The molecular magnetic Hamiltonian
If we want to take into account the effect of an external magnetic field \(\require{physics} \vb{B}_{\text{ext}}\) and the nuclear permanent magnetic dipole moments \(\qty{ \vb{M}_K }\), whose physical interactions are described in sections \(\ref{sec:Schrodinger_Hamiltonian}\) and \(\ref{sec:Pauli_Hamiltonian}\), we should, according to the discussion in section \(\ref{sec:perturbation_dependent_fock_spaces}\), quantize the corresponding Hamiltonian (T. Helgaker and Jørgensen 1991) as \[\begin{equation} \begin{split} \hat{\mathcal{H}} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K }, % \vb{G} % ) % = &\sum_{PQ}^M % \tilde{h}_{PQ} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K }, % \vb{G} % ) % \hat{E}_{PQ} \\ &+ \frac{1}{2} \sum_{PQRS}^M % \tilde{g}_{PQRS} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K }, % \vb{G} % ) % \hat{e}_{PQRS} \\ &+ h_{\text{nuc}}( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K }, % \vb{G} % ) % \thinspace , \end{split} \end{equation}\] in which the (modified) one- and two-electron integrals \(\tilde{h}_{PQ}\) and \(\tilde{g}_{PQRS}\) are without any further knowledge considered dependent on the uniform external magnetic field \(\vb{B}_{\text{ext}}\) and nuclear permanent magnetic dipole moments \(\qty{ \vb{M}_K }\), and the gauge origin of the external magnetic field \(\vb{G}\). By investigating the form of the interactions that are present in the system, we can now start turning our attention to the various terms that appear in the molecular magnetic Hamiltonian. The one-electron operator that is quantized is the one-electron Pauli Hamiltonian: \[\begin{equation} \begin{split} h^c ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K }, % \vb{G} % ) = % &\qty( % \frac{ % \norm{% \boldsymbol{\pi}^c ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K }, % \vb{G} % ) % }^2 % }{2} % - \phi(\vb{r}) % ) \vb{I}_2 \\ &+ \frac{1}{2} % \boldsymbol{\sigma} \vdot % \vb{B} ( % \vb{r}; % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K }, % \vb{G} % ) % \thinspace , \end{split} \tag{\ref{eq:Pauli_Hamiltonian_worked_out}} \end{equation}\] in which \(\boldsymbol{\sigma}\) are the Pauli matrices, \(\phi(\vb{r})\) is the molecular scalar potential generated by the nuclear charges: \[\begin{equation} \phi(\vb{r}) % = \sum_K^M % \frac{Z_K}{ \norm{\vb{r} - \vb{R}_K} } % \end{equation}\] and \(\boldsymbol{\pi}^c\) is the kinematic momentum operator that depends on the total vector potential \(\vb{A}\): \[\begin{equation} \boldsymbol{\pi}^c ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K }, % \vb{G} % ) = % -i \grad % + \vb{A} ( % \vb{r}; % \vb{B}_{\text{ext}}, \qty{ \vb{M}_K }, % \vb{G} % ) % % \tag{\ref{eq:kinematic_momentum}} % \thinspace . \end{equation}\]
The regular one-electron integrals \(h_{PQ}\) (not the ones with the tilde) then, most generally, depend on the external magnetic field \(\vb{B}_{\text{ext}}\) and its gauge origin \(\vb{G}\), and the nuclear permanent magnetic dipole moments \(\qty{ \vb{M}_K }\): \[\begin{equation} \begin{split} h_{PQ} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K }, % \vb{G} % ) % = \int \dd{\vb{r}} \thinspace % &\phi^\dagger_P ( % \vb{r}; % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K }, % \vb{G} % ) \\ &\hspace{6pt} \times h^c ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K }, % \vb{G} % ) \\ &\hspace{6pt} \times \phi_Q ( % \vb{r}; % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K }, % \vb{G} % ) % \thinspace , \end{split} \end{equation}\] in which the orthonormal spinors \(\qty{ \phi_P (\vb{r}; \vb{B}_{\text{ext}}, \qty{ \vb{M}_K }, \vb{G}) }\) are most generally dependent on both the external magnetic field \(\vb{B}_{\text{ext}}\) and its gauge origin \(\vb{G}\), and the nuclear permanent magnetic dipole moments \(\qty{ \vb{M}_K }\).
The two-electron operator that is quantized is still the Coulomb operator, whose integrals \(g_{PQRS}\) (without further investigation) depend on the external magnetic field \(\vb{B}_{\text{ext}}\), its gauge origin \(\vb{G}\) and the nuclear permanent magnetic dipole moments \(\qty{ \vb{M}_K }\): \[\begin{equation} \begin{split} &g_{PQRS} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K }, % \vb{G} % ) \\ & \hspace{12pt} = \int \int \dd{\vb{r}_1} \dd{\vb{r}_2} % \phi^\dagger_P ( % \vb{r}_1; % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K }, % \vb{G} % ) % \phi_Q ( % \vb{r}_1; % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K }, % \vb{G} % ) \\ & \hspace{96pt} % \times \frac{1}{\norm{\vb{r}_1 - \vb{r}_2}} \\ & \hspace{96pt} \times % \phi^\dagger_R ( % \vb{r}_2; % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K }, % \vb{G} % ) % \phi_S ( % \vb{r}_2; % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K }, % \vb{G} % ) % \thinspace . \end{split} \end{equation}\] Finally, the nuclear permanent magnetic dipole moments \(\qty{ \vb{M}_K }\) of the nuclear framework also interact with the external magnetic field \(\vb{B}_{\text{ext}}\) (T. Helgaker et al. 2012), leading to the nuclear contribution \[\begin{equation} h_{\text{nuc}} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } % ) % = - \sum_K^M % \vb{B}_{\text{ext}} \vdot \vb{M}_K % \thinspace . \end{equation}\]