Difference between revisions of "Atomic orbitals"

From apm
Jump to: navigation, search
(added link to page * Useful math)
(added: == External links ==)
Line 63: Line 63:
  
 
* [[Useful math]]
 
* [[Useful math]]
 +
 +
== External links ==
 +
 +
Inner electrons screening the attraction from the nucleus:
 +
* [https://en.wikipedia.org/wiki/Effective_nuclear_charge Effective nuclear charge] <= '''There's a table with screening constants.'''
 +
* [https://en.wikipedia.org/wiki/Shielding_effect Shielding effect] and [https://en.wikipedia.org/wiki/Electric-field_screening Electric-field screening]
 +
 +
Approximating orbitals:
 +
* [https://en.wikipedia.org/wiki/Slater-type_orbital Slater-type orbital]
 +
* [https://en.wikipedia.org/wiki/Gaussian_orbital Gaussian orbital]
 +
 +
Exact solutions to the Schrödinger equation:
 +
* angular part: [https://en.wikipedia.org/wiki/Spherical_harmonics Spherical harmonics] – [https://en.wikipedia.org/wiki/Table_of_spherical_harmonics Table of spherical harmonics]
 +
* radial part: [https://en.wikipedia.org/wiki/Laguerre_polynomials Laguerre polynomials]
 +
* [https://en.wikipedia.org/wiki/Separable_partial_differential_equation Separable partial differential equation]
 +
 +
Constants and basics:
 +
* Real valued helper orbitals: [https://en.wikipedia.org/wiki/Atomic_orbital#Real_orbitals Atomic orbital ~> Real orbitals]
 +
* a0 = 5.29 * 10^(-11) m — [https://en.wikipedia.org/wiki/Bohr_radius Bohr radius]
 +
* [https://en.wikipedia.org/wiki/Square_(algebra)#Absolute_square Square_(algebra)#Absolute_square]

Revision as of 23:36, 5 June 2021

Math for constructing orbitals

Raw solutions

Basic solutions of the Schrödinger equation for the one electron atomic orbitals (aka hydrogen-like atomic 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)

Related

External links

Inner electrons screening the attraction from the nucleus:

Approximating orbitals:

Exact solutions to the Schrödinger equation:

Constants and basics: