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

Single nucleus frequency tensor components¶

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) components 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

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

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 component from \({\Lambda'}_{0,0}^{(\sigma)}\) is stored.
Lambda_2: A pointer to a complex array of length 5, where the frequency components from \({\Lambda'}_{2,n}^{(\sigma)}\) is stored ordered as \(\left[{\Lambda'}_{2,n}^{(\sigma)}\right]_{n=-2}^2\).
omega_0_delta_iso_in_Hz: The isotropic chemical shift in Hz ( \(\omega_0\delta_\text{iso}/2\pi\)).
omega_0_zeta_sigma_in_Hz: The shielding anisotropy quantity in Hz ( \(\omega_0\zeta_\sigma/2\pi\)) defined using Haeberlen convention.
eta: The shielding asymmetry, \(\eta_\sigma \in [-1,1]\), defined using Haeberlen convention.
Theta: A pointer to an array of Euler angles, in radians, of length 3, ordered as \([\alpha, \beta, \gamma]\).
mf: The spin quantum number of the final energy state.
mi: 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 tensor (FT) components from the first-order perturbation expansion of electric quadrupole Hamiltonian, in a given frame, \(\mathcal{F}\), described by the Euler angles \(\Theta = [\alpha, \beta, \gamma]\) are

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

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 \({\Lambda'}_{2,n}^{(q)}\) is stored ordered as \(\left[{\Lambda'}_{2,n}^{(q)}\right]_{n=-2}^2\).
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 \([\alpha, \beta, \gamma]\).
mf: The spin quantum number of the final energy state.
mi: 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 tensor (FCF) components from the second-order perturbation expansion of electric quadrupole Hamiltonian, in a given frame, \(\mathcal{F}\), described by the Euler angles \(\Theta = [\alpha, \beta, \gamma]\), are

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

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 \(\mathbb{c}_k(i, j)\) 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 \({\Lambda'}_{0,0}^{(qq)}\) is stored.
Lambda_2: A pointer to a complex array of length 5, where the frequency components from \(\Lambda_{2,n}^{(qq)}\) are stored ordered as \(\left[{\Lambda'}_{2,n}^{(qq)}\right]_{n=-2}^2\).
Lambda_4: A pointer to a complex array of length 9, where the frequency components from \({\Lambda'}_{4,n}^{(qq)}\) are stored ordered as \(\left[{\Lambda'}_{4,n}^{(qq)}\right]_{n=-4}^4\).
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 Euler angles, in radians, of length 3, ordered as \([\alpha, \beta, \gamma]\).
mf: The spin quantum number of the final energy state.
mi: The spin quantum number of the initial energy state.

Two coupled nucleus frequency tensor components¶

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

void FCF_1st_order_weak_J_coupling_tensor_components(double *restrict Lambda_0, void *restrict Lambda_2, const double J_iso_in_Hz, const double J_aniso_in_Hz, const double J_eta, const double *Theta, const float mIf, const float mIi, const float mSf, const float mSi)¶

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

\[\begin{split} {\Lambda'}_{0,0}^{(J)} &= \mathcal{R'}_{0,0}^{(J)}(\Theta) ~~ \mathbb{d}_{IS}(m_{i_I}, m_{i_S}, m_{f_I}, m_{f_S}), ~\text{and} \\ {\Lambda'}_{2,n}^{(J)} &= \mathcal{R'}_{2,n}^{(J)}(\Theta) ~~ \mathbb{d}_{IS}(m_{i_I}, m_{i_S}, m_{f_I}, m_{f_S}), \end{split}\]

where \(\mathcal{R'}_{0,0}^{(J)}(\Theta)\) and \(\mathcal{R'}_{2,n}^{(J)}(\Theta)\) are the spatial orientation functions in frame \(\mathcal{F}\), and \(\mathbb{d}_{IS}(m_{i_I}, m_{i_S}, m_{f_I}, m_{f_S})\) is the spin transition function for \(\left|m_{i_I}, m_{i_S}\right> \rightarrow \left|m_{f_I}, m_{f_S}\right>\) transition.

Parameters

Lambda_0: A pointer to an array of length 1, where the frequency component from \({\Lambda'}_{0,0}^{(J)}\) is stored.
Lambda_2: A pointer to a complex array of length 5, where the frequency components from \({\Lambda'}_{2,n}^{(J)}\) is stored ordered as \(\left[{\Lambda'}_{2,n}^{(J)}\right]_{n=-2}^2\).
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 and defined using Haeberlen convention.
J_eta: The J-coupling anisotropy asymmetry parameter, \(\eta_J \in [-1,1]\), defined using Haeberlen convention.
Theta: A pointer to an array of Euler angles of length 3 ordered as \([\alpha, \beta, \gamma]\).
mIf: The spin quantum number of the final energy state of site \(I\).
mIi: The spin quantum number of the initial energy state of site \(I\).
mSf: The spin quantum number of the final energy state of site \(S\).
mSi: The spin quantum number of the initial energy state of site \(S\).

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

void FCF_1st_order_weak_dipolar_coupling_tensor_components(void *restrict Lambda_2, const double D_in_Hz, const double *Theta, const float mIf, const float mIi, const float mSf, const float mSi)¶

The frequency tensor (FT) components from the first-order perturbation expansion of the direct dipolar coupling Hamiltonian (weak coupling limit), in a given frame, \(\mathcal{F}\), described by the Euler angles \(\Theta = [\alpha, \beta, \gamma]\) are

\[ {\Lambda'}_{2,n}^{(d)} = \mathcal{R'}_{2,n}^{(d)}(\Theta) ~~ \mathbb{d}_{IS}(m_{i_I}, m_{i_S}, m_{f_I}, m_{f_S}), \]

where \(\mathcal{R'}_{2,n}^{(d)}(\Theta)\) are the spatial orientation functions in frame \(\mathcal{F}\), and \(\mathbb{d}_{IS}(m_{i_I}, m_{i_S}, m_{f_I}, m_{f_S})\) is the spin transition function for \(\left|m_{i_I}, m_{i_S}\right> \rightarrow \left|m_{f_I}, m_{f_S}\right>\) transition.

Parameters

Lambda_2: A pointer to a complex array of length 5, where the frequency components from \({\Lambda'}_{2,n}^{(d)}\) is stored ordered as \(\left[{\Lambda'}_{2,n}^{(d)}\right]_{n=-2}^2\).
D_in_Hz: The dipolar coupling, \(D\), in Hz.
Theta: A pointer to an array of Euler angles of length 3 ordered as \([\alpha, \beta, \gamma]\).
mIf: The spin quantum number of the final energy state of site \(I\).
mIi: The spin quantum number of the initial energy state of site \(I\).
mSf: The spin quantum number of the final energy state of site \(S\).
mSi: The spin quantum number of the initial energy state of site \(S\).