Symmetry and the geometric phase in ultracold hydrogen-exchange reactions

The hydrogen exchange reaction is referred to as “the simplest reaction.” Consisting of only 3 protons and 3 electrons, the forces on the nuclei can be accurately calculated from first-principles quantum mechanics. Consequently this reaction has been extensively studied, which has led to many advances in our understanding of chemical dynamics.^1–3 

In the ultracold regime, chemical reactions can be studied at the single quantum state level.^4–10 Reactions proceed through a single partial wave, and quantum mechanical effects are magnified. The essence of all chemical reactions is quantum mechanical; as such the ultracold regime is a window on reactions at their most fundamental level. A perfect illustration of this is the ultracold reaction between two fermionic KRb molecules. Ospelkaus et al. showed that the requirement that the total wavefunction be anti-symmetric with respect to the exchange of identical fermions suppresses the reaction between two KRb molecules in the same internal state.⁴ Taking advantage of this, they were able to turn the reaction on and off by changing the internal state of one of the molecules.

Studying the ultracold hydrogen-exchange reaction therefore offers the perfect testbed to study the fundamental aspects of chemical reactivity, such as symmetry effects, isotopic substitution, and the GP effect. The hydrogen exchange reaction becomes barrierless for vibrational levels $v$ = 3 and higher and therefore becomes amenable to ultracold collisional studies. For this reason, in recent years, there have been significant experimental efforts in creating vibrationally excited H₂ and HD molecules.^11,12 Indeed, the Zare group has recently reported population transfer (⁠$>1010$ molecules per pulse) of HD(⁠$v$ = 0, j = 0) to HD(⁠$v$ = 4, j = 0) with nearly 100% efficiency using the Stark induced adiabatic Raman passage (SARP), setting the stage for scattering experiments involving vibrationally excited HD at low temperatures.

The GP effect is purely quantum mechanical in origin, relating to the phase of the wavefunction encircling a conical intersection (CI).^13–17 Being quantum mechanical in origin, the GP does not readily manifest itself at higher collision energies where high partial waves lead to classical behaviour.¹⁸ However in the ultracold regime, it has been shown that the GP controls the $O$ + $OH→H$ + $O2$ reaction,¹⁹ just as the identical particle symmetry does in the KRb reaction.

In quantum mechanics, two wavefunctions can interfere either constructively or destructively, depending on their relative sign (or phase). A change of sign for either one of the wavefunctions will change destructive interference to constructive interference or vice versa. Often such a change of sign is a consequence of symmetry considerations. This is the case for the hydrogen exchange reaction where the inclusion of the GP introduces a change of relative sign between the dominant reaction pathways.²⁰ In the ultracold regime, phases are quantized, leading to maximally constructive or destructive interference between reaction pathways. Furthermore, when two reaction pathways are of similar magnitude, they will either cancel each other out or double up: the reaction can only be on or off. This is exactly the case for the ultracold hydrogen exchange reaction proceeding from $v$ = 4, j = 0, where the sign change associated with the GP turns the reaction on and off.^21,22 This is what is meant by the GP controlling ultracold reactions.¹⁹ 

In each of these ultracold reactions, the on/off character is due to a discrete symmetry. Discrete symmetries are of fundamental importance in quantum mechanics, but have no corresponding classical physical meaning. In this work, we examine the ultracold hydrogen exchange reaction, and its non-reactive atom-exchange isotopic counterparts, proceeding from excited rotational states. We show that just as for reactions proceeding from j = 0, large GP effects also appear for $j>0$⁠. In doing so, we highlight the important role that discrete symmetries play in ultracold chemistry and under what conditions we should expect them to control ultracold chemical reactions in general.

We use the atom-diatom scattering formalism as developed by Pack and Parker.^23,24 In the short range, we use adiabatically adjusting-principle-axis hyperspherical coordinates, an approach which ensures that all arrangements are treated fully equivalently, while in the long range, we use Delves hyperspherical coordinates for each arrangement channel. Calculations were performed on the BKMP2 potential energy surface,²⁵ and the vector potential approach of Mead and Truhler was used to include the GP effect.¹⁵ The coupled equations were propagated using the log-derivative method of Johnson.²⁶ Results are well converged in the ultracold regime with total angular momentum up to and including 4 used in all calculations. This approach has been used extensively in recent years to study the role of the GP in the ultracold hydrogen exchange reaction and its isotopic counterparts.^21,22,27,28

Ultracold reactions proceeding from $v$ = 4, j = 0 have been shown to be controlled by the GP.^21,22 The excited vibrational state is needed to overcome the reaction barrier while remaining in the ultracold regime. As such all reactions in this paper proceed from $v$ = 4, and $v$ is omitted from state-to-state labels for brevity. As discussed in the Introduction, the GP controls these reactions due to a number of factors: the quantization of phase shifts in the ultracold; the sign change due to the GP; and the similar magnitude of the two dominant reaction pathways. In this work, we will use the term “controlled by the GP” to mean state resolved rates which change by over an order of magnitude when the GP is included. Such state-to-state rates offer an excellent way to directly measure the GP effect in chemical reactions. We begin by studying the ultracold hydrogen exchange reaction proceeding from excited rotational states.

