Struct qsym2::symmetry::symmetry_element::symmetry_operation::SymmetryOperation

pub struct SymmetryOperation {
    pub generating_element: SymmetryElement,
    pub power: i32,
    pub positive_hemisphere: Option<PositiveHemisphere>,
    /* private fields */
}

Structure for managing symmetry operations generated from symmetry elements.

A symmetry element serves as a generator for symmetry operations. Thus, a symmetry element together with a signed integer indicating the number of times the symmetry element is applied (positively or negatively) specifies a symmetry operation.

Fields

generating_element: SymmetryElement

The generating symmetry element for this symmetry operation.

power: i32

The integral power indicating the number of times Self::generating_element is applied to form the symmetry operation.

positive_hemisphere: Option<PositiveHemisphere>

The positive hemisphere used for distinguishing positive and negative rotation poles. If None, the standard positive hemisphere as defined in S.L. Altmann, Rotations, Quaternions, and Double Groups (Dover Publications, Inc., New York, 2005) is used.

Implementations

impl SymmetryOperation

pub fn from_quaternion( qtn: (f64, Vector3<f64>), proper: bool, thresh: f64, max_trial_power: u32, tr: bool, su2: bool, poshem: Option<PositiveHemisphere>, ) -> Self

Constructs a finite-order-element-generated symmetry operation from a quaternion.

The rotation angle encoded in the quaternion is taken to be non-negative and assigned as the proper rotation angle associated with the element generating the operation.

If an improper operation is required, its generator will be constructed in the inversion-centre convention.

Arguments

• qtn - A quaternion encoding the proper rotation associated with the generating element of the operation to be constructed.
• proper - A boolean indicating if the operation is proper or improper.
• thresh - Threshold for comparisons.
• tr - A boolean indicating if the resulting symmetry operation should be accompanied by a time-reversal operation.
• su2 - A boolean indicating if the resulting symmetry operation is to contain a proper rotation in $\mathsf{SU}(2)$. The homotopy class of the operation will be deduced from the specified quaternion.
• poshem - An option containing any custom positive hemisphere used to distinguish positive and negative rotation poles.

Returns

The constructed symmetry operation.

Panics

Panics when the scalar part of the provided quaternion lies outside $[0, 1]$ by more than the specified threshold thresh, or when the rotation angle associated with the quaternion cannot be gracefully converted into an integer tuple of order and power.

pub fn calc_quaternion(&self) -> (f64, Vector3<f64>)

Finds the quaternion associated with this operation.

The rotation angle encoded in the quaternion is taken to be non-negative and assigned as the proper rotation angle associated with the element generating the operation.

If this is an operation generated from an improper element, the inversion-centre convention will be used to determine the angle of proper rotation.

Both $\mathsf{SO}(3)$ and $\mathsf{SU}(2)$ proper rotations are supported. For $\mathsf{SO}(3)$ proper rotations, only quaternions in the standardised form are returned.

See S.L. Altmann, Rotations, Quaternions, and Double Groups (Dover Publications, Inc., New York, 2005) (Chapter 9) for further information.

Returns

The quaternion associated with this operation.

Panics

Panics if the calculated scalar part of the quaternion lies outside the closed interval $[0, 1]$ by more than the threshold value stored in the generating element in self.

pub fn calc_pole(&self) -> Point3<f64>

Finds the pole associated with this operation with respect to the positive hemisphere defined in Self::positive_hemisphere.

This is the point on the unit sphere that is left invariant by the operation.

For improper operations, the inversion-centre convention is used to define the pole. This allows a proper rotation and its improper partner to have the same pole, thus facilitating the consistent specification of poles for the identity / inversion and binary rotations / reflections.

Note that binary rotations / reflections have unique poles on the positive hemisphere (i.e., $C_2(\hat{\mathbf{n}}) = C_2^{-1}(\hat{\mathbf{n}})$ and $\sigma(\hat{\mathbf{n}}) = \sigma^{-1}(\hat{\mathbf{n}})$).

See S.L. Altmann, Rotations, Quaternions, and Double Groups (Dover Publications, Inc., New York, 2005) (Chapter 9) for further information.

Returns

The pole associated with this operation.

Panics

Panics when no total proper fractions could be found for this operation.

pub fn calc_proper_rotation_pole(&self) -> Point3<f64>

Finds the pole associated with the proper rotation of this operation.

This is the point on the unit sphere that is left invariant by the proper rotation part of the operation.

