STUDIES IN THE ELECTRONIC STRUCTURE OF
SOME TRIATOMIC RADICALS
Douglas Alexander Brotchie
A Thesis Submitted for the Degree of PhD
at the
University of St Andrews
1973
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STUDIES IN THE ELECTRONIC STRUCTURE
OF SOME TRIATOM 1C RADICALS
A t h e s i s
p r e s e n t e d f o r t h e d e g r e e o f
D o c t o r o f Phi i os o ph y
in t h e F a c u l t y o f S c i e n c e o f t h e U n i v e r s i t y o f S t Andrews
by Dougl as A l e x a n d e r B r o t c h i e , B,Sc,
U n i t e d C o l l e g e o f St S a l v a t o r and S t Leona r d
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- 1 1 “
D e c l a r a t i o n
I d e c l a r e t h a t t h i s t h e s i s i s my own c o m p o s i t i o n , t h a t t h e
work o f whi ch i t i s a r e c o r d has been c a r r i e d o u t by m y s e l f , and t h a t i t has n o t been s u b m i t t e d in any p r e v i o u s a p p l i c a t i o n f o r a
h i g h e r d e g r e e .
The t h e s i s d e s c r i b e s t h e r e s u l t s o f r e s e a r c h c a r r i e d o u t a t
t h e C h e m i s t r y D e p a r t m e n t , The U n i t e d C o l l e g e o f St S a l v a t o r and S t Le o n a r d , U n i v e r s i t y o f S t Andrews, u n d e r t h e s u p e r v i s i o n o f
Dr C o l i n Thomson s i n c e t h e 1 s t O c t o b e r 1970, t h e d a t e o f my
a d m i s s i o n as a r e s e a r c h s t u d e n t .
I
“ I l l ”
;
I h e r e b y c e r t i f y t h a t Dougl as A l e x a n d e r B r o t c h i e has s p e n t t w e l v e t erms a t r e s e a r c h work u n d e r my s u p e r v i s i o n , has f u l f i l l e d
t h e c o n d i t i o n s o f O r d i n a n c e n o . 16 ( S t An dr e ws ) , and i s q u a l i f i e d t o s u b m i t t h e f o l l o w i n g t h e s i s In a p p l i c a t i o n f o r t h e d e g r e e o f
D o c t o r o f P h i l o s o p h y .
C o n t e n t s
- I V
D e c l a r a t i o n C o n t e n t s
Not e on u n i t s
0 A b s t r a c t 1 I n t r o d u c t i o n
2 T h e o r e t i c a l b a c kg r o u n d
S p e c i f i c r a d i c a l s
BF„
NHa HBF
HCO
HNF
8 FCO
9 NF^
10 F O ,
11 D i s c u s s i o n
12 Acknow1edgemen t s
13 B i b l i o g r a p h y
11
i V
V
1
2
7
30 46
57 65
75
80
85 95 100
110
Note on un i t s
The u n i t s empl oyed in t h i s t h e s i s do n o t co n f o r m t o t h e
Système I n t e r n a t i o n a l d ' U n i t é s , wh i c h i s now recommended as t h e
i n t e r n a t i o n a l s t a n d a r d f o r u s e in r e c o r d i n g s c i e n t i f i c d a t a .
I n s t e a d , f o r e a s e o f c o m p a r i s o n w i t h r e l a t e d wor k, we employ p r i n c i p a l l y t h e a t o m i c s y s t e m o f u n i t s ( H a r t r e e , P r o c . Ca mb. P h i 1.
S o c . , 24, 89 ( 1 9 2 6 ) . ) ,
C o n v e r s i o n f a c t o r s t o t h e SI s t a n d a r d s a r e g i v e n bel ow, îÿ
1 au o f d i s t a n c e ( = 1 b o h r ) . “ 0 . 5 2 9 177 x 10 m
1 au o f e n e r g y ( = 1 h a r t r e e ) ^ 4 , 3 5 9 828 aJ
1 au o f d i p o l e ( = 2 . 5 4 2 D ) - 3- 335 640 x 10 ^^ Cm
The Angst rom u n i t o f d i s t a n c e i s a l s o o c c a s i o n a l l y u s e d ; -10
1 A = 10 m.
All t e m p e r a t u r e s a r e q u o t e d i n d e g r e e s K e l v i n , and
m a g n e t i c f l u x d e n s i t i e s i n u n i t s o f Gauss ( = 10 ^ T ) .
- 1
0 A b s t r a c t
R e s t r i c t e d H a r t r e e - F o c k c a l c u l a t i o n s , w i t h e x p l i c i t n o n - e m p i r i c a l c a l c u l a t i o n o f a l l i n t e g r a l s , a r e p r e s e n t e d f o r t h e gro u nd s t a t e s and some
l o w - l y i n g e x c i t e d s t a t e s o f e i g h t t r i a t o m i c r a d i c a l s (BF^, NH2, HBF, HCO, HNF, FCO, NFj,, FO^ )• G e o m e t r i e s a r e c a l c u l a t e d f o r t h e s e s p e c i e s , and t h e r e s u l t s r e l a t e d t o e x p e r i m e n t wh e r e p o s s i b l e . The n a t u r e o f t h e
b o n d i n g In ea c h s p e c i e s i s d i s c u s s e d .
The e f f e c t o f v a r y i n g t h e s i z e o f t h e G a u s s i a n b a s i s s e t s empl oyed
was exami ned i n t e n s i v e l y , w i t h p a r t i c u l a r a t t e n t i o n b e i n g p a i d t o c h a n g e s |
in t h e t o t a l c a l c u l a t e d e n e r g y , t h e o p t i m a l g e o m e t r y , b o n d i n g and s p i n
d e n s i t y d i s t r i b u t i o n . I t was c o n c l u d e d t h a t f o r r e l i a b l e geo met r y
p r e d i c t i o n s a b a s i s o f 9 s and 5 P G a u s s i a n f u n c t i o n s was r e q u i r e d on e a c h
f i r s t row at o m, p r e f e r a b l y augment e d by p o l a r i s a t i o n f u n c t i o n s p o s s e s s i n g h i g h e r o r b i t a l a n g u l a r momentum quant um number. T h i s i s p a r t i c u l a r l y
n e c e s s a r y t o o b t a i n t h e c o r r e c t bond a n g l e s f o r h y d r i d e s . To p r e d i c t s p i n d e n s i t i e s a c c u r a t e l y a b a s i s s e t o f e s s e n t i a l l y H a r t r e e - F o c k q u a l i t y ,
p o s s e s s i n g a t l e a s t 11 s f u n c t i o n s , i s r e q u i r e d . For t h i s l a t t e r a p p l i c a t i o n c o n s i d e r a t i o n s o f b a l a n c i n g t h e s i z e o f t h e r e p r e s e n t a t i o n on t h e v a r i o u s
at oms in t h e m o l e c u l e i s p a r t i c u l a r l y c r u c i a l .
-■Æl ■* 2 “
\
1 I n t r o d u c t i o n
Any e x a m i n a t i o n o f t h e h i s t o r y o f c h e m i s t r y as a s c i e n c e , in t h e p r e s e n t - d a y s e n s e o f t h a t t e r m , w i l l r e v e a l a long l i n e o f w o r k e r s , e a c h s t r i v i n g t o r a t i o n a l i z e a c o l l e c t i o n o f e x p e r i m e n t a l o b s e r v a t i o n s by
r e c o u r s e t o some g e n e r a l t h e o r y . T h i s i s i n d eed t h e way t h a t s c i e n c e
p r o g r e s s e s , as s ee n by Karl P o p p e r ^ : f i r s t d e v e l o p an e x p l a n a t o r y
t h e o r y , a c c o u n t i n g f o r a l l p r e v i o u s o b s e r v a t i o n s , and u s e t h i s t h e o r y t o
make p r e d i c t i o n s . T e s t t h e p r e d i c t i o n s by e x p e r i m e n t ; i f e x p e r i m e n t a l
r e s u l t s a r e in a g r e e m e n t w i t h p r e d i c t i o n s , t h e n make more p r e d i c t i o n s ,
w h i l e i f e x p e r i m e n t c o n t r a d i c t s p r e d i c t i o n s , c o n s t r u c t a new t h e o r y . ' M
In t h i s p o i n t o f v i ew, no t h e o r y can e v e r be p r o v e d , b u t o n l y d i s p r o v e d .
A t h e o r y i s o n l y c o n s i d e r e d s c i e n t i f i c a l l y r e l e v a n t s o l ong as i t c o n t i n u e s t o make t e s t a b l e p r e d i c t i o n s .
