A generalization of the morse potential for diatomic molecules

JOURNAL OF MOLECULAR SPECTROSCOPY 23) 243 257 (1967)

A Generalization of the Morse Potential for Diatomic Molecules S. F L U G G E , P .

W A L G E R , AND A . W E I G U N Y

Physikalisches Institut der Universit~it Freiburg i. Breisgau, Germany To describe the vibrational spectra of diatomic molecules the potential curve V(r) is approximated by a sum of three Morse functions. This approach has the advantage of being more flexible than the simple Morse potential while preserving the correct asymptotical behavior: V(~) = const. Furthermore, such a sum potential can give rise to a small threshold above dissociation energy, as it occurs, e.g., in the ClIIu state of H2 due to dipole-dipole resonance forces. As the Schr6dinger equation cannot be solved exactly for this potential the energy eigenvalues are calculated by means of perturbation theory taking into account fourth-order corrections relative to the harmonic (zeroth) approximation. In connection with the approximate term formula a general procedure for fitting the parameters of the generalized Morse potential to spectroscopic constants is presented and applied to various molecules (N2, NO, O.~, and I2). The resulting potential curves are in good agreement with R K R curves, even for higher vibrational states where the simple Morse potential fails. I. INTRODUCTION One of t h e m a i n p r o b l e m s in t h e t h e o r y of d i a t o m i c molecules is to find an a p p r o p r i a t e a n s a t z for t h e p o t e n t i a l a c t i n g b e t w e e n t h e two nuclei. A v e r y conv e n i e n t a n d f r e q u e n t l y used r e p r e s e n t a t i o n of p o t e n t i a l curves is t h e M o r s e function. T h o u g h it shows qualitatively t h e r i g h t behf~vior over t h e whole range of t h e i n t e r n u c l e a r d i s t a n c e r, t h e M o r s e t e r m v~lues are in general too p o o r an a p p r o x i m a t i o n for t h e higher v i b r a t i o n a l states. C o r r e s p o n d i n g l y , t h e M o r s e c u r v e s coincide o n l y for low v i b r a t i o n a l s t a t e s w i t h t h e R K R curves w h i c h ~re c o m m o n l y r e g a r d e d as t h e " t r u e " p o t e n t i a l curves. T h e r e f o r e ~ m o r e flexible a n s a t z is w a n t e d p r e s e r v i n g t h e a s y m p t o t i c a l b e h a v i o r of t h e M o r s e p o t e n t i a l : V( ~ ) = const. I n this p a p e r we h a v e chosen t h e p o t e n t i a l to be a generalized M o r s e f u n c t i o n of t h e form 3

V(x)

= ~_, D , ( 1 -

--aix~2

e

) ;

a~ > 0,

(1)

i~1

x = (r -- r , ) / r , ,

(2)

w i t h r, d e n o t i n g t h e e q u i l i b r i u m d i s t a n c e of t h e nuclei. T h i s p o t e n t i a l e v i d e n t l y 243 Copyright © 1967by Academic Press Inc.

244

FLUGGE, "' ' ~ WALGER, AND WEIGUNY

converges in the limit x -+ ~ . (For other modifications of the Morse function see Ref. 1.) Only Di and ai are considered free parameters, whereas r~ is taken for known from rotational spectra. I t is the determination of D~ and a~ from vibrational t e r m values t h a t we shall be concerned with in the following. II. VIBRATIONAL STATES OF THE GENERALIZED MORSE FUNCTION Our starting point is the radial part of the SchrSdinger equation restricted to purely vibrational states:

td2 + ~2~ ( E -- V ) } x ( r )

=

(3)

As Eq. (3) with the potential (1) does not adapt itself to an exact solution, we shall proceed to arrive at the eigenvalues E b y perturbation theory: In the expansion ot V(r) about the equilibrium position r~ the anharmonic terms are considered small compared with the harmonic one. After introducing the abbreviations 3

D = ~D~, i=1 3

Dial 2,

a2D = ~

(4)

i=l 3 Cv-20t

"D = ~_, D

•

i(~ i v ,

~, =

3,4.-

"

,

i=1

the expansion of V(x) about x = 0 has the form 2~_2

V(x) = D

(--)~ - u=2

c~-~ (ax)";

Co = 1.

(5)

P !

