Coupled Hartree-Fock approximation for calculating frequency-dependent refractive index of helium ga

Dalgarno, A.

Victor, G. A.

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GCA Technical R e p o r t  No. 6 5 - 2 4 - N  
P H Y S I C S  OF PLANETARY A'IHOSPHERES 111: 
I 
TJB TIME-DEPENDENT COUPLED W T R E E - F O C K  APPROXIMATION 
G. A. V i c t o r  
A. D a l g a r n o  
GCA CORPORATION 
GCA TECHNOLOGY D I V I S I O N  
B e d f o r d ,  Massachusetts 
C o n t r a c t  N o .  NASW-1283 
- 
P r e p a r e d  f o r  
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 
HEADQUARTERS 
Washington, D. C .  
A u g u s t  19 65 
PHYSICS OF PLANETARY ATMOSPHERES 111: 
THE TIME-DEPENDENT COUPLED HARTREE-FOCK APPROXIMATICN 
By G. A. Victor and A. Dalgarno 
ABS TRACT 
A coupled Hartree-Fock approximation f o r  d e s c r i b i n g  t h e  e f f e c t s  of 
time-dependent p e r t u r b a t i o n s  on many-electron systems i s  presented .  It i s  
app l i ed  t o  t h e  c a l c u l a t i o n  of t h e  frequency-dependent r e f r a c t i v e  index of 
helium gas wi th  r e s u l t s  t h a t  d i f f e r  by between 4% and 8% from the  a c c u r a t e  
va lues  obtained by a r e f i n e d  v a r i a t i o n a l  c a l c u l a t i o n .  
INTRODUCTION 
There are  two gene ra l  schemes of approximation i n  t h e  time-independent 
t heo ry  of many e l e c t r o n  systems, t h e  uncoupled and t h e  coupled Hartree-Fock 
approximations [ 13". Time-dependent Hartree-Fock equa t ions  were der ived  
by Dirac [ 2 ]  and by Frenkel  [ 3 ]  and t h e i r  ex tens ion  t o  i n c l u d e  e x t e r n a l  
p e r t u r b a t i o n s  has been presented  by Karplus and Kolker [ 4 ]  i n  t h e  uncoupled 
approximation and by Thouless CS] and McLachlan and B a l l  161 i n  t h e  coupled 
approximation. The uncoupled approximation has been used by Karplus and 
Kolker [ 4 ]  t o  c a l c u l a t e  r e f r a c t i v e  i n d i c e s  and long range  fo rces  bu t  t h e  
equa t ions  presented  f o r  t h e  coupled approximation do n o t  lend themselves 
t o  ready  s o l u t i o n .  A s  i n  time-independent theory t h e  coupled approximation 
i s  t h e  more a c c u r a t e  Dalgarno [ 7 ]  and an a l t e r n a t i v e  formula t ion  of t h e  
coupled approximation i s  needed. 
Department of Applied Mathematics, The Queen's Un ive r s i ty  of B e l f a s t .  
Numbers i n  [ 3 throughout t h e  tex t  r e p r e s e n t  r e f e r e n c e  numbers. JC 
1 
2 
THE TIME-DEPENDENT HARTREE-FOCK APPROXIMATION 
The wave function Y (0) (zl, r , . . . r ) of a closed shell N 
electron system with electron pos?$ion vectors -N r is written in the Hartree-Fock approximation as a i  
where A is the normalized antisymmetrising operator and U!’) (2.) is the 
Hartree-Fock orbital. Y ( O )  satisfies the time-independent equation 
in which the Fock Hamiltonian operator is given by 
where 
2 being the nuclear charge, 
and 
cxik vanishes  unless  the  s p i n s  assoc ia ted  wi th  uk and Ui are p a r a l l e l  and 
i t  ope ra t e s  only on func t ions  of r. .  
t h e  equat ions  
