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Probability of vibrational excitations in high energy collisions
Volume 3. number 3
CHEMICAL PHYSICS LETTERS
PROBABILITY IN
OF HIGH
Department
VIBRATIONAL
ENERGY HYUNG WU
March
EXCITATIONS
COLLISIONS SHIN
of Chemistry, University Rena, Nevada, USA
1969
*
of Nevada,
Received 31 January 1969
Vibrational transition probability pnm for n-m is formulated from the solution of the time-independent Schr&iinger equation for colinear collision between an oscillator and a particle (an &om or a molecule). The resulting expression can he used for calculatingpnnl in high energy collisions.
At sufficiently high collision velocities, the usual perturbation calculation of vibrational transition probabilities p fails because the probabilities become too large [l, 21. For such high velocity collisions, however, we can formulate p from the solution of the time-independent SchrBdinger equation of the perturbed final oscillator (stationary) state considering the relative coordinate of the collision partners as ‘dynamic’ variables. In this letter we show such solutions for the colinear collision between an oscillator BC and an atom A starting with the Green function method. Morse and Feshbach [3] have shown the basic formulation of the Green function method for general perturbation and we extend it to the present molecular collision problem. The time-independent Schrijdinger equation for the collision system in which BC is hit along its line of oscillation by A with the perturbing potentiD V(<, x) which is a function of the distance x between centers of mass of the collision partners and the vibrational amplitude 4, is (tf2/2M) V(5) f [W- VCS>l q45) = V(5,x) q&), where &f= mBmC/(mB + 77Zc),v(t) is the iI’&raIIIOlecular potential, and W is the eigenvalue. We approximate BC by a harmonic osc;llator which isinane onential interaction wit2 A: i.e., V(5) = $K? , w h ere K is the force constant. For the colinear collision (BC-A) the instantaneous separation is z = x - ~5, where p = mg/(mg+mc). Then for an exponential interz&ion between C and A, we have V(z) =
= A exp( -z/a) = A exp(-x/a) exp(pS/u), where A and CLare potential parameters. When the range of the interacting force is large, we can write V(z) =A(1 +~
The solution of this equation can be found as arr eigenfunction szies,
We then see that, as a function of W. GW is analytic except for simple poles at W = W,. Thus, if it should happen that a Green function is known in a closed form, the eigenfunction @‘n and the corresponding eigesvalue W, may be found by investigating GW at its poles. In terms of Gw as kernel, the wave function is given by the integral equation
where Qn (5) is the unperturbed state. The sum represents the perturbed part of the wave function, and contains the simple harmonic osciIlator matrix elements:
work was carried out under Grant AFOSR-681354from the US Air Force Office of Scientific Research.
* ms
125
Volume 3. number 3
CHEMICAL
PHYSICS
The complete wave function is then a particular linear combination of the displaced functions obtained from eq. (2):
where E is the depression of the potential enqrgy caused by the perturbation. The term (2e/Xja is the displacement A of the equilibrium position of BC in the znth oscillator state. The normalization of e(5) has been adjusted so that Cn = 1. The probability of transition to the state m can then be derived from the coefficient C,. With the introdtlction of the harmonic oscillator wave functions and with the second-order corrections, we obtain from eq. (2)
{[ 4 (n+l)l+
&+1(5)
(4)
- (%)Q en-1 (51) ,
where w is the vibrational frequency of BC. With the recurrence formulae for the Hermite polynomials eq. (4) Fan be reduced to +(<) = = $+[5 + (2e/K)8], where the potential depres;;ion E is [F(x)12/2K. In the displaced functions, the center of the polynomials is sh-Xted fro-m 5-.& = 0 to (2t,/K)+, $0 that the Hermite polynomials H, [t + (2~./a~] can be expressed in terms of a series of K.-Z (5) with proper coefficients by choosinglthe density function to be exp[-52 - 2(2e/K)? <]. Such an expansion gives the following coefficients:
c::, =
I
x
F(tz+l ;n-nz+l( -Eo); >I2=zn
exp&o)
-l(m-n) eg
1 (m-n)!