Figure 1 shows j-resolved reaction rate coefficients for $H$ + $H2(j=1,2)→H$ + $H2(v′,j′)$ calculated at 1 $μ$K, in the Wigner threshold regime. These rates include contributions from both the exchange and non-exchange pathways.²⁰ It is seen that for reactions proceeding from j = 1, the reaction is, just as for j = 0, either on or off. The GP and NGP (no geometric phase) rates differ by about an order of magnitude, with the NGP reaction being on and the GP turning the reaction off. A couple of final states buck this trend, notably $v′=0$⁠, $j′=15$ where this is reversed. However these correspond to final states with rates that are orders of magnitude smaller than for the dominant final states. As such, the vibrational resolved rates also show this on/off character. Reactions proceeding from j = 2, however, show weaker influence of the GP. Here we find that while many channels do not exhibit a strong GP effect, many do, such as $v′=2$⁠, $j′=10$⁠.

H₂ exists in either para or ortho form; for j = 1, there is no pure rotational quenching, whereas for j = 2, there is pure rotational quenching to j = 0. We find that the rate for pure rotational quenching is around 2 orders of magnitude larger than the rates to inelastic channels and does not exhibit a strong GP effect. This is because when there is pure rotational quenching, the non-exchange pathway dominates the reaction, and the sign change along the exchange pathway due to the GP has a small effect.

We now move on to examine the non-reactive atom-exchange isotopic counterparts. These reactions proceeding from j = 0 have been studied in the ultracold regime and shown to exhibit large GP effects in the state-to-state rates.^27,28 Figure 2 shows j-resolved reaction rate coefficients for $D$ + $HD(j=1,2)→D$ + $HD(v′,j′)$ at 1 $μ$K. Just as for H₃, these rates include contributions from both the exchange and non-exchange pathways. Here we see that for j = 1, the GP still controls the reaction; for even symmetry, it turns the reaction on, while for odd symmetry, it turns the reaction off. For reactions proceeding from j = 2, the trend is mostly reversed and weaker; however, there are many final states where the GP changes the rate by over an order of magnitude.

Figure 3 shows the same data as Fig. 2 but for the H + HD(j = 1, 2) $→$ H + HD(⁠$v′,j′$⁠) reaction. In this case, there is not a consistent trend across all final states. However there are regions where the GP controls the reaction, most clearly for $v′=0$ $j′≲5$ where for j = 1 (2), the GP turns on (off) the even symmetry case and turns off (on) the odd symmetry case. Just as in the H₃ case, we find that pure rotational quenching dominates the rate to inelastic channels and does not exhibit a strong GP effect for either of the non-reactive atom-exchange isotopic counterparts (here $j=1→$ 0 and $j=2→$ 0 and 1 are allowed).

It is clear from Figs. 2 and 3 that the GP and exchange symmetry play complementary roles in the non-reactive atom-exchange isotopic counterparts to the hydrogen exchange reaction. The GP rates for each symmetry are well approximated by the NGP rate of the opposite symmetry; this is most clear when the GP controls the reaction but it is also true generally. This complementarity follows from the symmetry of the wavefunction when the GP is included. The characteristic of the GP is that the wavefunction around the CI is double valued and exhibits even symmetry on one side and odd symmetry on the other side. The double-valued GP wavefunction can therefore be accurately represented by a NGP wavefunction (of either even or odd symmetry) but only locally (i.e., only on one side of the CI or the other but not both simultaneously). A NGP wavefunction of suitable symmetry therefore accurately approximates the double-valued GP wavefunction in H₃ type systems as only one side of the CI is accessible. The other side, corresponding to the two transition state pathway, has negligible amplitude.^{20,24,29–32} This equivalence is shown explicitly in Fig. 4 which plots the GP rate vs the NGP rate of the opposite symmetry and is seen to be particularly good for the D + HD case. The points in the top right correspond to the rates for pure rotational quenching which are orders of magnitude larger than for the other inelastic channels. These rates will dominate the total rate and do not exhibit a strong GP effect.

In order to ascertain if the scatter seen at lower energies is due to numerical errors or the breakdown in the symmetry equivalence, we have performed gauge invariance checks for the H + HD case proceeding from j = 1, which shows the most scatter. The vector potential is given by $A=−mA2∇η(x)$⁠, where $∇$ is the gradient operator with respect to the nuclear coordinates x and $η(x)$ is the azimuthal angle around the CI.¹⁵ GP and NGP calculations correspond to odd and even integers for m_A, respectively. Figure 5 compares the GP and NGP calculations already presented (m_A = 0 and 1) to calculations with m_A = 2 and 3. It is seen that there is much less scatter here at low collision energies compared to Fig. 4, meaning the scatter seen is not primarily due to numerical noise but rather the physical breakdown in the symmetry equivalence. The equivalence is based on the assumption that the two transition state pathway is negligible compared to the exchange and non-exchange pathways, but this is not true for very small rates where all of the pathways have negligible contributions.

