Importance of Geometric Phase Effects in Ultracold Chemistry - The

Article pubs.acs.org/JPCA

Importance of Geometric Phase Effects in Ultracold Chemistry Jisha Hazra,† Brian K. Kendrick,*,‡ and Naduvalath Balakrishnan† †

Department of Chemistry, University of Nevada Las Vegas, Las Vegas, Nevada 89154, United States Theoretical Division (T-1, MS B221), Los Alamos National Laboratory, Los Alamos, New Mexico 87545, United States

‡

ABSTRACT: It is demonstrated that the inclusion of the geometric phase has an important effect on ultracold chemical reaction rates. The effect appears in rotationally and vibrationally resolved integral cross sections as well as cross sections summed over all product quantum states. The effect arises from interference between scattering amplitudes of two reaction pathways: a direct path and a looping path that encircle the conical intersection between the two lowest adiabatic electronic potential energy surfaces. It is magnified when the two scattering amplitudes have comparable magnitude and they scatter into the same angular region which occurs in the isotropic scattering characteristic of the ultracold regime (s-wave scattering). Results are presented for the O + OH → H + O2 reaction for total angular momentum quantum number J = 0−5. Large geometric phase effects occur for collision energies below 0.1 K, but the effect vanishes at higher energies when contributions from different partial waves are included. It is also qualitatively demonstrated that the geometric phase effect can be modulated by applying an external electric field allowing the possibility of quantum control of chemical reactions in the ultracold regime. In this case, the geometric phase plays the role of a “quantum switch” which can turn the reaction “on” or “off”.

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INTRODUCTION In molecules, the geometric phase (GP), also known as Berry’s phase,1,2 originates from the adiabatic transport of the electronic wave function when the nuclei follow a closed path encircling a conical intersection (CI) between two electronic potential energy surfaces (PESs). This was first pointed out by Longuet-Higgins3 in 1958 and Herzberg and Longuet-Higgins4 in 1963. They showed that in the presence of a conical intersection the real electronic wave function is no longer a single-valued function but becomes double-valued (i.e., it changes sign) when the corresponding nuclear-motion wave function traverses a closed path encircling the CI. The effect of this sign change may have implications for systems which possess a CI, and the standard Born−Oppenheimer (BO) treatment (where the electronic wave function is solved for fixed nuclear geometries) needs to be modified to account for the sign change. In particular, since the total BO wave function has to be single-valued, a corresponding sign change must also occur in the nuclear-motion wave function. Furthermore, the sign change must be considered appropriately for systems (both atom−molecule and molecule−molecule) which possess one or more identical nuclei in order to satisfy the Bose-Fermi statistics under an interchange of any two identical nuclei. The sign change occurs even when the energy of nuclear-motion lies well below the energy of the CI and the nuclear-motion is confined to just one adiabatic electronic PES. Mead and Truhlar5,6 generalized the BO method for systems with a CI including those with identical nuclei and showed that the sign change can be accounted for by introducing a vector potential or gauge potential (i.e., the momentum operator p → p − A © XXXX American Chemical Society

where A is the vector potential) in the Schrödinger equation for nuclear-motion. In Mead and Truhlar’s approach the real double-valued electronic wave function is multiplied by a complex phase factor which changes sign on encircling the CI so that the electronic wave function becomes complex and single-valued. In this process the nuclear-motion Schrödinger equation acquires the vector potential which originates from the nonzero diagonal derivative coupling. The resulting Schrödinger equation for nuclear-motion is mathematically identical to that of a charged particle moving in the presence of a “magnetic solenoid” centered at the CI. Mead initially called this the “Molecular Aharonov-Bohm effect”6 and found that when the geometric phase is incorporated in reactive scattering calculations, a sign change is introduced in the interference term between the purely reactive and quenching cross sections for the benchmark H + H27,8 system. The sign change led to oscillations in the angular dependence of its differential cross sections (DCS), but its effect on integral cross sections for chemical reactions has remained elusive until recently.9,10 The H + H2 reaction and its isotopologues have long served as benchmark systems for experimental verification of the GP effect in chemical reactions.11−18 Unfortunately, in all these studies the GP effects were found to be vanishingly small over a Special Issue: Dynamics of Molecular Collisions XXV: Fifty Years of Chemical Reaction Dynamics Received: July 3, 2015 Revised: August 20, 2015

