Controlling rotational quenching rates in cold molecular collisions

TL;DR

This paper demonstrates how the rotational quenching rates in cold molecular collisions can be controlled by adjusting the relative orientation of the molecules, with potential applications to similar systems.

Contribution

It introduces a method to manipulate rotational quenching rates through molecular alignment, based on helicity considerations, in cold molecular collision systems.

Findings

Quenching rate maximized when HD is aligned along the collision axis.

Quenching rate minimized at the magic angle.

Orientation control influences chemical process rates.

Abstract

The relative orientation of colliding molecules plays a key role in determining the rates of chemical processes. Here we examine in detail a prototypical example: rotational quenching of HD in cold collisions with H2. We show that the rotational quenching rate from j=2 -> 0, in the v=1 vibrational level, can be maximized by aligning the HD along the collision axis and can be minimized by aligning the HD at the so called magic angle. This follows from quite general helicity considerations and suggests that quenching rates for other similar systems can also be controlled in this manner.

Equations

$${\sigma }_{\alpha \to {\alpha }^{\prime }}=$$

$$q(\theta )=\frac{1}{2k_{\alpha }}$$

$$\frac{d{\sigma }_{\alpha m_{12}\to {\alpha }^{\prime }{m}_{12}^{\prime }}}{d\Omega }$$

$$\frac{d{\sigma }_{\alpha \to {\alpha }^{\prime }}}{d\Omega }=\sum _{m_{12}}\frac{d{\sigma }_{\alpha m_{12}\to {\alpha }^{\prime }{m}_{12}}}{d\Omega }.$$

$$\sum _{m=-j}^j{d}_{\tilde{m},m}^j(\beta )|j,m⟩,$$

$$\sqrt{\frac{3}{8}}|j_1=2,m_1=-2⟩$$

$$\frac{d{\sigma }_{\alpha ,m_{12}\to {\alpha }^{\prime },m_{12}^{\prime }}}{d\Omega }=\frac{1}{(2j_2+1)}$$

$$\times \sum _{m_2,m_1^{\prime },m_2^{\prime }}|\sum _{m_1}{d}_{0,m_1}^{j_1}(\beta )\times q_{\alpha ,m_1,m_2,m_{12}\to {\alpha }^{\prime },m_1^{\prime },m_2^{\prime },m_{12}^{\prime }}{|}^2.$$

$$\frac{d{\sigma }_{\alpha ,m_{12}\to {\alpha }^{\prime },m_{12}^{\prime }}}{d\theta }=\frac{2\pi \sin \theta }{(2j_2+1)}$$

$$\times \sum _{m_1,m_2,m_1^{\prime },m_2^{\prime }}|d_{0,m_1}^{j_1}(\beta ){|}^2|q_{\alpha ,m_1,m_2,m_{12}\to {\alpha }^{\prime },m_1^{\prime },m_2^{\prime },m_{12}^{\prime }}{|}^2.$$

$$\frac{d{\sigma }_{\alpha \to {\alpha }^{\prime }}}{d\theta }$$

$$\frac{1}{\sqrt{6}}|j_1=2,m_1=-2⟩$$

Full text

I Introduction

Across many areas of chemistry the dynamics of collision processes are determined to a large extent by the relative orientation of colliding molecules Bernstein et al. (1987); Levine and Bernstein (1987); Orr-Ewing and Zare (1994); Orr-Ewing (1996); de Miranda and Clary (1997); J. Alexander et al. (1998); Aldegunde et al. (2005); Jambrina et al. (2016). In the cold and ultracold regime we can study such dynamical effects with exquisite precision Ospelkaus et al. (2010); Knoop et al. (2010); Perreault et al. (2017); Wolf et al. (2017); Kendrick et al. (2015); Croft et al. (2017a, b); Balakrishnan (2016); Bohn et al. (2017).

