i o n a n d o p e r a t e s t h e n o r m a l c o o r d i n a t e a n a l y s i s t h r o u g h a n i m p l i e d f o r c e c o n s t a n t m a t r i x , F = L *"Aj_, \ w h e r e LL*" = G, t h e f a m i l i a r W i l s o n G - m a t r i x . T h e A - m a t r i x i s c o m p o s e d o f t h e e x p e r i m e n t a l v i b r a t i o n a l f r e q u e n c i e s f o r a s e l e c t e d b a s i s m o l e c u l e a n d t h e L - m a t r i x i s p a r a m e t e r i z e d i n a g e n e r a l w a y . T h e L - m a t r i x p a r a m e t e r s a r e v a r i e d u n t i l t h e i m p l i e d f o r c e c o n s t a n t m a t r i x g e n e r a t e s a n o p t i m u m m e c h a n i c a l p i c t u r e o f t h e b a s i s m o l e c u l e a n d i t s i s o t o p i c h o m o l o g s . How- e v e r t h i s t h e s i s e m p h a s i z e s t h e v i b r a t i o n a l f u n d a m e n t a l s o f i s o t o p i c h o m o l o g s i n s p e c i f y i n g t h e i m p l i e d f o r c e f i e l d . I n a p p l i c a t i o n s i x L - m a t r i x p a r a m e t e r s e n c o m p a s s t h e s i x t y - t h r e e p l a n a r v i b r a t i o n a l f r e q u e n c i e s o f e t h y l e n e a n d i t s d e u t e r o h o m o l o g s w i t h s l i g h t l y l e s s e r r o r t h a n t r a d i t i o n a l c a l c u l a t i o n s u s i n g a s many a s f i f t e e n p o t e n t i a l e n e r g y p a r a m e t e r s . A s w e l l , t h e i m p l i e d f o r c e c o n s t a n t s c o m p l y w i t h t h e e x i s t i n g p i c t u r e o f c h e m i c a l b o n d i n g w i t h o u t d e l i b e r a t e a p r i o r i r e f e r e n c e t o i t . I n p a r t i c u l a r , a s p e c t s o f t h e h y b r i d o r b i t a l f o r c e f i e l d a r e c o n f i r m e d w i t h o u t p r i o r c o n s t r a i n t s . i i i I n more d e t a i l e d computational s t u d i e s the i m p l i e d force f i e l d has r e v e a l e d a systematic t r e n d i n anharmofiic e f f e c t s which can he understood i n terms of d i f f e r e n t v i b r a t i o n a l amplitudes f o r d i f f e r e n t i s o t o p i c homologs. The i n f l u e n c e of v i b r a t i o n a l amplitude has been parameterized and i n c l u d e d w i t h i n the i m p l i c a t i o n method as a simple anharmonicity c o r r e c t i o n . For example, one L-matrix parameter and three v i b r a t i o n a l amplitude parameters encompass the nine observed v i b r a t i o n a l frequencies of water and i t s deuterohomologs w i t h -1 an average frequency e r r o r of 0.4 cm . Without amplitude c o r r e c t i o n s the average frequency e r r o r becomes 10.7 cm w i t h one L-matrix parameter or 12.8 cm w i t h four p o t e n t i a l energy parameters.. I t i s p a r t i c u l a r l y s i g n i f i c a n t t h a t t h i s simple p i c t u r e of anharmonicity employs the observed v i b r a t i o n a l frequencies r a t h e r t h a t the e m p i r i c a l l y d e r i v e d harmonic frequencies. As w e l l , the v i b r a t i o n a l amplitude parameters comply w i t h expected features of p o t e n t i a l energy surfaces such as the d i s s o c i a t i o n l i m i t . The p r i n c i p l e advantage of the i m p l i c a t i o n method i s t h a t there a fewer L-matrix parameters than F-matrix parameters. The p r i n c i p a l disadvantage i s t h a t approximations and i n t u i t i v e n o t a t i o n s are not e a s i l y b u i l t i n t o the i m p l i c a t i o n method. However, as experimental i n f o r m a t i o n becomes more complete and b e t t e r understood, the need f o r improved a n a l y t i c foundations dominates the need f o r handy approximations. i v TABLE OF CONTENTS PAGE CHAPTER ONE: I n t r o d u c t i o n 1 (1-1) Molecular S t r u c t u r e 1 (1-2) Molecular V i b r a t i o n s 2 (1-3) P e r t u r b a t i o n Methods 7 (1-4) I m p l i c a t i o n Methods 13 CHAPTER TWO: Simple Mixing I m p l i c a t i o n Methods 16 (2-1) The Implied f o r c e F i e l d 19 (2-2) A Simple Anharmonicity C o r r e c t i o n 23 (2-3) A p p l i c a t i o n s of Simple Mixing Methods 25 (2-3a) H 20, HDO and D 20: Harmonic Fundamentals 28 (2-3b) H 20, HDO and D 20: Observed Fundamentals 31 (2-3c) HCCH, HCCD, DCCD: Observed Fundamentals 35 (2-3d) Methane and Deuterohomologs: Observed Fundamentals 39 CHAPTER THREE General Mixing I m p l i c a t i o n Methods 44 (3 - D K Space 44 (3-2) I s o t o p i c Homologs i n K-Space 46 (3-3) Implied D i s p e r s i o n 48 (3-4) Selected A p p l i c a t i o n s i n K-Space 53 (3-4a) Formaldehyde and I t s Deuterohomologs 55 (3~4b) Ethylene and i t s Deuterohomologs 59 (3-4c) Chemical S i g n i f i c a n c e 68 v PAGE CHAPTER FOUR: Approximation Techniques 72 (4-1) The Weighted Trace Equation 72 (4-2) P a r t i c i p a t i o n and Molecular P a r t i t i o n i n g 76 EPILOGUE 86 Appendix I : The V i b r a t i o n a l Secular Equations 94 Appendix I I : Redundant Coordinate Systems 107 Appendix III:The E x p o n e n t i a l M a t r i x 113 References 88 v i LIST OF TABLES TABLE PAGE 1. C a l c u l a t e d Harmonic Fundamentals: HDO and D 2 0 29 2 . . Harmonic Force Constants: HgO, HDO and D 2 0 30 3- C a l c u l a t e d Fundamentals: HDO and D 2 0 32 4. Force Constants: HgO, HDO and' DO ' 33 5 . C a l c u l a t e d Fundamentals with Amplitude F a c t o r s : 34 HDO and D 2 0 6. V i b r a t i o n a l Fundamentals: Deuteroacetylenes 37 7. Force Constants: Acetylene 38 8. V i b r a t i o n a l Fundamentals: Deuteromethanes 4 l 9 . V i b r a t i o n a l Fundamentals: H2CO, HDCO and D2CO 56 10. The Force Constants of Formaldehyde 58 11-. The Force Constants' of Ethylene 64 1 2 . V i b r a t i o n a l Fundamentals: Ethylene and Deuterohomologs 65 1 3 . S i m i l a r Implied Force Constants 69 14. P a r t i c i p a t i o n Matrices for,Methylamine 82 ACKNOWLEDGEMENT To the novice, the l i n e s e p a r a t i n g p h y s i c a l p e r s p e c t i v e and mathematical a b s t r a c t i o n seems an unfocused domain w i t h unresolved g u i d e l i n e s . The k i n d l y l e a d e r s h i p and secure i n s i g h t of K.B. Harvey has been a welcoming guidepost throughout the course of these s t u d i e s . DEDICATION to Professor K. B. Harvey In Memoriam CHAPTER ONE: INTRODUCTION (1-1) Molecular S t r u c t u r e Several branches of chemistry meet i n the domain of molecular s t r u c t u r e ; here, chemists see an e l e c t r o n i c s t r u c t u r e superimposed on a r i g i d n u clear frame with s p e c i f i e d geometry. This p i c t u r e f u r n i s h e s a simple b a s i s f o r understanding chemical a c t i v i t y , f o r example, i n terms of simple molecular o r b i t a l theory. Thus, to some ex t e n t , a l l measures of molecular s t r u c t u r e belong to every branch of chemistry. Prom an i n t e r i o r p o s i t i o n , the study of molecular s t r u c t u r e j o i n s r i g o r o u s p h y s i c s , a b s t r a c t mathematics, and experimental i n f o r m a t i o n i n t o a s e l f - c o n t a i n e d body of knowledge. This j o i n i n g process, the a n a l y s i s of molecular s t r u c t u r e , should not be viewed as a c l o s e d r i n g of things to do with data and theory. Rather, e s p e c i a l l y i n chemistry, the a n a l y s i s of molecular s t r u c t u r e must c o n t r i b u t e to a perview of the t o p i c i n a way that encompasses the various c o n t r i b u t i n g branches. To achieve that o b j e c t i v e , the chemist adds an e x t e r i o r concept to h i s scheme of a n a l y s i s . In e f f e c t , the chemist expects the parameters of molecular s t r u c t u r e to e x h i b i t a reasonable p a t t e r n which complies with the b a s i c notions of chemical bonding. E x i s t i n g s t u d i e s confirm the e x p e c t a t i o n . - 2 - The e l e c t r o n i c s t r u c t u r e , when organized i n t o chemical bonds, provides b i n d i n g energy which i s a f u n c t i o n of the number of e l e c t r o n s , nuclear charge, and nuclear c o n f i g u r a t i o n . Nuclear mass does not s i g n i f i c a n t l y i n f l u e n c e the b i n d i n g energy. For s t a b l e molecules the bi n d i n g energy i s such that nuclear c o n f i g u r a t i o n i s co n s t r a i n e d to a neighbourhood of minimal p o t e n t i a l energy (maximal b i n d i n g energy). This p i c t u r e does not imply a r i g i d s t r u c t u r e . Rather, molecular s t r u c t u r e s are s e m i r i g i d geometrical e n t i t i e s embedded w i t h i n the p o t e n t i a l energy surface imposed by the e l e c t r o n i c s u p e r s t r u c t u r e . The r i g i d i t y depends upon the curvature of the surface and the i n e r t i a l mass of the n u c l e i . Consequently, the a n a l y s i s of molecular s t r u c t u r e w i l l i n v o l v e much more than simple geometry. F u r t h e r , i f chemical bonding i s recognized, more than pure physics w i l l appear. In p r a c t i c e , the a n a l y s i s i s s u f f i c i e n t l y cumbersome that some mathematical t o o l s which are not part of the physics w i l l enter the p i c t u r e . ( 1 - 2 ) Molecular V i b r a t i o n s Within l i m i t s , molecular i n c o n f i g u r a t i o n i s governed equations. The usual c l a s s i v i b r a t i o n s or the v a r i a t i o n by the v i b r a t i o n a l s e c u l a r c a l p r e s e n t a t i o n of these - 3 - equations ( 1 ; 2 ; 3 ) expresses p o t e n t i a l and k i n e t i c energy as q u a d r a t i c forms with t r a n s l a t i o n a l and r o t a t i o n a l k i n e t i c energy removed. A modest quantum mechanical p r e s e n t a t i o n , s u i t a b l e f o r teaching purposes can be found i n appendix one of t h i s t h e s i s . The dynamic v a r i a b l e s , c o n f i g u r a t i o n displacement c o o r d i n a t e s , measure the d i s t o r t i o n of the molecule from i t s e q u i l i b r i u m c o n f i g u r a t i o n - the p o i n t of minimum p o t e n t i a l energy. I f these coordinates are defined w i t h chemical bonding i n mind, the d e f i n i t i o n of p o t e n t i a l energy becomes g r e a t l y s i m p l i f i e d i n both approximative and i n t e r p r e t i v e aspects. C l e a r l y bond s t r e t c h i n g (an i n t e r n u c l e a r d i s t a n c e ) and valence angle bending w i l l be u s e f u l choices. In the harmonic o s c i l l a t o r approximation, dynamic d i s t o r t i o n s are assumed to be s u f f i c i e n t l y s m a l l that p o t e n t i a l and k i n e t i c energy can be expressed as q u a d r a t i c forms i n c o n f i g u r a t i o n displacement c o o r d i n a t e s , r ^ f , and t h e i r conjugate momenta, , r e s p e c t i v e l y . k i n e t i c energy = llz.HX-\ G\\V\.V[ (1.2.1) p o t e n t i a l energy (1.2.2) - 4 - It i s not p o s s i b l e to p r o p e r l y j u s t i f y the k i n e t i c energy expression i n a few l i n e s ; see Wilson Decius and Cross (1) e s p e c i a l l y t h e i r appendix V I I , or appendix one of t h i s t h e s i s f o r an account of the k i n e t i c energy. Present purposes r e q u i r e only a c l e a r d e f i n i t i o n of the G-matrix elements; these depend only upon the nuclear masses and the e q u i l i b r i u m geometry of the molecule. A l l p o s s i b l e G-matrix elements f o r bond s t r e t c h i n g and valance angle bending have been ta b u l a t e d i n appendix VI of Wilson, Decius and Cross ( 1 ) ; otherwise they may be c a l c u l a t e d from the f o l l o w i n g expression. Gij = ( \&/OiWtfc/aj) (1.2.3) jU/£ i s the r e c i p r o c a l mass of the DC — nucleus (or atom) and \b i s the c a r t e s i a n g r adient f o r the i n d i c a t e d atom. A f t e r the s e l e c t e d c o n f i g u r a t i o n coordinates have been w r i t t e n as f u n c t i o n s of the c a r t e s i a n coordinates of the i n d i v i d u a l atoms, the G -matrix f o l l o w s ; however, considerable labor i s i n v o l v e d . Prom the outset l i t t l e i s known about the p o t e n t i a l energy except that the concept of chemical bonding i s i n v o l v e d . C u r r e n t l y p o t e n t i a l energy i s parameterized i n various ways and experimental i n f o r m a t i o n i s used to - 5 - s p e c i f y the parameters. The p r i n c i p a l parameters are the q u a d r a t i c f o r c e constants - the F ,'s (F-matrix) of expression (1.2.2) above. Of the s e v e r a l types of experimental i n f o r m a t i o n dependent on p o t e n t i a l energy, fundamental v i b r a t i o n a l frequencies have provided the bulk of what i s known about p o t e n t i a l energy parameters. Quadratic f o r c e constants and fundamental v i b r a t i o n a l frequencies are r e l a t e d through the v i b r a t i o n a l s e c u l a r equations. i. Q C = X the i d e n t i t y matrix (1.2.4) f l_ ~ /\ a diagonal matrix (1.2.5) Here k i n e t i c and p o t e n t i a l energy have been simultaneously d i a g o n a l i z e d v i a the t r a n s f o r m a t i o n A*» ~ 21 & Li*. Q»e_ (1.2.6) where the are normal coordinates. In c l a s s i c a l mechanics each normal coordinate i s a p e r i o d i c f u n c t i o n of time w i t h p e r i o d )/% . (See Wilson, Decius and Cross (1) or appendix one of t h i s t h e s i s f o r a development w i t h p h y s i c a l substance.) - 6 - The diagonal A ~ matrix i s r e l a t e d to the various periods (or frequencies) of v i b r a t i o n ; 2. (1.2.7) where i^g i s the k' th v i b r a t i o n a l frequency i n wavenumber u n i t s , p o t e n t i a l energy i s expressed i n m i l l i d y n e Angstroms, mass i s expressed i n atomic mass u n i t s (carbon-twelve = 12.0000), and d i s t a n c e i s measured i n Angstroms. M a t r i x m u l t i p l i c a t i o n of (1.2.4) and (1.2.5), r e s p e c t i v e l y composed of the eigenvalues and eigenvectors of the matrix product GF; consequently the r e l a t i o n connecting p o t e n t i a l energy parameters and v i b r a t i o n a l frequencies i s g e n e r a l l y cumbersome. U s u a l l y the r e l a t i o n i s described through p e r t u r b a t i o n techniques (1, 4, 5, 6) l i k e those developed i n the f o l l o w i n g s e c t i o n (1-3), P e r t u r b a t i o n Methods. However, the bulk of t h i s t h e s i s seeks to e s t a b l i s h an improved understanding of the r e l a t i o n by more ab s t r a c t methods which are none the l e s s more d i r e c t l y r e l a t e d to the p h y s i c a l problems. L'GFL = A shows that the - 7 - (1-3) P e r t u r b a t i o n Methods In broad o u t l i n e the c a l c u l a t i o n of p o t e n t i a l energy- parameters by p e r t u r b a t i o n methods i n v o l v e s the v a r i a t i o n of p r e s e l e c t e d parameters u n t i l c a l c u l a t e d i n f o r m a t i o n agrees w i t h experimental i n f o r m a t i o n . The i n f o r m a t i o n to be f i t t e d i n c l u d e s v i b r a t i o n a l f r e q u e n c i e s , t h e i r symmetry type ( 1 , 6 ) , and when known, r e l a t e d mechanical q u a n t i t i e s such as mean square amplitudes ( 7 , 8 ) , c e n t r i f u g a l d i s t o r t i o n constants ( 8 ) , and C o r i o l i s c o e f f i c i e n t s (9»10). In r i g o r o u s a p p l i c a t i o n s , a given set of parameters span only the experimental i n f o r m a t i o n belonging to a s e r i e s of i s o t o p i c homologs; moreover t h i s i n f o r m a t i o n should be co r r e c t e d f o r anharmonic e f f e c t s . The work of Aldous and M i l l s (11) on the methyl h a l i d e s represents a guidepost i n ri g o r o u s a p p l i c a t i o n s f o r moderately complex molecules. (In t r i a t o m i c molecules, r i g o r may assume meaning w e l l beyond the scope of t h i s t h e s i s ; see Suzuki's a n a l y s i s (12) of carbon d i o x i d e ) . When r i g o r i s r e l a x e d , a given set of parameters may span the v i b r a t i o n a l frequencies f o r a s e r i e s of r e l a t e d molecules. The a n a l y s i s of the c h l o r i n a t e d benzenes d e l i n e a t e d by Scherer (14) as w e l l as the papers of Snyder and Schachtschneider (13,15,16,17) have added s u b s t a n t i a l credence to t h i s approach - e s p e c i a l l y when valence f o r c e f i e l d s are employed. Very b r i e f l y , the valence f o r c e - 8 - f i e l d i s b u i l t on patterns which describe the chemical bonding s t r u c t u r e of the molecule. In t h i s case, the p o t e n t i a l energy f u n c t i o n may be assembled from p o t e n t i a l energy parameters belonging to a few simple bonding u n i t s . The p e r t u r b a t i o n methods begin w i t h the s e l e c t i o n of p o t e n t i a l energy parameters, and some of these must be assigned nonzero i n i t i a l values. N e i t h e r task should be considered as t r i v i a l (18,19), but both the set of parameters, , and adequate i n i t i a l v a l u e s , , w i l l be assumed. Aldous and M i l l s (11) i l l u s t r a t e some of the problems. The G-matrix i s constructed from geometric i n f o r m a t i o n and the F-matrix i s constructed from the parameters. F = F ° * r „ (dF/d<t>m)°(<t>m-4>m) The matrix ( dF/dfy}) l s a convenient n o t a t i o n a l device intended to cover v a r i o u s kinds of p a r a m e t e r i z a t i o n (or model-building) i n qu a d r a t i c p o t e n t i a l energy f u n c t i o n s . F i r s t order p e r t