An Asymptotic Expression for the Energy Levels of the Asymmetric Rotor. III.Approximation for the Essentially Degenerate Levels of the Rigid RotorS. Golden and J. K. Bragg Citation: The Journal of Chemical Physics 17, 439 (1949); doi: 10.1063/1.1747285 View online: http://dx.doi.org/10.1063/1.1747285 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/17/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A semiclassical determination of the energy levels of a rigid asymmetric rotor J. Chem. Phys. 68, 745 (1978); 10.1063/1.435747 Energy Levels of an Asymmetric Rotor J. Chem. Phys. 31, 568 (1959); 10.1063/1.1730422 Erratum: An Asymptotic Expression for the Energy Levels of the Asymmetric Rotor. II. CentrifugalDistortion Correction J. Chem. Phys. 17, 586 (1949); 10.1063/1.1747335 An Asymptotic Expression for the Energy Levels of the Asymmetric Rotor. II. Centrifugal DistortionCorrection J. Chem. Phys. 16, 250 (1948); 10.1063/1.1746855 An Asymptotic Expression for the Energy Levels of the Rigid Asymmetric Rotor J. Chem. Phys. 16, 78 (1948); 10.1063/1.1746662

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THE JOURNAL OF CHEMICAL PHYSICS VOLUME 17. NUMBER 5 MAY. 1949

An Asymptotic Expression for the Energy Levels of the Asymmetric Rotor. m. Approximation for the Essentially Degenerate Levels of

the Rigid Rotor*

S. GOLDEN" AND J. K. BRAGG***

Mallinckrodt Chemical Laboratory, Harvard University, Cambridge, Massachusetts (Received November 15, 1948)

Formulas are given for an approximation to the energy values of the levels of the asymmetric rotor when the latter are essentially twofold degenerate. The method of arriving at these formulas follows that described recently, in which the energy matrix is examined under conditions where J-- 00. In contrast to the latter method, however, the present approximation considers those levels for which (in the limiting symmetric cases) I K I is very nearly equal to J. With these restrictions, the energy matrix of the asymmetric rotor assumes a form that can be related to that which may be obtained from a perturbed harmonic oscillator.

INTRODUCTION

T HE recent application of the correspondence principle! to the problem of the rigid asym­

metric rotor has simplified considerably the labor involved in approximating the energy values for the higher levels corresponding to large values of J. The correspondence principle gives results which may be expected to give good energy values for those asymmetric rotor levels which are essentially twofold degenerate.

In order to simplify further the computational work which is involved in the correspondence prin­ciple method, the energy matrix of the asymmetric rotor, evaluated in terms of a basis of symmetric rotor wave functions, has been examined under conditions where J is increased indefinitely. A simi­lar method has been described recently for those levels which correspond to small values of I K I (in the limiting symmetric cases), in which it was found that the energy matrix assumed an asymptotic form (J~ 00) similar to that which may be obtained from Mathieu's equation.2 In contrast, the present paper is concerned with those levels for which I K I assumes values very nearly equal to J. For this condition, the energy matrix assumes an asymptotic form (J~oo) which is similar to that which may be realized from the characteristic value problem of a perturbed harmonic oscillator.2a

As a consequence, it has been found possible to

* The support given by the Navy Department is gratefully acknowledged for some of the computational work reported here. It was carried out under Task Order V of Contract N5ori-76, Office of Naval Research, by Mrs. Grace C. Ek.

** National Research Council Predoctoral Fellow. Present address: Hydrocarbon Research, Inc., 115 Broadway, New York City.

*** Present address: Department of Chemistry, Cornell University, Ithaca, New York.

1 G. W. King, J. Chern. Phys. 15,820 (1947). 2 S. Golden, J. Chern. Phys. 16, 78 (1948), hereinafter re­

ferred to as (I). o. Professor J. H. Van Vleck has kindly pointed out that the

method employed here is similar to that employed by Luttinger and Kittel, "Note on the quantum theory of ferromagnetic resonance," to be published.

obtain an expression for the energy values of those rigid asymmetric rotor levels for which I K I is very nearly equal to J. The most significant portion of the expression involves the asymmetry parameter of the rotor in a closed form; some higher order corrections are given explicitly. An inherent short­coming of the present method is its inability to reveal the removal with increasing asymmetry of the twofold degeneracy in I K I (K ~O). Thus, as in the correspondence principle treatment, the present results must be restricted to those asymmetric rotor levels which are essentially twofold degenerate.

ASYMPTOTIC FORM OF THE ENERGY MATRIX

For the sake of brevity, use will be made of the results of (1). The problem then reduces to finding the diagonal elements of E'(K) (d. Eq. (5) of (1». The elements of the latter, evaluated in terms of a basis of symmetric rotor wave functions, are given by Eqs. (6) and (7) of (1), viz.,

(1) and

E'K:K+2=E'K+2;K=(H/G-F)[J(J, K+1)]!, (2)

where

f(J, n) = t[J(J + 1) -n(n+ 1)][J(J + 1) -n(n-1)].

