1666
Hyung Kyu Shin
Vibrational Transitions in Atom 4- Diatomic Systems. Use of the
-
Lenna rd Jones Potent ial a Hyung Kyu Shin Department of Chemistry,'
University of Nevada, Reno, Nevada 89507
(Received December 11, 1972)
Publication costs assisted by the U.S. Air Force Office of Scientific Research
An analytic expression for the vibrational transition probability has been formulated by use of the perturbing force derived from the Lennard-Jones potential for which potential parameters are obtained from crossed-beam experiments. The dependences of the probabilities on collision velocity, orientation angle, attractive forces, impact parameter, and temperature are shown in detail. Numerical results of the tran1, 0 2, 0 3, 1 2, 1 3, 2 3) for Nz Ar show a complicated strucsition probabilities (0 tured dependence on collisions velocity in the range of 4 x lo5 to 4 X 106 cm/sec. All transitions under consideration are quite effective in the velocity range. The temperature dependence of the deexcitation probability Plo(T) is also discussed; it is shown that log Plo(T) varies almost linearly with T-7I19, particularly at higher temperatures.
- - - - - -
I. Introduction
In developing a theory of molecular energy transfer we often encounter serious difficulties connected with the form of the interaction potential energies when the colliding molecules are interacting at short range and also with the mathematical treatment of the translational motion.24 Some of the potentials frequently used in the theory are described in terms of powers of the separation r between the centers of mass of the colliding particles; of these the Lennard-Jones (12-6) function is most commonly sed.^^,^^ Probably this function is the most realistic two-parameter potential. Sometimes, exponential potentials are used to calculate transition probabilities, but the potential parameters are usually determined by fitting such potentials to the Lennard-Jones (LJ) function,Za,7a The quantitative knowledge that is available on the LJ potential has been derived from a knowledge of macroscopic properties of the gas, such as the viscosity and the second virial coefficient; i. e., the knowledge obtained from the properties depends on the behavior of a gas in near-equilibrium.9 Therefore, use of such knowledge to calculate transition probabilities or collision cross sections is unsatisfactory since we are now interested in the properties of a gas in short range which is far removed from equilibrium. In recent years, important progress, however, has been made in determining intermolecular potentials from molecular beam experiments.10-12 Tully and Lee11 have determined the potential parameters for the LJ (20-6), (12-6), and (8-6) functions from crossed-beam experiments for NZ + Ar, Nz + Kr, 0 2 Ar, and 0 2 Kr systems. High-velocity beam scattering experiments by Colgate, et al., reveal that simpler potentials of the form K/r", where K and s are the potential parameters, are appropriate for 0 2 + Ar, NZ + Ar, and CO Ar. The LJ functions contain the effect of molecular attraction, while the latter forms do not. It should be mentioned that with the advent of large memory high-speed computers, accurate a priori interaction potentials have become available for simple molecular systems.13-15 In this paper we use the LJ (12-6) potential for which potential parameters are obtained from crossed-beam ex-
+
+
+
The Journal of Physical Chemistry, Vol. 77, No. 13, 1973
+
perimentll to derive an analytical expression for the vibrational transition probability of atom diatomic system, which provides a framework for discussing the dependences of vibrational transitions on collision velocity, orientation angle, attractive forces, impact parameter, and temperature. For this purpose, we start with analysis of the dynamics of an encounter of the colliding system, from which we determine the collision trajectory and collision time. The formulation will be applied to Nz + Ar; application to other collision systems, such as Nz Kr, is straightforward.
+
+
11. Atom
+ Diatomic Interaction
A. Development of the Overall Interaction Potential. In the interaction of rare gas atoms with the homonuclear diatomic molecules such as Nz and 0 2 the anisotropy of the interaction potential is not strong so that the angular locations of the rainbow extrema can provide information on the potential. From differential cross section measurements of Nz Ar, Nz + Kr, 0 2 Ar, and 0 2 + Kr, Tully and Lee11 have obtained potential parameters for various
+
+
(a) This work was supported by the U. S. Air Force Office of Scientific Research, Grant AFOSR-72-2231. (b) Theoretical Chemistry Group Contribution No. 1043. (a) K. F. Herzfeld and T. A. Litovitz, "Absorption and Dispersion of Ultrasonic Waves," Academic Press, New York, N. Y . , 1959, Chapter'VII; (b) T. L. Cottrell and J. C. McCoubrey, "Mo4ecular Energy Transfer in Gases," Butterworths, London, 1961, Chapter 6. K. Takayanagi, Progr. Theoret. Phys., 25, 1 (1963); Advan. At. Mol. Phys., 1, 149 (1965). D. Rapp and T. Kassal, Chem. Rev., 69,61 (1969). B. Widom and S. H. Bauer, J. Chem. Phys., 21, 1670 (1953). E. E. Nikitin, Opt. Spektrosk., 6, 141 (1959); Opt. Spectrosc. (USSR), 6, 93 (1959). (a) H. Shin, J. Chem. Phys., 41, 2864 (1964); (b) 42, 59 (1965). B. Hartmann and Z.I. Slawsky, J. Chem. Phys., 47: 2491 (1967). J. 0. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids," Wiley, New York, N. Y., 1967. S. 0. Colgate, J. E. Jordan, I . Amdur, and E. A. Mason, J. Chem. Phys., 51, 968 (1969). E. P. Tully and Y. T. Lee, J. Chem. Phys., 57, 866 (1972); also see references therein. For earlier work, see E. A. Mason and J. T. Vanderslice in "Atomic and Molecular Processes," D. R. Bates, Ed.. Academic Press, New York, N. Y., 1962, pp 663-695. M. Kraussand F. H. Mies, J. Chem. Phys., 42, 2703 (1965). M. D. Gordon and D. Secrest, J. Chem. Phys., 52, 120 (1970). W. A. Lester, J. Chem. Phys., 53, 1511, 1611 (1970); 54, e171 (1971).