s,+
(6)
( n! /
x F,nz+l] m-n+1 1 -co);
?722 32,
where co is the energy stored in a classical oscillator subjected to the perturbing force F(X), divided by one quantum of energy for the true oscillator; i.e., E/Ew. Then the transition probability that the oscillator is in the state nz after it is perturbed by F(x) is given by pttm = 1CyPi 2. Replacing the confluent hypergeometric function by a series we have
Pam
= expkO)Eo
)m-nl
where k is the larger 126
2
1 nz’jt:
[
j_)
(-)j(k+j)!E: 2 (Im-n[+j):j:
of zu. ?z. This formula
1
,
(6)
sat-
1969
isfies detailed balancing. To apply this expression to molecular collisions, the magnitude of the transfer of energy to the oscillator needed to be calculated for the assumed form of F(X). For this purpose we consider the following classical picture. If we considered the time-dependent perturbing energy V(4, t), then the solution of the timedependent Schrrjdinger equation should be obtained for the problem. Treanor [S] has shown, based on Kerner’s solution [S], that the timedependent wave functions for a forced harmonic oscillator can be used to obtain an approximate expression of Pnm, which is essentially identical to eq. (6). Although the connecting formula may appear to be a complicated expression, there is in principle a simple correspondence between v(<,t) and V(<,x). In the solution of the timedependent SchrBdinger equation for V(t, t) = tF(t), the displacement A is a function of time and has a simple reIation to F(t) showing that the center of the wave packet moves as a classical oscillator would move if driven by F(t). The perturbing potential v(<,x) can then be transformed into V(<, f) by solving the equation of motion for x(t). With the resulting time-distance relationship we can identify E as the energy that would have been absorbed by a classical oscillator which is driven by v[<,x(t)]. In our model, therefore, V(< , x) is used as a transition-inducing perturbation acting upon a quantum oscillator, and relative motion in ‘he coordinate x is treated classically. Then the magnitude of the energy transferred is, in the limit t _ 03, E
x
March
LETTEPS
=&lw
2 [&
7%) -co
exp(iwt) ti]”
,
(7)
where cr is the reduced mass of the collision partners. The explicit solutions of this equation have been known for sevesal simple interaction models [1,7,8]. If the energy transferred to the oscillator is very small compared with a single quantum, the confluent hypergeometric series approaches unity, so that we obtain
l;-m
[(n-,n)!]-2
(n:/m!) ;
M c 72 (8)
_bnm = I
~~-?z[(n2-n):]-2(m!/,t!)
;
m 3 n.
Then, for n+n-1, p,r n_l NTZ~~, which is the usual perturbation re< for the simple harmonic oscillator. For n + n+l, pn, n+l = (n+l) lo . An interesting result is obtained from eq. (6), if the oscillator is initially in the ground state:
Volume 3. number 3
P om
=
CHEMICAL PHYSICS LETTERS
E: exp(-c,)/m!
,
(9)
which is the same expression that Treanor [5] obtained from the solution of the time-dependent SchrGdinger equation. For lo << 1, this expression becomes porn N ~r/rn!. Therefore, porn increases from zero to a maximum value at Eo = nz and then decreases as e. increases (i.e., as the perturbation increases). Note that on adding Porn for all possible values of 1)~we obtain exp(-e,) times the Taylor series for exp(e,) so that for any fixed energy e. the sum of the individual transition probabilities is unity. ln a recent paper on vibration-vibration-translation energy transfer between diatomic molecules, Zelechow, Rapp and Sharp [9]-showed that if both oscillators are initially in the ground state, the‘ total probability of excitation to 1)~. including all nz + 1 degenerate states of level ?n. has the same form as that given above for 0 _ 112except that e. is now replaced by 2~~. We also note that poln appears in the form of the well-known Poisson distribution. Therefore, we could alternatively obtain porn by treating the transition process as the time-dependent stochastic process [lo] in terms of the dynamic variable E . The essential step of the present approach is to obtain a set of the wave functions brn from eq. (2) followed by a particular linear combination of them with the determination of appropriate coefficients. The solution for pnm obtained from such an approach is interesting because the perturbation approach to the vibrational motion of BC with the second-order corrections gives the formula which may be used to describe vibration-translation and vibrationvibration-translation energy transfer processes in high energy collisions. Treanor obtained an expression essentially identical to eq. (6) and showed, by comparing it with Rapp and Sharp’s classical results [I], that it provides correcr
March 1969
answers for the energy transfer to a simple harmonic oscillator for collision velocities many times greater than those that can properly be used in the usual first-order perturbation treatment. Therefore, pnnI obtained by considering the relative coordinate as a dynamic variable is essentially identical with that formulated by considering the coordinates as a function of time, and has a potential applicability to high velocity collisions. A numerical check shows that the calculated values of pnm for various n and izz from eq. (6) with an explicit solution of eq. (7) agree reasonably well with Secrest and Johnson’s exact quantum mechanical (numerical) results [4] particularly at high collision velocities. A detailed comparison with the exact results for physically reasonable collision systems w-ill be pubIished in a future report.