The s p i n  o r b i t a l s  Ui ( E ~ )  s a t i s f y  
-1 
Equation ( 2 )  can be w r i t t e n  a l t e r n a t i v e l y  i n  t h e  time-dependent 
form 
w i t h  s o l u t i o n  
W e  now apply a n  e x t e r n a l  time-dependent p e r t u r b a t i o n  
and proceed t o  so lve  t h e  per turbed time-dependent equat ion  
3 
wi th in  the  Hartree-'Fock approximation scheme. We expand accord ing  t o  
and i n  order  t h a t  (13)  be normalized t o  f i r s t  o rder  i n  k w e  r e q u i r e  t h a t  
and 
To s o l v e  ( 1 2 ) ,  w e  c o n s t r u c t  the f u n c t i o n a l  
N N  
where + e ,  
w e  may e$ fec t  a u n i t a r y  transformation such t h a t  
v i  and si a r e  Lagrange m u l t i p l i e r s .  For c losed  s h e l l  systems 
a. = 6 i j  xij - 6  x ij x (17) 
l j  
4 
and we may write 
A 7:;) exp ( i w t )  t v!’) exp (-iwt) + . . . 
“i = {  1- } 
{ } 
i i+ 1- 1 
ci = x 
x = (x(1) exp ( i w t )  + x!l) exp ( - i w t >  + . . . 
exp ( i w t )  + exp ( - i w t >  + . . . 
(0) 
V r i n g  t h  
equat ing  t h e  c o e f f i c i e n t s  of exp ( i w t )  and exp ( - i w t )  t o  zero  y i e l d s  t h e  
unperturbed equat ion  
unc t ion  (16) independently wi th  r e s p e c t  t o  t h e  o r b i t a l s  Ui , 
U{$y and U,, e l f  t o  f ind  a s t a t i o n a r y  value,  expanding i n  powers of X and 
and the  coupled f i r s t  o rde r  perturbed equat ions  
5 
where 
and 
ilct a 
and V ( I )  (zi) 
on thk-subscr ip ts .  
t h e  pe r tu rb ing  f i e l d  and t h e  
The m u l t i p l i e r s  X(1) and X ( y  a r e  given immediately by (22) a&$ (23) a$- 
l+ 1- 
i s  given by a s imi la r  express ion  with t h e  s i g n s  interchanged 
The p o t e n t i a l s  Vi;) and V!:) r e f l e c t  t h e  in f luence  of 
couple toge the r  t h e  s o l u t i o n s  U ( l )  and U ( 1 ) .  
N 
and (22) and (23) s impl i fy  t o  
I n  t h e  uncoupled Hartree-Fock approximation, t h e  p o t e n t i a l  t e r m s  
V ( l )  a re  assumed t o  vanish. i+ 
THE REFRACTIVE INDEX OF HELIUM 
The pe r tu rb ing  p o t e n t i a l  app ropr i a t e  t o  t h e  c a l c u l a t i o n  of t h e  r e f r a c t -  
i v e  index o r  dynamic d i p o l e  p o l a r i z a b i l i t y  of helium i s  
r l  
2 ) u ( r . )  = r1 cos e + r cos 0 i -1 1 2 w 
i 
and (28) reduces t o  
0 
For t h e  unperturbed o r b i t a l  U 
of Roothan, Sachs and Weiss [8]  and wrote U$1) i n  t h e  form 
( r )  w e  adopted t h e  twelve t e r m  r e p r e s e n t a t i o n  
- 
We solved (30) by f i n d i n g  t h e  s t a t i o n a r y  va lues  of t h e  func t iona l  
7 
for trial functions 
m 
a procedure which leads to the set of equations 
+ -  where a 
2 - <Uo1ri+'l@> and X+, X- and L are m by m matrices with elements 3 
a and c are column vectors with elements ai+' ai- and 
and 
L = L  ij ij 
J r  J 
8 
Expressing d i s t a n c e  i n  u n i t s  of t h e  Bohr r ad ius  a t h e  dynamic d i p o l e  
0' 
p o l a r i z a b i l i t y  of helium may be w r i t t e n  
a = 2  (J' + J' 3 a 
0 
(37) 
A comparison of t h e  va lues  of t he  dynamic p o l a r i z a b i l i t y  der ived us ing  
