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//! Implementation of symmetry transformations for orbitals.
use std::ops::Mul;
use approx;
use ndarray::{array, concatenate, s, Array2, Axis, LinalgScalar, ScalarOperand};
use ndarray_linalg::types::Lapack;
use num_complex::{Complex, ComplexFloat};
use crate::angmom::spinor_rotation_3d::SpinConstraint;
use crate::permutation::{IntoPermutation, PermutableCollection, Permutation};
use crate::symmetry::symmetry_element::SymmetryOperation;
use crate::symmetry::symmetry_transformation::{
assemble_sh_rotation_3d_matrices, permute_array_by_atoms, ComplexConjugationTransformable,
DefaultTimeReversalTransformable, SpatialUnitaryTransformable, SpinUnitaryTransformable,
SymmetryTransformable, TimeReversalTransformable, TransformationError,
};
use crate::target::orbital::MolecularOrbital;
// ---------------------------
// SpatialUnitaryTransformable
// ---------------------------
impl<'a, T> SpatialUnitaryTransformable for MolecularOrbital<'a, T>
where
T: ComplexFloat + LinalgScalar + ScalarOperand + Copy + Lapack,
f64: Into<T>,
{
fn transform_spatial_mut(
&mut self,
rmat: &Array2<f64>,
perm: Option<&Permutation<usize>>,
) -> Result<&mut Self, TransformationError> {
let tmats: Vec<Array2<T>> = assemble_sh_rotation_3d_matrices(self.bao, rmat, perm)
.map_err(|err| TransformationError(err.to_string()))?
.iter()
.map(|tmat| tmat.map(|&x| x.into()))
.collect();
let pbao = if let Some(p) = perm {
self.bao
.permute(p)
.map_err(|err| TransformationError(err.to_string()))?
} else {
self.bao.clone()
};
let old_coeff = &self.coefficients;
let new_coefficients = match self.spin_constraint {
SpinConstraint::Restricted(_) | SpinConstraint::Unrestricted(_, _) => {
let p_coeff = if let Some(p) = perm {
permute_array_by_atoms(old_coeff, p, &[Axis(0)], self.bao)
} else {
old_coeff.clone()
};
let t_p_blocks = pbao
.shell_boundary_indices()
.into_iter()
.zip(tmats.iter())
.map(|((shl_start, shl_end), tmat)| {
tmat.dot(&p_coeff.slice(s![shl_start..shl_end]))
})
.collect::<Vec<_>>();
concatenate(
Axis(0),
&t_p_blocks
.iter()
.map(|t_p_block| t_p_block.view())
.collect::<Vec<_>>(),
)
.expect("Unable to concatenate the transformed rows for the various shells.")
}
SpinConstraint::Generalised(nspins, _) => {
let nspatial = self.bao.n_funcs();
let t_p_spin_blocks =
(0..nspins)
.map(|ispin| {
// Extract spin block ispin.
let spin_start = usize::from(ispin) * nspatial;
let spin_end = (usize::from(ispin) + 1) * nspatial;
let spin_block = old_coeff.slice(s![spin_start..spin_end]).to_owned();
// Permute within spin block ispin.
let p_spin_block = if let Some(p) = perm {
permute_array_by_atoms(&spin_block, p, &[Axis(0)], self.bao)
} else {
spin_block
};
// Transform within spin block ispin.
let t_p_blocks = pbao
.shell_boundary_indices()
.into_iter()
.zip(tmats.iter())
.map(|((shl_start, shl_end), tmat)| {
tmat.dot(&p_spin_block.slice(s![shl_start..shl_end]))
})
.collect::<Vec<_>>();
// Concatenate blocks for various shells within spin block ispin.
concatenate(
Axis(0),
&t_p_blocks.iter().map(|t_p_block| t_p_block.view()).collect::<Vec<_>>(),
)
.expect("Unable to concatenate the transformed rows for the various shells.")
})
.collect::<Vec<_>>();
// Concatenate spin blocks.
concatenate(
Axis(0),
&t_p_spin_blocks
.iter()
.map(|t_p_spin_block| t_p_spin_block.view())
.collect::<Vec<_>>(),
)
.expect("Unable to concatenate the transformed spin blocks.")