u r b a t i o n theory (1,4,5) provides a system o f l i n e a r equations v a l i d f o r s m a l l p e r t u r b a t i o n s . where L l GF"La - A0 - 9 - F; L0and A are cons t r u c t e d from the parameters e i t h e r d i r e c t l y or by s o l v i n g the above v i b r a t i o n a l s e c u l a r equation. (This l a t t e r problem, s o l v i n g the v i b r a t i o n a l s e c u l a r equation by computer methods has been described by Shimanouchi and Suzuki ( 5 ) . Schachtschneider's t e c h n i c a l report ( 4 ) provides a l l r e l e v a n t d e t a i l s f o r the a p p l i c a t i o n of p e r t u r b a t i o n methods .) The expressions (1.3.1) represent one u s e f u l l i n e a r equation f o r each secure assignment of an experimental fundamental frequency; c l e a r l y more than one molecule may be i n v o l v e d i n the set of u s e f u l l i n e a r equations. I t i s assumed that experimental frequencies can be unambiguously assigned to the i n d i v i d u a l l i n e a r equations. F u r t h e r i t i s assumed that the number of u s e f u l l i n e a r equations i s l a r g e r than the number of p o t e n t i a l energy parameters to be determined. In equation (1.3.1), a p e r f e c t f i t of experimental f r e q u e n c i e s , A RR. ) ^ S n°t expected; consequently the e r r o r s , £ R , d e f i n e d by the i d e n t i t y are minimized as a weighted sum of squares, 21 K W R £ R with respect to the parameter c o r r e c t i o n s , - 10 - The l e a s t squares s o l u t i o n ( 4 , 5 , 6 , 1 1 ) , <j> = £ + zK \ (JVJ/'JVJ (A«-/&) (1.3.2) where JJ^ = ( (Of/d^JLc )**. and W i s a diagonal matrix of p o s i t i v e weighting f a c t o r s , provides improved estimates of the p o t e n t i a l energy parameters; however, because higher order p e r t u r b a t i o n terms have been ignored, the above c y c l e of c a l c u l a t i o n s r e q u i r e s i t e r a t i o n . The p e r t u r b a t i o n method o u t l i n e d above s u f f e r s one major d i f f i c u l t y ; both i n i t i a l and t a r g e t p a r a m e t e r i z a t i o n must e x h i b i t a s a t i s f a c t o r y f o r m u l a t i o n from the outset. Quite g e n e r a l l y , i n i t i a l p a r a m e t e r i z a t i o n i s o v e r s i m p l i f i e d and t a r g e t p a r a m e t e r i z a t i o n i n v o l v e s the i d e n t i f i c a t i o n of next-most s i g n i f i c a n t f a c t o r s . Except (perhaps) f o r the h y b r i d o r b i t a l f o r c e f i e l d of M i l l s (11,20,21,22), the g u i d e l i n e s f o r ab i n i t i o t a r g e t p a r a m e t e r i z a t i o n are s e r i o u s l y l i m i t e d ; see Aldous and M i l l s ( 1 1 ) . - 11 - The v a l i d i t y of i n i t i a l parameters i s r e a d i l y t e s t e d through the c a l c u l a t e d v i b r a t i o n a l frequencies and normal modes; u n f o r t u n a t e l y the value of a t a r g e t p a r a m e t e r i z a t i o n i s not e a s i l y t e s t e d . Schachtschneider's m u l t i p l e r e g r e s s i o n a n a l y s i s (4) has been a p p l i e d w i t h c o n v i n c i n g success; see Gayles, King and Schachtschneider ( 1 9 ) . To understand the r e s u l t s of a f a u l t y t a r g e t p a r a m e t e r i z a t i o n , l e t us de f i n e simple circumstances f o r i l l u s t r a t i v e purposes. Consider three l i n e a r equations i n two parameters; each l i n e a r equation provides a s t r a i g h t l i n e i n parameter space (with allowance f o r e r r o r , each l i n e becomes a band or s t r i p ) . The i n t e r s e c t i o n p o i n t ( s ) of the l i n e s or bands s p e c i f y the parameters; however a genuinely unique s p e c i f i c a t i o n need not appear, as i s i l l u s t r a t e d i n cases B and C below. CASE A CASE B CASE C - 12 - Case A, i n c l u d e d f o r v i s u a l r e f e r e n c e , i n d i c a t e s a p h y s i c a l l y s i g n i f i c a n t s p e c i f i c a t i o n of the parameters; the three l i n e s i n t e r s e c t near a common p o i n t . Case B f a i l s to s p e c i f y s i g n i f i c a n t parameters; the f a i l u r e may be due to a poor choice of parameters or i t may be due to more deeply rooted e f f e c t s such as f a i l u r e of the harmonic o s c i l l a t o r approximation i t s e l f . Here equation (1.3-2) provides a s o l u t i o n w i t h l i t t l e p h y s i c a l s i g n i f i c a n c e . In case B s i t u a t i o n , the weighting f a c t o r s w i l l unduly i n f l u e n c e the s p e c i f i c a t i o n of parameters as has been shown by N i b l e r and Pimentel (23). Case C, i l l - c o n d i t i o n i n g , provides no unique s o l u t i o n ; w ith e r r o r bands, the three l i n e s become a s i n g l e hand - the e q u i v a l e n t of one equation. More experimental i n f o r m a t i o n may s p e c i f y a unique s o l u t i o n ; thus the parameters need not be i n c o r r e c t l y chosen. However, w i t h i n the l i m i t a t i o n s imposed by the given experimental i n f o r m a t i o n , the p a r a m e t e r i z a t i o n must be considered f a u l t y . Both cases B and C f a i l to provide a unique s p e c i f i c a t i o n of the parameters, but only case C, i l l - c o n d i t i o n i n g , has r e c e i v e d d e l i b e r a t e a t t e n t i o n i n the l i t e r a t u r e (5,6,11,24) - probably due to the f a i l u r e of equations (1.3-2). - 13 - In more general examples, s i m i l a r uniqueness problems w i l l appear but with complications that q u i c k l y become l e s s and l e s s t r a c t a b l e . In short summary, p e r t u r b a t i o n methods expect more fore-knowledge than i s provided by the e x i s t i n g g u i d e l i n e s to chemical bonding. (1-4) I m p l i c a t i o n Methods I f the normal coordinates or the L-matrix of equation 1.2.6 were s p e c i f i e d by some means, then the complete set of q u a d r a t i c f o r c e constants, the F-matrix would be s p e c i f i e d by i n v e r s i o n of equation 1.2.5. F = L*A L" ( i . 4 . i ) where A i s determined by experimental v i b r a t i o n a l f r e q u e n c i e s . This approach, the b a s i s of i m p l i c a t i o n methods was noted by W i l l i a m J . Taylor i n 1950 ( 2 5 ) . I t i s the o b j e c t i v e of t h i s t h e s i s to devise both r i g o r o u s methods and i n t u i t i v e grounds f o r s p e c i f y i n g the L-matrix. The i m p l i c a t i o n methods, developed i n subsequent chapters have re v e a l e d aspects of p o t e n t i a l energy, experimental i n f o r m a t i o n , and even geometric d e t a i l s (25) which have not or could not be determined by p e r t u r b a t i o n methods. These - I l l - methods appear to f u r n i s h a more d e f i n i t i v e t o o l f o r the a n a l y s i s of small molecule i n f o r m a t i o n ; moreover, i m p l i c a t i o n methods have i n d i c a t e d a f r e s h approach, p a r t i t i o n i n g and p a r t i c i p a t i o n , to the a n a l y s i s of l a r g e r molecules. I m p l i c a t i o n methods are not e n t i r e l y new; Pulay and Torok ( 2 7 ) , Freeman (28) and others (29,30,31) have discussed s p e c i f i c forms of the L-matrix which may serve u s e f u l purposes. In p a r t i c u l a r Strey's minimized bending f o r c e constants (32) i n d i c a t e u s e f u l i n i t i a l p a r a m e t e r i z a t i o n , but h i s technique i s l i m i t e d to simple molecules where i n i t i a l p a r a m e t e r i z a t i o n i s not needed. Though s t i l l i n the formative stages, the i m p l i e d f o r c e f i e l d obtained by i m p l i c a t i o n methods o f f e r s s e v e r a l s i g n i f i c a n t advantages over the parameterized f o r c e f i e l d obtained by p e r t u r b a t i o n methods. (1) I m p l i c a t i o n methods are con s t r a i n e d to p o s i t i v e d e f i n i t e f o r c e constant matrices as r e q u i r e d by a minimum i n the p o t e n t i a l energy surface ( 3 ) ; p e r t u r b a t i o n methods need not obey t h i s c o n s t r a i n t . (2) The p o s i t i v e d e f i n i t e c o n s t r a i n t improves uniqueness i n a s p e c i f i c a t i o n of the f o r c e constants. (3) An improved d e s c r i p t i o n of nonuniqueness can be obtained i n that the range of p o s s i b l e s o l u t i o n s can be examined. - 15 - (4) I n i t i a l p a r a m e t e r i z a t i o n of the p o t e n t i a l energy can be h e l p f u l , but i t i s not e s s e n t i a l . (5) I m p l i c a t i o n methods help to i d e n t i f y e s s e n t i a l f o r c e constants when t a r g e t p a r a m e t e r i z a t i o n i s i n v o l v e d . (6) The 'mixing parameters' of i m p l i c a t i o n methods are fewer i n number than the p o t e n t i a l energy parameters of e q u i v a l e n t form. When a p p l i e d to •.-••the molecules s e l e c t e d i n t h i s t h e s i s i m p l i c a t i o n methods have; (7) presented a simple account of major anharmonic e f f e c t s ; (8) confirmed some aspects of the h y b r i d o r b i t a l f o r c e f i e l d of M i l l s (21) without assuming i t ; ( 9 ) shown the harmonic bending fundamental of HOD ( 3 3 ) to be i n c o n s i s t a n t w i t h respect to the other harmonic fundamentals of water; (10) shown geometric d i s t o r t i o n s to be an important f a c t o r i n the a n a l y s i s of s o l i d s t a t e s p e c t r a (see McQuaker (26) f o r a d e s c r i p t i o n of t h i s a p p l i c a t i o n ) . However i m p l i c a t i o n methods s u f f e r one se r i o u s disadvantage. I t i s d i f f i c u l t , but not e n t i r e l y i mpossible to invoke approximate d e s c r i p t i o n s i n v o l v i n g the p r i n c i p l e s of chemical bonding. This i s a n a t u r a l f e a t u r e of the p e r t u r b a t i o n methods. CHAPTER TWO: SIMPLE MIXING IMPLICATION METHODS I f point group theory p r e d i c t s n v i b r a t i o n s i n the k — symmetry s p e c i e s , then the Ny d i s t i n c t v i b r a t i o n a l f r e q u e n c i e s , the N d i s t i n c t q u a d r a t i c f o r c e constants, F and the N T d i s t i n c t L-matrix elements are numbered as l i f o l l o w s ( 1 ) : N V 5 z k n R N F ='/z1.K n R (n K +f) The d i s t i n c t L-matrix elements are con s t r a i n e d by N p equations of the form, thus the L-matrix e x h i b i t s N^ fewer degrees of freedom than the F-matrix. The b a s i c hypothesis o v e r l a y i n g t h i s e n t i r e t h e s i s i s that molecular mechanics can be securely analyzed i n terms of the L-matrix belonging to a p r e s e l e c t e d b a s i s molecule. The - 17 - main support f o r t h i s hypothesis stems from the reduced number of parameters i n v o l v e d i n the mechanical p i c t u r e . The q u a d r a t i c f o r c e constants are formed by i m p l i c a t i o n through the observed v i b r a t i o n a l frequencies and L-matrix belonging to the b a s i s molecule. F = FA C Any mechanical q u a n t i t y that i s a f u n c t i o n of the q u a d r a t i c f o r c e constants i s a l s o a f u n c t i o n of the L- matrix belonging to the b a s i s molecule. These r e l a t e d mechanical q u a n t i t i e s i n c l u d e the v i b r a t i o n a l frequencies of i s o t o p i c homologs of the b a s i s molecule ( 1 ) , c e n t r i f u g a l d i s t o r t i o n constants ( 8 ) , C o r i o l i s c o u p l i n g constants (9, 10), and mean square amplitudes of v i b r a t i o n (7, 8). I n - e f f e c t the N v v i b r a t i o n a l frequencies of the b a s i s molecule e l i m i n a t e N v parameters from the mechanical p i c t u r e . However, the L-matrix, f o r the b a s i s molecule o n l y , needs to be expressed i n terms of the N„ independent parameters i t contains as w i l l be done i n the f o l l o w i n g s e c t i o n . The case where = 1 has been designated as simple mixing; here the i m p l i c a t i o n method i s somewhat l e s s a b s t r a c t than the more general case d e a l t with i n chapter three. - 18 - As yet the mechanical p i c t u r e has been l i m i t e d to i s o t o p i c homologs; here only deuterium s u b s t i t u t i o n provides s u f f i c i e n t i n f o r m a t i o n to s p e c i f y the L-matrix. Other i s o t o p i c s u b s t i t u t i o n s appear to have too small an e f f e c t on v i b r a t i o n a l frequencies to a l l o w a proper i n t e r p r e t a t i o n of the mechanical p i c t u r e by e i t h e r i m p l i c a t i o n or p e r t u r b a t i o n methods (3*0 without the a i d of c o n s t r a i n t s ( 3 5 ) or mechanical i n f o r m a t i o n other than v i b r a t i o n a l f r e q u e n c i e s . However, the i s o t o p i c homologs of atoms heavier than deuterium have not been as. thoroughly s t u d i e d by i m p l i c a t i o n methods as the hydrogenic homologs. The f a i l u r e of the harmonic o s c i l l a t o r approximation i s already known through a p p l i c a t i o n of the product r u l e s (1) and many previous fo r c e constant c a l c u l a t i o n s . In s e c t i o n two of t h i s chapter, we suggest a novel anharmonicity c o r r e c t i o n with an apparent p h y s i c a l b a s i s which allows us to use the observed v i b r a t i o n a l frequencies w i t h almost as much success as d e r i v e d harmonic frequencies ( 2 ) . The f i n a l s e c t i o n of t h i s chapter i s given over to c a l c u l a t i o n s i n v o l v i n g the simple mixing i m p l i c a t i o n method. - 19 - (2-1) The Implied Force F i e l d When the L-matrix i s expressed i n the form, \A and P are orthogonal matrices i f ^ i s diagonal such that U6GU = r (2.1.2) The P-matrix encompasses e x a c t l y the N K parameters needed to span the f a m i l y . Equation (1.2.4) i s obeyed f o r a l l orthogonal P- matrices . C ' G C = P Y " l U L O U r ^ P = P C P = I (2.1.3) But the i m p l i e d f o r c e constant m a t r i x , F - ii*AC = ur'*PAPtr*-ut li.i.o) w i l l not e x h i b i t i t s proper p o i n t group symmetry (1) unless the P-matrix e x h i b i t s the property: - 20 - "Pij ~ 0 unless An and designate v i b r a t i o n s belong- i n g to the same symmetry s p e c i e s . (2.1.5) The proof of t h i s f a c t depends upon the a p p l i c a t i o n of point group theory i n molecular v i b r a t i o n s ( 1 ) . I f a molecule e x h i b i t s symmetry, there e x i s t s a simple orthogonal transform- a t i o n m a t r i x , S , such that and where (3* and F ; are i n a block diagonal form c o n s i s t a n t w i t h the symmetry of the molecule. Each d i s t i n c t block belongs to a d i f f e r e n t symmetry sp e c i e s . C l e a r l y there e x i s t s an orthogonal matrix ^JX a l s o i n block form, such t h a t JJLGJJL Both (3 and G* have the same eigenvalues ( 1 ) , and, because both and S are orthogonal, Consequently the equal m a t r i c e s , - 21 - are a l s o i n block form c o n s i s t a n t with the symmetry of the molecule. Notice the e s s e n t i a l block form of the P-matrix. In combination (2.1.4) and (2.1.5) provide an N R para- meter f a m i l y of i m p l i e d force constant matrices i f P i s not s p e c i f i e d . The r o l e of the P-matrix i s to mix the A;'» belonging to the same symmetry s p e c i e s . Here the P-matrix contains N T elements c o n s t r a i n e d by N„ c o n d i t i o n s of orthog- L r o n a l i t y . N̂ . degrees of freedom remain. In general a p p l i c a t i o n s , the f a m i l y i s generated by the orthogonal matrix <2^ where K i s a skew symmetric matrix. This w i l l be developed i n l a t e r chapters. However when e x a c t l y two fundamentals, 1/', and J belong to the same symmetry s p e c i e s , the mixing i s simple. In simple mixing the f a m i l y i s generated by a s i n g l e mixing para- meter; here an a l e g r a i c form i s p r e f e r r e d by the author. Pii = Pjj = (\+Xl) - pji = = xin-x1)'"2' P= I elsewhere (2.1.6) S u b s t i t u t i o n of (2.1.6) i n t o (2.1.4) gives a one para- meter f a m i l y of f o r c e constant matrices generated by the mixing parameter X - 22 - 0 = 0 e l sewhere b~ 0 elsewhere (2.1.7) I f the mathematical formalism of equations (2.1.6) and (2.1.7) were reduced to the form of 2 x 2 m a t r i c e s , then i s o - t o p i c homologs of l e s s e r point group symmetry could not be t r e a t e d by convenient means. When the mixing fundamentals are degenerate, the CL and h matrix elements are repeated so as to comply w i t h the degeneracy. As w e l l , the corresponding columns of the U-matrix must be formed i n t o p r o p e r l y o r i e n t e d l i n e a r combinations so that the optimum symmetry f a c t o r i z a t i o n i s obtained (1). I t has been assumed that each normal coordinate defined by the L-matrix can be assigned an experimental v i b r a t i o n a l frequency. With the i m p l i e d f o r c e constant matrix now expressed i n terms of an u n s p e c i f i e d mixing parameter, i t i s p o s s i b l e to c a l c u l a t e the v i b r a t i o n a l frequencies of any i s o t o p i c homolog of the b a s i s molecule as a f u n c t i o n of the mixing parameter. F = F° +- A Xd+X2-)"' F° = ur'Au* B -= u f ' W Qij= Oji = Ajj-Aii - by = bv. = Ajj - An - 23 - The corresponding experimental frequencies then s p e c i f y the mixing parameter and by i m p l i c a t i o n , the f o r c e constants. However, various