Geometric Phase and Interference Effects in Ultracold Chemical Reactions
267
sections [14–23]. Thus, it appears that an energy regime where only a single partial wave contributes is the most relevant regime to explore GP effects in a chemical
reaction. This regime, referred to as the cold and ultracold regime, has gained much
interest in recent years, thanks to the dramatic progress in cooling and trapping of
molecules in the mK and í µí¼K regimes. Here, we will focus on our recent studies
of the GP effect in chemical reactions in the ultracold regime taking the hydrogen
exchange reaction as an illustrative example.
The ultracold regime [34–38] provides a fascinating domain to explore quantum effects in chemical reactions. Because s-wave scattering dominates at ultracold
temperatures (for bosons and distinguishable particles), only the l = 0 partial wave
contributes and the GP effect is not smeared out by partial wave summation. Furthermore, isotropic scattering in the s-wave regime allows for maximum constructive or
destructive interference between wave functions along alternative paths around the
CI (direct and looping/exchange paths). These properties combined with an effective quantization of the scattering phase-shift in the ultracold regime (Levinson’s
theorem [39] í µí»¿(0) = ní µí¼ where í µí»¿ is the phase shift and n is the number of bound
states supported by the potential well) entail maximum constructive or destructive interference between the direct and exchange/looping scattering amplitudes.
This leads to a large enhancement or suppression of reactivity, as recently demonstrated for O+OH(v = 0, 1) → H+O 2 (v
′
, j
′
) [40, 41] and the hydrogen exchange
processes in H+H 2 (v = 4, j = 0), H+HD(v = 4, j = 0) and D+HD(v = 4, j = 0) reactions [42–45]. The H+H 2 reaction has an energy barrier for vibrational levels v < 3
but becomes barrierless for v > 3 [43, 46–48]. Indeed, vibrationally adiabatic potentials for the H+H 2 reaction for v = 4 and higher vibrational levels depict an effective
potential well. The bound state structure of this potential well has a dramatic effect
on the scattering process at ultracold temperatures as discussed below. Also, barrierless reactions occur with much larger rate coefficients at ultracold temperatures and
are more amenable to experiments than barrier reactions that proceed via tunneling.
The chapter is organized as follows. In Sect. 2 we briefly discuss the mechanism of the GP effect in ultracold reactions. Section 3 outlines the coupled channel
method employed in the scattering calculations. Illustrative results of GP effects in
H+H 2 /H+HD/D+HD reactions are presented in Sect. 4 followed by conclusions in
Sect. 5.
2 Mechanism of the GP Effect in the Ultracold Regime
In our previous work [40, 42] we showed that to observe the GP effect in reactive
and inelastic collisions two criteria should be satisfied: (i) the adiabatic PES must
exhibit a conical intersection; (ii) the scattering amplitudes along the two scattering
pathways (direct and exchange/looping) must have comparable magnitude and scatter into the same angular region. Isotropic scattering in the ultracold regime and the
effective quantization of the scattering phase shift as required by Levinson’s theorem
provide the criterion for maximum constructive and destructive interference between
267
sections [14–23]. Thus, it appears that an energy regime where only a single partial wave contributes is the most relevant regime to explore GP effects in a chemical
reaction. This regime, referred to as the cold and ultracold regime, has gained much
interest in recent years, thanks to the dramatic progress in cooling and trapping of
molecules in the mK and í µí¼K regimes. Here, we will focus on our recent studies
of the GP effect in chemical reactions in the ultracold regime taking the hydrogen
exchange reaction as an illustrative example.
The ultracold regime [34–38] provides a fascinating domain to explore quantum effects in chemical reactions. Because s-wave scattering dominates at ultracold
temperatures (for bosons and distinguishable particles), only the l = 0 partial wave
contributes and the GP effect is not smeared out by partial wave summation. Furthermore, isotropic scattering in the s-wave regime allows for maximum constructive or
destructive interference between wave functions along alternative paths around the
CI (direct and looping/exchange paths). These properties combined with an effective quantization of the scattering phase-shift in the ultracold regime (Levinson’s
theorem [39] í µí»¿(0) = ní µí¼ where í µí»¿ is the phase shift and n is the number of bound
states supported by the potential well) entail maximum constructive or destructive interference between the direct and exchange/looping scattering amplitudes.
This leads to a large enhancement or suppression of reactivity, as recently demonstrated for O+OH(v = 0, 1) → H+O 2 (v
′
, j
′
) [40, 41] and the hydrogen exchange
processes in H+H 2 (v = 4, j = 0), H+HD(v = 4, j = 0) and D+HD(v = 4, j = 0) reactions [42–45]. The H+H 2 reaction has an energy barrier for vibrational levels v < 3
but becomes barrierless for v > 3 [43, 46–48]. Indeed, vibrationally adiabatic potentials for the H+H 2 reaction for v = 4 and higher vibrational levels depict an effective
potential well. The bound state structure of this potential well has a dramatic effect
on the scattering process at ultracold temperatures as discussed below. Also, barrierless reactions occur with much larger rate coefficients at ultracold temperatures and
are more amenable to experiments than barrier reactions that proceed via tunneling.
The chapter is organized as follows. In Sect. 2 we briefly discuss the mechanism of the GP effect in ultracold reactions. Section 3 outlines the coupled channel
method employed in the scattering calculations. Illustrative results of GP effects in
H+H 2 /H+HD/D+HD reactions are presented in Sect. 4 followed by conclusions in
Sect. 5.
2 Mechanism of the GP Effect in the Ultracold Regime
In our previous work [40, 42] we showed that to observe the GP effect in reactive
and inelastic collisions two criteria should be satisfied: (i) the adiabatic PES must
exhibit a conical intersection; (ii) the scattering amplitudes along the two scattering
pathways (direct and exchange/looping) must have comparable magnitude and scatter into the same angular region. Isotropic scattering in the ultracold regime and the
effective quantization of the scattering phase shift as required by Levinson’s theorem
provide the criterion for maximum constructive and destructive interference between