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Opacity analysis of inelastic molecular collisions: I. The quantum mechanical sudden approximation
Volume 4. number 4
CREMICAL PHYSICS LETTERS
OPACITY I.
OF
ANALYSIS
THE
QUANTUM
INELASTIC
MECHANICAL
I_November 1969
MOLECULAR SUDDEN
COLLISIONS:
APPROXIMATION
R. D. LEVINE * Department of Chemistry. The Ohio State Uniwrsity Cohmbus. Ohio #3X0. USA
.
Received 15 September 1969
The quantum mechanical “sudden” approximation, valid in the limit of smali momentum transfer, is discussed and applied to the quenching of glory extrema in rotational excitation. Physically the approximation corresponds to the rotaticnal degrees of freedom being “frozen” during the mutual (elastic) scattering.
1. INTRODUCTION
2. THE SUDDEN APPKOXIMATION
The “sudden” approximation, designed in particular to handle rotational transitions in heavy molecules, is valid in the impulsive limit Ap <
The Hamiltonian for a non-reactive molecular collision, on an electronically adiabatic potentiaL
the semiclassical, short wavelength regime, o, the range of the potential, is much larger than the wavelength, po >> A, and the quantum mechanical approximation reduces to the well studied [l-3 J semi-classical Qa
sudden
approximation,
Ln terms of the velocity
for
?Jand
the energy AE converted into internal excitation the sudden limit is (u/v)
+ Also at the Depsrtmer;t of Physical Chemistry. Hebrew University,
Jerusalem.
Israel.
The
energy surface ** can be written as
Here K is the kinetic energy operator for the relative motion, h is the Hamiltonian of the internal degrees of freedom of the non-interacting molecules and V( r,R) is the mutual interaction (which vanishes at large relative separations). R and r are the relative separation and the set of internal degrees of freedom, respectively_ lixitiaiand
final states in the collision are characterized by the relative momentum Kk,2and a specified inter-
nal state n. The scattering amplitude between two states is given by [SJ, f l,r,n(knr,kn)
= -(a/2i3~2,(~~~k~,) flk&-
W
Here J.Lis the reduced mass and T is the transition operator, T =
ql + (E+-K-h-
V)-%],
(2)
where E is the total energy, Ei = E i ie, E > 0 and E -t 0 after the evaluation of the matrix elements. In the adiabatic limit [lo] one approximates T by assuming that the internal states of the
** These restrictions simpler notation.
are not essential,
but do Iead to
211
Volume 4. number 4
molecules
CHEMICAL PHYSICSLETTERS
adjust locally
to the perturbation,
one solves for the internal motion at a given.
i.e.
“clamped” separation and then solves for the relative separation_ Enthe impulsive limit one solves the problem of the relative motion for a given (fixed) internal configuration and then solves for the internal rearrangement *. In the impulsive limit, when the energy converted from the kinetic energy into internal excitation is small, one replaces the operator E-h by an effective kinetic energy E-E,,, where En is the internal energy of the state rz, Ts = y1+
(E+-,?z,-K-
V)-l V].
In eq_ (31, r is regarded as a parameter the solution for R is considered**. In terms of the R solution we obtain 7W,.
k&
= (k,
while
(4)
V)‘1 = VJ-l=(E+-El,,-K-
y)-l,
v)-‘[l
+ (E,,-lz)(E+-It-K-
I’)-l]. (6)
Using ey. (6) in
eq. (2) we get
* The description is only schematic. One can include some of the coordinates in the set I (i.e.. vibrations) in the set of coordinates Rfor which the solution !s obtained first. ** In the opposite (adiabatic) limit, R is regarded as the parameter when the solution for t is obtained. .212
=
x
x(E,+z)(E+-E,-K)-~T~+...
,
(7)
where (E+-E,-K-~,-‘V=(E+-E,-K)-‘Ts.
Introducing the spectral
(8)
resolution
each excitation tion probability
section is essen-
- E~z, where energy is weighted by its transifrom the initial state.
In the lowest (Born) approximation. if we repIace Ts by V in eq. (4) we obtain an approximation analogous to the impulse approximation of For molecular
convenient (and desirable) V(r.R)
problems
to resolve
= V#?)
+ V&R)
it is often V( r. A?) as
*
(9)
where Vl(lp) is independent of r?. In this case one can rewrite eq. (4) astt TSUi,,
, k,,i r) = (k,
1T"zlk& ,
[9a]
(10)
where T”z = V&l
+ (E+-E/-
V)-’
V2]
(11)
and /k;> = [I + (E*-EB
-K-
V#
Vl]jk,,).