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DOI: 10.1021/acs.jpca.5b06410 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A

wave resolved DCSs vanish when integrated over the scattering angles and summed over all partial waves to obtain integral cross sections. The O + OH reaction appears to be an excellent candidate for the study of the GP effect on ultracold chemical reactions owing to its barrierless pathway and favorable encirclement of the CI even at low collision energies.9 It continues to be the topic of active theoretical and experimental investigations both in the ultracold and thermal energy regimes.30−36 The OH radical has also been successfully cooled and trapped by buffer gas and stark deceleration techniques,37−39 and promising experiments on evaporative cooling of OH have recently been reported.40 Though experimental measurements of its rate coefficients at temperatures lower than 30 K have not been reported, we believe current progress in cooling and trapping of polar molecules and magnetic atoms will allow experimental investigations of the O + OH reaction at sub-Kelvin temperatures in the near future. In this paper, we extend our previous study of the O + OH(v = 0,j = 0) → H + O2(v′,j′) reaction for J = 0−39 by including total angular momentum quantum numbers up to J = 5. Also, results are presented for vibrationally excited OH in the v = 1 vibrational level. We analyze sensitivity of the GP effect to details of the interaction potential by varying both the shortrange and long-range intermolecular forces. The GP seems to alter the number of bound states in the effective potentials traversed by the looping and direct pathways when the threebody (short-range) interaction is tuned. This leads to strong modulations of the GP effect. We also discuss scenarios of controlling ultracold reaction rates by switching the GP effect “on” or “off” with an applied electric field. The paper is organized as follows: The Theory section presents a brief account of the theoretical approach adopted for the scattering calculations and the interference mechanism leading to the GP effect. The Results and Discussion section presents the scattering results, including the PES scaling studies, and the key results are summarized in the Conclusions.

wide range of energies. All of the experimental results reported thus far involve an average over a large number of partial waves or impact parameters. Though, the effect may manifest in a partial wave resolved calculation, it washes out when a summation/average over all contributing partial waves is made for comparison with experiment. This was explicitly demonstrated by Kendrick through accurate calculations of the H + D2 and D + H2 reactions using a time-independent quantum approach in hyperspherical coordinates.11,13 His findings were subsequently confirmed by Juanes-Marcos and Althorpe15−17 who employed a time-dependent quantum wave packet method to solve the Schrödinger equation. A very recent combined experimental and theoretical investigation by Jankunas et al.18 was unsuccessful in confirming the GP effect in the H + HD hydrogen exchange reaction. For systems such as H + H2 which proceed via a direct collinear mechanism, the scattering amplitudes for the looping path that encircles the CI via two transition states are small. Consequently, very little interference occurs between direct and looping pathways, and no observable GP effect is present. Theoretical studies of GP effects in other experimentally relevant chemical systems such as H + O2 have been performed by Kendrick et al.19−21 Pronounced GP effects on the bound-state spectrum of HO2, Na3, and N3 have also been shown to occur.21−23 All previous investigations of GP effects in chemical reactions have been done at thermal energies where the contributions from many values of total angular momentum tend to wash out the effect. The situation changes dramatically when the GP effect is explored in an energy regime where only a single partial wave contributes to the reaction. This is the case for ultracold chemical reactions where s-wave (i.e., isotropic) scattering dominates and the scattering phase-shift becomes effectively quantized (for systems with an attractive potential well). Recent experimental progress in cooling and trapping of molecules in the cold ( 0.9. For v′ = 1 (red bars in panel a) nearly 60% of the rotational levels correspond to |cos Δ| > 0.75 and half of these lie above 0.9. In panel b about 50% of the product states of v′ = 2 (black bars) correspond to |cos Δ| values above 0.75 and 70% of these lie above 0.9. Similarly, for v′ = 3 (red bars) nearly 45% of the rotational states have |cos Δ| > 0.75 of which 70% correspond to |cos Δ| > 0.9. Differential Cross Sections. All of the results presented above correspond to integral cross sections. However, most previous observations of GP effects at thermal energies have been restricted to differential cross sections. Though scattering is isotopic in the s-wave limit where the GP effect dominates, it would still be interesting to examine differential cross sections for selected transitions in the energy regime where a few higher partial waves begin to contribute. Figure 6 displays threedimensional plots of differential cross sections as a function of both energy and scattering angle for the O + OH(v = 0,j = 0) → H + O2(v′ = 2,j′) reaction for j′ = 1, (top panels a and b), j′ = 3 (middle panels c and d), and j′ = 5 (bottom panels e and f). Cross sections for these three final states were reported in our previous work (see Figures 3 and 4 of Kendrick et al.9). The NGP results are shown in the left panels, and the GP results are in the right panels. It is seen that for a given j′ state the GP and NGP results show distinctly different energy and angular dependence. When the GP rate dominates (as in j′ = 1 and 5) the DCSs display an isotropic distribution both in scattering angle (s-wave scattering) and scattering energy (Wigner threshold behavior). However, since the NGP rate is suppressed for these cases, the corresponding DCS is smaller in magnitude and the angle and energy dependence is more pronounced (because of contributions from higher partial waves that are not masked by the s-wave contribution). The same occurs when the NGP rate dominates (j′ = 3). In this case, the GP result becomes similar to the NGP result for j′ = 1 and 5. F


