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Reduced basis set expansions for the iodine ion orbitals
Volume 28A, n u m b e r 5
REDUCED
P H Y SIC S L E T T E R S
BASIS
SET
EXPANSIONS
FOR
16 D e c e m b e r 1968
THE
IODINE
ION
ORBITALS
M. S Y N E K a n d A. D A M O M M I O Department o f Physics, Texas Christian University, F o r t Worth, Texas, USA Received 2 November 1968
Analytical expansions, with s h o r t and easily manageable b a s i s sets, are available for the orbital wave functions of I-.
W a v e f u n c t i o n s f o r t h e n e g a t i v e i o n of i o d i n e are needed for certain applications concerning silver iodide crystals. Previously, orientational wave functions for A g + h a v e b e e n p r o v i d e d [1]. T o s u p p l e m e n t t h i s w o r k we c a l c u l a t e d a n a l y t i c a l e x p a n s i o n s f o r t h e o r b i t a l w a v e f u n c t i o n s of I - , ( Z = 5 3 ) , 5 s 2 5 p 6, 1S, using the Roothaan self-consistent field method w i t h a r e d u c e d s e t of S l a t e r - t y p e b a s i s f u n c t i o n s . T h e o r b i t a l s of t h e s a n d p s y m m e t r y a r e t r e a t e d in t h e m i n i m u m b a s i s s e t r e p r e s e n t a t i o n . B y c o n t r a s t t h e o r b i t a l s of t h e d s y m m e t r y a r e r e p r e s e n t e d b y f o u r b~s'Ls f u n c t i o n s , a s a p a r t i a l c o m -
promise with the saturated basis set in the usual s e n s e [2]. A l l t h e o r b i t a l e x p o n e n t s ~Xp of t h e individual basis functions were optimized. A p o r t i o n of t h e r e s u l t s i s d i s p l a y e d i n t a b l e 1. The value obtained for the total energy, namely E = - 6 9 1 0 . 4 9 6 (a.u.), c o u l d b e l o w e r e d s i g n i f i c a n t . ly b y e m p l o y i n g s t a t u r a t e d b a s i s s e t s f o r t h e s and p symmetries. However, the expansions presented are designed to be sufficiently simple for a p p l i c a t i o n s of a p r e l i m i n a r y o r o r i e n t a t i o n a l n a ture. The virial theorem was satisfied quite well; E p / E k = - 1.9999805.
Table 1. The optimized exponents ~)tp of the b a s i s functions and the elgenvectors of coefficients Cik p for I-, 5s25p 6, 1S. B a s i s function
~kp
Eigenvectors
ls
51.93962
1.00096
2s
19.51462
3s
11.60841
4s 5s
ls
3s 0.18730
-0.00316
1.13011
0.00233
-0.11902
6.36064
-0.00090
2.67574
0.00018 2p
4s
5s
-0.09145
0.02595
-0.70864
0.36314
-0.10373
1.24865
-0.78837
0.23120
0.03219
-0.00539
1.17943
-0.39443
-0.00602
0.00237
0.03989
1.04139
3p
4p
5p
2p
24.41035
0.98246
-0.49265
0.21967
3p
11.61955
0.04123
1.09262
-0.60124
0.13518
4p
6.04787
-0.00991
0.01698
1.12449
-0.28078
5p
2.27113
0.00143
-0.00105
0.04200
1.02028
3d
19.19818
0.22139
-0.09386
3d
344
2s -0.37189
4d
3d
11.23610
0.79821
-0.32782
4d
6.57594
4d
3.90877
0.04169 -0.00773
0.60265 0.54237
-0.04831
Volume 28A. number 5
P H Y SIC S L E T T E R S
T h e a s s i s t a n c e of the T e x a s C h r i s t i a n U n i v e r s i t y R e s e a r c h F o u n d a t i o n and of the W r i g h t - P a t terson Air Force Base is appreciated.
ON
THE
VALIDITY OF THE
16 December 1968
1. M. Synek and F. Schmitz, Phys. Letters 27A (1968) 349. 2. M. Synek, A . E . Rainis and E. A. Peterson, J. Chem. Phys. 46 (1967) 2039.
OF RECENT PHENOMENOLOGICAL ISING MODEL AND SUPERFLUIDITY
THEORIES
R. H S I A N G - T A O YEH
Department of Physics, State University of New York at Buffalo, Buffalo, New York, USA Received 2 November 1968
We show explicitly that, the recent phenomenological theories of Wong (for the Ising model) and Mamaladze (for superfluid), are in conflict with exact theory or with experiment.
