Derivatives over London orbitals

Since the theory of perturbation-dependent Fock spaces requires the first- and second-order partial derivatives of the spinors and ultimately of the underlying scalar functions, we will proceed by providing the perturbation partial derivatives of the integrals over the London orbitals.

The partial derivative of a London orbital with respect to the external magnetic field \(\require{physics} \vb{B}_{\text{ext}}\) is given by: \[\begin{equation} \eval{ % \pdv{ % \omega^\sigma_\mu( % \vb{r}; \vb{R}_K, \vb{B}_{\text{ext}}, \vb{G} % ) % }{ B_{\text{ext}, \thinspace m} } % }_{ \vb{B}_{\text{ext}, \thinspace 0} } % = % - \frac{i}{2} \qty[ % (\vb{R}_K - \vb{G}) \cross \vb{r} % ]_m % \thinspace % \chi^\sigma_\mu(\vb{r}; \vb{R}_K) % \thinspace . \end{equation}\]

For the overlap integrals, we find for the first-order partial derivative: \[\begin{equation} \eval{ % \pdv{ % S^{\sigma \tau}_{\mu \nu}(\vb{B}_{\text{ext}}) % }{ B_{\text{ext}, \thinspace m} } % }_{ \vb{B}_{\text{ext}, \thinspace 0} } % = \frac{i}{2} \int \dd{\vb{r}} % \chi^{\sigma *}_\mu(\vb{r}; \vb{R}_K) % \thinspace % \qty[ (\vb{R}_K - \vb{R}_L) \cross \vb{r} ]_m % \thinspace % \chi^\tau_\nu(\vb{r}; \vb{R}_L) % \end{equation}\] and for the second-order derivative: \[\begin{multline} \eval{ % \pdv{ % S^{\sigma \tau}_{\mu \nu}(\vb{B}_{\text{ext}}) % }{ B_{\text{ext}, \thinspace m} } % { B_{\text{ext}, \thinspace n} } % }_{ \vb{B}_{\text{ext}, \thinspace 0} } \\ = - \frac{1}{4} \int \dd{\vb{r}} % \chi^{\sigma *}_\mu(\vb{r}; \vb{R}_K) % \thinspace % \qty[ (\vb{R}_K - \vb{R}_L) \cross \vb{r} ]_m % \qty[ (\vb{R}_K - \vb{R}_L) \cross \vb{r} ]_n % \thinspace % \chi^\tau_\nu(\vb{r}; \vb{R}_L) % \thinspace . \end{multline}\]

