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Internal excitation in non-reactive molecular collissions: resonances in elastic scattering of atoms by diatomic molecules
CHEMICAL PHYSICS LETTERS
INTERNAL
1 (1968) 517-520. NORTH-HOLLAND
PUBLISHING COMPANY.
AMSTERDAM
EXCITATION IN NON-REACTIVE MOLECULAR COLLISIONS: RESONANCES IN ELASTIC SCATTERING OF ATOMS BY DIATOMIC MOLECULES l R. D. LEVINE **,
B. R. JOHNSON, J. T. MUCKEZMAN
Theoretical
Ckemistvy
Instittie
and R. Bz BERNSTEIN
and Ckemi.st?y Department
University of WSCORS~~, Madison.
Wisconsin.
USA
Received 11 December 1967
Resonances in slow molecular collisions are demonstrated by exact numerical solution of the closecoupled’equations for r&&ion4 excitation. including closed channels. The breakdown of the adiabatic approximation is exhibited. Resonance energies can be essentially predicte-a by a “best local” approximation. describin the collision expanded ‘i
The formal theory of collisions of systems with internal degrees of freedom has been the subject of considerable discussion [1,2]. In particular, for the case of low (sub-excitation) energy molecular collisions one expects the formation of temporarily bound systems due to internal excitation and de-excitation during the collision [3]. For collisions involving diatomic molecules at energies below the threshold for rotational excitation, Chen [4] and Micha [5] have used the Breit-Wigner type parametrization of the general theory to suggest the occurrence of resonances in the energy dependence of the cross section. As yet, no exact solution of the problem of coupling between open and closed channels appears to have been carried out in the molecular case. We report here exact numerical results for a model problem involving the collision of a structureless atom with a rigid rotor at sub-threshold energies, which exhibits the main features to be expected in a real system. These results then serve as a reference against which we compare various approximate theoretical predictions of the resonance energies and several variational principles [6,7] for the phase shift. The theory may be briefly outlined as follows. The total wave function ?Z$R, 2) for the system
2 As the size N of the expansion basis is enIarged. a better description of the physical situation is ob-
tained. From a formal point of view the truncated expansion can be derived from a variztiond principle 171. However, a given basis set determfnes a (finite dimensional) Hilbert space for the internal states of the rotor, within which our sob&ion is exact. *t This representation is discussed by Blatt and Biedenham [S]. It has been applied to the differentiaI equations formulation of the present problem by Arthurs and Dalgarno [9]. The integral equations form is discussed in section 2.31 of ref. [2] snd in ref. [lo].
National Aeronautics znd Space Administration Grant NsG-275-62 and the National Science Fourdation Grant GP-7409. ** Present address: Department of Physical Chemistry,
The Hebrew University, Jerusalem, Israel. 1968
_
where the &‘s are the set of orthonormal internal states of the rotor (spherical harmonics). Here Ip is the separation between atom and centre-of-mass of the rotor and r^ the orientation coordinate of the rotor. Substituting (1) in the Lippmann-Schwinger equation one obtains a set of coupled integral equations for the F,(R) that describe the relative motion. Upon introduction of the total angular momentum representation $$ these equations can be reduced to sets of coupfed radial integral equations, each characterized by a given channel designation (i and I, the rotational and orbital angu1a.r momenta) and each set corresponding to a given value of the (conserved) total angular momentum J (where d = i + I). A channel is closed when the rebtive kinetic energy is below the excitation threshold and the soktion of the relevant integral equation vanishes as
* This research received financial support from the
January
of the atom and rotor is
517
R-D. LEVINE, B. R. JOHNSON, J.T. MUCEERMAN and R. B. BERNSTEIN
518
R y 00. The numerical solution is carried out using the method of amplitude density functions
PI-
In the model problem the interaction potential was chosen to be the simplest realistic form exhibiting short-range anisotropy (for an atomhomonuclear diatomic svstem): V(R,
r’) = Vo(R) + u - I’,(@.
where Vo(R) is a L.-J.