REFERENCES [l]
D. Rapp and T. E. Sharp. J. Chem. Phys. 35 (L963) 2161. [2] Also see K.Takayanagi. Adv. At. Mol. Phys. 1 (1965) 149. 13) P. M. Morse and H. Feshbach. Methods of theoretical physics (McGraw-:Jill. New York. 1953) Ch. 7 of Vol. I and pp. 1645-1650 of Vol. f1. [Gl D. Secrest and B. R. Johnson, J. Chem. Phys. 45 (1966) 4556. 151C. E. Trcanor. J. Chem. Phys. 43 (1565) 532: 44 (1966) 2220. Can. J. Phys. 36 (1958) 371. I61 E.Kerner, [7l T. L. Cottrell and N. Ream. Trans. Faraday Sot. 51 (1955) 159; also see T. L. Cottrell and J. C. McCoubrey. Molecular energy transfer in gases (Butterworths, London. 1961) pp. 126-L29. I81 D. Rapp. J. Chem. Phys. 32 (1960) 735. I91 A. Zelechow. D, Rapp and T. E-Sharp, 6. Chem. Phys. 49 (1968) 286. II91 W. Feller, Introduction to probability theory and its applications, Vol. I (Wiley. New York. 1959) Ch. 17.
127
CHEMICAL PHYSICS LETTERS
PROBABILITY IN
OF HIGH
Department
VIBRATIONAL
ENERGY HYUNG WU
March
EXCITATIONS
COLLISIONS SHIN
of Chemistry, University Rena, Nevada, USA
1969
*
of Nevada,
Received 31 January 1969
Vibrational transition probability pnm for n-m is formulated from the solution of the time-independent Schr&iinger equation for colinear collision between an oscillator and a particle (an &om or a molecule). The resulting expression can he used for calculatingpnnl in high energy collisions.
At sufficiently high collision velocities, the usual perturbation calculation of vibrational transition probabilities p fails because the probabilities become too large [l, 21. For such high velocity collisions, however, we can formulate p from the solution of the time-independent SchrBdinger equation of the perturbed final oscillator (stationary) state considering the relative coordinate of the collision partners as ‘dynamic’ variables. In this letter we show such solutions for the colinear collision between an oscillator BC and an atom A starting with the Green function method. Morse and Feshbach [3] have shown the basic formulation of the Green function method for general perturbation and we extend it to the present molecular collision problem. The time-independent Schrijdinger equation for the collision system in which BC is hit along its line of oscillation by A with the perturbing potentiD V(<, x) which is a function of the distance x between centers of mass of the collision partners and the vibrational amplitude 4, is (tf2/2M) V(5) f [W- VCS>l q45) = V(5,x) q&), where &f= mBmC/(mB + 77Zc),v(t) is the iI’&raIIIOlecular potential, and W is the eigenvalue. We approximate BC by a harmonic osc;llator which isinane onential interaction wit2 A: i.e., V(5) = $K? , w h ere K is the force constant. For the colinear collision (BC-A) the instantaneous separation is z = x - ~5, where p = mg/(mg+mc). Then for an exponential interz&ion between C and A, we have V(z) =
= A exp( -z/a) = A exp(-x/a) exp(pS/u), where A and CLare potential parameters. When the range of the interacting force is large, we can write V(z) =A(1 +~
The solution of this equation can be found as arr eigenfunction szies,
We then see that, as a function of W. GW is analytic except for simple poles at W = W,. Thus, if it should happen that a Green function is known in a closed form, the eigenfunction @‘n and the corresponding eigesvalue W, may be found by investigating GW at its poles. In terms of Gw as kernel, the wave function is given by the integral equation
where Qn (5) is the unperturbed state. The sum represents the perturbed part of the wave function, and contains the simple harmonic osciIlator matrix elements:
work was carried out under Grant AFOSR-681354from the US Air Force Office of Scientific Research.