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Figure 6. Differential cross sections for O + OH(v = 0,j = 0) → H + O2(v′ = 2,j′) reaction for j′ = 1 (top panel a and b), j′ = 3 (middle panel c and d), and j′ = 5 (bottom panel e and f) as a function of the collision energy and scattering angle. In each panel the figures on the left correspond to NGP results (a, c, e) and those on the right correspond to GP results (b, d, f). Results include contributions from J = 0−5.

factors of 25 (v′ = 2,j′ = 11 in panel b), 15 (v′ = 3,j′ = 15 in panel d) and 4 (v′ = 4,j′ = 3 in panel f). For these cases, cos Δ ≈ +1 and |f NGP|2 = 2f 2 and |f GP|2 = 0. The variation of ⟨cos Δ⟩ and the ratio

|f loop |2 |f direct |2


and cos Δ are also given. For an easy comparison, the

ratio between the GP and NGP rate is also given in the last column. Most notable cases of large GP effects are j′ = 31 of v′ = 0 and j′ = 11 of v′ = 3 where GP rates exceed NGP rates by a factor of 27 or more. There are a few other states that show about a factor of 10 enhancement of the GP rates. Cases that show suppression of the GP effect correspond to j′ = 15 and 31 within v′ = 2. In both cases, cos Δ ≈ +1 and the ratio of GP/ NGP rates is 0.04, a suppression of the GP effect by a factor of 25. Among the 15 states listed in Table 2, 10 correspond to cos Δ ≈ −1 which leads to enhanced GP effects. In Figure 10, the rotationally resolved quenching rates computed with and without the GP effect for the O + OH (v = 1,j = 0) → O + OH(v′ = 0,j′) inelastic process are shown as a

vs collision energy behave similar to that

described in Figure 2 and Figure 3 for J = 0 (black) and J = 1 (red). To avoid repetition we do not show them here. As the collision energy increases above 10 mK, J = 1 (i.e., the l = 1 partial wave) begins to contribute and the GP and NGP rates approach each other. In addition to the results presented in Figure 9, significant GP effects are also found in other open rotational states of the O2 product. The GP and NGP rates of these rotational levels are presented in Table 2 at 1 μK. The corresponding values of G


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Figure 9. Rotationally resolved reaction rates for O + OH (v = 1,j = 0) → H + O2(v′,j′) reaction are plotted as a function of incident kinetic energy for j′ = 7 (a) and 11 (b) of v′ = 2; j′ = 9 (c) and 15 (d) of v′ = 3; and j′ = 1 (e) and 3 (f) of v′ = 4. The rates are summed over J = 0 and 1. At 1 μK, in panels a, c, and e, the GP (red) rates lie above the NGP (black) rates, whereas in panels b, d, and f, the GP (red) rates lie below the NGP (black) rates.

