The interaction between Ea, Gdr or Lu atoms and carbon monoxide has been studied by means of the density functional method with gradient correction and quasi-relativistic correction. Three linkage isomers, Ln-CO (I), Ln-OC (II), and Ln-(eta(2)-CO) (III) have been obtained through full geometric optimization. The vibrational frequencies and the dissociation energies (D-e) for decomposition into Ln atoms and CO have been calculated, The calculated frequency related to the C-O bond (v(C-O)) of Gd-CO is in good agreement with the experimental value. The ordering of the dissociation energy is D-e(La) > D-e(Gd) > D-e(Lu) for the same isomer with different lanthanoid atom, and it is D-e(I) > D-e(II) > D-e(II) for different isomers with the same lanthanoid. However, isomer III has much lower v(C-O) than does isomer I, showing that the traditional assumption used cis estimate the binding strength between the metal and CO by the lowering of VC-O is not always valid, For La-CO and Gd-CO, the relativistic effect produces little changes in the calculated bond lengths and vibrational frequencies, but if significantly reduces the dissociation energy. For Lu-CO the relativistic effect lowers the spin multiplicity of the molecule in the ground state, significantly enlarges the bond lengths between Lu and CO, reduces the dissociation energy, and raises the v(C-O). It is found that the bond formation can be described as the donation of the CCP 5 sigma electrons to the Ln 5d sigma orbitals in synergy with a feedback of the Ln 5d pi electrons to the CO 2 pi* orbitals. For isomer II the CO 5 sigma donation is very weak, and for isomer III both the CO 5 sigma and 1 pi electrons participate in donation. The donation of the bonding CO 1 pi electrons to Ln 5d orbitals may be the major cause for the large lowering of v(C-O) of isomer III. Accompanying the bond formation, there is transfer of roughly one electron from Ln 6s to Sd orbitals., With the increase of the Ln atomic number, the energy gap between the Ln tis and 5d levels enlarges rapidly, leading to decrease of the bonding energy for the heavier lanthanoid-carbon monoxide compounds. The relativistic effect lowers the Gs level and raises the 5d level and thus enhances this trend.