12 June 1998
Chemical Physics Letters 289 Ž1998. 219–223
The geometric phase effect on differential cross sections in chemical reactions: a classical mechanical approach Satrajit Adhikari, Gert D. Billing Department of Chemistry, H.C. Ørsted Institute, UniÕersity of Copenhagen, DK-2100 Ø Copenhagen, Denmark Received 29 December 1997; in final form 16 March 1998
Abstract The considerable agreement between experimental and theoretical product rotational state distributions for the D q H 2 reaction obtained with the inclusion of the geometric phase effect has created interest to see the influence of it on differential cross sections. Using the general hyperspherical formulation of the vector potential arising due to an arbitrary position of the conical intersection of the adiabatic potential energy hypersurfaces, we have calculated the distribution of scattering angles. This study has been carried out by quasi-classical trajectory calculations which require little computational effort. The features obtained by our study are qualitatively the same as those obtained quantum mechanically by Kuppermann and Wu. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction In the initial applications of the Born–Oppenheimer treatment of molecules, the parametric dependence of the adiabatic electronic wavefunction on the nuclear coordinates has been ignored with the assumption that the effects of nuclear motion can be included in a perturbative manner. However, this scenario suffered from some complications due to the presence of conical intersections w1–9x. A real electronic wavefunction changes sign w10–13x when the nuclear coordinates travel through a closed path around the conical intersection. The motivation of the generalized Born–Oppenheimer ŽBO. treatment came in order to avoid the double-valued electronic wavefunction. Mead and Truhlar w14–16x generalized the BO method by introducing a vector potential into the nuclear Schrodinger equation to ensure a ¨ single valued and continuous total wavefunction. In this approach, the real double-valued electronic
wavefunction is multiplied by a complex phase factor and the nuclear Schrodinger equation conse¨ quently requires a vector potential. When the nuclear coordinates travel through a closed path around the conical intersection, the phase factor changes sign and makes the complex electronic wavefunction single valued. As these effective nuclear Schrodinger equations are similar to those differential equations which can mimic a charged particle moving in the presence of a magnetic solenoid, this phenomenon is called ‘‘molecular Aharonov–Bohm effect’’ w17x and it is a special case of Berry’s phase w18x. In recent years, it has been observed that the effect is small on chemical reactions for energies well below the conical intersection but near the conical intersection it becomes prominent. At least this appears to be so for integral cross sections. However, for differential cross sections the effect of the conical intersection can be substantial even at energies well below the actual intersection. In order
0009-2614r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 3 0 0 - 5
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S. Adhikari, G.D. Billingr Chemical Physics Letters 289 (1998) 219–223
to have agreement between theoretically calculated differential and state-resolved cross sections and experimental data at all energies, this effect must be included in the theory. Using accurate quantum scattering calculations, Kuppermann and coworkers w19– 21x have identified this effect for the first time in a chemical reaction ŽH 3 and its isotopic variants.. As the conical intersection of the ground and excited
states of the H 3 system occurs at the symmetric triangular configuration, it is possible to incorporate the effect into the basis functions directly such that the nuclear Schrodinger equation itself does not need ¨ a vector potential. Though this is a reasonably good approximation for isotopic variants of the X 3 system, it is not general. In order to properly include the geometric phase effects in more complicated
˚ 2rdeg. at E s 1.8 eV for the D q H 2 Ž Õ s 1, j . ™ DH Ž ÕX s 1, jX . q H reaction as a function Fig. 1. Ža–e. Differential cross sections in ŽA X of scattering angle Ždeg. are shown. Ža. through Že. display the results for j s 9 through to 13, respectively, where in each figure, the solid lines with a star and a black box indicate the results with and without the geometric phase, respectively.
S. Adhikari, G.D. Billingr Chemical Physics Letters 289 (1998) 219–223
221
Fig. 1 Žcontinued..
molecules Žwhere the conical intersection does not occur at a special symmetry point of the coordinate system. the introduction of a vector potential becomes an alternative. Although the effect is quantum mechanical in nature, one can also see it in a simplified manner where the theory is semi-classical w22,23x. In our approach, the vector potential is treated quantum mechanically whereas the nuclear motion is treated classically. Thus, the classical
Hamiltonian has additional potential terms of the order " and " 2 . We have presented equations for the quantum potential for the general case where the position of the conical intersection is arbitrary. Using the general expression of the vector potential for an A q B 2 system, the rotational state resolved cross sections Ž ÕX s 1, jX . for D q H 2 reaction Ž u 0 s 11.58. have been compared with the available experimental distribution obtained by Zare and co-workers w24–26x
S. Adhikari, G.D. Billingr Chemical Physics Letters 289 (1998) 219–223
222
Fig. 1 Žcontinued..
and also our previous quasi-classical trajectory results Ž u s 08. in Fig. 1Ža. of Ref. w23x. In this article, we have investigated the effect of the geometric phase on differential cross sections for the same reaction and with the same general expression of the vector potential w23x.