2). 4~(o/R)~~
,
(2)
(12,6) potential:
Vo(R) = 46[(0/R)~~
- (u/Rj6:
(3)
and Q is the “asymmetry parameter”. The matrix elements of V in the total-angular momentum representation are readily evaluated 1111. Vo(R) has only diagonal elements, while the off-diagonal elements of the anisotropic part leads to excitation, with a playing the role of a coupling parameter. In the present communication we have limited consideration to J = j = 0 for energies E < Eth, where Eth is the threshold energy for the lowest allowed transition j -j’ (j’ = 2): Eth = B,hc[j’(
j’+l)
- j(j+l)]
,
(4)
where B,hc has its usual significance. Thus the SJ matrix is here just a single number, e216, where 6 is the phase shift. In the computations the clcsed channel (here j’=2, 2’=2) is retained in the expansion. It is this feature which gives rise to the resonance behavior. Fig; 1 shows the energy dependence of 6 for the stated parameters. The *static” approximation curve is based on a calculation in which only
one term in the expansion (1) is retained, SO no internal excitation during the collision is passible. The “adiabatic” curve is obtained in a way to be described below. The “exact” curve refers to the numerical integral equation solution as outlined above, in which the closed channel j’ = 2 is retained. The 3 sharp resonances, ZJ= 0, 1,2, clearly seen in the e&t curve, are, of course, absent in the approximate curves. As is -theoretically expected [6], adding terms to the expansion (1) increases the phase shift *, and the relative ordering 6exact
>
badiabatic
>
(5)
5static
is as predicted I?‘]. Fig. 2 shows the variation of the v = 1 resonance energy as a function of the asymmetry parameter a. The exact results are shown as points, with bars corresponding to the widths (=2/(dG/dE)) of th e resonances, which vary in a non-monotonic fashion over a range of more than 103. For the example chosen it is seen that the resonances are essentially “isolated”. Another interesting feature of the results is the existence of a maximum in the dependence of the resonance energy upon a in the region of negative a (to be discussed below). * This behavior has been confirmed also by consideration of the phase shift computed using an additional term (j’=4) in the expansion (1).
oe-
LEGEND C: Cmlml F%taliol Acpox. D: Ditloriim ktcntiil Appmr El_: Best Locot Potmtbi I
1
0.5
~:ExnctRescmce
1
ApPma.
Em?qy 6
Width
1.0
EfElh
Fig. 1. The phase shift B(E). The top curve is the result of the exact numerical computations, 0th~._ curves are approximations discussed in the text. The pgrameters in eqs. (2) and (3) are E = 6.3 B,hc, U= 3A. where Be = 1.2111 x 1O-14/hccm-1 is the rotational constant ~8 Hz. (Thus Z+h = GB,hc). The reduced mass for the relative motion that of He-Hz.
Fig. 2. The resonance
spectrum
as a function of the
asymmetry parameter ~1, for the v = I “bound state ” of the internally
excited
rotor
and the atom.
The three
curves are the results of approximate computations of
the resonance positions parameters
as discussed in the text; same as for fig. 1.
INTERNAL EXCITATION IN NON-REACTIVE MOLECULAR COLLISIONS Three approximations for the resonance energy are shown. In the “central” or zero-order approximation it is assumed that the potential, Vs, which supports the resonance is the isotropic mutual interaction between the excited (j’=2) rotor and the atom: Vi’)(R)
= Vo(R) + Etll
= v 0 (R) +
(7)
+ a_f2(j’, Z’,j’, Z’; J). 4e(o/Z2)12 + Eth , where the f2 coefficient is that of Percival and Seaton [12], here $. The rigenvalues of Vs(l)(R) are seen to have a significant dependence on the asymmetry parameter in the obvious direction. It is of interest to note that for negative values of a “curve crossing” can occur, i.e. V,(l)(R) can cross the (static) potential obtained by averaging V over the rotor in the ground (j = 0, so f2(j,Z,j, Z;J) =0) state. It seems that such curve crcssing is responsible for the failure of the distortion approximation for negative a and for the maximum in the resonance energies as function of Q. This behavior is also found for the other vibrational states (v =0,2). Other validations of this behavior are discussed elsewhere [13]. To obtain a still better approximation to V, one may attempt to construct a “best local” supporting potential Vs(R), as follows. In place of eq. (l), expand 9 as W?,
i) = E G,(R)x,(;[J?) n=l
(8)
such that, as R -m x,(PIW
-#,(F)
(9)
and hence * This approximation technique
is suggested by the partitioning 11.21 and corresponds to neglecting the
“level shift” which is a measure of the error of the assumption that the resonance corresponds to the rotor in the j’ =2 state.
