The one-electron Schrödinger Hamiltonian
When the classical Hamiltonian of the system is known, we can use the quantum mechanical correspondence principle to construct the corresponding quantum mechanical Hamiltonian. Thee Schrödinger Hamiltonian act on 1-component (i.e. scalar) wave functions. This formalism is sufficient to describe non-relativistic interactions in abscence of the interaction of the electron spin and the electromagnetic field.
The one-electron Schrödinger Hamiltonian for a free electron
The simplest system is of course that of a free electron. The corresponding Hamiltonian only consists of the field-free kinetic operator \(T^c\) \[\begin{equation} \require{physics} \mathcal{H}^c = T^c % \thinspace , \end{equation}\] where the field-free kinetic operator \(T^c\) is given by: \[\begin{equation} T^c = - \frac{1}{2} \laplacian % \thinspace . \end{equation}\]
The one-electron Schrödinger Hamiltonian for an electron in an electromagnetic field
In order to describe an electron in an electromagnetic field, we must use the minimal coupling (G. Dyall and Faegri 2007) (Bransden and Joachain 2000) relation for the momentum, together with the interaction of the charged particle with the scalar potential, leading to the non-relativistic time-independent Hamiltonian \[\begin{equation} \mathcal{H}^c(\phi, \vb{A}) = k^c(\vb{A}) - \phi(\vb{r}) \thinspace , \end{equation}\] in which \(k^c\) is the (scalar) kinetic energy operator: \[\begin{equation} k^c(\vb{A}) = \frac{1}{2} \norm{\boldsymbol{\pi}^c(\vb{A})}^2 \thinspace , \end{equation}\] where \(\boldsymbol{\pi}^c\) is the coordinate representation of the kinetic momentum operator (for an electron): \[\begin{equation} \boldsymbol{\pi}^c(\vb{A}) = -i \grad + \vb{A}(\vb{r}) \thinspace , \end{equation}\] where we have filled in the charge of the electron \(q=-1\). The scalar kinetic operator then consists of three parts: \[\begin{equation} k^c(\vb{A}) = T^c + P^c(\vb{A}) + D(\vb{A}) \thinspace , \end{equation}\] where \(T^c\) represents the field-free kinetic operator, \(P^c(\vb{A})\) is called the paramagnetic term, which is first order in the vector potential, and \(D(\vb{A})\) is called the diamagnetic term, which is second order in the vector potential. Realizing that the operators always act on a function, we can rewrite the paramagnetic term as \[\begin{align} P^c(\vb{A}) &= - i \vb{A}(\vb{r}) \vdot \grad - \frac{i}{2} \boldsymbol{\nabla} \vdot{\vb{A}(\vb{r})} \\ &=- i \vb{A}(\vb{r}) \vdot \grad \thinspace , \end{align}\] if the vector potential \(\vb{A}\) satisfies the Coulomb gauge condition. The diamagnetic term is just a constant term: \[\begin{equation} D(\vb{A}) = \frac{1}{2} \norm{\vb{A}(\vb{r})}^2 \thinspace . \end{equation}\]