For many d e c a d e s t h e t h e o r i e s d e v e l o p e d t o “ e x p l a i n " c h e mi c a l phenomena
we r e q u a l i t a t i v e , o r s i m p l y p i c t o r i a l ; t h i s was p a r t i c u l a r l y e v i d e n t in
o r g a n i c c h e m i s t r y . G r a d u a l l y however t h e need was f e l t t o p u r s u e t h e e x p l a n a t i o n t o a d e e p e r l e v e l , and in a more q u a n t i t a t i v e manner . I t was
f e l t t h a t a b e t t e r u n d e r s t a n d i n g o f ch e mi c a l b e h a v i o u r c o u l d be a r r i v e d a t
by c o n s i d e r i n g i n t e r a c t i o n s , n o t on a m a c r o s c o p i c o r eve n m i c r o s c o p i c s c a l e , b u t r a t h e r a t an a t o m i c and s u b - a t o m i c l e v e l . The w a v e - m e c h a n i c a l met hods
wh i c h c o u l d f u r n i s h an i n v e s t i g a t i v e t o o l f o r s uch an e n q u i r y had been
d e v e l o p e d by p h y s i c i s t s in t h e m i d - n i n e t e e n - t w e n t i e s , b u t t h e p r a c t i c a l ,
a p p l i c a t i o n o f t h i s scheme t o c h e mi c a l s i t u a t i o n s was n o t s t r a i g h t - f o r w a r d . 3
r J '
. 3,
I t became p o s s i b l e f o l l o w i n g t h e d e v e l o p m e n t o f t h e m o l e c u l a r o r b i t a l
■ : c o n c e p t a few y e a r s l a t e r , wh i c h was t o p r o v i d e a r o u t e t o t h e s o l u t i o n o f
t h e S c h r ô d i n g e r e q u a t i o n f o r p o l y - n u c l e a r s y s t e m s .
For r e a s o n s o f c o m p u t a t i o n a l and concept ual , s i m p l i c i t y , e a r l y c a l c u l a t i o n s
“ 3 "
\
l i m i t e d t h e i r c o n s i d e r a t i o n t o o r g a n i c m o l e c u l e s c o n t a i n i n g c o n j u g a t e d
s y s t e m s o f n - e l e c t r o n s , and u t i l i s e d t h e s i m p l i f y i n g c o n c e p t o f a-ir
s e p a r a b i l i t y . T h i s method was d e v e l o p e d by HUckel in 1931, and f o r many
y e a r s i t r emai ned t h e o n l y t h e o r e t i c a l t o o l used by c h e m i s t s , even a l t h o u g h
i t s f i e l d o f a p p l i c a t i o n was s o s e v e r e l y l i m i t e d . I t s i m p o r t a n c e can be
s e e n from t h e f a c t t h a t In 1961 i t was s t i l l p o s s i b l e f o r S t r e i t w i e s e r t o p u b l i s h " M o l e c u l a r o r b i t a l t h e o r y f o r o r g a n i c c h e m i s t s " , wh e r e r e a c t i v i t y
and s p e c t r a a r e a n a l y s e d e x c l u s i v e l y in t erms o f HLickel t h e o r y .
S u b s e q u e n t e f f o r t s w e r e o f c o u r s e made t o d e v e l o p HUckel t h e o r y i n t o s o m e t h i n g o f w i d e r g e n e r a l i t y - in p a r t i c u l a r t o i n c l u d e some c o n s i d e r a t i o n
o f i n t e r - e l e c t r o n r e p u l s i o n . A n o t h e r m a j o r e x t e n s i o n was t o i n c l u d e in a c a l c u l a t i o n a l l t h e v a l e n c e e l e c t r o n s o f a m o l e c u l e , and n o t s o l e l y t h e
p^ o r b i t a l s o f c a r b o n at oms in u n s a t u r a t e d s y s t e m s . T h i s a p p r o a c h
c u l m i n a t e d in t h e " e x t e n d e d HUckel t h e o r y " o f Hoffmann.
In 19 51 t h e r e had been p u b l i s h e d by Root haan t h e f i r s t o f two p a p e r s , in wh i ch he d e v e l o p e d a v e r y e l e g a n t method f o r t h e s o l u t i o n o f t h e
S c h r ô d i n g e r e q u a t i o n f o r a m a n y - e l e c t r o n s y s t e m , in t h e i n d e p e n d e n t
p a r t i c l e model o f H a r t r e e and Fock. ( Thi s method i s r e v i e we d in s e c t i o n
2 . 1 . ) The s e l f - c o n s i s t e n t f i e l d p r o c e d u r e employed - whi ch i s o n l y on e o f
s e v e r a l p o s s i b l e r o u t e s t o t h e s o l u t i o n o f t h e S c h r Ü d i n g e r e q u a t i o n - g i v e s
o n l y an a p p r o x i m a t i o n t o t h e e x a c t s o l u t i o n , s i n c e t h e method i g n o r e s
s i m u l t a n e o u s t w o - e l e c t r o n I n t e r a c t i o n s . The e x a c t s o l u t i o n can be
a p p r o a c h e d by t h e method o f c o n f i g u r a t i o n i n t e r a c t i o n , whi ch in t h e l i m i t o f an i n f i n i t e e x p a n s i o n w i l l y i e l d t h e e x a c t wave f u n c t i o n .
A l t h o u g h t h e t h e o r y was w e l 1- d e v e l o p e d , t h e p r a c t i c a l i m p l e m e n t a t i o n
o f t h e Root haan scheme f o r t h e g e n e r a l , a l l - e l e c t r o n c a s e had t o a w a i t t h e d e v e l o p m e n t o f p o we r f u l e l e c t r o n i c c o m p u t e r s . By I9 6 3, v a r i o u s
— 4 —
R o o t h a a n ' s s e l f - c o n s i s t e n t f i e l d p r o c e d u r e were a v a i l a b l e . Some, f o r
r e a s o n s o f s p e e d , u t i l i s e d a p p r o x i m a t i o n s in t h e i n t e g r a l e v a l u a t i o n s .
Th e s e a p p r o x i m a t i o n s we r e g e n e r a l l y c h o s e n t o g i v e t h e b e s t p o s s i b l e f i t
w i t h e x p e r i m e n t a l d a t a , and so a r e d e s c r i b e d as s e m i - e m p i r i c a l
c a l c u l a t i o n s .
O t h e r c a l c u l a t i o n s s e t o u t t o e v a l u a t e a l l i n t e g r a l s e x p l i c i t l y
in t er ms o f f u n d a me n t al p h y s i c a l q u a n t i t i e s , i n t r o d u c i n g no e m p i r i c a l t er ms a t a l l . The name ab i n i t i o ( o r a p r i o r i ) was used t o d e s c r i b e
t h e s e c a l c u l a t i o n s . In vi ew o f t h e v e r y l a r g e amount s o f c o m p u t e r t i me
r e q u i r e d f o r s uch ab i n i t i o c a l c u l a t i o n s , a t i me whi ch i s d e p e n d e n t on t h e number o f at oms in t h e m o l e c u l e c o n s i d e r e d , and a l s o t h e number o f
e l e c t r o n s b e l o n g i n g t o e a c h c o n s t i t u e n t at om, t h e m a j o r i t y o f ab i n i t i o
c a l c u l a t i o n s have been c a r r i e d o u t on s ma l l m o l e c u l e s , w i t h s i x o r l e s s
a t o m s . Most o f t h e s i m p l e common o r g a n i c m o l e c u l e s hav e been t r e a t e d , and as a bench mark f o r f u t u r e c a l c u l a t i o n s d e m e n t i , Mehl and von
(2)
N i e s s e n have c a r r i e d o u t c a l c u l a t i o n s on t h e g u a n i n e - c y t o s i n e b a s e p a i r : f o r a s y s t e m o f 29 at oms a l m o s t s e v e n t y b i l l i o n t w o - e l e c t r o n
i n t e g r a l s had t o be e v a l u a t e d , and t h e c o m p l e t e c a l c u l a t i o n t o o k e i g h t
days on an IBM S y s t e m/ 36 0 Model 195* The l a r g e s t ab i n i t i o c a l c u l a t i o n r e p o r t e d t o d a t e i s by B a t r a and S e k i , on a c a r b a z o l e - ( N O ^ ) ^ f l u o r e n o n e
compl ex - a f i f t y at om s y s t e m ^ ^ ^ ^ ^ .
The c u r r e n t " s t a t e o f t h e a r t " in ab i n i t i o c a l c u l a t i o n s up t o t h e (3)
end o f 1971 has been s u r v e y e d c o m p r e h e n s i v e l y by S c h a e f e r
One s t r i k i n g f e a t u r e , a p p a r e n t e i t h e r from S c h a e f e r ' s monograph, o r
t h e b i b l i o g r a p h y o f R i c h a r d s , Wa l ke r and H i n k l e y ^ ^ ^ , wh i c h r e c o r d s ( a l m o s t )
a l l t h e ab i n i t i o c a l c u l a t i o n s p u b l i s h e d u n t i l December 1969, i s t h e
5
-\
O p e n - s h e l l d i a t o m ! c s and h y d r i d e s a r e r e p r e s e n t e d , b u t v i r t u a l l y n o t h i n g
l a r g e r . Thi s p a r t l y r e f l e c t s t h e p r e d o mi n a n t i n t e r e s t o f c h e m i s t s in c l o s e d - s h e l l s p e c i e s - r a d i c a l s a r e , a f t e r a l l , a m i n o r i t y i n t e r e s t -
b u t i t i s a l s o a c o n s e q u e n c e o f t h e much g r e a t e r d i f f i c u l t y found In a c h i e v i n g c o n v e r g e n c e in t h e o p e n - s h e l l c a s e s .