i s o t o p i c v i b r a t i o n a l frequencies s p e c i f y various mixing parameters and i m p l i e d f o r c e constants. The range of s p e c i f i e d values i n d i c a t e s the e r r o r or d i s p e r s i o n to be a s s o c i a t e d with the i m p l i e d f o r c e constants. These e r r o r s are to be a s s o c i a t e d w i t h the harmonic o s c i l l a t o r approximation r a t h e r than the i m p l i c a t i o n method. (2-2) A Simple Anharmonicity C o r r e c t i o n According to the Born-Oppenheimer approximation p o t e n t i a l energy i s independent of nuclear mass (2); consequently quad- r a t i c (and higher) force constants are considered to be i s o t o p i c i n v a r i a n t s . But the harmonic o s c i l l a t o r approximation i n c l u d e s only q u a d r a t i c f o r c e constants and at l e a s t one very important f e a t - ure of the p o t e n t i a l energy surface i s ignored. A l l s t r e t c h i n g c o o r d i n a t e s are expected to e x h i b i t a d i s s o c i a t i o n l i m i t . In t h i s case the p o t e n t i a l energy surface i s expected to e x h i b i t l e s s - t h a n - q u a d r a t i c curvature. Consequently, f o r i s o t o p i c homolog.: s t u d i e s w i t h i n the harmonic o s c i l l a t o r approximation i t i s n a t u r a l to a s s o c i a t e s m a l l e r e f f e c t i v e f o r c e constants w i t h l a r g e r amplitudes of v i b r a t i o n . The same argument a p p l i e s to valence angle bending - 24 - coordinates as w e l l . One needs to consider the l i m i t i n g values of p o t e n t i a l energy f o r l a r g e d i s t o r t i o n s of the molecule. We have i n c o r p o r a t e d these q u a l i t a t i v e f e atures of p o t e n t i a l energy i n t o the harmonic o s c i l l a t o r approximation as f o l l o w s . The e f f e c t i v e q u a d r a t i c f o r c e constants f o r d i f f e r e n t i s o t o p i c homologs are r e l a t e d through amplitude f a c t o r s , the f^'s below. p j j ( i s o t o p i c homolog) = ^ ' i ^ j Fij (basis molecule) (2.2.1) The j3 'S can be t r e a t e d as a diagonal matrix. To a f i r s t approximation, a d i s t i n c t f̂ > i s needed f o r each d i s t i n c t i s o t o p i c a l l y s u b s t i t u t e d coordinate. For water and i t s deuterohomologs, only three amplitude f a c t o r s are needed: |6>(0D), ^>(HOD) and j3(D0D). Here (OD) designates the OD s t r e t c h i n g c o o r d i n a t e ; (HOD) and (DOD) designate valence angle bending c o o r d i n a t e s . The amplitude f a c t o r s are e a s i l y c a l c u l a t e d v i a the determinants of the v i b r a t i o n a l s e c u l a r equations f o r the i s o t o p i c homologs. The homolog i s noted w i t h a t i l d e . IGFIs Ml and \GF\= lAl Here both /\ and A are composed of experimental v i b r a t i o n a l f r e q u e n c i e s . S u b s t i t u t i o n of (2.2.1) f o r f gives the simple - 25 - equation: J A M S ! IMIS1 (2.2.2) Each symmetry species of each i s o t o p i c homolog f u r n - i shes an equation of the type (2.2.2); i t i s u s u a l l y p o s s i b l e to o b t a i n the i n d i v i d u a l amplitude f a c t o r s by c o n s i d e r i n g a l l equations of the type (2.2.2) i n combination. C a l c u l a t i o n s presented i n the f o l l o w i n g s e c t i o n show that : (1) Expected trends f o r l e s s - t h a n - q u a d r a t i c curvature are confirmed. (2) A few amplitude f a c t o r s enable the use of observed frequencies w i t h n e a r l y as much success as the harmonic f r e q - uencies (2) which are not g e n e r a l l y a v a i l a b l e . (3) The bending coordinates c o n t r i b u t e almost as much to the anharmonicity as do the s t r e t c h i n g c o o rdinates. (2-3) A p p l i c a t i o n s of Simple Mixing Methods In our i n i t i a l work, simple mixing was to be no more than a prologue to general s t u d i e s ; however, simple mixing i t s e l f began to grow i n t o a powerful a n a l y t i c method. The advantage of simple mixing appeared from the o u t s e t . - 26 - Using p e r t u r b a t i o n methods Shimanouchi and Suzuki (5) report f o r c e constants f o r the harmonic frequencies of H^O, HDO, and D 20 determined by Benedict, G a i l e r and P l y e r ( 3 3 ) . Unfortu n a t e l y the symmetric s t r e t c h i n g fundamental f o r D 20 was m i s p r i n t e d . In t h e i r c a l c u l a t i o n Shimanouchi and Suzuki could not detect the r a t h e r l a r g e m i s p r i n t e r r o r ; they use the c o r r e c t value i n a l a t e r note ( 3 6 ) . This simple o v e r s i g h t i s very important; i t unambiguously demonstrates the inadequacy of p e r t u r b a t i o n methods i n the a n a l y s i s of q u a n t i t a t i v e experimental i n f o r m a t i o n . By c o n t r a s t , the simple mixing method showed a gross e r r o r which proved to be no more than the m i s p r i n t already noted. More*over simple mixing methods show that the harmonic bending fundamental of HOD i s uniquely i n c o n s i s t e n t (see t a b l e one). This f a c t has not been p r e v i o u s l y noted elsewhere; a minor r e v i s i o n of the anharmonicity constants a s s o c i a t e d w i t h t h i s v i b r a t i o n i s suggested. Harmonic frequencies are r a r e l y a v a i l a b l e . A comparison of i m p l i e d f o r c e f i e l d s f o r d i f f e r e n t b a s i s molecules using the observed frequencies or zero-one t r a n s i t i o n s , and D̂ O f o r example, r e v e a l e d the anharmonicity p i c t u r e d e s c r i b e d i n the previous s e c t i o n . C a l c u l a t i o n s w i l l show that zero-one t r a n s i t i o n s can be used with n e a r l y as much confidence as the harmonic f r e q u e n c i e s . F i n a l l y , w i t h i n our research group, simple mixing methods became s u f f i c i e n t l y w e l l understood that molecular d i s t o r t i o n s i n the s o l i d s t a t e were explored ( 2 6 ) . U s u a l l y geometric para- - 27 - meters are not i n v e s t i g a t e d by means of v i b r a t i o n s p e c t r o s - copy . Though a s u b s t a n t i a l number of simple mixing molecules have been s t u d i e d with various o b j e c t i v e s i n mind, the pres- e n t a t i o n here s h a l l be l i m i t e d to genuinely new i n f o r m a t i o n . In p a r t i c u l a r , the anharmonicity p i c t u r e of the previous s e c t i o n w i l l be e s t a b l i s h e d . A few p r e l i m i n a r y notes and n o t a t i o n a l devices w i l l s i m p l i f y the p r e s e n t a t i o n of the computations. (1) A l l frequencies are expressed i n cm (2) A s t r e t c h i n g coordinate i s designated as (XY) and the a s s o c i a t e d bond length as r(XY) . Both are i n A. ( 3 ) A valence angle bending coordinate i s noted as (XYZ) and i s expressed i n r a d i a n measure. The e q u i l i b r i u m bond angle, ©(XYZ), w i l l be expressed i n degrees as i s u s u a l . Y designates the c e n t r a l atom. (4) Force constant u n i t s are as f o l l o w s : S t r e t c h - s t r e t c h m i l l i d y n e s / A Stretch-bend m i l l i d y n e s / r a d i a n 2 bend-bend m i l l i d y n e - A/(radian) ( 5 ) Force constants are noted with the appropriate co- or d i n a t e p a i r i n parentheses and separated by a colon. Common atoms w i t h i n the coordinate p a i r are u n d e r l i n e d . Sometimes t h i s provides a unique n o t a t i o n f o r a l l q u a d r a t i c f o r c e constants. - 28 - F o r e x a m p l e , i n m e t h a n e , t h e s t r e t c h - s t r e t c h f o r c e c o n s t a n t s a r e F(CH:CI1) a n d F ( C H : C H ) , t h e s t r e t c h - b e n d c o n s t a n t s a r e F ( C H : H C H ) a n d F ( C H : H C H ) , a n d t h e b e n d - b e n d c o n s t a n t s a r e F ( H ^ : H C H ) , F ( H C H : H C H ) a n d F ( H C H : H C H ) . F o r d i a g o n a l o r p r i n - c i p a l f o r c e c o n s t a n t s , a l l a t o m s a r e u n d e r l i n e d . I f some a t o m s a r e u n d e r l i n e d ^ "Hi* f o r c e c o n s t a n t w i l l be s a i d t o be c o n n e c t e d . I f no a t o m s a r e u n d e r l i n e d , t h e f o r c e c o n s t a n t i s u n c o n n e c t e d . T h i s n o t a t i o n d e v i s e d h e r e s a v e s c o n s t a n t r e f e r e n c e t o r e l a t e d f i g u r e s a n d d i a g r a m s ; h o w e v e r , i t i s n o t y e t s u f f i c i e n t l y w e l l d e f i n e d f o r g e n e r a l u s a g e . (2--3a) H^O, HDO a n d D 2 0 : H a r m o n i c F u n d a m e n t a l s B e n e d i c t , G a i l e r , a n d P l y e r ( 3 3 ) h a v e a n a l y z e d t h e v i b - r a t i o n - r o t a t i o n b a n d s f o r w a t e r a n d i t s d e u t e r o h o m o l o g s ; t h e h a r m o n i c f r e q u e n c i e s w e r e e s t a b l i s h e d b y e m p i r i c a l m e t h o d s ( 2 ) w h i c h i n v o l v e o n l y e x p e r i m e n t a l d a t a . F o r H 2 0 , t h e b a s i s m o l e c u l e , t h e t w o s y m m e t r i c f r e q u e n c - i e s a r e , U)i = 3 8 3 2 . 2 i \ a n d t h e a n t i s y m m e t r i c f r e q u e n c y i s I C J 3 = 3 9 4 2 . 5 1 A n a l y s e s o f t h e r o t a t i o n a l b a n d s f u r n i s h t h e g e o - m e t r i c p a r a m e t e r s ••' '. - 29 - r(OH) = 0.9572 A Cos 0(HOH) = -0.25210 or 8(H0H) = 104.6° This i n f o r m a t i o n enables us to c a l c u l a t e the qu a d r a t i c f o r c e constants and the harmonic frequencies of the i s o t o p i c homologs as a f u n c t i o n of the mixing parameter. These quan- t i t l e s were c a l c u l a t e d over a larg e range, but only the i n t e r - e s t i n g p o r t i o n s s h a l l be reporte d here. TABLE 1: C a l c u l a t e d Harmonic Fundamentals: HDO and D n0. X HDO D ?0* 0.000 1445.9 2822.7 3888.4 1208.8 2757.3 -0.050 1444.7 2824.1 3889.9 1205.8 2764.3 -0.100 1442.9 2827.2 3890.3 1202.6 2771.5 -0.150 1440.7 • 2832.0 3889.7 1199.5 2778.7 -0.200 1438.0 2838.4 3888.2 1196.4 2785.8 EXPTL. 1440.2 2824.3 3889.8 1206.4 2763-8** X=-0.045 1444.8 2824.0 3889.8 1206.1 2763.6 ERROR -4.6 0.3 0.0 0.3 0.2 * U ) j f o r D 20 i s not a f u n c t i o n of the mixing parameter. The c a l c u l a t e d and experimental values agree e x a c t l y 2888.8. ** M i s p r i n t e d as 2783.8 i n ( 3 3 ) . See ( 3 6 ) . - 30 - Of the f i v e fundamentals t a b u l a t e d i n t a b l e one, four of the experimental values appear i n the domain p r e d i c t e d by a mixing parameter i n the f o l l o w i n g i n t e r v a l . - 0.051$: X ^ - 0.037 The bending fundamental f o r HOD s i t s by i t s e l f at X -O.I56. I t i s c l e a r l y i n c o n s i s t e n t w i t h respect to the remaining fundamentals i n c l u d i n g those of the ba s i s molecule. E x c l u d i n g the o u t l i e r and i n c l u d i n g .the three b a s i s f r e q u e n c i e s , the mean e r r o r i n f i t t i n g the experimental data i s 0.1 cm \ Shimanouchi and Suzuki, who d i d not detect an o u t l i e r , r e p o r t a mean e r r o r 1.3 cm"1 ( 3 6 ) . The above i n t e r v a l f o r the mixing parameter places the i m p l i e d f o r c e constants i n the domain s p e c i f i e d by t a b l e 2. TABLE 2: Harmonic Force Constants H^p, HDO, Dg 0 f o r c e constants THIS WORK SHIMANOUCHI and SUZUKI F(0H:0H) 8.4534 ± 0.0002 8.4522 + 0.0079 F(OH:OH) -0.0999 ± 0.0002 -0.1053 ± 0.0079 F(0H:H0H) 0.2276 + 0.0160 0.1608 ± 0.0606 F(HOH:HOH) 0.6977 ± 0.0016 0.6929 ± 0.0019 Mean frequency e r r o r 0.1 m ^ 1,3 cm ̂ Though the forc e constants of Shimanouchi and Suzuki (36) do not s u b s t a n t i a l l y d i f f e r from the i m p l i e d f o r c e constants of t h i s work, t h e i r d i s p e r s i o n s do. The l a r g e r f o r c e constant - 31 - d i s p e r s i o n s and l a r g e r mean frequency e r r o r are due to the f a c t that p e r t u r b a t i o n methods f a i l to detect i n c o n s i s t e n t data. The i m p l i c a t i o n method c l e a r l y i d e n t i f i e s the i n c o n s i s t - ent frequency i n t h i s case. (2-3b) H^O, HDO, D^O: Observed Fundamentals Benedict, G a i l e r and P l y e r (33) have reporte d the observed fundamentals, zero-one t r a n s i t i o n s , f o r H^O, HDO and D 20. For HgO, the b a s i s molecule, the two symmetric frequencies are = 3656.7 \ ) L = 1594.6 and the antisymmetric fundamental i s I/3 = 3755-8 The bond length and valence angle are as before. Again the fund- amental frequencies of HDO and D 20 'are c a l c u l a t e d as a f u n c t i o n of the mixing parameter; however, the corresponding experimental values do not place the mixing parameter i n a small i n t e r v a l . - 32 - TABLE 3: C a l c u l a t e d Fundamentals: HDO andD 0 X HDO D„0* 0.30 1395.1 2718.2 3679.1 d 1183.7 2599.0 0 . 20 1398.2 2703-2 3691.9 1179.8 2607 .6 0.10 1399-5 2694.0 3101.2 1174.9 2618.5 0.00 1398.6 2691.5 3707.0 1169-3 2631.0 -0.10 1395-7 2695.8 : 3708.8 1163.3 2644.6 -0.20 1391.0 2706.5 3706.8 1157.3 2658.3 -0. 30 1384.9 2722.4 3701.2 1151.6 2671.4 -0.40 1378.1 • 2 742.3 3692.3 1146.5 2683.2 EXPTL. 1402.2 2726.7 3707.5 1178.3 2671.5 X=-0.25 1388.0 2713.9 3704.4 1154. 4 2664 .9 e r r o r (cm 14.2 12.8 3.o 23-9 6.6 * \) ( c a l c u l a t e d ) = 3 = 2752.1 1̂  ( e x p t l . ) = 2788.1 Beyond the f a c t that Table 3 f a i l s to i n d i c a t e a c l e a r l y s i g n i f i c a n t mixing parameter, l i t t l e can be s a i d except that the e r r o r s appear to be systematic i n that the bending frequencies f a l l together at one end of the range and the OD s t r e t c h i n g fundamentals f a l l at the other end. The l e a s t squares f i t , X= -0.25, gives e r r o r s of the same s i g n . - 33 - The i m p l i e d f o r c e constants s p e c i f i e d by the i n t e r v a l -0.30 £ X ± 0.10 are poorly d e f i n e d but comparable to those reported by Shimanouchi and Suzuki (5) • TABLE 4: Force Constants: H„0, HDO, D̂ O F(0H:0H) F(0H OH) F(0H:H0H) F(HOH:HOH) mean frequency e r r o r Implied (H^O b a s i s ) 7.59 ± 0.10 -0.10 ± 0.10 0.13 ± 0.66 0.70 ± 0.06 10.7 cm Shimanouchi and Suzuki 7.67 ± 0.11 -0.15 ± 0.11 -0.17 ± 0.42 0.67 ± 0.04 12.8 cm The anharmonicity c o r r e c t i o n d escribed i n s e c t i o n two of t h i s chapter improves the c a l c u l a t i o n by more than one order of magnitude. ^ (OD) can be c a l c u l a t e d from the antisymmetric fundamentals of H 20 and D̂ O through equation (2.2.2). Next ^ (DOD) i s c a l c u l a t e d from the symmetric frequencies of Ĥ O and D._,0 and ^3(0D) c a l c u l a t e d p r e v i o u s l y . F i n a l l y ^(HOD) i s c a l c u l a t e d from a l l of the v i b r a t i o n a l frequencies of Ĥ O and HDO. These amplitude f a c t o r s (3(0D) = 1.013133 (3(D0D)= 1.009937 A(H0D)= 1.002664 - 34 - f u r n i s h the force constant trends F (OH:OH) < F(0J3:0D) F(HOT:HOH)< F(HOD:HOD) < F(DOD:DOD) which are c o n s i s t e n t with l e s s than q u a d r a t i c curvature f o r both bending and s t r e t c h i n g d i s t o r t i o n s of water. When v i b r a t i o n a l frequencies are c a l c u l a t e d v i a equation (2.2J.) as a f u n c t i o n of the mixing parameter, the observed frequencies f u r n i s h a c l e a r and d i s t i n c t s p e c i f i c a t i o n of the mixing parameter and subsequently i m p l i e d f o r c e constants. TABLE 5: C a l c u l a t e d Fundamentals with Amplitude Factors HDO and D" 0 . X HDO D 20* 0.05 1403.2 2726.8 3704.8 1183.8 2659- 0 0.00 1402.4 2726.6 3707.1 1180.9 2665. 6 -0.05 1 4 0 1 .1 2728.1 3708.5 1177.9 2672. 4 -0.10 1399-4 2731.1 3709-9 1174.9 2679. 3 EXPTL. 1402.2 2726.7 3707.5 1178.3 2671. 5 X=-0.04 1401.4 2727.8 3708.2 1178.5 2671. 0 e r r o r 0.8 - 1 . 1 -0.7 -0.2 -0 . 5 * The experimental and c a l c u l a t e d values f o r 1̂3 agree e x a c t l y . These are used to c a l c u l a t e /3(OD) . - 35 - Table Five shows that the mixing parameter f a l l s i n the i n t e r v a l -0.05 < X< 0.00 which i n t u r n provides i m p l i e d f o r c e constants F(OH:'OH) = 7-681 ± 0.003 F(0H:0H) = -0.080 + 0.003 F(0H:H0H) = 0.274 ± 0.063 F(HOH:HOT)= O.663 ± 0.009 mean frequency e r r o r = 0.4 cm 1 In t h i s c a l c u l a t i o n F(OH:HOH) i s c l e a r l y p o s i t i v e as p r e d i c t e d by M i l l s ' (21) h y b r i d o r b i t a l f orce f i e l d . Without anharmonicity c o r r e c t i o n s , the s i g n of t h i s constant i s not c l e a r l y defined (see t a b l e Four). ( 2 - 3 c ) HCCH, HCCD, AND DCCD -.Observed Fundamentals According to the symmetry of l i n e a r molecules, the l o n g i t u d i n a l and trans v e r s e modes do not mix i n the harmonic osc- i l l a t o r approximation, Here only the l o n g i t u d i n a l modes w i l l be considered. For the b a s i s molecule, HCCH, the two symmetric long- i t u d i n a l fundamentals have been placed at = 3372.9 )JL = 1974.0 and the antisymmetric fundamental has been placed at ] ) 5 = 3285.8 - 36 - These v a l u e s , and those of the i s o t o p i c homologs, were s e l e c t e d out of the l i t e r a t u r e by Pimentel and N i b l e r ( 2 3 ) ; the combin- a t i o n , +- \)s j i s very close to and reported values f o r j ^ j vary to some extent. Without amplitude f a c t o r s , the observed fundament- a l s of HCCD and DCCD place the mixing parameter i n the i n t e r v a l , 0.09 < X < 0.16 In the i n t e r v a l the minimum e r r o r s a t t a i n a b l e cover- age as 10.5 cm ^. This compared with the average e r r o r obtained by N i b l e r and Pimentel (23) which ranges from 8.2 to 10.1 cm ̂ depending, on t h e i r choice of weighting f a c t o r s . As w e l l , the i m p l i e d f o r c e constants agree with t h e i r f o r c e constants. The i n t r o d u c t i o n of amplitude f a c t o r s reduces f r e q - uency e r r o r s and f o r c e constant d i s p e r s i o n s by nearly an order of magnitude; however, amplitude c o r r e c t i o n s f o r the n o n i s o t o p i c a l l y s u b s t i t u t e d coordinate (CC) are r e q u i r e d . 