How does an uncharged non-polar molecule that has a quadrupole moment (such as carbon dioxide) behave in an electric field? I know that in a homogeneous electric field, ions travel while dipoles orient along the field (rotate) and non-polar molecules are not affected.
What kind of electrical field, if any, would exert a force on a molecule with a quadrupole moment?
To determine the force on an arbitrary multipole moment, we first expand the E-field in a Taylor-series around the point $\vec{r}=0$:
$$\vec{E}(\vec{r})=\vec{E}_0+\vec{r}\cdot(\nabla\vec{E})_0+\frac{1}{2}\vec{r}\vec{r}:(\nabla\nabla\vec{E})_0+...$$
where the number of dots in the product denotes how many indices to contract over. We can then determine the force using this expansion, since it is just the charge multiplied by the E-field:
$$\vec{F}=q\vec{E}_0+\vec{\mu}\cdot(\nabla\vec{E})_0+\frac{1}{2}\mathbf{\Theta}:(\nabla\nabla\vec{E})_0+...$$ Typically, people use the traceless quadrupole, which would add an additional factor of $\frac{1}{3}$ in the third term, but I'm going to work with the basic definition throughout. So this shows that a quadrupole subject to a large electric field double gradient (not sure of the correct terminology for this) will experience a force even if it is charge neutral and has no net dipole.
For completeness, we can also write the energy since the force is just the gradient of the energy:
$$W=q\phi-\mu\cdot\vec{E}-\frac{1}{2}\mathbf{\Theta}:(\nabla\vec{E})+...$$ where we see, as TryHard said, that the E-field gradient interacts with the quadrupole.
As an aside, one can also obtain the torque (cross product of force and r) on an arbitrary multipole: $$\vec{T}=\vec{\mu}\times\vec{E}_0+\frac{1}{2}\mathbf{\Theta}\dot{\times}(\nabla\vec{E})_0+...$$
(I'm using the nonstandard notation $A\dot{\times}B$ to mean $\sum_{ijkl}\epsilon_{jkl}e_iA_{jl}B_{lk}$, where $\epsilon_{jkl}$ is the Levi-Civita operator and the $e_i$ are a set of orthonormal unit vectors). This suggests that a quadrupole will experience a torque when interacting with a field gradient.
For more information on this, I would read chapter 3 of Jeanne McHale's Molecular Spectroscopy. There are some typos and notational quirks(e.g. in the equation for torque, McHale just writes a cross product for the 2nd term, which isn't defined between matrices), but its overall a good book to give background in classical electrostatics and a quantum mechanical description of spectroscopy.
Different moments of a charge distribution couple to different components of the external electric field. In the case of the quadrupole moment, the coupling is to the gradient of the electric field (EFG). Such an interaction is for instance relevant in NMR of quadrupolar nuclei, NQR (not to be confused with naked quad run, according to the wikipedia - o tempora, o mores) and Mössbauer spectroscopy, although those techniques consider the nuclear quadrupole moment.
In the case of atoms and molecules without permanent electric monopoles or dipoles, EFG interactions with permanent quadrupoles are the leading field-multipole interaction energy term (ignoring dispersion terms, ie induced dipole-induced dipole). In some cases such interactions can be of particular importance, for instance in aromatic compounds (see Kocman et al. cited below).
The quadrupole moment has been measured in $\ce{CO_2}$, see eg Chetty cited below, which includes experimental methods.
References
Electric quadrupole moment of graphene and its effect on intermolecular interactions M. Kocman, M. Pykal and P. Jurecka Physical Chemistry Chemical Physics Vol. 16, 2014
N. Chetty and V.W. Couling Measurement of the electric quadrupole moments of CO2 and OCS Molecular Physics Vol. 109 (5), 2011, 655–666
