  Fusion reactions in the crust of an accreting neutron star are an important
source of heat, and the depth at which these reactions occur is important for
determining the temperature profile of the star. Fusion reactions depend
strongly on the nuclear charge $Z$. Nuclei with $Z\le 6$ can fuse at low
densities in a liquid ocean. However, nuclei with Z=8 or 10 may not burn until
higher densities where the crust is solid and electron capture has made the
nuclei neutron rich. We calculate the $S$ factor for fusion reactions of
neutron rich nuclei including $^{24}$O + $^{24}$O and $^{28}$Ne + $^{28}$Ne. We
use a simple barrier penetration model. The $S$ factor could be further
enhanced by dynamical effects involving the neutron rich skin. This possible
enhancement in $S$ should be studied in the laboratory with neutron rich
radioactive beams. We model the structure of the crust with molecular dynamics
simulations. We find that the crust of accreting neutron stars may contain
micro-crystals or regions of phase separation. Nevertheless, the screening
factors that we determine for the enhancement of the rate of thermonuclear
reactions are insensitive to these features. Finally, we calculate the rate of
thermonuclear $^{24}$O + $^{24}$O fusion and find that $^{24}$O should burn at
densities near $10^{11}$ g/cm$^3$. The energy released from this and similar
reactions may be important for the temperature profile of the star.
