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A comparison of classical and quantal transition probabilities for a non-adiabatic atom-atom collision
Volume 120.
number
2
4 October 1985
CHEMICAL PHYSICS LEl-lXRS
A COMPARISON OF CLASSICAL AND QUASTAId TRANSITION PROBABILITIES FOR A NON-ADIABATIC ATOM-ATOM COLLISION J.P. BRAGA, LJ. DUNNE 1 and J-N. MURRJXLL School of Chenris~q~ and MolecuIar Sdmces. Sussex U~ir:cr.rig.. Falmer. Bri_qhron BNI 9QJ_ Suuex. UK Received 7 June 1985; in final form I1 July 1985
A comparison is made bewren Ihe results or rull quanta1 non-adiabalic LransiCon probahiliI_v cnlculnlions [or He’-N collisions and those obtained Cram a recently published model which ~reaw the nuclear mcxion clnssicnlly and thr ekclronic
We have recently published a model for calculating non-adiabatic transition probabilities for molecular collisions which couples the classical motion of the nuclei with quantal motion of the electrons [ 11. The electronic state is defimed by a density matrix p, whose variation with time is obtained from the quantum-mechanical Liouville equation, with additional damping terms to produce eigenstates of the electronic Hamiltonian in the exit channels_ These terms are required to prevent population of closed channels, and to give physical significance to the distribution of vibrational and rotational energies amongst the electronic states of the products. Trajectories are calculated which describe the time dependence of the nuclear positions and momenta and the elements of the density matrix. The equations governing these, which are given in detail in ref. [I], conserve energy, angular momentum, and the positive definite character of the density matrix. Although the method has been established primarily for the treatment of non-adiabatic reactive atommolecule collisions, for which full quantal calculations are at present not feasible, it can be tested for simpler systems against accurate quantal calculations. In this paper we present the results of one such test, on the non-adiabatic reaction He+(2S)+Ne(2p6,1S)+He+(2S)+Ne(2ps3s,2P)
.(l)
This system has been studied previously by Olson and 1
Resent address: Polytechnic of the South Bank, landon
SE1 QAA, UK
0 OO$2614/85/g 03.30 0 Elsevier Science Publishers B-V. (North-Holland Physics Publishing Diviion)
Smith [2] who gave a diabatic representation of the potentials involved and showed that they produced a reasonable account of the experimental elastic and inelastic cross section data. We have extended some of Olson and Smith’s quanta1 calculations with the object of testing our “classical” model, and have given particular attention to alternative approaches to the damping terms in the Liouville equations, eqs. (16)(23) in ref. [l]_ The Olson-Smith potential is defined by the relations VII = 21_lr-J exp(--r/0.678),
Vz2= 0.618+(21&V12 =
12.3) exp(-r/0.678),
0.170 exp(--r/0.667) _
(3
Channel 1 is the reactant and channel 2 the product of reaction (1). All bond lengths are in bohr and energies in hartree. The diabatic states (2) have a crossing at 2.0&r,,, at an energy 0.530 Eh above the asymptotic energy of channel 1 (which is taken as zero). Note that VI2 is an exponentially decaying function of r (the internuclear separation) and from perturbation theory we know that a key quantity determining the transition probability is the value of VI2 at the crossing point; this is 0.0082 Eh_ The close-coupled equations of the full quantal treatment have been solved for values of the angular momentum I by the log-derivative method 131. The classical analogue to an I-partial wave is a trajectory 147
Volume 120, number 2
CHEMICAL PHYSICS LETTERS
4 October 1985
the I cut-off (the value of I beyond which the tran-
sition probability drops to zero) and we examine the reason for this later. These oscillations are essentially the Stueckelberg oscillations [4], and they arise in our trajectory calculations from the propagation of the real and imagi-
nary parts of p12 along that part of the trajectory which is between the incoming and outgoing traversals
Fig L Quanta1 transition probabilities as a function of nng~lar momentum Tar reaction (1) at an cncrgy of 2.61 Eh_ The full lint is the transition probability ckulatcd from the LandauZencr formula (3).