For improper operations, no conversions will be performed, unlike in Self::calc_pole.

Note that binary rotations have unique poles on the positive hemisphere (i.e., $C_2(\hat{\mathbf{n}}) = C_2^{-1}(\hat{\mathbf{n}})$ and $\sigma(\hat{\mathbf{n}}) = \sigma^{-1}(\hat{\mathbf{n}})$).

See S.L. Altmann, Rotations, Quaternions, and Double Groups (Dover Publications, Inc., New York, 2005) (Chapter 9) for further information.

Returns

The pole associated with the proper rotation of this operation.

Panics

Panics when no total proper fractions could be found for this operation.

pub fn calc_pole_angle(&self) -> f64

Finds the pole angle associated with this operation.

This is the non-negative angle that, together with the pole, uniquely determines the proper part of this operation. This angle lies in the interval $[0, \pi]$.

For improper operations, the inversion-centre convention is used to define the pole angle. This allows a proper rotation and its improper partner to have the same pole angle, thus facilitating the consistent specification of poles for the identity / inversion and binary rotations / reflections.

Returns

The pole angle associated with this operation.

Panics

Panics when no total proper fractions could be found for this operation.

pub fn convert_to_improper_kind( &self, improper_kind: &SymmetryElementKind, ) -> Self

Returns a copy of the current symmetry operation with the generating element converted to the requested improper kind (power-preserving), provided that it is an improper element.

Arguments

• improper_kind - The improper kind to which self is to be converted. There is no need to make sure the time reversal specification in improper_kind matches that of the generating element of self as the conversion will take care of this.

Panics

Panics if the converted symmetry operation cannot be constructed.

pub fn to_symmetry_element(&self) -> SymmetryElement

Converts the current symmetry operation $O$ to an equivalent symmetry element $E$ such that $O = E^1$.

The proper rotation axis of $E$ is the proper rotation pole (not the overall pole) of $O$, and the proper rotation angle of $E$ is the total proper rotation angle of $O$, either as an (order, power) integer tuple or an angle floating-point number.

If $O$ is improper, then the improper generating element for $E$ is the same as that in the generating element of $O$.

Returns

The equivalent symmetry element $E$.

pub fn get_abbreviated_symbol(&self) -> String

Generates the abbreviated symbol for this symmetry operation.

pub fn get_3d_spatial_matrix(&self) -> Array2<f64>

Returns the representation matrix for the spatial part of this symmetry operation.

This representation matrix is in the basis of coordinate functions $(y, z, x)$.

pub fn to_su2_class_0(&self) -> Self

Convert the proper rotation of the current operation to one in hopotopy class 0 of $\mathsf{SU}(2)$.

Returns

A symmetry element in $\mathsf{SU}(2)$.

pub fn set_positive_hemisphere(&mut self, poshem: Option<&PositiveHemisphere>)

Sets the positive hemisphere governing this symmetry operation.

Arguments

• poshem - An Option containing a custom positive hemisphere, if any.

Trait Implementations

impl Clone for SymmetryOperation

fn clone(&self) -> SymmetryOperation

Returns a copy of the value.

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source.

impl Debug for SymmetryOperation

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

impl<'de> Deserialize<'de> for SymmetryOperation

fn deserialize<__D>(__deserializer: __D) -> Result<Self, __D::Error>

where __D: Deserializer<'de>,

Deserialize this value from the given Serde deserializer.

impl Display for SymmetryOperation

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

impl FiniteOrder for SymmetryOperation

fn order(&self) -> Self::Int

Calculates the order of this symmetry operation.

type Int = u32

The integer type for the order of the element.

impl Hash for SymmetryOperation

fn hash<H: Hasher>(&self, state: &mut H)

Feeds this value into the given Hasher.

fn hash_slice<H>(data: &[Self], state: &mut H)

where H: Hasher, Self: Sized,

Feeds a slice of this type into the given Hasher.

impl<M> IntoPermutation<M> for SymmetryOperation

where M: Transform + PermutableCollection<Rank = usize>,

fn act_permute(&self, rhs: &M) -> Option<Permutation<usize>>

Determines the permutation of rhs considered as a collection induced by the action of self on rhs considered as an element in its domain. If no such permutation could be found, None is returned.

impl Inv for &SymmetryOperation

type Output = SymmetryOperation

The result after applying the operator.

fn inv(self) -> Self::Output

Returns the multiplicative inverse of self.

