B R I E F C O M M U N I C A T I O N S

C A L C U L A T I O N O F T H E E N E R G Y L E V E L S O F A S Y S T E M

F O R M E D B Y T W O N U C L E I W I T H O N E E L E C T R O N *

A. A. A b r a m o v a n d V . I . U l ' y a n o v a UDC 539.192

1. One has to f ind the non t r iv ia l so lu t ions u (x), 1 _< x < ~ and v(y), - 1 - y -< 1 to a s y s t e m of the fol lowing f o r m in o r d e r to d e t e r m i n e the e n e r g y l eve l s of a s y s t e m cons i s t ing of two nucle i and one e l e c - t r on [1] :

d ~ p2 (i)

H e r e X and ~t

[ u ( 1 ) l < ~ , lu(~-)l < ~ ,

d 2 dv p~ ~yy[ (Y- 1) - ~ 1 - - (~'Y2 + ~ + ~ - ' ] ' - ) v - - O '

I ~ ( - 1)1 < ~ ,

Iv(l) [ < r

a r e cons t an t s to be d e t e r m i n e d and a and p a r e g iven n u m b e r s , with p -- 0.

(2)

(3)

(4)

(5)

(6)

The problem has been so lved s e v e r a l t imes , e .g . , [2], w h e r e r e s u l t s a r e g iven fo r s e v e r a l s t a t e s and a b ib l i og raphy is given.

We have [3] d e s c r i b e d a method b a s e d on r e p e a t e d so lu t ion of the Cauchy p r o b l e m for the o r d i n a r y d i f f e ren t i a l equat ion, which is r ead i ly done by c o m p u t e r . The me thod a l so a l lows one to de r i ve so lu t ions c o r r e s p o n d i n g to a g iven type of s ta te , without r e f e r e n c e to o the r s t a t e s , which i s e s p e c i a l l y i m p o r t a n t in ca lcu la t ing the h ighe r e igenfunct ions . The method is a s fol lows. We spec i fy n u m b e r s k and n, the n u m b e r s of sign changes in u(x) and v(y) r e s p e c t i v e l y (0 -< n, 0 -< k), and the choice of k and n def ines a def ini te so lu - tion. We put X = )t 1 and solve (4) A (5) A (6) fo r ~t and v(y), s eek ing a solut ion fo r which v(y) has the n e c e s - s a r y n u m b e r of s ign changes . We subs t i tu te the r e su l t g = g l into (1) and solve (1) A (2) A (3) fo r X and u(x), taking the solut ion w h e r e u(x) has the n e c e s s a r y n u m b e r of s ign changes . The r e su l t X =X 2 is used with (4) A (5) A (6) etc. It ha s been shown [3] that th is p r o c e s s conve rges .

2. It ha s been sugges t ed [3] that the fol lowing subs t i tu t ion should be made to so lve (1) A (2) A (3) and (4) A (5) A (6):

x = (1 + ~2)1/2, d [~ 1/2 (1 + ~2)1/4 u]

y = sin % d (cos ~I2 ~1 -v) - - tg 0 (~). (n + 1) cos'/2~ I,vd,1

* R e a d at the Second Ukra in i an S y m p o s i u m on Quan tum C h e m i s t r y , Kiev, D e c e m b e r 1968. The equa t ions h e r e a r e g iven fo r the c a s e of equa l c h a r g e s , but the m e t h o d i s r ead i ly ex tended to unequal c h a r g e s .

Comput ing Cente r , A c a d e m y of Sc iences of the USSR, Moscow. T r a n s l a t e d f r o m T e o r e t i c h e s k a y a i ~ s p e r i m e n t a l ' n a y a Khimiya , Vol. 6, No. 3, pp. 384-386, M a y - J u n e , 1970. Or ig ina l a r t i c l e submi t t ed M a r c h 5, 1969.

�9 1973 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15.00.

