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Vibrational relaxation processes and molecular form factors
Volume
46A, number
PHYSICS
4
VIBRATIONAL
LETTERS
RELAXATION
MOLECULAR
31 December
PROCESSES
1973
AND
FORM FACTORS*
D. ROGOVIN Department of Physics and Optical Science Center, University of Arizona, Tucson, Arizona 85721. USA Received A theory that relates collision matrix elements systems to molecular form factors is presented.
11 October
for vibrational/rotational
In this letter we present a simple technique for calculating the matrix elements of the intermolecular potential between colliding molecules undergoing vibrational energy transfer or conversion processes. The theory of vibrational relaxation processes in diatomic or polyatomic molecular gases [ 1,2] depends critically on these matrix elements and to date, most calculations of (molecular) inelastic collision cross-sections have employed a set of questionable approximations in computing the relevant matrix elements [3-51. While these approximations greatly simplify the calculations, they lack sound justification and in some cases are not valid. For collisions between two molecular systems most calculations consider only 1) collinear collisions and 2) interactions between nearest neighbors. Finally, 3) the colliding potential is approximated by a simple repulsive exponential. The first ‘two approximations are kinematic and their effect is to eliminate all of the angular factors that would otherwise arise in the collision process. In order to simulate the full set of possible initial collision-orientations a steric factor PO = l/3 for each molecule is introduced into the transition amplitude. The second approximation is justified on the grounds that it is the short-range portion of the collision potential that dominates the dynamics of vibrational transfer and conversion processes. Unfortunately, the steric factor tends to vary with the colliding molecular species and there does not exist a reliable quantitative theory for calculating this factor for collisions involving non-linear polyatomic molecules [6]. The second approximation is clearly invalid for initial col-
%ork supported by Advanced and Kirtland Air Force Base.
Research
Projects
Agency
1973 transfer
processes
between
molecular
lision-orientations in which one atom of one molecule is very close to a number of atoms in the second molecule. Furthermore, one cannot investigate the possibility of rotational transfer or conversion occurring simultaneously with vibrational processes. Finally, in regards to these kinematic approximations, we note that Mies [7] has carried out a first-principle’s calculation using the full-angle-dependent potential for the excitation of hydrogen molecules by helium atoms. His results have raised grave doubts about the validity of 1) and 2). The dynamic approximation 3) enables one to write the intermolecular potential as a function of the center-of-mass distance times a function of the internal vibrational coordinates. There exists no sound justification of 3) and Mies [7] has shown that the assumption of additive exponential potentials acting between the atomic centers does not tit the results of his exact calculation. We assume that the potential can be approximated by u=zUii(r-Ri(“+Rp)). i,j ’
Here Ui,i is the interaction between atom i in polyatomic molecule A and atomj in polyatomic molecule B, r is the vector between the molecular center-ofmasses, Rp)(Rp) is the position vector of atom i(i) from the center-of-mass of molecule A(B). Eq. (1) assumes that the electronic degrees of freedom are not excited by the collision and in addition, that s = r - Ri(a) + Ri(b) is the only vector direction that is significant. Thus, one neglects [orientational] dipoledipole forces, etc., which are believed to dominate on259
Volume
46A, number
4
PHYSICS
ly the long-range portion of the intermolecular potential [8]. This part of the intermolecular force gives rise only to rotational processes and plays no role in vibrational relaxation. For situations in which the Born approximation applies we require the matrix elements of eq. (2) between the states Ik; Q, 0) and Ik’; (Y’,$)..Hcre k[k’] is the relative center-of-mass wave-vector before (after) the collision, CY(CX’) is the initial (final) rotation-vibration state of molecule A and /3(/I’) the same, but for molecule B. The relevant matrix element is then (k’;cu’, P’IUlk; a, P) =c Ui,Jq)F;_,( i. i
31 December
where Aq is the momentum transfer vector and Fk,,(-q) = (cu’lexp(--iq.R$%) is the form factor of atom i between molecular states ICI)and IQ’), at wave-vector 4. These quantities which also appear in neutron scattering depend only on the molecular structure, the initial and final rotation-vibration states and the wave-vector Q. They are independent of the colliding potential and for some transitions may be obtainable from neutron scattering [9] experiments. The factors U,i[q) are the Fourier transform of the atom-atom interaction. We note that this approach separates the internal dynamics of the molecule (i.e., the form factors Fi,, 1(-q)] from the center-ofmass motion (i.e., the Uii(4)) and is formally similar to the approach taken for inelastic collisions between fast electrons and atoms [lo]. Finally, we note that we have avoided all of the approximations mentioned above. For most molecular systems one cannot use the Born approximation for collisions occurring at thermal velocities and one must resort to other techniques. One approach is the Born approximation to the reactance matrix. This technique was used by
1973
Burke and Seaton [ 1 I ] to study e-H collision crosssections. Their efforts were quite successful and can be readily adapted to our problem. In general, one expects that many partial waves will be required to adequately describe the cross-section for inelastic processes. Consequently, one should avoid a partial wave analysis and use the following expression (k’. 01’.$ f Ik. a. /3 =
~~2i(?mk/h2)(k’, 1 i(2mk/h2)(k’.cr’,
a’, $1 L/Ik,cu.@ $lc/l k. a, 0) (3)
-q)F,‘+,n(q) (2)
260
LETTERS
for the transmission
matrix.
