Tree-level g-tensor model

by Nikolai V. Shokhirev

Part 1

Introduction

Systems with unpaired electron (radicals) have double-degenerate states (Kramers's doublets). The levels split in an external magnetic field.

The splitting depends on the nature of a radical and the magnitude and orientation of the external magnetic field

In the appropriate coordinate system the g-tensor (matrix) is diagonal

This coordinate system is formed by the principal axes. The diagonal values are called g-factors (g-values).

Crystal Field splitting of the d-type Atomic Orbitals

Typically in transition metal complexes the d-electrons are mostly localized at the central metal. In the Crystal Field Theory, a complex is modeled by a pure metal ion (no delocalization) in the electrostatic field of the rest of the molecule. It is usually assumed that the potential field is formed by the point charges located at ligands (as in ionic crystals).

Octahedral Complex

In the free ion (or highly symmetrical fild) the d-AO are 5-fold degenerate. In the pure octahedral Crystal Field (CF) the energy levels split into the e_g doublet (d_z², d_x² - d_y²)

z², x²-y²

(Created by Orbital Viewer)

and the t_2g triplet (d_xy, d_xz, d_yz):

d_xy, d_xz, d_yz

The e_g subset is shifted upward in energy, is not populated for low-spin (strong CF) complexes and is not taken into account in subsequent consideration.

Lowing the complex symmetry lead to further splitting in the triplet. In C_4v or D_4h symmetry one state split from the two others. Further lowing of symmetry lifts the degeneracy at all.