Lower bounds to energy eigenvalues for rigid rotator in electric field Stark effect calculation - Schroedinger equatio

Choi, J. H.

Smith, D. W.

English

NASA Technical Reports Server

. ,  
. 
.t 
Lower Bounds t o  Energy Eigenvalues for the Stark E f f e c t  
in a Rigid Rotator 
* 
, 
(CODE) 
l C A T E 4 0 R Y l  
by 
Jong H. Choi and Darwin W. Smith 
Department of Chemistry 
and 
Quantum Theory Project 
University of  Florida 
Gainesville, 32603 
GPO PRICE $ 
OTS PRICE(S) $ 
https://ntrs.nasa.gov/search.jsp?R=19650014062 2020-03-17T00:05:11+00:00Z
I ..~ . 
a t b 
E" -1- 
L 
i '  
I. INTRODUCTION 
In  t h i s  paper, the lower bounds t o  the  energy eigenvalues f o r  
the r i g i d  ro t a to r  i n  an e l e c t r i c  f i e l d  ( the  S tark  e f f e c t )  are calcu- 
l a t ed  using the method of Ltfwdin, reported i n  these proceedings.' 
notat ion i s  cons is ten t  with h i s ;  a convenient form of the lower bound 
t o  an eigenvalue being 
Our 
2 
In a problem where V 
e f f i c i e n t  K as a s t rength  parameter, one can write 
is  a pos i t ive  per turbat ion with a constant co- 
V = K U .  
For t h i s  case we obtain 
where 
.. . . 
e I 
- -  
-2- 
If Cf' is an eigenfunction of &. , then 
11. STARK EFFECT IN THE R I G I D  ROTATUR 
The wave equation f o r  the rigid r o t a t o r  cons is t ing  of two mass 
ml and rn which are  separated by fixed distance R, is 2' points  
where W i s  the ro t a t iona l  energy and M i s  the reduced mass of m and 
m2, i.e., 
1 
L e t  
and 
where i s  the Hamiltonian i n  dimensionless u n i t s  and E o  is the  
-3 -. 
corresponding energy. Then (2.1) reads simply 
- m & 2 + ~ ~ [ m I e , + ) =  E> Ct;% (e ,+> ,  
or 
The rigid rotator of dipole moment ,U i n  a uniform e lectr ic  
f i e l d  F i s  characterized by the wave equation of the form 
with 
and 
(2.5) 
In order that the perturbation V 
we shall write 
in (1.2) shall be positive definite,  
(2.9) 
(2. lo) 
and 
-4- 
t 
(2.11) u = 1 + u s e .  
Then this  division of 8 
gives 
into an unperturbed and a perturbed part 
with 
(2.12) 
v 7 ' 0 .  
For this case the SchrBdinger equation reads 
[ a(@,+) K us81 q ( e 2 + ~  = E ( ? ( e J q > .  (2.14) 
e 
This equation is eas i ly  seen t o  be separable with respect to  the variables 
8 and I+ , so we can write the eigenfunction i n  the product form 
(2.16) 
1 -5 - 
where P:( 8 ) is  an associated Lengendre polynomial. 
Since the space with which we are concerned can be subdivided 
according t o  the value of m, 
each subspace. The normalized reference funct ion ‘p i n  (1.1) i s  
a rb i t r a ry ;  however, i n  order t o  make t he  f i rs t  order i t e r a t i o n  of (1.1) 
convergent, the condition 
we can t reat  the problem separately f o r  
4 
. 
nust be sa t i s f i ed .  The l e f t  s ide  of t h i s  inequal i ty  approaches zero a s  
K approaches zero f o r  a f ixed reference function only i f  i s  an 
