3386
J . Phys. Chem. 1989, 93, 3386-3392
laxed TICT state leading to increased solvation rates. A similar acceleration of the C T rate in the TICT formation process is found In this case, the for pretwisted dialkylaminobenzonitriles.20~21 acceleration is due mainly to the preexponential factor and not to the activation energy. The C T in BA and derivatives helps in understanding the primary charge separation in photosynthesis. The time-resolved measurements on reaction centers of photosynthetic bacteria show that ET occurs preferentially to one side (the L branch).8 Thus, nature has built in a similar symmetry-disturbing bias as occurs for BACl and C9A. The present experiments reveal that such a bias can accelerate charge-transfer rate constants. From an evolutionary aspect, it has been assumedl that earlier photosynthetic systems possessed a symmetry-undisturbed R C and that
the invention of a slightly different s u r r ~ u n d i n grepresents ~~,~~ a step forward on the ladder of evolution. This can thus be understood in terms of accelerated charge transfer. Acknowledgment. W.R. thanks the Deutsche Forschungsgemeinschaft for support through a Heisenberg fellowship. This work is part of Project No. 05 3 14 FA I5 of the Bundesministerium fur Forschung und Technologie. N.M. acknowledges the support by a grant-in-aid (6265006) from the Japanese Ministry of Education, Science and Culture. (57) Michel, H.; Deisenhofer, J. Chem. Scr. 1987, 278, 173. (58) Michel, H.; Deisenhofer, J. In Progress in Photosynthesis Research; Biggins, J., Ed.; Martinus Nijhoff Dordrecht, 1987; Vol. 1, p 1.4.353.
FEATURE ARTICLE Recent Developments in the Local Mode Theory of Overtone Spectra Lauri Halonen Department of Physical Chemistry, University of Helsinki, Meritullinkatu 1 C, SF-001 70 Helsinki, Finland (Received: November 14, 1988)
An extension of a simple local mode model for stretching vibrations of polyatomic molecules to include bending degrees of freedom is discussed. In this model kinetic and potential energy operators expressed in terms of curvilinear internal coordinates are expanded and operators describing local modes and Fermi resonances between bending and stretching vibrations are retained in the final Hamiltonian. An excellent agreement with previous anharmonic force field calculations can be obtained in the case of well-bent triatomic molecules. The rotational energy level structure of stretching vibrational states in small symmetrical molecules is discussed in the context of localized vibrations. Simple relations can be found to exist between different vibration-rotation parameters in the local mode limit.
Introduction The vibrational and rotational energy level structure of highly excited vibrational states in polyatomic molecules is of current theoretical and experimental interest. The intensive experimental work in this area is understandable in the light of rapid developments in interferometric and various laser techniques that have made it possible to probe excited The aim of the theoretical work is in the first place to construct preferably simple and physically clear models to describe these states. This is important when one wants to understand, for example, problems related to intramolecular energy transfer and multiphoton absorption pathways. The stretching vibrations are particularly relevant in this context because a bond must break in a chemical reaction. The customary way to model vibrational energy level structures of polyatomic molecules is to express the vibrational term values as a power series in normal mode vibrational quantum numbers (1) Long, M. E.; Swofford, R. L.; Albrecht, A. C. Science (Washington, D.C.)1976, 191, 183.
(2) Scherer, G. J.; Lehmann, K. K.; Klemperer, W. J . Chem. Phys. 1983, 78, 2817; Ibid. 1984, 81, 5319. (3) Wong, J. S.; Green, W. H.; Lawrence, W. D.; Moore, C. B. J . Chem. Phys. 1987, 86, 5994. (4) Douketis, C.; Anex, D.; Ewing, G.; Reilly, J. P. J . Phys. Chem. 1985, 89,4173. Page, R.H.; Shen, Y. R.; Lee, Y.T. J . Chem. Phys. 1988,88.4621. (5) Coy, S. L.; Lehmann, K. K. J . Chem. Phys. 1986, 84, 5239
0022-3654/89/2093-3386$01.50/0
involving both harmonic and anharmonic terms. In the case of stretching vibrations only, in bent XY2 molecules the vibrational term values are expressed as6