Note t h a t the differences (1 -- c,) represent the deviations of V(x) from the simple Morse potential! I n view of perturbation theory Eq. (3) is rearranged as q-v--Y

×=

~=3

( - - ) " 2~" --~!2 2 -e

c ~ - 2 y X,

(6)

with x being substituted b y

y = (a~/)l12x

(7)

and = (a/~/)a,2;

~ = 2t, r~2D/h2;

a~,.n = 2ur,~E/h 2.

(8)

plays the role of an ordering parameter for the perturbational calculation. I n zeroth approximation we m a y neglect the anharmonic terms on the right-

A GENERALIZATION OF THE MORSE POTENTIAL

245

h a n d side of (6) and arrive at the zero-order wave functions 0

X,, = Cn exp ( - y 2 / 2 ) U , ~ ( y ) ;

H,~(y)

-~-

(

--

\n

y2/~n/~

n\

) e ( a / a y )e

_y2

Cn = ( 1 / 2 n n ! %//_)1/2,

,

(9) (10)

and the corresponding elgenvalues ' 0

v~ -- 2n ~ 1,

(11)

T h e corrections to ~,0, including the f o u r t h order, are (see Appendix for details of the calculation) A ~ = (~2/16)[c2(7 + 14n + 14n 2) -- c12(11 + 30n + 30n2)] -t- (e4/9"2S)[c4(372 + 992n -t- 744n 2 -t- 496n 3) -

-

-

-

c22(1029 -~ 2891n + 2499n 2 -t- 1666n 3)

(12)

c14(4185 + 14 715n ~- 19 035n 2 ~- 12 690n 3)

-- clc3(2340 -t- 7200n + 7560n 2 -I- 5040n 3) -t- c12c2(7182 -t- 23 814n ~- 28 350n 2 -{- 18 900n~)]. There are two remarkable points in this result: First, all contributions are of even order, i.e., only integer powers of e2 occur. This result holds in a n y order of p e r t u r b a t i o n t h e o r y (see relation g of the Appendix). I n consequence, the effective ordering p a r a m e t e r for the energy corrections is e2 instead of e. Values of e2 for the molecules N 2 , NO, 0 2 , and I2 can be found in Table V. Second, specializing to the pure Morse potential: c, = 1, it is easy to see t h a t the second-order terms of (12) c o m p r e h e n d the full a n h a r m o n i c i t y of the Morse energy formula An,~ = --e2(n + 1/~)2,

(13)

while the contributions of order e4 cancel each other. Hence we m a y presume t h a t the p e r t u r b a t i o n a l t r e a t m e n t of our generalized Morse potential will converge rapidly as long as c, ~ 1 (and, of course, 2 << 1).2 I n other words, it is the deviation of our s u m potential from the Morse function rather t h a n from the harmonic potential t h a t plays the role of the p e r t u r b a t i o n in the higher-order corrections so t h a t our formula (12) will be a good a p p r o x i m a t i o n even at high values of n. This result implies that, in the normalization condition, we have replaced the exact limits of integration: y~ = _(a~),/2 and yt. = :c by y~,~ = ~:~¢ just as Morse has done in the derivation of his "exact" term formula (2). For the molecules we have investigated and several others, too, this approximation is highly justified. 2 Strictly speaking, this presumption is only valid, if any order beyond the second vanishes in the limit c~ -~ 1. Beside the order 0 we have also found the contributions ~e 6 to cancel each other exactly so that a general proof for any order seems to be dispensable.

246

FL{J~GGE, WALGER, AND WEIGUNY III. CALCULATION OF POTENTIAL PARAMETERS

T h e problem of calculating the potential parameters D i , a~ from vibrational spectra is a t t a c k e d in two steps. ( a ) Fit of the c,'s to vibrational term values

C o m p a r i n g the coefficients of powers of the q u a n t u m n u m b e r n in Eq. (12) with those of the familiar t e r m formula Go(n) = ¢~on -

(14)

¢ooxon2 + ¢~oyon3 + . . .

and the zero-point energy G(0) = 1/4~¢0e-~- 1/~o~0x0 -t- 1/~w0y0,

(15)

we find, after some obvious algebraic manipulations, the following relations between the parameters c, and the spectroscopic constants 3 ~0, ~0xc, ~0Y0 : 4

2 -~- ~

" 4 cl 2c2 + 1185cl 4) = ¢o0 -}- woXo -- ~0Y0 (320cl c3 + 49c2~ - - lo5 - -2 ' ( 1 6 )

4

9" 256

(496c4 -- 5040clc~ -- 1666c22 -4- 18 900c12c~ -- 12 690cl 4) = ¢ooyu 2

cl

2

=

¢~0x0 +

2

3 ~¢~0yo = e c2.

