# Solutions for the Schrodinger equation with multiple particles

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## Solutions of the Schrodinger equation for two electrons

TODO. Can't find it easily. Anyone?
This is closely linked to the Pauli exclusion principle.
What does a particle even mean, right? Especially in quantum field theory, where two electrons are just vibrations of a single electron field.
Another issue is that if we consider magnetism, things only make sense if we add special relativity, since Maxwell's equations require special relativity, so a non approximate solution for this will necessarily require full quantum electrodynamics.
As mentioned at lecture 1 youtube.com/watch?video=H3AFzbrqH68&t=555, relativistic quantum mechanical theories like the Dirac equation and Klein-Gordon equation make no sense for a "single particle": they must imply that particles can pop in out of existence.
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## Orbital approximation

Just ignore the electron electron interactions.

## Schrödinger equation solution for the helium atom

No closed form solution, but good approximation that can be calculated by hand with the Hartree-Fock method, see hartree-Fock method for the helium atom.

## Why do multiple electrons occupy the same orbital if electrons repel each other?

That is, two electrons per atomic orbital, each with a different spin.
As shown at Schrödinger equation solution for the helium atom, they do repel each other, and that affects their measurable energy.
However, this energy is still lower than going up to the next orbital. TODO numbers.
This changes however at higher orbitals, notably as approximately described by the aufbau principle.

## Aufbau principle

Boring rule that says that less energetic atomic orbitals are filled first.
Much more interesting is actually determining that order, which the Madelung energy ordering rule is a reasonable approximation to.

## Electron configuration notation (, 1s2 2s2 2p6)

We will sometimes just write them without superscript, as it saves typing and is useless.

## Why does 2s have less energy than 1s if they have the same principal quantum number?

The principal quantum number thing fully determining energy is only true for the hydrogen emission spectrum for which we can solve the Schrödinger equation explicitly.
For other atoms with more than one electron, the orbital names are just a very good approximation/perturbation, as we don't have an explicit solution. And the internal electrons do change energy levels.
Note however that due to the more complex effect of the Lamb shift from QED, there is actually a very small 2p/2s shift even in hydrogen.

Looking at the energy level of the Schrödinger equation solution for the hydrogen atom, you would guess that for multi-electron atoms that only the principal quantum number would matter, azimuthal quantum number getting filled randomly.
However, orbitals energies for large atoms don't increase in energy like those of hydrogen due to electron-electron interactions.
In particular, the following would not be naively expected:
• 2s fills up before 2p. From the hydrogen solution, you might guess that they would randomly go into either one as they'd have the same energy
• in potassium fills up before 3d, even though it has a higher principal quantum number!
This rule is only an approximation, there exist exceptions to the Madelung energy ordering rule.

## Exception to the Madelung energy ordering rule

The most notable exception is the borrowing of 3d-orbital electrons to 4s as in chromium, leading to a 3d5 4s1 configuration instead of the 3d4 4s2 we would have with the rule. TODO how is that observed observed experimentally?

## Term symbol (, , )

This notation is so confusing! People often don't manage to explain the intuition behind it, why this is an useful notation. When you see Indian university entry exam level memorization classes about this, it makes you want to cry.
The key reason why term symbols matter are Hund's rules, which allow us to predict with some accuracy which electron configurations of those states has more energy than the other.
web.chem.ucsb.edu/~devries/chem218/Term%20symbols.pdf puts it well: electron configuration notation is not specific enough, as each such notation e.g. 1s2 2s2 2p2 contains several options of spins and z angular momentum. And those affect energy.
This is why those symbols are often used when talking about energy differences: they specify more precisely which levels you are talking about.
Basically, each term symbol appears to represent a group of possible electron configurations with a given quantum angular momentum.
We first fix the energy level by saying at which orbital each electron can be (hyperfine structure is ignored). It doesn't even have to be the ground state: we can make some electrons excited at will.
The best thing to learn this is likely to draw out all the possible configurations explicitly, and then understand what is the term symbol for each possible configuration, see e.g. term symbols for carbon ground state.
It also confusing how uppercase letters S, P and D are used, when they do not refer to orbitals s, p and d, but rather to states which have the same angular momentum as individual electrons in those states.
It is also very confusing how extremelly close it looks to spectroscopic notation!
The form of the term symbol is: $$2S+1LJ​ (1)$$
The can be understood directly as the degeneracy, how many configurations we have in that state.

## Hund's rules (1927)

Allow us to determine with good approximation in a multi-electron atom which electron configuration have more energy. It is a bit like the Aufbau principle, but at a finer resolution.
Note that this is not trivial since there is no explicit solution to the Schrödinger equation for multi-electron atoms like there is for hydrogen.
For example, consider carbon which has electron configuration 1s2 2s2 2p2.
If we were to populate the 3 p-orbitals with two electrons with spins either up or down, which has more energy? E.g. of the following two:
m_L -1  0  1
u_ u_ __
u_ __ u_
__ ud __

## Hund's first rule

Higher spin multiplicity means lower energy. I.e.: you want to keep all spins pointin in the same direction.