where w, is the harmonic wavenumber of the rth mode, the x constants describe the anharmonicity, and v I and v3 are vibrational quantum numbers for the symmetric and antisymmetric stretching vibrations, respectively. However, it has turned out that in H2X and D2X molecules (X = 0, S, or Se) this traditional approach gives a poor description of the energy levels due to a strong 2,2 resonance (Darling-Dennison resonance) between the normal modes.7 In water this resonance couples, for example, overtone and combination levels 2u1 with 2ujr 3u1 with u1 2u3, and 2ul + uj with 3u3, where u l and u3 are symmetric and antisymmetric stretching fundamentals, respectively. Although this modification of the standard treatment produces accurate results, this perspective does not provide the best possible physical picture because the resonance effects are so profound here that the resulting eigenvalues possess no obvious connection to the zereorder normal mode model. On the other hand, there is a beautiful alternative
+
(6) Herzberg, G. Infrared and Raman Spectra;Van Nostrand: New York, 1945. (7) Darling, B. T.; Dennison, D. M. Phys. Reu. 1940, 57, 128
0 1989 American Chemical Society
Feature Article perspective that has emerged in which the zero-order picture remains excellent throughout a large energy regime, the local mode model. In this alternative approach the stretching vibrational problem is formulated as two anharmonic bond oscillators that are coupled by kinetic and potential energy terms (see ref 8-13 for reviews of the local mode model). The theory for water in its simplest form contains only three parameters, w the harmonic wavenumber and w x the bond anharmonicity parameter, which characterize the properties of the decoupled bond oscillators, and a coupling parameter which describes the strength of the net coupling between the bond oscillators. The local mode model and its extensions to larger molecules give a satisfactory picture of many molecules. They predict correctly the energy level structures of excited stretching vibrational states: the diagonal anharmonic terms in w x become dominant at large stretching quantum numbers when compared with the coupling terms. Thus at high energies the bond oscillators become effectively decoupled from each other and an energy level structure with close local mode degeneracies is ~ b t a i n e d . ’ ~ * ~ ~ J ~ A nice feature of the local mode model is the clear and physically simple picture it offers of stretching vibrational states. It has worked surprisingly well for many different kinds of molecules in spite of the limitation that bending degrees of freedom have been neglected. For molecules with strong interactions such as Fermi resonances between bends and stretches this theory breaks down. For these cases it is desirable to develop models that are in spirit close to the local mode model for stretching vibrations only. It seems that for these purposes curvilinear internal coordinates, which describe bond and angle displacements, would be a natural choice. Indeed for triatomic molecules and for some tetraatomic molecules exact curvilinear internal coordinate Hamiltonians have been developed.16 This approach has been successful although it suffers from complexity particularly in the case of tetraatomic (and larger) molecules. Part of this article discusses recent development in including bending vibrations in a simple fashion to the local mode model. The rotational structure, which can be resolved or unresolved, forms an important part of experimental vibrational overtone spectra in the gas phase. Detailed analyses of vibration-rotation bands yield useful information about molecular structures and potential energy surfaces. A large amount of experimental and theoretical work has been carried out in this area during many decades.” The exceptional stretching vibrational energy level patterns with close local mode degeneracies near the local mode limit have given rise to questions about rotational levels of these vibrational levels. Do they possess any unusual features? Part of this article discusses this aspect of the local mode theory. Stretching Vibrations in Polyatomic Molecules The local mode Hamiltonians for polyatomic molecules are expressed in terms of curvilinear internal coordinates rj, which describe bond and angle displacements,’* and their conjugate momenta p j = -ih(d/dr,). An exact quantum mechanical vibrational Hamiltonian with total angular momentum J = 0 can be expressed in the formlg
The Journal of Physical Chemistry, Vol. 93, No. 9, 1989 3387
