J. Phys. Chem. 1083, 87, 2277-2279
2277
Effects of Dissociative Intermediate States on the Vibronic Dephasing in Visible/Ultraviolet Multiphoton Transitions of Gaseous Molecules Y. Fujlmura;
T. NakaJima,
Department of Chemistry, Faculty of Sclence, Tohoku Universlty, Sendal980, Japan
M. Kawasaki,
H. Sato,
Chemistry Department of Resources. Mie' Unlverslty, Kamihama-cho, Tsu 514, Japan
and I. Tanaka Department of Chemistry. Tokyo Institute of Technology. Ohokayama, Meguro, Tokyo 152, Japan (Received: Merch 14, 1983)
Vibronic dephasing bringing about sequential multiphoton transition via dissociative intermediate states is studied. It is shown that a linear interaction potential in the repulsive manifold makes a significant contribution to the frequency-dependentdephasing constant. This constant is calculated for the I2multiphoton transition via the lnlustate.
Introduction In recent years, much effort has been paid to the clarification of the mechanism of vibronic dephasing in molecules in gases, solids, and condensed phases.' One of the prominent effects of vibronic dephasing on visible/UV multiphoton processes is that sequential multiphoton processes are induced by an (elastic) interaction between the molecule and the heat bath.2 A similar effect is well-known in resonant light scattering: resonance fluorescence takes place as a result of the elastic scattering pro~ess.~-~ Kasatani et have recently reported two- and threephoton absorptions of I2 in gases, and from the results of polarization measurements they have concluded that when I2 molecules are excited in the energy region beyond the attractive potential of the 3110+u(labeled B) state, its absorption process can still be interpreted in terms of a sequential rather than a simultaneous mechanism, i.e., the photon absorption process proceeds via real repulsive intermediate states. The transition probability for the sequential two-photon process is expressed as2
Here, since the population decay constant I', is usually negli ibly small, the transition probability is proportional to rma. In this paper, we are concerned with the mechanism of vibronic dephasing bringing about the sequential multiphoton transition via repulsive intermediate states. I t is shown that the perturbation inducing the vibronic dephasing is expressed in terms of a intramolecular coordinate displacement of first order as well as those of quadratic and higher order, and the first-order term may make a significant contribution to the frequency-dependent dephasing constant. Most of the theories of vibronic dephasing have so far been developed for cases in which optical transitions take place between bound molecular states.' It is well-known that in these cases the quadratic term mainly contributes to the pure dephasing.
(9)
Structure of r g The dephasing constant associated with the optical transition from bound state a to repulsive state m is given as's8
I's
where and ji are the polarizations of photons, fiLm the transition matrix element, and ram = 1/2(I'aa rmm) in which I' and rmm denote the population decay constants and ram (8is a so-called pure dephasing constant. In the case of r m k 0,after averaging over all the orientations of the molecule, the above expression can be reduced to
+
+
-
(1)K. E.Jones and A. H. Zewail in 'Advances in Laser Chemistry", A. H. %wail, Ed., Springer Series in Chemical Physics, Springer, Berlin, 1978, p 196;D. J. Diestler and A. H. Zewail, J. Chem. Phys., 71, 3103 (1979). (2)Y.Fujimura and S. H. Lin, J. Chem. Phys., 74,3726 (1981). (3)D.L. Rouaseau and P. F. Williams, J. Chem. Phys., 64,3519(1976). (4)S.Mukamel, A. Ben-Reuven, and J. Jortner, J. Chem. Phys., 64, 3971 (1976). ( 5 ) R. M. Hochstrasser and C. A. Nyi, J.Chem. Phys., 70,1112(1979), and references therein. (6)K. Kasatani, T. Tanaka, K. Shibuya, M. Kawasaki, K. Obi, H. Sato, and I. Tanaka, J. Chem. Phys., 74,895 (1981).
(3)
where the &function approximation has been used to evaluate the continuum-continuum matrix element, t; represents the energy of the manifold of the repulsive state, i and f denote the initial and final states of perturbers, and pii is the population of the perturbers in the initial state. For the pure dephasing constant associated with the optical transition between two bound states, the integration of the coupling matrix element over t, in eq 3 is replaced by Vmm. The perturbation iqducing the dephasing is defined as V = H - Ho, where H and Ho represent the total system Hamiltonian and the sum of the molecular and perturber Hamiltonians, respectively. (7)Y.Fujimura, H. Kono, T. Nakajima, and S. H. Lin, J. Chem. Phys., 75,99 (1981). (8) To be published.
0 1983 American Chemical Society
2270
The Journal of Physical Chemlstty, Vol. 87, No. 13, 1983
8 Re
RA L
RC
4
Flgwe 1. A collinear collision model for vibronic dephasing. m, denote the mass of atom C. 9 represents the internuclear distance of molecule AB, and x the distance between atom C and the center of mass of the diatomic molecule.
To clarify the perturbation, we shall use a collinear collision model shown in Figure 1 for simplicity: diatomic molecule AB collides with perturber C.9 In the BornOppenheimer approximation, the wave function of the total system is given by *i(r,R) = @i(r,RWi(R) (4) where and O1 are the electronic and nuclear wave functions, respectively, and O1 satisfies the equation [TN+ EI(R)]OI(R) = EIOI(R) (5) The nuclear kinetic energy operator TN and the potential energy q(R) are expressed as tt2 a2 - h2 a2 - h2 a2 yN= - (6) 2mA aRA2 2mE aRB2 2mc dRc2
n(R) = €1(Rc- RB)+
- RA) +
ei(R~ - RB) (7)
After separating the motion of the center of mass of the whole system from that of the relative coordinates, the equation for the motion of the relative coordinates, q = Rg - RA, and X = Rc - (mARA mBRg)/(mA+ mB) is expressed as
J
Vi(q,x) 8i(q1x) = EiOl(q,x) (8)
where pAB = mAmB/(mA + mg) and PC = (mA + mB)mc/ (mA+ mB + mc), and q(Rc - RA) has been neglected. In eq 8, q(x) with 1 = i or f is the potential energy of the perturber, and Vl(q,x) with 1 = a or m, the perturbation inducing the dephasing, is expressed as V1(q,x) = ' I h - hq) - &) =
+ ...
