Atomic orbitals
Raw solutions
Basic solutions of the Schrödinger equation for the one electron atom orbitals: (Demtröder 3 – page 149)
First shell s orbital:
- phi(n=1, l=0, m=0) = 1/sqrt(pi) * (Z/a_0)^(3/2) * exp(-(Z*r)/a_0)
Second shell s orbital:
- phi(n=2, l=0, m=0) = 1/(4*sqrt(2*pi)) * (Z/a_0)^(3/2) * (2-(Z*r)/a_0) * exp(-(Z*r)/(2*a_0))
Second shell three p orbitals:
- phi(n=2, l=1, m=0) = 1/(4*sqrt(2*pi)) * (Z/a_0)^(3/2) * (Z*r)/a_0 * exp(-(Z*r)/(2*a_0)) * cos(theta)
- phi(n=2, l=1, m=+-1) = 1/(8*sqrt(pi)) * (Z/a_0)^(3/2) * (Z*r)/a_0 * exp(-(Z*r)/(2*a_0)) * sin(theta) * exp(+-i*phi)
Third shell s orbital:
- phi(n=3, l=0, m=0) = ...
Shorthands for the basic solutions for the p orbitals:
- phi_pz = phi(n=2, l=1, m=0)
- phi_pa = phi(n=2, l=1, m=+1)
- phi_pb = phi(n=2, l=1, m=-1)
Real valued helper orbitals
All what follows below is (for copy paste purposes) in a syntax that is
compatible with most programming languages (e.g. python)
Adding two counter-rotating wave functions together in two different ways
to get two static wave functions pointing in two static orthogonal directions.
https://en.wikipedia.org/wiki/Atomic_orbital#Real_orbitals
- phi_px = 1/sqrt(2) * (phi_pa + phi_pb)
- phi_py = -i/sqrt(2)* (phi_pa - phi_pb)
Building the hybrid orbitals
sp1 orbitals:
- phi_spa = 1/sqrt(2) * (phi_2s + phi_2pz)
- phi_spb = 1/sqrt(2) * (phi_2s - phi_2pz)
sp2 orbitals:
- phi_sp20 = 1/sqrt(3) * (phi_2s + sqrt(2) * phi_2pz)
- phi_sp2p = 1/sqrt(3) * (phi_2s - sqrt(1/2) * phi_2px + 1/sqrt(3/2) * phi_2py)
- phi_sp2n = 1/sqrt(3) * (phi_2s - sqrt(1/2) * phi_2px - 1/sqrt(3/2) * phi_2py)
TODO In which direction do these orbitals point relative to the axes?
sp3 orbitals:
The sp3 orbitals are oriented in the 111 directions (which is natural since highest symmetry)
- ① phi_sp3ppp = 1/2 * (phi_2s + phi_2px + phi_2py + phi_2pz)
- ② phi_sp3pnn = 1/2 * (phi_2s + phi_2px - phi_2py - phi_2pz)
- ③ phi_sp3npn = 1/2 * (phi_2s - phi_2px + phi_2py - phi_2pz)
- ④ phi_sp3nnp = 1/2 * (phi_2s - phi_2px - phi_2py + phi_2pz)
