Density Embedding

For fragment-based methods, density embedding was introduced in the context of the Pair-Interaction Molecular Orbital Method Kitaura et al.⁴². In density embedding, \(\widehat{V}_{IJ}\) is the potential fragment \(I\) feels because of the electron density and nuclei of fragment \(J\). Mathematically this is:

\[\widehat{V}_{IJ} = \sum_{i=1}^{n_I}\left[ \int dr \frac{\rho_J(r)}{|r-r_i|} - \sum_{A}^{N_A}\frac{Z_A}{|r_i-r_A|}\right]\]

where \(i\) runs over the \(n_I\) electrons in fragment \(I\), \(A\) runs over the \(N_A\) nuclei in fragment \(J\), and \(\rho_J(r)\) is the electron density of fragment \(J\) evaluated at \(r\). Since the Hamiltoniann for each fragment depends on the densities of the remaining fragments, the densities must be solved for iteratively.

For embedding higher-order \(n\)-mers, one typically uses the converged fragment densities and does not iterate using the \(n\)-mer Hamiltonians.

FMO02 Embedding

The FMO02 embedding method Nakano et al.⁷⁰: introduces two approximations into traditional density embedding based on the separation between \(I\) and \(J\). For moderately separated \(I\) and \(J\):

\[V^{IJ}_{\mu\nu} = \sum_{\lambda\in J} \left(\mathbf{P}^{J}\mathbf{S}\right)_{\lambda\lambda} \left(\mu\nu \middle| \lambda\lambda\right)\]

for \(\mu,\nu\in I\). The quantity \(\left(\mathbf{P}^{J}\mathbf{S}\right)_{\lambda\lambda}\) is the Mulliken population of orbital \(\lambda\). For distant \(I\) and \(J\):

\[V^{IJ}_{\mu\nu} \approx \sum_{A\in J} \Braket{\mu | \frac{Q_A}{\mid r_i - r_A\mid} | \nu}\]

where \(A\) indexes atoms, and \(Q_A\) is the Mulliken charge of atom \(A\) (the sum of the Mulliken populations on atom \(A\)).