1.011654 1.001905 1.004785 * Again the trend expected f o r a p o t e n t i a l with a d i s s o c i a t i o n l i m i t i s obeyed. The e f f e c t i v e q u a d r a t i c force constant decreases w i t h i n c r e a s i n g amplitude. With these amplitude f a c t o r s the observed fundamentals f o r HCCD f o r DCCD (3 (CD) = ^(CC) = ^ ( c c ) = - 37 - of HCCD and DCCD place the mixing parameter i n the i n t e r v a l : 0.112 < X < 0.118. — "i The minimum mean frequency e r r o r a t t a i n a b l e i s reduced to 0.9 cm TABLE 6 : V i b r a t i o n a l Fundamentals: deutero-acetylenes observed c a l c u l a t e d w i t h l ^ ( H C C D ) = 3335.6 J4(HCCD) = 1853-8 V^(HCCD) = 2583.6 i^(DCCD) = 2705.3 i^(DCCD) = 1769.6 l/.(DCCD) = 2439.2 mean frequency e r r o r No amplitude f a c t o r s 3333.1 1 8 4 5 . 5 2562 . 4 2 6 8 7 . 4 1 7 5 2 . 6 2 4 1 1 . 1 1 0 . 5 cm Amplitude f a c t o r s 3 3 3 3 . 4 1 8 5 5 • 1 2 5 8 3 . 6 2 7 0 8 . 0 1 7 6 8 . 0 2 4 3 9 - 2 0 . 9 cm The i m p l i e d f o r c e constants and t h e i r d i s p e r s i o n s are given i n t a b l e Seven along with the forc e constants c a l c u l a t e d v i a p e r t u r b a t i o n methods (without anharmonicity c o r r e c t i o n s ) by N i b l e r and Pimentel ( 2 3 ) . Notice that the i m p l i e d i n t e r a c t i o n constants are not c l e a r l y d i f f e r e n t from zero but the constants obtained by usual methods i n d i c a t e , w i t h l i m i t e d confidence, otherwise f o r F(CH:CC). - 38 - TABLE 7: Force Constants: Acetylene Implied*: HCCH BASIS NIBLER & PIMENTEL, F(CH:CH) 5-920 ± 0.005 5.906 + 0.043 F(CH:CH) 0.014 ± 0.005 -0.033 ± 0.043 F(CC:CC) 15.677 ± 0.026 16.066 ± 0.222 F(CH:CC) -0.005 ± 0.020 -0.109 ± 0.069 mean frequency e r r o r 0.9 cm * The amplitude f a c t o r s employed here w i l l be found i n the pre- ceding t e x t . This c a l c u l a t i o n shows that anharmonic e f f e c t s of the type under c o n s i d e r a t i o n need not be confined t o n o n i s o t o p i c a l l y s u b s t i t u t e d c o ordinates. I m p l i c a t i o n methods confirm t h i s w i t h a c e r t a i n t y equal to the c e r t a i n t y i n the assigned fundamentals. To be p h y s i c a l l y s i g n i f i c a n t , amplitude f a c t o r s should be subject to unambiguous i n t e r p r e t a t i o n . The amplitude f a c t o r s a s s o c i a t e d with the (CC) - s t r e t c h i n g f o r c e constants should not be i n t e r p r e t e d v i a the amplitude of the (CC) - s t r e t c h i n g c o o r d i n - ate. In ac e t y l e n e , the amplitude f a c t o r s which modify F(CC:CC) belong more pro p e r l y to the transverse modes of v i b r a t i o n . The appropriate l i m i t to be considered i s designated by l a r g e transverse displacements of acetylene. In t h i s case i t i s reasonable to a s s o c i a t e a (CC) - double bond wi t h l a r g e t r a n s - verse displacements. For smaller displacements something l e s s 8.2 cm - 39 - than t r i p l e bond stren g t h can be assumed. Consequently i t i s p l a u s i b l e to expect F(CC_:CC_) to decrease as the average t r a n s v e r s e displacements i n c r e a s e . The expected t r e n d F(CC i C C ) ^ . < F ( C C : C C ) H c c o <. F ( C C : C C ) 0 c c f i i s confirmed by the c a l c u l a t e d amplitude f a c t o r s . M u l t i p l e bond anharmonicity of the k i n d d escribed here may be expected f o r the m u l t i p l e bonds contained i n planar molecules. In the planar molecule, the amplitude of the out- of-plane modes w i l l determine the m u l t i p l e bond amplitude f a c t o r s (2-3d) Methane and Deuterohomologs: Observed Fundamentals I f the 29 d i s t i n c t fundamentals of methane and i t s deuterohomologs can be encompassed by a s i n g l e mixing parameter and three amplitude f a c t o r s , (3 (CD), ^(HCD), and L3 (DCD) , then the power of i m p l i c a t i o n methods and the simple p i c t u r e of anharmonic e f f e c t s w i l l have withstood a very severe t e s t . - 40 - Jones and McDowell ( 3 7 ) have reviewed, measured and assigned the fundamentals of methane and i t s deuterohomologs. The four fundamentals of CH^, the b a s i s molecule, are ^(A..) = 2 9 1 6 . 5 2/£(E) = 1 5 3 4 . 0 2 ^ ( P 2 ) = 3 0 1 8 . 7 ]^(F 2) = 1 3 0 6 . 0 R o t a t i o n a l a n a l y s i s provides the bond le n g t h . o r (CH) = 1 . 0 9 3 6 A The t e t r a h e d r a l symmetry defines the bond angles. Without amplitude c o r r e c t i o n s , the 25 observed fundamentals of the four deuterohomologs f a i l to imply a w e l l d e f i n e d s o l u t i o n . A l e a s t squares s o l u t i o n i s i n c l u d e d i n t a b l e e i g h t as a reference f o r the frequency e r r o r s . (S (DCD) was c a l c u l a t e d from f o r C H i | a n d CEV 5̂ (CD) was c a l c u l a t e d from j/5 and llj f o r CH^ and CD^. 3̂ (HCD) was c a l u a l a t e d as an average from the observed fundam- e n t a l s of the three mixed deuterohomologs (and methane). P (CD) = 1 . 0 1 4 1 7 (DCD) = 1 . 0 0 6 3 5 ^(HCD) = 1 . 0 0 4 2 1 These amplitude f a c t o r s resemble those determined f o r water and they comply with the expected t r e n d . - M l - TABLE 8: V i b r a t i o n a l Fundamentals: Deuteromethanes Observed C a l c u l a t e d C a l c u l a t e d CDH3 1155 (E) 1155 1159 1300 (A ) 1303 1306 1471 (E) 1472 1 4 7 3 2200 (A ) 2185 + 2215 + 2945 (A ) 2946 2946 3021 (E) 3017 3017 mean frequency e r r o r 4.0 4.7 CD 2H 2 1033 (A ) 1026 1032 1090 (B ) 1084 1089 1234 (B ) 1232 1237 1329 (A 2) 1329 1334 1 4 3 6 (A ) 1 4 3 4 1 4 3 6 2202 (A ) 2 1 4 2 + 2172 + 2 2 3 4 (B 2) 2229 2260 2976 (A ) 2971 2972 3013 (B ) 3016 3016 mean frequency e r r o r 10.0 8.1 CD^H 1003 (A 1) 997 1002 1036 (E) 1029 1036 1291 (E) 1289 1295 2 1 4 2 (A ) 2101 + 2131 + 2263 (E) 2229 + 2259 2993 (A 1) 2994 2994 mean frequency e r r o r 15-2 3-5 CD^ 997 (F 2) 991 997 1092 (E) 1085 1092 2108 (A x) 2063 + 2092 + 2259 (P 2) 2226 + 2258 mean frequency e r r o r 22.9 4.3 o v e r a l l mean frequency e r r o r 10.4 4 .1 * with amplitude c o r r e c t i o n s given i n the t e x t , •f" e r r o r l a r g e r than 10 cm - 42 - Comparison of c a l c u l a t e d frequencies given i n t a b l e Eight i n d i c a t e s a s u b s t a n t i a l r e d u c t i o n i n the e r r o r f o l l o w i n g the i n c l u s i o n of amplitude c o r r e c t i o n s but not as much as was obtained i n the previous examples. Most of the p e r s i s t i n g e r r o r i s contained i n the (CD) - s t r e t c h i n g f r e q u e n c i e s ; apparently each homolog needs i t s own amplitude f a c t o r f o r t h i s coordinate. A f a c t more s i g n i f i c a n t than the e r r o r r e d u c t i o n i s that most of the observed frequencies imply a mixing parameter i n a small i n t e r v a l , - 0.18 < X< - 0.24, when the amplitude f a c t o r s are employed. In t u r n t h i s mixing parameter i n t e r v a l i m p l i e s f o r c e constants as f o l l o w s : F(CH:CH) = 4 .966 + 0 . 009 F(CH:CH) = 0 .028 + 0 .009 P(HCH:HCH)= 0 .443 + 0 .003 P(CH:HCH) = 0 .104 + 0 .020 F(CH:HCH) = -0 . 104 + 0 .020 F(HCH:HCH)= -0 .093 P(HCH:HCH)= -0 .072 + 0 . 003 The redundant coordinate system employed to describe methane, b r i e f l y d e scribed as 4(CH) -+• 6 (HCH), e x h i b i t s the symmetry c o o r d i n a t e s , 2A + E + 2F 2 ( 1 ) . The genuine v i b r a t i o n s are A + E + 2F ; thus-two of the A force constants, _ 2,3 - F(CH:HCH,A1) = F(CH:HCH) + F(CH:HCH) F(HCH:HCH,A ) = F(H^H:HCH) + 4(HCH:HCH) + F (HCH:HCH), are o f t e n s a i d to be indeterminate ( 1 , 38, 39). We have reviewed the redundant coordinate system and have added new arguments i n favour of the d e t e r m i n i s t i c school of thought (40,41). These arguments (and counter arguments) are c o l l e c t e d i n appendix two of t h i s t h e s i s where i t i s shown that both of the for c e constants i n q u e s t i o n here are zero. In the present case F(CH:HCH) = - F(CH:HCH) which agrees w i t h the h y b r i d o r b i t a l f o r c e f i e l d proposed by M i l l s (21). In simple mixing s i t u a t i o n s i t i s p o s s i b l e to express the v i b r a t i o n a l frequencies belonging to i s o t o p i c homologs of a basi s molecule as a f u n c t i o n of a s i n g l e mixing parameter. The c o r r e s - ponding observed frequencies l o c a t e the mixing parameter i n a s p e c i f i e d i n t e r v a l which i n t u r n places the i m p l i e d f o r c e constants i n s p e c i f i e d i n t e r v a l s . As shown through the preceding examples, i m p l i c a t i o n methods prove to be a powerful technique f o r the a n a l y s i s of experimental i n f o r m a t i o n . CHAPTER THREE: GENERAL MIXING IMPLICATION METHODS G e n e r a l i z a t i o n of the simple mixing method depends on the completion of s e v e r a l t a s k s . ( 1 ) The orthogonal P-matrix of equation ( 2 . 1 . 4 ) must be g e n e r a l i z e d and parameterized. ( 2 ) The N^ parameters need to be expressed i n terms of r e l a t e d mechanical i n f o r m a t i o n such as the v i b r a t i o n a l f r e q - uencies of i s o t o p i c homologs. (3) An estimate of e r r o r or d i s p e r s i o n w i t h i n the i m p l i e d f o r c e constants needs to be formulated. These aspects of the general problem are developed i n the f o l l o w i n g three s e c t i o n s of t h i s chapter; the f o u r t h s e c t i o n i s given over to a p p l i c a t i o n s . ( 3 - D K- Space The d e s i r e d p a r a m e t e r i z a t i o n of a g e n e r a l i z e d orthogonal matrix i s achieved by the orthogonal form where Jo i s orthogonal and denotes a convenient expansion p o i n t . The exponential matrix d e f i n e d and reviewed i n appendix t h r e e , i s The task of determining a u s e f u l fo depends upon u s e f u l approximations. This t o p i c i s explored i n chapter f o u r of (3-1 - 0 orthogonal when the K-matrix i s skew symmetric, Kii = t h i s t h e s i s ; however, the i d e n t i t y m a t r i x , i s sometimes _ 2,5 - adequate. On s u b s t i t u t i o n of (3.1.1) i n t o ( 2 . 1 . 1 ) , (2.1.4) and ( 2 . 1 . 5 ) , the g e n e r a l i z e d i m p l i c a t i o n equations are obtained. 1IT. K (Fo \j ~ ~ ^ unless Al'\ and Ajj designate genuine v i b r a t i o n s belonging to the same symmetry species (3.1.4) As i n the simple mixing case, these equations r e f e r only t o ^ p e c i f i e d b a s i s molecule with w e l l known v i b r a t i o n a l funda- mentals. Again, parameters belonging to the b a s i s molecule are developed from r e l a t e d mechanical information,most conveniently the v i b r a t i o n a l frequencies of i t s i s o t o p i c homologs. For present purposes, assume that the m a t r i x , -l/z. Lo ~~ U P R> ( 3 . 1 . 5 ) , d i c t a t e s a s p e c i f i e d o r d e r i n g of the /\ - matrix. I f symmetry species are i d e n t i f i e d by experimental means, then the problem i s reduced. I f q u a l i t a t i v e normal coordinates are known then the - H6 - problem of ord e r i n g the f\- matrix i s s o l v e d ; however, such knowledge can be a a f e l y assumed only i n the case of s m a l l or very symmetrical molecules. This problem Is s t u d i e d i n gr e a t e r depth i n chapter fou r . As noted i n chapter two, a K-space expansion contains fewer parameters than the F-space expansion; thus the magnitude of a general f o r c e f i e l d s p e c i f i c a t i o n i s reduced from the outset. (3-2) I s o t o p i c Homologs i n K-space The v i b r a t i o n a l s e c u l a r equation f o r an i s o t o p i c homolog can be arranged i n a form most s u i t a b l e f o r the problem i n mind, where (AJCJ A and P belong to the b a s i s molecule and Pj & and A belong to the i s o t o p i c homolog. However, P i s not r e l a t e d to the i s o t o p i c homolog i n the same way that P i s r e l a t e d t o the ba s i s molecule. Rather i t expresses the i s o t o p i c l j - m a t r i x i n terms of the ba s i s L-matrix. L = L A P A (3.2.1) Equation (3-2.1) can be formulated from the v i b r a t i o n a l s e c u l a r equation i n determinant form, J G F - * i \ = o - 47 - by s u b s t i t u t i n g (2.1.4) and re a r r a n g i n g to y i e l d l A V r V S u r W 1 - ^ 0 a contained matrix which i s symmetric and d i a g o n a l i z e d by an orthogonal P. A small p e r t u r b a t i o n i n K-space i s p r e d i c t e d by the p a r t i a l d e r i v a t i v e d̂mn (o y/lrtrt/'rvim (3.2.3) w h e r e f u r n i s h e s the K-space p e r t u r b a t i o n and P„/\ ? r a su f £/i r?=A ( 3 . 2 . 4 ) represents the expansion p o i n t . Equation (3-2.3) depends on p r o p e r t i e s of the expon- e n t i a l matrix designated i n appendix t h r e e . In a p p l i c a t i o n , equations (3-2.3) and (3.-2.4) provide the nucleus of an extremely e f f i c i e n t computational scheme. The U-matrix provides automatic symmetry f a c t o r i z a t i o n of' the G-matrix (see s e c t i o n 1 of chapter two). An equ i v a l e n t f o r m u l a t i o n i n F- space r e q u i r e s 2N^ matrix d i a g o n a l i z a t i o n s per i s o t o p i c homolog (N homologs) or la r g e storage c a p a b i l i t i e s ; the K-space formul- a t i o n r e q u i r e d N - l matrix d i a g o n a l i z a t i o n s with p a r t i a l pre- d i a g o n a l i z a t i o n provided by the U-matrix. - 48 - What remains i s the s o l u t i o n of l i n e a r equations of the form (3.2.5) which has already been discussed i n equations (1.3-1) and ( 1 . 3 - 2 ) . i s formed. But higher terms i n the expansion of (3.2.1) are ignored; thus the P -matrix i s r e p l a c e d by ^(2* and the c a l c u l - a t i o n i s i t e r a t e d u n t i l convergence i s a t t a i n e d . (3-3) Implied D i s p e r s i o n In simple mixing, the i m p l i c a t i o n methods provide an e a s i l y understood p r o j e c t i o n scheme f o r determining the e r r o r a s s o c i a t e d w i t h i m p l i e d f o r c e constants. In general mixing s i t u a t i o n s , e r r o r or d i s p e r s i o n can be e s t a b l i s h e d by more general methods. Consider a f o r c e constant m a t r i x , F 0 , such that f o r a s e r i e s of i s o t o p i c homologs: Equation (3.2.5) i s a Taylor s e r i e s expansion of A tffc - Given a s u f f i c i e n t body of experimental i n f o r m a t i o n , the K-matrix elements are c a l c u l a t e d and the e x p o n e n t i a l matrix Q£* ( 3 - 3 - D the corresponding i m p l i e d f o r c e f i e l d F = L * A C i s composed from the experimental f r e q u e n c i e s . The s c a t t e r i n the - 49 - i m p l i e d f o r c e constants f o r a s e r i e s of i s o t o p i c homologs defines our measure of i m p l i e d d i s p e r s i o n . The d i s p e r s i o n can be c o n c i s e l y expressed i n terms of the maximum frequency e r r o r . A = max I Pie-pal and •j "J The proof f o l l o w s : (3.3-2) (3.3.3) (3-3.4) set where i s c a l c u l a t e d and i s experimental. l F r R | l = 2(»«T]zR(in)k-.^)«j M Terms of the order Ap» (42) provides are ignored. The Cauchy i n e q u a l i t y l'/2. 2. . i . A l l L\Q^ L\ consequently equation (3-3-3) i s obtained. R e c a l l - 50 - Polo ( 4 3 ) has w r i t t e n e x p l i c i t expressions f o r G - matrix elements; however exact expressions are not needed to estimate f o r c e constant d i s p e r s i o n s . Study of Polo's work w i l l j u s t i f y the f o l l o w i n g approximate expressions. For a s t r e t c h i n g c oordinate: where £ - f\ 6f 13 The sets of atoms >A and i n the molecule. B are d i s j o i n t and cover a l l atoms For a valence angle bending coordinate: - 51 - where YQC i s the distance from the c e n t r a l atom i n the bending coordinate to the CC — atom. The set A designates a l l atoms connected to one branch of the angle and & designates a l l of the atoms connected to the second branch of the angle. N e i t h e r set in c l u d e s the apex atom or atoms belonging to the other branches. In combination with i m p l i e d f o r c e constants and a frequency e r r o r , the approximate i n v e r s e G-matrix elements give an estimate of the e r r o r or d i s p e r s i o n to be associated".with the i m p l i e d force constants when i s o t o p i c frequency i n f o r m a t i o n i s the determining f a c t o r . The i m p l i e d d i s p e r s i o n s of equations (3-3.2), (3-3-3) and (3-3-4) w i l l not always agree wi t h s t a t i s t i c a l d i s p e r s i o n s c a l c u l a t e d by means of the f o r m u l a t i o n of Overend and Scherer (46). Q u a l i t a t i v e agreement between the two sets of d i s p e r s i o n s can be expected f o r the p r i n c i p a l f o r c e constants, but f o r some of the o f f diagonal f o r c e constants d i s p e r s i o n s w i l l be very s i g n i f i c a n t l y d i f f e r e n t . Most of the d i f f e r e n c e s are due to the d i f f e r e n t meaning that the two expressions of d i s p e r s i o n c a r r y . The i m p l i e d d i s p e r s - ions t e l l how much the i m p l i e d f o r c e constants must be changed t o cover a l l v i b r a t i o n a l f r e q u e n c i e s ; by c o n t r a s t , the d i s p e r s i o n s of Overend and Scherer t e l l how much the forc e constants can be v a r i e d without exceeding a s p e c i f i e d frequency e r r o r . However the s t a t i s t i c a l d i s p e r s i o n s of Overend and Scherer may i n some cases s e r i o u s l y overestimate the d i s p e r s i o n as w i l l be shown by the f o l l o w i n g , argument. Since the p o t e n t i a l - 52 - energy surface e x h i b i t s a minimum the f o r c e constant matrix i s p o s i t i v e d e f i n i t e ( 3 , 4 4 ) . In t u r n , t h i s r e q u i r e s the f o r c e constants to obey i n e q u a l i t i e s of the k i n d ( 4 5 ) . F u > 0 < F" Fjj ( 3 . 