The ∑2u+ state of the negative molecular ion H2−

Gupta, B . K . 1959

Physica 25 190-194

T H E ~X;~ + STATE OF T H E N E G A T I V E M O L E C U L A R ION H f b y B. K. G U P T A Physics Department, Osmania University, Hyderabad Deccan, India The variational method is applied to the stable state of the negative ion of hydrogen molecule. With the same effective charge for the electrons in H - and in H, the values of the dissociation energy De and of the equilibrium internuclear distance Re are 0.068 and 3.56, in atomic units, respectively as found by E y r i n g et al. 1) When two effective charges are used for H - and another for H atom, the values obtained for De and Re are 0.0125 and 5.76 respectively. In each case Moffitt's method 4) is also used to compute the same quantities, giving a large modification in the calculated De in the former and ~mly a slight change irt the latter case. 1. Introduction. I n a t h r e e electron b o n d h k e He2 +, the b o n d i n g orbitals are not cancelled quite c o m p l e t e l y b y t h e a n t i b o n d i n g ortbials, t h a t is, the addition of an a n t i b o n d i n g orbital leaves a slight excess of b o n d i n g over a n t i b o n d i n g ; the t h r e e electrons i n v o l v e d arise out of similar-atomic orbitals in the isolated a t o m s (in this case Is). H 2 - has a similar t h r e e electron b o n d ; b u t q u a n t i t a t i v e work on Hg.- is still lacking a n d this is an a t t e m p t to fill this gap. E y r i n g , H i r s c h f e l d e r and T a y l o r 1) (E.H.T.), using v e r y a p p r o x i m a t e wave functions, discussed the existence of stable states of H 2 - using the m e t h o d e m p l o y e d b y P a u l i n g 2) for Heg.+; this consists in displacing the calculated p o t e n t i a l e n e r g y c u r v e b o d i l y downwards. B u t W e i n b a u r a 3) using two effective charges, one for the H e a t o m a n d a n o t h e r for H e + ion got a v a l u e of the dissociation e n e r g y which agrees fairly well with the e x p e r i m e n t a l v a l u e o b t a i n e d f r o m the excited states of the diatomic He2 molecule. T h e w o r k on H 2 - corresponding to W e i n b a u m ' s on He2 + has not y e t been done; b u t it is to be e x p e c t e d t h a t the b o n d i n g will be v e r y weak. In the present p a p e r the work of E . H . T . is r e p e a t e d for Hg.-. C o m p u t a t i o n s are also m a d e for H 2 - corresponding to W e i n b a u m ' s w o r k for He2 +, b u t in place of W e i n b a u m ' s two effective charges, one for the a t o m a n d a n o t h e r for t h e ion, we h a v e used 3 effecti-de charges, one for H a t o m a n d two for the ion H - . I n addition to this, we h a v e also used Moffitt's 4) m e t h o d of ' A t o m s in Molecules' in each case to s t u d y the t r e n d of the values w h e n calculated b y this m e t h o d . -

-

190

-

-

T H E 2 ~ ' u + S T A T E OF T H E N E G A T I V E M O L E C U L A R I O N

Hg.-

191

I

2. The Variational Functions. According to the variational principle, an upper b o u n d to the energy of the ground state is

in which ~v is a function of the configuration and spin coordinates, and H is the complete Harniltonian of the system. The H a m i l t o n i a n for the system in atomic units *) is H

= - - Z i3= l V~ - - Z i3= l (21ra, + 2/rbf) + Zi
2

R

(2)

the nuclei being fixed at m u t u a l distance R ; rg is the distance between the two electrons i a n d j; a and b denote the two nuclei. The wave functions used in the present t r e a t m e n t are ~Vl = 1/~/3!{s1~1/'~8(~2/~3 -- ~3fl~.) + cyclic terms in 1, 2, 3) and

~v2 = 1/~/3!{Sl'O~l[9.s(o~2~8 -- o~3fl2) + cyclic terms in 1, 2, 3} ill which e and/~ are the spin functions, s = e-8"a,

11. =

+

s' = e-8"b,

/':. =

+

(3)

s a n d / are b o t h unnormalized. The linear combination of right s y m m e t r y for the state ~ + is ~Vl -- ~02. W h e n ~ = ¢ = r, the above functions reduce to the functions used b y E.H.T.

