# 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), include contributions from the zeroth and second-rank irreducible tensors which follow,

$\left. \varsigma_{0,0}^{(\sigma)} = \omega_0\delta_\text{iso} \right\} \text{Rank-0},$
\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 applied 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 tensor components in the frame defined by the Euler angles $$\Theta$$.

Note

• The method accepts frequency physical quantities, that is, $$\omega_0\delta_\text{iso}/2\pi$$ and $$\omega_0\zeta_\sigma/2\pi$$, as the isotropic chemical shift 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 components of the zeroth-rank irreducible tensor, $$\mathcal{R'}_{0,0}^{(\sigma)}(\Theta)/2\pi$$, is stored.

• R_2: A pointer to a complex array of length 5, where the components of the second-rank irreducible tensor, $$\mathcal{R'}_{2,n}^{(\sigma)}(\Theta)/2\pi$$, is stored ordered as $$\left[\mathcal{R'}_{2,n}^{(\sigma)}(\Theta)/2\pi\right]_{n=-2}^2$$.

• omega_0_delta_iso_in_Hz: The isotropic chemical shift in Hz, $$\omega_0\sigma_\text{iso}/2\pi$$.

• omega_0_zeta_sigma_in_Hz: The shielding anisotropy in Hz, $$\omega_0\zeta_\sigma/2\pi$$.

• eta: The shielding asymmetry, $$\eta_\sigma \in [0, 1]$$.

• Theta: A pointer to an array of Euler angles, in radians, of length 3, ordered as $$[\alpha, \beta, \gamma]$$.

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), include contributions from the second-rank irreducible tensor which follow,

\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 applied 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 tensor components 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 components of the second-rank irreducible tensor, $$\mathcal{R'}_{2,n}^{(q)}(\Theta)/2\pi$$, is stored ordered as $$\left[\mathcal{R'}_{2,n}^{(q)}(\Theta)/2\pi\right]_{n=-2}^2$$.

• 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 Euler angles, in radians, of length 3, ordered as $$[\alpha, \beta, \gamma]$$.

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), include contributions from the zeroth, second, and fourth-rank irreducible tensors which follow,

$\left. \varsigma_{0,0}^{(qq)} = \frac{\omega_q^2}{\omega_0} \frac{1}{6\sqrt{5}} \left(\frac{\eta_q^2}{3} + 1 \right) \right\} \text{Rank-0},$
\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 applied 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 second and fourth-rank tensor components in the frame defined by the Euler angles $$\Theta$$, respectively.

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 Euler angles, in radians, of length 3, ordered as $$[\alpha, \beta, \gamma]$$.

### First-order J-coupling (weak coupling limit)¶

void sSOT_1st_order_weakly_coupled_J_tensor_components(double *restrict R_0, void *restrict R_2, const double J_iso_in_Hz, const double J_aniso_in_Hz, const double J_eta, const double *Theta)

The scaled spatial orientation tensors (sSOT) from the first-order perturbation expansion of the $$J$$-coupling Hamiltonian under weak-coupling limit, in the principal axis system (PAS), include contributions from the zeroth and second-rank irreducible tensors which follow,

$\left. \varsigma_{0,0}^{(J)} = 2\pi J_\text{iso} \right\} \text{Rank-0},$
\begin{split} \left. \begin{aligned} \varsigma_{2,0}^{(J)} &= 2 \pi \zeta_J, \\ \varsigma_{2,\pm1}^{(J)} &= 0, \\ \varsigma_{2,\pm2}^{(J)} &= 2\pi \frac{1}{\sqrt{6}} \eta_J \zeta_J, \end{aligned} \right\} \text{Rank-2}, \end{split}
where $$J_\text{iso}$$ is the isotropic $$J$$-coupling, and $$\zeta_J$$, $$\eta_J$$ are the $$J$$-coupling tensor anisotropy and asymmetry parameters from the symmetric second-rank irreducible $$J$$ tensor, defined using Haeberlen convention.