3 . 5 ) For dichloromethane Shimanouchi and Suzuki (5) r e p o r t f o r c e constants and d i s p e r s i o n s c a l c u l a t e d by the Overend-Scherer method as f o l l o w s : (only the magnitude of the numbers i s import- ant here. ) F 2 2 = 3.8 ± 10.0 F 2 i | = 0.2 ± 32.1 F 4 4 = 1.3 ± 23.3 C l e a r l y the p o s i t i v e d e f i n i t e r u l e s , equations ( 3 - 3 - 5 ) , cannot be obeyed over the f u l l range of the r e p o r t e d d i s p e r s i o n s . B r i e f l y p e r t u r b a t i o n methods assume for c e constants to be independent v a r i a b l e s not c o n s t r a i n e d by the p o s i t i v e d e f i n i t e r u l e s ; consequently d i s p e r s i o n can be i n f l a t e d . By c o n t r a s t , i m p l i e d f o r c e constants are always p o s i t i v e d e f i n i t e i f the •matrix i s p o s i t i v e d e f i n i t e ( 3 ) . In the current context, p o s i t i v e d e f i n i t e means that every d i s t o r t i o n from e q u i l i b r i u m c o n f i g u r a t i o n generates an increase i n p o t e n t i a l energy; otherwise e q u i l i b r i u m - 53 - c o n f i g u r a t i o n would not correspond to a minimum i n p o t e n t i a l energy. In summary n e i t h e r measure of d i s p e r s i o n provides e x a c t l y what i s wanted. Implied d i s p e r s i o n may be too small but the Overend-Scherer d i s p e r s i o n s may be too l a r g e . N e i t h e r t e l l s how unique the s p e c i f i e d f o r c e constants are. Continued work i n t h i s area w i l l r e v e a l more appropriate measures of s i g n i f i c a n c e ; the i m p l i e d force f i e l d seems to be more subject to development along the l i n e s of uniqueness. (3-4) Selected A p p l i c a t i o n s i n K-Space The i n t e r p l a y of c a l c u l a t i o n of experimental inform- a t i o n and of the connecting mathematical s t r u c t u r e g e n e r a l l y i n f l u e n c e s the development of the mathematical s t r u c t u r e i n a f a v o r a b l e manner. From the beginning of t h i s p r o j e c t , c a l c u l a t - ions and mathematical s t r u c t u r e were b u i l t i n p a r a l l e l . In t h i s way s e v e r a l u n i n t e r e s t i n g notions were e l i m i n a t e d , and s e v e r a l i n t e r e s t i n g notions were uncovered. In the present s i t u a t i o n the need f o r p a r a l l e l c a l c u l - a t i o n s r e s u l t e d i n two l i m i t a t i o n s . C o n f i g u r a t i o n coordinates were l i m i t e d to bond s t r e t c h i n g and valence angle bending. When i s o t o p i c s u b s t i t u t i o n lowers the p o i n t group symmetry w i t h i n a s e r i e s of i s o t o p i c homologs s p e c i a l c o n t r o l mechanisms f o r degener- ate fundamentals are r e q u i r e d f o r the a p p l i c a t i o n of i m p l i c a t i o n methods; these have not yet been i n c l u d e d i n the general mixing program. In e f f e c t , symmetric tops are excluded u n t i l a modified e d i t i o n of the program i s w r i t t e n . - 54 - A program designed f o r general a p p l i c a t i o n has not yet been w r i t t e n ; i t awaits the completion of s e v e r a l added t a s k s . The K-space p i c t u r e s o f c e n t r i f u g a l d i s t o r t i o n constants ( 8 ) , C o r i o l i s c o u p l i n g c o e f f i c i e n t s ( 9 , 10), and the mean square amplitudes of e l e c t r o n d i f f r a c t i o n experiments ( 7 , 8) need to be developed and cast i n t o the language of automatic computing. F u r t h e r , the K-space p i c t u r e awaits the development of various approximation techniques. To some exte n t , the groundwork f o r these tasks i s e s t a b l i s h e d i n the f o l l o w i n g chapter. Thus f a r only the v i b r a t i o n a l fundamentals of i s o t o p i c homologs - as a means to s p e c i f y the i m p l i e d f o r c e f i e l d - have been discussed. Of three i s o t o p i c s e r i e s s t u d i e d , a) formaldehyde and i t s deuterohomologs b) ethylene and i t s deuterohomologs c) dichloromethane and i t s deuterohomologs, only formaldehyde and ethylene s p e c i f y w e l l d e f i n e d i m p l i e d f o r c e f i e l d s of the most general q u a d r a t i c form. As could be expected i n dichloromethane, the mixing of V (CC1, A ) and J^(C1CC1, A±) i s not s p e c i f i e d by the v i b r a t i o n s of the deuterohomologs. Per- t u r b a t i o n methods a l s o f a i l to s p e c i f y a general f o r c e f i e l d f o r dichloromethane as has been shown by Shimanouchi and Suzuki ( 5 ) Though a complete mechanical p i c t u r e of i m p l i c a t i o n methods i s not yet a v a i l a b l e , the present c a l c u l a t i o n s demonstrate the e f f e c t i v e n e s s of more general I m p l i c a t i o n methods. The c a l - c u l a t i o n s f o r formaldehyde and ethylene i l l u s t r a t e the p o s i t i v e cases. - 5 5 - In the case of dichloromethane, c a l c u l a t i o n s w i l l not be presented; here i t i s important that i m p l i c a t i o n methods do not i n d i c a t e uniqueness when the experimental i n f o r m a t i o n employed does not provide i t . S everal d i s t i n c t l y d i f f e r e n t f o r c e f i e l d s which provided very small frequency e r r o r s were determined f o r dichloromethane; a d d i t i o n a l experimental i n f o r - mation must be i n c l u d e d to s p e c i f y a s i n g l e f o r c e f i e l d . (3-4a) Formaldehyde and i t s Deuterohomologs Shimanouchi and Suzuki ( 5 ) have reviewed the fundam- e n t a l s of formaldehyde and i t s Deuterohomologs and c a l c u l a t e d the general q u a d r a t i c f o r c e constants. T h e i r c a r e f u l l y executed p e r t u r b a t i o n study forms a reference f o r comparison w i t h the i m p l i c a t i o n method. The i n - p l a n e normal coordinates of the C 2y molecules, b r i e f l y represented i n the form 3A + 2B (see f o o t n o t e ) , i n d i c - ate nine q u a d r a t i c f o r c e constants but only four mixing para- meters. Consequently, the i m p l i c a t i o n method i s e a s i e r to apply, f a s t e r to converge and i n i t i a l estimates of the f o r c e constants are not needed. The c a l c u l a t e d f o r c e constants and frequency e r r o r s d i f f e r somewhat from those of Shimanouchi and Suzuki ( 5 ) . The symmetry species n o t a t i o n used here f o r formaldehyde and l a t e r f o r ethylene i s used c o n s i s t e n t l y throughout the l i t e r - ature even a f t e r the J o i n t Commission on Spectroscopy p u b l i s h e d i t s recommendations (47) f o r the s e l e c t i o n of molecular axes. For s m a l l molecules the e a r l i e r r u l e ( 2 ) , T T^, < » seems u s e f u l and c l e a r enough. A 1 1 - 56 - TABLE 9: V i b r a t i o n a l Frequencies: H^CO, HDCO, DnCO. Observed CALCULATED Shimanouchi & Suzuki (10) This Work H2CO 2780 (A x) 2796. 3 (-16.3)* 2780. 0 (0.0) 17^3. 6 (A 1) 1752. 7 (-9.D 1743. 6 (0.0) 1503 (A 2) 1510. 6 (-7.6) 1503. 0 (0.0) 2874 (B 1) 2871. 9 (2.1) 2874. 0 (0.0) 1280 ( V 1273. 2 (6.8) 1280. 0 (0.0) D2CO 2055- 8 (A 1) 2049. 0 (6.8) 2045. 7 (10.1) 1700 (A 1) 1694. 3 (5.6) 1679. 7 (20.3) 1105. 7 (A 1) 1102. 0 (3.6) 1095. 8 (9.9) 2159. 7 (B x) 2167. 6 (-7-9) 2178. 0 (18.3) 990 (B 1) 994. 4 (-4.4) 995. 9 (-5.9) HDCO 2844. 1 (A r) 2833. 7 (10.3) 2831. 8 (12.3) 2120. 7 (A' ) 2116. 6 (4.1) 2105 • 0 (15-7) 1723. 4 (A R ) 1720. 0 (3.4) 1717. 0 (6.4) 1400 (A ' ) 1397. 1 (2.9) 1399. 1 (0.9) 1041 (A' ) ?,0 42. 1 (-1.1) 1038. 6 (2.4) mean e r r o r H 2C0 8. 4 0 .0 HDCO + 5. 8 7 .5 D2CO 4.4 O v e r a l l mean e r r o r 6.2 12.9 6.8 cm -1 * (observed - c a l c u l a t e d ) - 57 - The i m p l i e d force constants belong to the basis molecule, H^CO, and the corresponding v i b r a t i o n a l frequencies are c a l c u l a t e d without e r r o r . In terms of the mean e r r o r i n the fr e q u e n c i e s , the four parameter K-space f i t i s very n e a r l y as good as the nine parameter F-space f i t . The f i v e c o n f i g u r a t i o n coordinates r e q u i r e d to span the i n - p l a n e modes of formaldehyde can be represented i n the form: 2(CH) + (CO) + (HCH) + (OCHH) Though appendix three shows that a redundant coordinate system can be used, non-redundant coordinate systems are used whenever p o s s i b l e . Consequently an in-plane-wag has been defined as f o l l o w s : (OCXY) = (OCX) - (OCY). The geometric parameters s e l e c t e d by Shimanouchi and Suzuki (5) have been employed so that the c a l c u l a t i o n s w i l l be comparable i n every way. o r(CH) = 1.1139 A o r(CO) = 1.2078 A Q(HCH) = 116.56° The i m p l i e d d'l spersions f o r the i m p l i e d f o r c e constants are based on the maximum frequency e r r o r , 20 cm - 1. By c o n t r a s t , the d i s p e r s i o n s r e p o r t e d by Shimanouchi and Suzuki are, i n s e v e r a l cases, much l a r g e r . The l a r g e r i n t e r v a l s s p e c i f y the range of v a r i a t i o n that an i n d i v i d u a l f o r c e constant can d i s p l a y without exceeding a s p e c i f i e d frequency e r r o r . The sma l l e r i n t e r v a l s s p e c i f y the range of v a r i a t i o n necessary to match e x a c t l y a l l of the frequencies employed. - 58 - (OCHH) c o n t r i b u t e s only to the modes while (HCH) and (CO) c o n t r i b u t e only to the A-̂  modes; consequently p o i n t group theory (1) r e q u i r e s that both F(OCHH:CO) and F(OCHH:HCH) be i d e n t i c a l l y zero. As w e l l , symmetry p r o p e r t i e s r e q u i r e F(OCHH:CH) = - F(OCHH:CH). TABLE 10: The for c e Constants of Formaldehyde. Force Constant I m p l i c a t i o n P e r t u r b a t i o n F(CH:CH) 4.320 + 0.063 4.361 ± 0.084 F(CH:CH) 0.089 ± 0.063 0.092 ± 0.084 F(C0:C0) 13-415 ± 0-309 12.577 ± 0.271 F(C0:CH) 0.295 ± 0.140 0.704 ± 0.409 F(HCH:HCH) 0.819 + 0.022 0.840 ± 0.034 F(HCH:CH) 0.156 ± 0.037 -0.115 ± 0.255 F(HCH:C0) -0.920 + 0.082 -0.448 ± 0.146 F(0CHH:0CHH) 0.445 ± 0.032 0.432 ± 0.014 F(0CHH:CH) -0.115 ± 0.046 -0.071 ± 0.126 N e i t h e r the i m p l i e d f o r c e constants nor the frequency e r r o r s c a l c u l a t e d by i m p l i c a t i o n methods d i f f e r g r e a t l y from those obtained by Shimanouchi and Suzuki ( 5 ) . Consequently the four parameter i m p l i c a t i o n method appears to be eq u i v a l e n t to the more cumbersome p e r t u r b a t i o n method. The i m p l i e d f o r c e constants are h e a v i l y biased to.the b a s i s molecule; whether or not t h i s bias can be used to advan- tage awaits f u r t h e r a n a l y s i s . - 59 - Toward the end of t h i s chapter i t w i l l be shown that i m p l i e d force constants d i s p l a y a p a t t e r n of consistency which agrees with some of the p r i n c i p l e s of the h y b r i d o r b i t a l f o r c e f i e l d of M i l l s ( 2 1 ) . The i m p l i c a t i o n method g e n e r a l i z e s without l o s s of s i g n i f i c a n c e - i n terms of frequency e r r o r s or molecular s t r u c t u r e . ( 3 - 4 b ) Ethylene and i t s Deuterohomologs. In c o n j unction with experimental work Crawford, L a n c a s t e r , and Inskeep have c a l c u l a t e d the q u a d r a t i c f o r c e constants of ethylene (48) from the v i b r a t i o n a l frequencies of the two homologs, C 2 ^4 a n o " ^ 2 D 4 ' T n e ^ imposed one c o n s t r a i n t , F(HCH:CH) = 0 , and d i d not attempt to adjust t h e i r f o r c e con- st a n t s to minimize frequency e r r o r s . L a t er Brodersen (49) repeated the determination with a d e l i b e r a t e attempt to minimize frequency e r r o r s and without c o n s t r a i n t s . Again the f u l l symmetry homologs dominated the c a l c u l a t i o n ; intermediate homologs were used to define one of the f o r c e constants. Though Brodersen's choice of fundamentals d i f f e r e d l i t t l e from those of Crawford, Lancaster and Inskeep, the r e p o r t e d f o r c e constants d i f f e r s i g n i f i c a n t l y (see t a b l e e l e v e n ) ; however, the Brodersen f o r c e constants can be viewed as a refinement of the e a r l i e r work. Since automatic computing became a v a i l a b l e , a general valence f o r c e f i e l d f o r ethylene has not been reported. Scherer and Overend have reported a s i x parameter Urey Bradley force f i e l d (UBFF) ( 5 0 ) . More r e c e n t l y F l e t c h e r and Thompson have reported a ten parameter Hybrid O r b i t a l f o r c e f i e l d (HOFF) ( 2 2 ) . - 60 - Automatic computing would enable the use of data from a l l the deuterohomologs and provide minimum frequency e r r o r s i n the l e a s t squares sense. In e f f e c t the i m p l i e d f o r c e f i e l d r e p o r t e d here completes the refinement of the qu a d r a t i c f o r c e constants f o r ethylene. The f i f t y - f o u r fundamentals of the s i x deutero- ethylenes (51) are used to f i x the s i x mixing parameters f o r the b a s i s molecule ^C^H^. Though s i g n i f i c a n t d i f f e r e n c e s appear, the i m p l i e d f o r c e constants appear to be a refinement of'the e a r l i e r work reported by Brodersen. I t i s p o s s i b l e to c l a s s i f y the v i b r a t i o n s of the deuteroethylenes under the point group of the p o t e n t i a l energy, °2h ' ]/^aen i s o t o p i c s u b s t i t u t i o n reduces the po i n t group symmetry, ~~ species are scrambled but i n accord with the lower p o i n t group. Consequently a l l v i b r a t i o n s can be c l a s s i f i e d by e n c l o s i n g the scrambled D - species i n parentheses. 2h H 2C 2H 2 + D 2C 2D 2 : 3 A e + 2B n l g + 2 B 2 u + 2B_ 3u HDC 2H 2 + HDC 2D 2 : ( 3 A 8 + l g + 2 B 2 u + 2B_ ) 3u t r a n s - HDC2HD ( 3 A g + 2 B l g ) + (2B 2u + 2 V c i s -HDC2HD : < 3 A g + 2 B2u> l g + 2 V 1,1-dideuteroethylene : ( 3 A G + B 3 u > l g + 2 B2U> This n o t a t i o n immediately r e v e a l s the i n f o r m a t i o n needed to formulate the product r u l e s . F u r t h e r i t shows at a glance the dependence of v i b r a t i o n a l frequencies on symmetry f a c t o r e d f o r c e constants. - 61 - For example, i n trans d i d e u t e r o e t h y l e n e , the f i v e symmetric v i b r a t i o n s , (3A + 2 EL ) , deoend on the s i x A - f o r c e constants g l g g and the three EL - f o r c e constants. In K-space these f i v e l g v i b r a t i o n s depend on the three A and the one B mixing para- g -Lg meters. This s t r u c t u r e c a r r i e s over to the K-space p e r t u r b - a t i o n equations, (3-2.5), and i s of great help i n a s s i g n i n g experimental frequencies to the i n d i v i d u a l l i n e a r equations. I t should be r e a l i z e d that the c l a s s i f i c a t i o n under Dz\r\ , (3A + 2B-, ) + (2B~ + 2B_ ) , means e x a c t l y the same t h i n g g l g 2u 3u as the c l a s s i f i c a t i o n , 5A + 4B , obtained under C~, ; however, 5 g u 2h' ' the D g n n o t a t i o n emphasizes the u n d e r l y i n g s t r u c t u r e . The v i b r a t i o n s of the b a s i s molecule, H^C^H^, i n d i c a t e f i f t e e n independent q u a d r a t i c f o r c e constants but only s i x mixing parameters. The planar c o n f i g u r a t i o n coordinates adopted f o r t h i s study, MCH) + (CC) + 2(HCH) + 2(CCHH), c o n t r i b u t e to the v i b r a t i o n s as f o l l o w s : 4(CH) A + B., + B 0 + B_ g l g 2u 3u (CC) A g 2(HCH) A g + B ^ 2(CCHH) B 0 + B, 2u l g F o r t u n a t e l y the CH-stretching v i b r a t i o n s and the (CC) - s t r e t c h i n g v i b r a t i o n are known to be c h a r a c t e r i s t i c ; thus no problems are expected i n s t r u c t u r i n g the i m p l i e d f o r c e f i e l d . - 62 - For c l a r i t y l e t us note that (CCHH) 1 = 2(CCHH) (CCHH) 2 = fa-fa The symmetry p r o p e r t i e s of the c o n f i g u r a t i o n coord- i n a t e impose f i v e symmetry r e s t r i c t i o n s on the forc e constants, F(CCHH.-CC) = 0 F(CCHH:HCH) = 0 F(CCHH:HCH) = 0 F(CCHH:CH)- + .F(CCHH:CH) = 0 F(CCHH:CH, c i s ) + F(CCHH:CH, t r a n s ) = 0 In t h i s c a l c u l a t i o n the geometric parameters s e l e c t e d by Herzberg (2) have been employed. o r(CH) = 1.086 A o r(CC) = 1.339 A 9(HCH) = 117.6° Beginning with PQ — I i n equation ( 3 . 2 . 