eigenfunction of ; therefore  it s e e m s  reasonable t o  make t h i s  choice 
f o r  Q: provided K i s  s m a l l .  Thus i n  the subspace m, we have f o r  
the ,i! t h  exci ted state,  
(2.18) 
111 where 
nomial ,,“(cos t? ) given by 
N,L is  the normalization fac tor  f o r  the associated Lengendre poly- 
Lower bounds E,’ f o r  the energy leve ls  can be ca lcu la ted  by use 
of (1.3) through (1.7) with (5 i n  To obtained from a va r i a t ion  calcu- 
la t ion .  A simple but useful choice f o r  the a r b i t r a r y  l i n e a r  manifold 
B = (g1, g2’ * - -  gi 0’. 8,) is given by -
-6- 
Since U = 1 + COS 8 , 
Using the recurrence relation 
we obtain 
(2.21) 
(2.22) 
theref ore 
J?+l-m> (a+l*> for i = , (+2-m , B; =[( ( 2  +1)(2 +3) 
8 ; -  1 for i = 1 +I-m , 
for  i = JZ -m , 
( L 2 9  
and 
@;= 0 
(cpf (p> -- 
In order to have 
can see from (2.23), n 
E,' = R (X+ 
otherwise, 
at least one non-zero element for 
m u s t  be at l eas t  as large as 
, as we 
( 2 -m); otherwise 
2'  
f , 
which is t r i v i a l .  
For the ca lcu la t ion  of matrix elements of &, we consider 
Using (2.22), we obtain 
w i t h  - 1 i-1) (i-2m-1 
Di-l { E -  (i-tm-2) (i+m-l.)+K]{ I 2 i i 2 r n - 1 )  (2 i z2m-3)  
= 0 ,  ... ( i f l -  i-cm-2 ) ; 
1 - 
I f D i  - ) t;-(i+m-1) ( i+n )+K 
0 ,  ... (if I= is.m) . 
(2.26) 
(2.28) 
Note that the reason Di vanishes for certain values of i is t ha t  Fi 
is orthogonal to ~ 2 .  ' 
Introducing (2.~) i n to  {2..%), 
B = B; 2+2,1 &,i-k2 
4 
C 
-9- 
I f  we change the Hamiltonian i n  (2.8) t o  
= a - , K C 6 2 6 8  (2.35) 
we would expect the  same r e s u l t  since we only reversed the uniform electric 
f i e l d .  
as a p a r t i a l  check on the algebra, 
The algebraic  iden t i ty  of these two cases has been demonstrated. 
L 
111- UPPER AND LawER BOUNDS FOR THE RIGID ROTATOR--NUMERICAL RESULTS 
The e n t i r e  ca lcu la t ion  was done on the  IBM 709 a t  the  University 
Upper Bound energies w e r e  f i r s t  obtained by of F lor ida  Computing Center. 
the Rayleigh-Ritz va r i a t iona l  method using subroutine Givens5 (s ingle  
precis ion) .  
lower bound were so close t h a t  it was d i f f i c u l t  t o  say which w a s  lower, 
the upper bounds w e r e  ref ined using a double-precision i t e r a t i o n  method, 
and the remainder of the ca lcu la t ion  w a s  car r ied  out i n  double prec is ion  
2.lSO. 
I n  order t o  c l a r i f y  t h e  cases  where the upper bound and 
6 
Except for the  level rn = 0, 
However, the  Hamiltonian $f c&tes with Lz and therefore  one can 
separately solve the energy eigenvalue problem f o r  each subspace 
$or a given value m of Lz. Calculations f o r  the f i r s t  six subspaces 
(So t o  S ) have been car r ied  out for the  twenty lowest eigenvalues i n  
each subspace with the  per turbat ion coe f f i c i en t  
1.0. 
a l l  the o ther  levels are degenerate. 
Sm 
- 5  
K ranging from 0.1 t o  
