Toward Calculations of the ¹²⁹Xe Chemical Shift in Xe@C₆₀ at Experimental Conditions:  Relativity, Correlation, and Dynamics

We calculate the ¹²⁹Xe chemical shift in endohedral Xe@C₆₀ with systematic inclusion of the contributing physical effects to model the real experimental conditions. These are relativistic effects, electron correlation, the temperature-dependent dynamics, and solvent effects. The ultimate task is to obtain the right result for the right reason and to develop a physically justified methodological model for calculations and simulations of endohedral Xe fullerenes and other confined Xe systems. We use the smaller Xe···C₆H₆ model to calibrate density functional theory approaches against accurate correlated wave function methods. Relativistic effects as well as the coupling of relativity and electron correlation are evaluated using the leading-order Breit−Pauli perturbation theory. The dynamic effects are treated in two ways. In the first approximation, quantum dynamics of the Xe atom in a rigid cage takes advantage of the centrosymmetric potential for Xe within the thermally accessible distance range from the center of the cage. This reduces the problem of obtaining the solution of a diatomic rovibrational problem. In the second approach, first-principles classical molecular dynamics on the density functional potential energy hypersurface is used to produce the dynamical trajectory for the whole system, including the dynamic cage. Snapshots from the trajectory are used for calculations of the dynamic contribution to the absorption ¹²⁹Xe chemical shift. The calculated nonrelativistic Xe shift is found to be highly sensitive to the optimized molecular structure and to the choice of the exchange−correlation functional. Relativistic and dynamic effects are significant and represent each about 10% of the nonrelativistic static shift at the minimum structure. While the role of the Xe dynamics inside of the rigid cage is negligible, the cage dynamics turns out to be responsible for most of the dynamical correction to the ¹²⁹Xe shift. Solvent effects evaluated with a polarized continuum model are found to be very small.