R a d i c a l s h a v e , howe ver , an i n t r i n s i c i n t e r e s t o f t h e i r own. Under
normal l a b o r a t o r y c o n d i t i o n s most a r e e x t r e m e l y s h o r t - l i v e d ; t h e y can o f t e n o n l y be s t u d i e d e i t h e r when s t a b i l i z e d a t low t e m p e r a t u r e s (as when
t h e y a r e f r o z e n in gas m a t r i c e s ) o r e l s e as t r a n s i e n t s in f l a m e s . T h i s n o r m a l l y makes t h e d e t e r m i n a t i o n o f t h e p h y s i c a l p r o p e r t i e s v e r y d i f f i c u l t .
O p t i c a l and e l e c t r o n s p i n r e s o n a n c e (ESR) s p e c t r a can u s u a l l y be o b s e r v e d
and some p a r a m e t e r s d e d u c e d , w i t h v a r y i n g d e g r e e s o f r e l i a b i l i t y , from t h e s e . On r a r e r o c c a s i o n s mass and p h o t o e l e c t r o n s p e c t r a and e l e c t r o n
d i f f r a c t i o n p a t t e r n s can be d e t e r m i n e d . Yet r a d i c a l s a r e o f t e n p o s t u l a t e d
as i n t e r m e d i a r y s p e c i e s in r e a c t i o n s , p a r t i c u l a r l y in o r g a n i c c h e m i s t r y ,
and t h e i r p r o p e r t i e s a r e a c c o r d i n g l y o f c o n s i d e r a b l e i n t e r e s t t o t h e c h e m i s t .
T h i s , t h e n , i s t h e m o t i v a t i o n f o r t h e f o l l o w i n g s t u d y , in whi ch we
e n d e a v o u r t o e l u c i d a t e t h e e l e c t r o n i c and g e o m e t r i c s t r u c t u r e o f s e v e r a l
known and one unknown r a d i c a l . The r e s u l t s d e a l i n g w i t h s p e c i f i c r a d i c a l s
a r e d e a l t w i t h s e p a r a t e l y in s e c t i o n s 3 “ 10. We a l s o s e t o u t t o
s y s t e m a t i z e t h e p r a c t i c e o f ab i n i t i o c a l c u l a t i o n s on t r i a t o m i c s p e c i e s ,
m o t i v a t e d by c o n s i d e r a t i o n s s i m i l a r t o t h o s e t h a t i m p e l l e d Cade, Huo, Wahl and c o - w o r k e r s in t h e i r d e t a i l e d c a l c u l a t i o n s on d i a t o m i c s ( d i s c u s s e d in
c h a p t e r 3 o f r e f e r e n c e (3) ) • So f a r no s t u d i e s c o m p a r a b l e w i t h t h e s e
" 6 *"
we wi s h t o f i l l t h i s g a p , by e x a m i n i n g p a r t i c u l a r l y and in d e t a i l t h e e f f e c t t h a t c h o i c e o f b a s i s s e t s i z e has on v a r i o u s computed p r o p e r t i e s -
p r i n c i p a l l y t h e o p t i m i s e d g e o m e t r y , b u t a l s o t h e e l e c t r o n i c e n e r g y and t h e
form o f t h e wave f u n c t i o n , as i t i s r e v e a l e d by Mull I ken p o p u l a t i o n
a n a l y s i s and c a l c u l a t i o n o f s p i n d e n s i t i e s a t n u c l e i . By so d o i n g we hope t o be a b l e t o e s t i m a t e t h e r e l i a b i l i t y o f s i m i l a r c a l c u l a t i o n s on unknown s p e c i e s . The g e n e r a l c o n c l u s i o n s a r r i v e d a t w i t h r e g a r d t o t h e s e
q u e s t i o n s a p p e a r in s e c t i o n 11.
~ 7
2 T h e o r e t i c a l b ac k g r o u n d
2. 1 C a l c u l a t i o n o f m o l e c u l a r wave f u n c t i o n
The method empl oyed f o r t h e c a l c u l a t i o n s r e p o r t e d h e r e i s by now v e r y w e l l known, and h as been d i s c u s s e d by s o many d i f f e r e n t a u t h o r s ,
f r e q u e n t l y e mp l o y i n g r a t h e r d i f f e r e n t f o r m a l i s m s , t h a t I t seems
u n n e c e s s a r y and i nd e e d u n d e s i r a b l e t o add t o t h e i r number. I n s t e a d an
o u t l i n e o n l y o f t h e method w i l l be p r e s e n t e d , w i t h r e f e r e n c e b e i n g made
t o t h e most i m p o r t a n t o r i l l u m i n a t i n g p a p e r s on t h e s u b j e c t .
The m a t h e m a t i c a l r e l a t i o n s h i p s o f quantum m e c h a n i cs c a o be d e r i v e d
in two ways. They ca n be o b t a i n e d from c l a s s i c a l wave m e c h a n i c s , by i n t r o d u c i n g E i n s t e i n ' s r e l a t i o n s h i p (E = hv) and de B r o g l i e ' s c o n d i t i o n
(A « h / p ) , and u s i n g t h e s e t o o b t a i n S c h r O d i n g e r ' s e q u a t i o n . A l t e r n a t i v e l y
quant um m e c h a n i cs can be d e r i v e d a s t h e l o g i c a l c o n s e q u e n c e o f a s ma l l number o f f u n d a me n t al p o s t u l a t e s . Th es e p o s t u l a t e s a r e a b s t r a c t and
c o n c e p t u a l , and can be n e i t h e r pr o v e d n o r d i s p r o v e d . T h e i r j u s t i f i c a t i o n
and v a l u e l i e s in t h e a g r e e m e n t bet we en t h e d e r i v e d c o n c l u s i o n s and
e x p e r i m e n t .
Four p o s t u l a t e s o n l y a r e r e q u i r e d t o d e t e r m i n e t h e c o m p l e t e s t r u c t u r e
o f quant um m e c h a n i c s . For a s y s t e m o f p a r t i c l e s p o s s e s s i n g n d e g r e e s o f
f r e e d o m we s t a t e t h a t :
1. Any s t a t e o f t h e s y s t e m can be f u l l y d e s c r i b e d by a f u n c t i o n
V ( q ^ , q2 , . . . q ^ , t ) o f t h e n c o o r d i n a t e s and t i m e . T h i s f u n c t i o n o f s t a t e d e t e r m i n e s t h e v a l u e o f a l l p h y s i c a l o b s e r v a b l e s o f t h e s y s t e m , and
can i t s e l f be i n t e r p r e t e d in t e r ms o f p r o b a b i l i t y , by s a y i n g t h a t Y*Y.dT
(where d t d e n o t e s a volume e l e m e n t dq.j . d q ^ . . . . dq^ in c o n f i g u r a t i o n
s p a c e ) g i v e s t h e p r o b a b i l i t y t h a t t h e v a r i a b l e s q ^ , q2, . . . q ^ l i e in t h e volume e l e m e n t d% a t t i me t .
In o r d e r t h a t t h e p r o b a b i l i t y may be p h y s i c a l l y r e a s o n a b l e , t h e s t a t e f u n c t i o n is c o n s t r a i n e d t o s a t i s f y c e r t a i n w e l l - d e f i n e d c r i t e r i a , and i s
s a i d t o be a f u n c t i o n b e l o n g i n g t o c l a s s Q,.
2. To e v e r y p h y s i c a l o b s e r v a b l e t h e r e c o r r e s p o n d s a l i n e a r o p e r a t o r , whi ch may be o b t a i n e d from t h e c l a s s i c a l e x p r e s s i o n f o r t h e o b s e r v a b l e by
s i m p l e r u l e s . In o r d e r t h a t t h e v a l u e s f o r t h e o b s e r v a b l e s s h o u l d be
r e a l , t h e o p e r a t o r s a r e r e q u i r e d t o be H e r m l t i a n .
3. The s t a t e f u n c t i o n T i s r e q u i r e d t o s a t i s f y t h e e q u a t i o n
H(q, t ) T ( q , t ) = ! . . . A
whi ch i s S c h r d d i n g e r ' s t i m e - d e p e n d e n t e q u a t i o n . Here A i s t h e quant um
m e c h a n i ca l H a m i l t o n i a n o p e r a t o r , and H = h/2w.
4. The e x p e c t a t i o n v a l u e o f a d y nami ca l o b s e r v a b l e i s d e t e r m i n e d by t h e wave f u n c t i o n and t h e o p e r a t o r c o r r e s p o n d i n g t o t h e p a r t i c u l a r
o b s e r v a b l e (See p o s t u l a t e 2 ) .
<M> = /Y*MT.dT = < ¥ |m|¥>
f o r n o r m a l i z e d f u n c t i o n T. Here <M> d e n o t e s t h e e x p e c t a t i o n v a l u e o f t h e
dynami ca l o b s e r v a b l e M, and M i t s o p e r a t o r a n a l o g u e . We i n t r o d u c e a b o v e
w i t h o u t f u r t h e r comment D i r a c ' s b r a - k e t n o t a t i o n , wh i c h we s h a l l employ
l a t e r in t h i s s e c t i o n .