It is determined
We now require that the functions x, be constructed as linear combination of the Qnfe
(6)
whose eigenvalues are a-independent. (This approximation was used in &he parametrization in ref. [5].) In the distortion or first order approximation the influence of the anisotropic part of the interaction is taken into account by averaging V over the internal coordinates of the rotor in thej’=2 state, i.e. * V(l)(R) S
519
by a non-local
potential and hence often assumed to be negligible.
so tbz! the integral /df
xi@]‘?)
IV?, 3) x,(a/f?)
(11)
added to Eth is the best local potential. it has been shown [2,3] that this is equivalent to determining x7t as eigenstates of h,,t + V, subject to (9). In the present case N= 2 and on diagonalizing Itrot + V one obtains two “potential energy curves” the upper of which is the “bestrr IocaL V,(R). Eigenvalues of this V,(R) are seen to be in fairly good agreement with the exactly computed resonance positions. In the conventional [2,3] adiabatic approximation only one term in eq. (8) is used and the lowest of the two eigenvalues of h,,t c V is the scattering (adiabatic) potential. Although the phase shifts calculated for scattering by the adiabatic potential are in fair agreement with the exact values (fig. l), and clearly superior to the static approximation resuIts, the adiabatic calculations cannot, in principle, yield an internal excitation resonance. A full report [13] includes a detailed discussion of these and further results obtained with larger
basis
sets.
REFERENCES [l] H. M. Mittleman and K. M. Watson. Phvs. Rev. 113 (1959) 198; U. Fano, Phys. Rev. 124 (1961) 1866; H. Feshbach, Ann. Phys. (N-Y.) 19 (1962) 287. PI For a review, see: R. D. Levine. Quantum Mechanits of Molecular Rate Processes (Oxford Universitv Press, oxford 1968) chapter 3.2. [31 R. D. Levine. J. Chem. Phys. 46-(1967) 331. For a simplified account see R. D. Levine. University of Wisconsin, Theoretical Chemistry Institute, Keport WIS-TCI-203 (1966). I41 J. C.Y. Chen, Phys. Rev. 146 (1966) 61. 139; ISI D.A. Micha. Chem. Phys. Letters l(l967) Phys. Rev. 162 (1967) 88. t61 Y. H&n, T. F. O’Malley and L.Spruch, Phys. Rev. 134 (1964) B397; B911; W-A. McKinley and J. H. Macek. Phys. Letters 10 (1964) 210. 171 See ref. [ZJ section 3.32. Rev. Mod. Phys. 181 J. M. Blatt and L. C. Biedenharn. 24 (1952) 258.
520
,
R. D. LEVIN--.
[9] A.M. Arthurs and A. D. Dalgarno,
B. R. JOHNSON, J. T. MUCKRRMAN
Proc. Roy. Sot. (London) A256 (1960) 540. [lo] B. R. Johnson, A New Method for Computing Inelastic Molecular Ssattering with Application to Atom-Diatomic and Diatomic-Diatomic Collisicliriz, Ph. D. Thesis, University of Illinois (1966). Also B. R. Johnson and D. R. Secrest, to be published.
[ll]
ard R-B. BERNSTEIN
See R. B. Bernstein, A. Dalgarno, H. S. W. Massey and I. C. Percival, Proc. Roy. Sot. (London) A274 (1963) 427. [12] I. C. Percival and M. J. Seaton, Proc. Cambridge Phil. Sot. 63 (1967) 654. [13J R.D. Levine, B.R. Johnson, J.T. Muckermau and R. B. Bernstein, to be published.
