Chapter 2: Semidasskal Theory of Fermi Resonance Between Stretching and Bending Modes in Polyatomic Molecules
VII. CONCLUDING REMARKS
The semiclassical methods presented in this paper all involve the use of the Fourier components of the perturba- tion, some of which exist in analytic form33•34•48 or are straightforward to evaluate in the typical case by numerical quadrature. For the calculation described in Sec. V, the se- miclassical matrix elements (i.e., Fourier components) were analytic and could be evaluated by the use of a hand calcula- tor. The quantum mechanical Morse matrix elements of, for instance r or
r2,
while analytic,49 are more complicated to compute. The semiclassical techniques can therefore be par- ticularly useful when one wishes to use relatively simple methods for comparison with experimental absorption spec- tra: In actual experiments, the rovibrational structure may only be partially resolved (e.g., Ref. 35), and so a quick and approximate estimation of the Fermi resonance splittings and relative intensities can be helpful in fitting the data to various models.ACKNOWLEDGMENT
This work was supported by a grant from the National Science Foundation.
-32-
APPENDIX A: APPROXIMATE EVALUATION Of THE RESONANT ACTIONS
The solution of Eqs. (3.9) and (3.12) to calculate the resonant actions I~ and I
2
involves the determination of the angle-independent effective oscillator frequencies from Eqs.(3.5H3.7). One recalls that these frequencies are nonlinearly dependent on the resonant actions I; and I
2,
so the follow- ing approximate root finding procedure was used to deter- mine them: (1) A value for the zeroth order resonant actionI';
was determined from the zeroth order resonance condi- tionw7-2I-:'w7x=2.a.J~, (Al) and/: was found from Eq. (3.12) (one recalls that Ip is taken as a constant of the motion). (2) The zeroth order values of
I-;'
and
I:
were used to calculate the constantsg
and f"' from Eq. (3.7). (3) Approximate values for1;
and12
were calcu- lated from Eqs. (3.9) and (3.12) usingg and/"' in determin- ing the frequencies [Eqs. (3.5) and (3.6)] for the effective ze- roth order Hamiltonian [Eq. (3.81]. If desired, this procedure may then be iterated. However, for the model Hamiltonian given by Eq. (5.1), one iteration was sufficient to determine I~ and I~ to within 5% of the exact numerically calculated values and was the procedure used in the present paper.APPENDIX B: PHASE PLANE BEHAVIOR Of THE RESONANCE HAMILTONIAN
The analysis of the (Ia, a) phase plane behavior for the resonance Hamiltonian [Eq. (3.17)] is standard. For a given energy ER of the a motion in Eq. (3.17), Ia is given by
la =I: ±2[il2+wiX(ER - V0cos2a)]112/wiX, (Bl) where, from Eqs. (3.9), (3.10), (3.12), and (3.14), I~ denotes 2f1 lwiX. Since/a = 211 and,semiclassically,11 = (n1 +!),a phase plane portrait for the Morse "quantum number" n1 may be generated from Eq. (Bi) as a function of a:
n, = n~
± [
n 2+
lLliX(ER - Vo cos 2a)] 112/WiX, (B2) where n~ is given by Eq. (3.19).Figure I shows an (n1, a) phase plane plot on the inter- val (0, 11') for the three' different types of motion of Eq. (B2).
This idealized behavior for the full Hamiltonian (2.1) is iden- tical to that for a pendulum or "rotor" Hamiltonian. l-6 The curves that pass through a single point at a
=
0 and 1T corre- spond to the separatrix trajectory. The phase plane curves in Fig. l above and below the separatrix correspond to "rota- tions" in the (n" a) space. These are motions in which the action/1 = Ia/2(orthequantumnumbern1=11 - ~)variesonly slightly over a cycle of motion. Thereby, there is rela- tively little classical energy transfer between the Morse and harmonic oscillators. The phase plane curves inside the se- paratrix represent motions in which n1 varies greatly over a cycle of motion and thus reflects a large transfer of energy.
Such a large variation in n1 is expected for any initial n1 within the resonance width
.:111 1
=
.:1/1=
2(2V ofwiX)112=
2,fq, (B3) defined by the separatrix trajectory. i-6 The width of the reso- J. Chem. Phys., Vol. 82, No. 9, 1 May 1985nance increases with increasing coupling element V0 and with decreasing effective anharmonicity WtX of the Morse oscillator. The less the anharmonicity, the less the states in the progression In, 0), In - 1,2), etc. pass out of resonance.
