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Effect of Confinement on the Translation-Rotation Motion of Molecules: The Inelastic Neutron Scattering Selection Rule

L. William Poirier*

Texas Tech University, USA

1.1 Introduction

One of the core theoretical ideas used to understand the dynamics of free molecules is the simplifying notion that the overall (i.e., center-of-mass) translational motion can be cleanly separated from internal vibrations and rotations. Indeed, this separation is so universally applied - and the translational motion so easily dealt with - that it can be easy to "forget" that the latter even exists! On the other hand, if you place that same molecule in a nanoconfined environment, the situation can be vastly different. First and foremost, the continuum of states that characterizes translational motion for free molecules necessarily becomes quantized in the confined context. The quantum translational states for the trapped "guest" molecule can be "particle-in-a-box like" or more complicated, depending on the nature of the external field provided by the cage structure. Of course, larger cages give rise to smaller translational level spacings - which are in any event generally smaller than the rotational level spacings (and much smaller than the vibrational level spacings). If the confinement is very severe, however, then the translational and rotational level spacings can become comparable to each other - and even strongly coupled.

This is the situation for small molecules trapped inside fullerene cages - e.g., @ [1-16], HD@ [4, 7, 12, 15, 17, 18], HF@ [14-16, 19], [14-16, 20, 21], and @ [15, 22], all of which will be considered in this chapter. Based on the physical size of the fullerene (diameter 7 Å), one might well imagine that a number of guest molecules could be crammed into a single cage. In reality, the guest molecules are trapped via long-range van der Waals interactions that prevent them from getting closer than a few Å from the cage wall. Consequently, the effective cage size is much smaller - on the order of the size of the guest molecule itself. This implies that: (a) only one guest molecule can fit inside a single cage; (b) the corresponding translational level spacing is comparable to the rotational level spacing. It is hardly surprising, then, to also find that translation and rotation are indeed strongly coupled in these systems. This complicates life from a theoretical/computational standpoint, for which one must adopt an exact, coupled quantum dynamical treatment, encompassing all relevant translation-rotation (TR) guest molecule degrees of freedom [3-5, 15, 16, 19, 23-25]. That said, the combined TR states that result are not necessarily entirely devoid of structure either - it is just not that of the standard form that one expects in terms of TR separability.

In addition to providing a fundamentally different spectroscopic picture, the strong confinement-induced TR coupling also gives rise to a remarkable physical effect that was once thought to be impossible - selection rules for inelastic neutron scattering (INS) [26-28]. INS is an experimental technique in which a beam of neutrons is scattered through nuclear force interactions with the nuclei of the target sample. It is an extremely useful tool for probing nanoconfined hydrogen, in large part because the H atom nucleus provides the largest neutron scattering cross section across the entire periodic table. Even that of the "second place" contender - i.e., deuterium - is more than an order of magnitude smaller. INS offers other advantages as well, such as the ability to examine transitions between individual quantum states - including ortho-para nuclear spin transitions in , which would be forbidden, e.g., in an electric dipole or even Raman far-infrared (IR) spectrum. However, whereas such optically forbidden spectroscopic transitions certainly are allowed in INS, this is no guarantee that all other transitions are also necessarily allowed.

That INS spectroscopy does indeed have forbidden transitions, was first proposed and experimentally verified for @, in a stunning set of papers by Bacic, Horsewill, and coworkers [9, 10, 29, 30]. In this earlier work, in addition to computing the TR quantum eigenstates themselves (energy levels and wavefunctions) [3-5], an explicit numerical simulation of the experimental INS spectrum was also performed [9, 12, 29, 30]. These represent heroic calculations, applied to each TR transition individually, which also take various experimental circumstances into proper account. The end result is a reasonably accurate prediction of both INS transition energies and intensities (both "stick spectra" and more experimentally-relevant convolved spectra can be obtained). In this manner, it was discovered that some transition intensities for @ are vastly smaller than others - by four or more orders of magnitude. This provided excellent numerical evidence for a selection rule - which, indeed, was subsequently confirmed through actual INS experiments [7, 8, 10].

As impressive and unexpected as this discovery proved to be, one of the drawbacks of the above approach is that one must infer a general pattern for the selection rule, from amongst a necessarily limited set of specific transitions. Indeed, this led Bacic and coworkers to initially assume a selection rule for @ of the following form:

restricted INS selection rule for @(from p-ground state): Transitions are forbidden to all states for which [note: all terms will be explained].

In fact, the correct rule for transitions starting from the @ ground state is more general:

correct INS selection rule for @(from p-ground state): Transitions are forbidden to all states for which odd.

To the author's knowledge, the latter form above was first proposed by the author himself, at a scientific meeting in May, 2015. Since confirmed assignments for the experimental @ data did not exist beyond , the theoretical simulations were not performed beyond this point either, and so the available information at the time was consistent with either of the two rules above. This state of affairs motivated the present author to develop a group theoretical derivation of the general INS selection rule [11]. In parallel with this effort, Bacic and coworkers extended the analytical part of their calculations of the transition integrals [12], to encompass all possible initial and final states. Both approaches then gave rise to the following,

most general INS selection rule for @: Transitions are forbidden between states for which changes from even to odd (or vice-versa), and at least one .

As will be described in this chapter, the group theory approach provides physical understanding, and also has the great advantage that it leads to the correct and completely general INS selection rule for the appropriate symmetry group, "all at once." However, this approach has some limitations and issues, as well, which will also be addressed. To begin with, it does not provide any intensity information for allowed transitions; for this, it is necessary to calculate transition integrals explicitly, as per Bacic et. al. Secondly there is the question of choosing the most correct symmetry group to work with, in terms of experimental relevance. This is a particularly important question for INS spectroscopy, for which there is a preferred direction or orientation - i.e., that of the momentum transfer vector, , of the scattered neutron beam. Thirdly, there are some group theoretical complications that arise, due to the fact that the INS interaction operator itself is incommensurate with most of the standard molecular point groups - a key difference, e.g., from optical spectroscopy. Finally, all true group-theory-based selection rules are expressed in terms of the irreducible representations (irreps) of the appropriate symmetry group. While this form of a selection rule is truly universal, for specific systems it is still necessary to assign irrep labels to individual quantum states and/or basis functions - thereby making an association with the pertinent quantum numbers for those systems. On balance, it is clear that both the group theory and explicit integral approaches are necessary for interpreting and predicting INS experiments, as they provide complementary understanding.

Since 2015, Bacic, Horsewill, Felker, and others have continued to explore the INS selection rule - together with other TR effects of nanoconfinement - through a fruitful synergy that has emerged between theory and experiment. These highly interesting TR effects have now been observed and/or predicted across a range of nanoconfined systems. Regarding the INS selection rule itself, the irrep-based version of [11] is in principle valid for any molecule in any spherical environment (section 1.3.2). In terms of specific basis sets and quantum states,...