The second-quantized Pauli Hamiltonian
Within the Born-Oppenheimer approxmation, we can quantize the one-electron non-relativistic Pauli molecular Hamiltonian, together with the Coulomb repulsion: \[\begin{equation} \require{physics} \hat{\mathcal{H}} = \sum_{PQ}^M h_{PQ} \hat{E}_{PQ} + \frac{1}{2} \sum_{PQRS}^M g_{PQRS} \thinspace \hat{e}_{PQRS} + h_{\text{nuc}} \thinspace . \end{equation}\] In this expression, the one-electron integrals are: \[\begin{align} h_{PQ} &= \int \dd{\vb{r}} \phi_P^\dagger(\vb{r}) \thinspace \mathcal{H}^c(\vb{r}) \thinspace \phi_Q(\vb{r}) \\ &= h_{P \alpha, Q \alpha} + h_{P \alpha, Q \beta} + h_{P \beta, Q \alpha} + h_{P \beta, Q \beta} \thinspace , \end{align}\] with \(\mathcal{H}^c(\vb{r})\) the one-electron Pauli Hamiltonian. The contributing integrals over the scalar components \(h_{P \sigma, Q \tau}\) are then given by: \[\begin{align} & h_{P \alpha, Q \alpha} = \int \dd{\vb{r}} \phi_{P \alpha}^*(\vb{r}) \qty[ \frac{ \norm{\boldsymbol{\pi}^c}^2 }{2} - \phi_{\text{ext}}(\vb{r}) + \frac{ B_{\text{ext}, \thinspace z}(\vb{r}) }{2} ] \phi_{P \alpha}(\vb{r}) \\ % & h_{P \alpha, Q \beta} = \frac{1}{2} \int \dd{\vb{r}} \phi_{P \alpha}^*(\vb{r}) \thinspace \qty( B_{\text{ext}, \thinspace x}(\vb{r}) - i B_{\text{ext}, \thinspace y}(\vb{r}) ) \thinspace \phi_{Q \beta}(\vb{r}) \\ % & h_{P \beta, Q \alpha} = \frac{1}{2} \int \dd{\vb{r}} \phi_{P \beta}^*(\vb{r}) \thinspace \qty( B_{\text{ext}, \thinspace x}(\vb{r}) + i B_{\text{ext}, \thinspace y}(\vb{r}) ) \thinspace \phi_{Q \alpha}(\vb{r}) \\ % & h_{P \beta, Q \beta} = \int \dd{\vb{r}} \phi_{P \beta}^*(\vb{r}) \qty[ \frac{ \norm{\boldsymbol{\pi}^c}^2 }{2} - \phi_{\text{ext}}(\vb{r}) - \frac{ B_{\text{ext}, \thinspace z}(\vb{r}) }{2} ] \phi_{Q \beta}(\vb{r}) \thinspace . \end{align}\]
The two-electron Coulomb-repulsion integrals are: \[\begin{equation} g_{PQRS} = \int \int \dd{\vb{r}_1} \dd{\vb{r}_2} \phi_P^\dagger(\vb{r}_1) \phi_Q(\vb{r}_1) \thinspace \frac{1}{\norm{\vb{r}_1 - \vb{r}_2}} \thinspace \phi_R^\dagger(\vb{r}_2) \phi_S(\vb{r}_2) \thinspace . \end{equation}\]
The internuclear repulsion term \(h_{\text{nuc}}\) is then the constant term that arises due to the nuclear framework.
In a spin-orbital basis, we can write the one-electron part of the Pauli Hamiltonian as \[\begin{equation} \hat{h} = \sum_{\sigma \tau} \sum_{pq} h_{p \sigma, q \tau} \hat{a}^\dagger_{p \sigma} \hat{a}_{q \tau} = \sum_{pq} h^{0,0}_{pq} \hat{E}_{pq} + \sum_{M=-1}^1 \sum_{pq} h^{1, M}_{pq} \hat{T}^{1,M}_{pq} \end{equation}\] using the spin tensor operators and the reparametrization \[\begin{align} & h^{0,0}_{pq} = \frac{1}{2} ( h_{p \alpha, q \alpha} + h_{p \beta, q \beta} ) \\ & h^{1, -1}_{pq} = h_{p \beta, q \alpha} \\ & h^{1, 0}_{pq} = \frac{1}{\sqrt{2}} ( h_{p \alpha, q \alpha} - h_{p \beta, q \beta} ) \\ & h^{1, +1}_{pq} = - h_{p \alpha, q \beta} \thinspace . \end{align}\]
In the case of a magnetic field that only has a component in the \(z\)-direction, the mixed-spin one-electron integrals vanish, so we obtain \[\begin{equation} \hat{h} = \sum_{pq} h^{0,0}_{pq} \hat{E}_{pq} + \frac{1}{2} \sum_{pq} h^{1, 0}_{pq} \hat{T}^{1,0}_{pq} \thinspace , \end{equation}\] which eventually reduces to \[\begin{equation} \hat{h} = \sum_{pq} h^{0,0}_{pq} \hat{E}_{pq} + \frac{1}{2} B_z (\hat{N}_\alpha - \hat{N}_\beta) \thinspace . \end{equation}\]
This is exactly the result that is found in Helgaker (T. Helgaker, Jørgensen, and Olsen 2000), but we have also described how magnetic field effects can be described in a restricted formalism.
In the presence of an arbitrarily oriented magnetic field, the spin Zeeman term \[\begin{equation} \frac{1}{2} \vb{B} \cdot \boldsymbol{\sigma} \end{equation}\] constitutes an important interaction of the electrons’ spin with the magnetic field. The addition of this term breaks the symmetry (i.e. degeneracy) of the \(\alpha\) and \(\beta\) one-electron states. [[Symmetry breaking definitions]]
When a uniform magnetic field is applied in the \(z\)-axis, the spin Zeeman term becomes \[\begin{equation} \frac{1}{2} B_z \cdot \sigma_z \thinspace . \end{equation}\] In this case, projected spin (along the \(z\)-axis) is a symmetry of the Hamiltonian and thus spin-orbitals constitute symmetry-adapted one-particle states.