2. Results and discussion We have used the same numerical values of the Morse potential parameters as in Ref. w22x during initialization and final analysis of the quasi-classical trajectory ŽQCT. calculations and the semiclassical formula, based on the Bohr–Sommerfeld quantization, X
Õ sy
1
1 q
2
h
Ep d r r
Ž 1.
in the final analysis of the QCT calculations ŽScheme B w22x.. The computation of differential cross sections are carried out by QCT calculations for the D q H 2 Ž Õ s j s 1. ™ DH Ž ÕX s 1, jX . qH reaction at the total energy of 1.8 eV Žinitial kinetic energy 1.0 eV. with the LSTH w27,28x potential energy parameters. We have calculated the scattering angle distributions for different final states Ž ÕX s 1, jX . with or without
inclusion of the geometric phase starting from the initial state Ž Õ s 1, j s 1.. The convergence of these distributions has appeared when there are a sufficient number of trajectories in each scattering angle. Nearly 1.0 = 10 6 trajectories have been computed to obtain converged distributions for all the final jX states. Of course, convergence really came with 0.5 = 10 6 trajectories. The rotationally resolved differential cross section are subsequently smoothed by the moments expansion ŽM. in cosines w29–31x: d s R Ž jX , u . dv ck s
2 NR Ž jX .
s
s R Ž jX . 4p
M
1q
Ý c k cos Ž kp aŽ u . .
,
ks1
X NR Ž j .
Ý
cos Ž kp a Ž us . . ,
ss1
2 sR Ž jX . s p bmax NR Ž jX . rN,
a Ž u . s 12 Ž 1 y cos u .
Ž 2.
where N is the total number of trajectories and NR Ž jX . is the number of reactive trajectories leading to the DHŽ jX . product. Also, u is the scattering angle, s labels the reactive trajectories leading to the same product and bmax is the maximum impact parameter. In Figs. 1Ža–e., we have displayed rotationally resolved differential cross sections for jX equal to 9 through to 13 where each figure shows results ob-
S. Adhikari, G.D. Billingr Chemical Physics Letters 289 (1998) 219–223
tained with or without the geometric phase included. It is interesting to see the change of these distributions due to the presence of the geometric phase effect. Figs. 1Ža, b. Ž jX s 9,10. clearly indicate that distributions, with the inclusion of geometric phase, have higher peaks compared to ‘‘without’’ geometric phase situations. Similarly, one can see in Figs. 1Žd, e. Ž jX s 12,13. that non-geometric phase cases have predominance over geometric phase cases. Lastly, in Fig. 1Žc. Ž jX s 11., there are crossings between these distributions. The rotationally resolved differential cross section results as shown in Fig. 1 are expected if one will follow the integral cross section distributions displayed in Fig. 1Ža. of Ref. w23x. Kuppermann and Wu w20,21x have shown differential cross sections at Etot s 1.8 eV Žinitial kinetic energy 1.0 eV. for the D q H 2 Ž Õ s j s 1. ™ DHŽ ÕX s 1, jX . q H reaction with or without considering geometric phase. In their calculations, the differential cross section distributions representing with or without geometric phase cases have crossings at jX s 9 where for lower jX s with geometric phase and for higher jX s without geometric phase cases have predominance. Qualitatively, we have found the same feature for differential cross section distributions as they have obtained, except the crossing position Žour case jX s 11, but their case jX s 9.. This overestimation of the crossing position comes from the use of classical mechanics. The rotationally resolved differential cross sections in the presence of a vector potential indicate clearly the effect of the geometric phase in the D q H 2 reaction with a conical intersection and the use of classical mechanics is the most important aspect of our calculations where the computational cost is little.
Acknowledgements This research is supported by the Danish Natural Science Research Council.
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