  We propose a gauge-invariant formulation of the channel orbital-based
time-dependent configuration interaction singles (TDCIS) method [Phys. Rev. A
74, 043420 (2006)], one of the powerful ab initio methods to investigate
electron dynamics in atoms and molecules subject to an external laser field. In
the present formulation, we derive the equations of motion (EOMs) in the
velocity gauge using gauge-transformed orbitals, not fixed orbitals, that are
equivalent to the conventional EOMs in the length gauge using fixed orbitals.
The new velocity-gauge EOMs avoid the use of the length-gauge dipole operator,
which diverges at large distance, and allows to exploit computational
advantages of the velocity-gauge treatment over the length-gauge one, e.g, a
faster convergence in simulations with intense and long-wavelength lasers, and
the feasibility of exterior complex scaling as an absorbing boundary. The
reformulated TDCIS method is applied to an exactly solvable model of
one-dimensional helium atom in an intense laser field to numerically
demonstrate the gauge invariance. We also discuss the consistent method for
evaluating the time derivative of an observable, relevant e.g, in simulating
high-harmonic generation.
