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

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// -*- coding: utf-8 -*-
//
//  frequency_tensor.h
//
//  @copyright Deepansh J. Srivastava, 2019-2021.
//  Created by Deepansh J. Srivastava, Apr 11, 2019.
//  Contact email = srivastava.89@osu.edu
//

#include "frequency/spatial_orientation_tensor_components.h"
#include "frequency/spin_transition_functions.h"

// =====================================================================================
//               First-order nuclear shielding frequency tensor components
// =====================================================================================

/**
 * The frequency tensors (FT) components from the first-order perturbation expansion of
 * the nuclear shielding Hamiltonian, in a given frame, @f$\mathcal{F}@f$, described by
 * the Euler angles @f$\Theta = [\alpha, \beta, \gamma]@f$ are
 * @f[
 *    {\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),
 * @f]
 * where @f$\mathcal{R'}_{0,0}^{(\sigma)}(\Theta)@f$ and
 * @f$\mathcal{R'}_{2,n}^{(\sigma)}(\Theta)@f$ are the spatial orientation functions in
 * frame @f$\mathcal{F}@f$, and @f$\mathbb{p}(i, j)@f$ is the spin transition function
 * for @f$\left|i\right> \rightarrow \left|j\right>@f$ transition.
 *
 * @param Lambda_0 A pointer to an array of length 1, where the frequency component from
 *      @f${\Lambda'}_{0,0}^{(\sigma)}@f$ is stored.
 * @param Lambda_2 A pointer to a complex array of length 5, where the frequency
 *      components from @f${\Lambda'}_{2,n}^{(\sigma)}@f$ is stored ordered as
 *      @f$\left[{\Lambda'}_{2,n}^{(\sigma)}\right]_{n=-2}^2@f$.
 * @param omega_0_delta_iso_in_Hz The isotropic chemical shift in Hz
 *      (@f$\omega_0\delta_\text{iso}/2\pi@f$).
 * @param omega_0_zeta_sigma_in_Hz The shielding anisotropy quantity in Hz
 *      (@f$\omega_0\zeta_\sigma/2\pi@f$) defined using Haeberlen convention.
 * @param eta The shielding asymmetry, @f$\eta_\sigma \in [-1,1]@f$, defined using
 *      Haeberlen convention.
 * @param Theta A pointer to an array of Euler angles, in radians, of length 3, ordered
 *      as @f$[\alpha, \beta, \gamma]@f$.
 * @param mf The spin quantum number of the final energy state.
 * @param mi The spin quantum number of the initial energy state.
 */
static inline 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) {
  // Spin transition function
  double transition_fn = STF_p(mf, mi);

  // Return if the transition is zero
  if (transition_fn == 0.0) {
    // zero the R0 and R2 components before populating with shielding components
    *Lambda_0 = 0.0;
    vm_double_zeros(10, (double *)Lambda_2);
    return;
  }

  // Spatial orientation function
  sSOT_1st_order_nuclear_shielding_tensor_components(
      Lambda_0, Lambda_2, omega_0_delta_iso_in_Hz, omega_0_zeta_sigma_in_Hz, eta,
      Theta);

  // frequency component function from the zeroth-rank irreducible tensor.
  *Lambda_0 *= transition_fn;

  // frequency component function from the second-rank irreducible tensor.
  cblas_dscal(10, transition_fn, (double *)Lambda_2, 1);
}

// =====================================================================================
//             First-order electric quadrupolar frequency tensor components
// =====================================================================================

/**
 * The frequency tensor (FT) components from the first-order perturbation expansion of
 * electric quadrupole Hamiltonian, in a given frame, @f$\mathcal{F}@f$, described by
 * the Euler angles @f$\Theta = [\alpha, \beta, \gamma]@f$ are
 * @f[
 *    {\Lambda'}_{2,n}^{(q)} = \mathcal{R'}_{2,n}^{(q)}(\Theta) ~~ \mathbb{d}(i, j),
 * @f]
 * where @f$\mathcal{R}_{2,n}^{(q)}(\Theta)@f$ are the spatial orientation functions in
 * frame @f$\mathcal{F}@f$, and @f$\mathbb{d}(i, j)@f$ is the spin transition function
 * for @f$\left|i\right> \rightarrow \left|j\right>@f$ transition.
 *
 * @param Lambda_2 A pointer to a complex array of length 5, where the frequency
 *      components from @f${\Lambda'}_{2,n}^{(q)}@f$ is stored ordered as
 *      @f$\left[{\Lambda'}_{2,n}^{(q)}\right]_{n=-2}^2@f$.
 * @param spin The spin quantum number, @f$I@f$.
 * @param Cq_in_Hz The quadrupole coupling constant, @f$C_q@f$, in Hz.
 * @param eta The quadrupole asymmetry parameter, @f$\eta_q \in [0, 1]@f$.
 * @param Theta A pointer to an array of Euler angles, in radians, of length 3, ordered
 *      as @f$[\alpha, \beta, \gamma]@f$.
 * @param mf The spin quantum number of the final energy state.
 * @param mi The spin quantum number of the initial energy state.
 */
static inline 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) {
  // Spin transition function
  double transition_fn = STF_d(mf, mi);