Within the Born-Oppenheimer approximation, the nuclei with charges \(Z_K\) and positions \(\vb{R}_K\) are fixed in space and thus generate a scalar potential, in addition to true external scalar potential \(\phi_{\text{ext}}(\vb{r})\): \[\begin{equation} \phi(\vb{r}) = \phi_{\text{nuc}}(\vb{r}) + \phi_{\text{ext}}(\vb{r}) \thinspace . \end{equation}\] The potential generated by the nuclei at a point \(\vb{r}\) in space is given by the the Coulomb potential: \[\begin{equation} \phi_{\text{nuc}}(\vb{r}) = \sum_K^M \frac{Z_K}{\norm{\vb{r} - \vb{R}_K}} \thinspace , \end{equation}\] If the external electric field \(\vb{F}_{\text{ext}}\) is uniform, the resulting scalar potential can be rewritten as \[\begin{equation} \phi_{\text{ext}}(\vb{r}; \vb{F}_{\text{ext}}) = - \vb{F}_{\text{ext}} \vdot \vb{r} \thinspace . \end{equation}\]
If we were to apply a uniform external magnetic field \(\vb{B}_{\text{ext}}\), the corresponding vector potential \(\vb{A}_{\text{ext}}\) can always be rewritten as \[\begin{equation} \vb{A}_{\text{ext}} ( \vb{r}; \vb{B}_{\text{ext}}, \vb{G} ) = \frac{1}{2} \vb{B}_{\text{ext}} \cross ( \vb{r} - \vb{G} ) \thinspace , \end{equation}\] with \(\vb{G}\) the gauge origin. We note that this vector potential automatically satisfies the Coulomb gauge condition. We can derive that the following property holds for the difference of two vector potentials evaluated relative to the same gauge origin: \[\begin{equation} \vb{A}_{\text{ext}} ( \vb{a}; \vb{B}_{\text{ext}}, \vb{G} ) - \vb{A}_{\text{ext}} ( \vb{b}; \vb{B}_{\text{ext}}, \vb{G} ) = \vb{A}_{\text{ext}} ( \vb{a}; \vb{B}_{\text{ext}}, \vb{b} ) \thinspace . \end{equation}\]
The paramagnetic and diamagnetic terms that appear in the scalar kinetic operator \[\begin{equation} k^c(\vb{B}_{\text{ext}}, \vb{G}) = T^c + P^c_{\text{ext}}(\vb{B}_{\text{ext}}, \vb{G}) + D_{\text{ext}}(\vb{B}_{\text{ext}}, \vb{G}) \end{equation}\] can then be rewritten as follows. For the paramagnetic term, we initially find: \[\begin{align} P^c_{\text{ext}}(\vb{B}_{\text{ext}}, \vb{G}) &=- i \vb{A}_{\text{ext}}(\vb{r}) \vdot \grad \\ &= - \frac{i}{2} \vb{B}_{\text{ext}} \vdot (\vb{r} - \vb{G}) \cross \grad \thinspace . \end{align}\] Recognizing the orbital angular momentum operator (about the origin \(\vb{O}\), which is in nuclear context often a nuclear position \(\vb{R}_K\)) \[\begin{equation} \vb{L}^c(\vb{O}) = -i (\vb{r} - \vb{O}) \cross \grad \thinspace , \end{equation}\] we can rewrite the paramagnetic term as: \[\begin{equation} P^c_{\text{ext}}(\vb{B}_{\text{ext}}, \vb{G}) = \frac{1}{2} \vb{B}_{\text{ext}} \vdot \vb{L}^c(\vb{O}) - \frac{i}{2} \vb{B}_{\text{ext}} \cross (\vb{O} - \vb{G}) \vdot \grad \thinspace . \end{equation}\] The diamagnetic term becomes: \[\begin{align} D_{\text{ext}}(\vb{B}_{\text{ext}}, \vb{G}) &= \frac{1}{2} \norm{ \vb{A}_{\text{ext}} ( \vb{r}; \vb{B}_{\text{ext}}, \vb{G} ) }^2 \\ &= \frac{1}{8} \qty( \norm{\vb{B}_{\text{ext}}}^2 \norm{ \vb{r} - \vb{G} }^2 - \qty[ \vb{B}_{\text{ext}} \cdot (\vb{r} - \vb{G}) ]^2 ) \thinspace . \end{align}\]