T h e c r i t i c a l i n d i c e s , a s p r e d i c t e d by the L a n dau t h e o r y of the s e c o n d o r d e r p h a s e t r a n s i t i o n , a r e in c o n f l i c t with e x p e r i m e n t a l r e s u l t s in s u p e r fluid, o r e x a c t c a l c u l a t i o n of the I s i n g - M o d e l [1]. L a n d a u ' s t h e o r y m a k e s u s e of p o w e r s e r i e s e x p a n s i o n of the f r e e e n e r g y in the o r d e r p a r a m e t e r . R e c e n t l y , s e v e r a l a t t e m p t s [2-5] h a v e b e e n m a d e to d e r i v e the c o r r e c t c r i t i c a l i n d i c e s f r o m L a n d a u ' s t h e o r y , by c h o o s i n g a p p r o p r i a t e t e m p e r a t u r e d e p e n d e n c e of the c o e f f i c i e n t in the f r e e e n e r g y e x p a n s i o n . H o w e v e r , all t h e s e t h e o r i e s m a d e u s e of the p r i n c i p l e of m i n i m i z a t i o n of the f r e e e n e r g y , and e q u a t e the m o s t p r o b a b l e v a l u e of the o r d e r p a r a m e t e r to the a v e r a g e v a l u e . G e n e r a l a r g u m e n t s [1] h a v e a l r e a d y shown that t h i s p r o c e d u r e n e g l e c t s the e f f e c t of the f l u c t u a t i o n in the o r d e r p a r a m e t e r , which i s i m p o r t a n t n e a r the t r a n s i t i o n point, e s p e c i a l l y f o r s y s t e m s with s h o r t r a n g e i n t e r a c t i o n s . H e n c e t h i s a p p r o a c h i s not a p p r o p r i a t e f o r s u p e r f l u i d o r I s i n g - M o d e l . It i s o u r p u r p o s e to show t h i s e x p l i c i t l y . F o r the t w o - d i m e n s i o n a l I s i n g - M o d e l , Wong [2] s u g g e s t e d the f o l l o w i n g e x p r e s s i o n f o r the f r e e e n e r g y d e n s i t y d i f f e r e n c e f - f o , in t e r m s of the reduced temperature t = T/Tc:
+ C4(1-t)]~P) 4 + C6(1-t)¼(p)6 +
+ C 1 2 ( 1 - t ) ½ ( p ) 12 + C 1 4 ( 1 - t ) ¼ ( p ) 14 + + c 1 6 ( p ) 16 - h i p ) .
i kBT(1 - t)~ g(r,r')-
8A °
~I Jo(ir,~o
I-fiR)
for T > T c
(2) R = I r - r ' I, J o i s the z e r o t h o r d e r B e s s e l f u n c tion. F o r T < T c, g(Jr, r' ) has the s a m e f o r m , e x c e p t that B o i s r e p l a c e d by a d i f f e r e n t c o n s t a n t . So, in the n o t a t i o n of r e f . 1, g(R) ~ R ° a s ]l-tiRe0, a n d g ( R ) ~ R-~ f o r ] l - t i R >>1. In t e r m s of the c r i t i c a l i n d e x , t h i s i m p l i e s that 77 = 0. Eq. (1) a l s o i m p l i e s that 7 = ~ • S i n c e the d i m e n s i o n of the s y s t e m d = 2, t h i s m o d e l d o e s not a g r e e with one of the c o n s e q u e n c e s of s c a l i n g l a w s [1], n a m e l y , d 7 / ( 2 - 7 / ) = 2. M o r e o v e r , the e x a c t c a l c u l a t i o n of the I s i n g m o d e l [1] y i e l d s f o r I I - t l R >>1,
l exp[-Ii-tln]
( ( t - 1)R) ~ J
r rc
I (i:t) ll J 2
f - f o = Ao(1 - t ) - " [ V(P)] 2 + S o ( 1 - t)¼(P )2 +
+ C 8 ( 1 - t ) i p ) 8 + C l O ( 1 - t ) ~ i p ) 10 +
W h e r e Ao, Bo, Cn'S a r e c o n s t a n t s , (P) is the o r d e r p a r a m e t e r and h is the t h e r m a l d y n a m i c c o n j u g a t e of (p). F o l l o w i n g the m e t h o d of r e f . 1, the s p i n - s p i n c o r r e l a t i o n f u n c t i o n g ( r , r ' ) i s obt a i n e d as:
(1)
and f o r (1 - t)R << 1
g ( n ) ~ R-¼
T < Tc .