## Term symbols for carbon ground state

This example covered for example at Video 1. "Term Symbols Example 1 by TMP Chem (2015)".
Carbon has electronic structure 1s2 2s2 2p2.
For term symbols we only care about unfilled layers, because in every filled layer the total z angular momentum is 0, as one electron necessarily cancels out each other:
• magnetic quantum number varies from -l to +l, each with z angular momentum to and so each cancels the other out
• spin quantum number is either + or minus half, and so each pair of electron cancels the other out
So in this case, we only care about the 2 electrons in 2p2. Let's list out all possible ways in which the 2p2 electrons can be.
There are 3 p orbitals, with three different magnetic quantum numbers, each representing a different possible z quantum angular momentum.
We are going to distribute 2 electrons with 2 different spins across them. All the possible distributions that don't violate the Pauli exclusion principle are:
m_l  +1  0 -1  m_L  m_S
u_ u_ __    1    1
u_ __ u_    0    1
__ u_ u_   -1    1
d_ d_ __    1   -1
d_ __ d_    0   -1
__ d_ d_   -1   -1
u_ d_ __    1    0
d_ u_ __    1    0
u_ __ d_    0    0
d_ __ u_    0    0
__ u_ d_   -1    0
__ d_ u_   -1    0
ud __ __    2    0
__ ud __    0    0
__ __ ud   -2    0
where:
• m_l is , the magnetic quantum number of each electron. Remember that this gives a orbital (non-spin) quantum angular momentum of to each such electron
• m_L with a capital L is the sum of the of each electron
• m_S with a capital S is the sum of the spin angular momentum of each electron
For example, on the first line:
m_l  +1  0 -1  m_L  m_S
u_ u_ __    1    1
we have:
• one electron with , and so angular momentum
• one electron with , and so angular momentum 0
and so the sum of them has angular momentum . So the value of is 1, we just omit the .
TODO now I don't understand the logic behind the next steps... I understand how to mechanically do them, but what do they mean? Can you determine the term symbol for individual microstates at all? Or do you have to group them to get the answer? Since there are multiple choices in some steps, it appears that you can't assign a specific term symbol to an individual microstate. And it has something to do with the Slater determinant. The previous lecture mentions it: www.youtube.com/watch?v=7_8n1TS-8Y0 more precisely youtu.be/7_8n1TS-8Y0?t=2268 about carbon.
youtu.be/DAgEmLWpYjs?t=2675 mentions that is not allowed because it would imply , which would be a state uu __ __ which violates the Pauli exclusion principle, and so was not listed on our list of 15 states.
He then goes for and mentions:
• S = 1 so can only be 0
• L = 2 (D) so ranges in -2, -1, 0, 1, 2
and so that corresponds to states on our list:
ud __ __    2    0
u_ d_ __    1    0
u_ __ d_    0    0
__ u_ d_   -1    0
__ __ ud   -2    0
Note that for some we had a two choices, so we just pick any one of them and tick them off off from the table, which now looks like:
 +1  0 -1  m_L  m_S
u_ u_ __    1    1
u_ __ u_    0    1
__ u_ u_   -1    1
d_ d_ __    1   -1
d_ __ d_    0   -1
__ d_ d_   -1   -1
d_ u_ __    1    0
d_ __ u_    0    0
__ d_ u_   -1    0
__ ud __    0    0
Then for the choices are:
• S = 2 so is either -1, 0 or 1
• L = 1 (P) so ranges in -1, 0, 1
so we have 9 possibilities for both together. We again verify that 9 such states are left matching those criteria, and tick them off, and so on.
For the , we have two electrons with spin up. The angular momentum of each electron is , and so given that we have two, the total is , so again we omit and is 1.

## Schrödinger equation solution for the hydrogen molecule

Can we make any ab initio predictions about it all?

## Isomer

Isomers were quite confusing for early chemists, before atomic theory was widely accepted, and people where thinking mostly in terms of proportions of equations, related: Section "Isomers suggest that atoms exist (1874)".

## Cis-trans isomerism

Exist because double bonds don't rotate freely. Have different properties of course, unlike enantiomer.

## Enantiomer

Mirror images.
Key exmaple: d and L amino acids. Enantiomers have identical physico-chemical properties. But their biological roles can be very different, because an enzyme might only be able to act on one of them.

## Polymorphism (materials science)

TODO definition. Appears to be isomers
Example:

## Stereochemistry

Molecules that are the same if you just look at "what atom is linked to what atom", they are only different if you consider the relative spacial positions of atoms.