(17) (18a, b)

If we take the potential depth D from the dissociation energy Do and the zeropoint energy G(0), we are left with only f o u r relations connecting the five unk n o w n parameters a, c~. Hence it is possible to impose a further condition on the parameters: I n view of p e r t u r b a t i o n theory, we require all c~'s to be as near as possible to 1. After separating Eq. (16) into two parts, o~o "-F o~oxo "-~ p'woyo = 2, - - ( p + 1/~)~oy¢. = (~/256)(320c~e3 + 49c~2 -- 1554c~2c2 +

(16a) 1185c~),

(16b)

we proceed to determine a and the c,'s b y an iterative procedure: If, for small values of p, we neglect the term p'o~oyo relative to o~0 and o~0xo, Eq. (16a) renders 4 the p a r a m e t e r a = aM ..... W i t h this value of a, we immediately obtain, from Eq. (18a, b), approximate values for cl and c~. B y variation of p we then t r y to find values for c3 and c4 which lie as near as possible to 1. W i t h the resulting p-value we recalculate a from Eq. (16a) and repeat the whole procedure. T h e corrections of a, c~, and c2 being small for each iteration, our procedure converges v e r y rapidly. 3 For the sake of simplicity, we have divided the spectroscopic constants ¢o0,~oxo , ~oy0 (which usually are given in cm -~ units) by the factorf = ~a(hD/~rc~) ~ to make them dimensionless as are n,2 and An,, in Eqs. (11) and (12). t Note that the dimension factor f (footnote 3) contains a!

A GENERALIZATION OF THE MORSE POTENTIAL

247

(b) Calculation of D~ and a~from D, ~, and c, Being given the parameters D, a, and c, it remains to invert (for ~ =< 6) the system (4) which, by the definitions x~ = D~/D;

y, = a~/a,

(19)

m a y be simplified to 3

Z xiyi ~=1

~ =

b~ ;

v = O, 2, 3, 4, 5, 6 (20)

b0 = b2 = 1;

b. = c._~ for

u = 3, 4, 5, 6.

The solution of the system (20) is achieved by formal addition of the equation 3

Z

xiYi

=

bl,

(21)

i=1

with b~ being not specified for the present. The symmetrized system can now he solved b y the following trick:~ Any four subsequent equations of the complete system are considered as a subsystem of inhomogeneous equations linear in the three unknown quantities x~. For each subsystem to have a solution it is ne('essary and sufficient t h a t

(Yl Y2 Y3) ~

1

1

1

-- be

Yl

y2

yu

- - bK+l

2

2

2

Yl 3

Yl

Y2

3

y~

y3

3

y3

=

0;

K =

0 , 1, 2 , 3.

(22)

-- b,+2 -- b,+3

If and only if these conditions are fulfilled simultaneously, we arrive at a solution of the complete system. With reference to the ansatz (1) we are only interested in solutions with yi ~ 0 ;

Yi ~ Y,,

for

i ~ m.

(23)

The four conditions (22) are then equivalent to the linear system b~u3 + b~+lu2 + b~+2ul +4- b~+3 = 0;

K = 0, 1, 2, 3,

(24)

with the abbreviations --u3 = YlY2Ya, us = YlY2 + Y2y~ + y3yl, --ul

= yl +

Y2 +

(25)

Y3.

5 The authors are greatly indebted to Dipl. Phys. A. Keller who suggested this trick for solving the nonlinear system (20).

248

FLtTGGE, WALGER, AND WEIGUNY

Equations (24) have a solution, if and only if the coefficient determinant vanishes. We are thus left with a quadratic equation for the determination of bl as a function of the remaining b~. There are in general two solutions bl and, therefore, two sets of solutions u~ of the system (24). From the quantities ui we can compute the yi's as the roots of the cubic equation (26)

y3 -t- u l y 2 "Jr u2y -t- u3 = O.

Taking for known the roots y~ there is no difficulty in finding the remaining parameters xi from Eq. (20). We have thus proven that the system (20) can always be inverted with respect to x i , yi and, at the same time, have given a procedure for actually computing the solutions. There are some points, however, which must be regarded carefully: To ensure the potential V to be real, the values of (x~, y~) must either all be real [case (a)] or one pair (x3, ya) must be real and the others conjugate complex [case (b)]. To this end the coefficients u~ of Eq. (25) and hence also bl must be real. In case (b), there will occur oscillations of the potential in consequence of the imaginary part of a~. If the imaginary part, however, is about three times smaller than the real one, these oscillations are negligible. For the real solutions [case (a)] we may find a negative weight factor for one

V(r)

.'//

[CpI "~] .. /I 30000

•.y' iI III

20000

RKR .......