where Wilson’s g matrix elements are functions of internal coordinates and atomic masses of the molecule in question. These matrix elements are easily formed for any molecule with the method described in the book of Wilson, Decius, and V(r) is the potential energy function. V’(r) is a purely quantum mechanical kinetic energy term as it does not have a classical counterpart. It does not involve momentum operators. If only stretching vibrations are included in the model, V’(r) disappears and the g matrix elements become constants that depend only on atomic masses and equilibrium structure. In the case of bent XY2 molecules a useful stretching vibrational local mode Hamiltonian in terms of the Morse variable yi = 1 - exp(-mi) (i = I , 2) is12*21
is the Morse oscillator Hamiltonian of the ith bond oscillator with eigenvalues
E p = o ( n + X ) - wx(n + 1/2)*
where n is the vibrational quantum number, the harmonic wavenumber w = h ( 2 ~ ~ D & ) ~ / * / and h c , the bond anharmonicity h cMorse . dissociation energy De, parameter wx = ~ ~ h ~ g : ~ / 2The the Morse parameter a, and the coupling force constant f r , = [d2V/(drl dr2)Ieare potential energy parameters. g:r = I/m, + 1/ m yand g$ = cos p,/m, are kinetic energy coefficients. m, is the mass of the ith atom and cpc is the equilibrium valence angle. The potential energy coupling term is expressed in terms of the Morse variable y , = 1 - exp(-ar,) in order to ensure correct asymptotic limits at large amplitude displacements. When the y , variables in this term are expanded as a Taylor series and only the first term in the y , expansion is retained, the coupling term reduces to the more familiar form fr,rlr2. The physical picture of the Hamiltonian given is simple: anharmonic bond oscillators are coupled by kinetic and potential energy terms in g:rt and fit. The strength of this coupling together with bond anharmonicity parameter w x determines the structure of the energy level structure. The eigenvalues of the Hamiltonian in eq 3 are obtained best variationally with the local mode product basis Irn,n) = Im)ln), where Irn) and In) are the eigenfunctions of the Morse oscillator Hamiltonian i$‘. A simpler model, harmonically coupled anharmonic oscillator mode1,12J4-15 mentioned in the Introduction is obtained from the given local mode Hamiltonian by approximating the kinetic and potential energy coupling matrix elements (due to terms in g:rt andf,,) by the harmonic oscillator matrix elements and by restricting these couplings within the states uta, = m + n = constant. The selection rules are then such as the state Im,n) is coupled to the states Im* 1,nr1 ). A given diagonal term is just the sum of two Morse oscillator eigenvalues given in eq 5, and the coupling matrix elements turn out to be
(m+l,n-lIH/hclm,n) = X[(m (8) Henry, B. R. Acc. Chem. Res. 1977, I O , 207. (9) Henry, B. R. In Vibrational Spectra and Structure; Durig, J. R., Ed.; Elsevier: New York, 1981; Vol. 10. (10) Henry, B. R. Acc. Chem. Res. 1987, 20, 429. (11) Sage, M. L.; Jortner, J. Adu. Chem. Phys. 1981, 47, 293. (12) Child, M. S.; Halonen, L. Adu. Chem. Phys. 1984, 57, 1. (13) Child, M. S. Acc. Chem. Res. 1985, 18, 45. (14) Child, M. S . ; Lawton, R. T. Faraday Discuss. Chem. SOC.1981, 71, 273. ( I S ) Mortensen, 0. S.;Henry, B. R.; Mohammadi, M. A. J . Chem. Phys. 1981, 75, 4800. (16) Carter, S . ; Handy, N. C. Comput. Phys. Rep. 1986,5, 115; J . Chem. Phys. 1987, 87, 4294 Handy, N. C. Mol. Phys. 1987, 61, 207. (17) I. M. Mills In Specialist Periodical Reports, Theoretical Chemistry; Dixon, R. N., Ed.; The Chemical Society: London, 1974; Vol. 1. Papousek, D.; Aliev, M. R. Molecular Vibrational Rotational Spectra; Academia: Prague, 1982. (18) Hoy, A. R.; Mills, I. M.; Strey, G. Mol. Phys. 1972, 24, 1265.
(5)
+ l)n]1/2
(m-l,n+llH/hclrn,n) = A[m(n
+ 1)]1/2
(6)
X is a coupling parameter which describes the strength of the net
coupling between the bond oscillators. When no corrections due to interactions with other vibrations are taken into account, it is related to the kinetic and potential energy coefficients of the local model Hamiltonian as follow^:'^^^^^^^ =
Y~(d’rl/&
+ fir./&)
(7)
(19) Meyer, R.; Giinthard, Hs. H. J. Chem. Phys. 1968,49,1510. Pickett, H. M. J . Chem. Phys. 1972, 56, 1715. (20) Wilson, E. B.; Decius, J. C.; Cross, P. C. Molecular Vibrations; Dover: New York, 1984. (21) Halonen, L. J . Mol. Spectrosc. 1986, 120, 175. (22) Mills, I. M.; Robiette, A. G. Mol. Phys. 1985, 56, 743.
