Atomic orbitals

Math for constructing orbitals

Raw solutions

Basic solutions of the Schrödinger equation for the one electron atom orbitals:
(source – 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)

All what follows below is (for copy paste purposes) in a syntax that is
compatible with most programming languages (e.g. python)

Real valued helper orbitals

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)

For a better understanding of what is going on here:
When separating the exp(+-i*phi) part into cos(+-i*pi) + i*sin(+-i*phi)
One can see a phase shift of 90° between real and imaginary part of the wave function.
The direction of the phase shift determined the direction of the rotation.
That works for electrons travelling as wave packets in free space too.
Here the electron is delocalized over the whole 360° though.
So the rotation is not no observable as a moving packet of electron density.

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)