1667
Vibrational Transitions in A t o m 4- D i a t o m i c Systems
forms of the LJ potential; they have found that the LJ (20-6) and (12-6) potentials give best agreement to the angular locations of the rainbow extrema. However, the experimental data provide information only on the spherically symmetric portion of the overall interaction potential; i.e., the potential is a function only of r, U ( r ) . This potential can be used to determine the collision time and/ or the collision trajectory of the relative motion. In studying molecular energy transfer problems, in addition to the r dependence, we also need to know the dependence of the interaction energy on the vibrational coordinate(s) and the orientation angle(s). As far as these additional dependences are concerned, it seems that a t present we cannot hope for any information from scattering measurements; precise information concerning the angle dependence of the potential can be obtained if rotational energy transfer is measured as a function of angle. On the other hand, from the knowledge of U ( r ) obtained from scattering experiments, with some reasonably reliable procedures, we can construct a form of the potential, which is appropriate for the study of vibrational transitions. For this purpose, we take the LJ (12-6) potential to represent the spherically symmetric portion of the overall interaction V ( r ) = 4 0 [ ( u / r ) 1 2(2.1) It should be pointed out that the method which will be developed below for the derivation of vibrational transition probability is applicable to any power of r including fractional values. To derive the perturbing force for the atom diatomic system, we express the overall potential energy as the sum of two atom-atom interactions. Using the procedure shown in ref 16, we obtain the overall energy as
+
U(r,O,x) = 4 0
4D{
[(y- ($7 +
(4)s [(6cos20 U(r)
(q>i] -
+):
-a) (yr+
(15 cos4 8 -
d+x
$ cos' 8 + );
E
-4 0
[(g)(21 cos28 -;) (:>"
(e) (126 c0s48 a 4
+
8)(:r
COS^^ 4-
(2)
(6cos2B-~)
(T)
-
(s)d -
15 (15 cos4 8 - 2 cos2 8
+
Then, our immediate problem is the following: an oscillator in a definite, steady state has applied to it a t time t a perturbing force, F(r,B), t o find the subsequent behavior of the oscillator. The next section is devoted to this problem.
111. Vibrational Transition Probability A . Perturbed Wave Function. In this section we shall derive the expression for vibrational transition probabilities as a function of the collision velocity, impact parameter, and molecular orientation angle. For this purpose we solve the time-dependent Schrodinger equation for the perturbed oscillator
+ V(r,O,x)
+
+ +
where M and w are the reduced mass and vibrational fre) the oscillator, respectively, and the perquency ( 2 ~ of turbing force is parameterized in time. In this equation, the position variable x and the momentum p are connected by the commutation relation [x,p] = ih. To facilitate the solution of eq 3.1 we introduce the operators17 a+ and a which are related to these two variables as
(FYI}
(2.2) where d is the equilibrium bond djstance of the molecule, x is the displacement of the bond distance from the equilibrium value, and B is the angle between the r and the internuclear axis of the molecule. Unlike the potential function given in ref 16, eq 2.2 is obtained by including all terms u p to the fourth order in ( d x ) / r which can make a significant contribution to the overall value of transition probabilities. The second term V(r,B,x) of eq 2.2 represents the perturbation energy that is responsible for vibrational transitions. Since the molecule is homonuclear, the angle dependence in the perturbation energy always appears as even powers of cos 0, and it is seen that the anisotropy of the perturbation energy is important. We now propose eq 2.2 to represent the overall interaction potential of the atom diatomic systems N2 + Ar, Nz Kr, 0 2 Ar, and 0 2 Kr.
