London orbitals
The question now remains what a `good’ form of the perturbation-dependent orthonormal spinors \(\require{physics} \qty{ \phi_P ( \vb{r}; \vb{B}_{\text{ext}}, \qty{ \vb{M}_K } ) }\) would be. We know that we will have to expand their \(\alpha\)- and \(\beta\)-components in underlying scalar bases, so we should try to incorporate some of the physics of the problem inside the scalar bases.
In order to do so, we will first take a look at a one-electron field-free atomic system with a nucleus located at \(\vb{R}_K\), whose Schrödinger eigenvalue problem can be solved: \[\begin{equation} \mathcal{H}^c(\vb{B}_{\text{ext}, 0}) % \psi_{nlm}(\vb{r}; \vb{R}_K) % = E^{(0)} \psi_{nlm}(\vb{r}; \vb{R}_K) % \thinspace , \end{equation}\] in which \(\mathcal{H}^c(\vb{B}_{\text{ext}, 0})\) is the field-free Schrödinger (scalar) Hamiltonian: \[\begin{equation} \mathcal{H}^c(\vb{B}_{\text{ext}, 0}) % = - \frac{1}{2} \laplacian - \phi(\vb{r}) % \thinspace , \end{equation}\] where \(\phi(\vb{r})\) is the scalar potential in the system. Applying a uniform external magnetic field \(\vb{B}_{\text{ext}}\) along the \(z\)-axis, i.e. \[\begin{equation} \vb{B}_{\text{ext}} = B_{\text{ext}} \thinspace \vb{e}_z % \thinspace , \end{equation}\] the field-dependent Schrödinger Hamiltonian can be written as: \[\begin{equation} \mathcal{H}^c(\vb{B}_{\text{ext}}, \vb{G}) % = \frac{ % \norm{\boldsymbol{\pi}^c(\vb{B}_{\text{ext}}, \vb{G})}^2 % }{2} % - \phi(\vb{r}) % \thinspace , \end{equation}\] in which the square of the kinematic momentum operator becomes up to first order: \[\begin{equation} \label{eq:kinematic_momentum_first_order} \norm{\boldsymbol{\pi}^c(\vb{B}_{\text{ext}}, \vb{G})}^2 % = -\laplacian % + B_{\text{ext}} L^c_z(\vb{R}_K) % - 2 i \vb{A}_{\text{ext}}(\vb{R}_K; \vb{G}) \vdot \grad % + \mathcal{O}(\norm{\vb{B}_{\text{ext}}}^2) % \thinspace , \end{equation}\] where \(\vb{L}^c(\vb{R}_K)\) represents the angular momentum operator about \(\vb{R}_K\) in coordinate representation: \[\begin{equation} \vb{L}^c(\vb{R}_K) % = (\vb{r} - \vb{R}_K) \cross \grad % \end{equation}\] and \(\vb{G}\) is the gauge origin of the external magnetic field, i.e. the reference to which the vector potential is calculated: \[\begin{equation} \vb{A}_{\text{ext}}(\vb{r}; \vb{G}) % = \frac{1}{2} \vb{B}_{\text{ext}} \cross (\vb{r} - \vb{G}) % \thinspace . \end{equation}\] We can now see that if there gauge origin was located exactly at the position of the nucleus \(\vb{R}_K\), the original atomic eigenfunctions would also be eigenfunctions of the magnetically perturbed Hamiltonian, since the atomic eigenfunctions \(\psi_{nlm}\) are eigenfunctions of \(l^c_z\): \[\begin{equation} l^c_z \psi_{nlm}(\vb{r}; \vb{R}_K) = m \psi_{nlm}(\vb{r}; \vb{R}_K) \end{equation}\] and to the term that includes the gauge origin \(\vb{G}\) would vanish. However, if the gauge origin \(\vb{G}\) is not chosen at the position of the nucleus \(\vb{R}_K\) (which can’t be done for all nuclei in a molecular system, for example), the atomic eigenfunctions wouldn’t be eigenfunctions of the magnetically perturbed Hamiltonian.
