// -*- coding: utf-8 -*-
//
// frequency_component_function.h
//
// @copyright Deepansh J. Srivastava, 2019-2020.
// Created by Deepansh J. Srivastava, Apr 11, 2019.
// Contact email = srivastava.89@osu.edu
//
#include "frequency_component/spatial_orientation_tensor.h"
#include "frequency_component/spin_transition_function.h"
/**
* The frequency tensors (FT) 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)}(\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),
* @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
* components from @f${\Lambda'}_{0,0}^{(\sigma)}(\Theta, i,j)@f$ will be
* stored.
* @param Lambda_2 A pointer to a complex array of length 5 where the frequency
* components from @f${\Lambda'}_{2,n}^{(\sigma)}(\Theta, i,j)@f$ will be
* stored ordered according to
* @f$\left[{\Lambda'}_{2,n}^{(\sigma)}(\Theta, i,j)\right]_{n=-2}^2@f$.
* @param omega_0_delta_iso_in_Hz The quantity,
* @f$2\pi\omega_0\delta_\text{iso}@f$, given in Hz.
* @param omega_0_zeta_sigma_in_Hz The quantity, @f$2\pi\omega_0\zeta_sigma@f$,
* representing the strength of the nuclear shielding anisotropy, given in
* Hz, defined using Haeberlen convention.
* @param eta The nuclear shielding asymmetry parameter,
* @f$\eta_\sigma \in [-1,1]@f$, defined using Haeberlen convention.
* @param Theta A pointer to an array of length 3 where Euler angles,
* ordered as @f$[\alpha, \beta, \gamma]@f$, are stored.
* @param mf A float containing the spin quantum number of the final energy
* state.
* @param mi A float containing 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 zeroth-rank irreducible tensor
*Lambda_0 *= transition_fn;
// frequency component function from second-rank irreducible tensor
cblas_dscal(10, transition_fn, (double *)Lambda_2, 1);
}
/**
* The frequency component function (FCF) from the first-order electric
* quadrupole Hamiltonian, in a given frame, @f$\mathcal{F}@f$, described
* by the Euler angles @f$\Theta = [\alpha, \beta, \gamma]@f$, is
* @f[
* {\Lambda'}_{2,n}^{(q)}(\Theta,i,j) =
* \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${\Lambda'}_{2,n}^{(q)}(\Theta,i,j)@f$ will be stored
* ordered according to
* @f$\left[{\Lambda'}_{2,n}^{(q)}(\Theta,i,j)\right]_{n=-2}^2@f$.
* @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 length 3 where Euler angles,
* ordered as @f$[\alpha, \beta, \gamma]@f$, are stored.
* @param mf A float containing the spin quantum number of the final energy
* state.
* @param mi A float containing 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 second-rank irreducible tensor
cblas_dscal(10, transition_fn, (double *)Lambda_2, 1);
}
/**
* The frequency component functions (FCF) from the second-order 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)}(\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),
* @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$\mathbb{c}_i(i, j)@f$ 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${\Lambda'}_{0,0}^{(qq)}(\Theta, i,j)@f$ will be
* stored.
* @param Lambda_2 A pointer to a complex array of length 5 where the frequency
* components from @f$\Lambda_{2,n}^{(qq)}(\Theta, i,j)@f$ will be stored
* ordered according to
* @f$\left[{\Lambda'}_{2,n}^{(qq)}(\Theta, i,j)\right]_{n=-2}^2@f$.
* @param Lambda_4 A pointer to a complex array of length 5 where the frequency
* components from @f${\Lambda'}_{4,n}^{(qq)}(\Theta, i,j)@f$ will be
* stored ordered according to
* @f$\left[{\Lambda'}_{4,n}^{(qq)}(\Theta, i,j)\right]_{n=-4}^4@f$.
* @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 length 3 where Euler angles,
* ordered as @f$[\alpha, \beta, \gamma]@f$, are stored.
* @param mf A float containing the spin quantum number of the final energy
* state.
* @param mi A float containing 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 zeroth-rank irreducible tensor
*Lambda_0 *= *cl_value++;
// frequency component function from second-rank irreducible tensor
cblas_dscal(10, *cl_value++, (double *)Lambda_2, 1);
// frequency component function from fourth-rank irreducible tensor
cblas_dscal(18, *cl_value, (double *)Lambda_4, 1);
}
/*
===============================================================================
First order Weakly coupled Magnetic Dipole frequency in the PAS.
-------------------------------------------------------------------------------
The frequency includes the product of second rank tensor and the
spin transition functions in the weak coupling limit.
*/
static inline void weakly_coupled_direct_dipole_frequencies_to_first_order(
double *restrict Lambda_0, void *restrict Lambda_2, const double D,
const float mIf, const float mIi, const float mSf, const float mSi) {
// Spin transition contribution
double transition_fn = STF_dIS(mIf, mIi, mSf, mSi);
// Scaled R00
*Lambda_0 += 0.0;
/* Scaled R2m containing the components of the magnetic dipole second rank
tensor in its principal axis frame. */
vm_double_zeros(10, (double *)Lambda_2);
double *Lambda_2_ = (double *)Lambda_2;
Lambda_2_[4] = 2.0 * D * transition_fn; // Lambda_2 0 real
}