In a recent series of papers Perreault et al. have examined how changing the relative orientation of HD molecules in cold collisions with H2 and D2 affects the angular distribution of inelastically scattered HD Perreault et al. (2017, 2018a). In the experiments HD molecules were prepared in the $v=1,j=2$ state and the angular distribution of the scattered HD for the $v=1,j=0$ final state was measured. The low collision energy and light masses of the collision partners combined with the choice of the initial and final states limit the number of partial waves involved in the collision process. This was confirmed by subsequent theoretical studies which revealed that the angular distribution of the scattered HD in collisions with H2 was dominated by a single ($l=2$) partial-wave shape resonance as around 1 K Croft et al. (2018). Collisions of HD with H2 are also of current astrophysical interest as the HD molecule is believed to have played an important role in the cooling of the primordial gas in the formation of the first stars and galaxies Galli and Palla (2013); Balakrishnan et al. (2018); Desrousseaux et al. (2018).

Here we examine how rotational quenching rates in cold molecular collisions can be controlled by changing the relative alignment of one of the collisions partners, taking collisions of HD with H2 as a prototypical example. We show that in order to maximize the rate of rotational quenching of HD from $j=2\to 0$ the HD should be aligned along the collision axis wheres to minimize the quenching rate the HD should be aligned at the so called magic angle.

II Methodology

The H4 system and its isotopologues contain just 4 electrons, as such high quality ab initio potential energy surfaces are available. In this work we have used the full-dimensional H2-H2 potential of Hinde Hinde (2008), which is in good agreement with other available surfaces Boothroyd et al. (1991); Patkowski et al. (2008). Collisions of H2 dimers and their isotopologues are also amenable to quantum scattering calculations, due to the relatively shallow interaction potential and low density of states of their energy level structure. Schaefer and Meyer (1979); Lin and Guo (2002); Pogrebnya and Clary (2002); Gatti et al. (2005); Quéméner et al. (2008).

Scattering calculations for collisions of HD with H2 were performed in full-dimensionality using a modified version of the TwoBC code Krems . This methodology has been applied to many other similar systems Yang et al. (2015, 2018, 2016); dos Santos et al. (2013); Croft et al. (2018), and is outlined in detail elsewhere Quéméner et al. (2008); Quéméner and Balakrishnan (2009); dos Santos et al. (2011). Here we briefly review the methodology in order to define notation. The scattering calculations are performed within the time-independent close-coupling formalism which yields the scattering $S$ matrix Arthurs and Dalgarno (1960). For convenience, we label each asymptotic channel by the combined molecular state $\alpha=v_{1}j_{1}v_{2}j_{2}$, where $v$ and $j$ are vibrational and rotational quantum numbers respectively, in this work the subscript 1 refers to HD and 2 to H2. The integral cross section for state-to-state rovibrationally inelastic scattering is given by,

[TABLE]

where $k^{2}=2\mu E/\hbar^{2}$ is the square of the wave vector, $E$ is the collision energy, $\mu$ is the reduced mass, $T^{J}=1-S^{J}$, j12 = j1 + j2, $l$ is the orbital angular momentum quantum number, and $J$ the total angular momentum quantum number J = l + j12. To compute differential cross sections we also need the scattering amplitude as a function of the scattering angle $\theta$, which has previously been given by Schaefer et al. Schaefer and Meyer (1979) in the helicity representation:

[TABLE]

where $d^{J}_{m_{12},m^{\prime}_{12}}(\theta)$ is Wigner’s reduced rotation matrix. The rovibrational helicity resolved differential cross section is then obtained by summing over $m^{\prime}_{1}$ and $m^{\prime}_{2}$ and averaging over $m_{1}$ and $m_{2}$,

[TABLE]

where $d\Omega$ is the infinitesimal solid angle $\sin\theta d\theta d\phi$. Helicity resolved cross sections are useful as $m_{12}\to m_{12}$ transitions conserve $j_{z}$ in the body-fixed frame. This is the assumption made in the coupled-states approximation Pack (1974); McGuire and Kouri (1974); McGuire (1975) where the differential cross section is given by

[TABLE]

The coupled-states approximation has been shown to be generally valid for inelastic and reactive collisions away from resonances Krems and Nordholm (2001); Quéméner and Balakrishnan (2008); Schaefer and Meyer (1979).

