Frequency Tensors (FT), \(\Lambda_{L, n}^{(k)}(i,j)\)¶

Single nucleus frequency tensors¶

First order Nuclear shielding¶

void FCF_1st_order_nuclear_shielding_tensor_components(double *restrict Lambda_0, void *restrict Lambda_2, const double omega_0_delta_iso_in_Hz, const double omega_0_zeta_sigma_in_Hz, const double eta, const double *Theta, const float mf, const float mi)¶

The frequency tensors (FT) from the first-order perturbation expansion of the nuclear shielding Hamiltonian, in a given frame, \(\mathcal{F}\), described by the Euler angles \(\Theta = [\alpha, \beta, \gamma]\) are

\[\begin{split} {\Lambda'}_{0,0}^{(\sigma)}(\Theta, i,j) &= \mathcal{R'}_{0,0}^{(\sigma)}(\Theta) ~~ \mathbb{p}(i, j),~\text{and} \\ {\Lambda'}_{2,n}^{(\sigma)}(\Theta, i,j) &= \mathcal{R'}_{2,n}^{(\sigma)}(\Theta) ~~ \mathbb{p}(i, j), \end{split}\]

where \(\mathcal{R'}_{0,0}^{(\sigma)}(\Theta)\) and \(\mathcal{R'}_{2,n}^{(\sigma)}(\Theta)\) are the spatial orientation functions in frame \(\mathcal{F}\), and \(\mathbb{p}(i, j)\) is the spin transition function for \(\left|i\right> \rightarrow \left|j\right>\) transition.

Parameters

Lambda_0: A pointer to an array of length 1 where the frequency components from \({\Lambda'}_{0,0}^{(\sigma)}(\Theta, i,j)\) will be stored.
Lambda_2: A pointer to a complex array of length 5 where the frequency components from \({\Lambda'}_{2,n}^{(\sigma)}(\Theta, i,j)\) will be stored ordered according to \(\left[{\Lambda'}_{2,n}^{(\sigma)}(\Theta, i,j)\right]_{n=-2}^2\).
omega_0_delta_iso_in_Hz: The quantity, \(2\pi\omega_0\delta_\text{iso}\), given in Hz.
omega_0_zeta_sigma_in_Hz: The quantity, \(2\pi\omega_0\zeta_sigma\), representing the strength of the nuclear shielding anisotropy, given in Hz, defined using Haeberlen convention.
eta: The nuclear shielding asymmetry parameter, \(\eta_\sigma \in [-1,1]\), defined using Haeberlen convention.
Theta: A pointer to an array of length 3 where Euler angles, ordered as \([\alpha, \beta, \gamma]\), are stored.
mf: A float containing the spin quantum number of the final energy state.
mi: A float containing the spin quantum number of the initial energy state.

First order Electric Quadrupole¶

void FCF_1st_order_electric_quadrupole_tensor_components(void *restrict Lambda_2, const double spin, const double Cq_in_Hz, const double eta, const double *Theta, const float mf, const float mi)¶

The frequency component function (FCF) from the first-order electric quadrupole Hamiltonian, in a given frame, \(\mathcal{F}\), described by the Euler angles \(\Theta = [\alpha, \beta, \gamma]\), is

\[ {\Lambda'}_{2,n}^{(q)}(\Theta,i,j) = \mathcal{R'}_{2,n}^{(q)}(\Theta) ~~ \mathbb{d}(i, j), \]

where \(\mathcal{R}_{2,n}^{(q)}(\Theta)\) are the spatial orientation functions in frame \(\mathcal{F}\), and \(\mathbb{d}(i, j)\) is the spin transition function for \(\left|i\right> \rightarrow \left|j\right>\) transition.

Parameters

Lambda_2: A pointer to a complex array of length 5 where the frequency components from \({\Lambda'}_{2,n}^{(q)}(\Theta,i,j)\) will be stored ordered according to \(\left[{\Lambda'}_{2,n}^{(q)}(\Theta,i,j)\right]_{n=-2}^2\).
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.
mf: A float containing the spin quantum number of the final energy state.
mi: A float containing the spin quantum number of the initial energy state.

Second order Electric Quadrupole¶

void FCF_2nd_order_electric_quadrupole_tensor_components(double *restrict Lambda_0, void *restrict Lambda_2, void *restrict Lambda_4, const double spin, const double v0_in_Hz, const double Cq_in_Hz, const double eta, const double *Theta, const float mf, const float mi)¶

The frequency component functions (FCF) from the second-order electric quadrupole Hamiltonian, in a given frame, \(\mathcal{F}\), described by the Euler angles \(\Theta = [\alpha, \beta, \gamma]\), are

\[\begin{split} {\Lambda'}_{0,0}^{(qq)}(\Theta, i,j) &= \mathcal{R'}_{0,0}^{(qq)}(\Theta) ~~ \mathbb{c}_0(i, j), \\ {\Lambda'}_{2,n}^{(qq)}(\Theta, i,j) &= \mathcal{R'}_{2,n}^{(qq)}(\Theta) ~~ \mathbb{c}_2(i, j),~\text{and} \\ {\Lambda'}_{4,n}^{(qq)}(\Theta, i,j) &= \mathcal{R'}_{4,n}^{(qq)}(\Theta) ~~ \mathbb{c}_4(i, j), \end{split}\]

where \(\mathcal{R'}_{0,0}^{(qq)}(\Theta)\), \(\mathcal{R'}_{2,n}^{(qq)}(\Theta)\), and, \(\mathcal{R'}_{4,n}^{(qq)}(\Theta)\) are the spatial orientation functions in frame \(\mathcal{F}\), and \(\mathbb{c}_i(i, j)\) are the composite spin transition functions for \(\left|i\right> \rightarrow \left|j\right>\) transition.

Parameters

Lambda_0: A pointer to an array of length 1 where the frequency component from \({\Lambda'}_{0,0}^{(qq)}(\Theta, i,j)\) will be stored.
Lambda_2: A pointer to a complex array of length 5 where the frequency components from \(\Lambda_{2,n}^{(qq)}(\Theta, i,j)\) will be stored ordered according to \(\left[{\Lambda'}_{2,n}^{(qq)}(\Theta, i,j)\right]_{n=-2}^2\).
Lambda_4: A pointer to a complex array of length 5 where the frequency components from \({\Lambda'}_{4,n}^{(qq)}(\Theta, i,j)\) will be stored ordered according to \(\left[{\Lambda'}_{4,n}^{(qq)}(\Theta, i,j)\right]_{n=-4}^4\).
spin: The spin quantum number, \(I\).
Cq_in_Hz: The quadrupole coupling constant, \(C_q\), in Hz.
eta: The quadrupole asymmetry parameter, \(\eta_q \in [0, 1]\).
v0_in_Hz: The Larmor frequency, \(\nu_0\), in Hz.
Theta: A pointer to an array of length 3 where Euler angles, ordered as \([\alpha, \beta, \gamma]\), are stored.
mf: A float containing the spin quantum number of the final energy state.
mi: A float containing the spin quantum number of the initial energy state.