Stability of high and low spin states

Octahedral CoL$_6$ complexes exhibit high or low spin states, depending on ligand L. We present an explicitly correlated first principles calculation of CoL$_6$ with five different ligands, and show that the total energy difference $\Delta E$ between the high and low spin states is variationally determined in an intricate interplay of the interelectron repulsion $V_{ee}$, internuclear repulsion $V_{nn}$, and electronuclear attraction $V_{ne}$. This is in stark contrast to ``ligand field theory'' [1,2], where $\Delta E$ is approximated as $\Delta E \approx \Delta V_{ee}$ in a first order perturbation theory. Moreover, we show that $\Delta V_{ee}$ exhibits the opposite trend to $\Delta E$ and is three or four orders of magnitude greater than $\Delta E$, which demonstrates the failure of ligand field theory both quantitatively and qualitatively. Correctly, the crossover of high and low spin states is a consequence of different Co--L bondings, ionic or covalent, which is found by an accurate treatment of Coulomb correlation between ligand $p$ and cobalt $d$ electrons in the present calculation. [1] J. H. Van Vleck, J. Chem Phys {\bf 3}, 807 (1935). [2] Y. Tanabe and S. Sugano, J. Phys. Soc. Jpn. {\bf 9}, 766 (1954).