The cross section formulas presented so far are derived assuming that the relative orientation of the colliding molecules is uncontrolled — by averaging over initial projection quantum numbers ($m_{1}$, $m_{2}$, and $m_{12}$) and summing over final projection quantum numbers ($m^{\prime}_{1}$, $m^{\prime}_{2}$, and $m^{\prime}_{12}$). Here, however, we are interested in the cross section when the relative orientation of the collision partners is controlled in the manner described in the experiments of Perreault et al. on rotational quenching of HD in cold collisions with H2 and D2 Perreault et al. (2017, 2018a, 2018b). In those experiments the HD molecule was prepared in a state $|j=2,\tilde{m}=0\rangle$ by Stark-induced adiabatic Raman passage (SARP) with the quantization axis of $\tilde{m}$ determined by the orientation of the linear polarization of the SARP laser Zare (1982); Mukherjee et al. (2018). Choosing $\beta$ as the angle between the linear polarization of the SARP laser and the beam velocity in the laboratory frame, the state of a molecule prepared in a rotational state $|j,\tilde{m}\rangle$ can be expressed as

[TABLE]

in terms of projections $m$ onto the relative velocity axis. In the work of Perreault et al. the HD was prepared in the $v=1,j=2$ level with two different orientations, $\beta=0$ and $\frac{\pi}{2}$ referred to as HSARP and VSARP respectively, where the H and V refer to the horizontal and vertical alignment of the laser polarization respectively. HSARP therefore corresponds to initial state $|j_{1}=2,m_{1}=0\rangle$ while VSARP corresponds to,

[TABLE]

The helicity-resolved differential cross section for molecular collisions of molecules prepared using the SARP method is therefore given by

[TABLE]

Integrating over $\phi$ and taking advantage of the cylindrical symmetry of the problem we obtain the cross section as a function of the scattering angle $\theta$,

[TABLE]

We can now see that the effect of orienting the collision partners is to change the relative weighting of the helicity resolved cross sections.

III Results

We first consider off resonant collisions between HD and para-H2 due to its relative simplicity. As discussed helicity is approximately conserved for off resonant collisions and the angular momentum algebra is simplified since $j_{2}=0$ and so $m_{12}=m_{1}$. Fig. 1 shows the helicity resolved differential rate for collisions of HD($v=1$, $j=2$) with para-H2 at 1 K which is off-resonant Croft et al. (2018). As expected the dominant contribution is from $0\to 0$ which conserves helicity. In this case Eq. (8) is simplified as $m_{2}=0$ and so $m_{12}=m_{1}$ and $m^{\prime}_{1}=m^{\prime}_{2}=m^{\prime}_{12}=0$

[TABLE]

The effect of changing the relative alignment of the HD by $\beta$ is to change the relative contribution from each helicity - which gives us a handle with which to control the outcome. For example if we were interested in maximizing the cross sections we would want to maximize the helicity conserving contribution $0\to 0$. Fig. 2 shows the square of Wigner’s reduced rotation matrix $|d^{2}_{0,m}(\beta)|^{2}$ as a function of the polar angle $\beta$. In order to maximize the cross section we choose $\beta=0$ as the only contribution to the integral cross section that conserves helicity ($m_{12}=m_{1}=0\to m^{\prime}_{12}=m^{\prime}_{1}=0$) - which corresponds to HSARP in the work of Perreault et al. Whereas to minimize the integral cross section we choose $\beta=\beta_{m}=\arccos\frac{1}{\sqrt{3}}\approx 54.7\degree$ the so called magic angle, which corresponds to an initial state

[TABLE]

as then none of the contributions to the integral cross section conserve helicity ($m_{12}=m_{1}=\pm 1,\pm 2\to m^{\prime}_{12}=m^{\prime}_{1}=0$).

Fig. 3 shows the integral cross section as a function of the collision energy for different values of $\beta$ para-H2 collisions. It is seen that as expected $\beta=0$ leads to the largest cross section and the magic angle leads to the one of the smallest cross sections at any given energy. We would expect this approach to work for rotational quenching quite generally, at least away from resonances, as the contribution from $0\to 0$ will always be one of the helicity conserving contributions and to work especially well when quenching to a state where $j^{\prime}_{1}=j^{\prime}_{2}=0$, as we have here, where it would be the only helicity conserving contribution. The angle $\beta$ where $|d^{j}_{0,0}(\beta)|^{2}=0$ would however change depending on the initial rotational state $j$ such that $P_{j}(\cos\beta)=0$ where $P_{j}$ is the Legendre polynomial of degree $j$.