As suggested by London T. Helgaker and Jørgensen (1991), we will now take a closer look at what happens when we modify the original atomic eigenfunctions with a gauge-origin including plane wave phase factor. The corresponding functions are then called London orbitals: \[\begin{equation} \omega_{nlm}(\vb{r}; \vb{R}_K, \vb{B}_{\text{ext}}, \vb{G}) % = \exp( % -i \vb{A}_{\text{ext}}(\vb{R}_K; \vb{G}) \vdot \vb{r} % ) \psi_{nlm}(\vb{r}; \vb{R}_K) % \thinspace , \end{equation}\] where the phase factor depends on the value of the external vector potential evaluated at the nucleus: \(\vb{A}_{\text{ext}}(\vb{R}_K; \vb{G})\), which is related to the gauge-including term that occurs up to first order in the squared kinematic momentum operator in equation \(\eqref{eq:kinematic_momentum_first_order}\). We then have a very important property related to these gauge-including phase factors: the kinematic momentum operator (with an external vector potential calculated relative to the gauge origin \(\vb{G}\)) acting on this gauge-including phase factor only retains the external vector potential, but whose gauge origin is shifted to the position of the nucleus: \[\begin{equation} \boldsymbol{\pi}^c(\vb{B}_{\text{ext}}, \vb{G}) % \exp( % -i \vb{A}_{\text{ext}}(\vb{R}_K; \vb{G}) \vdot \vb{r} % ) % = \exp( % -i \vb{A}_{\text{ext}}(\vb{R}_K; \vb{G}) \vdot \vb{r} % ) % \vb{A}_{\text{ext}}(\vb{r}; \vb{R}_K) % \thinspace , \end{equation}\] and therefore if the kinematic momentum operator (with a vector potential calculated relative to the gauge origin \(\vb{G}\)) acts on a London orbital, we essentially only have to let the kinematic momentum operator act on the original atomic eigenfunction: \[\begin{multline} \boldsymbol{\pi}^c(\vb{B}_{\text{ext}}, \vb{G}) % \thinspace % \omega_{nlm}(\vb{r}; \vb{R}_K, \vb{B}_{\text{ext}}, \vb{G}) \\ = \exp( % -i \vb{A}_{\text{ext}}(\vb{R}_K; \vb{G}) \vdot \vb{r} % ) \thinspace % \boldsymbol{\pi}^c(\vb{B}_{\text{ext}}, \vb{R}_K) % \thinspace % \psi_{nlm}(\vb{r}; \vb{R}_K) % \thinspace , \end{multline}\] but we note that the external vector potential is now calculated relative to the position of the nucleus \(\vb{R}_K\). Applying this same property twice and recognizing that \[\begin{equation} \vb{A}_{\text{ext}}(\vb{R}_K; \vb{R}_K) = \vb{0} % \thinspace , \end{equation}\] we eventually find that the London-modified atomic functions are, up to first order, eigenfunctions of the perturbed Hamiltonian: \[\begin{multline} \mathcal{H}^c(\vb{B}_{\text{ext}}, \vb{G}) % \thinspace % \omega_{nlm}(\vb{r}; \vb{R}_K, \vb{B}_{\text{ext}}, \vb{G}) \\ = ( % E^{(0)} % + \frac{1}{2} B_{\text{ext}} \thinspace m % ) % \thinspace % \omega_{nlm}(\vb{r}; \vb{R}_K, \vb{B}_{\text{ext}}, \vb{G}) % + \mathcal{O}(\norm{\vb{B}_{\text{ext}}}^2) % \thinspace . \end{multline}\] This discussion therefore provides physical grounds to use the modified London orbitals as scalar functions for the expansion of the components of the spinors.
