Hirschfelder-Silbey perturbation theory applied to positive hydrogen io

Hake, R. B.

Hirschfelder, J. O.

Silbey, R. J.

English

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AN APPLICATION OF HIRSCHFELDER-SILBEY PERTURBATION THEORY TO THE H; ION* 
SY 4, 
i- R. B, Hake, Robert 3. Silbey and Joseph 0. Hirschfelder 
Theoretical Chemistry Institute, University of Wisconsin 
Madison Wisconsin 
-ABSTRACT 
The Hirschfelder-Silbey perturbation theory") has been applied to the 
b 
H: molecular ion. The first order perturbation equation has been solved 
by means of a variational method and the resulting gerade and ungerade 
energies are given through third order. 
The ungerade energy is in very good agreement with the exact energy, 
but the corresponding gerade energy is relatively poor. These results are 
currently under investigation. 
* 
This research was supported by National Aeronautics and Space Admini- 
stration Grant NsG-275-62. The authors also acknowledge support by 
the National Science Foundation for computer time and facilities at 
the University of Wisconsin Computing Center. 
i- Present address: Department of Chemistry, Massachusetts Institute of 
Technology, Cambridge, Massachusetts. 
https://ntrs.nasa.gov/search.jsp?R=19660028614 2020-03-16T17:28:20+00:00Z
4 
r' 
Introduction 
In a recent report HirschfePder and Silbey") have developed a new 
perturbation scheme for the invesrigation of atomic and molecular systems. 
This method uses a product of atomic or molecular orbitals as the zeroth 
order wave function and has the feature that all quantum states arising 
from a single electron configuration are treated simultaneously. 
Here we report some results obtained from the application of this 
+ 
method to the simple case of the H2 molecular ion. 
Theory for H, 
1 + 
+ The lowest gerade and ungerade states of H2 have wave equations 
(see figure 1) +- '  = I  \ zv - - - -  -G T b  
We write the wave functions in the form 
\ 
where a and b are atomic orbitals centred on atoms A and B 
respectively (see figure 1) and N and NU are normalization constants. g 
(3) 
figure 1 
Substituting equations (2) into (1) and adding the resulting equa 
gives 
which is the equation we wish to solve by perturbation methods. 
We write 
a =  a(')+ a(') + a(') + ---- 
b = b(')+ b(l) + b(') + ----  
E = E(')+ E (') + E (') + ---- 
(1) (2)  + ---- 
g g g 
U + EU 
E = E(')+ E 
U 
H = Ha+Va=H,,+Vb 
where 
- 1 2  1 Ha - - 2 V  - -  r a 
- 1  
r 
1 2  
Hb = - 2 v  - 
va = - lhb 
Vb = - l lr ,  
b 
ions 
3 
Substituting these expressioss i c t o  ( 3 )  and equating terms of like order 
we obtain the set of perturbation equations. 
Ha is the Hamiltonian of a hydrogen atom centred on A and we 
choose a to be the ground state eigenfunctions of the hydrogen atom; 
&O\ 
is hence the ground state energy of the hydrogen atom (= -0.5 a.u) 
A knowledge of \cI and$ through nth order determines the energy 
U 
through (2ni-1)st order. SO we require 
which means t h a t  we have 
be normalized or or thogonal  t o  h igher  o r d e r  wave func t ions .  i .e.  
A l s o  from the  above r e l a t i o n s h i p s  w e  o b t a i n  t h e  fol lowing express ions  
f o r  t h e  energy t e r m s  up t o  t h i r d  o rde r .  
It should be noted t h a t  when the  exac t  a i s  known t h e  express ion  
However, when so lv ing  f o r  a 
working wi th  t h e  exac t  a (" 
by a v a r i a t i o n a l  method w e  a r e  n o t  
so (6) should be used t o  ob ta in  
So lu t ion  of t h e  f i r s t  o rder  equat ion by a v a r i a t i o n  method 
I f  w e  cons t ruc t  t he  func t iona l  
w e  see t h a t  when J is s t a t i o n a r y  with r e s p e c t  t o  f i r s t  o rde r  v a r i a t i o n s  
of x then x i s  a s o l u t i o n  of t h e  f i r s t  o rder  equat ion (4). 
1 
6 
1.f we expacd in a complete set of functions 