(12)
Thus, the states /kE) are distorted waves, obtained from the plane waves under the action of
the distorting potential VI_ To first order, one can replace by V2.
I”2 in eq (10)
@(&*k,lIr)
list order = (k&l V2lkli. (13) limit Ts(t) can be computed using the short wavelength approximation in the semiclassical
x[l+(E,-h)(E+-E,z--K-V)-l+...]= =(E+-E,,-K-
= T” + TS(Ef - En -K)-1
Chew [IlJ
k,/r)(n)r.
In general, the coordinates in the set r should include only those internal degrees of freedom for which the impulsive limit obtains, i.e.. those for which the periods of motion are. large compared with the duration of the collision (that is oAfi<< E). This condition is often valid for rotational degrees of freedom for heavy (no hydrogen) diatomics at kinetic energies of chemical interest. The interpretation of Ts(k,,z, knl r) as an “elastic” (k, = kn) scattering amplitude for a fixed r is valid provided the impulsive limit obtains for all internal coordinates (as is ihe case for rotational excitation of a heavy rigid rotor). Correction terms to eq. (3) are obtainable from the expansion
=[E+-E,z+(E,,-it)-K-
x (En -h) (E+ -En-K-V)++...
and using the optical Theorem (eq. (15)) one finds
(5)
(E+-h-K-
n-lx
that the error in the total cross tially a weighted average of E,
tfit k&,
= -(~/2r&(mfZ?k,,
+ V(E+-S-K-
(3)
where the Dirac brackets denote integration over R only. The “sudden” scattering amplitude is thus given by f$Jk,,k,J
T =TS
1 November 1969
[12J. Here, Ss( r) = exp[-(i/k,)
$ V( r. Z+) dR]
Where the integration is over the classical
,
(14)
tra-
? More generally, VI is a Wdistorting” potential [S]. r.e., a potential that csnnot induce n - ?rztransitions. #‘t The proof of eq. (10) is essentially identical to that of eq. 2.5.65 of ref. [8]. The impulsive condition has
been invoked in the proof by requiring that E, =E,
Volume 4. numher 4
CHEMICAL
PHYSICS
jectory and yv,, = tZk,z_ The short wavelength approximation thus reduces eq. (5) to the usual sudden approximation. An example of the use of eq. (14) in eq. (5) was presented in detail in ref. [l]. The semiclassical limit of the procedure based on using eq. (10) has been previously discussed [13]. Differential and total cross sections are obtained from eq. (5) in the usual fashion. In particular, the total cross section from a given initial state is determined by the optical theorem
PI as
1 November
1969
the total cross section is linear in the amplitude (cf. eq. (15)). The degree of quenching 1 - cl/u, (notation of ref. [5]) can be obtained to order c/E by assuming that the 0 dependence oE Ao is essentially contained in ~in(26~(0) - 3ir/4). As an example we consider a long range anisotropy (Cl2 = I, cg = 1 f 92 (cos 0)) for an initial state of j = 0. EZxpanding 6g(f3) - 6 (0) to lowest order, we obtain in agreement w;lt -% previous work [5.6] l-
i&o)
= 0.14 a&R&v)2
(201
Our solution for rotationai excitation of a rigid rotor, using a partial wave expansion. results in an approximation that is equivalent to the Lowest order approximation of Curtis~ [15] and to the intermediate coupling approximation of Levine, Johnson and Bernstein [x6].
3. ROTATIONAL
EXCITATION
For the specific case oE rotational transitions in a collision of a (structureless) atom with a heavy (rigid rotor) diatomic molecule, we take T to be the orientation coordinate of the rigid rotor. Assuming as usual a potential of the type V(R, 6), Ts( kjr, k&) is simply the elastic (forward) transition amplitude for scattering by the potential V(R, 6) (which depends parametrically on 0). The total cross section (for a given 0) can then be written, as usual [14], as a sum of a Schiff- Landau- Lifshitz contribution and a glory term. Taking the potential as v = ~[C12(tWR,12/&2
- 2C6(Q) (R,&@] (15)
we get u=(e) = a;LL(e)
t Aan(
(17)
where Au is the oscillatory (in terms of velocity dependence) contribution to an and both terms are evaluated in the usual way. with a potential that is 0 dependent. Thus
and the contribution of the potential well is in the glory term Aan( determined, as is the case in ordinary potential scattering, by the (here 0 dependent) glory phase, 6g(0) = 0.473(2R,,I/fi~)~(0)
+ 0+/E).