In Fig. 2, an(n1> a) surface of section (e.g., Ref. l 7)ofthe Hamiltonian (5.1) is shown for actual classical trajectories having initial conditions corresponding to Ip
=
7.5 and n1=
I to 3.6. For this model Hamiltonian, this plot shows the "rotor" or "pendulum" behavior, although it is some- what distorted from Fig. l. For larger perturbations, the surface of section becomes increasingly distorted from the idealized behavior shown in Fig. l (cf. discussion in Ref. 25).A few remarks on the rate of classical and quantum energy exchange among the oscillators are perhaps in order.
For a classical resonance Hamiltonian [e.g., Eq. (3.17)), there is extensive classical energy exchange when the system is within the cosine well. (la changes considerably during the latter motion.) The frequency of the oscillatory energy ex- change is then obtained by expanding the cos 2a term about its minimum and is found to be porportional to the square root of the coefficient of the cosine term in Eq. (3.17) [i.e., it is proportional to the square root of the ( l, - 2) Fourier com- ponent of the perturbation]. Quantum mechanically, this type of energy exchange is expected to be approached when the initial wave packet consists of many eigenstates. When there are only two or three states, as in the present analysis, the frequency of energy exchange between zeroth order states is, in the case of an exact zeroth order degeneracy, proportional to the coupling matrix elements between the states. In the semiclassical limit, these matrix elements cor- respond to the Fourier components of the cosine perturba- tion term in Eq. (3.17). As a result, the quantum energy ex- change frequency may be thought of, in the case of an exact resonance, as being proportional to the coefficient rather than, as in the purely classical case, the square root of the coefficient of the cosine function. Thus, it is expected that, in any wave packet analysis, one must distinguish between the classical (i.e., many quantum states) and highly quantum (i.e., few quantum states) cases.
APPENDIX C: NORMALIZATION OF THE WAVE FUNCTION (4. 17)
A normalized wave function N.,
t/J.,
(a) satisfies-33-
small relative to q2m for that state (m = l, 2). In that case, one can determine the other overlaps < t/}~11t/J1) of nonnegli- gible magnitude and then use the normalization condition for the Fermi resonance
(C3) to determine the absolute value of the unknown overlap.
This approximation was tested in quantum mechanical cal- culations discussed in Sec. V and found to agree with the exact results in the system chosen to within 2%. The semi- classical wave functions are not suitable for determining the overlaps with the zeroth order states when the normalization series (C2) does not converge for several of the semiclassical eigenstates.
'B. V. Chirikov, E. Heil, and A. M. Sessler,J. Stat. Phys. 3, 307(1971); B. V.
Chirikov, Phys. Rep. 52, 263 ( 1979).
2G. H. Walker and J. Ford. Phys. Rev. 188, 416 (1969).
'E. V. Shuryak, Sov. Phys. iETP 44, 1070 (1976).
'See the review in P. Brumer, Adv. Chem. Phys. 47, 201 (1981).
'See the review in M. Tabor, Adv. Chem. Phys. 46, 73 (1981).
6E. F. Jaeger and A. J. Lichtenberg, Ann. Phys. (NY) 71, 319 (1972).
7T. Uzer, Chem. Phys. Lett. no. 356 (1984).
'E. Fermi, Z. Phys. 71, 250 ( 1931 ).
"W. Kaye, Spectrochim. Acta6, 257 (1954).
"'These resonances have been observed, for example, in CH2X2 (with X = D. F. Cl. Br, I) [B. R. Henry and I. Hung, Chem. Phys. 29, 465 ( 1978), and Refs. 12, 35, and 44), CHCl3 and CHBr 3 (Ref. 44), I. l, 2. 2-tetrachlor- oethane and I, I, l, 1-tetrabromoethane [B. R. Henry and M. A. Moham- madi, Chem. Phys. 55. 385 (1981)), tetramethylsilicon. tetramethylger- manium, and tetramethyltin [B. R. Henry, M. A. Mohammadi, I.
Hanazaki, and R. Nakagaki, J. Phys. Chem. 87, 4827 ( 1983)], CHD3 (Ref.
12), neopentane [B. R. Henry, 0. Sonnich Mortensen, W. F. Murphy, and D. A. C. Compton, J. Chem. Phys. 79, 2583 (1983)], and CHF, [H. R.
Dubai, M. Lewerenz, and M. Quack, Faraday Discuss. Chem. Soc. 75, 358 (1983); K. von Puttkamer, H. R. Dubai, and M. Quack, ibid. 75, 197 (1983), and references cited therein].
