Hydrogenoid orbitals

Hyperspherical harmonics are now explicitly considered as expansion basis sets for atomic and molecular orbitals. In this treatment the key role is played by a generalization of the famous Fock projection [5] for hydrogen atom in momentum space, leading to the connection between hydrogenic orbitals and four-dimensional harmonics, as we have seen in the previous section. It is well known that the hyperspherical harmonics are a basis for the irreducible representations of the rotational group on the four-dimensional hypersphere from this viewpoint hydrogenoid orbitals can be looked at as representations of the four-dimensional hyperspherical symmetry [14]. 

In this last expression (1.12), tio is the Bohr radius, equal to 0.529 A, and Z is the nuclear charge. To what extent are these hydrogenoid orbitals suitable to describe the d orbitals of transition metals In polyelectronic atoms, it is only the radial part of the orhitals that is different from hydrogenoid orbitals it is modified to take account of the charge on the nucleus and the screening effect created by the other electrons. Since the angular part of the orbitals is conserved, the expressions that are obtained for the 3d orbitals of hydrogenoid atoms enable us to analyse the symmetry properties of the d orbitals of all the transition metals. 

The shapes of those orbitals were calculated for the hydrogen atom (by solving the so-called time-independent Schrodinger equation). For polyelectronic atoms, because the calculation is too complex to yield an exact solution, the solutions of the hydrogen atom are used as approximations of the orbitals. Owing to this approximation, the AOs are usually called hydrogenoid orbitals. 

The hydrogenoid orbitals have precise shapes that depend on the level of quantization of the angular momentum of the electron inside those orbitals. The angular momentum is a vector quantity. The quantization of a vector quantity is dual its intensity is quantized (takes noncontinuous absolute values), and its direction in space is quantized (takes noncontinuous orientation in space). The orbitals inside which... 

Assuming hydrogenoid orbitals for these electrons, the contribution, to the field gradient, of an electron with quantum numbers n, I, m, with, for instance, m = Q, I 0), is the following ... 

The energy of the atomic orbitals in the hydrogenoid approximation is proportional to the square of the effective charge, and is (numerically) equivalent to the ionization potential... 

As is well known, conventional hydrogenoid spherical orbitals are strictly linked to tetradimensional harmonics when the atomic orbitals for the tridimensional hydrogen atom are considered in momentum space. We have therefore studied an alternative representation, providing the Stark and Zeeman basis sets, related to the spherical one by orthogonal transformation, see eqs. (12) and (15). The latter can also be interpreted as suitable timber coefficients relating different tree structures of hyperspherical harmonics for R (Fig. 1). 

Fig. 1 Top Shape of the one-electrrai (hydrogenoid) 4f orbitals in a Cartesian space. From top to bottom and left to right 4fj(x 3y2>, 4f y(3y 4fxyz, 4fz(x2 yi), 4fxz , 4fyz2, and 4fzi (combinations of Cartesian coordinates represent the angular functions). Bottom Radial wavefunction of the three 4f electrons of Nd compared with the radial wavefunctirai of the xenon core (a.u. = atomic units) redrawn after [1]...

Nevertheless, the hydrogenoid energy levels ate only valid in absence of electronic repulsion. As soon as more titan one electron is present in the atom, the electronic repulsion shifts the energy levels according to the orbital angular momentum of the electrons. The energy of tiie electronic levels thus depends on tiie distribution of the electrons in the different orbitals. 

In the case of the lanthanide ions, the electronic repulsion induces the higher shift relative to the hydrogenoid energy level of the subshell. The shift due to the spin-orbit coupling comes next ( 10 times lower than the electronic repulsion), whereas the shift due to the ligand field is weak ( a thousand times lower than the electronic repulsion). As a result, a remarkable property of the lanthanide ions is that the sphtting of these electronic levels remains fairly constant whatever the enviromnent around the lanthanide ion. The reason for this behavior is the inner character of the 4f orbitals. 

In the case of the lanthanide ions, the spectroscopic term is associated with the energy shift of the hydrogenoid energy level of the subshell due to the electronic repnlsion. The degeneracy of a spectroscopic term is given by the prodnct of the spin and orbital mnltipUcities (2-5 +l)(2L + l). 

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