4 ) , mean - 63 - e r r o r i n the frequencies of 20.6 cm the K-space i t e r a t i o n sequence described i n s e c t i o n (3-2) proceeds smoothly to the i m p l i e d f o r c e constants given i n t a b l e eleven and c a l c u l a t e d frequencies given i n t a b l e twelve. The l a r g e s t frequency e r r o r appears i n the (CC)- s t r e t c h i n g frequency f o r 1,1-dideuteroethylene (the c a l c u l a t e d frequency f a l l s 33 cm ^ below the observed frequency). In f a c t a l l of the i m p l i e d (CC) s t r e t c h i n g fundamentals f a l l below the observed fundamental. The same discrepancy appears i n formaldehyde f o r the (CO ) - s t r e t c h i n g fundamentals. In these cases; m u l t i p l e bond anharmonicity l i k e that d e s c r i b e d f o r acetylene i s suspected. The remaining l a r g e frequency e r r o r s i n the deutero- ethylenes appear, as expected, i n the ( C D ) - s t r e t c h i n g f r e q u e n c i e s . The f o r c e constant d i s p e r s i o n s , c a l c u l a t e d by the i m p l i c a t i o n technique of s e c t i o n ( 3 - 3 ) , have not been i n c l u d e d i n t a b l e eleven. D i s p e r s i o n s based on twice the mean frequency e r r o r are reported s e p a r a t e l y below. A l l but eleven o f / c a l c u l - ated frequencies f a l l w i t h i n t h i s range. For the p r i n c i p a l f o r c e constants the i m p l i e d d i s p e r s i o n s are comparable to those i m p l i e d F l e t c h e r & Thompson (22) F(CC:CC) +0.18 ±0.15 F(CH:CH) ±0.044 ±0.036 F(HCH:HCH) +0.013 ±0.008 F(CCHH:CCHH) +0.018 ±0.009 determined by F l e t c h e r and Thompson. The remaining i m p l i e d d i s - persions are geometric means/the above v a l u e s , see equation (3-3-3) TABLE 11: The Force Constants of Ethylene - - Crawford Brodersen Scherer & F l e t c h e r & This Lancaster & ( 4 9 ) Overend Thompson work Inskeep ( 4 8 ) (50) (22) F (CC_: CC_) 10.896 11.08 9.038 9- 305 11.184 F(CH:CH) 6.126 4.77 5 . 1 4 9 5.168 5 . 0 0 4 F(HCH:HCH) 0.731 0.708 0.661 0.683 0.725 F(CCHH:CCHH) 0.373 0.334 0 . 2 4 8 0.269 0.332 F(HCH:CC) -0.920 -0.826 -0.264 -0.273 -0.943 F(HCH:CH) 0.369 0.130 -0.123 0.087 0.182 F(HCH:CH) 0.000 0 . 1 4 1 0.000 0.000 0 . 2 4 1 F(HCH:HCH) 0.035 -0.012 0.000 0.022 0 .066 F(CCHH:CH) 0 . 5 H -0.163 0.123 0.098 -0.199 F(CCHH:CH, c i s ) 0.234 -0.332 0.000 0 .000 -0.313 F(CC_HH:CCHH) 0.035 0.006 - 0 . 0 4 8 - 0 . 0 4 8 0 . 0 1 4 F(CH:CC) 0.000 0.00 0.367 0.000 -0.154 F(CH:CH) 0 . 0 4 3 0.02 0.000 -0.012 0.031 F(CH:CH1 C i s ) -0.020 ' -0.06 0. 000 -0.018 - 0 . 1 0 4 F(CH:CHj t r a n s ) 0.050 0 . 1 4 0.000 -0.018 0.089 C o n s t r a i n t s F(CH:HCH) = = 0 NONE UBFF* HOFF NONE Parameters 14 15 6 10 6 mean e r r o r (cm-"'") 13.4 8.1 12.0 8.8 6.5 This i s the valence bonding image of the Urey Bradley force f i e l d ( 1 4 ) . - 65 - TABLE 12: V i b r a t i o n a l Fundamentals: ethylene and deuterohomologs SYMMETRY OBSERVED IMPLIED BRODERSEN MOLECULE 3026 3026 3026 1623 1623 1630 g 13^2 1342 1350 3103 3103 3110 2B l g 1236 1236 1238 3106 3106 3110 2 B 2 u 810 810 815 2990 2990 3001 2 B 3 u 1444 1444 1455 mean frequency e r r o r 0.0 6.0 H 2 C 2 H 2 2660 2238 2253 1518 1508 1511 3A g 985 977 978 2310 2312 2308 1011 1007 1009 2345 2336 2340 584 576 580 2200 2169 2189 2B_ 3u 1078 1065 1067 mean frequency e r r o r 11.8 6.2 D 2C 2D 2 3017 2998 3009 2230 2231 2236 1585 1552 1561 1384 1387 1402 1031 1024 . 1022. . . 3093 3096 3100 2334 2334 2336 1150 1153 1155 ( 2 B l g + 2 B 2 u ' 660 664 668 mean frequency e r r o r 8.1 9.7 ASYM-H C D„ 2 2 2 - 66 - SYMMETRY OBSERVED IMPLIED BRODERSEN MOLECULE 3059 3069 3066 2299 2283 2300 1571 1562 1574 1218 1213 1209 (3A g + 2 B 2 u 646 647 652 3054 3051 3057 2254 2238 2253 1342 1336 1335 (2B n + 2B 0 ) 1039 1043 1048 l g 3u mean frequency e r r o r ... 7.8 5 ..8 CIS-HDC2HD 3045 3043 3069 2285 2285 2279 1571 1562 1572 1286 1281 1284 (3A + 2 B ) 1004 1000 1007 § l g 3065 3059 3040 2273 2268 2298 1299 1288 1280 (2B_ + 2B 0 ) 678 659 664 mean frequency e r r o r 6.9 13.2 TRANS-HDC2HD - 67 - H 2C 2HD (3A g + 2B n . 2B 0 + 2B 0 ) l g + 2u 3u D 2C 2HD OBSERVED IMPLIED IMPLIED OBSERVED 3096 3101 3051 3049 3061 3062 2331 2332 3002 3000 2269 2281 2276 2278 2205 2222 1606 1589 1532 15^7 1401 1401 1280 1289 1290 1288 1043 1045 1129 1128 994 999 713 716 612 610 mean frequency e r r o r 3-7 7.2 Neith e r Crawford, Lancaster and Inskeep nor Brodersen pres- ent c a l c u l a t e d frequencies f o r the low symmetry homologs. F l e t c h e r and Thompson c a l c u l a t e frequencies f o r H~2C2HD and report a mean frequency e r r o r 9«1 cm 1 f o r t h i s molecule. The s i x parameter Urey Bradley force f i e l d Is too h i g h l y c o n s t r a i n e d to compare c a l c u l a t e d f r e q u e n c i e s . I f the f i v e c a l c u l a t i o n s are intercompared by m u l t i p l y i n g the mean frequency e r r o r and the number of ad j u s t a b l e parameters, then the Implied force f i e l d takes f i r s t rank. Moreover by t h i s measure of me r i t , the h y b r i d o r b i t a l f o r c e f i e l d f o l l o w s the Urey Bradley f o r c e f i e l d . - 68 - ( 3 - 4 c ) Chemical S i g n i f i c a n c e On one hand q u a d r a t i c f o r c e constants are mechanical parameters c o n s i s t a n t with mechanical i n f o r m a t i o n of exper- imental o r i g i n . On the other hand, as measures of bonding for c e s and i n t e r a c t i o n s , quadratic f o r c e constants should comply with chemical bonding s t r u c t u r e . Consequently the q u a d r a t i c f o r c e constant serves both p h y s i c a l and chemical purposes. P h y s i c a l s i g n i f i c a n c e , though subject to w e l l d e f i n e d measurement, i s l i m i t e d to pure mechanics. Chemical s i g n i f i c a n c e e s t a b l i s h e s i t s e l f through chemical bonding and i s subject to a wider i n t e r p r e t a t i o n , but chemical s i g - n i f i c a n c e i s not subject to w e l l defined measurement. Because i m p l i e d f o r c e constants are formulated w i t h minimal reference to chemical bonding s t r u c t u r e s , t h e i r chemical s i g n i f i c a n c e needs d e l i b e r a t e emphasis. In some r e s p e c t s , i t i s remarkable that i m p l i e d f o r c e constants e x h i b i t any degree of chemical s i g n i f i c a n c e at a l l . Ethylene and formaldehyde are i n f a c t c l o s e l y r e l a t e d molecules; i f one CH^-group of ethylene i s regarded as a s i n g l e atom, the r e s u l t a n t molecule would be an i s o t o p i c homolog of formaldehyde. Table t h i r t e e n shows tha t f o r c e constants f o r the two r e l a t e d molecules e x h i b i t unexpected s i g n and s i z e agreement. However the bonding u n i t , H 2 C = ^ does not possess a set of f o r c e constants which are l a r g e l y independent of the s u b s t i t u e n t as has been shown (17,20) f o r the bonding u n i t H C- . - 69 - TABLE 13: S i m i l a r Implied Force Constants F(CH:CH) F(CX:CX) F(CH:CH) F(CH:CK) F(HCH:HCH) F(CH:HC_H) F(CX:HCH) F(XCHH:XCHH) F(XCHH:CH) Formaldehyde (X= oxygen) 4 . 320 13.415 0.089 0.295 0.819 0.156 -0.920 0.445 -0.115 Ethylene (X= Carbon) 5.004 11.184 0.031 -0.154 0.725 0 .182 -0.943 0.332 -0.199 The s i m i l a r i t i e s confirm some of the p r i n c i p l e s used by M i l l s i n the h y b r i d o r b i t a l f o r c e f i e l d ( 2 1 ) . He suggests that F(HCH:CH) - k F(CH:CH) F(HCH:CX) = k 1F(CX:CX) where the constants k and k 1 depend mostly upon h y b r i d i z a t i o n and to some extent on the nature of the s u b s t i t u e n t . I f sp2" h y b r i d - i z a t i o n dominates other e f f e c t s , then k 1 = -2k. In both formaldehyde and ethylene the c e n t r a l atom i s h y b r i d i z e d and the h y b r i d i z a t i o n constants should be ne a r l y the same. - 70 - HYBRIDIZATION CONSTANTS formaldehyde ethylene k 0.0361 0.0364 k 1 -0.0686 -0 .0841 -1.90 -2. 31 The h y b r i d i z a t i o n constants, k , are n e a r l y equal f o r 1 the two molecules. Those i n v o l v i n g the double bonds, k , are comparable and r e l a t e d to the s i n g l e bond h y b r i d i z a t i o n constant, k, the expected way. I t i s g r a t i f y i n g to confirm these aspects of the h y b r i d o r b i t a l f o r c e f i e l d (21) without assuming i t . However, the h y b r i d o r b i t a l f o r c e f i e l d p r e d i c t s F(CCHH:CH) = F (HCH:CH), but the i m p l i e d f o r c e constants agree more c l o s e l y with a Urey Bradley force f i e l d here (14), As w e l l , n e i t h e r of the model f o r c e f i e l d s i n c l u d e s s t r e t c h i n g I n t e r a c t i o n s but the i m p l i e d f o r c e f i e l d i n d i c a t e s d e f i n i t e s t r e t c h i n g i n t e r a c t i o n s . As shown by Heath and L i n n e t t ( 5 2 ) , i n t e r a c t i o n f o r c e constants can be i n t e r p r e t e d on s t r i c t l y g e o m e t r i c a l grounds. Let a l l coordinates except /O-ii and P-y be f i x e d at t h e i r e q u i l - i b r i u m values. Let ,Oj be assigned a d e f i n i t e displacement, &j These co n s t r a i n e d c o n d i t i o n s imply a pseudo-equilibrium v a l u e , /Qf- } f o r the i n t e r n a l coordinate a s s o c i a t e d w i t h OA i n that the p o t e n t i a l energy i s minimized. F(CCHH:CH) = - F(HCH:CH). - 71 - In formaldehyde, i f the (CO) bond i s s t r e t c h e d , the (CH) bond i s shortened. In ethylene, i f the (CC) bond i s shortened the (CH) bond i s lengthened. Consequently the p s u e d o - e q u i l i b - rium geometry of formaldehyde tends toward the e q u i l i b r i u m geometry of ethylene and conversely. In t h i s sense the only signa t u r e d i s p a r i t y i n t a b l e t h i r t e e n i s q u i t e c r e d i t a b l e . The very few i m p l i e d f o r c e constants now i n hand do not provide a s u f f i c i e n t foundation f o r a general d i s c u s s i o n of t h e i r s p e c i a l chemical s i g n i f i c a n c e ; however, these examples i n d i c a t e the d e f i n i t e value of continued s t u d i e s w i t h i m p l i c - a t i o n methods. CHAPTER POUR: APPROXIMATION TECHNIQUES Thus f a r the i m p l i c a t i o n method has been l i m i t e d t o the case where there e x i s t s s u f f i c i e n t experimental i n f o r m a t i o n to s p e c i f y a l l q u a d r a t i c f o r c e constants. C l e a r l y l a r g e r molecules with low symmetry, methylamine f o r example, need not submit to such a general approach. Consequently, f u r t h e r progress w i t h i m p l i c a t i o n methods w i l l depend upon approximation techniques which e i t h e r operate e n t i r e l y w i t h i n the i m p l i c a t i o n scheme or cooperate with the more t r a d i t i o n a l methods of parameterized p o t e n t i a l energy. The weighted t r a c e equations d e l i n e a t e d i n s e c t i o n one of t h i s chapter provide some i n f o r m a t i o n about the l a t t e r o b j e c t i v e : cooperation of i m p l i c a t i o n methods and t r a d i t i o n a l p a r a m e t e r i z a t i o n of molecular p o t e n t i a l energy. F o r t u n a t e l y the i m p l i c a t i o n scheme lends i t s e l f to a unique approximative technique whenever some v i b r a t i o n s of a molecule can be s a i d to be c h a r a c t e r i s t i c v i b r a t i o n s . The second s e c t i o n of t h i s chapter describes the r o l e of c h a r a c t e r i s t i c v i b r a t i o n s i n the a n a l y s i s of the v i b r a t i o n a l s e c u l a r equations e n t i r e l y w i t h i n the i m p l i c a t i o n scheme. (4-1) The Weighted Trace Equations Consider the f u n c t i o n (4.1.1) - 73 - where F=ur\<£lAd*P?rILUT (4.1.2) as defined i n equations (3-1-3) and S i s an o r d e r i n g parameter - see appendix three. The weighting matrix W i s some s p e c i f i e d matrix w i t h the same symmetry as the F- matrix. In essence's/forms various l i n e a r combinations of force constants; e x p l i c i t choices f o r the V V - m a t r i x w i l l f o l l o w from the general development. Notice that Thus /"L i s defined such t h a t where JUT i s a dia g o n a l matrix. The diagonal.JfoP-matrix e n t r i e s are the eigenvalues of the matrix product , (3 W The techniques described i n appendix three give the general weighted t r a c e equation: J^B A^R. Jjf^ = Xcife/c + higher terms (4.1.4) - 74 - Consequently Trace Wf- i s maximum i f AJX- < py- Trace lA/P i s minimum i f J/X\ ^> j^j". Trace W~ i s saddle p o i n t i f JjS^ ~JjXi with respect to the mixing parameter y^. Suppose that the three force constants P P and ^ mm, mn ^nn dominate the two v i b r a t i o n a l frequencies Xv\ a n d (or the mixing parameter K i s most impo r t a n t ) . Then mn equations (4.1.3) and (4.1.4) provide bounds f o r the force constants. ( /{^ > ^ _ ( 4 . 1 . 5 ) £ Finn £ Gwn( /U + 'M~ Am)~yGmm G w Equations (4.1.5) are formed from three d i f f e r e n t W-matrices - one to s e l e c t each d i f f e r e n t f o r c e constant. The i n e q u a l i t i e s (4.1.5) are exact f o r the v i b r a t i o n a l s e c u l a r equation of order two. Otherwise they are not exact but they may i n d i c a t e important i n t e r a c t i o n f o r c e constants. For example, the two B 2 u modes of benzene and the inverse G-matrix elements (symmetry factored) (1) give the f o l l o w i n g bounds f o r the symmetry f a c t o r e d f o r c e constants. - 75 - 3-353 - F(CC:CC,B ) £ 4.375 O.916 ^ F(HCCC :HCCC ,B ) 6z 1.195 -I.658 ^ F(HCCC:CC,B 2 u) ± -0.593 Here the a p p l i c a t i o n of (4.1.5) i s exact and l i m i t s f o r an e s s e n t i a l i n t e r a c t i o n constant have been set by simple means. Thus, F(HCCC:CC,B 2 u) has not been s p e c i f i e d but i t cannot be zero. The Importance of t h i s i n t e r a c t i o n constant has already been e s t a b l i s h e d ( 5 3 ) . Equations (4.1.5) may prove most u s e f u l i n s e l e c t i n g important i n t e r a c t i o n constants. Another way of i d e n t i f y i n g important i n t e r a c t i o n f o r c e constants by i m p l i c a t i o n methods f o l l o w s . Assume an approximate f o r c e constant matrix f-Q and solve the. s e c u l a r equations where A a represents c a l c u l a t e d f r e q u e n c i e s . The a s s o c i a t e d i m p l i e d f o r c e f i e l d F = JL~* A C may w e l l i n d i c a t e important c o r r e c t i o n s to the approximate f o r c e constants used to assemble The A -matrix i s composed of experimental f r e q u e n c i e s . In t h i s a p p l i c a t i o n the d i s p e r s i o n equations, (3.3-3) and ( 3 - 3 - 4 ) , show that i f the c a l c u l a t e d and observed frequencies are close, so are the t r i a l and i m p l i e d f o r c e constants c l o s e . Consequently l i k e - 7 6 - the p e r t u r b a t i o n method i t s e l f , t h i s approach may assume more fore-knowledge than i s provided by the e x i s t i n g g u i d e l i n e s to chemical bonding. Yet i t i s b e l i e v e d that the i m p l i e d f o r c e constant matrix a s s o c i a t e d with approximate fo r c e constants w i l l provide a l e s s cumbersome approach than p e r t u r b a t i o n methods when the problem i s too complex to solve e n t i r e l y w i t h i n the p e r t u r b a t i o n scheme. (4-2) P a r t i c i p a t i o n and Molecular P a r t i t i o n i n g I t i s o f t e n p o s s i b l e to a s s o c i a t e s e l e c t e d v i b r a t i o n a l frequencies with s e l e c t e d p a r t s of the whole molecule - u s u a l l y chemical groups such as -NH^ or -CH^ I f a v i b r a t i o n a l frequency belongs to a chemical group, then the remaining atoms p a r t i c i p a t e to a much l e s s e r extent i n that v i b r a t i o n . This s e c t i o n considers the p a r t i t i o n i n g of a molecule i n t o two sets of atoms and des c r i b e s the p a r t i c i p a t i o n of the i n d i v i d u a l sets i n the various v i b r a t i o n s . C h a r a c t e r i s t i c frequencies are c l o s e l y r e l a t e d to mass dependence w i t h i n the v i b r a t i o n a l s e c u l a r equations. C h a r a c t e r i s t i c frequencies w i l l depend mostly on s e l e c t e d atoms and consequently t h e i r masses. D i f f e r e n t i a t i o n of the matrix equations G - L l S and L?FL. =A - 77 - th with respect to the r e c i p r o c a l mass of the ^ — atom leads to the equations: Both /I and QA/c^j^are d i a g o n a l . Consequently (4.2.1) (4.2.2) (4.2.3) Where i s a skew symmetric matrix . In combination (4.2.1) and (4.2.3) provide the fundamental mass equation. jj Because CA, KAT. = O / > ' 3 £ = /C = The G-matrix i s of the form (4.2.4) - 78 - t h e r e f o r e (see appendix one) which i s p o s i t i v e semi d e f i n i t e ( 4 4 ) , and moreover £> = Za; (QS/d/U<) ( 1 . 2 . 5 ) Thus the equation gives Zx /Joe = i ( 4.2.6) Consequently the q u a n t i t y Sin Aii th can be viewed as the p a r t i c i p a t i o n of the QC atom i n the . th »J v i b r a t i o n a l frequency. - 79 - As w e l l , i t can be e s t a b l i s h e d that ( 4 . 2 . 8 ) The l a t t e r equation f o l l o w s from the i d e n t i t y ( 5 4 ) . P a r t i t i o n i n g . J Let a molecule be p a r t i t i o n e d such that each of i t s atoms belongs to e i t h e r the set J\ or the set B Define the p a r t i c i p a t i o n m a t r i ces: yUcC ( / TTfA) = £^{L m«L ) ( 4 . 2 . 