According to the discussion on quantizing one- and two-electron operators in a spinor basis and its underlying scalar bases, we must introduce two scalar function sets to expand each component of the spinors in. Let us denote the original scalar function sets by \(\qty{ \chi^\alpha_\mu, \chi^\beta_\mu }\). We will then modify these underlying scalar bases according to the London gauge phase factor orbitals London (1937): \[\begin{equation} \omega^\sigma_\mu(\vb{r}; \vb{R}_K, \vb{B}_{\text{ext}}, \vb{G}) % = \exp( % - i \vb{A}_{\text{ext}}(\vb{R}_K; \vb{B}_{\text{ext}}, \vb{G}) % \vdot \vb{r} % ) % \thinspace % \chi^\sigma_\mu(\vb{r}; \vb{R}_K) % \thinspace . \end{equation}\] In this London orbital, \(\vb{R}_K\) is the position on which the scalar function is centered and \(\vb{A}_{\text{ext}}(\vb{R}_K; \vb{G})\) is the value of the external vector potential at \(\vb{R}_K\), relative to the gauge origin \(\vb{G}\). We then find that the overlap of any two London orbitals is dependent on the value of the external uniform magnetic field \(\vb{B}_{\text{ext}}\): \[\begin{align} S^{\sigma \tau}_{\mu \nu}(\vb{B}_{\text{ext}}) % &= \int \dd{\vb{r}} % \omega^{\sigma *}_\mu(\vb{r}; \vb{R}_K, \vb{B}_{\text{ext}}, \vb{G}) % \thinspace % \omega^\tau_\nu(\vb{r}; \vb{R}_L, \vb{B}_{\text{ext}}, \vb{G}) \\ &= \int \dd{\vb{r}} % \chi^{\sigma *}_\mu(\vb{r}; \vb{R}_K) % \thinspace \exp( % \frac{i}{2} \vb{B}_{\text{ext}} % \cross (\vb{R}_K - \vb{R}_L) % \vdot \vb{r} % ) % \chi^\tau_\nu(\vb{r}; \vb{R}_L) % \thinspace , \end{align}\] but is manifestly independent of the gauge origin \(\vb{G}\). Therefore, if we use these London orbitals to expand both components of the orthonormal spinors, the spinors dependent on the external magnetic field \(\vb{B}_{\text{ext}}\) and we must use the theory on perturbation-dependent Fock spaces (cfr. section \(\ref{sec:perturbation_dependent_fock_spaces}\)) in which the scalar metric is dependent on the perturbation.
According to the discussion of one-electron and two-electron operators in section \(\ref{sec:quantizing_operators}\), the required integrals over the underlying scalar bases (consisting of the London-modified orbitals) are then the following. The one-electron integrals are: \[\begin{align} & h^{\alpha \alpha}_{\mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } ) % = k^{\alpha \alpha}_{\mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } ) % + V^{\alpha \alpha}_{\mu \nu} ( \vb{B}_{\text{ext}} ) % + M^{\alpha \alpha}_{z, \thinspace \mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } ) \\ % & h^{\alpha \beta}_{\mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } ) % = M^{\alpha \beta}_{x, \thinspace \mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } ) % - i M^{\alpha \beta}_{y, \thinspace \mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } ) \\ % & h^{\beta \alpha}_{\mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } ) % = M^{\beta \alpha}_{x, \thinspace \mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } ) % + i M^{\beta \alpha}_{y, \thinspace \mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } ) \\ & h^{\beta \beta}_{\mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } ) % = k^{\beta \beta}_{\mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } ) % + V^{\beta \beta}_{\mu \nu} ( \vb{B}_{\text{ext}} ) % - M^{\beta \beta}_{z, \thinspace \mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } ) \end{align}\] and the two-electron integrals are: \[\begin{equation} g^{\sigma \sigma \tau \tau}_{\mu \nu \rho \lambda} ( % \vb{B}_{\text{ext}} % ) % = g_{\mu \nu \rho \lambda} ( \vb{B}_{\text{ext}} ) % \thinspace . \end{equation}\]
The second-quantized Hamiltonian, when using London orbitals for the underlying expansion of the spinor basis, then is gauge-independent: \[\begin{equation} \begin{split} \hat{\mathcal{H}} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } % ) % = &\sum_{PQ}^M % \tilde{h}_{PQ} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } % ) % \hat{E}_{PQ} % + \frac{1}{2} \sum_{PQRS}^M \tilde{g}_{PQRS} ( \vb{B}_{\text{ext}} ) % \hat{e}_{PQRS} \\ &+ h_{\text{nuc}} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } % ) % \end{split} \end{equation}\] and depends on the permanent nuclear dipole moments \(\qty{ \vb{M}_K }\) only through the one-electron integrals and the nuclear contribution.