1
Letters
pulsive exponentials as q(x) = A exp[-x/L], where A and L depend on states 1.l0 We assume a linear repulsive intramolecular potential in the resonant intermediate state and a harmonic potential in the ground state. The equilibrium point and the frequency of the vibrational mode are denoted by qo and w , respectively. A &function which is the eigenfunction of the Hamiltonian with the linear potential neglecting the nuclear kinetic energy operator is adopted as the nuclear wave function in the repulsive state. In terms of the dimensionless normal coordinates Q and 5 relative to the equilibrium point qo in the ground state defined as
where p is the reduced mass of the molecule AB,the matrix element associated with the repulsive states m, V,,, can be expressed as
V-(x) = (xfilVm(Q,~)lxm) = [hadc+ '/z(X~rd-,)~ + f/6(X~d,.)~ + ...I tf(5)
- t,) (14)
where Q, which depends on the optical excitation frequency W R is determined by using the following equations ~ w = R tm(Qc)
Q, = O for
for h w R ~
w
> emo
I R tmo
(15)
in which emo = tm(qo). In deriving eq 14, we have assumed that the intermolecular repulsive potential is cf(R) = A; exp[-af5] with af = ( h / ~ w ) l / ~ / 2The L ~ diagonal matrix element for the ground state is simply given by Vaa(x) = (xalVa(Q,?)lxa) =
where ua denotes the vibrational quantum number of the intramolecular vibration in the ground state, ai = ( h / ~ w ) l / ~ / 2 and L ~ , ( ~ ~ 1 4 =~ u,1 +~ 112 ~ ) in the harmonic oscillator approximation. Substituting eq 14 and 16 into eq 3, we can express the dephasing constant in the product form r (am d)
= KB
(17)
where K is given by
cI(x) (9)
with X = m A / ( m A+ mB). The eigenfunctions of the total system are given by *1(r?R) = wr,q,x)X1(q)h(~) (10) where xl(q) and & ( x ) satisfy
with 1 = a or m, and
with 1 = i or f, respectively. The intermolecular potential functions can approximately be expressed in terms of re(9) D. Secrest and B. R. Johnson, J . Chem. Phys., 45, 4556 (1966).
(18) and B, proportional to the pressure of the perturber at low density, takes the form
In deriving eq 17, we have assumed the same repulsive intermolecular interaction potential for the ground and intermediate states: Li = Lf, 42) = ti(?) = €,(e). Application to the Iz Multiphoton Transition via the lnluState We are in a position to apply the expression for the dephasing constant to the I2 multiphoton absorption observed by Kasatani et a16 This absorption process involves (10) K. F. Herzfeld and T. A. Litovitz, "Absorptionand Dispersion of Ultrasonic Waves", Academic Press, New York, 1959.
The Journal of Physical Chemistry, Vol. 87, No. 13, 1983 2279
Letters
tm(4) = -32004
1
/
I
30
I
/
v,=2
/
' / //,
--- ---- --- - _ _ _ _ _ _ _ _ _ _ _ _
(11) J. Tellinghuisen, J. Chem. Phys., 58, 2821 (1973). (12) J. Tellinehuisen. J. Chem. Phvs.. 57. 2397 (1972). (13) P. F. Wifiiams, A.Fernindez, 'nnd D.'L. Rousseau, Chem. Phys. Lett., 47, 150 (1977).
+ 20275
(20)
which has been derived to reproduce the potential energy and its slope at the equilibrium nuclear position in the ground state.12 Using L = 0.2 A for the intermolecular repulsive potential parameter, and (h/pw)'f2 = 0.123 A, we obtain Xa = 0.31. From this value it can be noted that terms higher than the quadratic terms of 4, and 0 in eq 14 and 16 can safely be omitted in calculating the dephasing constant. In Figure 2, the values of l$/B are plotted as a function of uR - (em(qO)/h - u,w). The broken and solid lines represent the values calculated by using only the quadratic terms, and those calculated by using the linear term involving 0, in addition to the quadratic ones, respectively. From this figure we can see that the linear term makes a dominant contribution to the frequency-dependent dephasing constant for wR > emo/ h - uaw, and, on the other band, the contribution only from the quadratic terms are approximately independent of the excitation frequency for 'JR 1 tmO/h- u,w. Figure 2 shows that the dephasing constant is strongly effected by the quantum number u, of the vibronic state in the ground state of 12. This indicates temperature effects on the sequential multiphoton process. In summary, in this paper, we have restricted ourselves to the derivation of an expression for the so-called pure dephasing constant related to the multiphoton transition via the repulsive intermediate state. The model calculation indicates the important contribution of the first-order term as well as the second-order term. In order to determine whether the simultaneous or the sequential process is dominant, we must evaluate the transition matrix elements including their energy denominators together with the dephasing constants discussed in this paper. In a subsequent paper, we shall discuss the mechanism for the multiphoton transition of I2 in detail.
Acknowledgment. Y. F. expresses his gratitude to Professor J. Jortner for the fruitful discussions. Registry No. Iodine, 7553-56-2.