+
F(r,O)
(5)1' [(21 cos28-?)2 (%>' 4-
(126 cos48 - 81 cos28
=
B. Perturbing Force. To derive the perturbing force in a form which can facilitate the explicit formulation of vibrational transition probabilities, we transform the perturbation energy V(r,B,x) into the form -F(r,B)x, where F(r,B) is the orientation-dependent perturbing force which acts on the molecule to cause vibrational transitions. The force can be derived by expanding V(r,B,x) in a power series of x f d and taking the terms that are linear in x / d . The result is
+
(A) 1/2
x =
(a+ + a ) ; p =
(Mhw)"2 -
i -
(a+
a)
(3.2)
where a and a + are hermitian conjugates of each other satisfying the relation [a,a+] = 1. In terms of these operators, we can express the Hamiltonian as
H
= fiu
(.+ i) - ~ ( t ) ( f i / 2 ~ w ) ' / ~+(a+) a
(3.3)
where we put N = a + a . We look for a solution of the Schrodinger equation in the form18
(16) H. Shin, J. Phys. Chem., 73,4321 (1969). (17) A. Messiah, "Quantum Mechanics," Vol. I, North-Holland, Amsterdam, 1968, Chapter 12. (18) I. I. Goldman and V. D. Krivchenkov, "Problems in Quantum Mechanics," Addison-Wesley, Reading, Mass., 1961, p 103.
The Journal of Physical Chemistry, VoL 77, No. 13, 1973
1668
Hyung Kyu Shin
*(x,t) =
c ( t ) exp[f(t)a+I e x p [ g ( t ) a l exp[h(t)NI @ ' k ( ~ , - m )
(3.4) where J / k is the wave function for the initial state, and c ( t ) , f ( t ) ,g ( t ) , and h ( t ) are functions of time to be determined. Equation 3.4 may be transformed into a form which is some linear combination of the unperturbed oscillator wave functions with expansion coefficients being determined by the nature of the perturbation. T o obtain the perturbed wave function, we note the following wellknown recursion relations17 a+qk = ( k + ~ ) " ' q h + l ; a$o = O (3.5) a$h = k l / ' $ k - l ; ( h 0); N $ k = h$k
+
Performing the first exponential operation of we obtain 31 [h(t)I2N exp[h(t)N]$, = 1 h(t)N
{+
+
J/k
k
2
+ -.} q k =
(3.15) The amount of vibrational energy transfer which appears in eq 3.16 is a function of u, 0, and b, and takes the form 1 AE = F[r(t),81 exp(iwt)dt )* (3.16)
-iJm
2M
x
1=0 r=O
In these expressions, the limiting integral [J-" F ( t ) exp(iot)dt]is real because F ( t ) is an even function of t. I t is then important to recognize that the quantity limt, If(t)l 2 is the magnitude of vibrational energy transfer measured in units of hw; i,e., AE/hw. By denoting this quantity simply by t and by using the relation g ( t ) = -f*(t), we finally write the transition probability as
in eq 3.4
eXP[kh(t)l$k (3.6) Then, two subsequent operations by exp[g(t)a] and expCf(t)a+]give the wave function in the form + ( x , t ) = ( ~ / ' c ( t exp[hh(t)l )
jugate of f( t ) . Therefore, the limiting quantities appearing in the transition probability are
(3.7)
-m
in which the time-dependent perturbing force is obtained by parameterizing r in time t. Substitution of eq 2.3 in this expression gives
Once the time-dependent function J / ( x , t ) is explicitly determined, an analytic expression of the vibrational n) a t t = + m can be obtained transition probability ( h from the integral representation
-
P k n = lim t+m
11:
$n*(x)+(x,t)dx
i2
(3.8)
On the basis of the orthogonality relation
1:
$n*(x)$m(x>dx =
6nm
we then obtain the transition probability, from eq 3.7 and 3.8, as
which contains the integrals of the type (3.18)
Pkn =
Note that the orthogonality requires n = m = k - 1 + r for the nonvanishing term and that r in eq 3.7 is replaced by n - k 1. B. A m o u n t of Vibrational Energy Transfer. By substituting eq 3.2-3.4 in the Schrodinger equation, and by equating the coefficients of the same operators on the right- and left-hand sides, we can readily find the differential equationsl6Js which eventually gives the solutions h(t) = -iwt (3.10)
+
f(t) =
c(t)
= exp
iexp(-iwt) (2Mfiw)l/'
F ( t ' ) exp(iwt')dt'
(3.11)
s_": F ( t") exp(iwt")dt"]
(3.12)
iwt -~1 -
[- 2
F(t') exp(-iwt')dt'
1.. 1: f
2Mliw
The solution g( t ) is the negative value of the complex conThe Journal of Physical Chemistry, Vol. 77, No. 73, 7973
We would now like an explicit expression for AE in order to calculate transition probabilities. A standard treatment of the dynamics of an encounter starts with the determination of the trajectory r ( t ) from the equation of motion. We find it convenient to write the equation of' motion in the form
dr
(3.19)
where the centrifugal energy term is represented by E( b / r)2, b being the impact parameter, p is the reduced mass of the collision system, and r* is the distance of closest approach. Using the procedure presented in ref 16, we derive the trajectory r ( t ) in the form t = iz - i ( ~ / 2 0 ) " ~~ (/ l 4 ) ( r / a ) ~ (3.20) In this expression 7 is defined as the collision time
where V = U ( r )
+ E ( b / r ) 2 . When
eq 3.20 is introduced,
Vibrational Transitions in Atom
-+ Diatomic Systems
1669
v(r)b )