3. Choice o/the effective charges. For each value of R, the effective charges a n d E, ~ m u s t obviously be chosen so as to m a k e the energy a m i n i m u m at t h a t R. The c o m p u t a t i o n a l work t h e n becomes prohibitive a n d hence it is reasonable to choose their values so as to m a k e the energy at R = cx~ a m i n i m u m **). The best values of the effective charges 5) for H - for a wave function of the t y p e shown in equation (3) are E = 0.28309, and ~ = 1.03925 giving the energy of H - equal to -- 1.0266, instead of the correct value -- 1.05512; and of course the best value of the effective charge for the H a t o m in the wave function e - ~ r i s ~ =~ 1. F o r simplifying the c o m p u t a t i o n s the values of the effective charges were t a k e n as ~ = 1.0, and ~---- 1/3.5 = 0.28571, y = 1.0. I t is found t h a t in t a k i n g the values of ~ and ~ as 0.28571 and 1.0 respectively not m u c h is lost *) Ato mic units are used through out this paper; the Bohr radius is the uni t of length while the unit of energy is the Rydberg = 13.54 eV. **) I n case we t a k e ~ -----~ = ~ a calculation of the energy at each R w i t h the exponents varying from one value of R to another is practicable and was made for HeB + b y P a u l i n g a).

192

B.K. GUI'TA .p

in accuracy (these values give energy of H - a s - - 1.025 I), but it definitely simplifies the computations a great deal.

4. Results. a) The energy is minimized in the usual way with E.H.T. function (i.e. ~ = ~ = 7), the best value of the effective charge turns out to be 0.8. In table I the first row shows the computed values of the energy at various internuclear distances. In the case of Hg.- the equilibrium internuclear distance being rather large it is reasonable to expect t h a t the method of Moffitt will give an improvem e n t on the values calculated directly from equation (1) with the E.H.T. function. The result of applying Moffitt's method with the E.H.T. function turns out to be exceptionally simple and is as follows: Re = 3.149/(5 and De = 0.02788~ Where 0 = ~ = 7 is the effective charge. Hence Re and De can be adjusted arbitrarily by a choice of ~. However the energy has been calculated b y this method at various values of R with ~ = 0.8. Table I shows the energy computed from the E.H.T. function by the variational integral (with ~ = 0.8) and b y using the same function in Moffitt's method. TABLE I

.R=)

[

E b) (variation)

E~) (Moffitt)

3.125 I --1.9448

'

~

-

3.750 I

4.375 I

S.00 [

S.S2S [

--1.9470

--1.9383

~

--2.0764 - ' - ~ - ~ ' - ~ T 6 - ~

a) R in units of ao the Bohr radius *) correct value.

--1.9274

oo --1.8800

~

](H+H-)*) --2.05512

--2.05512

b) E in Rydbergs

From table 1 we get values of Re, Em,n and De shown in table 3. It will be observed t h a t Moffitt's method tends to give a much shallower energy curve and a slightly higher Re value. Later we will see t h a t this is a trend in the right direction. b) The function e -~(~+r*) is not really a very good function for H - ; with = 0.8 it gives the energy -- 0.920 which indicates instability since the energy of H atom in these units is -- 1.0. Also for the H atom e -~r with = 0.8 is not really the ideal one can use. However, a function of the type given in equation (3) with ~, 7 adjusted to minimise the energy gives a much better value for the energy of H - as stated already; for the sake of convenience ~ and 7 have been modified slightly from their best values and have been taken respectively as 1/3.5 and 1.0. As before this function with three exponents ~, ~, 7 has also been used in calculating the energy b y Moffitt's m e t h o d : Table II shows the vMues of the energy computed from equation (1) with