3388 The Journal of Physical Chemistry, Vol. 93, No. 9, 1989
Halonen
TABLE I: Observed and Calculated Vibrational Term Values for %iH4 and “GeH,”
28SiH, u
1
normal Ul
u3
2
2UI VI
3
4
5 6 7 8 9
+
u3
2u3 34 2Vl u3 u I 2u3 u I + ?.us uI 219 3u3 3V3
+ + +
3U’
+ U) + +
4Ul + U 3 5q uj 6 ~ 1+ ~3 7Ul u j 8 ~+ 1 ~3
local
r
uoM/cm-I
1000 1000 2000 2000 1100 3000 3000 2100 2100 2100 2100 2100 4000 5000 6000 7000
AI F2 AI F2 F2 AI F2 AI F2
8000
F2
9000
F2
21 86.87 2189.19 4308.38 4309.35 4378.40 6362.10 6362.0 6496.13 6497.48 6500.6 6500.2 6502.87 8347.4 10267.22 12121.2 13914.4 15625.4 17266.6
E F2 Fl F2 F2 F2
F2
(obsd
-
calcd)/cm-l
uoM/cm-l
0.27 0.38 -0.70 0.18 1.11 -0.63 -0.73 -0.53 -0.07 2.16 0.45 2.31 -1.57 -0.65 1.78 10.76 4.86 -3.57
74GeH4 (obsd - calcd)/cm-’
2110.71 21 11.14 4153.55 4153.83
0.13 0.46 -0.26 0.02
11650 13357 15000 16574
-1.41 -0.05 4.8 1 8.15
“Observations are taken from ref 23
The beauty of the harmonically coupled anharmonic oscillator model is its simplicity. The Hamiltonian matrix elements are easy to calculate without the need to use more complicated Morse oscillator matrix elements. The ratio IX/wxl determines the energy level structure within a particular overtone manifold (utOt= m n = constant. These are the close-lying states that are coupled with each other in this model. The successive overtone manifolds are separated by w when the anharmonicity is neglected).12 Two different limits are readily identified: in the local model limit coupling between bond oscillators is small when compared with the bond anharmonicity (1x1 x22and v I 5 2v2). It is important to note that in treating resonances between overtone and combination states one has to include in the model all close-lying states that are in resonance. Thus, for example, in CHF3 at the third C H stretching overtone chromophore utot = U, + = 4 (0, is the C H stretching quantum number and u b is the C H bending quantum number), the strong Fermi couplings between the C H stretches and the C H bends are 4Vs 3V, 2vb 2V, 4Vb V, 6Vb 8Vb
-
-
+
+
-
+
-
where us(Al) is the C H stretching fundamental and vb(E) is the doubly degenerate C H bend. This idea has been used to construct empirical effective Hamiltonian matrices for different chromophores in CHF3.27 The diagonal terms are customary anharmonic expansiod (U,,U$lH/h+,,U$) xss(us
=
W,(U,
+ yd2 + Xbb(Ub +
+ 72) + wb(Ub + 1) + + Xsb(Os +
+ l ) + gbblb2 (8)
and the off-diagonal Fermi resonance couplings are from standard normal mode theory:I7
When the coupling is absent, eq 8 gives just the usual energy level (27) Dubal, H.-R.: Quack, M. J . Chem. Phys. 1984, 81, 3779.
The Journal of Physical Chemistry, Vol. 93, No. 9, 1989 3389
Feature Article formula for this system. us and ub are vibrational quantum numbers for the C H stretching and bending vibration in the normal 2, ..., Ob) is the coordinate representation, and lb (=-vb, -ub vibrational angular momentum quantum number. ws and wb are harmonic fundamental wavenumbers for the stretch and bend, x and g constants are anharmonicity constants, and @sbb is a parameter that describes the strength of the Fermi resonance coupling. This model is in spirit of the harmonically coupled anharmonic oscillator model for stretching states only, where all couplings within a particular overtone manifold (utOt= m + n = constant) are included. This empirical model has been successful in treating Fermi resonances, but unfortunately it lacks a clear and simple physical interpretation. Another major drawback is the limitation that different sets of spectroscopic parameters (w, x, @, etc.) are needed for different isotopic species. It is also difficult to relate these parameters to more fundamental concepts as to the molecular structures and potential energy surfaces. The exact internal coordinate Hamiltonian given in eq 2 can be used to calculate energy levels variationally. In practice these calculations can become numerically heavy because numerical integration is needed in setting up Hamiltonian matrices. An approximative approach that also uses internal coordinates has appeared as discussed below (see ref 28 and references cited therein for earlier work). The purely quantum mechanical kinetic energy term V'(r) is neglected because it gives an almost constant contribution to energy levels (see ref 29 and 28 for C 0 2 and H 2 0 , respectively). The g matrix elements and the potential energy function are expanded in terms of the Morse variable y i = 1 exp(-ari) for stretches and in terms of curvilinear bending coordinates for the bends. Important local mode and Fermi resonance terms are included in the final Hamiltonian from these expansions. This idea has been applied to HzO, H2S, and HzSe,2s to S02,30to CHF3,31and to CH2C12and CDzC1z.32 The most important part of the Hamiltonian used for H 2 0 , H2S, and H2Se molecules takes the form28
+
3
H=T+V=CHi i=O
(10)
where Ho is given in eq 3 and
and H3 = xu-,[
21
CvI
+ Y2)PBz +
e
All terms of this Hamiltonian possess a clear physical interpretation. Ho is the local model Hamiltonian already discussed. H , is a harmonic oscillator Hamiltonian for the bending vibration. H2 contains harmonic coupling terms between stretching and bending vibrations. These coupling terms are not present when normal coordinates are used due to the definition of these coordinates. H3 is the Fermi resonance Hamiltonian. gto and gfe appearing in the harmonic terms are g matrix elements from eq 2 evaluated at the equilibrium configuration. The first two of the Fermi resonance terms are kinetic energy contributions. They come from the first-derivative terms of the Taylor series expansion of geeand g8. When the harmonic approximation is adopted in (28) Halonen, L.; Carrington, T.,Jr. J . Chem. Phys. 1988, 88, 4171. (29) Sibert, E. L., 111; Hynes, J. T.; Reinhardt, W. P. J. Phys. Chem. 1983, 87, 2032. Sibert, E. L., 111; Reinhardt, W. P.; Hynes, J. T. J . Chem. Phys. 1982, 77, 3583. (30) Halonen, L. J . Chem. SOC.,Faraday Trans. 2 1988, 84, 1573. (31) Halonen, L.; Carrington, T.,Jr.; Quack, M. J . Chem. SOC.,Faraday Trans. 2 1988,84, 1371. Carrington, T.; Halonen, L.; Quack, M. Chem. Phys. Lerr. 1987, 140, 512. (32) Halonen, L. J . Chem. Phys. 1988, 88, 7599.