}
};
self.coefficients = new_coefficients;
Ok(self)
}
}
// ------------------------
// SpinUnitaryTransformable
// ------------------------
// ~~~~~~~~~~~~~~~~~
// For real orbitals
// ~~~~~~~~~~~~~~~~~
impl<'a> SpinUnitaryTransformable for MolecularOrbital<'a, f64> {
fn transform_spin_mut(
&mut self,
dmat: &Array2<Complex<f64>>,
) -> Result<&mut Self, TransformationError> {
let cdmat = dmat.view().split_complex();
if approx::relative_ne!(
cdmat.im.map(|x| x.powi(2)).sum().sqrt(),
0.0,
epsilon = 1e-14,
max_relative = 1e-14,
) {
log::error!("Spin transformation matrix is complex-valued:\n{dmat}");
Err(TransformationError(
"Complex spin transformations cannot be performed with real coefficients."
.to_string(),
))
} else {
let rdmat = cdmat.re.to_owned();
match self.spin_constraint {
SpinConstraint::Restricted(_) => {
if approx::relative_eq!(
(&rdmat - Array2::<f64>::eye(2))
.map(|x| x.abs().powi(2))
.sum()
.sqrt(),
0.0,
epsilon = 1e-14,
max_relative = 1e-14,
) {
// Identity spin rotation
Ok(self)
} else if approx::relative_eq!(
(&rdmat + Array2::<f64>::eye(2))
.map(|x| x.abs().powi(2))
.sum()
.sqrt(),
0.0,
epsilon = 1e-14,
max_relative = 1e-14,
) {
// Negative identity spin rotation
self.coefficients *= -1.0;
Ok(self)
} else {
log::error!("Unsupported spin transformation matrix:\n{}", &rdmat);
Err(TransformationError(
"Only the identity or negative identity spin transformations are possible with restricted spin constraint."
.to_string(),
))
}
}
SpinConstraint::Unrestricted(nspins, increasingm) => {
// Only spin flip possible, so the order of the basis in which `dmat` is
// expressed and the order of the spin blocks do not need to match.
if nspins != 2 {
return Err(TransformationError(
"Only two-component spinor transformations are supported for now."
.to_string(),
));
}
let dmat_y = array![[0.0, -1.0], [1.0, 0.0]];
if approx::relative_eq!(
(&rdmat - Array2::<f64>::eye(2))
.map(|x| x.abs().powi(2))
.sum()
.sqrt(),
0.0,
epsilon = 1e-14,
max_relative = 1e-14,
) {
// Identity spin rotation
Ok(self)
} else if approx::relative_eq!(
(&rdmat + Array2::<f64>::eye(2))
.map(|x| x.abs().powi(2))
.sum()
.sqrt(),
0.0,
epsilon = 1e-14,
max_relative = 1e-14,
) {
// Negative identity spin rotation
self.coefficients *= -1.0;
Ok(self)
} else if approx::relative_eq!(
(&rdmat - &dmat_y).map(|x| x.abs().powi(2)).sum().sqrt(),
0.0,
epsilon = 1e-14,
max_relative = 1e-14,
) {
// π-rotation about y-axis, effectively spin-flip
if increasingm {
if self.spin_index == 0 {
self.spin_index = 1;
self.coefficients *= -1.0;
} else {
assert_eq!(self.spin_index, 1);
self.spin_index = 0;
}
} else if self.spin_index == 0 {
self.spin_index = 1;
} else {
assert_eq!(self.spin_index, 1);
self.spin_index = 0;
self.coefficients *= -1.0;
}
Ok(self)
} else if approx::relative_eq!(
(&rdmat + &dmat_y).map(|x| x.abs().powi(2)).sum().sqrt(),
0.0,
epsilon = 1e-14,
max_relative = 1e-14,
) {
// 3π-rotation about y-axis, effectively negative spin-flip
if increasingm {
if self.spin_index == 0 {
self.spin_index = 1;
} else {
assert_eq!(self.spin_index, 1);
self.spin_index = 0;
self.coefficients *= -1.0;
}
} else if self.spin_index == 0 {
self.spin_index = 1;
self.coefficients *= -1.0;
} else {
assert_eq!(self.spin_index, 1);
self.spin_index = 0;
}
Ok(self)
} else {
log::error!("Unsupported spin transformation matrix:\n{rdmat}");
Err(TransformationError(
"Only the identity or πy spin transformations are possible with unrestricted spin constraint."
.to_string(),
))
}
}
SpinConstraint::Generalised(nspins, increasingm) => {
if nspins != 2 {
return Err(TransformationError(
"Only two-component spinor transformations are supported for now."