Figure 7. Differential cross sections for O + OH(v = 0,j = 0) → H + O2(v′ = 2,j′) reaction for j′ = 1 (top panel (a)), j′ = 3 (middle panel (b)), and j′ = 5 (bottom panel panel (c)) multiplied by sin θ and plotted as a function of the scattering angle θ. The scattering energy is 10−5 eV = 0.116 K and contributions from J = 0−5 are included.

Table 2. Ultracold (1 μ K) Rate Coefficients for the O + OH(v = 1, j = 0) → O + OH(v′, j′) Reaction with and without Geometric Phase Effectsa

Figure 8. Rate coefficient for O + OH (v = 1,j = 0) → H + O2(v′) reaction plotted as a function of incident kinetic energy: v′ = 2 (panel a), v′ = 3 (panel b), and v′ = 4 (panel c). The rates include contributions from J = 0 and 1. In panels b and c, the GP rates (red) lie above the NGP rates (black), whereas the opposite occurs in panel a.

v′

j′

0 0 0 0 1 1 1 2 2 2 3 3 4 4 4

17 19 27 31 5 11 17 15 19 31 11 23 5 7 11

NGP rate (cm3/s) 6.50 5.08 1.22 5.58 1.60 1.22 1.00 5.37 2.65 1.64 1.83 4.04 5.23 2.71 9.59

× × × × × × × × × × × × × × ×

10−14 10−13 10−14 10−16 10−15 10−14 10−13 10−13 10−13 10−14 10−14 10−14 10−13 10−13 10−14

GP rate (cm3/s) 4.17 3.81 7.97 1.64 2.34 1.62 3.85 1.95 3.51 5.96 5.82 4.19 1.22 7.37 3.62

× × × × × × × × × × × × × × ×

10−13 10−14 10−14 10−14 10−14 10−13 10−13 10−14 10−14 10−16 10−13 10−13 10−12 10−14 10−13


0.29 2.86 0.31 0.53 0.43 1.51 0.11 0.46 0.43 1.22 0.73 0.41 0.10 0.15 0.19

cos(Δ) −0.87 0.98 −0.87 −0.98 −0.95 −0.88 −0.96 0.99 0.97 0.93 −0.95 −0.91 −0.69 0.85 −0.79

GP rate NGP rate

6.42 0.08 6.55 29.49 14.63 13.32 3.83 0.04 0.13 0.04 31.72 10.34 2.34 0.27 3.77

a

The ratio of the average square modulus between the looping and direct pathways, cos Δ, and the ratio of the GP and NGP rates are listed.

modulation in the magnitudes of the GP and NGP rates for different rotational levels as in the H + O2 channel. To illustrate the behavior of cos Δ between various product states of the O + OH (v = 1,j = 0) → H + O2 (v′,j′) reaction and O + OH (v = 1,j = 0) → O + OH(v′ = 0, j′) inelastic process we plot two histograms at 1 μK in Figure 11, where panel a corresponds to the reactive process and panel b corresponds to the quenching process. The percentage occurrence of all open product states within a given |cos Δ| interval is presented here. Like in Figure 5 the interval spacing or “bin size” along the horizontal axis is 0.05. It is strikingly clear from the two panels that the distributions are biased toward the right edge where |cos Δ| ≈ 1. Approximately, 50% of the open rotational states of O2 in the O + OH (v = 1,j = 0)

function of incident kinetic energy. For the inelastic reaction, the GP and NGP rates arise from the competition between the oxygen exchange reaction and pure inelastic quenching (as reported in ref 10 for H + H2 exchange reaction). The panels a, b, c, d, e, and f correspond to the rotational levels j′ = 0, j′ = 3, j′ = 9, j′ = 10, j′ = 11 and j′ = 13 of OH(v′ = 0), respectively. The results include contributions from J = 0−1. Like in reactive scattering, large GP effects appear in the quenching rates, particularly for the cases in panels d, e, and f. For these rotational levels, the NGP rates are greater than the GP rates by factors of 5, 7, and 9, respectively. On the other hand, the difference between the GP and NGP rates in panels a, b, and c is moderate, that is, a factor of approximately 2. The GP effect is negligible in the total quenching rate coefficient due to H


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potential to qualitatively account for the effect of an external electric field on ultracold reactivity. This third modification affects the entire PES (both the short and long-range). Scaling of the Short-Range Potential. The sensitivity of rotationally resolved GP and NGP rate coefficients to the inner part of the PES is presented in Figure 12 and Figure 13 for O +

Figure 10. The rotationally resolved quenching rates for O + OH (v = 1,j = 0) → O + OH(v′ = 0, j′) collisions as a function of incident kinetic energy for j′ = 0 (a), 3 (b), 9 (c), 10 (d), 11 (e), and 13 (f) of v′ = 0. The results include contributions from J = 0−1.