For I K I nearly equal to J, the off-diagonal ele­ments of E'(K) may be put into a form suitable for expansion. To this end let3

K=-J+m, (3)

where m is a positive integer or zero. On substi­tuting Eq. (3) into the expression for [J(J, K + 1) ]1, there results

[J(J, K+1)]1=![(m+1)(m+2)(2J -m) X(2J-m-1)]1 (4)

= J[(m+ 1)(m+2)]1(1- (m/2J» X (1- (l/2J -m»l.

3 The results for K = J - m are identical with those derived from Eq. (3). The present choice is essentially a matter of convenience in notation.

439

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440 S. GOLDEN AND J. K. BRAGG

For m/2J«1«2J, the following expansion may be obtained:

[f(J, K+1)J! = J[(m+ 1)(m+2)J!/1 + 1/2J -1/32J2

+1/64J3+··· -«m+3/2)/2J) X (1 + 1/32J2+· .. ) +O(m2/128J4) I. (5)

Hence the off-diagonal elements of E'(K) may be written

E'K; K+2=E'm; m+2 '" J(H/G - F) (1 + 1/2J -1/32J2

+ l/64J3)[(m+ 1)(m+2)J! ~ t(H/G - F) (1 + 1/32J2)

X(m+3/2)[(m+1)(m+2)J!. (6)

The diagonal elements are

E'K;K=E'm; m=J2-2Jm+m2. (7)

Now, the diagonal term J2 may be subtracted from E'(K), and the resulting matrix may be divided by (-2J), so that

(8)

where E"(K) has the following non-vanishing ele­ments.

E"m; m=m-m2/2J, (9)

and

E" m; m+2'" -fj[(m+ 1)(m+2) J! +(fj' /2J)(m+3/2)

X[(m+1)(m+2)J!, (10)

with

fj =t(H/G- F)(l + 1/2J -1/32J2+ 1/64J3) , fj' = t(H/G - F) (1 + 1/32J2).

If terms O(m2/ J) are neglected m comparison with terms O(m), as J-'>oo,

E" m; m-'>m, E" m; m+2-,>-fj[(m+ 1)(m+2) J!. (11)

The matrix constructed from the elements of Eq. (11) may be recognized as involving the square

TABLE I. Operators and non-vanishing matrix elements used with the harmonic oscillator representation.

Operator Definition Matrix elements

<PI d2

<PI(m; m) =2m+1 --+x· dx2

<P2 d2

<P2 (m; m+2) = [(m+ 1) (m+2) J+ -+x· dx2

<P3 d' <P3(m; m+2) = (2m+3)[(m+1)(m+2)J+ --+x, dx'

<P. d' dx'+x' <P,(m; m) =3(m2+m+!)

<P,(m; m+4) = [(m+1)(m+2)(m+3)(m+4)J+

of the momentum and the square of the displace­ment of a linear harmonic oscillator in the Heisen­berg matrix representation. This suggests the use of a complete set of normalized Hermite orthogonal functions (i.e., harmonic oscillator wave functions) as a basis to develop the matrix defined by Eqs. (9) and (10). The operators required to accomplish this will be developed in the following section.

OPERATORS FOR THE HARMONIC OSCILLATOR REPRESENTATION

In terms of a basis of Hermite orthogonal func­tions,4

Ym(X) = (2mm hr!)-!Hm exp( - tx2) ,

m=O, 1, "', (12) the operator

£ = t(1+ 1/2J)<P I - (1/8J) <P12_t(1+ 1/4J) -fj<P2+W/4J)<Pa (13)

gives rise to the matrix E" (K). The various operators are defined in Table I, which gives also their non-' vanishing matrix elements. The table may be verified readily from the p~operties of the Hermite orthogonal functions. 5

The approximate characteristic values of the matrix E"(K) are now given by the characteristic values of the differential equation

£y=}.y, (14)

where £ is given by Eq. (13). When fj and fj' vanish, the functions Ym of Eq. (12) are the solutions of Eq. (14).

APPROXIMATE SOLUTION OF THE DIFFERENTIAL EQUATION

In Eq. (14) let

a=}.+t(1+1/4J).

The equation to be solved is

/ t(1 + 1/2J) <PI - (1/8J) <P12

(15)

-fj<P2+(fj'/4J)<Pa}y=ay. (16)

The operator <P2 gives rise to off-diagonal elements of order fj. These may be eliminated by the change of variable

x = ~[(1 + 1/2J +2fj)/ (1 + 1/2J - 2m Ji. Substitution of this expression into Eq. (16) gives, after some algebraic manipulation,

£'y = a<PI' - (1/8Jw) <p/2 -A<Pa'-B<P4'ly=a'y, (17)

'For a discussion of these functions see, for example, L. Pauling and E. B. Wilson, Jr., Introduction to Quantum Me­cha?tics (McGraw-Hill Book Company, Inc., New York, 1935), p.77.