with an impact parameter, b = 1/(2~E)~/3, where p is the reduced mass of the system and E is the collision energy. The classical trajectories were calculated with a potential given by Tr (Vp) (eq. (1 l), ref. [ 1)). Fig. 1 shows the quanta1 transition probability for an impact energy of 2.6 1 E, _ The results agree with earlier calculations by Olson and Smith [2]; mainly distorted wave calculations but at a few values of I, the full close-coupled results were obtained. Fig. 2 shows the results from classical trajectories at the same collisional energy without the addition of damping terms to the Liouville equations (E = 0 in eq. (22) of ref. [l]). The similarity between figs. 1 and 2 is quite striking, particularly in respect to the oscillatory behaviour. There is a small difference in the position of
of the crossing region. It should be stressed that interference is absent from any classical model in which one integrates probabilities rather than amplitudes and phases. For example, in the popular Tulley-Preston surfacehopping model [S], a transition probability is accumulated for every passage through a diabatic crossing point and there is no interference between the incoming and out going passes. If, in this model, the transition probability is calculated by the LandauZener approximation [6], which implies no momentum coupling, and it is a weak transition (i.e. second crossings can be ignored) then the Tulley-Preston result is identical to the Landau-Zener formula [7] which for a double passage is
p = 2. e-286(1 _ e-2n6),
(3)
where %rVj$(R,) ’ = Jru(d VI r /dR - dvz2/dR)R,
(4)
and IJ is the velocity at the crossing point which depends on 1. This probability is shown as the continuous line in fig. 1. We attribute the difference in the 1 cut-off of the quanta1 and classical models to tunneling in the former. Tunneling allows the nuclear wavefunctions to penetrate to the diabatic crossing point even when the classical turning point is at a greater value of r. To confum this interpretation we have made calculations at the same impact energy and with the same potential (1) for a model system in which the mass of He is replaced Ar (mass 40) as this will behave more classically_ The values of 2 at the maximum of the last oscillation for the quantal and the classical calculations differ by
only 5% (compared with 17% for the original system)_ Fig. 2. Transition probabilities obtained rrom the classical trajeclorics and the Liouvillc equation. Energy as for iii. 1.
148
If the impact energy for reaction (1) is less than 0.618 E, then channel 2 is closed and the the quanta1 transition probability is zero. However, because the
Volume 120, number 2
CHEMICAL PHYSICS LE-I-IXRS
nuclear motion in our classical model occurs over an average potential, Tr(Vp), a sufficient condition for a trajectory to emerge from a collision is that the total energy be greater than the asymptotic value of this potential, (plrJ’L1 + pz2V22) (r= -)_ This trajectory can emerge in the closed region with a non-zero value of p22 unless a constraint is applied to the equations of motion so as to produce eigenstates of the electronic Hamiltonian on exit (pi1 = 0 or l)_ It is for this reason (plus the wish to make ro-vibrational state analyses for molecular collisions) that the model proposed in ref. [l] has damping terms added to the Liouville equations_ The procedure suggested in ref. [l] for damping the density matrix has terms which are even handed to the two states; they damp towards state 1 if ~22 C 0.5 and to state 2 if k1 > 0.5 (pl1 + e7 = 1). It can be seen from fig. 2, which shows the results obtained from the undamped equations, that for aJl values of I, p22 on exit isalwaysless than 0.5 and we have confirmed that at no point in the trajectories does p2? exceed this value. Thus our even-handed recipe will for this system, which has very low non-adiabatic probabilities, lead to trajectories which for aJl impact parameters exit on channel 1. To show that our original recipe gives sensible results when applied to a strongly non-adiabatic system we have repeated the calculations setting VIZ to five times the Olson-Smith value. Solution of the quanta1 close-coupling equations give a cross section for the non-adiabatic process of 5.01 ai. The even-handed damping terms are characterized by a strength parameter and from a batch of 160 trajectories evenly spaced over the range of 0 < I <320, and a value E = 0.1, we obtained 103 emerging in pure state 1 and 57 emerging in pure state 2. This gives a cross section of 5.4 f 0.5 a;, which shows that the model is SatiShCtory. The Landau-Zener model gives a cross section of 4.62 a$ and is, therefore, of similar accuracy. For large values of Vi2 the transition probability should approach zero. This is because the dynamics is controlled by the adiabatic potentials which are the eigenvalues of the diabatic potential matrix, and if Yr2 is large these adiabatic potentials are very far apart. We have confirmed this for both the quanta1 and classical calculations with a VI2 set to fifty times the OlsonSmith value. It is illuminating to follow the evolution of the density matrix during the classical collision.