INTERNAL
1 (1968) 517-520. NORTH-HOLLAND
PUBLISHING COMPANY.
AMSTERDAM
EXCITATION IN NON-REACTIVE MOLECULAR COLLISIONS: RESONANCES IN ELASTIC SCATTERING OF ATOMS BY DIATOMIC MOLECULES l R. D. LEVINE **,
B. R. JOHNSON, J. T. MUCKEZMAN
Theoretical
Ckemistvy
Instittie
and R. Bz BERNSTEIN
and Ckemi.st?y Department
University of WSCORS~~, Madison.
Wisconsin.
USA
Received 11 December 1967
Resonances in slow molecular collisions are demonstrated by exact numerical solution of the closecoupled’equations for r&&ion4 excitation. including closed channels. The breakdown of the adiabatic approximation is exhibited. Resonance energies can be essentially predicte-a by a “best local” approximation. describin the collision expanded ‘i
The formal theory of collisions of systems with internal degrees of freedom has been the subject of considerable discussion [1,2]. In particular, for the case of low (sub-excitation) energy molecular collisions one expects the formation of temporarily bound systems due to internal excitation and de-excitation during the collision [3]. For collisions involving diatomic molecules at energies below the threshold for rotational excitation, Chen [4] and Micha [5] have used the Breit-Wigner type parametrization of the general theory to suggest the occurrence of resonances in the energy dependence of the cross section. As yet, no exact solution of the problem of coupling between open and closed channels appears to have been carried out in the molecular case. We report here exact numerical results for a model problem involving the collision of a structureless atom with a rigid rotor at sub-threshold energies, which exhibits the main features to be expected in a real system. These results then serve as a reference against which we compare various approximate theoretical predictions of the resonance energies and several variational principles [6,7] for the phase shift. The theory may be briefly outlined as follows. The total wave function ?Z$R, 2) for the system
2 As the size N of the expansion basis is enIarged. a better description of the physical situation is ob-
tained. From a formal point of view the truncated expansion can be derived from a variztiond principle 171. However, a given basis set determfnes a (finite dimensional) Hilbert space for the internal states of the rotor, within which our sob&ion is exact. *t This representation is discussed by Blatt and Biedenham [S]. It has been applied to the differentiaI equations formulation of the present problem by Arthurs and Dalgarno [9]. The integral equations form is discussed in section 2.31 of ref. [2] snd in ref. [lo].
National Aeronautics znd Space Administration Grant NsG-275-62 and the National Science Fourdation Grant GP-7409. ** Present address: Department of Physical Chemistry,
The Hebrew University, Jerusalem, Israel. 1968
_
where the &‘s are the set of orthonormal internal states of the rotor (spherical harmonics). Here Ip is the separation between atom and centre-of-mass of the rotor and r^ the orientation coordinate of the rotor. Substituting (1) in the Lippmann-Schwinger equation one obtains a set of coupled integral equations for the F,(R) that describe the relative motion. Upon introduction of the total angular momentum representation $$ these equations can be reduced to sets of coupfed radial integral equations, each characterized by a given channel designation (i and I, the rotational and orbital angu1a.r momenta) and each set corresponding to a given value of the (conserved) total angular momentum J (where d = i + I). A channel is closed when the rebtive kinetic energy is below the excitation threshold and the soktion of the relevant integral equation vanishes as
* This research received financial support from the
January
of the atom and rotor is
517
R-D. LEVINE, B. R. JOHNSON, J.T. MUCEERMAN and R. B. BERNSTEIN
518
R y 00. The numerical solution is carried out using the method of amplitude density functions
PI-
In the model problem the interaction potential was chosen to be the simplest realistic form exhibiting short-range anisotropy (for an atomhomonuclear diatomic svstem): V(R,
r’) = Vo(R) + u - I’,(@.
where Vo(R) is a L.-J.
