J. Phys. Chem. 1903, 87, 727-729
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High-Lying Levels of Ozone via an Algebraic Approach I. Benjamin, R. D. Levlne,' and J. L. Klnsey The Fritz Haber Molecular Dynamics Research Center, The Hebrew Universw, Jerusalem 9 1904, Israel and Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (Received: December 2 1, 1982)
An algebraic Hamiltonian for the vibrational motion in ozone, including a Darling-Dennison type coupling term, is used to fit the recently observed high overtones. The good quality of the fit is due to the coupling of anharmonic normal modes by a quartic order (Darling-Dennison type) coupling term. The algebraic Hamiltonian contains six parameters and these suffice to provide a better fit than the six-parameter spectroscopic term formula which uses a Darling-Dennison type coupling among harmonic modes. The present results provide a hitherto unexplored quantitative application of the algebraic approach to perturbed high lying states of triatomics.
Introduction Ozone' and other molecules,2 where the first excited state of the symmetric and antisymmetric normal modes belong to different symmetry species but have comparable frequencies, are particularly prone to strong anharmonic coupling of their overtones. In general, any vibrational state3ul, u2, u3 is coupled to the state u1 f 2, u2, u3 T 2. The particularly strong coupling between such modes was first pointed out by Darling and Dennison2 for HzO. Recently? the higher lying vibrational states of ozone were measured4 essentially up to the dissociation limit. Tentative assignments were made by using the DarlingDennison perturbation term1q2and by assuming an unexcited bend mode. The explicit coupling term occurring in Darling-Dennison analysis2 ( ~ 1 1 ~ 2 , ~ 3 ~ v ~ ~ 1 - 2 , ~ 2=1 ~ 3 + 2 )
(7/2)[(01
- l ) u i ( u 3 + 1 ) ( u 3 + 2)1'/' (1)
is evaluated2 for coupled harmonic normal modes. y is the coupling constant. The purpose of this Letter is to present an algebraic formulation of the problem. In the algebraic approach, the Darling-Dennison type coupling is between anharmonic modes. An explicit evaluation of the coupling term is still possible, however, and leads to
better than that4 based on the original Darling-Dennison coupling term, particularly so for the higher vibrational states (Table I). Yet the (small) amount of computational labor is identical.
Algebraic Hamiltonian for Anharmonic Vibration The algebraic approach to the vibration-rotation spectrum of diatomic molecules has been discussed in detail6 and a preliminary account of similar applications to triatomics has been published.' Here, however, we are concerned with the purely vibrational spectrum and hence a simpler approach, based on the coupling of scalar boson^^^^ will suffice. An algebraic description of a single anharmonic vibrational mode is provided by two creation (at#) and annihilation (a,P) operators for harmonic motion.6i8 The simplest algebraic Hamiltonian for an anharmonic oscillator can be ~ r i t t e n ~ p ~ . ~
H = hw('/P2 + f/2Q2) = hw(A+A- + yzI0)
(3)
w is the vibrational frequency. In terms of the harmonic operatorsg
Q = (1/2N)1/2(af/3+ Pta)
(~1,~z,~3IVI~1-2,~z,~3+2) =
(y/2)([1- X l ( U 1 - 2)1[1 - X l ( U 1 - 1)1x (ul - l)ul[l - x3u3][l - x3(u3 + 1)](u3+ 1)(u3 + 2))'12 (2) When (1)and (2) are compared it is seen that coupling of anharmonic modes introduces a correction factor of the for each normal mode. (See type ([l- xu][l- x(u + 1)])1/2 also (9).) x i is not an independent parameter but is the anharmonicity constant of the ith normal mode, xi = 1/Ni where N/2 is the number of bound states of the anharmonic (but uncoupled) normal mode. Hence, no new parameters are introduced by the algebraic a p p r ~ a c h . ~ The fit to the spectrum with the algebraic approach is (1) A. Barbe, C. Secroun, and P. Jouve, J. Mol. Spectrosc., 49,171 (1974). (2) B. T. Darling and D. M. Dennison, Phys. Rev., 57, 128 (1940). (3)The notation ul,u2, and u3 is the usual spectroscopic one where the three quantum numbers refer to the symmetric stretch, bend, and asymmetric stretch, respectively. In this note attention is restricted to states for which the bend mode is unexcited. (4)D. G. Imre, J. L. Kinsey, R. W. Field, and D. H. Katayama, J. Phys. Chem., 86, 2564 (1982).