T H E 9'~'u+ STATE OF T H E N E G A T I V E MOLECULAR I O N

Hs-

193

this wave function as compared with those obtained b y using this ,wave function in Moffitt's method. The values of the effective charges used are = 1.0, and ~ = 1/3.5, ? = 1.0. TABLE IIa)

R

I

E (variation)

3"s° --2.0115

I

4.90

I

s.2s

--2.0356

I

--2.0367

s.60

I

6.30

--2.0376

I

oo

--2.0366

--2.0252

In+.- l ~-2.05512[

l

E (Moffitt)

"-

-

--'-2"~-'~- ~

"

~

~

~

~ - ~

a) units as in table 1.

From table II we get values for R6, Em, n and De shown in the last two rows of table I n . T A B L E TITa)

I

R,

l

E,n~.

I

D,

I. Variation; = • ~,

0.8

3.56

2. Moffitt; 6 = • ?

0.8

3.94

--2.077

0.022

1.0

8.76

--2.0377

0.0125

4. Moffitt ; = 1.0; • = 1/3.5, ? = 1.0

5.80

--2.0670

0.0119

3. Variation; = 1.0; • = I/3.5,? =

--1.948

0.068

a) units as in table I.

Table I I I shows t h a t there is a large change in the value of Re and D, between row 1 and 3. This is very different from the case of He9 + where successive improvement of the wave function ~) 3) produce a change of a few percent in the calculated R, and De. Thus it appears that in case the binding is exceptionally weak (for H2-, De is of the order of 0.15 e.v. in comparison with the total energy of the .dissociation products N 30 e.v.) very false results can be obtained from a straightforward variational calculation in which the wave function used is not accurate enough. Given the shallow potential curve and the large equilibrium internuclear distance, it appears as if Moffitt's method of 'Atoms in Molecules' can be used in this case to give an improvement on the values calculated by the variational method. Row 2 of table I I I shows t h a t even in the case of one exponential for both H and H - Moffitt's method tends to make the value of D6 more reasonable, but leaves Re less affected. On the other hand, comparison of rows 3 and 4 indicates t h a t once the function used is accurate enough the improvements brought about by using Moffitt's method are slight; also from table II it is seen t h a t the result is essentially equivalent to a bodily shift downwards of the calculated energy curve. This however is not so for the values in table I. With such a low dissociation energy, it is doubtful whether Hg.-, inspite

194

THE 2~'u+ STATE OF THE NEGATIVE MOLECULAR ION H 2 -

of the stability shown in these calculations, can be formed o r on being formed, can exist for an appreciable interval of time, under any physical circumstances.

5. Integrals. The most difticult part of these computations was the evaluation of the integrals written as H,~(m, n; 4) b y J a m e s and Cool i d g e 6); these were extended to H,V(rn, n; 2, #) because of two effective charges 7) and tables of these functions were computed for the appropriate values of 2 and/~. However all the integrals arising with the E.H.T. function, were evaluated from Kopineck's s) tables.

Acknowledgements. The author is thankful to Dr. A. R a h m a n , for suggesting him this problem and for m a n y useful discussions which he had with him during this work. The author is also thankful to the Government of India for the award of a Research training scholarship. Received 16-12-58 REFERENCES

1) 2) 3) 4) 5) 6) 7) 8)

E y r i n g , H., H i r s e h f e l d e r , J. O. and T a y l o r , H. S., J. chem. Phys. 4 (1936) 479. P a u l i n g , L., J. chem. Phys. I (1933) 56. W e i n b a u m , S., J. chem. Phys. 3 (1935) 547. M o f f i t t , W., Prom roy. Soc. A210 (1951) 224, 245. C h a n d r a s e k h a r , S., Astrophysical Journal 1 0 0 (1944) 176. J a m e s , I-I. M. and Coolidge, A. S., J. chem. Phys. I (1933) 825. Zener, C. and G u i l l e m i n , Jr., V., Phys. Rev. 34 (1928) 999. K o p i n e c k , H. J., Zeit. Naturforsch. 79 (1952) 785.