https://www.quanty.org/ 2026-04-29T06:56:54+00:00 https://www.quanty.org/ https://www.quanty.org/_media/wiki/dokuwiki.svg text/html 2025-11-20T01:45:33+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.quanty.org/physics_chemistry/orbitals/j?rev=1763603133&do=diff Spin-orbit coupled states (J) [Download a notebook that generates this page]The $j$-$j_z$ coupled states are the eigen-orbitals of the spin-orbit coupling Hamiltonian. For the basis sets on harmonics we did not discuss spin, as the basis states for spin up are the same as the basis states for spin down. The spin-orbit coupled states however have spin explicitly included. The $j$$j_z$$$ J_{l,j}^{(j_z)} = \left( \begin{array}{c} (-1)^{l-1/2+j_z} \sqrt{2j+1} \left(\begin{array}{ccc} l &\phantom{… text/html 2025-11-20T01:45:34+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.quanty.org/physics_chemistry/orbitals/k?rev=1763603134&do=diff Kubic Harmonics (K) [Download a notebook that generates this page]The kubic harmonics (also known as cubic harmonics) are linear combinations of the spherical harmonics and irreducible representations of the cubic ($O_h$) point group. For $s$, $p$ and $d$ wave-functions the kubic harmonics are besides a different order the same as the tesseral harmonics. For higher angular momentum they are different. $T_{Z\,\mathrm{to}\,K}$$\{Z_{l}^{(m=-l)},K_{l}^{(m=-l+1)},\dots,K_{l}^{(m=l)}\}$$T_{Z\,\mathr… text/html 2025-11-20T01:45:34+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.quanty.org/physics_chemistry/orbitals/pyramidal?rev=1763603134&do=diff Pyramidal Harmonics From the Tesseral Harmonics and for a given angular momentum $l$ one can construct an orbital basis where the basis orbitals have the same shape, but differ in orientation. This implies that this basis can be obtained by specifying one of these orbitals and obtain the others by rotating the first one. A possible realization is to align these orbitals along the slant edges of a pyramid with a regular polygon of order $2l+1$$l=0...3$$l$orbitals index text/html 2025-11-20T01:45:35+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.quanty.org/physics_chemistry/orbitals/start?rev=1763603135&do=diff One-particle orbitals The standard operators as used in Quanty require one to define a basis set for the orbitals. Most (but not all) standard operators implemented assume an atomic shell with spherical symmetry. The spin-orbitals are then given by a radial wave-function times a function depending on the angular coordinates: $\psi(x,y,z)=R(r)\Theta(\theta,\phi)$$Y$$Z$$K$$j-j_z$$j$orbitals index text/html 2025-11-20T01:45:34+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.quanty.org/physics_chemistry/orbitals/transformation_matrix?rev=1763603134&do=diff Orbital Transformation matrix alligned paragraph text Example description text Input -- some example code Result text produced as output Table of contents orbitals index text/html 2025-11-20T01:45:34+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.quanty.org/physics_chemistry/orbitals/y?rev=1763603134&do=diff Spherical Harmonic (Y) [Download a notebook that generates this page]The spherical harmonics are defined as $$ Y_l^{(m)} = \frac{\sqrt{2 l+1}}{2 \sqrt{\pi }} \sqrt{\frac{(l-m)!}{(l+m)!}} e^{i m \phi } P_l^{(m)}(\cos (\theta )), $$ with $l$ the angular momentum and $m$ the z projection of the angular momentum, $-l \leq m\leq l$. $P_l^{(m)}$ are the associated Legendre polynomials. For positive $m$ these are defined in terms of the unassociated Legendre polynomials as:$$ \begin{align} P_l^{(m)}(… text/html 2025-11-20T01:45:34+00:00 Anonymous (anonymous@undisclosed.example.com) https://www.quanty.org/physics_chemistry/orbitals/z?rev=1763603134&do=diff Tesseral Harmonics (Z) [Download a notebook that generates this page]The spherical harmonics are complex functions. For many cases one does not need to work with complex numbers and by making a suitable linear combination of the complex orbitals one can get a real basis. The tesseral harmonics are linear combinations of the spherical harmonics with $+m$$-m$$m>0$$\cos(m\phi)$$m<0$$\sin(m\phi)$$$ Z_l^{(m)}=\left\{\begin{array}{ll} Y_l^{(0)} & m=0\\ \frac{1}{\sqrt…