  We develop a biorthogonal linear-scaling algorithm for a transcorrelated
method based on the localized nature of transformed orbitals. The
transcorrelated method, which employs a similarity-transformed Hamiltonian
referred to as a transcorrelated Hamiltonian, enables highly accurate
first-principles condensed-matter calculations in principle. Meanwhile, the
transcorrelated Hamiltonian numerically prevents us from applying it to large
systems because the transcorrelated Hamiltonian is a non-Hermitian operator and
contains a 3-body electron-electron interaction term. Non-Hermiticity means
that left and right wave functions of the total energy expectation value are
different from each other. Namely, a biorthogonal form is required. Our new
method allows us to handle the non-Hermitian operator and exhibits a
linear-scaling behavior.
