Scaled spatial orientation tensors (sSOT), \(\varsigma_{L,n}^{(k)}\)¶

Single nucleus spatial orientation tensors¶

First order Nuclear shielding¶

void sSOT_1st_order_nuclear_shielding_tensor_components(double *restrict R_0, void *restrict R_2, const double omega_0_delta_iso_in_Hz, const double omega_0_zeta_sigma_in_Hz, const double eta, const double *Theta)¶

The scaled spatial orientation tensors (sSOT) from the first-order perturbation expansion of the nuclear shielding Hamiltonian, in the principal axis system (PAS), includes the zeroth and second-rank irreducible tensors which follows,

\[ \left. \varsigma_{0,0}^{(\sigma)} = \omega_0\delta_\text{iso} \right\} \text{Rank-0}, \]

\[\begin{split} \left. \begin{aligned} \varsigma_{2,0}^{(\sigma)} &= -\omega_0\zeta_\sigma, \\ \varsigma_{2,\pm1}^{(\sigma)} &= 0, \\ \varsigma_{2,\pm2}^{(\sigma)} &= \frac{1}{\sqrt{6}}\omega_0 \eta_\sigma \zeta_\sigma, \end{aligned} \right\} \text{Rank-2}, \end{split}\]

where \(\sigma_\text{iso}\) is the isotropic nuclear shielding, and, \(\zeta_\sigma\), \(\eta_\sigma\) are the shielding anisotropy and asymmetry parameters from the symmetric second-rank irreducible nuclear shielding tensor defined using Haeberlen convention. Here, \(\omega_0 = -\gamma_I B_0\) is the Larmor frequency where, \(\gamma_I\) and \(B_0\) are the gyromagnetic ratio of the nucleus and the magnetic flux density of the external magnetic field, respectively.

For non-zero Euler angles, \(\Theta = [\alpha, \beta, \gamma]\), Wigner rotation of \(\varsigma_{2,n}^{(\sigma)}\) is performed following,

\[ \mathcal{R'}_{2,n}^{(\sigma)}(\Theta) = \sum_{m = -2}^2 D^2_{m, n}(\Theta) \varsigma_{2,n}^{(\sigma)}, \]

where \(\mathcal{R'}_{2,n}^{(\sigma)}(\Theta)\) are the tensors in the frame defined by the Euler angles \(\Theta\).

Note

The method accepts a frequency physical quantity, that is, \(\omega_0\sigma_\text{iso}/2\pi\) and \(\omega_0\zeta_\sigma/2\pi\), as the isotropic nuclear shielding and nuclear shielding anisotropy, respectively.
When \(\Theta = [0,0,0]\), \(\mathcal{R'}_{2,n}^{(\sigma)}(\Theta) = \varsigma_{2,n}^{(\sigma)}\) where \( n \in [-2,2]\).
\(\mathcal{R'}_{0,0}^{(\sigma)}(\Theta) = \varsigma_{0,0}^{(\sigma)} ~~~ \forall ~ \Theta\).
The method returns \(\mathcal{R'}_{0,0}^{(\sigma)}(\Theta)/2\pi\) and \(\mathcal{R'}_{2,n}^{(\sigma)}(\Theta)/2\pi\), that is, in units of frequency.

Parameters

R_0: A pointer to an array of length 1 where the zeroth-rank irreducible tensor, \(\mathcal{R'}_{0,0}^{(\sigma)}(\Theta)/2\pi\), will be stored.
R_2: A pointer to a complex array of length 5 where the second-rank irreducible tensor, \(\mathcal{R'}_{2,n}^{(\sigma)}(\Theta)/2\pi\), will be stored ordered according to \(\left[\mathcal{R'}_{2,n}^{(\sigma)}(\Theta)/2\pi\right]_{n=-2}^2\).
omega_0_delta_iso_in_Hz: The quantity, \(\omega_0\sigma_\text{iso}/2\pi\), given in Hz.
omega_0_zeta_sigma_in_Hz: The quantity, \(\omega_0\zeta_\sigma/2\pi\), given in Hz.
eta: The nuclear shielding asymmetry, \(\eta_\sigma \in [0, 1]\).
Theta: A pointer to an array of length 3 where Euler angles, ordered as \([\alpha, \beta, \gamma]\), are stored in radians.