9 ) Both 777/1) and 7T/B) are square p o s i t i v e s e m i d e f i n i t e matrices and t h e i r diagonal elements are sums of a t o m - p a r t i c i p a t i o n s . Through ( 4 . 2 . 5 ) they sum to the i d e n t i t y matrix Ti(A) + 7 T ( B ) = I ( 4 . 2 . 1 0 ) - 80 - and they obey p o s i t i v e s e m i d e f i n i t e i n e q u a l i t i e s (45) hi For a nonplanar molecule riOMfl TTffii) $ 3A/A where i s the number of atoms i n A ( 4 4 , 4 5 ) . The q u a n t i t y defines the p a r t i c i p a t i o n of the A— set of atoms i n the J — v i b r a t i o n . I f the A — set of atoms does not p a r t i c i p a t e i n the ft — v i b r a t i o n , then TT(A)„n = O TT(B)nn = 1 and the p o s i t i v e s e m i d e f i n i t e i n e q u a l i t y gives TT(A)n^ - O f o r a l l j again TT(&>)flj = O f o r a l l j except j =n r e c a l l TT(A) + TJ(B) = I moreover note the strong i m p l i c a t i o n of n u l l p a r t i c i p a t i o n TTMnn=0 implies M * ^ ~ ^ = 0 f o r a l l CC contained i n the A-set. - 81 - Consequently I s o t o p i c s u b s t i t u t i o n w i t h i n the r\ -set leaves the c h a r a c t e r i s t i c v i b r a t i o n s of the E> - set i n v a r i a n t . This f a c t provides a t e s t f o r the v a l i d i t y of n u l l p a r t i c i p a t i o n ! As an example of p a r t i t i o n i n g and p a r t i c i p a t i o n l e t us consider methylamine. Let there be s i x modes c h a r a c t e r i s t i c of the methyl-set, three modes c h a r a c t e r i s t i c of the amine-set, and s i x mixed modes. The p r i n c i p l e s j u s t described r e q u i r e TTXCH^and 7T(NH2) to be of the form given by t a b l e f o u r teen. Unity designates the i d e n t i t y matrix and the arrows show the i n t e r - r e l a t i o n s h i p s used to formulate 77" ( C H ) and 7T(NH ). - 82 - TABLE 14 P a r t i c i p a t i o n Matrices f o r Methylamlne The d i s t r i b u t i o n of u n i t , nonzero, and zero blocks i n the p a r t i c i p a t i o n matrices f o r (CH ) - (NH^); the arrows summarize the d e r i v a t i o n of the zero-blocks f o r l i m i t e d p a r t i c i p a t i o n . TT(CH ) (CH^)-MODES UNITY kA' + 2 A" ZERO ZE i RO ZERO MIXED MODES NONZERO 3A» + 3A" ZE i RO RO H^)-MODES a£jR\J (N < 0 - ZERO — (CH^)-MODES ZERO ZERO ZERO MIXED MODES NONZERO 3A' + 3A" ZERO ZERO ZERO (NH )-MODES UNITY 2A' + A" Tt (NH 2) - 83 - The symmetry species show the f u r t h e r f a c t o r i z a t i o n obtained when methylamine i s given a plane of symmetry. The fundamentals assigned by Delle p i a n e and Ze r b i (55) support the hypothesis of complete s e p a r a t i o n . The three modes c h a r a c t e r i s t i c of amine are e a s i l y i d e n t i f i e d . CH 3NH 2 3361 A' 1623 A' 3427 cm - 1 A" I t i s important to note that these i n v a r i a n t fundamentals support the hypothesis of n u l l p a r t i c i p a t i o n over the methyl group. S i m i l a r l y f o r the s i x methyl modes: CH3NH~2 2961 A' 2820 A' 1473 A' 1430 A' 2985 A" 1476 cm - 1 A" And these fundamentals support the hypothesis of n u l l p a r t i c i p a t i o n over the amine group. Note the v i o l a t i o n of the r u l e of monotony (1) which s t a t e s that a l l v i b r a t i o n a l frequencies must decrease upon heavier i s o t o p i c s u b s t i t u t i o n (see equation 4 . 2 . 7 ) . CD 3NH 2 3361 1624 3427 cm CH 3ND 2 2961 2817 1468 1430 2985 1485 cm' - 84 - The question as to how l i m i t e d p a r t i c i p a t i o n i n f o r - mation can be used i n connection with i m p l i c a t i o n methods now a r i s e s . TT(A) and/T(&)can be w r i t t e n i n the form: 4 and <pA i s a diagonal matrix The matrices and sum to the i d e n t i t y m a t r i x , t h e r e f o r e they commute and are simultaneously d i a g o n a l i z e d by a s i n g l e orthogonal According to the p r i n c i p l e s set down i n appendix three f o r the ex p o n e n t i a l matrix and the p r i n c i p l e s of l i m i t e d par- t i c i p a t i o n , the skew-symmetric K-matrix of equation (4.2.11) e x h i b i t s the mixing p r o p e r t y ! - 85 - |^ •» —Q unless and represent genuine v i b r a t i o n s of the same symmetry species and cover the same set of atoms. In the present example, the s e l e c t e d forms of the TT — matrices cause the methyl-modes to mix only w i t h themselves, the amine-modes to mix only with themselves and the mixed-modes to mix only with themselves. The mixed modes are not coupled to any of the c h a r a c t e r i s t i c modes; t h i s i s a remarkable and unexpected property ! With a plane of symmetry methylamine e x h i b i t s 66 independent p o s s i b l e q u a d r a t i c f o r c e constants or 51 p o s s i b l e general mixing parameters. But i f l i m i t e d p a r t i c i p a t i o n i s assumed - and experimental evidence v a l i d a t e s the assumption - the number of mixing parameters i s reduced to 1 4 . Through molecular p a r t i t i o n i n g i t seems q u i t e l i k e l y that reasonable i m p l i e d f o r c e constants could be obtained f o r a c l a s s of more complex molecules. As w e l l p a r t i t i o n i n g a p p l i e s i n other senses. The G-matrix can be p a r t i t i o n e d i n t o two p a r t s , <3(A) and (3( B) such that Gift) +-G(B) = G w i t h much the same r e s u l t s . The a t o m - p a r t i c i p a t i o n d e n s i t i e s , foCj > provide an i n t e r e s t i n g d e s c r i p t i o n of v i b r a t i o n a l modes. The equation / / \ provides meaning i n that foij i s d i r e c t l v r e l a t e d to s m a l l i s o t o p i c s h i f t s . EPILOGUE Unfortunately the e x p l o r a t i o n of i m p l i c a t i o n methods i n the d e s c r i p t i o n of molecular mechanics and chemical bonding s t r u c t u r e i s not yet complete but t h i s t h e s i s has e s t a b l i s h e d the b a s i c approach to a cumbersome problem by new methods. The i n g r e d i e n t s of the i m p l i c a t i o n method i n c l u d e two important f a c t o r s . An experimentally w e l l known b a s i s molecule i s adopted and i t s v i b r a t i o n a l frequencies remove a c o r r e s - ponding number of parameters from the pure mechanical d e s c r i p t i o n . The reduced p a r a m e t e r i z a t i o n has been cast i n t o a mathematical form t h a t . i s handy i n computations as w e l l as a n a l y s i s , as d e l i n e a t e d i n chapter four. A p p l i c a t i o n of the I m p l i c a t i o n technique has l e d to a simple account of some dominating anharmonic e f f e c t s . I f continued a p p l i c a t i o n of the method i s as rewarding, then f u r t h e r s t u d i e s are d e f i n i t e l y warranted.^ To t h i s end, the concepts of p a r t i c i p a t i o n and molecular p a r t i t i o n i n g appear t o o f f e r the optimum prospects. I f molecules not r i c h i n hydrogen are to be considered i n more ri g o r o u s terms, the r o l e of mechanical i n f o r m a t i o n other than v i b r a t i o n a l frequencies must be i n c l u d e d i n the i m p l i c a t i o n scheme, Jones, Asprey and'Ryan (3*0 have shown that a complete p i c t u r e of mechanical i n f o r m a t i o n i s e s s e n t i a l i n the p e r t u r b a t i o n method. - 87 - Therefore, the l i m i t a t i o n to hydrogen r i c h molecules i s not s p e c i f i c f o r the i m p l i c a t i o n method. The l i m i t a t i o n i s merely one, of development w i t h i n the i m p l i c a t i o n method. When the molecular system precludes a more ri g o r o u s approach,it i s p l a u s i b l e that i m p l i c a t i o n t e c h - niques may c o n t r i b u t e s u b s t a n t i a l l y to a simple but proper p a r a m e t e r i z a t i o n of the p o t e n t i a l energy f u n c t i o n . BIBLIOGRAPHY 1. E.B. Wilson, J r . , J.C. Decius, and P.C. 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Cossee, The J o u r n a l of Chemical Physics 44_, 97 (1966). 19. J.H. Schachtschneider, J.N. Gayles, J r . , and W.T. King, Spectrochimica Acta 23, 703 (1967). - 90 - 20. J.L. Duncan, Spectrochimica Acta 20, 1197 (1964). 21. I.M. M i l l s , Spectrochimica Acta 19., 1585 (1963) 22. W.H. F l e t c h e r arid W.T. Thompson, J o u r n a l of Molecular Spectroscopy 25, 240 (1968). 23. J.W. N i b l e r and G.C. Pimentel, J o u r n a l of Molecular Spectroscopy 26, 294 (1968). 24. N.A. Narasimham and C.V.S. Ramachandraras, J o u r n a l of Molecular Spectroscopy 30, 192 (1969). 25- W.J. T a y l o r , The J o u r n a l of Chemical Physics 18, 1301 (1950). 26. N.R. McQuaker, " V i b r a t i o n a l Spectra of the Ammonium H a l i t e s and the A l k a l i - M e t a l B o r o h y d r i t e s " . Ph.D. T h e s i s , The U n i v e r s i t y of B r i t i s h Columbia, 1970. 27- F. Torok and P. Pulay, J o u r n a l of Molecular S t r u c t u r e 1, 1 (1969). 28. D.E. Freeman, J o u r n a l of Molecular Spectroscopy 2_7, 27 (1968). 29. F. B i l l e s , Acta Chim, Acad. S c i . Hung. 4_7, 53 (1966). 30. C.J. Peacock, U. Heidborn and A. M u l l e r , J o u r n a l of Molecular Spectroscopy 30., 338 (1969). 31. B.S. Averbukh, L.S. Mayants and G.B. Shaltuper, J o u r n a l Of Molecular Spectroscopy 30, 310 (1969). - 91 - 32. G. S t r e y , J o u r n a l of Molecular Spectroscopy 2_4, 87 (1967). 33- W.S. Benedict, N.Gailer and E.K. P l y l e r , The J o u r n a l of Chemical Physics 2_4, 1139 (1956). 34. L.H. Jones, L.B. Asprey, and R.R. Ryan, The J o u r n a l of Chemical Physics 47, 3371 (1967). 35. L.H. Jones, R.S. McDowell and M. G o l d b l a t t , The J o u r n a l Of Chemical P h y s i c s ' '48, 2663 (1968). 36. T. Shimanouchi and I . Suzuki, The J o u r n a l of Chemical Physics 43, 1854 (1965). 37- L.H. Jones and R.S. McDowell, J o u r n a l of Molecular Spectroscopy 3, 632 (1959). 38. I.M. M i l l s , Chemical Physics L e t t e r s 3, 267 ( 1 9 6 9 ) . 39- B. Crawford, J r . , and J . Overend, J o u r n a l of Molecular Spectroscopy 12, 307 (1964). 40. R.L. Hubbard, J o u r n a l of Molecular Spectroscopy 6_, 272 (1961). 41. R. Gold, J.M. Dowling and A.G. M e i s t e r , J o u r n a l of Molecular Soectroscopy 2, 9 (1958) 42. W. Kaplan, Advanced C a l c u l u s , Addison-Wesley P u b l i s h i n g Company, Inc., Reading, Massachusetts, 1952. 43. S.R. P o l o , The J o u r n a l of Chemical Physics 24., 1133 (1956) - 92 - 44. S. P e r l i s , Theory of M a t r i c e s , Addison-Wesley P u b l i s h i n g Company, Inc., Reading Massachusetts, 1 9 5 2 . 4 5 . M. Marcus, Basic Theorems i n Matr i x Theory, N a t i o n a l Bureau of Standards A p p l i e d Mathematics S e r i e s 5 7 , I 9 6 0 . 46 . J . Overend and J.R. Scherer, The J o u r n a l of Chemical Physics 3_2, 1289 ( I 9 6 0 ) . 4 7 . J o i n t Commission f o r Spectroscopy, The J o u r n a l of Chemical Physics 2 3 , 1997 ( 1 9 5 5 ) . 4 8 . B.L. Crawford, J r . , J.E. Lancaster, and R.G. Inskeep, The J o u r n a l of Chemical Physics '21, 678 ( 1 9 5 3 ) . 4 9 . S. Brodersen, Matematisk-fysiske s k r i f t e r udgivet af Det Kongelige Dansk Videnskabernes Selskab 1, no. 4 ( 1 9 5 7 ) . 5 0 . J.R. Scherer and J . Overend, The J o u r n a l of Chemical Physics 33., 1681 ( i 9 6 0 ) . 5 1 . S. Brodersen and A. Langseth, Matematisk-fysiske S k r i f t e r udgivet af Det Kongelige Danske Videnskabernes Selskab 1, no. 5 (1958) 5 2 . D.F. Heath and J.N. L i n n e t t , Transactions of the Faraday Society 4 4 , 556 ( 1 9 4 8 ) . 53- R.A. Kydd, "The V i b r a t i o n s of Some Aromatic Molecules", Ph.D. T h e s i s , The U n i v e r s i t y of B r i t i s h Columbia ( 1 9 6 9 ) . - 93 - 54. H. Margenau and G.M. Murphy , The Mathematics of Physics and Chemistry, Toronto, Canada, 19 43. 55. G. D e l l e p i a n e and G. Z e r b i , The J o u r n a l of Chemical Physics 4_8, 3573' (1968) . 56. L . J . B o d i , The Theory of V i b r a t i o n a l - R o t a t i o n a l I n t e r a c t i o n i n Polyatomic Molecules, U n i v e r s i t y of Wisconsin, Naval Research Laboratory, T e c h n i c a l Report Wis-OOR-11, Madison, Wisconsin, 1954. 57. ' E.C. Kemble, The Fundamental P r i n c i p l e s of Quantum Mechanics with Elementary A p p l i c a t i o n s , Dover P u b l i c a t i o n s , Inc., New York, New York, 1937- 58. R.J. Malhiot and S.M. F e r i g l e , The J o u r n a l of Chemical Physics 22, 717 ( 1 9 5 4 ) ; 23., 30 (1955). 59- L.L. Oden and J.C. Decius, Spectrochimica Acta 20, 667 (1963). 60. C. C h e v a l l y , Theory of L i e Groups, P r i n c e t o n U n i v e r s i t y P r e s s , P r i n c e t o n , New J e r s e y , 1946. APPENDIX ONE: THE VIBRATIONAL SECULAR EQUATIONS A S i m p l i f i e d Quantum Mechanical D e s c r i p t i o n - The concepts of c l a s s i c a l mechanics, mainly w e l l defined t r a j e c t o r i e s , f a i l to provide a genuine d e s c r i p t i o n of microsystems (56, 57)- In some examples such as the hydrogen atom, the f a i l u r e of c l a s s i c a l concepts i s sub- s t a n t i a l ; i n other cases such as the harmonic o s c i l l a t o r , c l a s s i c a l concepts remain adequate w i t h i n l i m i t s , e s p e c i a l l y f o r polyatomic molecules. Consequently, a c l a s s i c a l p i c t u r e of molecular v i b r a t i o n s has p e r s i s t e d while the necessary concepts of quantum mechanics grew to dominate the d e s c r i p t i o n of microsystems, e s p e c i a l l y molecular s t r u c t u r e . Though chemists are not quantum mechanicians, they are becoming more and more quantum o r i e n t e d ; c l a s s i c a l mechanics has i n f a c t almost e n t i r e l y disappeared from the l a t t i c e w o r k of chemical l o g i c . Thus, to the student of p h y s i c a l chemistry, a c l a s s i c a l p i c t u r e of molecular v i b r a t i o n s i s b u i l t upon u n f a m i l i a r , almost i r r e l e v a n t , foundations. A quantum mechanical d e s c r i p t i o n of molecular v i b r a t i o n s f a l l s w i t h i n the quantum p i c t u r e u s u a l l y presented to chemists; moreover both molecular v i b r a t i o n s and the e x i s t i n g quantum p i c t u r e would b e n e f i t from a connected development. - 95 - Quantum d e s c r i p t i o n s i n c l u d i n g molecular v i b r a t i o n s already e x i s t but these ( 1 , 5 6 , 57) are w r i t t e n f o r the experienced s p e c t r o s c o p i c s p e c i a l i s t s or the pure mechanicians. In s h o r t , a handy reference f o r molecular v i b r a t i o n s , w r i t t e n i n quantum mechanical language s u i t a b l e f o r i n t r o d u c t o r y purposes, i s not known (to the aut h o r ) . the above-stated purposes; i t i n c l u d e s a l l c l a s s i c a l i n f ormation^ captures the f l a v o r of r i g o r i n quantum mechanics and provides a f i r m foundation f o r f u r t h e r r i g o r and d e t a i l . ( I - l ) Transformation of the Schrodinger Equation The C a r t e s i a n coordinate Schrodinger equation f o r p o t e n t i a l energy f u n c t i o n V, governs the t o t a l energy of The f o l l o w i n g d e s c r i p t i o n i s intended to serve N p a r t i c l e s under a 0 ( T . 1 . 1 ) the system ;E, and the p r o b a b i l i t y , of a c o n f i g u r a t i o n s p e c i f i e d by the C a r t e s i a n coordinates (>Gĵ  Vct,^*) where CC counts p a r t i c l e s . In the C a r t e s i a n S'chrodinger equation (T.1.1) ~p] i s Planck's constant; 3jp the C a r t e s i a n wave f u n c t i o n , i s a f u n c t i o n of C a r t e s i a n c o o r d i n a t e s . - 96 - V the p o t e n t i a l energy, i s a f u n c t i o n of the c o n f i g u r a t i o n of the molecule ( i n the absence of s i g n i f i c a n t e x t e r n a l i n f l u e n c e s ) . I t i s i n v a r i a n t with respect to r o t a t i o n s and t r a n s l a t i o n s . For present purposes e l e c t r o n i c motion i s considered to be embedded w i t h i n the p o t e n t i a l energy f u n c t i o n ; see the Born- Oppenheimer s e p a r a t i o n of e l e c t r o n i c and nuclear motion (2) . At Whether or not the e l e c t r o n i c masses should be i n c l u d e d i n the masses of the i n d i v i d u a l p a r t i c l e s under c o n s i d e r a t i o n i s not completely c l e a r . The Born-Oppenheimer s e p a r a t i o n c a l l s f o r nuclear masses but d e t a i l e d s t u d i e s u s u a l l y employ atomic masses with s a t i s f a c t o r y agreement ( 2 ) . The i n v a r i a n c e of the p o t e n t i a l energy with respect to r o t a t i o n and t r a n s l a t i o n , as w e l l as the p r i n c i p l e s of chemical bonding, suggests a t r a n s f o r m a t i o n of c o o r d i n a t e s . New coordinates would i n c l u d e three t r a n s l a t i o n c o o r d i n a t e s , three r o t a t i o n coordinates and (3N-6 ) coordinates of c o n f i g u r a t i o n - bond l e n g t h s , valence angles, and r e l a t e d geometric measures. The t r a n s f o r m a t i o n i s c u r v i l i n e a r . - 97 - To t h i s end l e t the tra n s f o r m a t i o n p r o p e r t i e s of (T;l.l) be e s t a b l i s h e d ; l e t the new coordinates be designated as t ^ . The r u l e s of p a r t i a l d i f f e r e n t i a t i o n (42) e s t a b l i s h the i d e n t i t y : 55?̂ " 1 UxJlati (1.1.2) D i f f e r e n t i a t i o n of ( T . 