Since both the scalar one-electron operators of which the integrals are evaluated and the extra exponential factor depend on the external magnetic field \(\vb{B}_{\text{ext}}\), we must use the product rule for derivatives if we want to calculate partial derivatives with respect to the external magnetic field \(\vb{B}_{\text{ext}}\): \[\begin{align} & \eval{ % \pdv{ % h^{\sigma \tau}_{\mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } % ) % }{ B_{\text{ext}, \thinspace m} } % }_{ % \vb{B}_{\text{ext}, \thinspace 0}, % \thinspace % \qty{ \vb{M}_{K, \thinspace 0} } % } \notag \\ & \hspace{12pt} % = \frac{i}{2} \int \dd{\vb{r}} % \chi^{\sigma *}_\mu(\vb{r}; \vb{R}_K) % \thinspace % \qty[ (\vb{R}_K - \vb{R}_L) \cross \vb{r} ]_m % \thinspace % h^{c, \thinspace \sigma \tau} ( % \vb{B}_{\text{ext}, \thinspace 0}, % \qty{ \vb{M}_{K, \thinspace 0} } % ) \thinspace % \chi^\tau_\nu(\vb{r}; \vb{R}_L) \notag \\ & \hspace{24pt} % + \int \dd{\vb{r}} % \chi^{\sigma *}_\mu(\vb{r}; \vb{R}_K) % \eval{ % \pdv{ % h^{c, \thinspace \sigma \tau}_{\mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K }, % \vb{R}_L % ) % }{ B_{\text{ext}, \thinspace m} } % }_{ % \vb{B}_{\text{ext}, \thinspace 0}, % \thinspace % \qty{ \vb{M}_{K, \thinspace 0} } % } % \chi^\tau_\nu(\vb{r}; \vb{R}_L) \end{align}\] and \[\begin{align} & \eval{ % \pdv{ % h^{\sigma \tau}_{\mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } % ) % }{ B_{\text{ext}, \thinspace m} } % { B_{\text{ext}, \thinspace n} } % }_{ % \vb{B}_{\text{ext}, \thinspace 0}, % \thinspace % \qty{ \vb{M}_{K, \thinspace 0} } % } \notag \\ & \hspace{12pt} % = - \frac{1}{4} \int \dd{\vb{r}} % \chi^{\sigma *}_\mu(\vb{r}; \vb{R}_K) % \thinspace % \qty[ (\vb{R}_K - \vb{R}_L) \cross \vb{r} ]_m % \qty[ (\vb{R}_K - \vb{R}_L) \cross \vb{r} ]_n \notag \\ & \hspace{80pt} \times % h^{\sigma \tau}_{\mu \nu} ( % \vb{B}_{\text{ext}, 0}, % \qty{ \vb{M}_{K, 0} } % ) % \thinspace % \chi^\tau_\nu(\vb{r}; \vb{R}_L) \notag \\ & \hspace{24pt} % + \frac{i}{2} ( 1 + P_{mn} ) \int \dd{\vb{r}} % \chi^{\sigma *}_\mu(\vb{r}; \vb{R}_K) % \qty[ (\vb{R}_K - \vb{R}_L) \cross \vb{r} ]_m \notag \\ & \hspace{132pt} \times % \eval{ % \pdv{ % h^{c, \thinspace \sigma \tau}_{\mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K }, % \vb{R}_L % ) % }{ B_{\text{ext}, \thinspace m} } % }_{ % \vb{B}_{\text{ext}, \thinspace 0}, % \thinspace % \qty{ \vb{M}_{K, \thinspace 0} } % } % \chi^\tau_\nu(\vb{r}; \vb{R}_L) \notag \\ & \hspace{24pt} % + \int \dd{\vb{r}} % \chi^{\sigma *}_\mu(\vb{r}; \vb{R}_K) % \eval{ % \pdv{ % h^{c, \thinspace \sigma \tau}_{\mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K }, % \vb{R}_L % ) % }{ B_{\text{ext}, \thinspace m} } % { B_{\text{ext}, \thinspace n} } }_{ % \vb{B}_{\text{ext}, \thinspace 0}, % \thinspace % \qty{ \vb{M}_{K, \thinspace 0} } % } % \chi^\tau_\nu(\vb{r}; \vb{R}_L) \end{align}\]

We then find the following partial derivatives for the (total) one-electron integrals. The diagonal elements, i.e. \(h^{\alpha \alpha}_{\mu \nu}\) and \(h^{\beta \beta}_{\mu \nu}\) both contain contributions from the scalar operators and the magnetic field in the \(z\)-direction. For \(h^{\alpha \alpha}_{\mu \nu}\), we find: \[\begin{align} & \eval{ % \pdv{ % h^{\alpha \alpha}_{\mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } % ) % }{ B_{\text{ext}, \thinspace m} } % }_{ % \vb{B}_{\text{ext}, \thinspace 0}, % \thinspace % \qty{ \vb{M}_{K, \thinspace 0} } % } \notag \\ & \hspace{12pt} % = \frac{1}{2} \int \dd{\vb{r}} % \chi^{\alpha *}_\mu(\vb{r}; \vb{R}_K) % \thinspace % \qty[ % i \qty[ % (\vb{R}_K - \vb{R}_L) % \cross \vb{r} % ]_m \qty( % T^c - \phi(\vb{r}) % ) % + L^c_m(\vb{R}_L) % ] % \thinspace % \chi^\alpha_\nu(\vb{r}; \vb{R}_L) \notag \\ & \hspace{24pt} % + \frac{1}{2} \delta_{mz} % S^{\alpha \alpha}_{\mu \nu} ( % \vb{B}_{\text{ext}, \thinspace 0}, \qty{ \vb{M}_{K, \thinspace 0} } ) % \thinspace , \end{align}\] and for \(h^{\beta \beta}_{\mu \nu}\), we find an analogous result, but the spin-magnetic field interaction is flipped: \[\begin{align} & \eval{ % \pdv{ % h^{\beta \beta}_{\mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } % ) % }{ B_{\text{ext}, \thinspace m} } % }_{ % \vb{B}_{\text{ext}, \thinspace 0}, % \thinspace % \qty{ \vb{M}_{K, \thinspace 0} } % } \notag \\ & \hspace{12pt} % = \frac{1}{2} \int \dd{\vb{r}} % \chi^{\beta *}_\mu(\vb{r}; \vb{R}_K) % \thinspace % \qty[ % i \qty[ % (\vb{R}_K - \vb{R}_L) % \cross \vb{r} % ]_m \qty( % T^c - \phi(\vb{r}) % ) % + L^c_m(\vb{R}_L) % ] % \thinspace % \chi^\beta_\nu(\vb{r}; \vb{R}_L) \notag \\ & \hspace{24pt} % - \frac{1}{2} \delta_{mz} % S^{\beta \beta}_{\mu \nu} ( % \vb{B}_{\text{ext}, \thinspace 0}, \qty{ \vb{M}_{K, \thinspace 0} } ) % \thinspace . \end{align}\] For the off-diagonal elements, we only have spin-magnetic field contributions to the partial derivatives: \[\begin{align} & \eval{ % \pdv{ % h^{\alpha \beta}_{\mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } % ) % }{ B_{\text{ext}, \thinspace m} } % }_{ % \vb{B}_{\text{ext}, \thinspace 0}, % \thinspace % \qty{ \vb{M}_{K, \thinspace 0} } % } % = \frac{1}{2} \qty( % \delta_{mx} S^{\alpha \beta}_{\mu \nu} % - i \delta_{my} S^{\alpha \beta}_{\mu \nu} % ) \\ % & \eval{ % \pdv{ % h^{\beta \alpha}_{\mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } % ) % }{ B_{\text{ext}, \thinspace m} } % }_{ % \vb{B}_{\text{ext}, \thinspace 0}, % \thinspace % \qty{ \vb{M}_{K, \thinspace 0} } % } % = \frac{1}{2} \qty( % \delta_{mx} S^{\beta \alpha}_{\mu \nu} % +i \delta_{my} S^{\beta \alpha}_{\mu \nu} % ) % \thinspace . \end{align}\]

