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In molecular dynamics simulation, it is quite common to calculate a precise temperature profile with an atomic or sub-angstrom resolution. While this calculation typically requires the degrees of freedom (DOF) of individual atoms, those in a molecule subject to geometric constraints is undefined in general. Conventionally, the approximation of subtracting from the atom DOF by 1/2 per one distance constraint has been used, but this approximation is valid for spatially homogeneous systems only. In the present study, on the basis of statistical mechanics, we derived more general expressions for the DOF of atoms in fully or partially rigid molecules. The expressions ensure that in an equilibrium state, all constrained atoms have the same temperature as the equilibrium temperature of the whole system and are also applicable to inhomogeneous systems.

Degrees of freedom of atoms in a rigid molecule for local temperature calculation in molecular dynamics simulation