.to_string(),
));
}
let nspatial = self.bao.n_funcs();
let old_coeff = &self.coefficients;
let new_coefficients = if increasingm {
let b_coeff = old_coeff.slice(s![0..nspatial]).to_owned();
let a_coeff = old_coeff.slice(s![nspatial..2 * nspatial]).to_owned();
let t_a_coeff = &a_coeff * rdmat[[0, 0]] + &b_coeff * rdmat[[0, 1]];
let t_b_coeff = &a_coeff * rdmat[[1, 0]] + &b_coeff * rdmat[[1, 1]];
concatenate(Axis(0), &[t_b_coeff.view(), t_a_coeff.view()]).expect(
"Unable to concatenate the transformed rows for the various shells.",
)
} else {
let a_coeff = old_coeff.slice(s![0..nspatial]).to_owned();
let b_coeff = old_coeff.slice(s![nspatial..2 * nspatial]).to_owned();
let t_a_coeff = &a_coeff * rdmat[[0, 0]] + &b_coeff * rdmat[[0, 1]];
let t_b_coeff = &a_coeff * rdmat[[1, 0]] + &b_coeff * rdmat[[1, 1]];
concatenate(Axis(0), &[t_a_coeff.view(), t_b_coeff.view()]).expect(
"Unable to concatenate the transformed rows for the various shells.",
)
};
self.coefficients = new_coefficients;
Ok(self)
}
}
}
}
}
// ~~~~~~~~~~~~~~~~~~~~
// For complex orbitals
// ~~~~~~~~~~~~~~~~~~~~
impl<'a, T> SpinUnitaryTransformable for MolecularOrbital<'a, Complex<T>>
where
T: Clone,
Complex<T>: ComplexFloat<Real = T>
+ LinalgScalar
+ ScalarOperand
+ Lapack
+ Mul<Complex<T>, Output = Complex<T>>
+ Mul<Complex<f64>, Output = Complex<T>>,
{
fn transform_spin_mut(
&mut self,
dmat: &Array2<Complex<f64>>,
) -> Result<&mut Self, TransformationError> {
match self.spin_constraint {
SpinConstraint::Restricted(_) => {
if approx::relative_eq!(
(dmat - Array2::<Complex<f64>>::eye(2))
.map(|x| x.abs().powi(2))
.sum()
.sqrt(),
0.0,
epsilon = 1e-14,
max_relative = 1e-14,
) {
// Identity spin rotation
Ok(self)
} else if approx::relative_eq!(
(dmat + Array2::<Complex<f64>>::eye(2))
.map(|x| x.abs().powi(2))
.sum()
.sqrt(),
0.0,
epsilon = 1e-14,
max_relative = 1e-14,
) {
// Negative identity spin rotation
self.coefficients.map_inplace(|x| *x = -*x);
Ok(self)
} else {
log::error!("Unsupported spin transformation matrix:\n{}", dmat);
Err(TransformationError(
"Only the identity or negative identity spin transformations are possible with restricted spin constraint."
.to_string(),
))
}
}
SpinConstraint::Unrestricted(nspins, increasingm) => {
// Only spin flip possible, so the order of the basis in which `dmat` is
// expressed and the order of the spin blocks do not need to match.
if nspins != 2 {
return Err(TransformationError(
"Only two-component spinor transformations are supported for now."