First order Electric Quadrupole¶

void sSOT_1st_order_electric_quadrupole_tensor_components(void *restrict R_2, const double spin, const double Cq_in_Hz, const double eta, const double *Theta)¶

The scaled spatial orientation tensors (sSOT) from the first-order perturbation expansion of the electric quadrupole Hamiltonian, in the principal axis system (PAS), includes the second-rank irreducible tensor which follows,

\[\begin{split} \left. \begin{aligned} \varsigma_{2,0}^{(q)} &= \frac{1}{\sqrt{6}} \omega_q, \\ \varsigma_{2,\pm1}^{(q)} &= 0, \\ \varsigma_{2,\pm2}^{(q)} &= -\frac{1}{6} \eta_q \omega_q, \end{aligned} \right\} \text{Rank-2}, \end{split}\]

where \(\omega_q = \frac{6\pi C_q}{2I(2I-1)}\) is the quadrupole splitting frequency, and \(\eta_q\) is the quadrupole asymmetry parameter. Here, \(I\) is the spin quantum number of the quadrupole nucleus, and \(C_q\) is the quadrupole coupling constant.

As before, for non-zero Euler angles, \(\Theta = [\alpha,\beta,\gamma]\), a Wigner rotation of \(\varsigma_{2,n}^{(q)}\) is performed following,

\[ \mathcal{R'}_{2,n}^{(q)}(\Theta) = \sum_{m = -2}^2 D^2_{m, n}(\Theta) \varsigma_{2,n}^{(q)}. \]

where \(\mathcal{R'}_{2,n}^{(q)}(\Theta)\) are the tensors in the frame defined by the Euler angles \(\Theta\).

Note

When \(\Theta = [0,0,0]\), \(\mathcal{R'}_{2,n}^{(q)}(\Theta) = \varsigma_{2,n}^{(q)}\) where \( n \in [-2,2]\).
The method returns \(\mathcal{R'}_{2,0}^{(q)}(\Theta)/2\pi\), that is, in units of frequency.

Parameters

R_2: A pointer to a complex array of length 5 where the the second-rank irreducible tensor, \(\mathcal{R'}_{2,n}^{(q)}(\Theta)/2\pi\), will be stored ordered according to \(\left[\mathcal{R'}_{2,n}^{(q)}(\Theta)/2\pi\right]_{n=-2}^2\).
spin: The spin quantum number, \(I\).
Cq_in_Hz: The quadrupole coupling constant, \(C_q\), in Hz.
eta: The quadrupole asymmetry parameter, \(\eta_q \in [0, 1]\).
Theta: A pointer to an array of length 3 where Euler angles, ordered as \([\alpha, \beta, \gamma]\), are stored in radians.

Second order Electric Quadrupole¶

void sSOT_2nd_order_electric_quadrupole_tensor_components(double *restrict R_0, void *restrict R_2, void *restrict R_4, const double spin, const double v0_in_Hz, const double Cq_in_Hz, const double eta, const double *Theta)¶

The scaled spatial orientation tensors (sSOT) from the second-order perturbation expansion of the electric quadrupole Hamiltonian, in the principal axis system (PAS), includes the zeroth, second and fourth-rank irreducible tensors which follows,

\[\left. \varsigma_{0,0}^{(qq)} = \frac{\omega_q^2}{\omega_0} \frac{1}{6\sqrt{5}} \left(\frac{\eta_q^2}{3} + 1 \right) \right\} \text{Rank-0}, \]

\[\begin{split} \left. \begin{aligned} \varsigma_{2,0}^{(qq)} &= \frac{\omega_q^2}{\omega_0} \frac{\sqrt{2}}{6\sqrt{7}} \left(\frac{\eta_q^2}{3} - 1 \right), \\ \varsigma_{2,\pm1}^{(qq)} &= 0, \\ \varsigma_{2,\pm2}^{(qq)} &= -\frac{\omega_q^2}{\omega_0} \frac{1}{3\sqrt{21}} \eta_q, \end{aligned} \right\} \text{Rank-2}, \end{split}\]