Figure 12. Variation of the GP (red) and NGP (black) rotationally resolved rate coefficient of O + OH (v = 0,j = 0) → H + O2(v′ = 2,j′ = 1) reaction as a function of % change in the PES for J = 0 at 1 μK (panel a). The corresponding ⟨cosΔ⟩ and


are plotted in

panels b and c, respectively.

Figure 11. Same as in Figure 5 except that here panel a represents all open product states of O2 in the O + OH (v = 1,j = 0) → H + O2 (v′,j′) reaction, and panel b describes all open states of OH in the O + OH (v = 1,j = 0) → O + OH(v′ = 0, j′) inelastic process.

→ H + O2 (v′,j′) reaction correspond to |cos Δ| > 0.75, and 64% of these have |cos Δ| > 0.9. Similarly, about 70% of the open rotational states of OH in panel b correspond to |cos Δ| > 0.75, and half of these are characterized by |cos Δ| > 0.9. Sensitivity of GP and NGP Rates to the Potential Energy Surface. While the magnitude of the GP effects presented so far depends on the details of the PES, the dynamical origin (i.e., the interference mechanism) of the GP effect does not. Here we explore sensitivity of the GP effects to small changes in the PES and investigate whether the GP/NGP reaction rates can be controlled or manipulated by modifying the interaction potential. Three different modifications of the interaction potential are explored here. First, we slightly modify the inner part of the PES (where the GP effect is important) while keeping the rest of the PES unchanged. In the second case we vary only the long-range coefficient, C6, while keeping the inner part of the PES intact. In the last case, we have added energy shifts comparable to Stark shifts to the diatomic OH

Figure 13. Same as in Figure 12 but for the O + OH (v = 0,j = 0) → H + O2(v′ = 2,j′ = 3) reaction.

OH (v = 0,j = 0) collisions. The former corresponds to the H + O2(v′ = 2,j′ = 1) product state, whereas the latter refers to the H + O2(v′ = 2,j′ = 3) channel. In both cases results are presented for a collision energy of 1 μK where only s-wave (J = l = 0) contributes. In panel a of both figures, the GP (red) and NGP (black) rates are plotted as a function of the percentage change in the PES. The modification corresponds to a scaling of the PES within ±2%. A 0% PES scaling means no modification of the PES, and the results correspond to those reported in ref 9. Panels b and c present the corresponding cosine of the phase difference between the scattering I


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The Journal of Physical Chemistry A amplitudes for the looping and direct pathways (⟨cos Δ) and the ratio


, respectively. The sensitivity of the rate

coefficients to small changes in the PES is striking in both figures. Both of the GP and NGP rates are significantly modified by tuning the short-range part of the PES. As Figure 12 illustrates, the GP rates are suppressed by several orders of magnitude for scaling factors of −0.75 and −0.5% (i.e., they completely “turn off” the reactivity) whereas % scaling factors of −0.25, −0.10, 0, +0.10, + 0.25, and +0.5 enhance the GP rates (i.e., they “turn on” the reactivity). Figure 13 shows that the GP rates are suppressed by several orders of magnitude for scaling factors in the range −0.25 to +0.5%. From panels b and c of both figures it is clear that the large NGP/GP rates occur when either ⟨cos Δ⟩ ≈ ±1 or the ratio