In a molecular context, in addition to a uniform magnetic field \(\vb{B}_{\text{ext}}\), the nuclei with their associated permanent magnetic (dipole) moments \(\qty{ \vb{M}_K }\) also generate a vector potential up to a first-order multipole expansion (T. Helgaker et al. 2012) (T. Helgaker and Jørgensen 1991): \[\begin{equation} \vb{A}_{\text{nuc}} ( \vb{r}; \qty{ \vb{M}_K } ) = \frac{1}{c^2} \sum_K^M \frac{ \vb{M}_K \cross (\vb{r} - \vb{R}_K) }{ \norm{ \vb{r} - \vb{R}_K }^3 } \thinspace , \end{equation}\] which also satisfies the Coulomb gauge condition. (Note that will not include any gauge-modification to this term). The total vector potential \(\vb{A}\) then consists of both contributions: \[\begin{equation} \vb{A} ( \vb{r}; \vb{B}_{\text{ext}}, \qty{ \vb{M}_K }, \vb{G} ) = \vb{A}_{\text{ext}} ( \vb{r}; \vb{B}_{\text{ext}}, \vb{G} ) + \vb{A}_{\text{nuc}} ( \vb{r}; \qty{ \vb{M}_K } ) \thinspace , \end{equation}\] which means that the scalar kinetic operator must be written as: \[\begin{equation} k^c ( \vb{B}_{\text{ext}}, \qty{ \vb{M}_K }, \vb{G} ) = T^c + P^c ( \vb{B}_{\text{ext}}, \qty{ \vb{M}_K }, \vb{G} ) + D ( \vb{B}_{\text{ext}}, \qty{ \vb{M}_K }, \vb{G} ) \thinspace . \end{equation}\] The paramagnetic term then consists of two contributions: \[\begin{equation} P^c ( \vb{B}_{\text{ext}}, \qty{ \vb{M}_K }, \vb{G} ) = P^c_{\text{ext}} ( \vb{B}_{\text{ext}}, \vb{G} ) + P^c_{\text{nuc}} ( \qty{ \vb{M}_K } ) \thinspace , \end{equation}\] where the nuclear paramagnetic term can be written with the use of the orbital angular momentum operator \(\vb{L}^c(\vb{R}_K)\) about the nuclear position \(\vb{R}_K\) for each nucleus \(K\): \[\begin{equation} P^c_{\text{nuc}} ( \qty{ \vb{M}_K } ) = \frac{1}{c^2} \sum_K \frac{ \vb{M}_K \vdot \vb{L}^c(\vb{R}_K) }{ \norm{ \vb{r} - \vb{R}_K }^3 } \thinspace . \end{equation}\] Finally, the diamagnetic term itself consists of three contributions: \[\begin{equation} D ( \vb{B}_{\text{ext}}, \qty{ \vb{M}_K }, \vb{G} ) = D_{\text{ext}} ( \vb{B}_{\text{ext}}, \vb{G} ) + D_{\text{couple}} ( \vb{B}_{\text{ext}}, \qty{ \vb{M}_K }, \vb{G} ) + D_{\text{nuc}}(\qty{ \vb{M}_K }) \thinspace , \end{equation}\] in which \(D_{\text{couple}}\) is the diamagnetic term that arises from the coupling of the external magnetic field with the nuclear permanent magnetic dipole moments: \[\begin{align} & D_{\text{couple}} ( \vb{B}_{\text{ext}}, \qty{ \vb{M}_K }, \vb{G} ) = \vb{A}_{\text{ext}} ( \vb{r}; \vb{G} ) \vdot \vb{A}_{\text{nuc}} ( \vb{r}; \qty{ \vb{M}_K } ) \\ &\hspace{3pt} = \frac{1}{2c^2} \sum_K \frac{ (\vb{B}_{\text{ext}} \vdot \vb{M}_K ) \qty[ (\vb{r} - \vb{G}) \vdot (\vb{r} - \vb{R}_K) ] - \qty[ \vb{B}_{\text{ext}} \vdot (\vb{r} - \vb{R}_K) ] \qty[ \vb{M}_K \vdot (\vb{r} - \vb{G}) ] }{ \norm{ \vb{r} - \vb{R}_K }^3 } \end{align}\] and \(D_{\text{nuc}}\) is the diamagnetic term that arises from the interaction of the nuclear permanent magnetic dipole moments with themselves: \[\begin{align} & D_{\text{nuc}}(\qty{ \vb{M}_K }) = \frac{1}{2} \norm{ \vb{A}_{\text{nuc}} ( \vb{r}; \qty{ \vb{M}_K } ) }^2 \\ &\hspace{3pt} = \frac{1}{2c^4} \sum_{KL} \frac{ (\vb{M}_K \vdot \vb{M}_L) \qty[ (\vb{r} - \vb{R}_K) \vdot (\vb{r} - \vb{R}_L) ] - \qty[ \vb{M}_K \vdot (\vb{r} - \vb{R}_L) ] \qty[ \vb{M}_L \vdot (\vb{r} - \vb{R}_K) ] }{ \norm{\vb{r} - \vb{R}_K}^3 \norm{\vb{r} - \vb{R}_L}^3 } \thinspace . \end{align}\]