MORSE

.........

35

"10000

~I.00

~.26

~,60

d.75

2.00

r cA]

FIG. 1. Potential curves of the ground state of the N2 molecule (re = 1.0976.~)

A GENERALIZATION

OF THE MORSE

249

POTENTIAL

V(r) [ C M -~1

/"

#

1" 111 / j/

30000

•

#

/' /i /

20000

:I

I~,y

i/

it,

~0000

.

.

.

.

.I .

.

.

.

~12s

.

.

RKR MORSE

-- ......

3S

/

L /

1.oo

-.- .- . . . . . .

.

.

~16o

.

.

.

.

~:Ts

.

.

.

21oo>

.

r

[A]

FIG. 2. P o t e n t i a l c u r v e s of t h e g r o u n d s t a t e of t h e N O m o l e c u l e (r, = 1.1508 ~_)

of the Morse functions in the sum potential. I n this case the potential will exceed the dissociation energy D at finite values of r forming a threshold before falling down to D in the limit r --* ~ , and vibrational states m a y exist from which the molecule can go over to free atoms b y tunneling. Such m a x i m a above the dissociation energy D do occur in some molecules in consequence of the dipole-dipole resonance interaction being proportional to 1 / r 3 and thus exceeding the familiar vail der Waals attraction. Contrary to the latter, this interaction m a y be attractive as well as repulsive according to the electronic state of the molecule. In the C'IIu state of the H~-molecule such a potential m a x i m u m exists, as was first pointed out by King and van Vleck (3) in 1939. More accurate investigations b y Browne, and b y Kolos and Wolniewicz (4) have confirmed this result. The possibility of including such potential m a x i m a presents an essential advantage of our sum potential when compared with the simple Morse function. IV. NUMERICAL RESULTS FOR THE MOLECULES N~, NO, 02 AND Ie The formalism we have developed in the preceding has been applied to the molecules ~ N2 , NO, 02 , and I2. The calculation of the potential curves for the " W e h a v e also a p p l i e d o u r m e t h o d to v a r i o u s e l e c t r o n i c s t a t e s of t h e He m o l e c u l e . B e c a u s e of t h e s m a l l r e d u c e d m a s s , h o w e v e r , o u r t e r m f o r m u l a (14) is n o t q u i t e so g o o d a n

250

FLOGGE, WALGER, AND WEIGUNY

V(r) [CM'"] 30000

A .

. //1

/

///

20000

fl"

/4//

RKR

----

MORSe

..........

3S

iI 111 ~0000

I "1

. 0

.

.

.

.K26 .

.

.

.

"£~o .

.

.

.

,1:7~ .

.

.

.

2'oo )

FIG. 3. Potential curves of the ground state of the O~ molecule (r~ = 1.2074A) ground states of N2 and 02 meets with no difficulty: As far as the vibrational states are known (up to about half the dissociation energy), their term values m a y be described satisfactorily by the polynomial (14) with the values of wo, ~0x0, and o~0ye fixed over the whole range. The term differences AG and the R K R curves are taken from Vanderslice, Mason, and Lippincott (5) and from Vanderslice, Mason, and Maisch (6), respectively. The values of r~ and D can be found in Table V. For the ground state of the NO molecule, however, the extrapolation of the R K R curve of Vanderslice, Mason, and Maisch (7) leads to re = 1.159 A instead of 1.1508 A as has been determined from rotational transitions (see Ref. 8). I n consequence the R K R curve of Vanderslice et al. is slightly displaced to the right when compared with our potential curve (which is based on the correct value r~ = 1.1508 .~). This shift will be due mainly to some uncertainty in the R K R results: In the ground state X2II of NO there is a noticeable coupling of rotation and electronic motion introducing an error of about 1% into the r,.... and r m i . values (see Ref. 7). For the ground state of the I2 molecule, Verma (9) has verified vibrational states up to n = 114. To take account of all these data in the calculation of approximation in this case so that the resulting curves eannat be used for a quantitative test of the RKR method.