(3)
So the e x a c t s o l u t i o n of g ( R ) i s q u i t e d i f f e r e n t f r o m that g i v e n by W o n g ' s m o d e l . N o t e that 7/ = ¼ 345
REDUCED
P H Y SIC S L E T T E R S
BASIS
SET
EXPANSIONS
FOR
16 D e c e m b e r 1968
THE
IODINE
ION
ORBITALS
M. S Y N E K a n d A. D A M O M M I O Department o f Physics, Texas Christian University, F o r t Worth, Texas, USA Received 2 November 1968
Analytical expansions, with s h o r t and easily manageable b a s i s sets, are available for the orbital wave functions of I-.
W a v e f u n c t i o n s f o r t h e n e g a t i v e i o n of i o d i n e are needed for certain applications concerning silver iodide crystals. Previously, orientational wave functions for A g + h a v e b e e n p r o v i d e d [1]. T o s u p p l e m e n t t h i s w o r k we c a l c u l a t e d a n a l y t i c a l e x p a n s i o n s f o r t h e o r b i t a l w a v e f u n c t i o n s of I - , ( Z = 5 3 ) , 5 s 2 5 p 6, 1S, using the Roothaan self-consistent field method w i t h a r e d u c e d s e t of S l a t e r - t y p e b a s i s f u n c t i o n s . T h e o r b i t a l s of t h e s a n d p s y m m e t r y a r e t r e a t e d in t h e m i n i m u m b a s i s s e t r e p r e s e n t a t i o n . B y c o n t r a s t t h e o r b i t a l s of t h e d s y m m e t r y a r e r e p r e s e n t e d b y f o u r b~s'Ls f u n c t i o n s , a s a p a r t i a l c o m -
promise with the saturated basis set in the usual s e n s e [2]. A l l t h e o r b i t a l e x p o n e n t s ~Xp of t h e individual basis functions were optimized. A p o r t i o n of t h e r e s u l t s i s d i s p l a y e d i n t a b l e 1. The value obtained for the total energy, namely E = - 6 9 1 0 . 4 9 6 (a.u.), c o u l d b e l o w e r e d s i g n i f i c a n t . ly b y e m p l o y i n g s t a t u r a t e d b a s i s s e t s f o r t h e s and p symmetries. However, the expansions presented are designed to be sufficiently simple for a p p l i c a t i o n s of a p r e l i m i n a r y o r o r i e n t a t i o n a l n a ture. The virial theorem was satisfied quite well; E p / E k = - 1.9999805.
Table 1. The optimized exponents ~)tp of the b a s i s functions and the elgenvectors of coefficients Cik p for I-, 5s25p 6, 1S. B a s i s function
~kp
Eigenvectors
ls
51.93962
1.00096
2s
19.51462
3s
11.60841
4s 5s
ls
3s 0.18730
-0.00316
1.13011
0.00233
-0.11902
6.36064
-0.00090
2.67574
0.00018 2p
4s
5s
-0.09145
0.02595
-0.70864
0.36314
-0.10373
1.24865
-0.78837
0.23120
0.03219
-0.00539
1.17943
-0.39443
-0.00602
0.00237
0.03989
1.04139
3p
4p
5p
2p
24.41035
0.98246
-0.49265
0.21967
3p
11.61955
0.04123
1.09262
-0.60124
0.13518
4p
6.04787
-0.00991
0.01698
1.12449
-0.28078
5p
2.27113
0.00143
-0.00105
0.04200
1.02028
3d
19.19818
0.22139
-0.09386
3d
344
2s -0.37189
4d
3d
11.23610
0.79821
-0.32782
4d
6.57594
4d
3.90877
0.04169 -0.00773
0.60265 0.54237
-0.04831
Volume 28A. number 5
P H Y SIC S L E T T E R S
T h e a s s i s t a n c e of the T e x a s C h r i s t i a n U n i v e r s i t y R e s e a r c h F o u n d a t i o n and of the W r i g h t - P a t terson Air Force Base is appreciated.
ON
THE
VALIDITY OF THE
16 December 1968
1. M. Synek and F. Schmitz, Phys. Letters 27A (1968) 349. 2. M. Synek, A . E . Rainis and E. A. Peterson, J. Chem. Phys. 46 (1967) 2039.
OF RECENT PHENOMENOLOGICAL ISING MODEL AND SUPERFLUIDITY
THEORIES
R. H S I A N G - T A O YEH
Department of Physics, State University of New York at Buffalo, Buffalo, New York, USA Received 2 November 1968
We show explicitly that, the recent phenomenological theories of Wong (for the Ising model) and Mamaladze (for superfluid), are in conflict with exact theory or with experiment.
