  Metal-organic frameworks often display large positive or negative thermal
expansion coefficients. MOF-74, a material envisioned for many applications
does not display such a behavior. For this system, temperature-dependent x-ray
diffraction reveals particularly small negative thermal expansion coefficients
perpendicular and positive ones parallel to the hexagonally arranged pores. The
observed trends are explained by combining state-of-the-art density-functional
theory calculations with the Gr\"uneisen theory of thermal expansion, which
allows tracing back thermal expansion to contributions of individual phonons.
On the macroscopic level, the small thermal expansion coefficients arise from
two aspects: compensation effects caused by the large coupling between stress
and strain perpendicular to the pores and the small magnitudes of the mean
Gr\"uneisen tensor elements, $\langle\gamma\rangle$, which provide information
on how strains in the material influence its phonon frequencies. To understand
the small mean Gr\"uneisen tensor in MOF-74, the individual mode contributions
are analyzed based on the corresponding atomic motions. This reveals that only
the lowest frequency modes up to ~3 THz provide non-negligible contributions,
such that $\langle\gamma\rangle$ drops sharply at higher temperatures. These
considerations reveal how the details of the anharmonic properties of specific
phonon bands determine the magnitude and sign of thermal expansion in a
prototypical material like MOF-74.