Some s t a t e m e n t s o f t h e p o s t u l a t e s p r e f e r t o subsume 4 i n t o 2.
I t s h o u l d a l s o be p o i n t e d o u t t h a t t h e r e i s no u n i q u e s e t o f
p o s t u l a t e s ; o t h e r s e t s e q u i v a l e n t t o t h a t above c o u l d be c o n s t r u c t e d . For a m o l e c u l a r s y s t e m t h e wave f u n c t i o n w i l l be a f u n c t i o n o f a l l t h e c o o r d i n a t e s , b o t h s p a c e and s p i n , o f t h e p a r t i c l e s c o m p r i s i n g t h e m o l e c u l e .
To s a t i s f y p o s t u l a t e 4, t h e wave f u n c t i o n must be an e i g e n - f u n c t i o n o f t h e
quant um m e c h a n i ca l o p e r a t o r c o r r e s p o n d i n g t o e n e r g y , t h a t i s t o s a y o f t h e
H a m i l t o n i a n o p e r a t o r .
H T ( q , t ) = E Y ( q , t )
and t h e r e f o r e from e q u a t i o n A,
9
-E ï ( q , t ) = i K ~
whi ch g I ves
W( q , t ) = i|^(q)exp(“ î Et / K) •
For a c o n s e r v a t i v e s y s t e m A Is n o t a f u n c t i o n o f t i m e , s o t h a t il/(q)
s a t i s f i e s t h e t i m e - i n d e p e n d e n t S c h r Ü d i n g e r e q u a t i o n ,
Hij;(q) = Ei|i(q) . . . B
wh e r e i|j(q) i s t h e wave f u n c t i o n f o r some s t a t i o n a r y s t a t e o f t h e s y s t e m .
In t h i s s t u d y we s e t o u t t o c a l c u l a t e s uch wave f u n c t i o n s f o r some
s p e c i f i c ch e mi c a l s p e c i e s .
In e q u a t i o n B a b o v e , A is t h e o p e r a t o r a n a l o g u e o f t h e H a m i l t o n i a n
o f c l a s s i c a l p h y s i c s . I f we n e g l e c t r e l a t i v i s t i c and m a g n e t i c e f f e c t s ,
i t i s e x p r e s s e d in a t o m i c u n i t s by
fi = I ( " " ' 0
^ = " .
wh e r e i and j a r e i n d i c e s r u n n i n g o v e r t h e n e l e c t r o n s o f t h e s y s t e m , A and B r e f e r t o t h e N n u c l e i , and r e p r e s e n t s t h e c h a r g e on n u c l e u s A.
The f i r s t summation d ep e nd s o n l y on t h e n u c l e a r g e o me t r y and k i n e t i c
e n e r g y ; t h e n e x t i s c l e a r l y a c o l l e c t i o n o f o n e - e l e c t r o n t e r m s , and may
be r e f e r r e d t o as t h e o n e - e l e c t r o n H a m i l t o n i a n , and t h e f i n a l t er m i s t h e t w o - e l e c t r o n H a m i l t o n i a n , d e s c r i b i n g I n t e r - e l e c t r o n i c Coulomb r e p u l s i o n .
I t w i l l be found c o n v e n i e n t a t t i m e s t o r e f e r t o t h e component s o f t h e
H a m i l t o n i a n in t h e a bove t e r m s , wh i c h may be e x p r e s s e d s y m b o l i c a l l y by
H = Hq + H^ + H^ •
- 10
\
I t can t h u s be s e e n t h a t Y (which now r e p r e s e n t s o n l y t h e s p a t i a l p a r t
o f t h e wave f u n c t i o n ) w i l l be a f u n c t i o n o f b o t h t h e e l e c t r o n i c and
n u c l e a r c o o r d i n a t e s ;
' I = ■
2 . 1 . 1 Born- Oppe nhei mer a p p r o x i m a t i o n
A s i m p l i f i c a t i o n Is commonly made now by i n t r o d u c i n g t h e Born-
Oppenhei mer a p p r o x i m a t i o n , wh i ch as s umes t h e n u c l e i o f t h e s p e c i e s t o be
f i x e d in p o s i t i o n . Th i s a p p r o x i m a t i o n can be j u s t i f i e d in vi ew o f t h e
g r e a t d i f f e r e n c e b et we en t h e n u c l e a r and e l e c t r o n i c m a s s e s . ( I t s h o u l d
however be n o t e d t h a t a wave f u n c t i o n c a l c u l a t e d u s i n g t h e Bor n- Op p e nh ei mer
a p p r o x i m a t i o n c o r r e s p o n d s t o no p h y s i c a l r e a l i t y , s i n c e t h e p r o p o s a l o f
f i x e d n u c l e i v i o l a t e s t h e u n c e r t a i n t y p r i n c i p l e . )
As a c o n s e q u e n c e o f t h e a p p r o x i m a t i o n t h e t o t a l wave f u n c t i o n can be
f a c t o r ! s e d
wh e r e t h e e l e c t r o n i c wave f u n c t i o n Y w i l l depend p a r a m e t r i c a l l y on t h e
n u c l e a r c o o r d i n a t e s . Th i s Y w i l l be t h e s o l u t i o n o f t h e e l e c t r o n i ce
S c h r Ü d i n g e r e q u a t i o n
A Y = e e EY e
and i t i s t h i s e l e c t r o n i c wave f u n c t i o n t h a t we s h a l l r e f e r t o fr om now on .
, t h e e l e c t r o n i c H a m i l t o n i a n , o m i t s t h e t er m a r i s i n g in t h e f u l l
H a m i l t o n i a n from t h e n u c l e a r k i n e t i c e n e r g i e s . The e f f e c t o f t h e f i n i t e
n u c l e a r mas s es can be i n t r o d u c e d l a t e r i n t o t h e wave f u n c t i o n as a
p e r t u r b a t i o n .
2 . 1 . 2 S i n g l e p a r t i c l e model
The most compl ex s y s t e m f o r wh i c h i t has s o f a r p r o v e d p o s s i b l e t o
11
-J
l i k e ) a t om s , i . e . t w o - c e n t r e p r o b l e m s . For any s y s t e m w i t h a l a r g e r
number o f e l e c t r o n s t h e I n t e r a c t i o n e x p r e s s e d by t h e Coulomb r e p u l s i o n
t e r m p r o d u c e s p r o b l e m s in t h e a n a l y s i s whi ch c a n n o t be s o l v e d e x a c t l y ;
i n s t e a d a p p r o x i m a t e met hods have t o be a d o p t e d . One common a p p r o x i m a t i o n
whi ch i s used h e r e has t h e c o n s i d e r a b l e m e r i t o f c o n f o r m i n g t o n a i v e
p i c t u r e s o f e l e c t r o n b e h a v i o u r . T h i s Is t o a p p r o x i m a t e t h e wave
f u n c t i o n , whi ch i s a f u n c t i o n o f 4n i n d e p e n d e n t v a r i a b l e s f o r an n - e l e c t r o n
s y s t e m , by a H a r t r e e p r o d u c t o f o n e - e l e c t r o n f u n c t i o n s ea c h d e p e n d i n g
o n l y on t h e s p a c e - s p i n c o o r d i n a t e s o f one e l e c t r o n . Such i|<. a r e c a l l e d
m o l e c u l a r ( s p i n ) o r b i t a l s , by a n a l o g y w i t h B o h r ' s a t o m i c o r b i t a l s . in
t h i s model t h e i n t e r a c t i o n bet we en p a r t i c l e s i s n o t c o n s i d e r e d e x a c t l y ,
and i n s t e a d o f t h e s i m u l t a n e o u s i n t e r a c t i o n o f a l l p a r t i c l e s , ea c h
p a r t i c l e i s c o n s i d e r e d t o move in an " a v e r a g e " p o t e n t i a l f i e l d , wh i ch i s a
t i m e - a v e r a g e o f t h e d i s t r i b u t i o n o f a l l t h e o t h e r e l e c t r o n s in t h e s y s t e m .
In o r d e r t o s a t i s f y t h e P a u l i a n t i s y m m e t r y p r i n c i p l e f o r f e r m i o n s ,
and t o a l l o w f o r t h e i n d i s t i n g u i s h a b i 1Î t y o f e l e c t r o n s , t h e c o m p l e t e wave
f u n c t i o n i s p r e f e r a b l y w r i t t e n as a l i n e a r c o m b i n a t i o n o f H a r t r e e
p r o d u c t s , whi ch may be r e e x p r e s s e d e i t h e r in t h e w e l l - k n o w n form o f t h e
S l a t e r d e t e r m i n a n t ,
Ï = ...
<!<2 (2 ) ...
(n) » n ( " )
o r e q u i v a l e n t l y as t h e s i m p l e H a r t r e e p r o d u c t , o p e r a t e d on by a p r o j e c t i o n
o p e r a t o r :
12
-\
Here Â(n) is t h e n - e l e c t r o n a n t i - s y m m e t r i z e r ,
A(n) = ( n l ) ^ / ^ ( - l ) P p
wh e r e P is t h e o p e r a t o r p e r m u t i n g e l e c t r o n s bet ween t h e v a r i o u s o r b i t a l s ,
and p is t h e p a r i t y o f t h e p e r m u t a t i o n . Yj (j ’ ) r e p r e s e n t s t h e i - t h
m o l e c u l a r s p i n o r b i t a l ( m s o ) , o c c u p i e d by e l e c t r o n j . Each o f t h e s e
o r b i t a l s can be f a c t o r i s e d i n t o a s p a t i a l p a r t , whi ch i s t er med a
m o l e c u l a r o r b i t a l ( m . o . ) , and a s p i n f u n c t i o n ;
w h e r e n. can t a k e t h e v a l u e a o r 3, d e p e n d i n g upon m^ f o r t h e e l e c t r o n .
The pr ob l em o f c a l c u l a t i n g a wave f u n c t i o n now r e s o l v e s t o a
d e t e r m i n a t i o n o f t h e s e o n e - e l e c t r o n o r b i t a l s . Th i s i s n o r m a l l y t a c k l e d
by u s i n g t h e v a r i a t i o n a l t h e o r e m , wh i ch s t a t e s t h a t f o r an a p p r o x i m a t e
n o r m a l i z e d wave f u n c t i o n Y' ,
<Y « I AIY ' > = E ' > Eg
wh e r e E^ is t h e l o w e s t e i g e n - v a l u e o f t h e H a m i l t o n i a n H, and t h e e q u a l i t y
h o l d s o n l y when Y' = Y. Thus <Y*|H| Y' > y i e l d s an u p p e r bound t o t h e
t r u e e n e r g y o f t h e s y s t e m , and t h i s may be used w i t h t h e R l t z v a r i a t i o n a l
p r o c e d u r e t o d e t e r m i n e a wave f u n c t i o n . I f t h e wave f u n c t i o n i s e x p r e s s e d
in t e r ms o f some p a r a m e t e r A . ,
T' = )
t h e n t h e p a r a m e t e r s may v a r i e d s o t h a t < T ' | A | T ' > Is m i n i m i z e d , w h i l e
< Y ' | Y ' > = 1. T h i s i m p l i e s a s e t o f e q u a t i o n s t o be s a t i s f i e d ,
Ix.( ) = 0 .
i
G e n e r a l l y t h e method w i l l n o t l e a d t o t h e e x a c t e i g e n - f u n c t i o n Y, b u t
o n l y t o t h e b e s t a p p r o x i m a t i o n t o i t in t er ms o f t h e a v a i l a b l e p a r a m e t e r s .
" 1 3 “
I n c r e a s e s , t h e o p t i m i s e d wave f u n c t i o n w i l l a p p r o a c h more c l o s e l y t h e
e x a c t v a l u e .
2.1 . 3 S e l f - c o n s i s t e n t f i e l d p r o c e d u r e
The a c t u a l p r o c e d u r e by whi c h t h e v a r i a t i o n a l p r i n c i p l e i s a p p l i e d
t o t h e wave f u n c t i o n i s t er me d t h e s e l f - c o n s i s t e n t f i e l d (SCF) met hod.
O r i g i n a l l y d e v e l o p e d by H a r t r e e f o r t h e c a l c u l a t i o n o f a t o m i c s t r u c t u r e ,
i t i s r e a d i l y a d a p t a b l e t o t h e m o l e c u l a r c a s e . The f o r m a l i s m t h a t we
employ h e r e was b r i l l i a n t l y d e v e l o p e d by R o o t ha a n , in two f u n d a me n t al
p a p e r s .
When t h e e l e c t r o n c o n f i g u r a t i o n o f t h e s p e c i e s u n d e r s t u d y c o n f or ms
t o a c l o s e d s h e l l s t a t e , t h e n t h e d e r i v a t i o n is f a i r l y s t r a i g h t f o r w a r d .
By s u b s t i t u t i n g e x p l i c i t e x p r e s s i o n s f o r t h e wave f u n c t i o n and t h e
H a m i l t o n i a n i n t h e e x p r e s s i o n
E = <Y|H|Y>
one o b t a i n s
E = 2%H. + I ( 2 J , . - K , . ) . . . C
i ' i , j ' J ' J
wh e r e H. a r e o n e - e l e c t r o n i n t e g r a l s o v e r t h e o r b i t a l s :
Hj
-and J . j -and K. j a r e t w o - e l e c t r o n i n t e g r a l s , t h e Coulomb -and e x c h a n g e
i n t e g r a l s r e s p e c t i v e l y .
Kjj =
A l t e r n a t i v e l y , t h e l a t t e r two r e l a t i o n s h i p s may be e x p r e s s e d in t er ms o f ;;|
* A y
- 14
Kjj =
To e x p r e s s t h e c o n d i t i o n t h a t t h e e n e r g y i s s t a b l e w i t h r e s p e c t t o
v a r i a t i o n in t h e o r b i t a l s cf)., Root haan i n t r o d u c e s u n d e t e r m i n e d
( L a g r a n g i a n ) m u l t i p l i e r s e . . . A f t e r i n c l u d i n g t h e a d d i t i o n a l c o n s t r a i n t * J
t h a t
a s e r i e s o f e q u a t i o n s a r e o b t a i n e d , wh i ch r e d u c e t o
F'l-j = E S i j * ;
f = H, I ( 2 J , - K^) .
The t o t a l wave f u n c t i o n Y i s i n v a r i a n t w i t h r e s p e c t t o an a r b i t r a r y
u n i t a r y t r a n s f o r m a t i o n on t h e o r b i t a l s . Th i s may be c h o s e n s o a s t o make
t h e m a t r i x o f L a g r a n g i a n m u l t i p l i e r s d i a g o n a l ( i . e . e . j = e . . ô . j ) ,
in whi ch c a s e t h e e q u a t i o n s r e d u c e t o a s e t o f n p s e u d o - e i g e n - v a l u e
e q u a t i o n s ,
F4>j = e..(|). .
( P s e u d o - e i g e n - v a l u e s i n c e t h e o p e r a t o r F d epe nds upon t h e s e t o f
s o l u t i o n s ^ . . ) Th e s e e q u a t i o n s may be s o l v e d by a r e l a t i v e l y s t r a i g h t
f o r w a r d I t e r a t i v e p r o c e d u r e : a g u e s s i s made f o r , F i s c a l c u l a t e d ,
and t h e e q u a t i o n s s o l v e d f o r <j). and e . . . By c o m p a r i s o n w i t h t h e s t a r t i n g
o r b i t a l s a new s e t o f i s c o n s t r u c t e d . T h i s p r o c e d u r e Is r e p e a t e d
u n t i l t h e t r i a l and o u t p u t v e c t o r s a g r e e t o w i t h i n some s p e c i f i e d a c c u r a c y ,
15
-The o p e n - s h e l l c a s e i s c o n s i d e r a b l y more co mp l ex. An e x p r e s s i o n f o r
t h e e n e r g y can be o b t a i n e d , a n a l o g o u s t o C a bo v e :
E = 2 K + I ( 2 J „ - K „ )
K K y i
+ + I (ZaJmn ' ' "k*) )
m m,n k,m
wh e r e t h e c o n v e n t i o n i s i n t r o d u c e d t h a t s u b s c r i p t s k and 1 r e f e r t o c l o s e d -
s h e l l o r b i t a l s , m and n r e f e r t o o p e n - s h e l l o r b i t a l s (and i and j r e f e r t o
u n d i f f e r e n t i a t e d o r b i t a l s ) . a and b a r e n u m e r i c a l c o n s t a n t s , s p e c i f i c
f o r t h e p a r t i c u l a r e l e c t r o n c o n f i g u r a t i o n , and f i s a f r a c t i o n a l o c c u p a n c y
number.
P r o c e e d i n g as b e f o r e , L a g r a n g i a n m u l t i p l i e r s e . j a r e i n t r o d u c e d t o
e x p r e s s t h e c o n s t r a i n t t h a t t h e o r b i t a l s s t a y o r t h o g o n a l d u r i n g t h e
v a r i a t i o n . M a n i p u l a t i o n o f t h e r e l a t i o n s h i p s t o f i n d t h e c o n d i t i o n o f
minimum e n e r g y r e s u l t s in two d i s t i n c t s e t s o f e q u a t i o n s f o r t h e c l o s e d -
and o p e n - s h e l l o r b i t a l s r e s p e c t i v e l y .
(H, + 2 J ^ - H 2 J ^
-2 3 - ^ '
Here f o u r new o p e r a t o r s , t h e c l o s e d - and o p e n - s h e l l Coulomb and e x c h a n g e
o p e r a t o r s a r e i n t r o d u c e d .