In  Table I, upper bounds l$, = & and lower bounds EL = E,’ of 
energy eigenvalues are given to s ix teen  s ign i f i can t  d i g i t s  f o r  t yp ica l  
choices of K and m. The eigenvalues are labeled by the  e of the  
unperturbed state. 
i 
-10- 
% is  t h e  number of bas i s  ‘functions used f o r  f inding upper bounds 
and NL is the number of bas i s  functions (P,, P,+,, ... 
m m  
( these  functions are the associated Legendre polynomials pm, Pm+p - - -  
“ 1  m m m  ’m+NL- 1 
manifold A 
This method of evaluat ing lmer bounds, using the  bracketing 
property of (l.l), requires  good upper bounds, and the upper bounds are 
icproved by increasing the number of bas i s  functions 
va r i a t iona l  calculat ion.  
upper bounds are general ly  obtained by taking In order  t o  
obtain good lower bounds, we have t o  have a well-chosen bas is  f o r  the 
Aronzajn space7, and we can see from the r e s u l t s  indicated in  Table I t h a t  
i n  the present case the dimension of the Aronzajn space NL i s  s u f f i c i e n t  
to give good lower bounds provided % = J  + 5. 
agree t o  fourteen s ign i f i can t  digits’ i n  those cases where Nu = NL =A + 5.  
For the highest  energy levels l i s t e d ,  f o r  example 
N = 20, 
Nu. used i n  the  , 
The r e su l t s  i n  Table I indica te  t h a t  f a i r l y  good 
Nu =l+ 5.
Upper and lower bounds 
m = 1, 1 = 20, Nu = 20, 
the agreement is poorer, but i s  improved somewhat by increasing L 
The l imi t a t ion  here seems t o  be due t o  the poorness of t he  upper bound NL* 
r a the r  than the dimension of the Aronszajn space. 
Nv’ To see i n  more d e t a i l  the e f f ec t  of N f o r  a f ixed  value of L 
s e v e r d  exainples a re  given in  Table 11. 
l i n e a r  manifold -& is chosen as indicated in  (2.20), f o r  N less than 
( 1 - m),  the  lower bound % 
It is  seen from (2.23) t ha t ,  whenthe 
L 
is .a pers i s ten t  lower bound, given by 
I n  order t o  obtain h c t t c r  lower bounds, therefore ,  the  l i n e a r  
manifold 
elements of p i ;  namely ,+,~-m, 
Table 11 beyond the values indicated did not improve the lower bound. 
has t o  be chosen i n  such a way t h a t  there  are non-vanishing 
,Bf-m+1, a d  pp--m-~\jC Incre is ing  Nu i n  r n  our c a s t .  
I 
-11- 
REFERENCES 
* Supported in part by the National Science Foundation and in part 
by research grant NsG-512 from the National Aeronautics and Space 
Administration. 
2. Reference 1,'equation (43). 
3. Except f o r  the case m = 0, 
so we will consider onlynabsolute value of 
we have doubly degenerate eigenstates, 
tl*. 
m. 
4, Equation (15) in Reference 1, 
5. %igenvalues and Eigenvectors by Given's Method", Quantum Chemistry 
Program Exchange , QCPE 12C (1963) . 
6. C. E. Reid, "Eigenvalues by Llfwdin's Partitioning Method", Double 
Precision, QC?E 1 4 ~  (1963). 
P. 0. Llfwdin, Quantum Theory:Project, University of Florida, Report 
No, 64, 1964 (unpublished). 