1.2) w i t h respect to Xoc provides a second i d e n t i t y : d > a - t-z-i (axJi S J J aha* i r i 3 ) J dxi a*j When cast i n t o g e n e r a l i z e d c o o r d i n a t e s , the k i n e t i c part of the C a r t e s i a n coordinate Schrodinger equation becomes where the f o l l o w i n g q u a n t i t i e s have been d e f i n e d f o r b r e v i t y , ( T . 1 . 6 ) - 98 - With r e f l e c t i o n , paper, and p e n c i l , the f i r s t of these q u a n t i t i e s (T.1.5) w i l l be recognised as the f a m i l i a r G- matrix of Wilson, Decius and Cross ( 1 ) ; t h i s form, but not the d e r i v a t i o n , of the G-matrix has been p r e v i o u s l y w r i t t e n by Malhiot and F e r i g l e (58). However, i n the present con t e x t , the G-matrix elements may be dynamic v a r i a b l e s . From the ou t s e t , quantum mechanics f u r n i s h e s a n a t u r a l and e f f i c i e n t o r i g i n f o r the G-matrix; i n c l a s s i c a l mech- anics the o r i g i n of the G-matrix can be cumbersome. As could be expected, quantum mechanics leads to new terms without simple c l a s s i c a l analogs; the ^ 's of (1.1.6) - l o c a l angular momentum? - c o n t r i b u t e to the zero p o i n t energy and generate d i s t o r t i o n s i n the average geometry of the molecule depending on the v i b r a t i o n a l s t a t e of the molecule ( w i t h i n the harmonic o s c i l l a t o r approximation.) (We are not prepared to endow these higher order k i n e t i c energy terms with p h y s i c a l s i g n i f i c a n c e . ) Equation (1.1.4) i s nothing more than a g e n e r a l i z e d L a p l a c i a n operator; the simple d e r i v a t i o n and form of (1.1.4) gives i t some advantage, depending on o b j e c t i v e , over the eq u i v a l e n t expression found i n Margenau and Murphy (54) and the usual quantum mechanical textbooks (1, 57). In our n o t a t i o n , the usual g e n e r a l i z e d L a p l a c i a n reads: (1.1.7) - 99 - where \(3\ i s the determinant of the G-matrix ( i n c l u d i n g r o t a t i o n and t r a n s l a t i o n ) . The equivalence of (T.1.4) and (T.1.7) can be seen through the a c t i o n of the operator (T.1.7) on the g e n e r a l i z e d coordinates themselves. With r e f l e c t i o n , (T.1.8) brings (T.1.7) i n t o the form ( T . 1 . 4 ) ; the converse, (T.1.4) to (T.1.7) i s much more d i f f i c u l t . The t r a n s f o r m a t i o n i s not yet complete. The transformed wavefunction c a r r i e s a de n s i t y f a c t o r i f i t i s to preserve i t s meaning as a p r o b a b i l i t y d i s t r i b u t i o n when squared (57). in (T.1.8) (the i d e n t i t y matrix) Consequently the g e n e r a l i z e d Schrodinger equation i s f r e q u e n t l y expressed i n the form (1, 36). But the C a r t e s i a n wave f u n c t i o n can be r e t a i n e d i f i t s meaning and the r o l e of the density f a c t o r i s r e c a l l e d . - 100 - ( T - 2 ) Separation of R o t a t i o n , T r a n s l a t i o n and C o n f i g u r a t i o n The three f o l l o w i n g t r a n s l a t i o n a l coordinates d e f i n e the centre of mass of. the molecule; these are dynamic v a r i a b l e s . A ?' v ~ IAS. v Z o = /v\' Hoc / r i o c ^ o c ( 1 . 2 . 1 ) where and Z??^ i s the mass of the C C ~ p a r t i c l e . The three r o t a t i o n a l coordinates f o r a s e m i r i g i d body appear to be d i f f i c u l t to de f i n e (56); however the 3N C a r t e s i a n v e c t o r q u a n t i t i e s below w i l l be shown to e x h i b i t d e s i r e d mechanical p r o p e r t i e s . -- / f o { ( & - £ ) , ° ] Here, R , R , and R are presumed to be r o t a t i o n a l ' x y z ^ coordinates. ( T . 2 . 2 ) - 101 - With these r o t a t i o n a l and t r a n s l a t i o n a l c o o r d i n a t e s , the G-matrix takes the form: G = / G ( t r a n s l a t i o n ) 0 0 \ 1 .0 G ( r o t a t i o n ) 0 \ \ 0 0 G ( c o n f i g u r a t i o n ) I (1.2.3) When the p o t e n t i a l energy i s a f u n c t i o n of c o n f i g u r a t i o n o n l y , the Schrodinger equation, according to the above r e p r e s e n t a t i o n , separates i n t o t r a n s l a t i o n a l , r o t a t i o n a l and v i b r a t i o n a l p a r t s . The sep a r a t i o n of t r a n s l a t i o n i s rig o r o u s but the se p a r a t i o n of r o t a t i o n depends upon the r i g i d i t y of the molecule. For r o t a t i o n and t r a n s l a t i o n the higher order k i n e t i c terms, ( T . 1 . 6 ) , are a l l i d e n t i c a l l y zero; thus the decoupling i s complete. The matrix form ( T . 2 . 3 ) i s most e a s i l y d e r i v e d by co n s i d e r i n g the t r a n s l a t i o n a l (and r o t a t i o n a l ) G-matrix elements i n a C a r t e s i a n vector form. Let t denote a g e n e r a l i z e d coordinate. [G(Xo,t), G(y.,i)j 6(i.,t)] = M' z« v«t [GM , 6 M , G ( H ^ i ) ] = E* fot)x(A*) where /[^ = ( Xoc-X0 ^ & — Y0 ; 2.cx~ 2 d J - 102 - R o t a t i o n a l and a l l c o n f i g u r a t i o n a l coordinates e x h i b i t the property: ZK %t = (0,0,0) And a l l c o n f i g u r a t i o n coordinates e x h i b i t the property EJ&£WA*) = 10,0,0) These p r o p e r t i e s can be v e r i f i e d by c a r r y i n g out the i n d i c a t e d operations f o r the various coordinates. With very l i t t l e l abor i t can be shown that G ( t r a n s l a t i o n ) = K A ' G ( r o t a t i o n ) Ixx -I IxY I, " 2 £ and the most common c o n f i g u r a t i o n G-matrix elements have been t a b u l a t e d elsewhere; otherwise these can be w r i t t e n v i a (1.1.5). The moment of i n e r t i a t e n s o r , G ( r o t a t i o n ) , depends upon the o r i e n t a t i o n of the molecule and i t s instantaneous c o n f i g u r a t i o n ; thus the sep a r a t i o n of r o t a t i o n and c o n f i g u r - a t i o n (or v i b r a t i o n ) i s not complete. V i b r a t i o n - r o t a t i o n - 103 - i n t e r a c t i o n s are discussed by Wilson, Decius and Cross (1, see Chapter 11) and i n greater d e t a i l by Bodi (56); however, both references employ the Eckart c o n d i t i o n s and d i f f e r from t h i s development i n that r e s p e c t . Because the Eckart c o n d i t i o n s have not been invoked here, l i n e a r molecules are not e x c e p t i o n a l cases (from the o u t s e t ) ; moreover the equations d e r i v e d thus f a r need not be con s t r a i n e d to r i g i d systems. However, the p h y s i c a l substance of t h i s development, which depends on equations (T.2.2), the r o t a t i o n a l c o o r d i n a t e s , remains l a r g e l y unexplored. Equations (1.2.2) have not been p r e v i o u s l y w r i t t e n but here they i n d i c a t e a u s e f u l r o l e i n the a n a l y s i s of molecular mechanics. The author suggests that these equations be named "the Harvey c o n d i t i o n s " i n memory of K.B. Harvey, d i r e c t o r of t h i s research. (1-3) The V i b r a t i o n a l Schrodinger Equation A f t e r the s e p a r a t i o n of r o t a t i o n and t r a n s l a t i o n the v i b r a t i o n a l Schrodinger equation remains to be solved. An approximate s o l u t i o n adequate f o r many purposes can be obtained by r e p l a c i n g k i n e t i c energy c o e f f i c i e n t s by e i t h e r t h e i r average of e q u i l i b r i u m values and expressing the p o t e n t i a l energy as a quadratic form i n c o n f i g u r a t i o n - displacement c o o r d i n a t e s , /Ot . Here i t i s assumed that - 104 - the molecule remains near i t s e q u i l i b r i u m c o n f i g u r a t i o n while i t v i b r a t e s . To solve the v i b r a t i o n a l Schrodinger equation, L>L^xl dfrdft dps V J/ Yr.3.1) when <Jij } ^-i and are a l l constant c o e f f i c i e n t s l i n e a r t r a n s f o r m a t i o n of the form, A f t = Zj L * j (T.3-2) (T.3.3) such that (the i d e n t i t y matrix) (3*.3-4) f~ [_ — /\ ( a diagonal matrix) ( T . 3 - 5 ) reduces ( T . 3 - T ) to a sum of independent quantum o s c i l l a t o r s i n the normal c o o r d i n a t e s , Qj - 105 - Except f o r the terms J?j - \T.n $n Ljn d . 3 . 7 ) the s o l u t i o n of ( 1 . 3 - 6 ) i s a product of harmonic o s c i l l a t o r wavefunctions and the energy i s a sum of harmonic o s c i l l a t o r energies ( 1 ) . With nonzero S , standard methods f o r s o l v i n g d i f f e r e n t i a l equations (42) i n d i c a t e a s o l u t i o n as f o l l o w s : ( 1 . 3 . 8 ) _ M s~\ //. —^ n f h y i s the n o r m a l i z i n g f a c t o r i\y i s the ^ hermite polynomial ( 1 . 3 - 9 ) ( 1 . 3 - 1 0 ) (T. 3 . 1 D ( T . 3 . 1 2 ) - 106 - For the wavefunctions (T.3-10) the expected value of the normal coordinates i s not zero. For example. and the magnitude of the s h i f t depends on the quantum numbers. The added zero-point energy i s simply the p o t e n t i a l energy of the molecule when i t assumes i t s expected ground s t a t e c o n f i g u r a t i o n . However the p h y s i c a l s i g n i f i c a n c e of these higher order k i n e t i c terms, given the approximations invoked, i s not yet c e r t a i n . - 107 - APPENDIX TWO: REDUNDANT COORDINATE SYSTEMS Redundant coordinate systems are sometimes needed f o r the a n a l y s i s of molecular s t r u c t u r e i n f a m i l i a r g e o m e t r i c a l terms. This appendix deals with the question as to whether or not redundant coordinates imply r e l a t e d interdependencies f o r mechanical q u a n t i t i e s such as f o r c e constants as has been suggested by Hubbard (40) and Gold, Dowling and Me i s t e r ( 4 l ) . Crawford and Overend (39) have attempted to show that such r e l a t i o n s f o r f o r c e constants are i n f a c t a r b i t r a r y . A l l of these authors have t r e a t e d the redundant coordinate system as though the coordinates were independent v a r i a b l e s subject to c o n s t r a i n t s or redundancy c o n d i t i o n s . P o t e n t i a l energy i s expressed as a Taylor s e r i e s i n redundant coordinates. In t h e i r approach i t i s not c l e a r that the T a y l o r s e r i e s c o e f f i c i e n t s , formed by p a r t i a l d i f f e r e n t i a t i o n , can be c a l l e d upon without s u b s t a n t i a l j u s t i f i c a t i o n when the coordinates cannot be independent. Unfortunately the p r e v i o u s l y c i t e d authors have not considered the p r o p r i e t y of the Taylor s e r i e s c o e f f i c i e n t s when redundant coordinate systems are employed. F u r t h e r , there i s something i n t r i n s i c a l l y u n s a t i s f y i n g with an a b - i n i t i o expression of p o t e n t i a l energy i n terms of redundant coordinates. I f necessary, one can accept the - 108 - n o t i o n of p o t e n t i a l energy i n terms of redundant coordinates but only as a der i v e d or a u x i l i a r y q u a n t i t y - not as an i n i t i a l assumption or i n t r i n s i c property. In t h i s appendix, the redundant coordinate system and a l l r e l a t e d q u a n t i t i e s s h a l l be s t r i c t l y r e f e r r e d to a non redundant coordinate system. The s i g n i f i c a n t step here i s that the t r a n s f o r m a t i o n matrix i s not square and t h e r e f o r e not i n v e r t i b l e i n the usual simple way. Our approach provides the supplementary i n f o r m a t i o n needed to r e s o l v e the indeterminacy noted by Crawford and Overend ( 3 9 ) . The two approaches are i n f a c t complementary r a t h e r than c o n t r a d i c t o r y . ( I I -1) Transformations i n v o l v i n g redundant coordinates As geo m e t r i c a l q u a n t i t i e s d e f i n e d on a set of p o i n t s or atoms, the redundant coordinates ( p ) can be defined without question i n terms of nonredundant coordinates ( "fc ) . higher terms (II.1.1) C l e a r l y every c o n f i g u r a t i o n of the molecule s p e c i f i e s e x a c t l y a l l of the non .redundant coordinates and a l l of the redundant coordinates. Consequently the r e l a t i o n between redundant coordinates and nonredundant coordinates i s one- to-one, i n terms of the molecular c o n f i g u r a t i o n . This means - 109 - that the inve r s e t r a n s f o r m a t i o n e x i s t s even though the matrix (®^fc)0 n o ^ square! For i n f i n i t e s m a l d i s t o r t i o n s , equations (11.1.1) define an overdetermined system of l i n e a r equations; thus the nonredundant coordinates can be expressed i n terms of the redundant coordinates. Let The matrix i s the l e f t hand i n v e r s e of the matrix «S , j / s ' s )VJs - Gofers) - x ( S ^ s ) i s a square matrix of f u l l rank and (3N-6) x ( 3 N - 6 ) , where N i s the number of atoms. Therefore I f the p o t e n t i a l energy i s pr o p e r l y d e f i n e d i n terms of nonredundant c o o r d i n a t e s , 2 V - F . j t i t j s u b s t i t u t i o n of ( I T . 1 . 3 ) provides an equi v a l e n t expression i n terms of redundant coordinates. - 110 where the -matrix, ( i r . i . 4 ) represents the for c e constant matrix f o r redundant coordinates. As w e l l G * = S G ^ (II . 1 . 5 ) ^-K- ~~ S L (II. 1 . 6 ) where and are redundant coordinate r e p r e s e n t a t i o n s of L, and c3 f o r nonredundant coordinates. Pseudo inverses of F% 5 and i—£ , noted as P]?. and r e s p e c t i v e l y , can be w r i t t e n i n the form <3̂  - sistsyG (s%) s ( i r . 1 . 7 ) (II.1.8) (II. 1 . 9 ) -111- Decius (8) has shown that r gives the p o t e n t i a l energy i n terms of g e n e r a l i z e d forces' - a f a c t l a t e r i t e r a t e d by Cyvin ( 7 ) . Decius and Oden ( 5 9 ) , because the i n v e r s e t r a n s f o r m a t i o n i s not needed to formulate f~~/z , have used the above r e l a t i o n to s p e c i f y a l l of the constants i n v o l v e d i n F~/z The eigenvectors of which correspond to zero eigenvalues span the n u l l space of the S-transformation (impossible c o n f i g u r a t i o n s of the molecule). Let Ny = 3N-6 the number of genuine v i b r a t i o n s Np. be the number of redundant coordinates be the orthogonal matrix w i t h d i a g o n a l i z e s where j" - 1 i s diagonal but of the diagonal e n t r i e s are zero. Let XJ% be p a r t i t i o n e d as a matrix "Uk= ( U \ U « ^ such that G ^ U - U P and 0KUZ=0 where LA.̂ . are eigenvectors with zero eigenvalues and are eigenvectors with nonzero eigenvalues noted by the diagonal matrix ' (see reference 42) (II.1.10) t h e r e f o r e (II.1.11) a n d UtFtU^o ( i i . 1 . 1 2 ) - 112 - Equations OCT. 1 . 1 1 ) and ( I I . 1 . 1 2 ) which f o l l o w from ( 1 1 . 1 . 4 ) and OCT. 1 . 1 0 ) provide the r e l a t i o n s needed to define the so c a l l e d indeterminate f o r c e constants a r i s i n g from the redundant coordinate system. I t w i l l be observed that the equations of t h i s t h e s i s are c o r r e c t as w r i t t e n i f the U-matrix i s understood t o be composed of the eigenvectors of the C3^_ matrix w i t h nonzero eigenvalues. Here the U-matrix need not be square but i t s columns remain orthogonal. In summary once i t i s r e a l i z e d that a l l coordinate transformations covering the p o s s i b l e c o n f i g u r a t i o n s of a molecule are one-to-one whether the t r a n s f o r m a t i o n generates redundant coordinates or not, then the redundancy c o n d i t i o n s on f o r c e constants f o l l o w v i a the in v e r s e t r a n s f o r m a t i o n . - 113 - APPENDIX THREE: THE EXPONENTIAL MATRIX The e x p o n e n t i a l matrix i s defined l i k e the e x p o n e n t i a l f u n c t i o n , 0 K s = Z K n s r t /n i where K i s a square matrix and 5 i s an or d e r i n g parameter. The or d e r i n g parameter enables us to e s t a b l i s h the e s s e n t i a l p r o p e r t i e s of the exp o n e n t i a l matrix w i t h l i t t l e a l g e b r a i c l a b o r . For example, the formula ^ = K<Z*S = < Z K s K o) s i s e a s i l y d e r i v e d . More general p r o p e r t i e s can be e s t a b l i s h e d by c o n s i d e r i n g the matrix f u n c t i o n ^ ~Ks HB .<zr*_a<z where i s , f o r the present purpose, a square m a t r i x ; K and s are as defined above. M a t r i x d i f f e r e n t i a t i o n gives m = LKjH]= sue," The (H+l ) t h d e r i v a t i v e can be expressed i n terms of the Y\ — d e r i v a t i v e , - 114 - Consequently In our a p p l i c a t i o n s we are i n t e r e s t e d i n the H-matrix as a perturbed form of the _fL -matrix and conversely. The H and j f L matrices are r e l a t e d through the K- matrix and ord e r i n g parameter by the expansion about S=0. + higher terms r e c a l l t h a t : L i m i t H — J S - * O When i s the i d e n t i t y m a t r i x , the above equations show that (& — X" (the i d e n t i t y matrix) or (<z.K0"'= <2>*s - 115 - As w e l l , when the K-matrix i s skew symmetric (skew hermitian) , 4 - K = -K the e x p o n e n t i a l form i s orthogonal ( u n i t a r y ) . Consequently the __fl_-matrix can be viewed as a diagonal matrix composed of the eigenvalues of H; i n t h i s case the orthogonal matrix Ĉ / represents the eigenvectors of H. The e x p o n e n t i a l matrix may appear i n the l i t e r a t u r e of mathematics (60) u s u a l l y i n connection with the theory of l i e groups. A p p l i c a t i o n s of the e x p o n e n t i a l matrix have not been extensive i n chemistry nor i n any area where mathematics i s more a t o o l than a l o g i c a l a r t . Yet i t s p r o p e r t i e s i n d i c a t e that i t can be a u s e f u l t o o l .

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