"G. A. Voth, R. A. Marcus, and A. H. Zewail, J. Chem. Phys. 81, 5494 (1984).
"J. W. Perry, D. J. Moll, A. Kuppermann, and A. H. Zewail, J. Chem.
Phys. 82, 1195 ( 1985).
nE. L. Sibert Ill, W. P. Reinhardt, and J. T. Hynes, Chem. Phys. Lett. 92, 455 (1982); J. Chem. Phys. 81, 1115 (1984); E. L. Sibert III, J. T. Hynes, and W. P. Reinhardt, ibid. 81, 1135 (1984); E. L. Sibert III, Ph.D. thesis, University of Colorado, 1983; V. Buch, R. B. Gerber, and M.A. Ratner, J.
Chem. Phys. 81. 3393 ( 1984).
14R. A. Marcus, Ann. NY Acad. Sci. 357, 169 (1980).
"G. M. Zaslavsky, Phys. Rep. 80, 157 (1981).
'6K. G. Kay, J. Chem. Phys. 71., 5955 (1980).
N~. [If:, (a)t/J.,
(a)da=
l. (Cl) "For reviews, see (a) D. W. Noid, M. L. Koszykowski, and R. A. Marcus, Annu. Rev. Phys. Chem. 32, 267 (1981); (b) S. A. Rice, Adv. Chem. Phys.47, 11711981).
By truncating the expansion (4.17) at terms of order q2 and using Eq. (Cl), N~. is found to be given from
_1_'.::::1+.!L[ 1
+
1 ]N~,ir 16 [v(ni)+1]2 [v(ni)-1]2
q4 [ 1
+ 1024 [v(ni)+ 1]2[v(ni)+2]2
l ]
+
[v(ni)-1] 2 [v(n 2 •1)-2] (C2)
The normalization series in Eq. (C2) may be "slow" to converge for one of the semiclassical eigenstates ¢1(a) in- volved in the resonance, an example being when v(n i)
±
m is"D. W. Noid and R. A. Marcus, J. Chem. Phys. 67. 559 (1977).
'"D. W. Noid, M. L. Koszykowski, and R. A. Marcus, J. Chem. Phys. 71, 2864 ( 1979).
'°C. Jaffe' and P. Brumer, J. Chem. Phys. 73, 5646 (1980).
"E. L.Sibert III, W. P. Reinhardt. andJ. T. Hynes,J. Chem. Phys. 77, 3583 (1982).
22(a) F.. J. Hdler, E. B. Stechel. and M. J. Davis, J. Chem. Phys. 71, 4759 (1979); 73, 4720(1980); (b) N. De Leon, M.J. Davis, andE. J. Heller. ibid.
80, 794 ( l:l84).
"E. L. Sibert III, J. T. Hynes, and W. P. Reinhardt, J. Chem. Phys. 77, 3595 (1982); J. S. Hutchinson. E. L. Sibert III, andJ. T. Hynes, ibid. 81, 1314 (1984).
240. W. Noid, M. L. Koszykowsk.i, and R. A. Marcus, J. Chem. Phys. 78, 40!811983).
"T. Vzer, D. W. Noid, and R. A. Marcus, J. Chem. Phys. 79, 4412 (1983);
T. Uzer and R. A. Marcus, ibid. 81, 5013 ( 1984).
'0R. T. Swimm andJ. B. Delos, J. Chem. Phys. 71, 1706 (1979).
J. Chem. Phys., Vol. 82. No. 9. 1May1985
27C. Jaffe and W. P. Reinhardt, J. Chem. Phys. 71, 1862 (1979); 77, 5191 (1982).
28Thel, and !2 in Eq. (2.1) are the usual action variables divided by 2,,. [see, for example, H. Goldstein, Classical Mechanics (Addison-Wesley, Read·
ing, Mass., 1980), p. 457].
'"The derivation of the resonance Hamiltonian given in Ref. 7 is for a specif- ic perturbation. For completeness, the derivation for a general perturba- tion is given in the present paper.
30 A different treatment using semiclassical matrix elements is given in R. B.
Gerber and M.A. Ratner, Chem. Phys. Lett. 68, 195 ( 1979); R. B. Gerber, R. M. Roth, and M. A. Ratner, Mo!. Phys. 44, 1335 ( 1981 ).
31R. A. Marcus, Chem. Phys. Lett. 7, 525 ( 1970).