7. 
8. Except for one case, m = 0, = 0, K = 0.5, where it is only 
13 significant figures. 
CI m = 0 ,  K = 0.5 
NU NL 5 and 
. 
5 '. c 
6 
7 
8 
9 
10 
i i  
.- 
I C  
1 3  
14 
13 
16 
17 
18 
79 
20 
29 
2s 
20 
20 
J 
6 
7 
8 
9 
i o  
11 
12 
13 
I4 
15 
16 
17 
18 
19 
2C 
,- -, 
i" 
23 
23 
20 
54 
:r,. 
n 
- ,  -___ 
a 
t *,.t Nu "L U L E and E 
1 
2 
3 
4 
5 
6 . 
7 
a 
9 
10 
11 
12 
73 
il: 
15 
16 
17 
IS 
19 
20 
5 
6 
7 
3 
9 
10 
11 
12 
13 
i 4  
15 
16 
17 
18 
19 
18 
29 
23 
20 
20 
6 
. 7  
8 
9 
10 
7 i  
12 
13 
14 
15 
16 
17 
18 
19 
18 
29 
23 
'7 . ._ 
2c) 
23 
20 
24 
6.01 1577927242672 
6.01 1577927242670 
12,0083261 4435840 
12.0083261~~35839 
20.00551 9331 62358 
20.00551 9331 62356 
30 -003846267391 37 
30 003846267391 39 
42.00281 392300602 
42.002ai 392300600 
56-002141 27777589 
56.002141 27777586 
72.001 681 30591 093 
72.001 681 30591 091 
90 -001 3538855291 2 
90 -001 3536855291 0 
1 1 0.001 1 1 29663826 
11 0.001 1 129663825 
132.0009307395922 
132=0009307395921 
156.000789671 5423 
1 a2.00067828~6897 
1 &2.00067€2&36897 
156.000789671 5422 
21 0.0005868386502 
21 0.00056883s6500 
240.OOO5l59358681 
240.00051 59358679 
272.0004557472937 
272.0 0 Oq37472681 
306.0304054860207 
306.0001;055860206 
3'42.000363087451 4 
2k2.009;6;0874365 
Y;3_.0003670871c366 
gao. 000327470541 9 
3EO. 0003269?27098 
380.00032699581 56 
420.0052377999262 
.420.0002179695039 
420.0 OO29+25qO 5 9 
* Table I. ( T h i r d  section) 
I 
5 29.99359063893720 
29.9935906388371 8 
5 
5 
6 6 41.99761 861 361275 a 
41 -99761 E61 361 275 6 .  
7 7 55,9992321 3730367 
7 55-9792321 3730366 
3 90.00023337531 91 7 
?O.OO0233>7531912 
10 I(! 1 1 0. 000g640283063 
79 I 1  0.0003640283062 
9 
9 
7 ?  11 1 32. ooW11243634 
IO 132.00041 12436339 
12 12 1 55.00041 80542290 
12 1 56.00041 80542288 
13 
14 
35 
i 6  
17 
13 
c 
"? 
13 
14 
15 
i G  
17 
ia 
19 
.-. r. . -  
- ; i  
t 
Table 11. Effect  of on 5 i n  the convergence 
t o  the eigenvalue. 
N % and E 1 % m K 
0 110.000011U16r;83 
109.9477U361b3&3 
11o.oooo11 U16Ls2 
109.9000000000000* 
10 
13 
0 110.000286041G31~~ 
10~.5000000000000~- 
109 7412282838218 
110.0002860416817 
110.0002860416818 
0.5 10 15 
1-9 
10 
13 
15 
1. 0.7 16 20 
1-14 
15 
18 
272.0002233160454 
271.300000000000W 
272,0002233160452 
271.6219989276613 
15 6,0007b.3 219165 1 
1 ~ ~ . 0 0 0 G 0 0 0 0 0 0 0 ~  
155.4101468398438 
156.0007432191644 
15 6,000743 219 165 1 
1.0 12 15 
1-9 
10 
13 
15 
0-7 10 12 110.0004230273525 
10~.3000000000000* 
110.0004230273523 
110.0003141668453 
3 
1-6 
8 
10 
420.000244~1007~0 
419 ooooooooooOoo* 
419.3717170996393 
42o.0002449100768 
420.0002449100769 
1.0 20 20 
1-14 
5 
15 
18 
20 
%e notat ion 1-9 indicates t h a t  the value of N ranges 
from one through nine. 
bAsterisks indicate  pers i s ten t  lower bound, 
i 


Lower bounds to energy eigenvalues for the stark effect in a rigid rotator

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