The electronic structures of LiYF_4:Ce~(3+) and LiYF_4 crystals simulated by an embedded (in a microcrystal containing 1938 ions) cluster CeY_4Li_8F_(24), and Y_5Li_8F_(24) respectively, were computed by the ab initio self-consistent relativistic DV-X_a(discrete variational X_a) method. The ground-state calculation showed that only the lowest 5d level E_d of Ce~(3+) ion lies around the BCB (bottom of the conduction band) while the lowest 4f levels is 2.5 eV lower than BCB. The CB states consist of 4p of Y mixed with 5d of Ce, even for the wavefunctions (WFS) of E_d under BCB there are still 24% of Y-4p and 9% of F-2p as components. Furthermore, transition state (TS) calculation was performed in this work to obtain the 4f→5d transition energies E_(fd), to improve the calculation of Ref.([6]) in which a small CeF_8 cluster embedded in an array of point charge was used and the results of ground-state calculation were roughly used to compare directly with the observed 4f→5d transition energies. The ionic radius of Ce~(3+) is larger than that of Y~(3+), for modeling approximately the lattice relaxation, we simply let the eight fluorine ions of the nearest-neighbor and next-nearest-neighbor move out radially and simultaneously. As results, the CeY_4Li_8F_(24) cluster with 4.56% outward relaxation of the eight fluorines has the lowest total energy and gave satisfactory 4f→5d energies E_(fd), but the calculated ground-state E_d is 0.68 eV higher than BCB. For another cluster with 7.36% outward relaxation the E_d is 0.43 eV lower than BCB, which makes the observation of fine structure (including zero-phonon line) of the lowest 5d band understandable easier, but the splits between the transition energies E_(fd) were not as good as the former. Therefore, we consider the relaxation is somehow around 4.56%～7.36% outward, not as large as 10%. The electronic structures of LiYF_4: Ce ~ (3+) and LiYF_4 crystals simulated by an embedded (in a microcrystal containing 1938 ions) clusters CeY_4Li_8F_ (24), and Y_5Li_8F_ (24) respectively, were computed by the ab initio self-consistent relativistic DV-X_a (discrete variational X_a) method. The ground-state calculation showed that only the lowest 5d level E_d of Ce ~ (3+) ion lies around the BCB (bottom of the conduction band) while the lowest 4f levels are 2.5 eV The CB states consist of 4p of Y mixed with 5d of Ce, even for the wavefunctions (WFS) of E_d under BCB there are still 24% of Y-4p and 9% of F-2p as components. Further, transition state (TS) calculation was performed in this work to obtain the 4f → 5d transition energies E_ (fd), to improve the calculation of Ref. ([6]) in which a small CeF_8 cluster embedded in an array of point charge was used and the results of ground-state calculation were roughly used to compare directly with the observed 4f → 5d transition energies. The ionic radius of Ce ~ (3+) is larger than that of Y ~ (3+), for modeling approximately the lattice relaxation, we simply let the eight fluorine ions of the nearest-neighbor and next-nearest-neighbor move As results, the CeY_4Li_8F_ (24) cluster with 4.56% outward relaxation of the eight fluorines has the lowest total energy and gave satisfactory 4f → 5d energies E_ (fd), but the calculated ground-state E_d is 0.68 eV For another cluster with 7.36% outward relaxation the E_d is 0.43 eV lower than BCB, which makes the observation of fine structures (including zero-phonon line) of the lowest 5d band understandable easier, but the splits between the transition energies E_ (fd) were not as good as the former. Therefore, we consider the relaxation is somehow around 4.56% ~ 7.36% outward, not as large as 10%.