Magnetic molecular properties
Suppose our system of interest is subject to a uniform external magnetic field \(\require{physics} \vb{B}_{\text{ext}}\) and nuclear permanent magnetic dipole moments \(\qty{ \vb{M}_K }\) for each nucleus \(K\). If there are \(M\) nuclei in the molecule, there are in total \(3M\) perturbations related to the nuclear permanent magnetic dipole moments, which brings the total number of perturbations to \(3 + 3M\). Up to second order, we can expand the energy as \[\begin{equation} \begin{split} E(\vb{B}_{\text{ext}}, \qty{ \vb{M}_K }) = % & E(\vb{B}_{\text{ext}, \thinspace 0}, \qty{ \vb{M}_{K, 0} }) \\ & + \sum_m % E^{(1,0)}_m B_{\text{ext}, \thinspace m} % + \sum_{Ki} % E^{(0,1)}_{Ki} M_{Ki} \\ & + \frac{1}{2} \sum_{mn} % E^{(2,0)}_{mn} % B_{\text{ext}, \thinspace m} B_{\text{ext}, \thinspace n} % + \frac{1}{2} \sum_m \sum_{Ki} % E^{(1,1)}_{m,Ki} B_{\text{ext}, \thinspace m} M_{Ki} \\ & + \frac{1}{2} \sum_{Ki} \sum_{Lj} % E^{(0,2)}_{Ki,Lj} M_{Ki} M_{Lj} % \thinspace . \end{split} \end{equation}\] In this expression, we use indices \(m\) and \(n\) to label the \(x,y\)- and \(z\)-components of the external magnetic field \(\vb{B}_{\text{ext}}\) and we use compound indices \(Ki\) to refer to the \(i\)-th component of the nuclear permanent magnetic dipole moment of nucleus \(K\) and similarly for \(Lj\).
The expansion coefficients that appear in the Taylor expansion are more commonly referred to by their following names. (T. Helgaker et al. 2012) The first-order property for the external magnetic field is called the permanent magnetic dipole moment \(\vb{m}\): \[\begin{equation} m_m % = - E^{(1,0)}_m % = - \eval{ % \dv{ % E(\vb{B}_{\text{ext}}, \qty{ \vb{M}_K }) % }{ % B_{\text{ext}, \thinspace m} % } % }_{ \vb{B}_{\text{ext}, \thinspace 0}, \qty{ \vb{M}_{K, 0} } } % \end{equation}\] and the second-order property for the magnetic field is called the magnetizability \(\boldsymbol{\xi}\): \[\begin{equation} \xi_{mn} % = - E^{(2,0)}_{mn} % = - % \eval{ % \frac{ % \dd{^2}{E(\vb{B}_{\text{ext}}, \qty{ \vb{M}_K })} % }{ % \dd{B_{\text{ext}, \thinspace m}} % \dd{B_{\text{ext}, \thinspace n}} % } % }_{ \vb{B}_{\text{ext}, \thinspace 0}, \qty{ \vb{M}_{K, 0} } } % \thinspace . \end{equation}\] The first-order property related to the nuclear permanent magnetic dipole moments is called the hyperfine coupling constant \(\vb{A}\): \[\begin{equation} A_{Ki} % = E^{(0,1)}_{Ki} % = \eval{ % \dv{ % E(\vb{B}_{\text{ext}}, \qty{ \vb{M}_K }) % }{ % M_{Ki} % } % }_{ \vb{B}_{\text{ext}, \thinspace 0}, \qty{ \vb{M}_{K, 0} } } % \end{equation}\] and its second-order related property is called the nuclear spin-spin coupling \(\vb{T}\): \[\begin{equation} T_{Ki, Lj} % = E^{(0,2)}_{Ki,Lj} % = \eval{ % \frac{ % \dd{^2}{E(\vb{B}_{\text{ext}}, \qty{ \vb{M}_K })} % }{ % \dd{M_{Ki}} % \dd{M_{Lj}} % } % }_{ \vb{B}_{\text{ext}, \thinspace 0}, \qty{ \vb{M}_{K, 0} } } % \thinspace . \end{equation}\] The mixed property related to the magnetic field and the nuclear permanent magnetic dipole moment is finally related to the nuclear shielding tensor \(\boldsymbol{\sigma}\): \[\begin{equation} -\delta_{im} + \sigma_{Ki,m} % = E^{(1,1)}_{Ki,m} % = \eval{ % \frac{ % \dd{^2}{E(\vb{B}_{\text{ext}}, \qty{ \vb{M}_K })} % }{ % \dd{M_{Ki}} % \dd{B_{\text{ext}, \thinspace m}} % } % }_{ \vb{B}_{\text{ext}, \thinspace 0}, \qty{ \vb{M}_{K, 0} } } % \thinspace , \end{equation}\] in which the contribution \(-\delta_{im}\) is due to the nuclear contribution to the energy such that the nuclear shielding tensor \(\sigma_{Ki,m}\) naturally represents the the modification to the nuclear-magnetic field interaction due to the electrons. (T. Helgaker et al. 2012)