* ms
125
Volume 3. number 3
CHEMICAL
PHYSICS
The complete wave function is then a particular linear combination of the displaced functions obtained from eq. (2):
where E is the depression of the potential enqrgy caused by the perturbation. The term (2e/Xja is the displacement A of the equilibrium position of BC in the znth oscillator state. The normalization of e(5) has been adjusted so that Cn = 1. The probability of transition to the state m can then be derived from the coefficient C,. With the introdtlction of the harmonic oscillator wave functions and with the second-order corrections, we obtain from eq. (2)
{[ 4 (n+l)l+
&+1(5)
(4)
- (%)Q en-1 (51) ,
where w is the vibrational frequency of BC. With the recurrence formulae for the Hermite polynomials eq. (4) Fan be reduced to +(<) = = $+[5 + (2e/K)8], where the potential depres;;ion E is [F(x)12/2K. In the displaced functions, the center of the polynomials is sh-Xted fro-m 5-.& = 0 to (2t,/K)+, $0 that the Hermite polynomials H, [t + (2~./a~] can be expressed in terms of a series of K.-Z (5) with proper coefficients by choosinglthe density function to be exp[-52 - 2(2e/K)? <]. Such an expansion gives the following coefficients:
c::, =
I
x
F(tz+l ;n-nz+l( -Eo); >I2=zn
exp&o)
-l(m-n) eg
1 (m-n)!
s,+
(6)
( n! /
x F,nz+l] m-n+1 1 -co);
?722 32,
where co is the energy stored in a classical oscillator subjected to the perturbing force F(X), divided by one quantum of energy for the true oscillator; i.e., E/Ew. Then the transition probability that the oscillator is in the state nz after it is perturbed by F(x) is given by pttm = 1CyPi 2. Replacing the confluent hypergeometric function by a series we have
Pam
= expkO)Eo
)m-nl
where k is the larger 126
2
1 nz’jt:
[
j_)
(-)j(k+j)!E: 2 (Im-n[+j):j:
of zu. ?z. This formula
1
,
(6)
sat-
1969
isfies detailed balancing. To apply this expression to molecular collisions, the magnitude of the transfer of energy to the oscillator needed to be calculated for the assumed form of F(X). For this purpose we consider the following classical picture. If we considered the time-dependent perturbing energy V(4, t), then the solution of the timedependent Schrrjdinger equation should be obtained for the problem. Treanor [S] has shown, based on Kerner’s solution [S], that the timedependent wave functions for a forced harmonic oscillator can be used to obtain an approximate expression of Pnm, which is essentially identical to eq. (6). Although the connecting formula may appear to be a complicated expression, there is in principle a simple correspondence between v(<,t) and V(<,x). In the solution of the timedependent SchrBdinger equation for V(t, t) = tF(t), the displacement A is a function of time and has a simple reIation to F(t) showing that the center of the wave packet moves as a classical oscillator would move if driven by F(t). The perturbing potential v(<,x) can then be transformed into V(<, f) by solving the equation of motion for x(t). With the resulting time-distance relationship we can identify E as the energy that would have been absorbed by a classical oscillator which is driven by v[<,x(t)]. In our model, therefore, V(< , x) is used as a transition-inducing perturbation acting upon a quantum oscillator, and relative motion in ‘he coordinate x is treated classically. Then the magnitude of the energy transferred is, in the limit t _ 03, E
x
March
LETTEPS
=&lw
2 [&
7%) -co
exp(iwt) ti]”
,
(7)
where cr is the reduced mass of the collision partners. The explicit solutions of this equation have been known for sevesal simple interaction models [1,7,8]. If the energy transferred to the oscillator is very small compared with a single quantum, the confluent hypergeometric series approaches unity, so that we obtain
l;-m
[(n-,n)!]-2
(n:/m!) ;
M c 72 (8)
_bnm = I
~~-?z[(n2-n):]-2(m!/,t!)