TABLE 11: Potential Energy Parameters for Hydrogen Sulfide and Hydrogen Selenide"
parameter
DJaJ a/A-'
fJaJ A-2 fJaJ f,ll/aJ A4 fee/aJ
feee/aJ feeee/aJ
fr,/aJ A-z fJaJ A-' fM/aJ A-l j&/aJ A-2
H2S LMFR modelz8 anh ffI7 0.7722 (17) 1.6656 (21) 4.284 (14) -21.406 (75)b 83.19 (34)b 0.75866 (65) -0.1 -0.9 -0.01864 (94) 0.054 -0.2268 (69) -1.6
HzSe LMFR anh ffI7
0.7059 (1 3) 1.5752 (17) 4.284 (2) 3.503 (10) -23.4 (3) -16.556 (50)b 120. (6) 60.85 (22)b 0.758 (5) 0.71741 (59) -0.1 (1) -0.7 -0.9 (1) -0.09 -0.015 (5) -0.02268 (75) 0.054 (30) 0.130 -0.1901 (85) -0.2 (1) -1.6 (7) -1.8
3.507 (10) -16.7 (3) 63. (5) 0.710 (10) -0.70 (4) -0.09 (3) -0.024 (10) 0.130 (50) -0.2 ( 1 ) -1.8 (6)
" Uncertainties in parentheses are as quoted in given references. LMFR stands for local mode and Fermi resonance model. bCalculated from De and a . TABLE 111: Observed and Calculated Vibrational Term Values (cm-') for Hvdroeen Selenide" " - Sulfide and Hvdroeen " ,
I
H,3% obsd - calcd
normal
local
uIv,uI
mn*u
obsd
0 10 020 100 00 1 110 0 11 02 1 200 10 1 2 10 111 0 12 300 20 1 102 003 2 11 30 1 103 3 11
00'1 00'2 10'0 10-0 10'1 10-1 10-2 20'0 20-0 20'1 20-1 11+1 30'0 30-0 21'0 21-0 30-1 40-0 31-0 40-1
1182.57 2353.97 26 14.41 2628.46 3779.17 3789.28 4939.23 5145.12 5147.36 6288.15 6289.17 6388.73 7576.3 7576.3 7751.9 7779.2 8697.3 9911.05 10194.48 11008.78
normal uIu2uJ
local mn'u
0 10 100 00 1 110 0 11 111
00'1 10'0 10-0 10'1 10-1 20-1
. I ,
obsd
H?OSe obsd - calcd
-0.67 -1.01 0.16 0.38 1.22 -0.02 0.37 0.69 0.64 1.38 0.90 3.50 0.94 0.75 -1.70 -0.78 0.62 1.28 -1.53 -1.91
1034.17 2059.97 2344.36 2357.66 3361.72 3371.81
-0.51 0.88 -0.22 0.56 0.65 -0.68
4615.33 4617.40 5612.73 5613.72
0.28 0.15 0.29 -0.53
6798.15 6798.23 6953.6
0.37 0.24 -0.5 1
8894.6
-0.24
obsd
obsd - calcd
obsd
855.40 1896.38
-0.38 -0.56
741.42 1686.70 1697.36
2742.77 2754.44 4592.32
-0.09 0.06 0.14
DJzS
D;OSe obsd
- calcd
-1.40 -0.03 0.03
"Observations are from ref 28. calculating matrix elements, these terms together with the potential give rise to the resonance energy Fermi resonance term in fie# selection rule Am = f l , Au = 7 2 , or An = f l , Av = r 2 (u is the bending quantum number); Le., they give rise to similar coupling discussed in CHF3. It is interesting to see that there are two kinetic energy Fermi resonance operators although the effect of the second one is much smaller than that of the first one because the coefficient of the first term is much larger in magnitude than the coefficient of the second one. In the traditional approach where rectilinear normal coordinates are used all Fermi resonance coupling is due to the potential energy terms. The coefficients of the operators have been chosen such that the various kinetic and potential energy parameters possess their usual meanings, for example c
3390 The Journal of Physical Chemistry, Vol. 93, No. 9, 1989 When a symmetrized Morse oscillator basis set is used for the stretches and a harmonic oscillator basis set is used for the bend, all matrix elements of the Hamiltonian can be obtained analytically. This is important because it enables one to use least-squares optimization of the potential energy parameters together with high quantum levels in the calculation. Table I1 contains potential energy parameters for hydrogen sulfide and hydrogen selenide from this kind of calculation (LMFR together with a comparison approximative an anharmonic force field ca1~ulation.l~ The experimental levels of H2S, D2S, H2Se and DzSe used in the fits are in Table 111. The results are striking. The vibrational spectrum of both normal and fully deuterated species of hydrogen sulfide can be understood well with a single potential energy surface which agrees well with anharmonic force field calculations. The same conclusion holds for hydrogen selenide. An extension of the model to larger systems faces difficulties if all degrees of freedom are included. Often in practice it is necessary to include only strongly coupled motions as in recent calculations of CHF331and CH2C12.32It is also possible to couple only levels that are in strong resonance with each other and take weaker couplings into account by approximative methods such as perturbation theory. This approach has been adopted, for example, in CHD333and in benzene.24
