Hubbard-like Hamiltonians for interacting electrons in $s,p$, and $d$ orbitals

Hubbard-like Hamiltonians are widely used to describe on-site Coulomb interactions in magnetic and strongly-correlated solids, but there is much confusion in the literature about the form these Hamiltonians should take for shells of $p$ and $d$ orbitals. This paper derives the most general $s,p$, and $d$ orbital Hubbard-like Hamiltonians consistent with the relevant symmetries, and presents them in ways convenient for practical calculations. We use the full configuration interaction method to study $p$ and $d$ orbital dimers and compare results obtained using the correct Hamiltonian and the collinear and vector Stoner Hamiltonians. The Stoner Hamiltonians can fail to describe properly the nature of the ground state, the time evolution of excited states, and the electronic heat capacity.

Figure 1

The magnetic correlation between two $p$-shell atoms, each with two electrons, for a large range of parameters $U/\left|t\right|$ and $J/\left|t\right|$, where $t$ is the sigma bond hopping. The different regions of the graph are labeled by the symmetry of the ground state. The top graph is generated from our Hamiltonian and the bottom from the vector Stoner Hamiltonian. The bottom graph has a region with symmetry $\underset{g}{{}^3\mathrm{\Sigma }^{-}}$ extending a long way up the $J$ axis, which is not present in the ground state of our Hamiltonian. The bottom graph also includes a region with two degenerate states with symmetries ${}^1\mathrm{\Delta }_g$ and ${}^1{\mathrm{\Sigma }}_g^{+}$; this degeneracy is broken when our Hamiltonian is used.

Figure 2

The magnetic correlation between two $d$-shell atoms, each with six electrons, for a large range of parameters $U/\left|t\right|$ and $J/\left|t\right|$, where $t$ is the sigma bond hopping. The different regions of the graph are labeled by the symmetry of the ground state. The top graph is generated using our $d$-shell Hamiltonian with $\mathrm{\Delta }J/\left|t\right|=0.0$; the middle graph is generated using our Hamiltonian with a small value of $\mathrm{\Delta }J/\left|t\right|=0.1$; and the bottom graph is generated using the vector Stoner Hamiltonian.

Figure 3

The time evolution under our Hamiltonian (top) and the vector Stoner Hamiltonian (bottom) of a starting state with two electrons in $1x$, the $x$ orbital on atom 1, of a $p$-atom dimer with $U=5.0\left|t\right|$ and $J=0.7\left|t\right|$. We see that the two electrons in $1x$ are able to hop into $1y$ and $1z$ when the wave function is evolved using our Hamiltonian but not when it is evolved using the vector Stoner Hamiltonian.

Figure 4

The electronic heat capacity per atom of a dimer with four electrons split over two $p$-shell atoms with $U=5.0\left|t\right|$ and $J=0.7\left|t\right|$. The solid line is generated by diagonalising our Hamiltonian exactly and the dashed line by diagonalising the vector Stoner Hamiltonian exactly.