  // Return if the transition is zero
  if (transition_fn == 0.0) {
    // zero the R2 components before populating with quad components
    vm_double_zeros(10, (double *)Lambda_2);
    return;
  }

  // Spatial orientation function
  sSOT_1st_order_electric_quadrupole_tensor_components(Lambda_2, spin, Cq_in_Hz, eta,
                                                       Theta);

  // frequency component function from the second-rank irreducible tensor.
  cblas_dscal(10, transition_fn, (double *)Lambda_2, 1);
}

// =====================================================================================
//             Second-order electric quadrupolar frequency tensor components
// =====================================================================================

/**
 * The frequency tensor (FCF) components from the second-order perturbation expansion of
 * electric quadrupole Hamiltonian, in a given frame, @f$\mathcal{F}@f$, described by
 * the Euler angles @f$\Theta = [\alpha, \beta, \gamma]@f$, are
 * @f[
 *    {\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),
 * @f]
 * where @f$\mathcal{R'}_{0,0}^{(qq)}(\Theta)@f$,
 * @f$\mathcal{R'}_{2,n}^{(qq)}(\Theta)@f$, and, @f$\mathcal{R'}_{4,n}^{(qq)}(\Theta)@f$
 * are the spatial orientation functions in frame @f$\mathcal{F}@f$, and
 * @f$\mathbb{c}_k(i, j)@f$ are the composite spin transition functions for
 * @f$\left|i\right> \rightarrow \left|j\right>@f$ transition.
 *
 * @param Lambda_0 A pointer to an array of length 1, where the frequency
 *      component from @f${\Lambda'}_{0,0}^{(qq)}@f$ is stored.
 * @param Lambda_2 A pointer to a complex array of length 5, where the frequency
 *      components from @f$\Lambda_{2,n}^{(qq)}@f$ are stored ordered as
 *      @f$\left[{\Lambda'}_{2,n}^{(qq)}\right]_{n=-2}^2@f$.
 * @param Lambda_4 A pointer to a complex array of length 9, where the frequency
 *      components from @f${\Lambda'}_{4,n}^{(qq)}@f$ are stored ordered as
 *      @f$\left[{\Lambda'}_{4,n}^{(qq)}\right]_{n=-4}^4@f$.
 * @param spin The spin quantum number, @f$I@f$.
 * @param Cq_in_Hz The quadrupole coupling constant, @f$C_q@f$, in Hz.
 * @param eta The quadrupole asymmetry parameter, @f$\eta_q \in [0, 1]@f$.
 * @param v0_in_Hz The Larmor frequency, @f$\nu_0@f$, in Hz.
 * @param Theta A pointer to an array of Euler angles, in radians, of length 3,
 *      ordered as @f$[\alpha, \beta, \gamma]@f$.
 * @param mf The spin quantum number of the final energy state.
 * @param mi The spin quantum number of the initial energy state.
 */
static inline 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) {
  // Composite spin transition functions
  double *cl_value = malloc_double(3);
  STF_cL(cl_value, mf, mi, spin);

  // Spatial orientation function
  sSOT_2nd_order_electric_quadrupole_tensor_components(
      Lambda_0, Lambda_2, Lambda_4, spin, v0_in_Hz, Cq_in_Hz, eta, Theta);

  // frequency component function from the zeroth-rank irreducible tensor.
  *Lambda_0 *= *cl_value++;

  // frequency component function from the second-rank irreducible tensor.
  cblas_dscal(10, *cl_value++, (double *)Lambda_2, 1);

  // frequency component function from the fourth-rank irreducible tensor.
  cblas_dscal(18, *cl_value, (double *)Lambda_4, 1);
}

// =====================================================================================
//      First-order J-coupling frequency tensor components (weakly coupling limit)
// =====================================================================================