and
-
Here, x = 1/N is the anharmonicity parameter. In the limit x 0, Q and P reduce to the coordinates and momenta of the correspondingharmonic oscillator. The basis (5) The number of parameters in the algebraic Hamiltonian is six, including the Darling-Dennison coupling constant. These six parameters suffice to fit the entire spectrum of vibrational states. Ten parameters are required in the spectroscopic term formula (Table X of ref 1). Only six parameters are required, however, in the spectroscopic term formula for computing the levels for which u2 = 0. (6) F. Iachello and R. D. Levine, J. Chem. Phys., 77, 3046 (1982). (7)0. S.van Roosmalen, A. E. L. Dieperink, and F. Iachello, Chem. Phys. Lett., 86, 32 (1982). (8) R. D. Levine in "Intramolecular Dynamics", Proceedings of the 15th Jerusalem Conference, B. Pullman and J. Jortner, Ed., Reidel, Dordrecht, 1982. (9) R. D. Levine, Chem. Phys. Lett., in press.
0022-3654/83/2087-0727$01.50/00 1983 Amerlcan Chemlcal Society
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The Journal of Physical Chemistry, Vol. 87, No. 5, 1983
Letters
TABLE I : Vibrational Energy Levels (in cm-’) of Ozone expt levela
present resultsb 1040.94 1100.19 2054.26 2104.15 2196.35 303 7.83 3073.96 3177.78 3 282.71 3987.42 4009.33 4129.93 4238.35 4359.98 4898.03 4908.97 5058.49 5158.39 5292.95 5427.51 5766.89 5771.51 5964.55 6046.50 6188.91 6338.04 6485.22 6 594.24 6595.96 6844.50 6903.22 7055.15 7208.14 7374.09 7532.96
( u , ,u2,u,
(i 1 0
cm-’)
spectroscopic fitC
1042 1103 2058 2110 2201 3046 3084 3185 3 289 3967 4026 4136 4252 4357
Algebraic Model for Ozone v1,v3 Modes Since no excited bending modes (u2) were identified in the observed spectrum; we will treat O3as a two-oscillator problem. The algebraic Hamiltonian for two coupled anharmonic oscillators takes the form H = Ha + H, + Vas (10)
3291 4000 4142
H, and Haare Morse-type algebraic Hamiltonians for the symmetric and antisymmetric modes, respectively. Each mode is described in terms of a pair of creation and corresponding annihilation operators (i.e., a and p now carry the labels s or a). V, is the coupling between the two modes and contains, in principle, all possible cross terms in the generatorsloof the algebras used to construct Haand H,. A coupling term which is bilinear in the generators (e.g., Q,Q,) will directly couple only states which shift one quantum among the symmetric and antisymmetric modes. Hence, the coupling used here includes a term which is quartic in the generators. Thus, V, is taken to be the sum of two terms: (i) a cross-anharmonicity term which is bilinear in the generators, and (ii) a Darling-Dennison term quartic in the generators. Since the self-anharmonicity term in the Morse Hamiltonian is w ~ ( a t a )we ~ , take as the cross-anharmonicity
4372 4934
5145
5197
543 5 5749
5444 5797
5962
5984
6187
6219
6497
6 506
6987
6939
7207
7 244
7523
7560
v i = X(w,X,Waxa)’~2as+asaa~aa
set of states that diagonalize the Hamiltonian (3) is given by8,’ IN,u) = [u!(N- u)!]-’/2(at)”(p+)N-”lo)
(6)
where ( 0 ) is the vacuum state. Using the explicit expressions ( 4 ) in (3) and the familiar harmonic oscillator commutation relations leads to
+ l)]’i21N,~+l)
(7)
A-IN,u) = {[l- X ( U - l ) ] ~ ~ ~ / ~ l N , ~ - l )
HIN,u) = hw(1 - X U ) U ~ N , I J ) (8)
where x = 1 / N . Equation 8 provides the physical identification of w and x by noting that the spectrum is that of a Morse oscillator. The present results are based on the observation that since Q = 2-’12(A++ A J , where A+ and A- are creation and annihilation operators for the Morse oscillator, the action of Q2 on )N,u)either leaves u unchanged or shifts it by f 2 . Indeed, apart from an anharmonicity correction factor, 0, the matrix elements of Q2 which tends to unity as x for the Morse states (computed by use of (7)) are those of q2 for the harmonic states, e.g.
