Atomic orbitals

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Positive lobe of a p-orbital (single electron solution for the Schrödinger equation) this was rendered using the software "libfive". Related: F-Rep in constructive solid geometry.

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 (single electron hydrogen-like atom 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)

Next steps

Accounting for shielding/screening

Unfortunately the nice orbitals for one electron hydrogen-like atoms (which are nice exact analytic solutions of the Schrödinger equation) are not applicable anymore to atoms with two or more electrons (or molecules). The problem is the combination of ...

  • ... that electrons mutually repulse each other and
  • ... that electrons are not localized in a small enough space that they can be assumed to be point charges (static aka non-moving). Like done with the positively charged atomic core nucleus.

In effect this leads to a shielding/screening effect that (with distance from the nucleus increasingly) hides the charge of the core from the electrons. Particularly the higher the electron shell the more electrons are "below" and the more screening it experiences. Ignoring shielding/screening would lead to errors >100% so this absolutely must be dealt with.

A first approximation is to assume that electrons are repulsed by the electron density cloud of all the other electrons (the "mean field") This is ignoring short lived virtual particle states. (These are responsible e.g. for the rather small london disprsion forces.) Still this will give at least "chemical accuracies" that are reasonably in the ballpark.

In practice accounting for shielding/screening is done by choosing some sort of approximation orbitals that ...

  • ... can take a parameter for the shielding/screening effect
  • ... may be qualitatively different form the hydrogen-like atom orbitals

Unfortunately instead of choosing qualitatively different orbitals to better match the now qualitatively different electric potential (deviating from simple 1/r) qualitatively different orbitals are typically rather chosen to improve on computing efficiency (including making things manageably computable in the first place). There are at least two types of commonly used approximation orbitals

  • (1) Slater-type orbitals
  • (2) Gaussian-type orbitals
  • ... ???

Both (1) and (2) sweep nodes in the wave function generously under the rug (to check). Gaussian type orbitals are especially crude.

Next step: Construct these approximation orbitals that take into account the shielding/screening effect of the inner electrons.

Superposing Slater-type approximation orbitals while adhering to the Pauli exclusion principle (two electrons with opposing spin per orbital) should gives at least a crude electron density distribution.

Naive superposition of such approximation orbitals of adjacent atoms in a molecule or crystal (or crystolecule) leads to even more not insignificant errors though.

When it comes to visualization-only purposes this should already give some nice reasonable looking results. So maybe one can stop here if it's only for that.

Going further gets into serious business.

Ignoring entanglement between electrons in first approximation

For a more accurate modelling of the "real" situation the modelling would need to be done in an holistic way. That is: All electrons must be described by one single multi particle wave function that cannot be fully disentangled into individual electron states. That is because in the general situation there can be at least a bit of quantum entanglement between the electrons.

In practice as first approximation it's assumed though that the electrons are mostly non-entangled. That is that the multi particle wave function can be written as a product of single particle wave functions. Product states. Entangled states are exactly the ones that cannot be written as product states. Some modelling methods later on introduce ways to account for the error that originates from ignoring entanglement here.

Note: There is no entanglement in the case for one electron hydrogen-like atoms. These are fully disentangled.

Accounting for particle indistinguishability

At last for the Hartree-Fock method the product state for the multi particle wave function needs to be constructed such that the swapping two "particle" positions leads to a swap of sign of the multi particle wave function.

  • This must be modeled because this is an inherent property of fermions which include electrons. Details go into more fundamental physics.
  • This is called antisymmetrization
  • This is done by means of a Slater determinant (that come with some properties that are important to know for calculations)

(wiki-TODO: Find out how to construct an initial guess wave function for density functional theory (DFT) and note it here – this info is hard to find)

d-orbitals and f-orbitals

d-orbitals and f-orbitals (as they fall out of the Schrödinger equation math) do not match the real situation well.
What typically is used instead is crystal field theory or ligand field theory. This can be used for predicting:

  • presence of unpaired spins (magnetism) and
  • magnitude of low energy energy gaps translating into F- center color absorption and colors of gemstones

Relevant for practical nano-engineering:
Documented structure of natural minerals (and decently stable synthetic compounds) can often give a hint about which coordination d-block and f-block metal atoms are most happy with.

What are atomic orbitals useful for?

Hard math

Orbitals (and linear supperpositions of them) can serve as a basis for highly accurate quantumchemistry simulations.
Both in solid state physics and also in physics of individual molecules any physical phase.

For gem-gum-tec most relevant is perhaps the simulation tooltip chemistry in piezomechanosynthesis.

For metals often something as crude as linear combinations of gaussian bell distributions is used instead of atomic orbitals.

Artistic side

What should not be understated regarding science communication:
Orbitals can also be used for nice artistic visualization that (while still being an somewhat crude approximation)
are a bit more accurate than some painfully unphysical metaball blob nonsense. Less artistically pleasing the standard visualizations are of course
Ball-and-stick or space-filling CPK visualizaton (Corey Pauling Koltun).

Related

Math - how to find these solutions

There are plenty of resources describing this in excruciating detail. So just a rough outline here.

  • Schrödinger equation (SG) in 3D - the static version suffices HΨ = EΨ
    where H is the Hamiltonian a differential operator extracting infromation about both potential and kinetic energy from the wave function.
  • SG with the Hamiltonian containing the field of a nucleus as a point charge for the potential energy
  • separation of variables - to solve the radial and the two spherical parts as separate equations
    (equations remaining linked only via eigenvalue constants which are the quantum numbers)
  • do some smart linear super-positioning get to standing waves
    For an intuitive interpretation: A 90° phase shift between real and complex part indicates the direction of a moving/spinning wave.
    You want to get rid of that phase shift by superpositioning.
    Note that nothing that indicates direction of motion is visible in the absolute square.

As a side-note:
Beyond basic math for orbitals if there is need to project orbitals into a different basis there are the Klebsch Gordan coefficients.

External links

Inner electrons screening the attraction from the nucleus:

Approximating orbitals:

Exact solutions to the Schrödinger equation:

Constants and basics:

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