3 = p k k : 3 = m
« /V M vs ...Î
^ = I ^ k k : "o = m
3
16
-\
c o n t r i b u t i o n s :
(H , + 2 J ^ - r<^ + 2 J ^ - + % G n k * n
f ( H, + 2 J , - k + 2 a J „ - b K j * = l e , é, + l e è
. . . E
n
Root haan showed how i t i s p o s s i b l e , by t h e use o f a s u i t a b l e u n i t a r y
t r a n s f o r m a t i o n , t o d i a g o n a l i z e t h e m a t r i x o f e l e m e n t s e^|^, and a l s o
t h a t c o m p r i s i n g b u t no t r a n s f o r m a t i o n w i l l e l i m i n a t e c o e f f i c i e n t s
o f t h e form and t h o s e l i n k i n g t h e c l o s e d - and o p e n - s h e l l s e t s
T h i s d i f f i c u l t y seemed a m a j or s t u m b l i n g b l o c k in o p e n - s h e l l
c a l c u l a t i o n s .
I t was c i r c u m v e n t e d by R o o t h a a n , who, by an i n g e n i o u s s e r i e s o f
r e a r r a n g e m e n t s , managed t o e x p r e s s t h e o f f - d i a g o n a l L a g r a n g i a n
m u l t i p l i e r s in t e r ms o f new o p e r a t o r s , whi ch he c a l l e d Coulomb and
e x c h a n g e c o u p l i n g o p e r a t o r s , L^ and L^, and and Mg. He showed t h a t
- I ^ n ^ n k = n ( 2 0 ^ 0 ’
= f ( 2 a L ^ - BM^)
wh e r e a = and g ” s u b s t i t u t i n g t h e s e t e r ms in t h e
e q u a t i o n s E a b o v e , t h e t e r ms a r i s i n g from t h e o f f - d i a g o n a l m u l t i p l i e r s
can be a b s o r b e d i n t o e f f e c t i v e H a m i l t o n i a n s , y i e l d i n g u l t i m a t e l y two
e q u a t i o n s
Fc*k “ j>ik+i
F o^m = Ye 0 ^ nmrn n
17
-F d> o^m — E d> mm^m
f o r t h e c l o s e d and open s h e l l o r b i t a l s (<{)^ and r e s p e c t i v e l y . Not e
t h a t d i f f e r e n t e f f e c t i v e H a m i l t o n i a n s a r e employed f o r t h e two s e t s o f
e q u a t i o n s . J u s t as in t h e c l o s e d s h e l l c a s e , t h e s e can be s o l v e d by
an i t e r a t i v e p r o c e d u r e .
2 . 1 . 4 Beyond t h e wave f u n c t i o n
Along w i t h t h e wave f u n c t i o n , t h e s e l f - c o n s i s t e n t f i e l d p r o c e d u r e
o u t l i n e d above y i e l d s t h e t o t a l e n e r g y and t h e e i g e n - v a l u e s o f t h e
v a r i o u s o r b i t a l s . For t h e c l o s e d - s h e l l c a s e t h e e i g e n - v a l u e s can be
i d e n t i f i e d w i t h t h e o r b i t a l e n e r g i e s in t h e s e n s e o f Koopmans' t h e o r e m ,
whi c h s t a t e s t h a t t h e o r b i t a l e n e r g y can be e q u a t e d t o t h e n e g a t i v e o f
t h e i o n i z a t i o n p o t e n t i a l from t h a t l e v e l . Thes e o r b i t a l e n e r g i e s can
be c o r r e l a t e d w i t h e x p e r i m e n t a l d a t a o b t a i n e d from p h o t o e l e c t r o n s p e c t r o s c o p y
o r w i t h i o n i z a t i o n p o t e n t i a l s c a l c u l a t e d from t h e r m o c h e m i c a l d a t a .
U n f o r t u n a t e l y t h e Root haan o p e n - s h e l l p r o c e d u r e , e mp l o y i n g e f f e c t i v e
H a m i l t o n i a n s , does n o t g i v e e i g e n - v a l u e s whi ch can be i d e n t i f i e d w i t h
o r b i t a l e n e r g i e s . The most common c o n s e q u e n c e o f t h i s i s t h a t t h e s i n g l y -
o c c u p i e d o r b i t a l i s shown t o p o s s e s s an e i g e n - v a l u e l ower t h a n t h e h i g h e s t
d o u b l y - o c c u p i e d o r b i t a l . However i t i s c o m p a r a t i v e l y s t r a i g h t f o r w a r d t o
modi f y t h e t h e o r y s o t h a t t h e e i g e n - v a l u e s a r e p h y s i c a l l y s i g n i f i c a n t .
F u r t h e r m o r e we have shown in s e c t i o n 3*5, by c o n s i d e r a t i o n o f a p a r t i c u l a r
e x a mp l e , t h a t t h e e i g e n - v a l u e s c a l c u l a t e d d i r e c t l y by R o o t h a a n ' s o p e n - s h e l l
- 1
From t h e t o t a l e n e r g y , e x p r e s s e d as a f u n c t i o n o f t h e g e o m e t r i c a l
p a r a m e t e r s o f t h e m o l e c u l e , an e n e r g y s u r f a c e can be c o n s t r u c t e d from
whi ch t h e minimum e n e r g y g e o m e t r y can be e s t a b l i s h e d , and f o r c e c o n s t a n t s
c a l c u l a t e d ( a l t h o u g h t h i s l a t t e r a n a l y s i s i s n o n - t r i v i a l f o r p o l y a t o m i c s ) . (7)
Th es e p r o p e r t i e s b e l o n g t o t h e c l a s s i d e n t i f i e d by Boys and Cook as
d e r i v a t i v e p r o p e r t i e s .
Use is s omet i mes made o f t h e t o t a l m o l e c u l a r e n e r g y in a c o m p a r i s o n
w i t h t h e t o t a l e n e r g y o f t h e c o n s t i t u e n t atoms in o r d e r t o o b t a i n
e s t i m a t e s o f d i s s o c i a t i o n e n e r g i e s . However u n l e s s a c c o u n t i s t a k e n o f
c o r r e l a t i o n e f f e c t s , t h e r e s u l t o b t a i n e d from s uch a c a l c u l a t i o n i s
c o m p l e t e l y u n r e l i a b l e , and any a g r e e m e n t w i t h t h e r mo c h e mi c a l d a t a
f o r t u i t o u s .
Once t h e wave f u n c t i o n has been o b t a i n e d , i t can be a n a l y s e d in
v a r i o u s wa ys. As s t a t e d in p o s t u l a t e 4 , t h e v a l u e o f any dynami ca l
o b s e r v a b l e can be o b t a i n e d from t h e wave f u n c t i o n by u t i l i s i n g t h e
a p p r o p r i a t e o p e r a t o r . Thus t h e e x p e c t a t i o n v a l u e s o f t h e d i p o l e and
h i g h e r e l e c t r i c a l moments, q u a d r u p o l e and o c t u p o l e moments, e l e c t r i c
f i e l d g r a d i e n t s a t t h e n u c l e i and many o t h e r p r o p e r t i e s can a l l be
e v a l u a t e d r o u t i n e l y . In many c a s e s however t h e r e e x i s t s no e x p e r i m e n t a l
meas ur ement o f t h e s e p r o p e r t i e s f o r t h e p u r p o s e o f c o m p a r i s o n . Even t h e
d i p o l e moment has o n l y been meas u r ed f o r a v e r y s ma l l number o f r a d i c a l s ,
deduc ed from o b s e r v a t i o n s o f t h e S t a r k e f f e c t .
The r e l i a b i l i t y o f c a l c u l a t e d e x p e c t a t i o n v a l u e s Is i n d i c a t e d by t h e
t he o r e m o f M i l l e r and P l e s s e t , whi ch s t a t e s t h a t o n e - e l e c t r o n p r o p e r t i e s
c a l c u l a t e d from H a r t r e e - F o c k wave f u n c t i o n s a r e a c c u r a t e t o f i r s t o r d e r .
However t h e p r e c i s e i n t e r p r e t a t i o n a t t a c h e d t o " f i r s t o r d e r " in t h i s
19
A ver y-commonl y us ed method o f a n a l y s i s o f t h e wave f u n c t i o n i s t h a t
(81
due t o M u l l i k e n . In t h i s me t h od , b a s e d on t h e LCAO a s s u m p t i o n , t h e
c o n t r i b u t i o n o f t h e o r b i t a l s c e n t r e d on each atom t o t h e t o t a l wave
f u n c t i o n i s c o n s i d e r e d , and t h e e l e c t r o n d e n s i t y a s s i g n e d a c c o r d i n g l y t o
g i v e " g r o s s a t o m i c p o p u l a t i o n s " . A l t e r n a t i v e l y t h e o r b i t a l c o e f f i c i e n t s
ca n be i n t e r p r e t e d t o g i v e " n e t a t o m i c p o p u l a t i o n s " and o v e r l a p
p o p u l a t i o n s , w i t h t h e l a t t e r b e i n g c l o s e l y i d e n t i f i e d w i t h t h e ch e mi c a l
c o n c e p t o f an I h t e r - a t o m i c b ond. T h i s met hod, t hough w i d e l y u s e d , is
e x t r e m e l y c r u d e , and d e f i c i e n c i e s hav e o f t e n been p o i n t e d o u t , u s u a l l y
h i n g i n g upon t h e e q u i p a r t i t i o n o f t h e e l e c t r o n p o p u l a t i o n in b o n d i n g
o r b i t a l s bet we en d i s s i m i l a r at oms when c o n s t r u c t i n g t h e g r o s s a t o m i c
p o p u l a t i o n s . Improved schemes o f p o p u l a t i o n a n a l y s i s h a v e been p u t
f o r w a r d , e . g . by Lüwdln^^^ and by Pol I t z e r . In r e c e n t y e a r s I t
has become I n c r e a s i n g l y p o p u l a r , n o t t o t a b u l a t e some a r b i t r a r i l y d e f i n e d
p a r a m e t e r s , b u t r a t h e r t o p o r t r a y e l e c t r o n d i s t r i b u t i o n by g i v i n g c o n t o u r
d i a g r a ms o f t h e e l e c t r o n d e n s i t y .
