General relativistic quasiequilibrium states of black hole-neutron star binaries are computed in the moving-puncture framework .
We propose three conditions for determining the quasiequilibrium states and compare the numerical results with those obtained in the excision framework .
We find that the results obtained in the moving-puncture framework agree with those in the excision framework and with those in the third post-Newtonian approximation for the cases that ( i ) the mass ratio of the binary is close to unity irrespective of the orbital separation , and ( ii ) the orbital separation is large enough ( m _ { 0 } \Omega \lesssim 0.02 , where m _ { 0 } and \Omega are the total mass and the orbital angular velocity , respectively ) irrespective of the mass ratio .
For m _ { 0 } \Omega \gtrsim 0.03 , both of the results in the moving-puncture and excision frameworks deviate , more or less , from those in the third post-Newtonian approximation .
Thus the numerical results do not provide a quasicircular state , rather they seem to have a non-negligible eccentricity of order 0.01 –0.1 .
We show by numerical simulation that a method in the moving-puncture framework can provide approximately quasicircular states in which the eccentricity is by a factor of \sim 2 smaller than those in quasiequilibrium given by other approaches .