.to_string(),
));
}
let dmat_y = array![
[Complex::from(0.0), Complex::from(-1.0)],
[Complex::from(1.0), Complex::from(0.0)],
];
if approx::relative_eq!(
(dmat - Array2::<Complex<f64>>::eye(2))
.map(|x| x.abs().powi(2))
.sum()
.sqrt(),
0.0,
epsilon = 1e-14,
max_relative = 1e-14,
) {
// Identity spin rotation
Ok(self)
} else if approx::relative_eq!(
(dmat + Array2::<Complex<f64>>::eye(2))
.map(|x| x.abs().powi(2))
.sum()
.sqrt(),
0.0,
epsilon = 1e-14,
max_relative = 1e-14,
) {
// Negative identity spin rotation
self.coefficients.map_inplace(|x| *x = -*x);
Ok(self)
} else if approx::relative_eq!(
(dmat - &dmat_y).map(|x| x.abs().powi(2)).sum().sqrt(),
0.0,
epsilon = 1e-14,
max_relative = 1e-14,
) {
// π-rotation about y-axis, effectively spin-flip
if increasingm {
if self.spin_index == 0 {
self.spin_index = 1;
self.coefficients.map_inplace(|x| *x = -*x);
} else {
assert_eq!(self.spin_index, 1);
self.spin_index = 0;
}
} else if self.spin_index == 0 {
self.spin_index = 1;
} else {
assert_eq!(self.spin_index, 1);
self.spin_index = 0;
self.coefficients.map_inplace(|x| *x = -*x);
}
Ok(self)
} else if approx::relative_eq!(
(dmat + &dmat_y).map(|x| x.abs().powi(2)).sum().sqrt(),
0.0,
epsilon = 1e-14,
max_relative = 1e-14,
) {
// 3π-rotation about y-axis, effectively negative spin-flip
if increasingm {
if self.spin_index == 0 {
self.spin_index = 1;
} else {
assert_eq!(self.spin_index, 1);
self.spin_index = 0;
self.coefficients.map_inplace(|x| *x = -*x);
}
} else if self.spin_index == 0 {
self.spin_index = 1;
self.coefficients.map_inplace(|x| *x = -*x);
} else {
assert_eq!(self.spin_index, 1);
self.spin_index = 0;
}
Ok(self)
} else {
log::error!("Unsupported spin transformation matrix:\n{dmat}");
Err(TransformationError(
"Only the identity or πy spin transformations are possible with unrestricted spin constraint."
.to_string(),
))
}
}
SpinConstraint::Generalised(nspins, increasingm) => {
if nspins != 2 {
panic!("Only two-component spinor transformations are supported for now.");
}
let nspatial = self.bao.n_funcs();
let old_coeff = &self.coefficients;
let new_coefficients = if increasingm {
let b_coeff = old_coeff.slice(s![0..nspatial]).to_owned();
let a_coeff = old_coeff.slice(s![nspatial..2 * nspatial]).to_owned();
let t_a_coeff = &a_coeff * dmat[[0, 0]] + &b_coeff * dmat[[0, 1]];
let t_b_coeff = &a_coeff * dmat[[1, 0]] + &b_coeff * dmat[[1, 1]];
concatenate(Axis(0), &[t_b_coeff.view(), t_a_coeff.view()]).expect(
"Unable to concatenate the transformed rows for the various shells.",
)
} else {
let a_coeff = old_coeff.slice(s![0..nspatial]).to_owned();
let b_coeff = old_coeff.slice(s![nspatial..2 * nspatial]).to_owned();
let t_a_coeff = &a_coeff * dmat[[0, 0]] + &b_coeff * dmat[[0, 1]];
let t_b_coeff = &a_coeff * dmat[[1, 0]] + &b_coeff * dmat[[1, 1]];
concatenate(Axis(0), &[t_a_coeff.view(), t_b_coeff.view()]).expect(
"Unable to concatenate the transformed rows for the various shells.",
)
};
self.coefficients = new_coefficients;
Ok(self)
}
}
}
}
// -------------------------------
// ComplexConjugationTransformable
// -------------------------------
impl<'a, T> ComplexConjugationTransformable for MolecularOrbital<'a, T>
where
T: ComplexFloat + Lapack,
{
fn transform_cc_mut(&mut self) -> Result<&mut Self, TransformationError> {
self.coefficients.mapv_inplace(|x| x.conj());
self.complex_conjugated = !self.complex_conjugated;
Ok(self)
}
}
// --------------------------------
// DefaultTimeReversalTransformable
// --------------------------------
impl<'a, T> DefaultTimeReversalTransformable for MolecularOrbital<'a, T> where
T: ComplexFloat + Lapack
{
}
// ---------------------
// SymmetryTransformable
// ---------------------
impl<'a, T> SymmetryTransformable for MolecularOrbital<'a, T>
where
T: ComplexFloat + Lapack,
MolecularOrbital<'a, T>:
SpatialUnitaryTransformable + SpinUnitaryTransformable + TimeReversalTransformable,
{
fn sym_permute_sites_spatial(
&self,
symop: &SymmetryOperation,
) -> Result<Permutation<usize>, TransformationError> {
symop
.act_permute(&self.mol.molecule_ordinary_atoms())
.ok_or(TransformationError(format!(
"Unable to determine the atom permutation corresponding to the operation `{symop}`."
)))
}
}