Vol. II, part A, ed. Warren P. Mason (Academic Press, New York 1965) p. 133. 121 T.L. Contrell and J.C. McDoubrey, in Molecular ?nerg!’ transfer in gases (Ratterworths, London 1961). [31 J.L. Stretton, in Transfer and storage of energy by molecules, Vol. II, eds.G.M. Burnett and A. North (WileyInterscience, New York 1969) p. 58. 141 K. Takayanagi, Adv. At. Mol. Phys. 1 (1965) 149. (51 K. Takayanagi, Prog. Theoret. Phys. (Kyoto) Suppl. 25 (1963) 1. 161 K.F. Herzfeld has developed a theory for linear molecules. See K.F. Herzfeld, in Theories of relaxation times, in Dispersion and absorption of sound by molecular process, ed. D. Sette (Academic Press, London 1963). [71 F.H. Mies, .I. Chem. Phys. 42 (1965) 2709. [81 H.A. Rabitz and R.G. Gordon, J. Chem. Phys. 53 (1970) 1815. [91 Marshall and Lovesey, Theory of thermal neutron scattering (Oxford 197 1). IlO1 L. Landau and E.M. Lifshitz, Course in theoretical physics, Vol. III (Pergamon Press Ltd., London 1958). 1111 V.H. Burke and M.J. Seaton, Proc. Physical Society 77 (1961) 199.
[II H.O. Kneser, in Physical acoustics,
46A, number
PHYSICS
4
VIBRATIONAL
LETTERS
RELAXATION
MOLECULAR
31 December
PROCESSES
1973
AND
FORM FACTORS*
D. ROGOVIN Department of Physics and Optical Science Center, University of Arizona, Tucson, Arizona 85721. USA Received A theory that relates collision matrix elements systems to molecular form factors is presented.
11 October
for vibrational/rotational
In this letter we present a simple technique for calculating the matrix elements of the intermolecular potential between colliding molecules undergoing vibrational energy transfer or conversion processes. The theory of vibrational relaxation processes in diatomic or polyatomic molecular gases [ 1,2] depends critically on these matrix elements and to date, most calculations of (molecular) inelastic collision cross-sections have employed a set of questionable approximations in computing the relevant matrix elements [3-51. While these approximations greatly simplify the calculations, they lack sound justification and in some cases are not valid. For collisions between two molecular systems most calculations consider only 1) collinear collisions and 2) interactions between nearest neighbors. Finally, 3) the colliding potential is approximated by a simple repulsive exponential. The first ‘two approximations are kinematic and their effect is to eliminate all of the angular factors that would otherwise arise in the collision process. In order to simulate the full set of possible initial collision-orientations a steric factor PO = l/3 for each molecule is introduced into the transition amplitude. The second approximation is justified on the grounds that it is the short-range portion of the collision potential that dominates the dynamics of vibrational transfer and conversion processes. Unfortunately, the steric factor tends to vary with the colliding molecular species and there does not exist a reliable quantitative theory for calculating this factor for collisions involving non-linear polyatomic molecules [6]. The second approximation is clearly invalid for initial col-
%ork supported by Advanced and Kirtland Air Force Base.
