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String‐plucking model for vibrational excitation of molecules
1.For a recent review, see D. Rapp and T. Kassal, Chem. Rev. 69, 61 (1969).
2.A. Russek, Physica (Utr.) 48, 165 (1970).
3.B. H. Mahan, J. Chem. Phys. 52, 5221 (1970).
4.F. H. Heidrich, K. R. Wilson, and D. Rapp, J. Chem. Phys. 54, 3885 (1971).
5.R. D. Levine and B. R. Johnson, Chem. Phys. Lett. 8, 501 (1971).
6.R. I. Morse and R. J. Labreque, J. Chem. Phys. 55, 1552 (1971).
7.E. E. Nikitin, Opt. Spectrosc. 6, 93 (1959).
8.M. Attermeyer and R. A. Marcus, J. Chem. Phys. 52, 393 (1970).
9.E. Thiele and R. Katz, J. Chem. Phys. 55, 3195 (1971).
10.We refer to the latter as a quantum mechanical model, even though it is semiclassical in nature with the motion of the atom treated classically.
11.D. Secrest, J. Chem. Phys. 51, 421 (1969).
12.Quantum mechanically this is not true. The time independent Schrödinger equation is where and The quantum solution depends on and independently, while the classical result depends only on their product. The quantum result has an extra degree of freedom because for vibrational excitation is impossible quantum mechanically.
13.The reduced velocity is one‐half of the β parameter in Ref. 9.
14.P. J. Kuntz, M. H. Mok, and J. C. Polanyi, J. Chem. Phys. 50, 4623 (1969).
15.J. D. Kelley and M. Wolfsberg, J. Chem. Phys. 53, 2967 (1970).
16.D. Secrest and W. Eastes, J. Chem. Phys. 56, 2502 (1972).
17.A. F. Wagner and V. McKoy, J. Chem. Phys. 58, 2604 (1973).
18.M. A. Wartell and R. J. Cross, Jr., J. Chem. Phys. 55, 4983 (1971).
19.Ronald Razner, J. Chem. Phys. 51, 5602 (1969).
20.For this condition is typically satisfied by a rotational angular momentum of 50 ℏ and a vibrational energy of 1 eV ( in reduced units), with the minimum necessary vibrational energy inversely proportional to the rotational angular momentum.
21.It is possible that this correlation is due to the central field nature of the breathing sphere model.
22.D. J. Locker and D. J. Wilson, J. Chem. Phys. 52, 271 (1970). They calculate the exact classical trajectories to obtain the time dependent potential V(y,t), which is then used to solve Eq. (14) exactly.
23.Similar modifications have been used previously in the calculations of vibrational energy transfer. In the modified wavenumber approximation [K. Takayanagi, Prog. Theor. Phys. Suppl. 25, 1 (1963)]
23.noncollinear collisions are compared with corresponding collinear collisions by subtracting from the centrifugal kinetic energy. Also, Benson and Berend [J. Chem. Phys. 44, 4247 (1966)] have found that the vibrational energy transfer for collision with initial rotational energy can be approximated by for collisions with initial conditions and However, Kelley and Wolfsberg15 have shown that both of these approximations are inaccurate.
24.The quantum calculations were performed using the method of R. G. Gordon, J. Chem. Phys. 51, 14 (1969).
25.L. Landau and E. Teller, Phys. Z. Sowjetunion 10, 34 (1936).
26.S. J. Yao and H. Y. Yao, J. Chem. Phys. 54, 4424 (1971).
27.The discrepancy in Fig. 5 between Nikitin’s result and our own can be partly explained as follows. One step in Nikitin’s derivation is to approximate the factor by While this approximation is not needed mathematically, if it is not introduced the resulting energy transfer has more degrees of freedom than allowed by the exact classical equation’s of motion. That is, without changing the exponent to 2, the energy transfer cannot be expressed in terms of the minimum set of reduced parameters m, and Nikitin’s energy transfer can be brought into better harmony with the string‐plucking result by using instead the factor which increases his by a factor of 2.1.
28.Attermeyer and Marcus8 suggest that at lower energy the adiabatic approximation is more accurate. Thiele and Katz9 verified this prediction for an adabatic quantum mechanical calculation. They found that adiabatic transition probabilities smaller than 0.1 are within 10% of exact calculations.
29.Y. T. Lee, R. J. Gordon, and D. R. Herschbach, J. Chem. Phys. 54, 2410 (1971).
30.A. P. M. Baede and J. Los, Physica (Utr.) 52, 422 (1971);
30.K. Lachmann and D. R. Herschbach, Chem. Phys. Lett. 6, 106 (1970).
31.H. J. Loesch and D. R. Herschbach, J. Chem. Phys. 57, 2038 (1972).
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