A GENERALIZATION OF THE MORSE POTENTIAL

251

R K R curves, Verm~ (9) as well as Vanderslice, Weissmun, and Battino (10) were forced to use high-order polynomials for the t e r m values (up to n 7 !) with different sets of coefficients w0, w0x0, etc. for appropriately chosen regions of the potential curve. B y means of our formula (14) with only three free parameters, ¢o0, ~0x0, and ~0y0, and these having fixed values over the whole range, however, a reasonable fit has been achieved at least up to n = 45. Beyond, of course, i.e., when approaching the dissociation energy, our curve represents no more than a reasonable extrapolation. The potential curves resulting from our calculation have been compared with the simple Morse potential and the R K R curves in Figs. 1-4 or Tables I - I V , respectively; the values of x~, y~ m a y be found in Table VI. A p a r t from the above-mentioned restrictions for NO and I2 our curves confirm the R K R results, whereas the Morse function--even for low vibrational s t a t e s - - d e v i a t e s considerably from both, our sum potential and the R K R curve. I n critically examining our results one must keep in view, however, t h a t experimental information from vibrational spectra alone is not sufficient to determine the potential parameters c, in a unique way. [This question has been investigated thoroughly b y Jost and K o h n (11).] With regard to the convergence of the perturbational series, we have removed this ambiguity b y requiring the c,'s to be as near as possible to 1. To improve our method, it m a y be worth while

V(r)

[CH"~3 "12000

BOO0

// /z /~////

iii//

~000

....

2.60 =

,

,

,

....... MORSE ........ 9S

i

3.00

.

.

.

.

i

3.50

.

.

.

.

i

~.00

.

.

.

.

i

r

)

4.5.0

(A]

FIG. 4. Pot,entia] curves of the ground state of Lhe I2 molecule (r~ = 2.669 ~.)

252

FL~JTGGE, WALGER, AND W E I G U N Y TABLE I N2 POTENTIALS r[~.] 0.896 0.900 0.904 0.907 0.912 0.916 0.921 0.926 0.931 0.936 0.942 0.949 0.956 0.964 0.973 0.983 0.994 1.008 1.027 1.055 1.098 1.146 1.185 1.213 1.238 1.261 1.282 1.302 1.321 1.339 1.358 1.375 1.393 1.410 1.427 1.447 1.462 1.477 1.496 1.512 1.528

Morse 41 39 37 36 33 31 29 27 25 23 21 19 17 15 12 10 8 5 3 1 1 3 5 7 10 12 14 16 18 20 22 24 25 27 29 31 32 34 36 37

798.8 686.4 650.7 172.9 801.0 983.3 807.9 736.8 766.6 893.8 770.8 458.6 315.8 064.3 769.8 498.5 315.8 978.7 518.7 184.5 0.0 201.5 534.5 731.3 959.4 173.5 295.7 380.0 398.2 330.5 379.1 210.7 140.5 947.8 734.8 805.5 332.5 834.7 699.3 234.2 735.4

3S

39 37 35 34 32 30 28 26 24 22 20 18 16 14 12 10 8 5 3 1 1 3 5 8 10 12 15 17 19 21 23 25 27 29 32 33 35 37 39 40

548.7 586.2 693.0 317.5 107.3 411.4 379.2 441.5 595.4 837.9 842.1 663.7 640.1 508.4 329.3 164.8 076.5 829.7 449.7 170.5 0.0 217.3 616.5 904.1 246.9 594.9 862.0 103.5 286.9 388.8 628.5 640.5 769.6 771.6 758.4 069.0 778.4 464.3 562.1 293.1 989.2

RKR

40 38 36 34 33 31 29 27 25 23 21 19 16 14 12 10 8 5 3 1 1 3 5 8 10 12 14 16 19 21 23 25 27 29 31 33 34 36 38 40

493.0 688.0 859.0 988.0 096.0 171.0 218.0 235.0 223.0 181.0 111.0 011.0 882.0 725.0 538.0 323.0 079.0 806.0 505.0 175.0 0.0 175.0 505.0 806.0 079.0 323.0 538.0 725.0 882.0 011.0 111.0 181.0 223.0 235.0 218.0 171.0 094.0 988.0 853.0 688.0 493.0

T A B L E II NO POTENTIALS r[A] 0.929 0.933 0.937 0.940 0.945 0.949 0.953 0.957 0.962 0.966 0.971 0.976 0.982 0.987 0.994 1.001 1.008 1.017 1.026 1.036 1.049 1.063 1.082 1.113 1.151 1.210 1.253 1.284 1.312 1.337 1.360 1.383 1.404 1.425 1.445 1.465 1.486 1.505 1.525 1.545 1.564 1.584 1.604 1.623 1.643 1.663 1.684 1.704 1. 725