(19)
c(e) is the effective depth of the potential (16) near its equilibrium position. The averaging over 0 can be done in the final stage, since Here
LETTEPS
ACKNOWLEDGEMENT I
would like to acknowledge
cussions
with Dr.
the benefit of dis-
B. R. Johnson.
REFERENCES [I] K. H-Kramer and R. B. Bernstein, J. Chem. Phys. 40 (1964) 200; R. B. Bernstein and K. H.Kramer, J. Chem. P&s. 44 (1966) 4473. 121 _ _ H. Paulv and J. P. Toennies. Advan. .&om. UoI. Phys. i (1965) 195; J. P. Toennies, 2. Physik 193 (1966) 76. [31 R.J. Cross Jr., J. Chem. Phgs. 47 (i967) 3724; 49 (1968) 1753. [IJ R. K. Helbing and E. W. Rothe, J Chem. Phys. 46 (1968) 3915. [5J Ft. E.Olson and R. B. Bernstein, J. Chem. Phys. 50 (1969)246. [6) R.J. Cross Jr., J. Chem. Phys. 49 (1968) l976. [T] W. H. Miller, J. Chem. Phys. 50 (1969) 3224. [8] R. D. Levine, Quantum mechanics of molecu:ular rate processes (Otiord University Press. Oxford. 1969). [9j a. R: D. Levine, to be published: b. R. D. Levine and R. B. Bernstein. to be published. [lOI R. D. Levine, J. Chcm. Phys. 46 (1967) 331; 49 (1968) 51; 50 (1969) 1. [llJ G. Chew. Phys. Rev. SO (1950) 196. [12] R. J. Glauber, Lectures Theor. Phys. i (1959) 3L5. 113) R. D. Levine, Chem. Phys. Letters 2 (1968) 76. [I41 R. B. Bernstein, Advan. Chem. Phys. LO (L966) 75. [15] C. F. Curtiss. J. Chem. Phys. 49 (1966) 1952. [16] R. 13.Levine, B. R. Johnson and R. B-Bernstein. University of Wisconsin, Theoreticat Chemistry Institute Report WIS-TCI-305 (1968).
_ 213
CREMICAL PHYSICS LETTERS
OPACITY I.
OF
ANALYSIS
THE
QUANTUM
INELASTIC
MECHANICAL
I_November 1969
MOLECULAR SUDDEN
COLLISIONS:
APPROXIMATION
R. D. LEVINE * Department of Chemistry. The Ohio State Uniwrsity Cohmbus. Ohio #3X0. USA
.
Received 15 September 1969
The quantum mechanical “sudden” approximation, valid in the limit of smali momentum transfer, is discussed and applied to the quenching of glory extrema in rotational excitation. Physically the approximation corresponds to the rotaticnal degrees of freedom being “frozen” during the mutual (elastic) scattering.
1. INTRODUCTION
2. THE SUDDEN APPKOXIMATION
The “sudden” approximation, designed in particular to handle rotational transitions in heavy molecules, is valid in the impulsive limit Ap <
The Hamiltonian for a non-reactive molecular collision, on an electronically adiabatic potentiaL
the semiclassical, short wavelength regime, o, the range of the potential, is much larger than the wavelength, po >> A, and the quantum mechanical approximation reduces to the well studied [l-3 J semi-classical Qa
sudden
approximation,
Ln terms of the velocity
for
?Jand
the energy AE converted into internal excitation the sudden limit is (u/v)
+ Also at the Depsrtmer;t of Physical Chemistry. Hebrew University,
Jerusalem.
Israel.
The
energy surface ** can be written as
Here K is the kinetic energy operator for the relative motion, h is the Hamiltonian of the internal degrees of freedom of the non-interacting molecules and V( r,R) is the mutual interaction (which vanishes at large relative separations). R and r are the relative separation and the set of internal degrees of freedom, respectively_ lixitiaiand
final states in the collision are characterized by the relative momentum Kk,2and a specified inter-
nal state n. The scattering amplitude between two states is given by [SJ, f l,r,n(knr,kn)
= -(a/2i3~2,(~~~k~,) flk&-
W
Here J.Lis the reduced mass and T is the transition operator, T =
ql + (E+-K-h-
V)-%],
(2)
where E is the total energy, Ei = E i ie, E > 0 and E -t 0 after the evaluation of the matrix elements. In the adiabatic limit [lo] one approximates T by assuming that the internal states of the
** These restrictions simpler notation.
are not essential,
but do Iead to
211
Volume 4. number 4
molecules
CHEMICAL PHYSICSLETTERS
adjust locally
to the perturbation,
one solves for the internal motion at a given.
i.e.