Research
Projects
Agency
1973 transfer
processes
between
molecular
lision-orientations in which one atom of one molecule is very close to a number of atoms in the second molecule. Furthermore, one cannot investigate the possibility of rotational transfer or conversion occurring simultaneously with vibrational processes. Finally, in regards to these kinematic approximations, we note that Mies [7] has carried out a first-principle’s calculation using the full-angle-dependent potential for the excitation of hydrogen molecules by helium atoms. His results have raised grave doubts about the validity of 1) and 2). The dynamic approximation 3) enables one to write the intermolecular potential as a function of the center-of-mass distance times a function of the internal vibrational coordinates. There exists no sound justification of 3) and Mies [7] has shown that the assumption of additive exponential potentials acting between the atomic centers does not tit the results of his exact calculation. We assume that the potential can be approximated by u=zUii(r-Ri(“+Rp)). i,j ’
Here Ui,i is the interaction between atom i in polyatomic molecule A and atomj in polyatomic molecule B, r is the vector between the molecular center-ofmasses, Rp)(Rp) is the position vector of atom i(i) from the center-of-mass of molecule A(B). Eq. (1) assumes that the electronic degrees of freedom are not excited by the collision and in addition, that s = r - Ri(a) + Ri(b) is the only vector direction that is significant. Thus, one neglects [orientational] dipoledipole forces, etc., which are believed to dominate on259
Volume
46A, number
4
PHYSICS
ly the long-range portion of the intermolecular potential [8]. This part of the intermolecular force gives rise only to rotational processes and plays no role in vibrational relaxation. For situations in which the Born approximation applies we require the matrix elements of eq. (2) between the states Ik; Q, 0) and Ik’; (Y’,$)..Hcre k[k’] is the relative center-of-mass wave-vector before (after) the collision, CY(CX’) is the initial (final) rotation-vibration state of molecule A and /3(/I’) the same, but for molecule B. The relevant matrix element is then (k’;cu’, P’IUlk; a, P) =c Ui,Jq)F;_,( i. i
31 December
where Aq is the momentum transfer vector and Fk,,(-q) = (cu’lexp(--iq.R$%) is the form factor of atom i between molecular states ICI)and IQ’), at wave-vector 4. These quantities which also appear in neutron scattering depend only on the molecular structure, the initial and final rotation-vibration states and the wave-vector Q. They are independent of the colliding potential and for some transitions may be obtainable from neutron scattering [9] experiments. The factors U,i[q) are the Fourier transform of the atom-atom interaction. We note that this approach separates the internal dynamics of the molecule (i.e., the form factors Fi,, 1(-q)] from the center-ofmass motion (i.e., the Uii(4)) and is formally similar to the approach taken for inelastic collisions between fast electrons and atoms [lo]. Finally, we note that we have avoided all of the approximations mentioned above. For most molecular systems one cannot use the Born approximation for collisions occurring at thermal velocities and one must resort to other techniques. One approach is the Born approximation to the reactance matrix. This technique was used by
1973
Burke and Seaton [ 1 I ] to study e-H collision crosssections. Their efforts were quite successful and can be readily adapted to our problem. In general, one expects that many partial waves will be required to adequately describe the cross-section for inelastic processes. Consequently, one should avoid a partial wave analysis and use the following expression (k’. 01’.$ f Ik. a. /3 =
~~2i(?mk/h2)(k’, 1 i(2mk/h2)(k’.cr’,
a’, $1 L/Ik,cu.@ $lc/l k. a, 0) (3)
-q)F,‘+,n(q) (2)
260
LETTERS
for the transmission
matrix.
Vol. II, part A, ed. Warren P. Mason (Academic Press, New York 1965) p. 133. 121 T.L. Contrell and J.C. McDoubrey, in Molecular ?nerg!’ transfer in gases (Ratterworths, London 1961). [31 J.L. Stretton, in Transfer and storage of energy by molecules, Vol. II, eds.G.M. Burnett and A. North (WileyInterscience, New York 1969) p. 58. 141 K. Takayanagi, Adv. At. Mol. Phys. 1 (1965) 149. (51 K. Takayanagi, Prog. Theoret. Phys. (Kyoto) Suppl. 25 (1963) 1. 161 K.F. Herzfeld has developed a theory for linear molecules. See K.F. Herzfeld, in Theories of relaxation times, in Dispersion and absorption of sound by molecular process, ed. D. Sette (Academic Press, London 1963). [71 F.H. Mies, .I. Chem. Phys. 42 (1965) 2709. [81 H.A. Rabitz and R.G. Gordon, J. Chem. Phys. 53 (1970) 1815. [91 Marshall and Lovesey, Theory of thermal neutron scattering (Oxford 197 1). IlO1 L. Landau and E.M. Lifshitz, Course in theoretical physics, Vol. III (Pergamon Press Ltd., London 1958). 1111 V.H. Burke and M.J. Seaton, Proc. Physical Society 77 (1961) 199.
[II H.O. Kneser, in Physical acoustics,