T h e c r i t i c a l i n d i c e s , a s p r e d i c t e d by the L a n dau t h e o r y of the s e c o n d o r d e r p h a s e t r a n s i t i o n , a r e in c o n f l i c t with e x p e r i m e n t a l r e s u l t s in s u p e r fluid, o r e x a c t c a l c u l a t i o n of the I s i n g - M o d e l [1]. L a n d a u ' s t h e o r y m a k e s u s e of p o w e r s e r i e s e x p a n s i o n of the f r e e e n e r g y in the o r d e r p a r a m e t e r . R e c e n t l y , s e v e r a l a t t e m p t s [2-5] h a v e b e e n m a d e to d e r i v e the c o r r e c t c r i t i c a l i n d i c e s f r o m L a n d a u ' s t h e o r y , by c h o o s i n g a p p r o p r i a t e t e m p e r a t u r e d e p e n d e n c e of the c o e f f i c i e n t in the f r e e e n e r g y e x p a n s i o n . H o w e v e r , all t h e s e t h e o r i e s m a d e u s e of the p r i n c i p l e of m i n i m i z a t i o n of the f r e e e n e r g y , and e q u a t e the m o s t p r o b a b l e v a l u e of the o r d e r p a r a m e t e r to the a v e r a g e v a l u e . G e n e r a l a r g u m e n t s [1] h a v e a l r e a d y shown that t h i s p r o c e d u r e n e g l e c t s the e f f e c t of the f l u c t u a t i o n in the o r d e r p a r a m e t e r , which i s i m p o r t a n t n e a r the t r a n s i t i o n point, e s p e c i a l l y f o r s y s t e m s with s h o r t r a n g e i n t e r a c t i o n s . H e n c e t h i s a p p r o a c h i s not a p p r o p r i a t e f o r s u p e r f l u i d o r I s i n g - M o d e l . It i s o u r p u r p o s e to show t h i s e x p l i c i t l y . F o r the t w o - d i m e n s i o n a l I s i n g - M o d e l , Wong [2] s u g g e s t e d the f o l l o w i n g e x p r e s s i o n f o r the f r e e e n e r g y d e n s i t y d i f f e r e n c e f - f o , in t e r m s of the reduced temperature t = T/Tc:
+ C4(1-t)]~P) 4 + C6(1-t)¼(p)6 +
+ C 1 2 ( 1 - t ) ½ ( p ) 12 + C 1 4 ( 1 - t ) ¼ ( p ) 14 + + c 1 6 ( p ) 16 - h i p ) .
i kBT(1 - t)~ g(r,r')-
8A °
~I Jo(ir,~o
I-fiR)
for T > T c
(2) R = I r - r ' I, J o i s the z e r o t h o r d e r B e s s e l f u n c tion. F o r T < T c, g(Jr, r' ) has the s a m e f o r m , e x c e p t that B o i s r e p l a c e d by a d i f f e r e n t c o n s t a n t . So, in the n o t a t i o n of r e f . 1, g(R) ~ R ° a s ]l-tiRe0, a n d g ( R ) ~ R-~ f o r ] l - t i R >>1. In t e r m s of the c r i t i c a l i n d e x , t h i s i m p l i e s that 77 = 0. Eq. (1) a l s o i m p l i e s that 7 = ~ • S i n c e the d i m e n s i o n of the s y s t e m d = 2, t h i s m o d e l d o e s not a g r e e with one of the c o n s e q u e n c e s of s c a l i n g l a w s [1], n a m e l y , d 7 / ( 2 - 7 / ) = 2. M o r e o v e r , the e x a c t c a l c u l a t i o n of the I s i n g m o d e l [1] y i e l d s f o r I I - t l R >>1,
l exp[-Ii-tln]
( ( t - 1)R) ~ J
r rc
I (i:t) ll J 2
f - f o = Ao(1 - t ) - " [ V(P)] 2 + S o ( 1 - t)¼(P )2 +
+ C 8 ( 1 - t ) i p ) 8 + C l O ( 1 - t ) ~ i p ) 10 +
W h e r e Ao, Bo, Cn'S a r e c o n s t a n t s , (P) is the o r d e r p a r a m e t e r and h is the t h e r m a l d y n a m i c c o n j u g a t e of (p). F o l l o w i n g the m e t h o d of r e f . 1, the s p i n - s p i n c o r r e l a t i o n f u n c t i o n g ( r , r ' ) i s obt a i n e d as:
(1)
and f o r (1 - t)R << 1
g ( n ) ~ R-¼
T < Tc .
(3)
So the e x a c t s o l u t i o n of g ( R ) i s q u i t e d i f f e r e n t f r o m that g i v e n by W o n g ' s m o d e l . N o t e that 7/ = ¼ 345