312

TABLE i

State

3de 2 p~t 5go 4 f~ 7 irl 7 h~o 3 d5 7h6 7 hu 7 i~ 7 iu 7 i~9

R0

1,9972 8,83 7,93

23,9 19

70 18

69 77

E, a .u ,

--0,602634619 --0,175049 --0,1345138 --0,07824 --0,071223

Unstable I --0,01406

--0,0570334 Unstable Unstable Unstable

--0,01572 --0,02083

It i s r e a d i l y v e r i f i e d t ha t w e g e t f o r 0 (7/) t ha t

1 [ 1 1 - - 4p 2 0 ' - [ - (n-I- 1 ) s i n * 0 + c o s * O + ~ - ~ - + 4cos~n

0( 2 '

0 ~--- - - - - - - rt~. 2

t-~, sin~'~l cos~0=0, (4')

(5 ')

(6')

A s o l u t i o n v 01) i s e i t h e r e v e n o r odd in ~?, so (6') c a n b e r e p l a c e d b y

0 (0) = - - rex~2.

T h e p r o b l e m of (4') A (5') A (6 H) i s s o l v e d f r o m - r r / 2 to 0; in o r d e r to s t a r t out f r o m - 7r/2 w e p u t

(6 . )

) ~ ( n + I ) ~ 0 - - @ + ~ 1 ==- 7 - - arctg 1 2

1 - ~ ( p - - 1 ) ~ - - L -2- + P + 2p -k 2 ~2 + . . .

T h e r e s u l t f o r bt i s sough t in s u c h a w a y a s to s a t i s f y 0 ( o ) = - m r / 2 , and the s e a r c h i s f a c i l i t a t e d b y the f a c t t h a t 0(0) d e c r e a s e s m o n o t o n i c a l l y w i t h # .

S i m i l a r l y f o r w(~) w e h a v e

(o' -i- shl ~ o~ + [ - - ~ -+- 1 - - 2~2+4a~ 2 (~2 q_ 1)3/2 _ 4~2 (.~2 + 1) ~t - - 4 f (~2+ 1) cos%o - - O, (1') 4~2 (~2 + 1)2

~o (0) = n/2, (2')

~o (~_) : - - k n - - arctg 1 / L (3')

T0 s o l v e (1') A (2') A ( 3 ' ) we take ~0, 0 < ~0 < ~ and s o l v e (1') / \ (3') f r o m ~ to (0 and (1') A (2') f r o m 0 to ~0; w e g e t r e s p e c t i v e l y COright (~) and Wlef t (~), and f o r ~ l a r g e w e u s e the a s y m p t o t i c r e p r e s e n t a t i o n

co (~) = - - kn - - arctg IVY--- (bll~ + b~l~ 2 + bal~ a + . . . ) 1 ,

w h e r e

bi- - -a l2 l / -~ , b 2 = ( b ~ - - b , - - i x ) / 2 V - ~ , bs-b---(2b2--}-a]2)/2[/~.

F o r ~ s m a l l w e u s e

313

y - k - p + 2p-{- 2 ~ + " "

We search for ~, the sea rch being facil i tated by the fact that Wright (~0) -Wleft (~) dec rea se s monotonically in ~.

3. This method has been applied to some energy levels of H:. The energy E is minimized with re- spect to R 0 (distance between the nuclei) by calculation for various R 0 followed by interpolation (Table 1).

These results refine those previously obtained by other methods. Results for eigenfunctions of high number have not been obtained before, so far as we are aware.

We are indebted to Yu. A. Kruglyak for much advice.

~o 2. 3.

L I T E R A T U R E C I T E D

P. Gombash, The Many-Body Problem in Quantum Mechanics [Russian translation], Moscow IIL (1953). D. R. Bates, K. Ledsham, and/k L. Stewart, Trans . Roy. Soc., L., A246, 215 (1953). A. A. Abramov and V. I. Ul'yanova, ZhVMMF, 1__, 551 (1961).

314