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

$\mathcal{R'}_{2,n}^{(J)}(\Theta) = \sum_{m = -2}^2 D^2_{m, n}(\Theta) \varsigma_{2,n}^{(J)},$
where $$\mathcal{R'}_{2,n}^{(J)}(\Theta)$$ are the tensor components in the frame defined by the Euler angles, $$\Theta$$.

Note

• When $$\Theta = [0,0,0]$$, $$\mathcal{R'}_{2,n}^{(J)}(\Theta) = \varsigma_{2,n}^{(J)}$$ where $$n \in [-2,2]$$.

• $$\mathcal{R'}_{0,0}^{(J)}(\Theta) = \varsigma_{0,0}^{(J)} ~~~ \forall ~ \Theta$$.

• The method returns $$\mathcal{R'}_{0,0}^{(J)}(\Theta)/2\pi$$ and $$\mathcal{R'}_{2,n}^{(J)}(\Theta)/2\pi$$, that is, in units of frequency.

Parameters
• R_0: A pointer to an array of length 1, where the zeroth-rank irreducible tensor, $$\mathcal{R'}_{0,0}^{(J)}(\Theta)/2\pi$$, is stored.

• R_2: A pointer to a complex array of length 5, where the second-rank irreducible tensor, $$\mathcal{R'}_{2,n}^{(J)}(\Theta)/2\pi$$, is stored ordered as $$\left[\mathcal{R'}_{2,n}^{(J)}(\Theta)/2\pi\right]_{n=-2}^2$$.

• J_iso_in_Hz: The isotropic $$J$$-coupling, $$J_\text{iso}$$, in Hz.

• J_aniso_in_Hz: The $$J$$-coupling anisotropy, $$\zeta_J$$, in Hz.

• J_eta: The $$J$$-coupling asymmetry, $$\eta_J \in [0, 1]$$.

• Theta: A pointer to an array of Euler angles, in radians, of length 3, ordered as $$[\alpha, \beta, \gamma]$$.

### First-order dipolar-coupling (weak coupling limit)¶

void sSOT_1st_order_weakly_coupled_dipolar_tensor_components(void *restrict R_2, const double D_in_Hz, const double *Theta)

The scaled spatial orientation tensors (sSOT) from the first-order perturbation expansion of the dipolar-coupling Hamiltonian under weak-coupling limit, in the principal axis system (PAS), include contributions from the second-rank irreducible tensors which follow,

\begin{split} \left. \begin{aligned} \varsigma_{2,0}^{(d)} &= 4\pi D, \\ \varsigma_{2,\pm1}^{(d)} &= 0, \\ \varsigma_{2,\pm2}^{(d)} &= 0, \end{aligned} \right\} \text{Rank-2}, \end{split}
where $$D$$ is the dipolar-coupling.

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

$\mathcal{R'}_{2,n}^{(d)}(\Theta) = \sum_{m = -2}^2 D^2_{m, n}(\Theta) \varsigma_{2,n}^{(d)},$
where $$\mathcal{R'}_{2,n}^{(d)}(\Theta)$$ are the tensor components in the frame defined by the Euler angles, $$\Theta$$.

Note

• When $$\Theta = [0,0,0]$$, $$\mathcal{R'}_{2,n}^{(d)}(\Theta) = \varsigma_{2,n}^{(d)}$$ where $$n \in [-2,2]$$.

• The method returns $$\mathcal{R'}_{2,n}^{(d)}(\Theta)/2\pi$$, that is, in units of frequency.

Parameters
• R_2: A pointer to a complex array of length 5, where the second-rank irreducible tensor, $$\mathcal{R'}_{2,n}^{(d)}(\Theta)/2\pi$$, is stored ordered as $$\left[\mathcal{R'}_{2,n}^{(d)}(\Theta)/2\pi\right]_{n=-2}^2$$.

• D_in_Hz: The dipolar coupling, $$D$$, in Hz.

• Theta: A pointer to an array of Euler angles, in radians, of length 3, ordered as $$[\alpha, \beta, \gamma]$$.