= )'Cn$* 
then 
0 where a/* 
We require J to be stationary w.r.t. changes in the C's 
i.e. = 0 = X ' C ~ ~ , Q  + ).c, HQVW 
a=% h m 
- c Z C p \ - \ q p  + 2% 
The prime indicates that the n = term is omitted from, the summation, 
i 
Since Hnp = H t a  we have 
i 
7 
I 
2 L c n H n 4  t 2Lp = 0 
I n  gene ra l  we need 
- c a x  - = - - -  = o  a s  a s  - 
a CI acz ace 
- =  - - - - - - -  
which g ives  an i n f i n i t e  set of equations of t h e  form (10). I f  w e  
c cons ider  only a f i n i t e  number of bas i s  func t ions  i n  the  expansion (8) 
for X i.e. w r i t e  
U 
(Lt b 
equat ions  (10) may be w r i t t e n  a s  then t h e  set of N 
That i s  
c ' Hal=  -v 
so c -   +j-'p# 
- -  e 
For a given b a s i s  set t h e  mat r ices  
and hence 
and may be obtained 
i s  der ived by the  usual  mat r ix  inve r s ion  techniques.  
. 
The s e t  of b a s i s  func t ions  used were t h e  LBwdin o r b i t a l s ( * )  which 
have t h e  gene ra l  form 
where 
a r e  t h e  Laguerre polynomials, 
This se t  of func t ions  appear s u p e r f i c i a l l y  t o  be s i m i l a r  t o  the  
hydrogenic func t ions  however they  have only one func t ion  i n  common, t h a t  
i s  t h e  ground s t a t e  Is hydrogen e igenfunct ion .  The advantage gained 
by us ing  t h e  LUwdin o r b i t a l s  i s  t h a t  a denumerably i n f i n i t e  set of them 
i s  complete, whereas t h e  hydrogenic o r b i t a l s  form a complete set  only  
when continuum s t a t e s  a r e  included. It i s  hoped t h a t  an expansion in 
terms of Luwdin o r b i t a l s  w i l l  converge t o  t h e  s o l u t i o n  for 
r a p i d l y  than an expansion of hydrogenic o r b i t a l s  would. 
a ('1 more 
From an examination of t h e  f i r s t  o rde r  equat ion  
w e  see t h a t  t h e  equat ion  has a pole  a t  r = 0 ; w e  t h e r e f o r e  expec t  a 
s i n g u l a r i t y  i n  t h e  s o l u t i o n  a t  
b 
r b = O .  
Now near  r = 0 t h e  second t e r m  i n  (13) has  t h e  form b 
9 
" 
If we write b o  since r = R at r a 
So introduction of a term of the form C e -%(O) in a ('1 will 
remove the singularity at P = 0 . 
- P 
b 
In the light of this the trail a(') was constructed from a linear 
combination of e with a finite set of tthodin orbitals. The 
finite set chosen did not include the Is orbital hence each term in 
. In order to solve (11) for the matrix (0) the set was orthogonal to a 
6 it is necessary first to have the matrix elements H and Vn . nm 
The 
and L f j ~ d l n ' ~ )  . We have, 
Hnm elements may be obtained d i r e c t l y  from the  work of Hirsch fe lde r  
I n  t h e  de r iva t ion  of t h e  elements V t h e  func t ions  have been w r i t t e n  n 
i n  t e r m s  o f  t h e  p r o l a t e  sphe ro ida l  coord ina te s  def ined  by (see f i g u r e  1) 
which g ives  
c 
F r o m  (9)  w e  have 
where t h e  d i f f e r e n t  rig r e f e r  t o  d i f f e r e z t  va lues  of n and . 
W e  can express  t h e  f u n c t i o a s  i n  terms of sphe ro ida l  
coord ina tes  us i zg  che re1a;;or.s 
i 
where 
and 
When t h i s  i s  done we obtair!  2x~ressiozsfor 
which a r e  given i n  t h e  appe-ndlx. 
Hence using these  results and may be cons t ruc ted  
and a found. 
Once w e  have f o r  some f i n i t e  b a s i s  s e t  of func t ions  w e  can 
use t h e  approximate a (') they d e f i n e  t o  c a l c u l a t e  t h e  e n e r g i e s  
L 2) (3) 
a d  E($)  (1) . This has  been done (making use of t h e  
i n t e g r a l s  given i n  the  appendix) f o r  d i f f e r e n t  sets of b a s i s  func t ions  
and some of t h e  r e s u l t s  a r e  given i n  t a b l e  2. 
D i s cus s i. on 3 f Resul t s  
W e  should no te  two p o i n t s  i n  connection wi th  t h e s e  r e s u l t s .  F i r s t ,  
Y 