6 See, for example, V. Rojansky, Introductory Quantum Mechanics (Prentice-Hall, Inc., New York, 1946), pp. 26 and 41.

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ASYMMETRIC ROTOR 441

where the (p' are the operators listed in Table I with ~ as the variable instead of x. Here

w = [(1 + 1/2J)2-4/32Jt, A = (1/4Jw3)[2/3(1 + 1/2J -4/3/3') - /3'W2J, B = (/3/ J W 3) [/3 -/3'(1 + 1/2J) J, a' = a/w. (18)

The first two terms of Eq. (17) are diagonal in a new harmonic oscillator representation. The off­diagonal elements in this new representation are now of order (/3/ J). Even for moderately large values of J, therefore, a rapid convergence of the usual perturbation series may be expected.

Before giving this series, it is convenient to con­sider the coefficient B. Since the quantity

[/3 -/3'(1 + 1/2J) J'" - (1/32P)(H/C - F),

and for K = 0, I (H/C - F) I = 1/3, a representative value for B is (1/128J3). An examination of the matrix elements of (p/ shows that with this value of B second-order terms may safely be neglected.

Combination of the above results gives the con­ventional perturbation series, correct to and in­cluding second-order terms:

where

a(O) = (m+t)w- (1/2J)(m+t)2, a(l) = -3Bw(m2+m+t), a(2) = - (4A 2w)[(m+l)(m+2)

X (m+3/2)2/(2 - (2m+3)/ Jw)

(19)

-m(m-l)(m-t)2/(2 - (2m-l)/ Jw)]. (20)

PERTURBATION TECHNIQUE

The perturbation technique utilized in (I) may be employed here to account for the disparity in the magnitudes of the order of the asymmetric rotor matrix and the order of the matrix developed from Eq. (14). Formally, one augments the former by adding suitably to it rows and columns of zeros. A remainder matrix R is added to the latter to adjust for such inequalities that remain between the elements of the latter and the former. By construc­tion, the remainder matrix will give no appreciable contribution in the region being considered. From the practical viewpoint, therefore, its influence on the characteristic values being approximated here may be neglected.

When it is necessary to consider the influence of R, it will be necessary to determine the trans­formation matrix that diagonalizes the matrix ob­tained from Eq. (14). Since the characteristic values of the latter may be determined approximately from Eq. (20), the transformation matrix is, in principle, readily evaluated. The transformed R matrix may then be regarded as a perturbation to be added to the diagonal matrix obtained from

TABLE II. Computation of the energy values of essentially degenerate levels of the asymmetric rotor.

Harmonic Level.' oscillator Tables of

JK-l,K+l approximation reference 6

K= -0.8

1010,0-1 91.02780 91.02780 109, 1-2 54.98445 54.98444 108,2_3 22.74492 22.74489 107,3_, -5.68590 -5.68593 106, , -30.29838 -30.29839 106, • -30.29838 -30.29846

K= -0.6

1010,0_1 92.11766 92.11765 109, 1-2 58.15739 58.15739 108,2_3 27.81338 27.81341 107,3 1.10654 1.10694 107, , 1.10654 1.10677

K= -0.4

1010,0_1 93.28164 93.28164 109, 1 61.55563 61.55565 109, 2 61.55563 61.55564 108,2 33.26898 33.26924 108,3 33.26898 33.26914

K= -0.2

1010,0_1 94.53635 94.53636 109,1 65.22976 65.22987 109,2 65.22976 65.22984

K=O.O

1010,0_1 95.90572 95.90573

* The convention of reference (6) is used for the labeling of the energy levels.

Eq. (14). Its effect may be evaluated by using the conventional series-perturbation procedure.

When R may be safely neglected, the reduced energy values of the asymmetric rotor are given approximately by combining Eqs. (8), (IS), and (20) :

E'.J(K) = (J +t)2-2J(m+t)w +(m+t)2-2J(a(I)+a(2», (21)

where T=K_1-K+1,

m = J - K-I for type I representation, =J-K+I for type III representation.

For comparison with the values tabulated by King, Hainer, and Cross,6 use is made of the relation

E/(K)=FJ(J+1)+(C-F)E'.J(K). (22)

Such a comparison is given in Table II.

ACKNOWLEDGMENT

We wish to express our appreciation for the encouragement given us by Professor E. Bright Wilson, Jr.

6 G. W. King, R. M. Hainer, and P. C. Cross, J. Chern. Phys. 11, 27 (1943). See also, R. M. Hainer, P. C. Cross, and G. W. King, "The Asymmetric Rotor. VII", J. Chern. Phys. (to be published) where a careful comparison is made of several methods of approximating the energy values for J = 12.

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