Relativistic shrinking

The energy of a Is-electron in a hydrogen-like system (one nucleus and one electron) is —Z /2, and classically this is equal to minus the kinetic energy, 1/2 mv, due to the virial theorem E — —T = 1/2 V). In atomic units the classical velocity of a Is-electron is thus Z m= 1). The speed of light in these units is 137.036, and it is clear that relativistic effects cannot be neglected for the core electrons in heavy nuclei. For nuclei with large Z, the Is-electrons are relativistic and thus heavier, which has the effect that the 1 s-orbital shrinks in size, by the same factor by which the mass increases (eq. (8.2)). 

Qualitatively, internal orbitals are contracted towards the nucleus, and the Radon core shrinks and displays a higher electron density. External electrons are much more efficiently screened from the nucleus, and the repulsion described by the (Hartree-Fock) central potential U(rj) in Eq. (7) of the preceding subsection is greatly enhanced. In fact, wave functions and eigenstates of external electrons calculated with the relativistic and the non-relativistic wave equations differ greatly. 

However, deviations from straightforward extrapolations within the Periodic Table were also considered [19]. As a consequence of relativistic effects on the electronic structure, the, v- andp-orbitals of heavy elements should shrink whereas higher lying orbitals should expand. Consequently, the two, v electrons in element 112 and also the two pm electrons in element 114 could form closed electron shells, and eka-mercury and eka-lead could both be chemically inert gases like element 118, eka-radon. 

Martin " was the first to estimate the effects of relativity on the spectroscopic constants of Cu2. The scalar relativistic (mass-velocity and Darwin) terms were evaluated perturbatively using Hartree-Fock or GVB (Two configuration SCF (ffg -mTu)) wavefunctions. At these levels the relativistic corrections for r, cu and D, were found to be — 0.05 A, 15 cm and -h0.06eV for SCF, and —0.05 A, + 14 cm and -l-0.07eV for GVB. The shrinking of the bond length is less than half of the estimate based on the contraction of the 4s atomic orbital. 

One can minimize the energy of just a valence shell to obtain its wave function both in the H.F. part and the i,/s provided the trial functions are kept orthogonal to the inner orbitals [see Eqs. (86) and (99b)]. The relativistic shrinkage of the inner shells and the shell structure then causes the outer electrons to shrink too. Thus, even to calculate the Mon-relativistic valence shell, a knowledge of the inner orbitals is needed. These may probably be approximated by the relativistic H.F. orbitals obtained from the actual free ions corresponding to the cores. The medium potential V is also affected by them, and hence by relativistic effects. However, the m, parts of the outer wave function are not sensitive to changes in V. ... 

Relativistic corrections make significant impact on the electronic properties of heavy atoms and molecules containing heavy atoms. The inner s orbitals are the closest to the nucleus and thus experience the high nuclear charge of the heavy atoms. Thus, the inner s orbitals shrink as a result of mass-velocity correction. This, in turn, shrinks the outer s orbitals as a result of orthogonality. Consequently, the ionization potential is also raised. The p orbitals are iilso shrunk by mass-velocity correction but to a lesser extent since the angular momentum keeps the electrons away from the nucleus. However, the spin-orbit interaction splits the p shells into pi/2 and pj/2 subshells and expands the P3/2 subshells. The net result is that the mass-velocity and spin-orbit interactions tend to cancel for the P3/2 shell but reinforce for the Pi/2. 

We have to remember, however, that the relativistic effects also propagate from the inner shells to the valence shell through the orthogonalization condition, which has to be fulfilled after the relativistic contraction. This is why the gold valence orbital 6s shrinks, which has an immediate consequence in the relativistic shortening of the bond length in Au2, cited at the beginning of this chapter. 

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