2). 4~(o/R)~~
,
(2)
(12,6) potential:
Vo(R) = 46[(0/R)~~
- (u/Rj6:
(3)
and Q is the “asymmetry parameter”. The matrix elements of V in the total-angular momentum representation are readily evaluated 1111. Vo(R) has only diagonal elements, while the off-diagonal elements of the anisotropic part leads to excitation, with a playing the role of a coupling parameter. In the present communication we have limited consideration to J = j = 0 for energies E < Eth, where Eth is the threshold energy for the lowest allowed transition j -j’ (j’ = 2): Eth = B,hc[j’(
j’+l)
- j(j+l)]
,
(4)
where B,hc has its usual significance. Thus the SJ matrix is here just a single number, e216, where 6 is the phase shift. In the computations the clcsed channel (here j’=2, 2’=2) is retained in the expansion. It is this feature which gives rise to the resonance behavior. Fig; 1 shows the energy dependence of 6 for the stated parameters. The *static” approximation curve is based on a calculation in which only
one term in the expansion (1) is retained, SO no internal excitation during the collision is passible. The “adiabatic” curve is obtained in a way to be described below. The “exact” curve refers to the numerical integral equation solution as outlined above, in which the closed channel j’ = 2 is retained. The 3 sharp resonances, ZJ= 0, 1,2, clearly seen in the e&t curve, are, of course, absent in the approximate curves. As is -theoretically expected [6], adding terms to the expansion (1) increases the phase shift *, and the relative ordering 6exact
>
badiabatic
>
(5)
5static
is as predicted I?‘]. Fig. 2 shows the variation of the v = 1 resonance energy as a function of the asymmetry parameter a. The exact results are shown as points, with bars corresponding to the widths (=2/(dG/dE)) of th e resonances, which vary in a non-monotonic fashion over a range of more than 103. For the example chosen it is seen that the resonances are essentially “isolated”. Another interesting feature of the results is the existence of a maximum in the dependence of the resonance energy upon a in the region of negative a (to be discussed below). * This behavior has been confirmed also by consideration of the phase shift computed using an additional term (j’=4) in the expansion (1).
oe-
LEGEND C: Cmlml F%taliol Acpox. D: Ditloriim ktcntiil Appmr El_: Best Locot Potmtbi I
1
0.5
~:ExnctRescmce
1
ApPma.
Em?qy 6
Width
1.0
EfElh
Fig. 1. The phase shift B(E). The top curve is the result of the exact numerical computations, 0th~._ curves are approximations discussed in the text. The pgrameters in eqs. (2) and (3) are E = 6.3 B,hc, U= 3A. where Be = 1.2111 x 1O-14/hccm-1 is the rotational constant ~8 Hz. (Thus Z+h = GB,hc). The reduced mass for the relative motion that of He-Hz.
Fig. 2. The resonance
spectrum
as a function of the
asymmetry parameter ~1, for the v = I “bound state ” of the internally
excited
rotor
and the atom.
The three
curves are the results of approximate computations of
the resonance positions parameters
as discussed in the text; same as for fig. 1.
INTERNAL EXCITATION IN NON-REACTIVE MOLECULAR COLLISIONS Three approximations for the resonance energy are shown. In the “central” or zero-order approximation it is assumed that the potential, Vs, which supports the resonance is the isotropic mutual interaction between the excited (j’=2) rotor and the atom: Vi’)(R)
= Vo(R) + Etll
= v 0 (R) +
(7)
+ a_f2(j’, Z’,j’, Z’; J). 4e(o/Z2)12 + Eth , where the f2 coefficient is that of Percival and Seaton [12], here $. The rigenvalues of Vs(l)(R) are seen to have a significant dependence on the asymmetry parameter in the obvious direction. It is of interest to note that for negative values of a “curve crossing” can occur, i.e. V,(l)(R) can cross the (static) potential obtained by averaging V over the rotor in the ground (j = 0, so f2(j,Z,j, Z;J) =0) state. It seems that such curve crcssing is responsible for the failure of the distortion approximation for negative a and for the maximum in the resonance energies as function of Q. This behavior is also found for the other vibrational states (v =0,2). Other validations of this behavior are discussed elsewhere [13]. To obtain a still better approximation to V, one may attempt to construct a “best local” supporting potential Vs(R), as follows. In place of eq. (l), expand 9 as W?,
i) = E G,(R)x,(;[J?) n=l
(8)
such that, as R -m x,(PIW
-#,(F)
(9)
and hence * This approximation technique
is suggested by the partitioning 11.21 and corresponds to neglecting the
“level shift” which is a measure of the error of the assumption that the resonance corresponds to the rotor in the j’ =2 state.
