  In the nonrelativistic Schr\"{o}dinger equation, the total spin $S$ and spin
projection $M$ are good quantum numbers. In contrast, spin symmetry is lost in
the presence of spin-dependent interactions such as spin-orbit couplings in
relativistic Hamiltonians. Previous implementations of relativistic density
matrix renormalization group algorithm (R-DMRG) only employing particle number
symmetry are much more expensive than nonrelativistic DMRG. Besides, artificial
breaking of Kramers degeneracy can happen in the treatment of systems with odd
number of electrons. To overcome these issues, we introduce time-reversal
symmetry adaptation for R-DMRG. Since the time-reversal operator is
antiunitary, this cannot be simply achieved in the usual way. We define a
time-reversal symmetry-adapted renormalized basis and present strategies to
maintain the structure of basis functions during the sweep optimization. With
time-reversal symmetry adaptation, only half of the renormalized operators are
needed and the computational costs of Hamiltonian-wavefunction multiplication
and renormalization are reduced by half. The present construction of
time-reversal symmetry-adapted basis also directly applies to other tensor
network states without loops.
