Polar molecules engaged in pendular states captured by molecular-beam scattering experiments

We demonstrate that when two polar molecules as those of water, ammonia, and hydrogen sulfide encounter each other at a distance much larger than their dimensions they engage a synchronous motion that promotes the transformation of free rotations into coupled pendular states. This discovery has been prompted by high-resolution molecular beam scattering experiments presented here, addressed to the measure of the total integral cross section changes as a consequence of molecular rotation couplings. The experimental observations and the theoretical treatment developed to shed light on the details of the phenomenon suggest that the interplay among free rotations and pendular states depends on the relative velocity, on the rotational levels, and on the dipole moments of the interacting molecules. The features of this intriguing phenomenon may be crucial for the interpretation and the control of basic chemical and biological processes.

Figure 1

Measured cross sections $Q$ for ${\mathrm{D}}_2\mathrm{O}\text{−}{\mathrm{D}}_2\mathrm{O}$, ${\mathrm{D}}_2\mathrm{O}\text{−}{\mathrm{ND}}_3$, ${\mathrm{D}}_2\mathrm{O}\text{−}{\mathrm{H}}_2\mathrm{S}$, and ${\mathrm{ND}}_3\text{−}{\mathrm{H}}_2\mathrm{S}$ systems reported as a function of MB velocity $v$. Continuous lines represent cross sections, calculated in the center-of-mass system using the semiclassical JWKB method and then convoluted in the laboratory frame for the direct comparison with measured data.

Figure 2

The total interaction potential $V$ (solid) and the $V_{\mathrm{ILJ}}$ component (dash) plotted as function of the intermolecular distance $r$. (a) water-water, (b) water-ammonia, (c) water$-$hydrogen-sulfide, and (d) ammonia$-$hydrogen-sulfide systems. The insets emphasize the long-range behavior of the interaction in the mainly probed distance ranges (see text).

Figure 3

Comparison of $V\left(r\right)$ for the various systems. The inset emphasizes the differences in the long-range region.

Figure 4

Cross sections $Q$ calculated by using the total interaction $V_{\mathrm{ILJ}}+V_{\mathrm{elec}}$ for each polar molecule pair (see also Fig. 1) and by using the $V_{\mathrm{ILJ}}$ component for the reference systems. All the potential parameters are listed in Table 2. Note that for each pair the use of $V_{\mathrm{ILJ}}$ alone gives a cross section $Q_{\mathrm{ILJ}}$ within the range of the reference systems.

Figure 5

Average cross section ratio and average cross section difference (see text) plotted as function of the dipole product. Bars represent the standard deviation from the average values evaluated within the experimental velocity range (see Fig. 4).

Figure 6

The interaction potential $V\left(r\right)$ for the water-water system and examples of the corresponding partial contributions $Q_{\ell }$ (as a function of the classical turning point $r_c$) for two collision velocities $g$. In (a) $V\left(r\right)=V_{\mathrm{ILJ}}$ while in (b) $V\left(r\right)=V_{\mathrm{ILJ}}+V_{\mathrm{elec}}$. In the last case the partial contributions are accumulated at much larger distance $r$.

Figure 7

The $V_{\mathrm{elec}}/V_{\mathrm{ILJ}}$ ratio and the $V_{\mathrm{elec}}$ component at a common distance ($r=9$ Å) plotted as function of the dipole product (see Fig. 2).

Figure 8

Definitions of the coordinate system used in the molecular dynamics simulations. $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ are the local coordinates of the dipole moments ${\vec{\mu }}_1$ and ${\vec{\mu }}_2$, respectively, while $(r,\theta ,\varphi )$ are the relative coordinates of the collision. $\vec{g}$ is the relative velocity and $b$ the impact parameter. The turning point occurs at $t=0$, $r=b$, and $\theta ={90}^{\circ }$.

Figure 9

Examples of coplanar collisions at relative velocity $g=1.5$ km/s and impact parameter $b=10$ Å. Collision events are illustrated by evaluating both the local coordinates of the two dipole moments, $y_1{z}_1$ (orange/light gray) and $y_2{z}_2$ (blue/dark gray), and $U_{12}$ as a function of the time $t$. The system is defined before the collision by a negative $t$, after the collision by a positive $t$ and at the turning point by $t=0$. (a) At the beginning the molecules are both in $J=0$ and after the collision both molecules slowly rotate. (b) Collision between molecules both in $J=1$ and (c) collision between molecules in $J_1=3$ and $J_2=5$. In all cases the coupling induces a local modification of the rotational modes. The dynamics of the collision complex is driven by the interaction component $U_{12}$, well evident in the first two cases where it exhibits an average negative (attractive) contribution. Note that most part of its influence manifests in the time scale from $-1$ to 1 ps. In the third case the coupling tends to be less effective, since in the same time scale, $U_{12}$ exhibits several oscillations from positive to negative values, with an average effect tending to zero.

Figure 10

The case of two water molecules colliding at relative velocity $g$=1.5 km/s and impact parameter $b=10$ Å is illustrated. (a) In the time interval $-6$ ps $<t<$ $-2$ ps the molecules maintain their initial orientation and rotate with $J_1=J_2=1$ in perpendicular planes. When the relative distance is approximately $r=42$ Å ($t=-2$ ps), they begin to feel each other, as indicated by the slight distortion of their motions. The next time interval ($-2$ ps $<t<$ 2 ps), shows the insurgence of a pendular state and the associated coupled motion occurs in the neighborhood of the turning point ($r=10$ Å, $t=0$); further on (2 ps $<t<$ 6 ps), both molecules return to behave like free rotors but with a change in the relative orientation of $J_1$ and $J_2$. In (b) the molecules initially rotate with $J_1=3$ and $J_2=5$. In both cases the permanent dipole-permanent dipole interaction energy $U_{12}$ is plotted as function of time $t$ (and of relative distance $r$) along the collision. Note that most part of its influence manifests again in the time scale from $-1$ to 1 ps. In the last case the coupling tends to be less effective, since in the same time scale, $U_{12}$ exhibits several oscillations from positive to negative values, with an average effect tending to zero.

Figure 11

The variation of the $F\left(r_c\right)$ components with the rotational states and the effect of the average over the Boltzmann distribution for the collision of two water molecules. (a) reports the average anisotropy function $A_{J_1{J}_2}\left(r_c\right)$ at the turning point of collisions with impact parameter $b=9$. (b)–(d) plot the probabilities $P_{J_1{J}_2}(T_1,T_2)$ of bimolecular collisions involving two given rotational states considering $T_1=T_2=300$ K, $T_1=300$ K, and $T_2=100$ K, and $T_1=T_2=100$ K, respectively. (e)–(g) illustrate the corresponding weighted average anisotropy function ${\overline{A}}_{J_1{J}_2}\left(r_c\right)$.