t h e  uncoupled and the  coupled approximations i s  given i n  t a b l e  1. The 
uncoupled r e s u l t s  a r e  i n  c l o s e  agreement wi th  those obtained by Karplus and 
Kolker [ 4 ]  
o r b i t a l  Uo(r). Table 1 a l s o  includes t h e  accu ra t e  va lues  obtained by 
Chan and Dalgarno [ 9 ]  based upon a t en ty  parameter c o r r e l a t e d  r ep resen ta t ion  
of t h e  unperturbed wave func t ion  !T!(Oy. Table 1 shows t h e  coupled approximat- 
i on  i s  about 4% too small a t  t h e  la rger  wavelengths, i n c r e a s i n g  t o  about 8% 
f o r  t h e  s h o r t e r  wavelengths. The uncoupled approximation i s  everywhere 
about 8% too  la rge .  I f  t h e  va lues  a r e  co r rec t ed  by s c a l i n g  f a c t o r s  chosen 
t o  ensure  an accu ra t e  va lue  f o r  the s t a t i c  p o l a r i z a b i l i t y ,  t h e  e r r o r  i n  
t h e  sca l ed  va lues  i s  nowhere g rea t e r  than 4% fo r  t he  coupled r e s u l t s .  
based upon a l e s s  e labora te  r ep resen ta t ion  of t h e  unperturbed 
These r e s u l t s ,  taken toge ther  with c a l c u l a t i o n s  of s t a t i c  p o l a r i z a b i l i t i e s  
of heavier  systems [ l o ] ,  [ l l ]  suggest t h a t  t h e  coupled approximation may be 
s u c c e s s f u l  i n  p red ic t ing  frequency-dependent p o l a r i z a b i l i t i e s  a l so .  
ACKNOWLEDGMENT 
This  work has been p a r t l y  supported by GCA Corporation under c o n t r a c t s  
w i th  the  A i r  Force and t h e  National Aeronautics and Space Administration. 
9 
TABLE 1 
FREQUENCY-DEPENDENT POLARIZABILITY OF HELIUM ( cm3> 
Far:mncy 
0.00 
0.05 
0.10 
0.15 
0.20 
0.25 
0.30 
0.35 
0.40 
0.45 
0.50 
0.55 
0.60 
Wavelength 
9 112 
4556 
3037 
2278 
1822 
15 19 
1302 
1139 
10 12 
9 11 
828 
7 59 
Uncoupled 
0.220 
0.221 
0.223 
0.226 
0.231 
0.237 
0.246 
0.257 
0.271 
0.289 
0.314 
0.348 
0.398 
Accurate 
0.204 
0.205 
0.207 
0.210 
0.214 
0.220 
0.227 
0.237 
0.250 
0.267 
0.289 
0.320 
0.366 
Coup led 
0.196 
0.196 
0.198 
0.201 
0.205 
0.210 
0.217 
0.226 
0.237 
0.252 
0.272 
0.299 
0.337 
10 
REFERENCES 
1. Dalgarno, A. 1962 Adv. Phys. ll, 281. 
2. Dirac, P. A. M. 1930 Proc. Camb. P h i l .  SOC. 26, 376. 
3. Frenkel ,  J. 1934 Wave mechanics, advanced gene ra l  theory.  
(Clarendon P res s  : Oxford). 
4 .  Karplus, M. and Kolker, H. J. 1963 J .  Chem. Phys. 39, 2987. 
Karplus, M. and Kolker, H. J. 1964 J .  Chem. Phys. 41, 3955. 
5. Thouless, D. J. 1961 Quantum mechanics of many-body systems 
(Academic Press :  New Yorkl. 
6. McLachlan, A. D. and B a l l ,  M. A. 1964 Rev. Mod. Phys. 36, 844. 
7 .  Dalgarno, A. 1959 Proc. Roy. SOC. A251, 282. 
8. Roothan, C.  C. J .  , Sachs, L. M. and Weiss, A. 1960 Rev. 
Mod. Phys. 32,  186. 
9 .  Chan, Y. M. and Dalgarno, A. 1965 Proc. Phys. SOC. 85, 227. 
10. Dalgarno, A. and McIntyre, H. A. 1965 Proc. Phys. SOC. 85, 47. 
11. Cohen, H. D. 1965, Pr iva te  communication. 
11 


Physics of planetary atmospheres III - The time-dependent coupled Hartree-Fock approximation

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