  Evolutions of single-particle energies and Z=64 sub-shell along the isotonic
chain of N=82 are investigated in the density dependent relativistic
Hartree-Fock (DDRHF) theory in comparison with other commonly used mean field
models such as Skyrme HF, Gogny HFB and density dependent relativistic Hartree
model (DDRMF). The pairing is treated in the BCS scheme, except for Gogny HFB.
It is pointed out that DDRHF reproduces well characteristic features of
experimental $Z$-dependence of both spin orbital and pseudo-spin orbital
splittings around the sub-shell closure Z=64. Non-local exchange terms of the
isoscalar $\sigma$ and $\omega$ couplings play dominant roles in the
enhancements of the spin-orbit splitting of proton 2d states, which is the key
ingredient to give the Z=64 sub-shell closure properly. On the other hand, the
$\pi$ and $\rho$ tensor contributions for the spin-orbit splitting cancel each
other and the net effect becomes rather small. The enhancement of the sub-shell
gaps towards Z=64 is studied by the DDRHF, for which the local terms of the
scalar and vector meson couplings are found to be important.