“clamped” separation and then solves for the relative separation_ Enthe impulsive limit one solves the problem of the relative motion for a given (fixed) internal configuration and then solves for the internal rearrangement *. In the impulsive limit, when the energy converted from the kinetic energy into internal excitation is small, one replaces the operator E-h by an effective kinetic energy E-E,,, where En is the internal energy of the state rz, Ts = y1+
(E+-,?z,-K-
V)-l V].
In eq_ (31, r is regarded as a parameter the solution for R is considered**. In terms of the R solution we obtain 7W,.
k&
= (k,
while
(4)
V)‘1 = VJ-l=(E+-El,,-K-
y)-l,
v)-‘[l
+ (E,,-lz)(E+-It-K-
I’)-l]. (6)
Using ey. (6) in
eq. (2) we get
* The description is only schematic. One can include some of the coordinates in the set I (i.e.. vibrations) in the set of coordinates Rfor which the solution !s obtained first. ** In the opposite (adiabatic) limit, R is regarded as the parameter when the solution for t is obtained. .212
=
x
x(E,+z)(E+-E,-K)-~T~+...
,
(7)
where (E+-E,-K-~,-‘V=(E+-E,-K)-‘Ts.
Introducing the spectral
(8)
resolution
each excitation tion probability
section is essen-
- E~z, where energy is weighted by its transifrom the initial state.
In the lowest (Born) approximation. if we repIace Ts by V in eq. (4) we obtain an approximation analogous to the impulse approximation of For molecular
convenient (and desirable) V(r.R)
problems
to resolve
= V#?)
+ V&R)
it is often V( r. A?) as
*
(9)
where Vl(lp) is independent of r?. In this case one can rewrite eq. (4) astt TSUi,,
, k,,i r) = (k,
1T"zlk& ,
[9a]
(10)
where T”z = V&l
+ (E+-E/-
V)-’
V2]
(11)
and /k;> = [I + (E*-EB
-K-
V#
Vl]jk,,).
(12)
Thus, the states /kE) are distorted waves, obtained from the plane waves under the action of
the distorting potential VI_ To first order, one can replace by V2.
I”2 in eq (10)
@(&*k,lIr)
list order = (k&l V2lkli. (13) limit Ts(t) can be computed using the short wavelength approximation in the semiclassical
x[l+(E,-h)(E+-E,z--K-V)-l+...]= =(E+-E,,-K-
= T” + TS(Ef - En -K)-1
Chew [IlJ
k,/r)(n)r.
In general, the coordinates in the set r should include only those internal degrees of freedom for which the impulsive limit obtains, i.e.. those for which the periods of motion are. large compared with the duration of the collision (that is oAfi<< E). This condition is often valid for rotational degrees of freedom for heavy (no hydrogen) diatomics at kinetic energies of chemical interest. The interpretation of Ts(k,,z, knl r) as an “elastic” (k, = kn) scattering amplitude for a fixed r is valid provided the impulsive limit obtains for all internal coordinates (as is ihe case for rotational excitation of a heavy rigid rotor). Correction terms to eq. (3) are obtainable from the expansion
=[E+-E,z+(E,,-it)-K-
x (En -h) (E+ -En-K-V)++...