For the second-order derivatives with respect to the external magnetic field, we only retain the diagonal elements \(h^{\sigma \sigma}_{\mu \nu}\): \[\begin{align} & \eval{ % \pdv{ % h^{\sigma \sigma}_{\mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } % ) % }{ B_{\text{ext}, \thinspace m} } % { B_{\text{ext}, \thinspace n} } % }_{ % \vb{B}_{\text{ext}, \thinspace 0}, % \thinspace % \qty{ \vb{M}_{K, \thinspace 0} } % } \notag \\ & \hspace{12pt} % = \frac{1}{4} \int \dd{\vb{r}} % \chi^{\sigma *}_\mu(\vb{r}; \vb{R}_K) % \thinspace % \bigg[ % \qty[ (\vb{R}_K - \vb{R}_L) \cross \vb{r} ]_m % \qty[ (\vb{R}_K - \vb{R}_L) \cross \vb{r} ]_n % \qty( T^c - \phi(\vb{r}) ) \notag \\ & \hspace{120pt} % + i (1 + P_{mn}) \qty[ % (\vb{R}_K - \vb{R}_L) % \cross \vb{r} % ]_m L^c_n(\vb{R}_L) \notag \\ & \hspace{120pt} % + \delta_{mn} % \norm{ \vb{r} - \vb{R}_L }^2 % + \qty( \vb{r} - \vb{R}_L )_m % \qty( \vb{r} - \vb{R}_L )_n % \bigg] % \thinspace % \chi^\sigma_\nu(\vb{r}; \vb{R}_L) % \thinspace , \end{align}\] because the contribution to second-order derivative due to the spin-magnetic field interaction vanishes.