/**
 * The frequency tensor (FT) components from the first-order perturbation expansion of
 * the J-coupling Hamiltonian (weak coupling limit), in a given frame,
 * @f$\mathcal{F}@f$, described by the Euler angles @f$\Theta = [\alpha, \beta,
 * \gamma]@f$ are
 * @f[
 *    {\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}),
 * @f]
 * where @f$\mathcal{R'}_{0,0}^{(J)}(\Theta)@f$ and
 * @f$\mathcal{R'}_{2,n}^{(J)}(\Theta)@f$ are the spatial orientation functions in frame
 * @f$\mathcal{F}@f$, and @f$\mathbb{d}_{IS}(m_{i_I}, m_{i_S}, m_{f_I}, m_{f_S})@f$ is
 * the spin transition function for @f$\left|m_{i_I}, m_{i_S}\right> \rightarrow
 * \left|m_{f_I}, m_{f_S}\right>@f$ transition.
 *
 * @param Lambda_0 A pointer to an array of length 1, where the frequency
 *      component from @f${\Lambda'}_{0,0}^{(J)}@f$ is stored.
 * @param Lambda_2 A pointer to a complex array of length 5, where the frequency
 *      components from @f${\Lambda'}_{2,n}^{(J)}@f$ is stored ordered as
 *      @f$\left[{\Lambda'}_{2,n}^{(J)}\right]_{n=-2}^2@f$.
 * @param J_iso_in_Hz The isotropic J-coupling, @f$J_\text{iso}@f$, in Hz.
 * @param J_aniso_in_Hz The J-coupling anisotropy, @f$\zeta_J@f$, in Hz and
 *      defined using Haeberlen convention.
 * @param J_eta The J-coupling anisotropy asymmetry parameter, @f$\eta_J \in
 *      [-1,1]@f$, defined using Haeberlen convention.
 * @param Theta A pointer to an array of Euler angles of length 3 ordered as
 *      @f$[\alpha, \beta, \gamma]@f$.
 * @param mIf The spin quantum number of the final energy state of site @f$I@f$.
 * @param mIi The spin quantum number of the initial energy state of site
 *      @f$I@f$.
 * @param mSf The spin quantum number of the final energy state of site @f$S@f$.
 * @param mSi The spin quantum number of the initial energy state of site
 *      @f$S@f$.
 */
static inline 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) {
  // Spin transition function
  double transition_fn = STF_dIS(mIf, mIi, mSf, mSi);

  // Return if the transition is zero
  if (transition_fn == 0.0) {
    // zero the R0 and R2 components before populating with shielding components
    *Lambda_0 = 0.0;
    vm_double_zeros(10, (double *)Lambda_2);
    return;
  }

  // Spatial orientation function
  sSOT_1st_order_weakly_coupled_J_tensor_components(Lambda_0, Lambda_2, J_iso_in_Hz,
                                                    J_aniso_in_Hz, J_eta, Theta);

  // frequency component function from the zeroth-rank irreducible tensor.
  *Lambda_0 *= transition_fn;

  // frequency component function from the second-rank irreducible tensor.
  cblas_dscal(10, transition_fn, (double *)Lambda_2, 1);
}

// =====================================================================================
//           First-order Weakly coupled Magnetic Dipole frequency in the PAS.
// =====================================================================================

/**
 * The frequency tensor (FT) components from the first-order perturbation expansion of
 * the direct dipolar coupling Hamiltonian (weak coupling limit), in a given frame,
 * @f$\mathcal{F}@f$, described by the Euler angles @f$\Theta = [\alpha, \beta,
 * \gamma]@f$ are
 * @f[
 *    {\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}),
 * @f]
 * where @f$\mathcal{R'}_{2,n}^{(d)}(\Theta)@f$ are the spatial orientation functions in
 * frame @f$\mathcal{F}@f$, and @f$\mathbb{d}_{IS}(m_{i_I}, m_{i_S}, m_{f_I},
 * m_{f_S})@f$ is the spin transition function for @f$\left|m_{i_I}, m_{i_S}\right>
 * \rightarrow \left|m_{f_I}, m_{f_S}\right>@f$ transition.
 *
 * @param Lambda_2 A pointer to a complex array of length 5, where the frequency
 *      components from @f${\Lambda'}_{2,n}^{(d)}@f$ is stored ordered as
 *      @f$\left[{\Lambda'}_{2,n}^{(d)}\right]_{n=-2}^2@f$.
 * @param D_in_Hz The dipolar coupling, @f$D@f$, in Hz.
 * @param Theta A pointer to an array of Euler angles of length 3 ordered as
 *      @f$[\alpha, \beta, \gamma]@f$.
 * @param mIf The spin quantum number of the final energy state of site @f$I@f$.
 * @param mIi The spin quantum number of the initial energy state of site @f$I@f$.
 * @param mSf The spin quantum number of the final energy state of site @f$S@f$.
 * @param mSi The spin quantum number of the initial energy state of site @f$S@f$.
 */
static inline 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) {
  // Spin transition function
  double transition_fn = STF_dIS(mIf, mIi, mSf, mSi);

  // Return if the transition is zero
  if (transition_fn == 0.0) {
    // zero the R0 and R2 components before populating with shielding components
    vm_double_zeros(10, (double *)Lambda_2);
    return;
  }

  // Spatial orientation function
  sSOT_1st_order_weakly_coupled_dipolar_tensor_components(Lambda_2, D_in_Hz, Theta);
  // frequency component function from the second-rank irreducible tensor.
  cblas_dscal(10, transition_fn, (double *)Lambda_2, 1);
}