Integrals over London orbitals
Let now us turn to the evaluation of these integrals Ruud et al. (1993). The matrix elements of the scalar kinetic operator \(k^c\) between two gauge-including scalar functions can be calculated analogously to the process described in the discussion of physical grounds of the London orbitals. The only difference is that now, the vector potential has external and nuclear contributions (cfr. equation \(\eqref{eq:vector_potential_ext_nuc}\)), but no complications really arise from this, since initial property related to the London orbital still remains functionally the same: \[\begin{multline} \require{physics} \boldsymbol{\pi}^c ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K }, % \vb{G} % ) % \exp( % -i \vb{A}_{\text{ext}} ( % \vb{R}_K; % \vb{B}_{\text{ext}} % \vb{G}% ) \vdot \vb{r} % ) \\ = \exp( % -i \vb{A}_{\text{ext}} ( % \vb{R}_K; % \vb{B}_{\text{ext}} % \vb{G}% ) \vdot \vb{r} % ) \thinspace , \vb{A} ( % \vb{r}; % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } % \vb{R}_K) % \thinspace , \end{multline}\] such that we have: \[\begin{multline} \boldsymbol{\pi}^c ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K }, % \vb{G} % ) \thinspace % \omega_\mu(\vb{r}; \vb{R}_K, \vb{B}_{\text{ext}}, \vb{G}) \\ = % \exp( % - i \vb{A}_{\text{ext}}(\vb{R}_K; \vb{G}) % \vdot \vb{r} % ) \thinspace % \boldsymbol{\pi}^c ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K }, % \vb{R}_K % ) % \thinspace % \chi_\mu(\vb{r}; \vb{R}_K) % \thinspace . \end{multline}\] We then find that the integrals over the scalar kinetic operator are also gauge-independent: \[\begin{align} k^{\sigma \tau}_{\mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } % ) % &= \int \dd{\vb{r}} % \omega^{\sigma *}_\mu ( % \vb{r}; % \vb{R}_K, \vb{B}_{\text{ext}}, \vb{G} % ) % \thinspace % k^c ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K }, % \vb{G} % ) % \thinspace % \omega^\tau_\nu ( % \vb{r}; % \vb{R}_L, \vb{B}_{\text{ext}}, \vb{G} % ) \\ &= \int \dd{\vb{r}} % \chi^{\sigma *}_\mu(\vb{r}; \vb{R}_K) % \thinspace \exp( % \frac{i}{2} \vb{B}_{\text{ext}} % \cross (\vb{R}_K - \vb{R}_L) % \vdot \vb{r} % ) \notag \\ & \hspace{72pt} \times % k^c ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K }, % \vb{R}_L % ) % \thinspace % \chi^\tau_\nu(\vb{r}; \vb{R}_L) % \end{align}\] and thus only depend on the perturbations \(\vb{B}_{\text{ext}}\) and \(\qty{ \vb{M}_K }\). The kinetic matrix elements \(k^{\sigma \tau}_{\mu \nu}\) can be split up in the following contributions, according to the discussion in section \(\ref{sec:Schrodinger_Hamiltonian}\): \[\begin{align} k^{\sigma \tau}_{\mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } % ) % &= T^{\sigma \tau}_{\mu \nu}( \vb{B}_{\text{ext}} ) % + P^{\sigma \tau}_{\mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K }, % \vb{R}_K % ) % + D^{\sigma \tau}_{\mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K }, % \vb{R}_K % ) \\ &= T^{\sigma \tau}_{\mu \nu}( \vb{B}_{\text{ext}} ) % + P^{\sigma \tau}_{\text{ext}, \thinspace \mu \nu} ( % \vb{B}_{\text{ext}}, % \vb{R}_K % ) % + P^{\sigma \tau}_{\text{nuc}, \thinspace \mu \nu} ( % \qty{ \vb{M}_K } % ) \notag \\ & \hspace{12pt} % + D^{\sigma \tau}_{\text{ext}, \thinspace \mu \nu} ( % \vb{B}_{\text{ext}}, % \vb{R}_K % ) % + D^{\sigma \tau}_{\text{couple}, \thinspace \mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K }, % \vb{R}_K % ) \notag \\ & \hspace{12pt} % + D^{\sigma \tau}_{\text{nuc}, \thinspace \mu \nu} ( % \qty{ \vb{M}_K }, % \vb{R}_K % ) % \thinspace . \end{align}\] The integrals over the scalar potential energy