Molecular Rotations The stretching vibrational energy level structure of bent XY, molecules obtained with the harmonically coupled anharmonic oscillator model can also be obtained with the traditional approach based on normal coordinates if quartic anharmonic resonance terms, Darling-Dennison resonance operators, are included as discussed in the I n t r o d ~ c t i o n . ~The ~ * ~diagonal ~ matrix elements of this model are given by eq 1 and the coupling matrix elements, which obey the selection rules AuI = f 2 , Au3 = r 2 , are given by the formulas (~1+2,~3-21H/hclul,u)) = + 2)03(u3 - 1)11’2 j/4Kl133[(UI +
Halonen The simplification of the vibrational problem with x-K relations suggests that there might be similar simplifications in the rotational theory. This would be nice because a detailed analysis of the rotational states of highly excited vibrational states is difficult due to the high density of states. The understanding of the structures of these rotational levels is important because some new high-resolution techniques such as laser studies with optoacoustic detection rnethods,’s2laser spectroscopy of molecular beams: and microwave optical double resonance experiments5 have resolved rotational structures of many overtone bands. Rotational levels are of interest not only due to pure spectroscopic interest, but they might also play an important role in dynamics. Astrophysical applications of vibration-rotation spectroscopy is another area where more knowledge of rotational energy level structures is needed. Bent XY2 molecules are again taken as an example case in this article, but the theory has also been extended to ammonia, methane, hydrogen peroxide, ethene (C2H4), and propadiene (C3H4)type molecule^.^^-^^ The theory is best presented using normal coordinates with Darling-Dennison resonances included for the stretching vibrational states, although identical results could be obtained by transforming the vibration-rotation operators to the internal coordinate representation and by using the harmonically coupled anharmonic oscillator model for the stretching vibrational states. Thus we express the vibration-rotation operators in terms of dimensionless normal coordinates 9, = ( 4 ~ ~ c w , / h ) l / ~ their Q , , conjugate momentum operators p , = i(d/aq,) (where r = 0-r 3), a>d in terms of molecule fixed Cartesian components J,., J,,, and J, of the total angular momentum J. Q,is the usual rth normal coordinate and w, is the corresponding harmonic wavenumber. The vibration-rotation operators of greatest importance in the case of stretching vibrational states in bent XY2 molecules are ( r = 1 , 3; 6 = x, y , z; the molecule axes are chosen to be principal axes such that the z axis is the C2 axis and t h e y axis is perpendicular to the molecular
H,,/hc = -j/&(q~ H , I / hc = -2Bili3 [ (w3 / w 1 11/29lP3 where u1 and 0 ) are the symmetric and antisymmetric stretching quantum numbers in the normal coordinate picture. K l 1 3 3is a coupling parameter. Thus in water, for example, the symmetric and antisymmetric stretching overtone states 2ul and 2u3 are strongly coupled by this resonance. The two alternative approaches, this model based on Darling-Dennison resonances and the harmonically coupled anharmonic oscillator model discussed earlier, give identical eigenvalues if the following x-K relations between the vibrational parameters are applied22*34 (see also ref 23 and 35 for earlier work on relationships between normal mode anharmonicity constants and the diagonal local mode anharmonicity): wl=w+X,
w3=w-X
(15) xi1 = ~ 3 =3 ~ 1 3 / 4= Kll33/4 = - w x / ~ The equivalence of these two approaches has also been discussed for methane,22ethene (C2H4),36propadiene (C3H4).36 and benzene)’ type molecules, and it has been treated in a general way in a recent paper.38 A nice feature of this development is the simplification of the customary theory by the discovery of the x-K relations. These relations are approximative in the sense that only stretching vibrations are considered, but it has been found that they are quite accurately satisfied for many real molecules.22 The conventional vibration-rotation theory of high-resolution molecular spectroscopy is well studied but complex in detail.]’ (33) Sibert, E. L., 111 J . Chem. Phys. 1988, 88,4378. (34) Lehmann, K. K. J. Chem. Phys. 1983, 79, 1098. (35) Henry, B. R.; Siebrand, W. J . Chem. Phys. 1968, 49, 5369. Hayward, R.; Henry, B. R. J . Mol. Spectrosc. 1974, 50, 58. Hayward, R.; Henry, B. R. J. Mol. Spectrosc. 1975, 57, 221. (36) Mills, I. M.; Mompean, F. J. Chem. Phys. Lett. 1986, 124, 425. (37) Lehmann, K. K. J. Cbem. Phys. 1986, 84, 6524. (38) Della Valle, R. G.Mol. Phys. 1988, 63, 61 1.
