# Spin¶

Spin is a flag. As such it cannot be constructed; Spin() is an invalid command. Instead, Spin provides derived classes (flags) to represent spin components and projections:

 Spin.Up The ‘up’ component of a spinor (up-up component of a spin matrix). Spin.Down The ‘down’ component of a spinor (down-down component of a spin matrix). Spin.RealUpDown The real part of the ‘up-down’ component of a spinor (spin matrix.) Spin.ImagUpDown The imaginary part of the ‘up-down’ component of a spinor (spin matrix). Spin.All All spin components. Spin.Sum The sum ‘Spin.Up + Spin.Down’ Spin.X The spin projection along ‘x’ (Spin.X = 2*Spin.RealUpDown). Spin.Y The spin projection along ‘y’ (Spin.Y = -2*Spin.ImagUpDown). Spin.Z The spin projection along ‘z’ (Spin.Z = Spin.Up - Spin.Down). Spin.Unknown Unknown spin.

## Usage Example¶

Calculate the electron density for all spin and evaluate some components:

# Calculate the electron density for a given configuration.
ed_up = ElectronDensity(configuration, spin=Spin.All)

# Take some spin projections.
x = ed.spinProjection(spin=Spin.X)
y = ed.spinProjection(spin=Spin.Y)
z = ed.spinProjection(spin=Spin.Z)
s = ed.spinProjection(spin=Spin.Sum)
r = ed.spinProjection(spin=Spin.RealUpDown)
i = ed.spinProjection(spin=Spin.ImagUpDown)
u = ed.spinProjection(spin=Spin.Up)
d = ed.spinProjection(spin=Spin.Down)

# Evaluate for Spin.X at the origin.
data = ed.evaluate(0.0*Bohr, 0.0*Bohr, 0.0*Bohr, spin=Spin.X)


## Note about Spin.All¶

Precisely which spin components are returned when calling an objects query method with spin = Spin.All depends on the queried object. E.g. ElectronDensity.evaluate(x, y, z, spin=Spin.All) returns a list of four electron density values at the grid point (x, y, z) corresponding to Spin.Sum, Spin.X, Spin.Y, and Spin.Z. In other cases (e.g. ExchangeCorrelationPotential), the returned array contains the values corresponding to the spinor components Spin.Up, Spin.Down, Spin.RealUpDown, and Spin.ImagUpDown. Refer to the object’s documentation for details.

## Note on Spin in low level interface functions¶

In all low level interface functions such as calculateHamiltonianAndOverlap, calculateDensityMatrix, calculateSelfEnergy etc., the following rules for the spin parameter apply:

• UNPOLARIZED: Valid spin parameters are Spin.Up and Spin.All, which both yield the same result in this case, as there is no designated spin direction in UNPOLARIZED calculations.
• POLARIZED: Valid spin parameters are Spin.All, Spin.Up, and Spin.Down. The default is Spin.All, in which case the function returns a pair of matrices, one for the Spin.Up and one for the Spin.Down component. For Spin.Up or Spin.Down, only the respective spin component is returned.
• NONCOLLINEAR / SPINORBIT: In noncollinear calculations, only Spin.All is an accepted parameter. The returned matrix contains the spin components Spin.Up, Spin.Down, Spin.UpDown, and Spin.DownUp in an interleaved fashion, see below for an example.

Examples

# Calculate the density matrix for a polarized system.
D = calculateDensityMatrix(polarized_configuration, spin=Spin.All)
# Extract the Spin.Up component.
D_uu = D
# Extract the Spin.Down component.
D_dd = D

# Calculate the density matrix for a noncollinear system.
D = calculateDensityMatrix(noncollinear_configuration, spin=Spin.All)
# Get all up-up entries:
D_ud = D[::2,::2]
# Get all down-down entries:
D_du = D[1::2,1::2]
# Get all up-down entries:
D_ud = D[1::2,::2]
# Get all down-up entries:
D_du = D[::2,1::2]