Morse 38 36 34 33 31 29 28 26 24 23 22 20 18 17 15 13 12 10 9 7 5 4 2

1 3 5 6 8 10 11 13 15 16 17 19 20 22 23 24 25 27 28 29 30 31 32 33

3S

015.3 214.3 476.1 212.6 181.3 621.7 118.5 670.4 935.3 605.6 014.2 499.0 777.8 421.5 636.9 980.1 445.1 641.3 017.7 411.7 612.9 010.4 332.4 645.0 0.0 212.4 223.5 050.3 882.9 618.8 269.7 950.7 496.9 042.3 504.9 951.4 446.9 774.9 143.3 478.5 713.9 977.9 203.0 330.2 477.3 583.9 702.1 725.6 757.4

35 34 32 31 29 27 26 25 23 22 20 19 17 16 14 13 11 l0 8 7 5 3 2

1 3 5 7 9 10 12 14 16 17 19 20 22 24 25 26 28 29 31 32 33 35 36 37 253

681.0 020.7 417.1 250.7 374.2 932.0 540.7 199.1 589.8 355.1 875.6 464.8 859.6 592.5 922.1 367.7 923.9 222.4 685.8 160.2 443.7 906.0 284.8 637.8 0.0 233.0 316.0 235.9 183.8 046.9 833.7 667.1 365.2 073.1 698.9 315.4 995.2 493.7 044.2 563.0 973.1 420.2 826.8 123.8 446.3 724.0 016.5 200.5 394.1

RKR

36 35 34 33 31 30 29 27 26 24 23 21 20 18 16 15 13 11 10 8 6 4 2

2 4 6 8 10 11 13 15 16 18 20 21 23 24 26 27 29 30 31 33 34 35 36

953.0 714.0 443.0 141.0 805.0 434.0 034.0 605.0 149.0 665.0 152.0 613.0 044.0 448.0 824.0 172.0 492.0 784.0 048.0 285.0 493.0 673.0 825.0 948.5 0.0 948.5 825.0 673.0 493.0 285.0 048.0 784.0 492.0 172.0 824.0 448.0 044.0 613.0 152.0 665.0 149.0 605.0 034.0 434.0 805.0 141.0 443.0 714.0 953.0

F L t ) G G E , WALGER, AND W E I G U N Y

254

TABLE III

02 POTENTIALS r[A] 0.979 0.982 0.985 0.989 0.993 0.997 1.002 1.006 1.011 1.016 1.022 1.027 1.034 1.041 1.049 1.057 1.067 1.078 1.091 1.106 1.126 1.158 1.207 1.262 1.307 1.340 1.370 1.397 1.422 1.445 1.469 1.491 1.513 1.534 1.556 1.577 1.598 1.619 1.641 1.662 1.683 1.704 1. 725 1.747 1.768

Morse 29 28 27 26 25 23 22 21 19 18 17 16 14 13 11 10 8 7 5 4 2

2 3 5 6 7 9 10 11 13 14 15 16 17 18 19 20 21 22 23 24 25

588.2 563.5 564.8 272.8 025.0 820.3 373.4 261.8 928.0 654.2 202.1 053.7 536.4 120.4 620.3 240.1 673.8 142.4 571.8 054.6 471.8 833.8 0.0 772.2 288.4 731.1 206.2 624.0 986.0 263.7 607.9 839.9 063.6 218.1 408.3 522.1 610.9 672.5 753.1 753.2 721.5 657.4 560.5 471.2 306.9

3S

27 26 25 24 23 22 20 19 18 17 16 15 13 12 11 9 8 5 5 3 2

2 3 5 6 8 9 11 12 14 15 16 18 19 20 22 23 24 25 26 27 28

333.3 416.7 522.0 362.5 220.4 154.9 848.1 841.9 631.7 473.0 148.6 098.2 706.2 402.5 016.3 735.6 276.0 841.2 361.3 921.7 407.1 820.7 0.0 785.2 357,3 ~79.1 456,6 991.8 483,2 896,4 397,9 787,0 179.3 504.2 881.7 181.6 462.7 721.5 013.2 218.1 393.5 537.9 650.2 780.1 824.3