4 October 1985
During the incoming part of the trajectory p1 i changes from unity to a small value but this trend is totally reversed in the outgoing part of the trajectory. We return to the questjon of how to deal with very weak non-adiabatic systems if one requires the classical trajectories to emerge in pure states. What we propose is that the undamped trajectory is computed and when the asymptotic product region is reached (VI2 = 0) a random number, uniformly distributed between 0 and 1, is sampled and compared with the asymptotic value of pi1 _ If the random number is less than pl 1 it is damped towards state 1. If not, it is damped to state 2. These dampings are produced by the choice of sign, f 1 respectively, in the following expressions which replace eqs. (23) and (23) of ref. [I]:
The symbols are defined in ref. [I]. There are no phenomenological parameters (as e) in this procedure and it gives the same transition probability as the undamped equations (within statistical error). There will be zero population of closed channels; if the random number favours a closed channel then that trajectory will not emerge from the collision_ For the system under study a sample of 160 trajectories gives a cross section of 5.0 + 0.5 a;, in agreement with the quanta1 value. At low energies the dynamics is expected to be less well represented by the classical model. To test this we have carried out a series of calculations on the same potentials at an energy of 0.625 E,, which is only 0.2 eV above the threshold for opening channel 2. The classical results in this case show no oscillations and the cut-off is at a slightly higher I value t!tan obtained from the quanta1 calculations, unlike the highenergy situation. The cross sections calculated from these two figures differ by approximately a factor of two_ The quanta1 value is 033 ai, the undamped classical value is 0.65 0: and a value‘identical to the latter is obtained by applying the random damping method described above (subject to the usual statistical error of the Monte Carlo integration method)_ The Landau-Zener cross section at 0.44 ag is in this case rather better. Although the close correspondence which exists 149
Volume 120;number 2
CHEMICAL PHYSICS LETTERS
at high impact energy between the quanta1 and classical transition probabilities at each angular momentum is not shown near the energy threshold, the collision cross sections for the transition that are calculated by the two methods arc not in gross disagreement. The classical method, with damping to resolve the density matrix to pure states may therefore be usefully applied to systems for which the quanta1 calculations are in-tpossible_
References [l] L.J. Dunne, J.N. MyelI and J.G. Stamper. Chem. Phys
Letter5 112 (1984) 497_ R.F. Olson and F.T. Smith; Phys Rev. 3.(1971) 1607. B-R. Johnson, J. Comp. Phys. 6 (1970) 378. E.C.G. Stueckelberg, Helv. Phys Acta 5 (1933)‘369. 3-C. Tully and P.K. R&on, J. Chem.‘Phys.-55 (1971) 562. [6] L.D. Landau, Phys. 2. Sow. i (1932) 46; C. Zencr, Proc. Roy. Sot. Al37 (1932) 696. [7] hi. Child, Molecular collision theory (Academic Press. [2] [3] 143 [5]
New York, 1974) p. 167.
150
4 October. 1985
number
2
4 October 1985
CHEMICAL PHYSICS LEl-lXRS
A COMPARISON OF CLASSICAL AND QUASTAId TRANSITION PROBABILITIES FOR A NON-ADIABATIC ATOM-ATOM COLLISION J.P. BRAGA, LJ. DUNNE 1 and J-N. MURRJXLL School of Chenris~q~ and MolecuIar Sdmces. Sussex U~ir:cr.rig.. Falmer. Bri_qhron BNI 9QJ_ Suuex. UK Received 7 June 1985; in final form I1 July 1985
A comparison is made bewren Ihe results or rull quanta1 non-adiabalic LransiCon probahiliI_v cnlculnlions [or He’-N collisions and those obtained Cram a recently published model which ~reaw the nuclear mcxion clnssicnlly and thr ekclronic
We have recently published a model for calculating non-adiabatic transition probabilities for molecular collisions which couples the classical motion of the nuclei with quantal motion of the electrons [ 11. The electronic state is defimed by a density matrix p, whose variation with time is obtained from the quantum-mechanical Liouville equation, with additional damping terms to produce eigenstates of the electronic Hamiltonian in the exit channels_ These terms are required to prevent population of closed channels, and to give physical significance to the distribution of vibrational and rotational energies amongst the electronic states of the products. Trajectories are calculated which describe the time dependence of the nuclear positions and momenta and the elements of the density matrix. The equations governing these, which are given in detail in ref. [I], conserve energy, angular momentum, and the positive definite character of the density matrix. Although the method has been established primarily for the treatment of non-adiabatic reactive atommolecule collisions, for which full quantal calculations are at present not feasible, it can be tested for simpler systems against accurate quantal calculations. In this paper we present the results of one such test, on the non-adiabatic reaction He+(2S)+Ne(2p6,1S)+He+(2S)+Ne(2ps3s,2P)