;
m 3 n.
Then, for n+n-1, p,r n_l NTZ~~, which is the usual perturbation re< for the simple harmonic oscillator. For n + n+l, pn, n+l = (n+l) lo . An interesting result is obtained from eq. (6), if the oscillator is initially in the ground state:
Volume 3. number 3
P om
=
CHEMICAL PHYSICS LETTERS
E: exp(-c,)/m!
,
(9)
which is the same expression that Treanor [5] obtained from the solution of the time-dependent SchrGdinger equation. For lo << 1, this expression becomes porn N ~r/rn!. Therefore, porn increases from zero to a maximum value at Eo = nz and then decreases as e. increases (i.e., as the perturbation increases). Note that on adding Porn for all possible values of 1)~we obtain exp(-e,) times the Taylor series for exp(e,) so that for any fixed energy e. the sum of the individual transition probabilities is unity. ln a recent paper on vibration-vibration-translation energy transfer between diatomic molecules, Zelechow, Rapp and Sharp [9]-showed that if both oscillators are initially in the ground state, the‘ total probability of excitation to 1)~. including all nz + 1 degenerate states of level ?n. has the same form as that given above for 0 _ 112except that e. is now replaced by 2~~. We also note that poln appears in the form of the well-known Poisson distribution. Therefore, we could alternatively obtain porn by treating the transition process as the time-dependent stochastic process [lo] in terms of the dynamic variable E . The essential step of the present approach is to obtain a set of the wave functions brn from eq. (2) followed by a particular linear combination of them with the determination of appropriate coefficients. The solution for pnm obtained from such an approach is interesting because the perturbation approach to the vibrational motion of BC with the second-order corrections gives the formula which may be used to describe vibration-translation and vibrationvibration-translation energy transfer processes in high energy collisions. Treanor obtained an expression essentially identical to eq. (6) and showed, by comparing it with Rapp and Sharp’s classical results [I], that it provides correcr
March 1969
answers for the energy transfer to a simple harmonic oscillator for collision velocities many times greater than those that can properly be used in the usual first-order perturbation treatment. Therefore, pnnI obtained by considering the relative coordinate as a dynamic variable is essentially identical with that formulated by considering the coordinates as a function of time, and has a potential applicability to high velocity collisions. A numerical check shows that the calculated values of pnm for various n and izz from eq. (6) with an explicit solution of eq. (7) agree reasonably well with Secrest and Johnson’s exact quantum mechanical (numerical) results [4] particularly at high collision velocities. A detailed comparison with the exact results for physically reasonable collision systems w-ill be pubIished in a future report.
REFERENCES [l]
D. Rapp and T. E. Sharp. J. Chem. Phys. 35 (L963) 2161. [2] Also see K.Takayanagi. Adv. At. Mol. Phys. 1 (1965) 149. 13) P. M. Morse and H. Feshbach. Methods of theoretical physics (McGraw-:Jill. New York. 1953) Ch. 7 of Vol. I and pp. 1645-1650 of Vol. f1. [Gl D. Secrest and B. R. Johnson, J. Chem. Phys. 45 (1966) 4556. 151C. E. Trcanor. J. Chem. Phys. 43 (1565) 532: 44 (1966) 2220. Can. J. Phys. 36 (1958) 371. I61 E.Kerner, [7l T. L. Cottrell and N. Ream. Trans. Faraday Sot. 51 (1955) 159; also see T. L. Cottrell and J. C. McCoubrey. Molecular energy transfer in gases (Butterworths, London. 1961) pp. 126-L29. I81 D. Rapp. J. Chem. Phys. 32 (1960) 735. I91 A. Zelechow. D, Rapp and T. E-Sharp, 6. Chem. Phys. 49 (1968) 286. II91 W. Feller, Introduction to probability theory and its applications, Vol. I (Wiley. New York. 1959) Ch. 17.
127