RKR

28 27 26 25 24 22 21 20 19 17 16 15 13 12 11 9 8 6 5 3 2

2 3 5 6 8 9 11 12 13 15 16 17 19 20 21 22 24 25 26 27 28

639.0 546.0 429.0 287.0 122.0 934.0 722.0 486.0 227.0 945.0 640.0 313.0 963.0 591.0 196.0 778.0 337.0 873.0 387.0 878.0 344.0 787.2 0.0 787.2 344.0 878.0 387.0 873.0 337.0 778.0 196.0 591.0 963.0 313.0 640.0 945.0 227.0 486.0 722.0 934.0 122.0 287.0 429.0 546.0 639.0

T A B L E IV I2 POTENTIALS r[~,] 2.288 2.289 2.290 2.291 2.292 2.294 2.296 2.298 2.301 2.305 2.309 2.315 2.321 2.328 2.336 2.346 2. 358 2.371 2.386 2.403 2.423 2.446 2.474 2.510 2. 564 2.621 2.669 2.717 2.790 2.874 2.940 3.001 3.056 3.111 3.164 3.217 3.273 3.329 3.389 3.451 3.517 3.591 3.671 3.760 3.861 3.973 4.108 4.263 4.448 4.628 4.942 5.133

Morse 13 13 13 13 13 13 12 12 12 11 11 11 10 10 9 8 7 7 6 5 4 3 2 1

1 1 2 3 3 4 5 5 6 6 7 7 8 9 9 10 10 10 11 11 11 12 12

3S

597.6 498.0 399.0 300.6 202.6 008.5 816.4 626.5 345.6 978.5 619.5 096.1 590.5 022.3 400.8 664.1 836.6 006.0 128.0 230.1 296.0 370.9 437.6 509.4 592.5 111.0 0.0 92.7 515.9 275.6 988.2 690.6 336.3 978.3 583.5 168.4 759.4 318.8 881.0 420.6 948.8 486.1 005.0 512.2 007.0 467.6 918.3 320.8 676.8 926.1 205.6 311.5

11 11 11 11 11 11 11 11 10 10 10 9 9 8 8 7 7 6 5 4 3 3 2 1

1 2 2 3 4 5 5 6 7 7 8 9 9 10 10 11 11 11 11 12 12 12 12 255

951.0 866.7 782.9 699.5 616.5 451.9 289.0 127.9 889.4 577.3 271.9 826.0 394.5 908.7 376.0 743.0 029.2 309.9 545.6 759.4 935.6 112.4 272.8 425.5 570.6 109.1 0.0 94.3 537.4 362.3 160.6 965.4 717.4 473.9 192.1 888.4 590.4 250.6 905.7 522.0 108.8 683.0 208.8 687.8 112.9 462.2 751.8 960.0 101.9 181.0 266.1 308.6

RKR

12 12 12 12 12 11 11 11 10 10 10 9 9 8 7 7 6 5 5 4 3 2 2 1

1 2 2 3 4 5 5 6 7 7 8 9 9 10 10 10 11 11 11 12 12 12 12

509.8 427.1 302.3 170.5 039.7 824.5 562.1 251.2 893.5 490.7 045.2 559.1 034.5 474.7 882.8 260.5 608.9 930.5 226.6 498.5 747.3 973.8 179.1 363.6 527.4 102.1 0.0 102.1 527.4 363.6 179.1 973.8 747.3 498.5 226.6 930.5 608.9 260.5 882.8 474.7 034.5 559.1 045.2 490.7 893.5 251.2 562.1 824.5 039.7 170.5 302.3 352.3

256

FLIJ-GGE, W A L G E R , TABLE

AND WEIGUNY V

BASIC PARAMETERS OF MOLECULES DEALT WITI~ IN THIS PAPER

Molecule

r~[A]

D[cm~q

a

(a/y).lO 2

N~ NO O~ I2

1.09755 ~ 1.1508 d 1.20739 d 2.669"

79 896 4- 40 b 53 453 4- 40 b 42 050 4- 15 b 12 559.7"

2.9637 ~ 3.1731¢ 3.2201 ~ 4.9740 ¢

1.4822 1.7916 1.8878 0.85699

R e f e r e n c e 13. b R e f e r e n c e 14. ¢ Present investigution. d R e f e r e n c e 12. e R e f e r e n c e 9. TABLE