For the partial derivatives related to the nuclear permanent magnetic dipole moments, we have for the diagonal elements: \[\begin{align} \eval{ % \pdv{ % h^{\alpha \alpha}_{\mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } % ) % }{ M_{Mi} } % }_{ % \vb{B}_{\text{ext}, \thinspace 0}, % \thinspace % \qty{ \vb{M}_{K, \thinspace 0} } % } % &= \frac{1}{c^2} \int \dd{\vb{r}} % \chi^{\alpha *}_\mu(\vb{r}; \vb{R}_K) % \thinspace % \frac{ % L^c_i(\vb{R}_M) % } { % \norm{ \vb{r} - \vb{R}_M }^3 } \thinspace % \chi^\alpha_\nu(\vb{r}; \vb{R}_L) \notag \\ & \hspace{12pt} % - \frac{1}{2} % \eval{ % \pdv{ % B^{\alpha \alpha}_{z, \thinspace \mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } % ) % }{ M_{Mi} } % }_{ % \vb{B}_{\text{ext}, \thinspace 0}, % \thinspace % \qty{ \vb{M}_{K, \thinspace 0} } % } % \thinspace , \end{align}\] in which the spin-magnetic field contribution is given by: \[\begin{align} & \frac{1}{2} \eval{ % \pdv{ % B^{\sigma \tau}_{j, \thinspace \mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } % ) % }{ M_{Mi} } % }_{ % \vb{B}_{\text{ext}, \thinspace 0}, % \thinspace % \qty{ \vb{M}_{K, \thinspace 0} } % } \notag \\ & \hspace{12pt} % = \frac{\delta_{ij}}{2c^2} \Bigg[ % \frac{8 \pi}{3} % \chi^{\sigma *}_{\mu}(\vb{R}_L; \vb{R}_K) % \chi^{\tau}_{\nu}(\vb{R}_L; \vb{R}_L) \notag \\ & \hspace{48pt} % + \int \dd{\vb{r}} % \chi^{\sigma *}_\mu(\vb{r}; \vb{R}_K) % \thinspace % \frac{ % 3 [\vb{r} - \vb{R}_M]_j^2 % - \norm{ \vb{r} - \vb{R}_M }^2 % } { % \norm{ \vb{r} - \vb{R}_M }^5 % } \thinspace % \chi^\tau_\nu(\vb{r}; \vb{R}_L) % \Bigg] % \thinspace . \end{align}\] For \(h^{\beta \beta}_{\mu \nu}\), we have an analogous contribution, only the spin-magnetic field interaction is flipped: \[\begin{align} \eval{ % \pdv{ % h^{\beta \beta}_{\mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } % ) % }{ M_{Mi} } % }_{ % \vb{B}_{\text{ext}, \thinspace 0}, % \thinspace % \qty{ \vb{M}_{K, \thinspace 0} } % } % &= \frac{1}{c^2} \int \dd{\vb{r}} % \chi^{\beta *}_\mu(\vb{r}; \vb{R}_K) % \thinspace % \frac{ % L^c_i(\vb{R}_M) % } { % \norm{ \vb{r} - \vb{R}_M }^3 } \thinspace % \chi^\beta_\nu(\vb{r}; \vb{R}_L) \notag \\ & \hspace{12pt} % - \frac{1}{2} % \eval{ % \pdv{ % B^{\beta \beta}_{z, \thinspace \mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } % ) % }{ M_{Mi} } % }_{ % \vb{B}_{\text{ext}, \thinspace 0}, % \thinspace % \qty{ \vb{M}_{K, \thinspace 0} } % } % \thinspace . \end{align}\] The off-diagonal elements only have contributions from the spin-magnetic field interaction: \[\begin{align} & \eval{ % \pdv{ % h^{\alpha \beta}_{\mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } % ) % }{ M_{Mi} } % }_{ % \vb{B}_{\text{ext}, \thinspace 0}, % \thinspace % \qty{ \vb{M}_{K, \thinspace 0} } % } % = ( \delta_{jx} - i \delta_{jy} ) % \frac{1}{2} \eval{ % \pdv{ % B^{\alpha \beta}_{j, \thinspace \mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } % ) % }{ M_{Mi} } % }_{ % \vb{B}_{\text{ext}, \thinspace 0}, % \thinspace % \qty{ \vb{M}_{K, \thinspace 0} } % } \\ % & \eval{ % \pdv{ % h^{\beta \alpha}_{\mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } % ) % }{ M_{Mi} } % }_{ % \vb{B}_{\text{ext}, \thinspace 0}, % \thinspace % \qty{ \vb{M}_{K, \thinspace 0} } % } % = ( \delta_{jx} + i \delta_{jy} ) % \frac{1}{2} \eval{ % \pdv{ % B^{\beta \alpha}_{j, \thinspace \mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } % ) % }{ M_{Mi} } % }_{ % \vb{B}_{\text{ext}, \thinspace 0}, % \thinspace % \qty{ \vb{M}_{K, \thinspace 0} } % } % \thinspace . \end{align}\] For the second-order partial derivative, the only contributions are from the scalar kinetic operator, so we only retain the diagonal elements: \[\begin{align} & \eval{ % \pdv{ % h^{\sigma \sigma}_{\mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } % ) % }{ M_{Mi} } % {M_{Nj}} % }_{ % \vb{B}_{\text{ext}, \thinspace 0}, % \thinspace % \qty{ \vb{M}_{K, \thinspace 0} } % } \notag \\ & \hspace{12pt} % = \frac{1}{c^4} \int \dd{\vb{r}} % \chi^{\sigma *}_\mu(\vb{r}; \vb{R}_K) % \thinspace \notag \\ & \hspace{72pt} % \times \frac{ % \delta_{ij} (\vb{r} - \vb{R}_M) \vdot (\vb{r} - \vb{R}_N) % - [\vb{R} - \vb{R}_M]_i [\vb{R} - \vb{R}_N]_j }{ % \norm{ \vb{r} - \vb{R}_M }^3 % \norm{ \vb{r} - \vb{R}_N }^3 % } \thinspace \chi^\sigma_\nu(\vb{r}; \vb{R}_L) % \thinspace . \end{align}\] Note that for the partial derivatives with respect to only the nuclear permanent magnetic dipole moments, there are no contributions due to the London orbitals because the vector potential \(\vb{A}_{\text{ext}}\) in the gauge-including phase factor is only related to the external magnetic field.

