Visualization of spinors

Visualization of spinors is more difficult than visualization of orbitals. A spinor Ψ is a two-component complex wave function, which can be described with four real functions φ: real part α φ_α^R, imaginary part α φ_α^I, real part β φ_β^R, imaginary part β φ_β^I:

The density ρ is:

ρ = Ψ^† Ψ

The spin magnetization density m is:

m = Ψ^† σ Ψ

where σ is the vector of the Pauli spin matrices. A spinor is fully determined by the spin magnetization density and a phase factor e^iθ, which both are functions of spatial coordinates. In ADF one can visualize the (square root of the) density and spin magnetization density, however, the phase factor e^iθ is summarized only with a plus or minus sign. For this small molecule a fine grid is chosen for better visualization.

Select the ADFinput window with the name 'TlH_SO.adf.
Select 'View' from the SCM Menu.
Rotate the molecule, such that one can see both atoms.
Select 'Grid' -> 'Fine' from the 'Fields' Menu.
Select 'Spinor: spin magnetization density' from the 'Add' Menu.
In the new control line, press on the pull-down menu and
 Select 'Orbitals (occupied)' -> 'SCF_J1/2:1  22:  1.0  ..'.

The arrows in this picture are in the direction of the spin magnetization density m. All arrows are approximately in the same direction, which means that this spinor is an eigenfunction of spin in this direction of the arrows. In fact this 22 j1/2 spinor is almost a pure α orbital. The arrows are drawn staring from points in space where the square root of square root of the density is 0.03. The color of the arrows are default red or blue, indicating minus or plus for the phase factor e^iθ.

The (square root of the) density and the approximate phase vector e^iθ can also be viewed separately:

Select 'Isosurface: Double (+/-)' from the 'Add' Menu.
In the new control line, press on the pull-down menu and
 Select 'Orbitals (occupied)' -> 'SCF_J1/2:1  18:  1.0  ..'.
Select 'Show Molecule Details' from the 'View' Menu'.
In the new control line Deselect 'Show Atoms'.
In the control line with 'Spinor' deselect the visualization of this field.

This spinor 18j1/2 is almost a pure 5p1/2 Tl spinor. A p1/2 atomic orbital has a spherical atomic density, but a spin magnetization density which is not the same in each point in space.

In the control line with 'Spinor', press on the pull-down menu and
 Select 'Orbitals (occupied)' -> 'SCF_J1/2:1  18:  1.0  ..'.
In the control line with 'Spinor' select the visualization of this field.
In the control line with 'Isosurface: Double' deselect the visualization of this field.