and using the optical Theorem (eq. (15)) one finds
(5)
(E+-h-K-
n-lx
that the error in the total cross tially a weighted average of E,
tfit k&,
= -(~/2r&(mfZ?k,,
+ V(E+-S-K-
(3)
where the Dirac brackets denote integration over R only. The “sudden” scattering amplitude is thus given by f$Jk,,k,J
T =TS
1 November 1969
[12J. Here, Ss( r) = exp[-(i/k,)
$ V( r. Z+) dR]
Where the integration is over the classical
,
(14)
tra-
? More generally, VI is a Wdistorting” potential [S]. r.e., a potential that csnnot induce n - ?rztransitions. #‘t The proof of eq. (10) is essentially identical to that of eq. 2.5.65 of ref. [8]. The impulsive condition has
been invoked in the proof by requiring that E, =E,
Volume 4. numher 4
CHEMICAL
PHYSICS
jectory and yv,, = tZk,z_ The short wavelength approximation thus reduces eq. (5) to the usual sudden approximation. An example of the use of eq. (14) in eq. (5) was presented in detail in ref. [l]. The semiclassical limit of the procedure based on using eq. (10) has been previously discussed [13]. Differential and total cross sections are obtained from eq. (5) in the usual fashion. In particular, the total cross section from a given initial state is determined by the optical theorem
PI as
1 November
1969
the total cross section is linear in the amplitude (cf. eq. (15)). The degree of quenching 1 - cl/u, (notation of ref. [5]) can be obtained to order c/E by assuming that the 0 dependence oE Ao is essentially contained in ~in(26~(0) - 3ir/4). As an example we consider a long range anisotropy (Cl2 = I, cg = 1 f 92 (cos 0)) for an initial state of j = 0. EZxpanding 6g(f3) - 6 (0) to lowest order, we obtain in agreement w;lt -% previous work [5.6] l-
i&o)
= 0.14 a&R&v)2
(201
Our solution for rotationai excitation of a rigid rotor, using a partial wave expansion. results in an approximation that is equivalent to the Lowest order approximation of Curtis~ [15] and to the intermediate coupling approximation of Levine, Johnson and Bernstein [x6].
3. ROTATIONAL
EXCITATION
For the specific case oE rotational transitions in a collision of a (structureless) atom with a heavy (rigid rotor) diatomic molecule, we take T to be the orientation coordinate of the rigid rotor. Assuming as usual a potential of the type V(R, 6), Ts( kjr, k&) is simply the elastic (forward) transition amplitude for scattering by the potential V(R, 6) (which depends parametrically on 0). The total cross section (for a given 0) can then be written, as usual [14], as a sum of a Schiff- Landau- Lifshitz contribution and a glory term. Taking the potential as v = ~[C12(tWR,12/&2
- 2C6(Q) (R,&@] (15)
we get u=(e) = a;LL(e)
t Aan(
(17)
where Au is the oscillatory (in terms of velocity dependence) contribution to an and both terms are evaluated in the usual way. with a potential that is 0 dependent. Thus
and the contribution of the potential well is in the glory term Aan( determined, as is the case in ordinary potential scattering, by the (here 0 dependent) glory phase, 6g(0) = 0.473(2R,,I/fi~)~(0)
+ 0+/E).
(19)
c(e) is the effective depth of the potential (16) near its equilibrium position. The averaging over 0 can be done in the final stage, since Here
LETTEPS
ACKNOWLEDGEMENT I
would like to acknowledge
cussions
with Dr.
the benefit of dis-
B. R. Johnson.
REFERENCES [I] K. H-Kramer and R. B. Bernstein, J. Chem. Phys. 40 (1964) 200; R. B. Bernstein and K. H.Kramer, J. Chem. P&s. 44 (1966) 4473. 121 _ _ H. Paulv and J. P. Toennies. Advan. .&om. UoI. Phys. i (1965) 195; J. P. Toennies, 2. Physik 193 (1966) 76. [31 R.J. Cross Jr., J. Chem. Phgs. 47 (i967) 3724; 49 (1968) 1753. [IJ R. K. Helbing and E. W. Rothe, J Chem. Phys. 46 (1968) 3915. [5J Ft. E.Olson and R. B. Bernstein, J. Chem. Phys. 50 (1969)246. [6) R.J. Cross Jr., J. Chem. Phys. 49 (1968) l976. [T] W. H. Miller, J. Chem. Phys. 50 (1969) 3224. [8] R. D. Levine, Quantum mechanics of molecu:ular rate processes (Otiord University Press. Oxford. 1969). [9j a. R: D. Levine, to be published: b. R. D. Levine and R. B. Bernstein. to be published. [lOI R. D. Levine, J. Chcm. Phys. 46 (1967) 331; 49 (1968) 51; 50 (1969) 1. [llJ G. Chew. Phys. Rev. SO (1950) 196. [12] R. J. Glauber, Lectures Theor. Phys. i (1959) 3L5. 113) R. D. Levine, Chem. Phys. Letters 2 (1968) 76. [I41 R. B. Bernstein, Advan. Chem. Phys. LO (L966) 75. [15] C. F. Curtiss. J. Chem. Phys. 49 (1966) 1952. [16] R. 13.Levine, B. R. Johnson and R. B-Bernstein. University of Wisconsin, Theoreticat Chemistry Institute Report WIS-TCI-305 (1968).
_ 213