≈ 1. We note

that, in panel b of Figure 12, the rapid change in sign of cos Δ (+1 to −1) occurs when the scaling factor is varied from −0.5% to −0.25%. This is because the number of bound states supported by the two effective potentials, Vlooping/direct (as effct discussed in refs 9 and 10) along the looping and direct pathways, are changed. A change in sign of cos Δ occurs also in panel b of Figure 13 for scaling factors between −0.5% and −0.25% but the effect is not as dramatic as in Figure 12. Thus, the modulation of the GP/NGP rate can be attributed to the change in bound state structure of the HO2 well as the depth of the PES is altered by the scaling factors. For some values of the PES scaling factors the GP and NGP rates become comparable. This occurs for scaling factors −1.5, + 0.75, and +1.0% in Figure 12 and for −0.75, −0.50, and +1.5% in Figure 13, respectively. For the former cases (Figure 12), either the values of cos Δ ≈ 0 (for scaling factors −1.5 and +0.75) or


Figure 14. Effect of modifying the C6 coefficient on the GP (green and blue) and NGP (black and red) rotationally resolved reaction rate coefficient for the O + OH (v = 0,j = 0) → H + O2(v′ = 2,j′ = 1) reaction as a function of the collision energy. The effect of negative variations are shown in black (NGP) and blue (GP), whereas the positive variations are depicted by red (NGP) and green (GP) curves. The overall variation is monotonic in both cases.

long-range coefficient alone does not significantly alter the interference between the direct and looping pathways. The Effect of an External Electric Field. Since controlled studies of ultracold reactions with external electric and magnetic fields are actively being pursued in many research groups it would be interesting to explore how the GP effect is modified by an external electric field. An explicit calculation of the GP/NGP rate coefficient in the presence of an electric field is beyond the scope of this study. Instead, we will approximately include the Stark effect through an energy shift of the OH potential by an amount equivalent to the Stark splitting of the OH levels in an electric field. This energy shift ultimately affects the bound levels of HO2. Because of its permanent dipole moment (1.67 D) the Stark energy shift for OH in a typical laboratory external DC electric field of 100 kV/ cm is ∼4 K, which is comparable to the energy differences between the highest lying bound states of HO2, ∼5 K.21 Hence, the relative number of bound states between the direct and looping pathways could be modified by applying an external electric field. Figure 15 displays the rotationally resolved reaction rates for J = 0 at 1 μK as a function of the applied electric field. As before the red and black curves denote the GP and NGP rates, respectively. The results are presented for three rotationally resolved final states in O + OH(v = 0,j = 0) → H + O2(v′,j′) reaction: (a) v′ = 0,j′ = 19 (solid curves); (b) v′ = 1,j′ = 3 (dashed curves); (c) v′ = 1,j′ = 9 (dashed-dotted curves). The variation of the NGP rates for these three cases with the electric field is not dramatic, but is rather systematic. In contrast, the GP rates are strongly affected by the electric field. At low to moderate electric field strengths (below 200 kV/cm) the GP and NGP rates exhibit similar trends. However, for electric field strengths in excess of 300 kV/cm a drastic suppression of the GP rate is observed for all three cases. For v′ = 0,j′ = 19 and v′ = 1,j′ = 3, the lowest value of the GP rate is observed for an electric field of 500 kV/cm. For v′ = 1,j′ = 9 it occurs at 450 kV/cm. This can be explained by the ⟨cos Δ⟩ and

> 1 (for scaling factor +1.0). For the latter

case (Figure 13), cos Δ ≈ 0 and


< 1 for all three

values of the scaling factor. Here, the interference effect is less important because of the different magnitudes of the looping and direct scattering amplitudes, and from eq 2 one obtains 1 1 |f NGP/GP |2 = 2 |f direct |2 or |f NGP/GP |2 = 2 |f loop |2 . Thus, in these cases little or no GP effect is observed. Effect of Modifying the Long-Range Part of the Interaction Potential. Figure 14 describes how the rates computed with and without the GP are affected when the C6 coefficient is varied by ±4%. Only the long-range part of the PES is modified by this procedure. This is illustrated for the O + OH (v = 0,j = 0) → H + O2(v′ = 2,j′ = 1) reaction as a function of the incident kinetic energy for J = 0. The GP rates are denoted by the green and blue curves, whereas the NGP rates are shown in black and red curves. The black (NGP) and blue (GP) curves correspond to the negative variations, while the red (NGP) and green (GP) curves denote the positive variations. It is seen that both GP and NGP rates are comparably affected by the scaling, and the effect is more pronounced in the ultracold limit where results are generally more sensitive to long-range forces. The effect vanishes in both rates for energies above 0.1 K. The overall effect seems to be monotonic throughout the entire energy range and the rates computed with the GP are found to be greater than the NGP rates. These findings are not surprising because to see the large GP effect observed in the previous study one has to modify the inner part of the PES where the GP is important. Tuning the