VI

VALUES OF AUXILIARY QUANTITIES UNDERLYING CONSTRICTION OF POTENTIALS

xl

Molecule N2

xt.2 x3 xl,2 x~ xl x2 x3 x,,2 x3

NO 02

]2

trying

= = = = = = = = =

to replace this rather

from rotational

yi

0.498 T i0.724 4.2"10 -3 0.499 T i2.373 1 . 1 4 . 1 0 -5 6 . 5 ' 1 0 -7 1.310 -0.310 - - 0 . 0 1 7 =t= i0.032 1.035

mathematical

yl.~Ya yl.2 y~ Yl y: y3 yl.2 y3

argument

= = = = = = = = =

0.766 4- i0.194 2.320 0.709 4- i0.073 2.805 9.358 0.877 0.158 0.250 4- i l . 2 6 7 0.936

by additional

information

spectra. MATHEMATICAL

In the perturbational general properties

calculation

of the matrix

APPENDIX

o f Ann , ( 1 2 ) ,

we have

made

use of some

elements

a¢

J,~,,(p) = ( - ) " + m C , , C m l

dye-Y2yPH~(y)Hm(y); ,1

n,m,p

= O, 1 , 2 , 3 . . . ,

a¢

viz. : (a)

Jn,,(O) = ~.....

(b)

Jn,~(p) = J i n x ( p )

as a consequence replacing

of orthogonality

in the matrix

element

and

reality

of the Hermite

J~m(p) the first polynomial

polynomials.

By

Hn(y) according to

A GENERALIZATION OF THE MORSE POTENTIAL

257

Eq. (9) and subsequent integration by parts it is easy to verify that (c) J,,,,(p) = 0

if

In-

m l>p.

This relation causes all summations appearing in the perturbational calculus to be finite. For explicit calculation of J~,~(p) it is convenient to use the relation (d) J~,~(p) = ( ( n ~- 1)/2)l~2J,,+~,m(p -

1) + (n/2)I~2J~_~,m(p -

1)

in connection with ( a ) - ( c ) . The recursion (d) is due to the well-known recursion formula for Hermite polynomials, H~+~(y) = 2 y H ~ ( y )

-

2nHn_~(y).

( e ) J~,~(p) = 0 if n -t- m ~- p is an odd integer [because then the integrand in J ~ m ( p ) is an odd function]. It thence follows immediately

(f) J ~ , , ( p ) ' J , , m ( p

± 1) = 0

in all cases, or more generally, (g) J,~,,,(P~)'J,~,~2(P2) " " J . . . . . . . (P~) = 0, ifn + m ~ -

~p,

#=1

is an odd integer. Hence, all odd-order contributions to energy will vanish, because, according to the general formulas of perturbation theory, the odd-order corrections contain only products of the type involved in relation (g). It is to be noted that this relation is due to the symmetry of oscillator matrix elements and, there fore, holds independently of the special potential ansatz. RECEIVED: December 30, 1966 REFERENCES 1. H. M. HULBURT AN1) J. O. HIRSCHFELDER, J. Chem. Phys. 9, 61 (1941); A. S. COOLIDGE, H. M. JAMES, AND E. L. VERNON, Phys. Rev. 54, 726 (1938). 2. P. M. MORSE, Phys. Rev. 34, 57 (1929). 3. G. W. KING AND J. H. VAN VLECK, Phys. Rev. 55, 1165 (1939). 4. J. C. BROWNE, J. Chem. Phy,s. 40, 43 (1963); W. KOLOS AND L. WOLNIEWICZ, J. Chem. Phys. 43, 2429 (1965). 5. J. T. VANDERSLICE, E. A. MASON, AND E. R. LIPPINCOTT, J. Chem. Phys. gO, 129 (1959). 6. J. W. VANDERSLICE, E. A. MASON, AND W. G. MAISCH, J. Chem. Phys. 32, 515 (1960). 7. J. W. VANDERSLICE, E. A. MASON, AND W. G. MAISCH, J. Chem. Phys. 31, 738 (1959). 8. R. H. GILETTE AND E. H. EYSTER, Phys. Ret,. 56, 1113 (1939). 9. R. D. VERMA, J. Chem. Phys. 32, 738 (1960). 10. J. T. VANDERSLICE, S. WEISSMAN, AND R. BATTINO, J. Chem. Phys. 39, 2226 (1963). 11. R. Josw AND W. KOHN, Phys. Rev. 8B, 382 (1952).

12. G. HERZBERG,"Spectra of Diatomic Molecules," pp. 540-560. Van Nostrand, Princeto,~ ~ New Jersey, 1950. 13. A. LOFTHUS,Can. J. Phys. 34, 780 (1956). 14. F. R. GILMORE, J. Q~ant, Spectry. & Radiative Transfer 5, 369 (1965).