The total state resolved rates are computed by adding the rates for even and odd exchange symmetry multiplied by the appropriate nuclear spin statistical factor. This summation reduces the overall GP effect in the total rates since the rate is on for one of the symmetries and off for the other. For the D + HD case, since D is a spin 1 boson, the even and odd factors are 2/3 and 1/3, respectively. For the H + HD case, since H is a spin 1/2 fermion, the even and odd factors are 1/4 and 3/4, respectively. The difference between the GP and NGP total rates is thus primarily due to the nuclear spin weighting factor.

On the other hand with nuclear-spin final-state resolution, the even and odd symmetry GP rates can be measured directly. The rates shown in Figs. 2 and 3 are then obtained by multiplying by the appropriate nuclear spin weighting factors. For example, the total experimental rate for D + HD is $ktotexp=23kevnGP+13koddGP$⁠. If instead, the experiment measures the rate for a given symmetry, say even, then $kevnexp=23kevnGP$ and so $kevnGP=32kevnexp$⁠, while for the odd case $koddexp=13koddGP$ and so $koddGP=31koddexp$⁠. Due to the symmetry correspondence between the GP and NGP rates, the experimentally measured rates are well approximated by the NGP rates of opposite symmetry: $koddNGP=32kevnexp$ and $kevnNGP=31koddexp$⁠. By contrast, a calculation which ignores the GP entirely would predict $koddNGP=31koddexp$ and $kevnNGP=32kevnexp$⁠.

We can take advantage of this equivalence to define a constant which quantifies the GP effect expressed entirely in terms of experimentally obtainable rates. We define

where deviation from 1 quantifies the GP effect. It cannot however be obtained from purely experimental measurements as it contains NGP rates. We therefore take advantage of the symmetry equivalence to obtain

For the D + HD case, this yields

where $Δ$ is expressed entirely in terms of experimentally obtainable quantities.

While the examples given thus far are in the Wigner threshold regime, this symmetry-GP equivalence is in fact quite general. Figure 6 shows this equivalence as a function of collision energy for $D$ + $HD(j=1)→D$ + $HD(v′=0,j′=15)$⁠. It is seen that for the non-reactive atom-exchange isotopic counterparts to the hydrogen exchange reaction, any state resolved rate at any energy exhibiting a significant difference between the nuclear-spin resolved even and odd exchange symmetry is exhibiting a strong GP effect. In this case, the presence of the shape resonance at around 1 K leads to a significant GP effect of $Δ≈10$⁠.

We have examined the hydrogen exchange reaction proceeding from initial states $v$ = 4, j = 1 and 2, finding that the GP plays a crucial role in the reaction. For reactions proceeding from j = 1, the GP plays an important role in all final states, whereas for j = 2, the effect is reduced but there are still many final states where the GP has a strong effect.

For the non-reactive atom-exchange isotopic counterparts to the hydrogen exchange reaction, just as in the H₃ case, we find that there are always final states which exhibit a strong GP effect. Due to symmetry, for these reactions, state-to-state rates including the GP are well approximated by NGP rates of the opposite identical-particle exchange symmetry. This symmetry effect can be used to make a measurement of the GP effect. Experimentally this amounts to finding a final state with a large difference between even and odd symmetries, which requires nuclear-spin final-state resolution.

The importance of the GP and identical-particle symmetry shown here reflects the importance of discrete symmetries in ultracold chemical reactions generally, where their effect is magnified. Discrete symmetries have been shown to play a key role in a diverse range of ultracold reactions: $KRb$ + $KRb→K2$ + $Rb2$⁠,⁴ $O$ + $OH→O2$ + $H$⁠,¹⁹ and the hydrogen exchange reaction.²¹ Discrete symmetries are present in all quantum systems exhibiting reflection symmetry and this on/off character is expected to be ubiquitous across ultracold chemistry. This highlights the important role that ultracold reactions can play in understanding fundamental chemical processes more generally.

We acknowledge support from the U.S. Army Research Office, MURI Grant No. W911NF-12-1-0476 (N.B.), and the U.S. National Science Foundation, Grant No. PHY-1505557 (N.B.). B.K.K. acknowledges that part of this work was done under the auspices of the U.S. Department of Energy, Project No. 20170221ER of the Laboratory Directed Research and Development Program at Los Alamos National Laboratory. Los Alamos National Laboratory is operated by Los Alamos National Security, LLC, for the National Security Administration of the US Department of Energy under Contract No. DE-AC52-06NA25396. J.F.E.C. gratefully acknowledges support from the ITAMP visitors program.