It is determined
We now require that the functions x, be constructed as linear combination of the Qnfe
(6)
whose eigenvalues are a-independent. (This approximation was used in &he parametrization in ref. [5].) In the distortion or first order approximation the influence of the anisotropic part of the interaction is taken into account by averaging V over the internal coordinates of the rotor in thej’=2 state, i.e. * V(l)(R) S
519
by a non-local
potential and hence often assumed to be negligible.
so tbz! the integral /df
xi@]‘?)
IV?, 3) x,(a/f?)
(11)
added to Eth is the best local potential. it has been shown [2,3] that this is equivalent to determining x7t as eigenstates of h,,t + V, subject to (9). In the present case N= 2 and on diagonalizing Itrot + V one obtains two “potential energy curves” the upper of which is the “bestrr IocaL V,(R). Eigenvalues of this V,(R) are seen to be in fairly good agreement with the exactly computed resonance positions. In the conventional [2,3] adiabatic approximation only one term in eq. (8) is used and the lowest of the two eigenvalues of h,,t c V is the scattering (adiabatic) potential. Although the phase shifts calculated for scattering by the adiabatic potential are in fair agreement with the exact values (fig. l), and clearly superior to the static approximation resuIts, the adiabatic calculations cannot, in principle, yield an internal excitation resonance. A full report [13] includes a detailed discussion of these and further results obtained with larger
basis
sets.
REFERENCES [l] H. M. Mittleman and K. M. Watson. Phvs. Rev. 113 (1959) 198; U. Fano, Phys. Rev. 124 (1961) 1866; H. Feshbach, Ann. Phys. (N-Y.) 19 (1962) 287. PI For a review, see: R. D. Levine. Quantum Mechanits of Molecular Rate Processes (Oxford Universitv Press, oxford 1968) chapter 3.2. [31 R. D. Levine. J. Chem. Phys. 46-(1967) 331. For a simplified account see R. D. Levine. University of Wisconsin, Theoretical Chemistry Institute, Keport WIS-TCI-203 (1966). I41 J. C.Y. Chen, Phys. Rev. 146 (1966) 61. 139; ISI D.A. Micha. Chem. Phys. Letters l(l967) Phys. Rev. 162 (1967) 88. t61 Y. H&n, T. F. O’Malley and L.Spruch, Phys. Rev. 134 (1964) B397; B911; W-A. McKinley and J. H. Macek. Phys. Letters 10 (1964) 210. 171 See ref. [ZJ section 3.32. Rev. Mod. Phys. 181 J. M. Blatt and L. C. Biedenharn. 24 (1952) 258.
520
,
R. D. LEVIN--.
[9] A.M. Arthurs and A. D. Dalgarno,
B. R. JOHNSON, J. T. MUCKRRMAN
Proc. Roy. Sot. (London) A256 (1960) 540. [lo] B. R. Johnson, A New Method for Computing Inelastic Molecular Ssattering with Application to Atom-Diatomic and Diatomic-Diatomic Collisicliriz, Ph. D. Thesis, University of Illinois (1966). Also B. R. Johnson and D. R. Secrest, to be published.
[ll]
ard R-B. BERNSTEIN
See R. B. Bernstein, A. Dalgarno, H. S. W. Massey and I. C. Percival, Proc. Roy. Sot. (London) A274 (1963) 427. [12] I. C. Percival and M. J. Seaton, Proc. Cambridge Phil. Sot. 63 (1967) 654. [13J R.D. Levine, B.R. Johnson, J.T. Muckermau and R. B. Bernstein, to be published.