the ratios J


shown in panels a and b of Figure 16. For DOI: 10.1021/acs.jpca.5b06410 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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the suppression of the GP effect in this case is not due to the change in bound state structure of the HO2 complex. This is because ⟨cos Δ⟩ remains approximately +1 above 300 kV/cm and it does not switch between the extreme values (±1) as it does in the PES scaling in panel b of Figure 12. Therefore, the relative number of bound states between the looping and direct pathways are not modified by this simple model for the inclusion of an electric field. Instead, the suppression of the GP rate occurs due to the ratio


passing through unity. A

more detailed study with a rigorous account of the Stark effect is needed to investigate GP effects on chemical reactions in the presence of an electric field. Although fairly high electric field strengths are used in our illustrative calculations, field strengths beyond 200 kV/cm may be beyond current experimental capabilities.

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Figure 15. A comparison of the GP (red) and NGP (black) rates for O + OH (v = 0,j = 0) → H + O2(v′,j′) reaction as a function of the applied electric field at 1 μK for J = 0. The solid, dashed, and dasheddotted curves represent the rates for (a) O2(v′ = 0,j′ = 19), (b) O2(v′ = 1,j′ = 3), and (c) O2(v′ = 1,j′ = 9) products, respectively.

Figure 16. Variation of ⟨cos Δ⟩ (panel a) and the ratio

CONCLUSIONS We have carried out an extensive study of the geometric phase (GP) effect in chemical reactions at cold and ultracold collision energies by taking the O + OH(v,j)→ H + O2(v′,j′) reaction as an illustrative example. The geometric phase is included using a vector potential approach within a coupled-channel formulation of the Schrödinger equation in hyperspherical coordinates. The GP effect is investigated for reactions involving v = 0 and 1 of the OH molecule. For the v = 0 reaction, total angular momentum quantum numbers J = 0−5 are included, while for the vibrationally excited reaction, results are presented for J = 0−1. For the latter case, the GP effect is investigated both in the H + O2 reactive channel and the vibrational quenching channel leading to O + OH(v′ = 0) product. In all cases, the GP effect was found to occur most prominently in the rotationally resolved rate coefficients but it was also present at a smaller extent in vibrationally resolved and total reaction rates. A detailed comparison of results from calculations with (GP) and without the geometric phase (NGP) is presented, and the relative magnitude of the GP and NGP rate coefficients is explained through a simple model involving a direct path and looping path which encircle the conical intersection between the two lowest adiabatic potential energy surfaces. It was found that when the direct and looping contributions to the scattering amplitudes have comparable magnitude and when they scatter into the same angular region (s-wave, isotropic scattering), the GP/NGP reaction rate is maximized depending on the relative phase difference between the two scattering pathways. Because of the isotropic nature of the scattering in the ultracold regime, the scattering phase shift attains “quantized” values (i.e., it approaches nπ where n is an integer). The phase quantization and isotropic scattering result in maximum constructive or destructive interference which controls the chemical reactivity in the ultracold regime. We have also examined the differential cross sections as a function of the collision energy and scattering angle. It was found that when the GP/NGP rates dominate, the corresponding differential cross sections exhibit an isotropic behavior in energy and scattering angle, reflecting the dominance of isotropic scattering and Wigner threshold behavior in the ultracold limit (s-wave limit). When they are suppressed, a more anisotropic dependence on scattering angle and energy is observed due to contributions from nonzero partial waves. Finally, we have explored the sensitivity of GP/NGP results to the short-range and long-range part of the interaction


(panel b) as a function of the electric field at 1 μK for the three different reaction rates presented in Figure 15.

electric fields of 0−200 kV/cm, the magnitude of the direct and looping contributions to the scattering amplitudes is widely different, i.e. the ratios


are either ≫1 for reaction (b)

and (c) or

Importance of Geometric Phase Effects in Ultracold Chemistry - The

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