We now move on to the more complicated case of collisions of HD with ortho-H2 on resonance. Unlike in the off resonance case where we took advantage of the approximation that helicity is conserved here we do no expect that to be the case Quéméner and Balakrishnan (2008). In addition the extra angular momentum of the H2 complicates the analysis. Fig. 4 shows the dominant helicity resolved cross sections. Unlike in the para-H2 case there are significant contributions to the cross section from non-helicity conserving transitions, specifically $1\to 0$ and $3\to 0$. The contribution from $2\to 0$ on the other hand is relatively small. Just as for the para-H2 case $\beta$ gives a handle to vary the integral cross section: in this case however we take advantage of the relatively small cross section for $2\to 0$. To maximize the cross section we again choose $\beta=0$ as $m_{1}=0$ can couple with $m_{2}=0,\pm 1$ to make $m_{12}=m_{1}+m_{2}=\pm 1,0$ all of which have large cross sections. In order to minimize the cross section we want to minimize exactly those contributions and maximize the contribution from $2\to 0$ so again we choose the magic angle.

Fig. 5 shows the integral cross section for ortho-H2 collisions as a function of the collision energy for different values of $\beta$. Again it is seen that, as expected, $\beta=0$ leads to the largest cross section and the magic angle leads to the smallest cross section. It is not clear from this case if it is in general true that the magic angle minimizes the cross section on resonance. We do however expect that choosing the magic angle will always reduce the cross section compared to $\beta=0$ as $0\to 0$ is expected to be a significant (even if not dominant) contribution to the total cross section in the on resonant case.

The degree of control over the integral cross section can be quantified by the ratio of the integral cross section for $\beta=0$ to $\beta=\beta_{m}$ $\frac{\sigma(E,\beta=0)}{\sigma(E,\beta=\beta_{m})}$, which is shown in Fig. 6 as a function of collision energy. Away from resonance it is seen that greater control over the rate of rotational quenching can be exerted for collisions with para-H2 than for ortho-H2. This is because by choosing $\beta=0$ the only contribution to the total cross section conserves helicity ($m_{12}=m_{1}=0\to m^{\prime}_{12}=m^{\prime}_{1}=0$) whereas by choosing $\beta=\beta_{m}$ none of the contributions conserve helicity ($m_{12}=m_{1}=\pm 1,\pm 2\to m^{\prime}_{12}=m^{\prime}_{1}=0$). In collisions with ortho-H2 however contributions from helicity conserving transitions are unavoidable. Generally speaking we therefore expect that choosing $\beta=0$ as opposed to $\beta=\beta_{m}$ will have the biggest effect when quenching to a state where $j^{\prime}_{1}=j^{\prime}_{2}=0$. On the other hand the biggest effect on the integral cross section is seen on resonance for collisions with ortho-H2 where the integral cross section can be enhanced by up to a factor of 4. It is not clear if choosing $\beta=\beta_{m}$ will in general be as effective for other resonances or whether this is a consequence of the small contribution from $m_{12}=2\to m^{\prime}_{12}=0$ in this case.

IV Conclusion

Precise stereodynamic control of cold molecular collisions has been experimentally demonstrated by changing the relative alignment of the collision partners using the SARP method. In this work we have examined how rotational quenching rates in cold collisions of HD with H2 can be controlled using this technique, showing they can be enhanced by up to factor of 4: in order to maximize the rotational quenching rate of HD from $j=2\to 0$ the HD should be aligned along the relative collision axis, wheres to minimize the rate the HD should be aligned at the so called magic angle. This follows from quite general helicity considerations and as such we expect that rotational quenching rates for cold collisions of other similar molecules can also be controlled in this way. This work demonstrates the exquisite control that can be achieved in cold molecular collisions when the incoming molecules are prepared using the SARP method. In future work we intend to examine rotational quenching rates for other initial rotational states as well as other systems.

Acknowledgments

We acknowledge support from the US National Science Foundation, grant No. PHY-1806334. J.F.E.C gratefully acknowledges support from the Dodd-Walls Centre for Photonic and Quantum Technologies. We thank Dick Zare, Nandini Mukherjee, and William Perreault for many stimulating discussions, and Hua Guo for his careful reading of the manuscript.