Finally, the mixed partial derivative becomes: \[\begin{align} & \eval{ % \pdv{ % h^{\sigma \sigma}_{\mu \nu} ( % \vb{B}_{\text{ext}}, % \qty{ \vb{M}_K } % ) % }{ B_{\text{ext}, \thinspace m} } % {M_{Mi}} % }_{ % \vb{B}_{\text{ext}, \thinspace 0}, % \thinspace % \qty{ \vb{M}_{K, \thinspace 0} } % } \notag \\ & \hspace{12pt} % = \frac{1}{2c^2} \int \dd{\vb{r}} % \chi^{\sigma *}_\mu(\vb{r}; \vb{R}_K) \notag \\ & \hspace{72pt} \times % \Bigg[ % i \qty[ (\vb{R}_K - \vb{R}_L) \cross \vb{r} ]_m % \frac{ % L^c_i(\vb{R}_M) % } { % \norm{ \vb{r} - \vb{R}_M }^3 % } \notag \\ & \hspace{88pt} % + \frac{ % \delta_{im} (\vb{r} - \vb{R}_L) \vdot (\vb{r} - \vb{R}_M) % - [\vb{r} - \vb{R}_L]_i [\vb{r} - \vb{R}_M]_m % } { % \norm{ \vb{r} - \vb{R}_M }^3 % } % \Bigg] \thinspace % \chi^\sigma_\nu(\vb{r}; \vb{R}_L) % \thinspace . \end{align}\]

For the partial derivatives of the two-electron integrals, the only contributions arise from the differentiation of the London phase factor, so any partial derivatives with respect to the nuclear permanent magnetic dipole moments vanish. For the first-order partial derivative with respect to the external magnetic field, we have: \[\begin{align} \eval{ % \pdv{ % g^{\sigma \sigma \tau \tau}_{\mu \nu \rho \lambda} ( % \vb{B}_{\text{ext}} ) % }{ B_{\text{ext}, \thinspace m} } % }_{ % \vb{B}_{\text{ext}, \thinspace 0} % } % &= \frac{i}{2} \int \int \dd{\vb{r}_1} \dd{\vb{r}_2} % \chi^{\sigma *}_\mu(\vb{r}_1; \vb{R}_K) % \chi^\sigma_\nu(\vb{r}_1; \vb{R}_L) \notag \\ & \hspace{84pt} \times % \frac{ % [(\vb{R}_K - \vb{R}_L) \cross \vb{r}_1 % + % (\vb{R}_M - \vb{R}_N) \cross \vb{r}_2]_m % }{\norm{\vb{r}_1 - \vb{r}_2}} \notag \\ & \hspace{84pt} \times % \chi^{\tau *}_\rho(\vb{r}_2; \vb{R}_M) % \chi^\tau_\lambda(\vb{r}_2; \vb{R}_N) % \end{align}\] and the second-order term is given by: \[\begin{align} \eval{ % \pdv{ % g^{\sigma \sigma \tau \tau}_{\mu \nu \rho \lambda} ( % \vb{B}_{\text{ext}} ) % }{ B_{\text{ext}, \thinspace m} } % { B_{\text{ext}, \thinspace n} } % }_{ % \vb{B}_{\text{ext}, \thinspace 0} % } % &\hspace{12pt} = % - \frac{1}{4} \int \int \dd{\vb{r}_1} \dd{\vb{r}_2} % \chi^{\sigma *}_\mu(\vb{r}_1; \vb{R}_K) % \chi^\sigma_\nu(\vb{r}_1; \vb{R}_L) \notag \\ & \hspace{84pt} \times % [ % (\vb{R}_K - \vb{R}_L) \cross \vb{r}_1 % + % (\vb{R}_M - \vb{R}_N) \cross \vb{r}_2 % ]_m \notag \\ & \hspace{84pt} \times % [ % (\vb{R}_K - \vb{R}_L) \cross \vb{r}_1 % + % (\vb{R}_M - \vb{R}_N) \cross \vb{r}_2 % ]_n \notag \\ & \hspace{84pt} \times % \frac{1}{\norm{\vb{r}_1 - \vb{r}_2}} % \thinspace \chi^{\tau *}_\rho(\vb{r}_2; \vb{R}_M) % \chi^\tau_\lambda(\vb{r}_2; \vb{R}_N) % \thinspace . \end{align}\]