.(l)
This system has been studied previously by Olson and 1
Resent address: Polytechnic of the South Bank, landon
SE1 QAA, UK
0 OO$2614/85/g 03.30 0 Elsevier Science Publishers B-V. (North-Holland Physics Publishing Diviion)
Smith [2] who gave a diabatic representation of the potentials involved and showed that they produced a reasonable account of the experimental elastic and inelastic cross section data. We have extended some of Olson and Smith’s quanta1 calculations with the object of testing our “classical” model, and have given particular attention to alternative approaches to the damping terms in the Liouville equations, eqs. (16)(23) in ref. [l]_ The Olson-Smith potential is defined by the relations VII = 21_lr-J exp(--r/0.678),
Vz2= 0.618+(21&V12 =
12.3) exp(-r/0.678),
0.170 exp(--r/0.667) _
(3
Channel 1 is the reactant and channel 2 the product of reaction (1). All bond lengths are in bohr and energies in hartree. The diabatic states (2) have a crossing at 2.0&r,,, at an energy 0.530 Eh above the asymptotic energy of channel 1 (which is taken as zero). Note that VI2 is an exponentially decaying function of r (the internuclear separation) and from perturbation theory we know that a key quantity determining the transition probability is the value of VI2 at the crossing point; this is 0.0082 Eh_ The close-coupled equations of the full quantal treatment have been solved for values of the angular momentum I by the log-derivative method 131. The classical analogue to an I-partial wave is a trajectory 147
Volume 120, number 2
CHEMICAL PHYSICS LETTERS
4 October 1985
the I cut-off (the value of I beyond which the tran-
sition probability drops to zero) and we examine the reason for this later. These oscillations are essentially the Stueckelberg oscillations [4], and they arise in our trajectory calculations from the propagation of the real and imagi-
nary parts of p12 along that part of the trajectory which is between the incoming and outgoing traversals
Fig L Quanta1 transition probabilities as a function of nng~lar momentum Tar reaction (1) at an cncrgy of 2.61 Eh_ The full lint is the transition probability ckulatcd from the LandauZencr formula (3).
with an impact parameter, b = 1/(2~E)~/3, where p is the reduced mass of the system and E is the collision energy. The classical trajectories were calculated with a potential given by Tr (Vp) (eq. (1 l), ref. [ 1)). Fig. 1 shows the quanta1 transition probability for an impact energy of 2.6 1 E, _ The results agree with earlier calculations by Olson and Smith [2]; mainly distorted wave calculations but at a few values of I, the full close-coupled results were obtained. Fig. 2 shows the results from classical trajectories at the same collisional energy without the addition of damping terms to the Liouville equations (E = 0 in eq. (22) of ref. [l]). The similarity between figs. 1 and 2 is quite striking, particularly in respect to the oscillatory behaviour. There is a small difference in the position of
of the crossing region. It should be stressed that interference is absent from any classical model in which one integrates probabilities rather than amplitudes and phases. For example, in the popular Tulley-Preston surfacehopping model [S], a transition probability is accumulated for every passage through a diabatic crossing point and there is no interference between the incoming and out going passes. If, in this model, the transition probability is calculated by the LandauZener approximation [6], which implies no momentum coupling, and it is a weak transition (i.e. second crossings can be ignored) then the Tulley-Preston result is identical to the Landau-Zener formula [7] which for a double passage is
p = 2. e-286(1 _ e-2n6),
(3)
where %rVj$(R,) ’ = Jru(d VI r /dR - dvz2/dR)R,
(4)
and IJ is the velocity at the crossing point which depends on 1. This probability is shown as the continuous line in fig. 1. We attribute the difference in the 1 cut-off of the quanta1 and classical models to tunneling in the former. Tunneling allows the nuclear wavefunctions to penetrate to the diabatic crossing point even when the classical turning point is at a greater value of r. To confum this interpretation we have made calculations at the same impact energy and with the same potential (1) for a model system in which the mass of He is replaced Ar (mass 40) as this will behave more classically_ The values of 2 at the maximum of the last oscillation for the quantal and the classical calculations differ by
only 5% (compared with 17% for the original system)_ Fig. 2. Transition probabilities obtained rrom the classical trajeclorics and the Liouvillc equation. Energy as for iii. 1.