although t h e  v a r i a t i o n a l  method we a r e  us ing  t o  s o l v e  equat ion  ( 4 )  w i l l  
given an approximate a which improves a s  b a s i s  func t ions  a r e  added 
success ive ly ,  the  expec ta t ion  va lue  of t h e  Hamiltonian c a l c u l a t e d  us ing  
t h e  wave func t ion  through f i r s t  o rde r  ( i . e .  &(I) ) w i l l  n o t  
n e c e s s a r i l y  decrease  a s  each new b a s i s  f u n c t i o n  i s  added. That i s  t h e  
v a r i a t i o n  method does no t  s y s t e m a t i c a l l y  g ive  a s t a t i o n a r y  va lue  of 
The second p o i n t  i s  t h a t  t h e  second-order energy f o r  t h e  ungerade 
12. 
s t a t e ,  Eq 
Rayleigh-Schrudinger theory where E'" S o , always. 
i s  p o s i t i v e  i n  c o n t r a s t  t o  t h e  r e s u l t s  of t h e  u s u a l  
Suppose w e  t a k e  the  o r i g i n a l  wave equat ion  
and expand 
13 
Y 
Using t h e  f i r s t  or&; W Z V ~  frrzc'_ion obta ized  3y the  'new' pe r tu rba t ion  
theory  as  a t r i a l  wave fur-czrzn f o r  zi Rayleigh-SchrBdinger t rea tment  
w e  o b t a i n  
I f  w e  t ake  t h e  second order  energy given by t h e  Hirschfe lder -Si lbey  
theory  ( 6 )  and s u b s t i t u t e  i n  (I&> w e  have 
This  r e l a t i o n s h i p  does i l~ t  appear t o  glve any r e s t r i c t i o n s  t o  t h e  s i g n  of 
(a (27 
so  the  p o s i t i v e  -~ai:.ies of e,, need n o t  be a cause f o r  ($1 
concern. 
From an examination of Ta'aLe 2 we  see t h a t  us ing  a b a s i s  of e -% (0) 
t oge the r  wi th  t h e  2 ~ , 2 p , 3 ~ , 3 ? , 3 d , ~ s , 4 ~ , 4 d , 4 f , 5 g  Luwdin o r b i t a l s  t h e  
ungerade energy (i) i s  very  c l o s e  t o  t h e  ' e x a c t '  va lue  given by Peek (4) . 
However t h e  gerade energy €,(I) compares r e l a t i v e l y  poorly wi th  Peek ' s  
va lue  and i n  f a c t  w e  s ee  t h a t  t h e  b a s i s  set of 8 func t ions  gave a b e t t e r  
va lue  f o r  
b a s i s  se ts  should be  used i n  an attempt t o  o b t a i n  convergence of &(A) 
and 
we a l r eady  have f o r  t h e  ungerade energy it would appear t h q t  t h e  method 
i s  capable of g iv ing  f a i r l y  accu ra t e  r e s u l t s .  
&(I) . It seems c l e a r  from t h e s e  r e s u l t s  t h a t  l a r g e  
€,,(I) t o  some s t eady  va lue .  However, judging from t h e  r e s u l t s  
I b 
Y 
k 
0 '  
W 
m 
E: 
0 
E D :  
k '  
9) 
P 
z 9 
JJ 
c 
9) 
k '  
9) 
W 
W 
4 a ,  
k 
0 
W 
X 
rl 
k u z 
W 
0 '  
m u 
c 
9) 
rl w 
s 
.. 
d 
. P . . O . . .  
0 0 0 0 0 0 0 0  
I I  I 1  
15 
d d d o ' d d d  
I I I  
. .  d d d o o  
8 1  
0 
P 
16 
i 
3 
a 
N 
II 
d 
8 w 
+ c u  
X 
$4 
0 w 
m 
aJ 
7 
d 
(d 
3 
2 
2 
$4 
W 
d 
i- 
\ 
l-l 
n 
W 
d 
3 
w 
n 
m 
W 
n 
hl 
W 
rr: 
+ 
\ 
rl 
n 
d 
W 
WM 
n 
m 
W 
z 
0 
VI 0 4  
0 m u  
6\1 
0 
6d 
r\l 
0 
s m (T\ m 
b 
m 
ri 
0 
b 
\s ca 
a 
0 
m 
0 
0 
0 
rn m 
9 
0 
0 
0 
cn 
2 
d 
03 m 
I4 
m ra 
0 
m cn e 
0 
d e 
0 
2 
F-4 
a 
0 
lB 
0 
m 
a3 
0 
u3 
0 
a 
0 
N 
a 
hl 
m 
0 
u) 
0 
0 
N 
cr) m 
0 
W 
0 
V k  
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II 
0 * m co 
0 
a 
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p: 
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17 
Appendix 
L 
I 
\. 
The following integrals  are used i n  the so lut ion of the matrix 
equation (11) and i n  the evaluation of the energy terms. 
W e  de f ine  
I& I s  the molecular bond length. 
1. 
. 
18 
L 
191 
. , I  
. 
REFERENCES 
1. J. 0. Hirschfe lder  and R. J. Silbey, WIS-TCI-168A (1966). 
2. P. 0. LBwdin and H. Shul l ,  Phys. Rev. 101, 1730 (1956). 
3. J. 0. Hirschfe lder  and P. 0. LUwdin, Mol. Phys. 2, 229 (1959). 
4. J. M. Peek, Sandia Corporat ion Report SC-RR-65-77 (1966). 


An application of Hirschfelder-Silbey perturbation theory to the H2 plus ion

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