+ p:)jc2 (01
/ w3) l
(16) b 1 931j,
( 1 71
and
H?,/hc = j/249193 + P I P 3 ) ( ~ J + x j2A
(18)
The first operator is responsible for the vibrational dependence of rotational constants (the bending vibration has been neglected, i.e., its effect is included in the equilibrium rotational constant BE): ~ ‘ = ~ ~ - ~ f ( j / 2 ( q 1 2 + p 1 2 ) ) - ( r ~ ( j / 2 ( q 3 2 + (~I 39 )2 ) )
which reduces to the customary form6 when complications due to vibrational resonances are neglected. The coefficients cif and cis are spectroscopic parameters that can be calculated from the cubic force field. The two other operators cause interactions between rotational levels of different vibrational levels. H I 2is the Coriolis operator with the coupling coefficient f13 and H;2 is the a-resonance operator with the coupling coefficients ~ ~ 1 3Both . of these vibrationally off-diagonal operators can have significant first-order effects on the rotational energy level structure in the case of close-lying vibrational levels. Thus the resonances caused by these terms can be particularly significant in molecules close to the local mode limit ( w x >> 1x1) due to close degeneracies of vibrational levels. The Darling-Dennison resonance operator defined by its matrix elements in eq 14 together with these three vibration-rotation operators given above in eq 16-18 are responsible for major effects on the rotational energy level structure. Higher order terms most likely cause just local (39) Halonen, L.; Robiette, A. G.J . Chem. Phys. 1986, 84, 6861 (40) Halonen. L. J . Chem. Phys. 1987, 86, 588. (41) Halonen. L. J. Chem. Phys. 1987, 86, 3 1 1 5 .
The Journal of Physical Chemistry, Vol. 93, No. 9, 1989 3391
Feature Article
3060
3050
3040
3030
3020
A i+A2+2E
- Ai+A2+2E AitE
-EE tt FF ii tt FF 22 A1+F2
3010 Ai+A2+2E ApE
3000
-A i t E
Figure 1. Rotational energy level structure up to J = 4 for the degenerate v I / v 3pair of the aqmonia- and methane-type molecules in the special local mode limit. For numerical values of parameters used see ref 39. The figure is reproduced from ref 39 with permission of the publishers (copyright American Institute of Physics, 1986). (The error in this figure pointed out in ref 40 has been corrected.)
resonances or are important for levels with large rotational quantum numbers. Simple relations similar to x-K relations can be found between the rovibrational spectroscopic parameters in a special local mode limit with no potential or kinetic energy coupling between the bond 0, m, m (Le., g, 0, thus, X O)], with oscillators a special constraint on structure (LYXY a / 2 ) and with the bending fundamental much lower in wavenumber than the stretching ones ( w z 1x1) correlation diagrams for H,O-type molecules (A < 0) for expectation values I = (vlu311/2(q12+ q32))ulu3), where i = 1 or 3, u = ut + u, = rn + n = 2, 3, and 4 is the total stretching quantum number. Values between symmetric states (+) are drawn as full lines and values between antisymmetric states (-) as dashed lines. The figure is reproduced from ref 40 with permission of the publishers (copyright American Institute of Physics, 1987).
justified in a recent theoretical The first experimental evidence of these symmetrical level patterns comes from a highresolution infrared study of the 3v1/(2vl + v 3 ) band of GeH4.43 These infrared spectra show a typical structure of a symmetric top in excellent agreement with the theory. In order to calculate rotational constants for the stretching overtone and combination states, it is necessary to calculate the expectation values that appear in eq 19. They are evaluated by solving the vibrational problem with the Darling-Dennison resonance model and applying x-K relations given in eq 15. The vibrational eigenfunctions needed to calculate these expectation values are thus expressed as linear combinations of harmonic oscillator wave functions in the normal coordinate representation. These expectation values are plotted in Figure 2 as a function of Iwx/XI between the normal model limit, wx > 1x1. This procedure is in spirit similar to the one used to obtain correlation diagrams for vibrational l e v e l ~ . ~ ~ J ~ The different behavior of the matrix elements at the two limits is striking. At the local mode limit the expectation values for the close local mode pairs become equal (for example, for the states ~
~~
(42) Michelot, F.; Moret-Bailly, J.; De Martino, A. Chem. Phys. Lett. 1988, 248, 52. Chevalier, M.; De Martino, A,; Michelot, F. J. Mol. Spectrosc. 1988, 131, 382. (43) Zhu, Q.; Thrush, B. A.; Robiette, A. G. Chem. Phys. Lett. 1988, 250, 181.