operator are also gauge-independent, in fact, the matrix elements between two London orbitals of a scalar one-electron operator \(f^{c, \sigma \tau}\) that includes the gauge origin \(\vb{G}\) of the external uniform magnetic field \(\vb{B}_{\text{ext}}\) are always gauge independent and have the following form: \[\begin{align} f^{\sigma \tau}_{\mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } % ) % &= \int \dd{\vb{r}} % \omega^{\sigma *}_\mu ( % \vb{r}; % \vb{R}_K, \vb{B}_{\text{ext}}, \vb{G} % ) \notag \\ % & \hspace{72pt} \times % f^{c, \sigma \tau} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K }, % \vb{G} % ) \thinspace % \omega^\tau_\nu ( % \vb{r}; % \vb{R}_L, \vb{B}_{\text{ext}}, \vb{G} % ) \\ &= \int \dd{\vb{r}} % \chi^{\sigma *}_\mu(\vb{r}; \vb{R}_K) % \thinspace \exp( % \frac{i}{2} \vb{B}_{\text{ext}} % \cross (\vb{R}_K - \vb{R}_L) % \vdot \vb{r} % ) \notag \\ & \hspace{72pt} \times % f^{c, \sigma \tau} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K }, % \vb{R}_L % ) \thinspace % \chi^\tau_\nu(\vb{r}; \vb{R}_L) % \thinspace . \end{align}\] The two-electron integrals are also gauge-independent: \[\begin{align} & g^{\sigma \sigma \tau \tau}_{\mu \nu \rho \lambda} ( % \vb{B}_{\text{ext}} % ) \notag \\ & \hspace{12pt} % = \int \int \dd{\vb{r}_1} \dd{\vb{r}_2} \omega^{\sigma *}_\mu ( % \vb{r}_1; % \vb{R}_K, \vb{B}_{\text{ext}}, \vb{G} % ) % \omega^\sigma_\nu ( % \vb{r}_1; % \vb{R}_L, \vb{B}_{\text{ext}}, \vb{G} % ) \notag \\ & \hspace{96pt} \times % \frac{1}{\norm{\vb{r}_1 - \vb{r}_2}} \notag \\ & \hspace{96pt} \times % \omega^{\tau *}_\rho ( % \vb{r}_2; % \vb{R}_M, \vb{B}_{\text{ext}}, \vb{G} % ) % \omega^\tau_\lambda ( % \vb{r}_2; \vb{R}_N, \vb{B}_{\text{ext}}, \vb{G} % ) \\ &\hspace{12pt} = % \int \int \dd{\vb{r}_1} \dd{\vb{r}_2} % \chi^{\sigma *}_\mu(\vb{r}_1; \vb{R}_K) % \chi^\sigma_\nu(\vb{r}_1; \vb{R}_L) \notag \\ & \hspace{96pt} % \times \exp( % \frac{i}{2} \vb{B}_{\text{ext}} \cross \qty[ % (\vb{R}_K - \vb{R}_L) \vdot \vb{r}_1 % + % (\vb{R}_M - \vb{R}_N) \vdot \vb{r}_2 % ] % ) \notag \\ & \hspace{96pt} \times % \frac{1}{\norm{\vb{r}_1 - \vb{r}_2}} \notag \\ & \hspace{96pt} \times % \chi^{\tau *}_\rho(\vb{r}_2; \vb{R}_M) % \chi^\tau_\lambda(\vb{r}_2; \vb{R}_N) % \thinspace . \end{align}\] The integrals that arise from the spin-magnetic field interaction are those over the one-electron operator related to the total magnetic field \(\vb{B}\) and are according to the discussion in section \(\ref{sec:Pauli_Hamiltonian}\) split up into: \[\begin{equation} \vb{B}^{\sigma \tau}_{\mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } % ) % = \vb{B}^{\sigma \tau}_{\text{ext}, \thinspace \mu \nu} % + \vb{B}^{\sigma \tau}_{\text{FC}, \thinspace \mu \nu} ( % \qty{ \vb{M}_K } % ) % + \vb{B}^{\sigma \tau}_{\text{SD}, \thinspace \mu \nu} ( % \qty{ \vb{M}_K } % ) % \thinspace , \end{equation}\] in which the part related to the uniform external magnetic field \(\vb{B}_{\text{ext}}\) simplifies to: \[\begin{equation} \vb{B}^{\sigma \tau}_{\text{ext}, \thinspace \mu \nu} % = \vb{B}_{\text{ext}} \thinspace % S^{\sigma \tau}_{\mu \nu} ( % \vb{B}_{\text{ext}} ) % \thinspace . \end{equation}\]
We should note that the use of London orbitals has as a consequence that the second-quantized Hamiltonian does not depend on our choice of the gauge origin \(\vb{G}\) of the external vector potential \(\vb{A}_{\text{ext}}\). Since, computationally, we do not have access to a complete spinor basis (in which any calculation would be independent of the gauge origin), London orbitals provide an excellent tool to calculate magnetic properties.