(11)
Here, x is a free parameter, and the numerical coefficient in front of the operator in (11)will be denoted by x,. Had we included only this term in V, the energy levels would be given by (i = a or s)
E(ua,us)=
CWi(Ui i
+ 1 / 2 ) + i2j 1 Xij(Ui + 1 / 2 ) ( U j + 1 / 2 1 (12)
The w’s in (12) are the “infinitesimal amplitude” frequencies, and x,, = wa/Na,x,, = w,/N,, and x,, = x(x,x,)’12. The states which diagonalize the Hamiltonian (10) including the vc,” given by (11)are just the products I N a u a ) INsus ) *
The derivation of the spectroscopic term formula by algebraic means has been outlined bef01-e.~The novel feature here is the inclusion of a second contribution to V , which is quartic in the generators. It is a DarlingDennison2-type coupling.’l v(ii) aa =
and hence to
-
(9)
+
4913
= ( h w u - hwxu2)IN,u)
+ l ) ) ( u + l ) ( u + 2)]’/2
I t will also be useful to note that the Hamiltonian (3) can be written as H = (w/N)(afap@) or, since (uta + P+P acts as N on the basis states (6), H acts as (1 x)wcuta wxatcucutcu. The self-anharmonicity term is w x ( a f ~ ~ ) ~ .
a All bound levels for which u,(bend) = 0 and u , + uj < The optimal values of the parameters are 7 are listed. W , = 1051.67 cm-’, N , = 98, N, = 217, w s = 1105.28 Computed by using cm-’, x = 5.0033, y = -29.16 cm-I. the parameters (Table X ) and procedure of ref 1. Essentially the same entries as column four in Table I of ref 4.
A+IN,u)= [ ( l- X U ) ( U
(N,u+2)Q21N,u)= ( 1 / 2 ) [ ( 1- x u ) ( l - x ( u
r~a
s2
(13)
In practice, since both Q: and Qs2 have also diagonal elements we have included only the nondiagonal contribution. When (5) is used
(VG))nd= ( Y / ~ ) [ A + ; A + -,~ A-a2A+s2]
(14)
The matrix elements of this operator in the product states (IO) Here, the generators (of an SU(2) albegraa) are Pa,Qe,and I,, for
Haand Pe, Q8,and Io. for H,.
(11)This term is the anharmonic analogue of the & term in the original Darling-Dennison Hamiltonian. It is the only term which directly couples ,.u u, to v, f 2, u, i2. It is also possible to couple such states indirectly (i.e., in second-order perturbation theory) by coupling terms which are cubic in the generators. When this is done, we find (as did Darling and Dennison) that, to second-order in perturbation theory, all such terms have the same u,, u, dependence.
The Journal of Physical Chemistry, Vol. 87, No. 5, 1983 729
Letters
luau,)
= IN,u,)lN,v,) can be computed by using ( 5 ) or (9)
and are given in (2).
Computations The spectrum is computed by diagonalizing the Hamiltonian (10) in the products state basis. Note that here the basis states are anharmonic oscillators. The diagonal elements are given by (12) and the off-diagonal ones by (2). Since the Darling-Dennison coupling conserves the number of quanta (ul u3 u, + us) in the two stretch modes, only small submatrices need be diagonalized. Indeed, the computational procedure is identical with that suggested by Darling and Dennison.lV2 They, too,used (12) for the diagonal elements. However, the off-diagonal matrix elements were computed for harmonic oscillator product states, which leads to (1).As is evident from Table I, at the higher vibrational states there is a significant improvement by using an anharmonic basis set. There are six parameters in the Hamiltonian: q and x i , for i = a and s, the cross-anharmonicity x,, and the Darling-Dennison coupling parameter y. The same number of parameters is required in the conventional spectroscopic a p p r o a ~ h . ~
+
Concluding Remarks The present resulta and the recently obtained12good fit to the high vibrational states of HCN13 demonstrate the considerable accuracy that can be achieved by using algebraic Hamiltonians for higher overtones of triatomic molecules. Using the same number of parameters as the corresponding spectroscopic term formulas provides better levels of accuracy. Work is in progress on a number of additional molecules as well as on a more detailed application of the algebraic approach to triatomic molecules.
Acknowledgment. We thank Onno van Roosmalen for very many discussions of the algebraic approach to triatomic molecules and Y. M. Engel for the use of his fitting program. The Fritz Haber Research Center is supported by the Minerva Geselschaft fur die Forschung, mbH, Munchen, BRD. This work was supported by the Air Force Office of Scientific Research under Grant AFOSR81-0030. (12) 0.S. van Roosmalen, F. Iachello, R. D. Levine, and A. E. L. Dieprink, work in progress. (13)K. K. Lehmann, G. J. Scherer, and W. Klemperer, J. Chem. Phys., 77,2853 (1982).