impl Inv for SymmetryOperation

type Output = SymmetryOperation

The result after applying the operator.

fn inv(self) -> Self::Output

Returns the multiplicative inverse of self.

impl Mul<&SymmetryOperation> for &SymmetryOperation

type Output = SymmetryOperation

The resulting type after applying the * operator.

fn mul(self, rhs: &SymmetryOperation) -> Self::Output

Performs the * operation.

impl Mul<&SymmetryOperation> for SymmetryOperation

type Output = SymmetryOperation

The resulting type after applying the * operator.

fn mul(self, rhs: &SymmetryOperation) -> Self::Output

Performs the * operation.

impl Mul<SymmetryOperation> for &SymmetryOperation

type Output = SymmetryOperation

The resulting type after applying the * operator.

fn mul(self, rhs: SymmetryOperation) -> Self::Output

Performs the * operation.

impl Mul for SymmetryOperation

type Output = SymmetryOperation

The resulting type after applying the * operator.

fn mul(self, rhs: SymmetryOperation) -> Self::Output

Performs the * operation.

impl PartialEq for SymmetryOperation

fn eq(&self, other: &Self) -> bool

This method tests for self and other values to be equal, and is used by ==.

fn ne(&self, other: &Rhs) -> bool

This method tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl Pow<i32> for &SymmetryOperation

type Output = SymmetryOperation

The result after applying the operator.

fn pow(self, rhs: i32) -> Self::Output

Returns self to the power rhs.

impl Pow<i32> for SymmetryOperation

type Output = SymmetryOperation

The result after applying the operator.

fn pow(self, rhs: i32) -> Self::Output

Returns self to the power rhs.

impl Serialize for SymmetryOperation

fn serialize<__S>(&self, __serializer: __S) -> Result<__S::Ok, __S::Error>

where __S: Serializer,

Serialize this value into the given Serde serializer.

impl SpecialSymmetryTransformation for SymmetryOperation

fn is_proper(&self) -> bool

Checks if the spatial part of the symmetry operation is proper.

Returns

A boolean indicating if the spatial part of the symmetry operation is proper.

fn is_spatial_identity(&self) -> bool

Checks if the spatial part of the symmetry operation is the spatial identity.

Returns

A boolean indicating if the spatial part of the symmetry operation is the spatial identity.

fn is_spatial_binary_rotation(&self) -> bool

Checks if the spatial part of the symmetry operation is a spatial binary rotation.

Returns

A boolean indicating if the spatial part of the symmetry operation is a spatial binary rotation.

fn is_spatial_inversion(&self) -> bool

Checks if the spatial part of the symmetry operation is the spatial inversion.

Returns

A boolean indicating if the spatial part of the symmetry operation is the spatial inversion.

fn is_spatial_reflection(&self) -> bool

Checks if the spatial part of the symmetry operation is a spatial reflection.

Returns

A boolean indicating if the spatial part of the symmetry operation is a spatial reflection.

fn contains_time_reversal(&self) -> bool

Checks if the symmetry operation is antiunitary or not.

Returns

A boolean indicating if the symmetry oppperation is antiunitary.

fn is_su2(&self) -> bool

Checks if the proper rotation part of the symmetry operation is in $\mathsf{SU}(2)$.

Returns

A boolean indicating if this symmetry operation contains an $\mathsf{SU}(2)$ proper rotation.

fn is_su2_class_1(&self) -> bool

Checks if the proper rotation part of the symmetry operation is in $\mathsf{SU}(2)$ and connected to the identity via a homotopy path of class 1.

Returns

A boolean indicating if this symmetry operation contains an $\mathsf{SU}(2)$ proper rotation connected to the identity via a homotopy path of class 1.

fn is_identity(&self) -> bool

Checks if the symmetry operation is the identity in $\mathsf{O}(3)$, E, or in $\mathsf{SU}(2)$, E(Σ).

fn is_time_reversal(&self) -> bool

Checks if the symmetry operation is a pure time-reversal in $\mathsf{O}(3)$, θ, or in $\mathsf{SU}(2)$, θ(Σ).

fn is_inversion(&self) -> bool

Checks if the symmetry operation is an inversion in $\mathsf{O}(3)$, i, but not in $\mathsf{SU}(2)$, i(Σ).

fn is_reflection(&self) -> bool

Checks if the symmetry operation is a reflection in $\mathsf{O}(3)$, σ, but not in $\mathsf{SU}(2)$, σ(Σ).

impl Eq for SymmetryOperation