\[\begin{split} \left. \begin{aligned} \varsigma_{4,0}^{(qq)} &= \frac{\omega_q^2}{\omega_0} \frac{1}{\sqrt{70}} \left(\frac{\eta_q^2}{18} + 1 \right), \\ \varsigma_{4,\pm1}^{(qq)} &= 0, \\ \varsigma_{4,\pm2}^{(qq)} &= -\frac{\omega_q^2}{\omega_0} \frac{1}{6\sqrt{7}} \eta_q, \\ \varsigma_{4,\pm3}^{(qq)} &= 0, \\ \varsigma_{4,\pm4}^{(qq)} &= \frac{\omega_q^2}{\omega_0} \frac{1}{36} \eta_q^2, \end{aligned} \right\} \text{Rank-4}, \end{split}\]

where \(\omega_q = \frac{6\pi C_q}{2I(2I-1)}\) is the quadrupole splitting frequency, \(\omega_0\) is the Larmor angular frequency, and \(\eta_q\) is the quadrupole asymmetry parameter. Here, \(I\) is the spin quantum number, and \(C_q\) is the quadrupole coupling constant.

For non-zero Euler angles, \(\Theta = [\alpha,\beta,\gamma]\), Wigner rotation of \(\varsigma_{2,n}^{(qq)}\) and \(\varsigma_{4,n}^{(qq)}\) are performed following,

\[\begin{split} \mathcal{R'}_{2,n}^{(qq)}(\Theta) &= \sum_{m = -2}^2 D^2_{m, n}(\Theta) \varsigma_{2,n}^{(qq)}, \\ \mathcal{R'}_{4,n}^{(qq)}(\Theta) &= \sum_{m = -4}^4 D^4_{m, n}(\Theta) \varsigma_{4,n}^{(qq)}. \end{split}\]

where \(\mathcal{R'}_{2,n}^{(qq)}(\Theta)\) and \(\mathcal{R'}_{4,n}^{(qq)}(\Theta)\) are the tensors in the frame defined by the Euler angles \(\Theta\).

Note

When \(\Theta = [0,0,0]\), \(\mathcal{R'}_{2,n}^{(qq)}(\Theta) = \varsigma_{2,n}^{(qq)}\) where \( n \in [-2,2]\).
When \(\Theta = [0,0,0]\), \(\mathcal{R'}_{4,n}^{(qq)}(\Theta) = \varsigma_{4,n}^{(qq)}\) where \( n \in [-4,4]\).
\(\mathcal{R'}_{0,0}^{(qq)}(\Theta) = \varsigma_{0,0}^{(qq)} ~~~ \forall ~ \Theta\).
The method returns \(\mathcal{R'}_{0,0}^{(qq)}(\Theta)/2\pi\), \(\mathcal{R'}_{2,n}^{(qq)}(\Theta)/2\pi\), and \(\mathcal{R'}_{4,n}^{(qq)}(\Theta)/2\pi\), that is, in units of frequency.

Parameters

R_0: A pointer to an array of length 1 where the zeroth-rank irreducible tensor, \(\mathcal{R'}_{0,0}^{(qq)}(\Theta)/2\pi\), will be stored.
R_2: A pointer to a complex array of length 5 where the second-rank irreducible tensor, \(\mathcal{R'}_{2,n}^{(qq)}(\Theta)/2\pi\), will be stored ordered according to \(\left[\mathcal{R'}_{2,n}^{(qq)}(\Theta)/2\pi\right]_{n=-2}^2\).
R_4: A pointer to a complex array of length 9 where the fourth-rank irreducible tensor, \(\mathcal{R'}_{4,n}^{(qq)}(\Theta)/2\pi\), will be stored ordered according to \(\left[\mathcal{R'}_{4,n}^{(qq)}(\Theta)/2\pi\right]_{n=-4}^4\).
spin: The spin quantum number, \(I\).
v0_in_Hz: The Larmor frequency, \(\omega_0/2\pi\), in Hz.
Cq_in_Hz: The quadrupole coupling constant, \(C_q\), in Hz.
eta: The quadrupole asymmetry parameter, \(\eta_q \in [0, 1]\).
Theta: A pointer to an array of length 3 where Euler angles, ordered as \([\alpha, \beta, \gamma]\), are stored in radians.