For an o p e n - s h e l 1 s y s t e m one can a l s o o bt ai n, from t h e wave f u n c t i o n
a m e a s u r e o f t h e s p i n d e n s i t y , t h a t Is t h e e x c e s s o f t h e a v e r a g e number o f
e l e c t r o n s w i t h one s p i n o v e r t h e number w i t h t h e o p p o s i t e s p i n , a t ea c h
n u c l e u s In t h e m o l e c u l e . T h i s s p i n d e n s i t y , wh i c h can b e r e l a t e d t o t h e
o b s e r v a b l e I s o t r o p i c h y p e r f i n e c o u p l i n g by t h e Fermi c o n t a c t I n t e r a c t i o n ,
may be e x p r e s s e d a s
Pn =
wh e r e t h e summat ion Is o v e r a l l e l e c t r o n s , w h i l e t h e o p e r a t o r t a k e s
20
-2 . -2 Ab i n i t i o progr ams
To c a r r y o u t t h e SCF c a l c u l a t i o n o f m o l e c u l a r wave f u n c t i o n s as
d e c s r i b e d in t h e p r e v i o u s s e c t i o n w i t h o u t i n t r o d u c i n g any a r b i t r a r y
e m p i r i c a l o r s e m i - e m p i r i c a l p a r a m e t e r s in t h e i n t e g r a l e v a l u a t i o n i s a
v e r y l e n g t h y t a s k in vi ew o f t h e enormous amounts o f d a t a whi ch have t o
be h a n d l e d , and s u c h c a l c u l a t i o n s have o n l y r e a l l y become f e a s i b l e
f o l l o w i n g t h e a d v e n t o f l a r g e e l e c t r o n i c c o m p u t e r s . The i m p o r t a n c e o f
t h e s e ma c h i n e s can be d e m o n s t r a t e d by l o o k i n g a t on e o f t h e v e r y r a r e
a l l - e l e c t r o n c a l c u l a t i o n s wh i c h we r e p e r f o r me d by hand - on , u s i n g (11)
a minimal b a s i s s e t . In 1955 a c a l c u l a t i o n u s i n g d e s k c a l c u l a t o r s
t o o k t h r e e m a n - y e a r s . Ten y e a r s l a t e r t h e same c a l c u l a t i o n on a s e c o n d
g e n e r a t i o n c o mp ut e r t o o k 0 . 0 2 m i n u t e s : an i mprovement in s p e e d by a
•7
f a c t o r o f more t h a n 10 . Thus o n l y w i t h t h e d e v e l o p m e n t o f f a s t c o m p u t e r s
w i t h l a r g e c o r e memori es in t h e 1 9 6 0 ‘s have m. o. c a l c u l a t i o n s on p o l y a t o m i c
s p e c i e s become, f i r s t f e a s i b l e and t h e n r o u t i n e . Each a d v a n c e in c o m p u t er
t e c h n o l o g y , w h e t h e r s p e e d i n g up t h e ma c hi ne c y c l e t i m e , o r I n c r e a s i n g t h e
s p e e d o f a c c e s s t o d a t a in s l ow s t o r e , e n l a r g e s t h e r a n g e o f compounds
t h a t f a l l w i t h i n t h e s c o p e o f ab i n i t i o s t u d y .
I t Is c l e a r t h a t w r i t i n g c o m p u t er p r ogr ams t o c a r r y o u t t h e s e
c a l c u l a t i o n s i s a m a j o r t a s k , whi ch n e c e s s a r i l y i n v o l v e s t h e c o l l a b o r a t i o n
o f a l a r g e t eam. Many p r ogr ams have now been w r i t t e n ( o f wh i ch some o f
t h e b e t t e r known a r e IBMOL^^^^ POLYATOM^^^), MOSES^^^) and
REFLECT^^^)) a n d , t o r e d u c e d u p l i c a t i o n o f programming e f f o r t , many o f
t h e s e p rogr ams a r e r o u t i n e l y made a v a i l a b l e t o a l l i n t e r e s t e d s c i e n t i s t s
t h r o u g h t h e Quantum C h e m i s t r y Program Exchange. U s u a l l y ea ch p r o gr a m has
c h a r a c t e r i s t i c s t r e n g t h s and d r a w b a c k s , and a s ma l l s a mp l e have been
(17)
- 21
a f t e r t h e i r i n i t i a l a p p e a r a n c e , b o t h in t h e c o d i n g and in t h e a l g o r i t h m s
empl oyed. Th e s e h ave been known t o i n c r e a s e t h e s p e e d o f e x e c u t i o n by
an o r d e r o f m a g n i t u d e . As an e x a mp l e , t h e v a r i o u s s t a g e s in t h e
d e v e l o p m e n t o f IBMOL by Cl ement ! a r e p a r t i c u l a r l y w e l l document ed^
In t h e work r e p o r t e d h e r e on t r i a t o m i c r a d i c a l s we have m a i n l y made
u s e o f t h r e e w e l l - k n o w n and w e 11“ document ed p r o g r a m s ; IBMOL-4, IBMOL-5
and ATMOL.
Al most a l l t h e p r o d u c t i o n runs we r e c a r r i e d o u t w i t h t h e IBMOL s y s t e m . ( 1 2 )
Th i s was o r i g i n a l l y d e v e l o p e d by Cl ement i and Davis , and has
s u b s e q u e n t l y been m o d i f i e d and i mproved q u i t e e x t e n s i v e l y ; a t t h e t i me o f
w r i t i n g t h e d ev e l o p m e n t o f v e r s i o n 6 Is j u s t c o n c l u d i n g . At t h e
U n i v e r s i t y o f St Andrews a m o d i f i e d v e r s i o n - a p a r t i a l h y b r i d o f v e r s i o n
2 and v e r s i o n 4 - was d e v e l o p e d by Dr C o l i n Thomson f o r r u n n i n g on t h e
IBM Sy s t e m/ 3 6 0 Model 44 o f t h e U n i v e r s i t y Computing L a b o r a t o r y . The s ma l l
c o r e s i z e o f 128i< b y t e s n e c e s s i t a t e d c o n s i d e r a b l e r e s t r u c t u r i n g and
o v e r l a y i n g o f t h e p r o g r a m , and a r e d u c t i o n In t h e maximum s i z e o f
c a l c u l a t i o n i t c o u l d h a n d l e .
C o n s i d e r a b l e c h a n g e s w e r e made t o IBMOL by Cl ement i and Mehl, and a
new v e r s i o n r e l e a s e d In 1971 a s IBMOL-5^^^^. T h i s d i d away w i t h t h e
o p e r a t o r i n t e r v e n t i o n in t h e r u n n i n g o f t h e p r o g r a m , by s o d o i n g making I t
more s u i t a b l e f o r m u l t i - p r o g r a m m i n g e n v i r o n m e n t s . The symmetry t r a n s f o r
m a t i o n s e c t i o n f ound s e p a r a t e l y In IBMOL-4 was a b s o r b e d i n t o t h e I n t e g r a l
e v a l u a t i o n r o u t i n e s In IBMOL-5, s o t h a t I n t e g r a l s we r e e v a l u a t e d d i r e c t l y
o v e r s y m m e t r y - a d a p t e d f u n c t i o n s . F i n a l l y t h e form o f i n p u t was g r e a t l y
I mproved, and a l a r g e number o f p r o g r a m - s u p p l i e d d e f a u l t o p t i o n s s p e c i f i e d ;
T h i s g r e a t l y f a c i l i t a t e s j o b p r e p a r a t i o n .
F i n a l l y , us e has bee n made o f t h e ATMOL s y s t e m o f p r o g r a m s .
22
-as in IBMOL, b u t I t u s e s a d i f f e r e n t SCF method t o a v o i d t h o s e s i t u a t i o n s
wh e r e R o o t h a a n ' s method f a i l s t o c o n v e r g e . The a u t h o r s u s e a scheme
b a s e d on McWeeny's method o f s t e e p e s t d e s c e n t s , b u t w i t h m o d i f i c a t i o n s
i n s p i r e d by R o o t h a a n ' s method t o s pe e d c o n v e r g e n c e . By a p a r t i c u l a r
c h o i c e o f v a r i o u s e f f e c t i v e H a m i l t o n i a n s , d e f i n e d in t er ms o f d e n s i t y
m a t r i c e s , i t i s p o s s i b l e t o s i m u l a t e R o o t h a a n ' s met hod. A l t e r n a t i v e l y ,
H a m i l t o n i a n s can be c h o s e n s o t h a t t h e f i n a l e l g e n - v a l u e s a r e p h y s i c a l l y
s i g n i f i c a n t , in t h e s e n s e o f Koopmans' t heorem.
