Generalised potential for computing classical turning points of molecular oscillators

J. Quant. Spectrosc. Radiat. Transfer Vol. 29, No. I, pp. 93-95, 1983

0022.-40731831010093~)3503.0010

Printed in Great Britain.

© 1983 Pergamon Press Ltd.

NOTE

GENERALISED POTENTIAL FOR COMPUTING CLASSICAL TURNING POINTS OF MOLECULAR OSCILLATORS S. R. GOGAWALEt,A. M. GHODGAONKARK. , RAMANI,and A. D. TILLUt Western Regional Instrumentation Centre, University of Bombay, Bombay400 098, India

(Received 29 September 1981) Abstract--Expressions for classical turning points of the molecular oscillator are derived using a generalised first-order potential. Equivalence of the potential is established with results derived from the Morse, Dunham, and RKR potentials when the energy is a quadratic function of the vibrationalquantum number. ANALYSIS Understanding of classical turning points of molecular oscillators is of fundamental importance for constructing potential energy curves. Oldenberg, t Rydberg, 2 and Klein 3 were the first to evolve graphical procedures for constructing potential energy curves. However, a critical examination of these methods by Rees 4 for the classical turning points revealed problems at low quantum numbers. This led Rees to put them in closed form. Jarmain 5 further simplified the formulae for classical turning points in terms of vibrational quantum numbers and showed the equivalence of his results with those of Dunham. 6 All of these methods involve a large number of spectroscopic constants and therefore an attempt was made by us 7 to evolve simple expressions without an appreciable loss of accuracy. This goal was attained by expanding the Morse and Dunham potential functions and establishing their equivalence with those of Rees and Jarmain when the energy is quadratic in the vibrational quantum number. However, no generalised potential which could encompass all of these potentials was envisaged. Here, we described this treatment. The generalised potential function is given by l-

ao

L

(re~Pl2

\7] j'

(1)

where a0 = o~2[4Be and re is the equilibrium distance; p is a non-zero real number. For p = - 1, Eq. (1) becomes

U(r) = ao(r - re)2/re2.

(2)

Equation (2) is the first-order expansion of the Dunham 6 and Morse potentials. For p = + 1, Eq. (1) becomes

U(r) = ao[(r - re)/r] 2.

(3)

Equation (3) is a first-order Simons-Parr-Finlan s expansion. Setting p = - 1 and expressing it in the form of the dimensionless quantity a2re2 leads to the Kratzer potential, namely,

U(r) = De[(r- re)Ire] 2,

(4)

where De is the dissociation energy of the diatomic molecule. Thus, Eq. (1) is an adequate representation for first-order effects of a generalised potential function. We now show equivalence with the Rydberg-Klein-Rees potential for computing classical tDepartment of Physics, University of Poona, Poona-411 007, India. 93

S.R. GOGAWALEet

94

al.

turning points of molecular oscillators. For this purpose, we express Eq. (1) in the form of dissociation energy (De) and Morse parameter (a), respectively. The transformed Eq. (I) is U(F)

--- D e a 2 r f [ l

-

(relr)P] 2.

(5)

Rearranging Eq. (5) leads to

"U" ~/2.}=ln{l+c~r~[1-(rJr)P]}. In {1 +(-~)

(6)

Expanding the r.h.s, of Eq. (6), we obtain the following expressions for classical turning points of molecular oscillators: • U" (r'F=(r~)~P{1- l-ln[l+(~)ar,~

-

OZ

~/2~.

J}'

re

Following Rees, 4 the classical turnings points (rl.2) are given by

r,.2=

+f

-+f.

(81

The half width is given by

f

[(r:) p _(r0_P]/2 = ~

[r 1ln rD lz2+ U,/2]]

(9)

For p = - 1, we obtain equivalence with the results of Rees, Morse, and Dunham, respectively. The factor g may be similarly defined as

- ~

g=\r,/

.

(10)

Using Eq. (6), we have

(I~P \r~/ =

l l n [ l + "U" 1/2.. (re)V/{ 1 arc - -

J},

L

(1) P=(re)P~{l-~eln[1-(~e)1/2..J}. U"

(ll)

Equations (9) and (10) lead to

~(re)P(1/are) In

[(De 1/2 + U I/2)/(De~/2 - U

~/2)]

-

j

For most cases, are ~> 1. Hence, a slow variation of the logarithmic term is unlikely to contribute significantly to g. Therefore, in first approximation,

1 rep g = -~

In [(De1/:+ U,/2)l(Dem - U~;2)].

(13)

Classical turning points of molecular oscillators

95

For p = - 1, we find again the required equivalence with the RKR method. Furthermore, for p < 0, no new information is obtained about convergence of the series. However, for p > 0, Eq. (1) will converge rapidly for values of r contained in the interval (r~/21t/Pl,oo). This hypothesis is supported by the fact that, for r < rel2, the SPF expansion displayed oscillating behaviour and slow convergence. For p < 0, Eq. (1) at best converges in the interval 0 <- r <-211tPlre.9

1. 2. 3. 4. 5. 6. 7. 8. 9.

REFERENCES O. Oldenberg, Z. Phys. 56, 563 (1929). R. Rydberg, Z. Phys. 73, 376 (1931). O. Klein, Z. Phys. 76, 226 (1932). A. L. G. Rees, Proc. Phys. Soc. (London) A59, 998 (1947). W. R. Jarmain, Can. J. Phys. 38, 217 (1960). J. L. Dunham, Phys. Rev. 41, 713 (1932). A. M. Ghodgaonkar, K. Ramani, and S. R. Gogawale, JQSRT 27, 467 (1982). G. Simons, R. G. Parr, and J. M. Finlan,/. Chem. Phys. 59, 3229 (1973). A. J. Thakkar, J. Chem. Phys. 62, 1693 (1975).