# 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 $$S$$.