Orientation dependent spin density matrix of tumbling molecules in thermal equilibrium

The spin state of randomly moving paramagnetic molecules is described by a density matrix ρ(Ω), which is a function of the random molecular coordinate Ω. As a typical example, Ω denotes the orientation of molecules in a liquid. It is shown that in thermal equilibrium ρ(Ω) is equal to a statistical Boltzmann distribution, which is determined by the full Ω-dependent spin Hamiltonian ℋ(Ω). Thus the effect of the anisotropic part of ℋ(Ω) is not canceled in the case of rapid tumbling. This result is proved using a quantum mechanical description of the lattice, in a way similar to Van Vleck's derivation of the Langevin-Debye formula for susceptibilities. Based on the same formalism, an extension of the equation of motion of ρ(Ω) (the stochastic Liouville equation) is proposed. The modified equation guarantees the approach of ρ(Ω) to its thermal equilibrium value. The new term in the equation is also discussed in terms of Brownian motion.