Halonen
. _ -- ,-
Ob*-
I
I
Figure 3. Again the vibrational dependence of the various matrix elements at the two limits is quite different. The matrix elements of the vibrational part of the Coriolis operator disappear between the above-mentioned local mode pairs as energy levels become degenerate (energy increases). The a resonance, on the other hand, remains important. No real molecules satisfy exactly the drastic assumptions made about the special local mode limit, but many molecules are quite close to it and for these cases the relations obtained between cr parameters can be used as a starting point in analyzing highresolution vibration-rotation spectra. The results should help to obtain physically sound solutions and not just a set of fitting parameters. On the other hand, the model where effective vibration-rotation parameters are calculated can be directly applied to real molecules. Good results have been obtained for acetylene.40
ImO+) and ImG-) in the local mode notation or mul and ( m - l ) v l
ConcIusion The idea to treat part of the vibrations of a polyatomic molecule as local modes is old,44but only the developments during the past 10 years have made it a serious alternative to the customary normal model approach. To a certain extent it is a paradox that the local model has not become a generally accepted alternative to the customary theories of molecular vibrations. It offers a physically clearer picture than the models based on normal coordinates and quartic Darling-Dennison resonance couplings. It is possible that the slowness of scientists to adopt new ways of looking at problems is part of the reason for the relatively small popularity of local modes in the past, but also the difficulties in treating bending vibrations in a simple way have made local modes less useful for molecules with significant Fermi resonances between stretching and bending vibrations. Fortunately the recent developments of the theory as discussed in this article have introduced a simple, clear, and accurate approach to this problem. The applications so far include only relatively small molecules, but in the future hopefully larger molecules can be treated. The limitations in computer memory sizes will require that approximations are needed when these extensions are made. For example, only strongly coupled modes are included and the effects of other vibrations are treated by approximative methods. The first steps toward the understanding of the rotational energy level structure in the context of localized vibrations have been made. Much of this work is formulated in terms of normal coordinates, but some other contributions in this area show that it would also be possible to develop these models using curvilinear internal coordinate^.^^ It is likely that this side of the theory will progress further in the future.
are obtained from experimental spectra, also become equal for these vibrational states. The vibrational dependence of the effective Coriolis and a13 constants can be investigated in the same way by calculating the vibrational matrix elements of i(qlp3- plq3) for Coriolis effects and qlq3 pip3 for a-resonance effects as a function of Iwx/XI. These correlation diagrams are shown in
(44) Ellis, J. W. Phys. Rev. 1928, 32, 906; Ibid. 1927, 33, 21; Trans. Faraday SOC.1929, 25, 888. Mecke, R. Z . Phys. Chem. E 1932, 17, 1; Z . Phys. 1933, 81, 311; Ibid. 1936, 99, 217. Timm, B.; Mecke, R. Ibid. 1936, 98, 363. Rumpf, K.; Mecke, R. Z . Phys. Chem. E 1939, 44, 299. (45) Quade, C. R. J . Chem. Phys. 1976, 64,2783; Ibid. 1983, 79,4089. Clodius, W. B.; Quade, C. R. J . Chem. Phys. 1984,80, 3528; Ibid. 1985,82, 2365. Quade, C. R. Ibid. 1985,82,2509. Ovchinnikova, M. Ya. Chem. Phys. 1988, 120, 249. Claudius, W. B.; Shirts, R. B. J . Chem. Phys. 1984,81, 6224.
- - ... - 02+-
1
I
1
-2
I
0
-1
I
2
1
log,,lox/hl
Figure 3. Normal mode (w > 1x1) correlation diagrams for H20-type molecules (A < 0) for the integrals I = I(rnn+li(qg3-plq3)lrnn-) (full line) and I = I(rnn+l(qlq3+ pg,)lrnn-)l (dashed line) in the local mode labeling. v = uI + u3 = rn n = 2, 3 , and 4. The figure is reproduced from ref 40 with permission of the publishers (copyright American Institute of Physics, 1987).
+
+ v3 in the normal mode notation with m 1 2 ('/Z(q,' + p I 2 ) ) = ( 1/z(q32+ p : ) ) ) . Thus the effective rotational constants, which +
