  We investigate the singlet-triplet relaxation due to the spin-orbit coupling
together with the electron-phonon scattering in two-electron multivalley
silicon single quantum dots, using the exact diagonalization method and the
Fermi golden rule. The electron-electron Coulomb interaction, which is crucial
in the electronic structure, is explicitly included. The multivalley effect
induced by the interface scattering is also taken into account. We first study
  the configuration with a magnetic field in the Voigt configuration and
identify the relaxation channel of the experimental data by Xiao {\em et al.}
[Phys. Rev. Lett. {\bf 104}, 096801 (2010)]. Good agreement with the experiment
is obtained. Moreover, we
  predict a peak in the magnetic-field dependence of the singlet-triplet
relaxation rate induced by the anticrossing of the singlet and triplet states.
We then work on the system with a magnetic field in the Faraday configuration,
where the different values of the valley splitting are discussed. In the case
of large valley splitting, we find the transition rates can be effectively
manipulated by varying the external magnetic field and the dot size.
  The intriguing features of the singlet-triplet relaxation in the vicinity of
the anticrossing point are analyzed. In the case of small valley splitting, we
find that the transition rates are much smaller than those in the case of large
valley splitting, resulting from the different configurations of the triplet
states.