148
If the impact energy for reaction (1) is less than 0.618 E, then channel 2 is closed and the the quanta1 transition probability is zero. However, because the
Volume 120, number 2
CHEMICAL PHYSICS LE-I-IXRS
nuclear motion in our classical model occurs over an average potential, Tr(Vp), a sufficient condition for a trajectory to emerge from a collision is that the total energy be greater than the asymptotic value of this potential, (plrJ’L1 + pz2V22) (r= -)_ This trajectory can emerge in the closed region with a non-zero value of p22 unless a constraint is applied to the equations of motion so as to produce eigenstates of the electronic Hamiltonian on exit (pi1 = 0 or l)_ It is for this reason (plus the wish to make ro-vibrational state analyses for molecular collisions) that the model proposed in ref. [l] has damping terms added to the Liouville equations_ The procedure suggested in ref. [l] for damping the density matrix has terms which are even handed to the two states; they damp towards state 1 if ~22 C 0.5 and to state 2 if k1 > 0.5 (pl1 + e7 = 1). It can be seen from fig. 2, which shows the results obtained from the undamped equations, that for aJl values of I, p22 on exit isalwaysless than 0.5 and we have confirmed that at no point in the trajectories does p2? exceed this value. Thus our even-handed recipe will for this system, which has very low non-adiabatic probabilities, lead to trajectories which for aJl impact parameters exit on channel 1. To show that our original recipe gives sensible results when applied to a strongly non-adiabatic system we have repeated the calculations setting VIZ to five times the Olson-Smith value. Solution of the quanta1 close-coupling equations give a cross section for the non-adiabatic process of 5.01 ai. The even-handed damping terms are characterized by a strength parameter and from a batch of 160 trajectories evenly spaced over the range of 0 < I <320, and a value E = 0.1, we obtained 103 emerging in pure state 1 and 57 emerging in pure state 2. This gives a cross section of 5.4 f 0.5 a;, which shows that the model is SatiShCtory. The Landau-Zener model gives a cross section of 4.62 a$ and is, therefore, of similar accuracy. For large values of Vi2 the transition probability should approach zero. This is because the dynamics is controlled by the adiabatic potentials which are the eigenvalues of the diabatic potential matrix, and if Yr2 is large these adiabatic potentials are very far apart. We have confirmed this for both the quanta1 and classical calculations with a VI2 set to fifty times the OlsonSmith value. It is illuminating to follow the evolution of the density matrix during the classical collision.
4 October 1985
During the incoming part of the trajectory p1 i changes from unity to a small value but this trend is totally reversed in the outgoing part of the trajectory. We return to the questjon of how to deal with very weak non-adiabatic systems if one requires the classical trajectories to emerge in pure states. What we propose is that the undamped trajectory is computed and when the asymptotic product region is reached (VI2 = 0) a random number, uniformly distributed between 0 and 1, is sampled and compared with the asymptotic value of pi1 _ If the random number is less than pl 1 it is damped towards state 1. If not, it is damped to state 2. These dampings are produced by the choice of sign, f 1 respectively, in the following expressions which replace eqs. (23) and (23) of ref. [I]:
The symbols are defined in ref. [I]. There are no phenomenological parameters (as e) in this procedure and it gives the same transition probability as the undamped equations (within statistical error). There will be zero population of closed channels; if the random number favours a closed channel then that trajectory will not emerge from the collision_ For the system under study a sample of 160 trajectories gives a cross section of 5.0 + 0.5 a;, in agreement with the quanta1 value. At low energies the dynamics is expected to be less well represented by the classical model. To test this we have carried out a series of calculations on the same potentials at an energy of 0.625 E,, which is only 0.2 eV above the threshold for opening channel 2. The classical results in this case show no oscillations and the cut-off is at a slightly higher I value t!tan obtained from the quanta1 calculations, unlike the highenergy situation. The cross sections calculated from these two figures differ by approximately a factor of two_ The quanta1 value is 033 ai, the undamped classical value is 0.65 0: and a value‘identical to the latter is obtained by applying the random damping method described above (subject to the usual statistical error of the Monte Carlo integration method)_ The Landau-Zener cross section at 0.44 ag is in this case rather better. Although the close correspondence which exists 149
Volume 120;number 2
CHEMICAL PHYSICS LETTERS
at high impact energy between the quanta1 and classical transition probabilities at each angular momentum is not shown near the energy threshold, the collision cross sections for the transition that are calculated by the two methods arc not in gross disagreement. The classical method, with damping to resolve the density matrix to pure states may therefore be usefully applied to systems for which the quanta1 calculations are in-tpossible_
References [l] L.J. Dunne, J.N. MyelI and J.G. Stamper. Chem. Phys
Letter5 112 (1984) 497_ R.F. Olson and F.T. Smith; Phys Rev. 3.(1971) 1607. B-R. Johnson, J. Comp. Phys. 6 (1970) 378. E.C.G. Stueckelberg, Helv. Phys Acta 5 (1933)‘369. 3-C. Tully and P.K. R&on, J. Chem.‘Phys.-55 (1971) 562. [6] L.D. Landau, Phys. 2. Sow. i (1932) 46; C. Zencr, Proc. Roy. Sot. Al37 (1932) 696. [7] hi. Child, Molecular collision theory (Academic Press. [2] [3] 143 [5]
New York, 1974) p. 167.
150
4 October. 1985