  We present the theory of a density matrix renormalization group (DMRG)
algorithm which can solve for both the ground and excited states of
non-Hermitian transcorrelated Hamiltonians, and show applications in \emph{ab
initio} molecular systems. Transcorrelation (TC) accelerates the basis set
convergence rate by including known physics (such as, but not limited to, the
electron-electron cusp) in the Jastrow factor used for the similarity
transformation. It also improves the accuracy of approximate methods such as
coupled cluster singles and doubles (CCSD) as shown by some recent studies.
However, the non-Hermiticity of the TC Hamiltonians poses challenges for
variational methods like DMRG. Imaginary-time evolution on the matrix product
state (MPS) in the DMRG framework has been proposed to circumvent this problem;
but this is currently limited to treating the ground state, and has lower
efficiency than the time-independent DMRG (TI-DMRG), due to the need to
eliminate Trotter errors. In this work, we show that with minimal changes to
the existing TI-DMRG algorithm, namely replacing the original Davidson solver
with the general Davidson solver to solve the non-Hermitian effective
Hamiltonians at each site for a few low-lying right eigenstates, and following
the rest of the original DMRG recipe, one can find the ground and excited
states with improved efficiency compared to the original DMRG when
extrapolating to the infinite bond dimension limit in the same basis set.
Accelerated basis set convergence rate is also observed, as expected, within
the TC framework.