Some c a l c u l a t i o n s ha v e been c a r r i e d o u t w i t h t h e ATOMSCF p r o g r a m ,
(21
)
w r i t t e n by Roos, S a l e z , Ve i l l a r d and Cl ement i , wh i ch c a r r i e s o u t SCF
c a l c u l a t i o n s on atoms In t h e H a r t r e e - F o c k scheme.
2 . 3 Cho i ce o f I n p u t
When w i s h i n g t o c a r r y o u t a c a l c u l a t i o n u s i n g o n e o f t h e s e p rogr ams
t h e u s e r has t o s p e c i f y a l a r g e number o f p a r a m e t e r s . O b v i o u s l y t h e
g e o me t r y and e l e c t r o n c o n f i g u r a t i o n have t o be s p e c i f i e d , b u t a l s o a
s u i t a b l e b a s i s s e t o f a t o m i c o r b i t a l s must be c h o s e n , and t h i s l a t t e r
demands g r e a t c a r e .
Both t h e s i z e o f t h e b a s i s s e t and t h e n u m e r i c a l v a l u e o f t h e e x p o n e n t s
ch o s e n e x e r t a l a r g e I n f l u e n c e on t h e c a l c u l a t e d p r o p e r t i e s . ( Th i s a s p e c t
o f t h e c a l c u l a t i o n s Is s o I m p o r t a n t t h a t i t i s d i s c u s s e d s e p a r a t e l y In
s e c t i o n 11. ) I f t h e e x p o n e n t s c h o s e n a r e r e a s o n a b l e , t h e n g e n e r a l l y t h e
l a r g e r t h e b a s i s s e t . I . e . t h e more c l o s e l y I t a p p r o a c h e s c o m p l e t e n e s s , t h e
more r e l i a b l e t h e c a l c u l a t i o n s w i l l b e . However a n o t h e r f a c t o r mus t be
c o n s i d e r e d , whi ch p r e v e n t s t h e I n d i s c r i m i n a t e us e o f t h e l a r g e s t b a s i s s e t
p o s s i b l e . The t i me s p e n t on i n t e g r a l e v a l u a t i o n Is r o u g h l y p r o p o r t i o n a l
- 23
s t r o n g a r g u me n t s on t h e g r o un d s o f c o s t f o r u s i n g t h e s m a l l e s t p o s s i b l e
b a s i s s e t commens ur a t e w i t h t h e a c c u r a c y r e q u i r e d . Thus when s e l e c t i n g
t h e b a s i s , a d e l i c a t e b a l a n c e has t o be s t r u c k bet we en a c c u r a c y and
economy.
Normal l y t h e a t o m i c b a s i s s e t s s e l e c t e d hav e been o b t a i n e d by a
p r o c e s s o f e x p o n e n t o p t i m i z a t i o n on t h e at om. I t i s a l m o s t c e r t a i n
t h a t t h e same b a s i s w i l l n o t be o p t i m a l in t h e m o l e c u l a r s i t u a t i o n .
(The c l e a r e s t exa mp l e o f t h i s i s t h e h ydr oge n at om, wh e r e t h e a t o m i c
e x p o n e n t is e x a c t l y 1, w h i l e t h e same atom In m o l e c u l a r h y dr o g e n Is b e s t
r e p r e s e n t e d by a more compact o r b i t a l , w i t h an e x p o n e n t o f a b o u t 1 . 2 . )
in vi ew o f t h e c o mp u t i n g t i m e r e q u i r e d , v e r y l i t t l e work has been
done on t h e o p t i m i z a t i o n o f e x p o n e n t s In m o l e c u l e s . P o p l e and c o
-(2 2)
w o r k e r s have r e p o r t e d " s c a l i n g f a c t o r s " , whi ch r e l a t e m o l e c u l a r -
o p t i m i z e d e x p o n e n t s t o t h e at om e x p o n e n t s , f o r hy d r o g e n and t h e
f i r s t - r o w e l e m e n t s c a r b o n t o f l u o r i n e . Only f o r h y d r o g e n i s t h e
s c a l i n g f a c t o r f ound t o d i f f e r s i g n i f i c a n t l y from u n i t y . A c c o r d i n g l y
In m o l e c u l a r c a l c u l a t i o n s t h e h y d r o g e n a t o m i c b a s i s Is commonly " s c a l e d "
by m u l t i p l y i n g a l l t h e e x p o n e n t s by a c o n s t a n t f a c t o r . The p r a c t i c a l
e f f e c t t h i s has Is d e m o n s t r a t e d In o u r c a l c u l a t i o n s on NHg. (See
s e c t i o n 4 . 3 . )
In t h e normal c a l c u l a t i o n o f m o l e c u l a r wave f u n c t i o n s , I t Is more
e f f i c i e n t t o I n c r e a s e t h e s i z e o f t h e a t o m i c b a s i s , r a t h e r t h a n o p t i m i z e
t h e b a s i s s e t in t h e mo1 e c u 1 a r s i t u a t i o n . %
Most ab I n i t i o c a l c u l a t i o n s u s e b a s i s s e t s o f one o f two t y p e s ;
S l a t e r - t y p e o r b i t a l s , w i t h an e x p ( - r ) d e p e n d e n c e , o r G a u s s i a n o r b i t a l s ,
. %
h a v i n g an e x p ( - r ^ ) d e p e n d e n c e . The l a t t e r , w h i l e g i v i n g a c o n s i d e r a b l y |
24
-c o n v e n i e n -c e in t h a t t h e m a n y - -c e n t r e d i n t e g r a l s a r e e a s i e r t o e v a l u a t e
o v e r G a u s s i a n o r b i t a l s t h a n o v e r S l a t e r f u n c t i o n s , and a l l t h e p rogr ams
t h a t we employ and d i s c u s s h e r e u s e b a s i s s e t s o f t h i s , G a u s s i a n , for m.
Numerous t a b l e s o f o p t i m i z e d a t o m i c Ga u s s i a n f u n c t i o n s have been
p u b l i s h e d . In t h i s work we u s e t h e f o l l o w i n g s o u r c e s : f o r hyd r o g e n (2 3)
e x p o n e n t s , Huzinaga ; f o r ( 3 , 2 ) and ( 5 , 2 ) b a s e s , Whitman and Hornback
; f o r (7,3) b a s i s Roos and S i e g b a h n ^ ^ ^ ^ , and f o r ( 9 , 5 ) and ( 1 1 , 7 )
b a s e s Huzi naga and c o - w o r k e r s ‘
(When r e p r e s e n t i n g b a s i s s e t s we s h a l l use t h e n o t a t i o n ( a , b , c ) t o
r e p r e s e n t a b a s i s s e t c o m p r i s i n g a f u n c t i o n s o f s t y p e , b f u n c t i o n s o f
p t y p e and c f u n c t i o n s o f d t y p e . In t h e e v e n t o f a m o l e c u l e c o n t a i n i n g
h yd r og e n in a d d i t i o n t o f i r s t row e l e m e n t s , we s e p a r a t e d e t a i l s o f t h e
h y d r o g e n b a s i s s e t f r om t h e o t h e r s by a s l a s h ( / ) . A c o n t r a c t e d b a s i s
s e t w i l l be s i m i l a r l y r e p r e s e n t e d , b u t u s i n g b r a c k e t s in p l a c e o f
p a r e n t h e s e s , t h u s : < a , b , c / d , e > . )
2. 3*1 C o n t r a c t i o n c o e f f i c i e n t s
A f u r t h e r m o d i f i c a t i o n o f t h e b a s i s s e t i s o f t e n made, by g r o u p i n g
t o g e t h e r a number o f G a u s s i a n s h a v i n g t h e same c e n t r e and s ymmet r y, and ■
s p e c i f y i n g t h a t t h e i r c o e f f i c i e n t s in t h e m o l e c u l a r wave f u n c t i o n w i l l
remai n in a c o n s t a n t r a t i o t o e a c h o t h e r - t h e s e r a t i o s b e i n g s p e c i f i e d
by t h e pr ogr am u s e r . Thus I n s t e a d o f t h e c o e f f i c i e n t s o f ea c h b a s i s
f u n c t i o n b e i n g a l l o w e d t o v a r y I n d e p e n d e n t l y , o n l y one v a r i a b l e r e m a i n s :
a c o e f f i c i e n t d e t e r m i n i n g t h e c o n t r i b u t i o n o f t h e e n t i r e g r o u p . The
c o m p o s i t e f u n c t i o n , c a l l e d a c o n t r a c t e d G a u s s i a n , w i l l be h a n d l e d by t h e
SCF p r o c e d u r e s a s one f u n c t i o n , y e t I t f u r n i s h e s a b e t t e r d e s c r i p t i o n o f
t h e e l e c t r o n d i s t r i b u t i o n In an at om t h a n a s i n g l e , u n c o n t r a c t e d G a u s s i a n
c o u l d do. The u s e o f s uch c o n t r a c t e d f u n c t i o n s in a c a l c u l a t i o n r e d u c e s