32For the case of coupled states having widely varying quantum numbers, use of the geometric mean for the intermediate actions in the evaluation of the semiclassical matrix elements [Eq. (2.6)) may yield somewhat more accurate results [for example, P. F. Naccacbe, J. Phys. B 5, 1308 (1972) and Ref. 34]. The arithmetic mean is not well-suited for use in the semi- classical matrix elements
H,,
when the difference in quantum numbers between states i andj is large (Ref. 34)."M. L. Koszykowski, D. W. Noid, and R. A. Marcus, J. Phys. Chem. 86, 2113 (1982).
34D. M. Wardlaw, D. W. Noid, and R. A. Marcus, J. Phys. Chem. 88, 536 (1984); D. M. Wardlaw, D. W. Noid, and R. A. Marcus (to be published).
350. Sonnich Mortensen, B. R. Henry, and M. A. Mohammadi, J. Chem.
Phys. 75, 4800 (1981).
36M. L. Sage, J. Phys. Chem. 83, 1455 (1979).
37The resonance Hamiltonian [Eq. (3.17)) differs from the usual resonance Hamiltonia.-i (for example, Refs. 1-6) by the term linear in I. and the cos 2a (instead of cos a) angle dependence. This 2a angle term is intro- duced to obtain the standard Mathieu equation in Eq. (4.12).
381. C. Duncan, D. Ellis, and I. J. Wright, MoL Phys. 20, 673 (1971).
39(a) N. W. Mclachlan, Theory and Applications of Mathieu Functions (Clarendon, Oxford, 1947); (b) Expansion (6) in Sec. 2.16 of Ref. 39(a); (c) F(a) =Ace,(a,q) +Bse,(a, q) from Sec. 2.16 of Ref. 39(a), where A= 1 and B = i. The restriction v > 0 is unnecessary.
-34-
"'ro determine the value of v,, one uses the zeroth order primitive wave functions (Refs. 25 and 31) as a guide. Since a solution to the zeroth order Mathieu equation (when q = 0) is - exp [iva j, the corresponding zeroth order wave function (Eq. (4.11)] is -exp [i(S-I + v)a]. On the other hand, the primitive semiclassical wave function is - exp [in. a). We there- fore have "• equal to $ - I + v, and hence the order v is given by 211, + I - $.Upon introducing the expression given in Eq. (4.2) for$, one obtains Eq. (4.16) of the text for v. The notation v(n,) = 2(n0 - n;) may be used to distinguish between the different possible values of v as a function of n,. From Eq. (4.16), it is seen that the solutions F(a) of the Mathieu equation [Eq. (4.12)] are in general of fractional order [Ref. 39(a)].
41T. Tamir, Math. Comp. 16, 100(1962).
42Handbook of Mathematical Functions, edited by M. Abramowitz and I.
A. Stegun (Dover, New York, 1965), Chap. 20.
431. N. L. Connor, T. Uzer, R. A. Marcus, and A. D. Smith, J. Chem. Phys.
80, 5095 ( 1984).
44H. L. Fang and R. L. Swofford, J. Chem. Phys. 72, 6382 ( 1980).
"For the two states involved in an avoided crossing, the orders v, and v, of the Mathieu equation [Eq. (4.12)] become integers (Ref. 25). The appropri- ate solutions are then Mathieu functions of integer order [Ref. 39(a)]. Us- ing these functions, one would proceed as in Sec. IV B to obtain suitable overlap formulas. Since the avoided crossing point is such a special condi- tion, a treatment of the wave function for this case has been omitted in the inteiests of brevity.
46(a) E. L. Sibert III, J. T. Hynes, and W. P. Reinhardt, J. Phys. Chem. 87, 2032 (1983); (b) L.A. Gribov, Opt. Spectrosc. 31, 842 (1971); H. M. Pick- ett, J. Chem. Phys. 56, 1715 (1972); R. Meyer and H. H. Gunthard, ibid.
49, 1510 (1968); C.R. Quade, ibid. 64, 2783 (1976); 79, 4089 (!983); W. B.
Clodius and C.R. Quade, ibid. 80, 3528 (1984).
47E. B. Wilson, Jr., J.C. Decius, and P. C. Cross, Molecular Vibrations (Do- ver, New York, 1955), Chap. 4.
"I.E. Sazonov and N. I, Zhimov, Opt. Spectrosc. 34, 254(1973).
"'M. L. Sage, Chem. Phys. 35, 375 ( 1978); J. A. C. Gallas, Phys. Rev. A 21, 1829 (1980); V. S. Vasan and R. J. Cross, J. Chem. Phys. 78, 3869 (1983).
J. Chem. hys., Vol. 82, No. 9, 1May1985