Quantum mechanics/Many electron systems
Subject classification: this is a chemistry resource. 
Type classification: this is a lesson resource. 
Contents
Orbitals[edit]
The Hamiltonian of a system made by N particles (with coordinates r_{1}, r_{2}, ..., r_{N}) that do not interact is:
(Eq. 1)
According to the rules of separable Hamiltonians, the eigenfunctions Ψ(r_{1}, r_{2}, ..., r_{N}) of should be the product of the eigenfunctions of the individual Hamiltonians .
(Eq. 2)
So separation is very convenient, but is the separation possible in a system with many electrons? The answer is no; it is enough to look at the Hamiltonian for the helium atom.
(Eq. 3)
The five terms represent respectively the kinetic energy of electron 1, the attraction between electron 1 and the nucleus, the kinetic energy of electron 2, the attraction between electron 2 and the nucleus, the repulsion between the electrons (r_{12} is their distance). The repulsion between electrons makes the Hamiltonian nonseparable. But the separation is too good to be lost so we write an approximate Hamiltonian where the repulsion between electrons is substituted by an average repulsion due to the other electrons, i.e.
(Eq. 4)
U(r) is the average repulsion felt by an electron in position r due to the other electron(s). The approximate Hamiltonian is separable. The eigenfunctions of the operators are single electron eigenfunctions called orbitals. This approximation is the basis for the idea of orbital in multi electron systems and the idea of molecular orbitals in molecules. Many chemistry students are unable to provide a precise definition of orbital; the one below should be remembered forever:
An orbital is a single electron wavefunction. Describing a multi electron system in terms of its orbitals is an approximation. The approximation consists of neglecting the detail of the electronelectron repulsion and considering only an average repulsion with the other electrons. 
Multi electron wavefunction[edit]
Spin must be included[edit]
If we make the Hamiltonian separable as the sum of single oneelectron Hamiltonians (like Eq. 1) the wavefunction cannot be just Ψ(r_{1}, r_{2}, ..., r_{N}) = φ_{1}(r_{1})φ_{2}(r_{2})...φ_{N}(r_{N}). Even though the spin does not appear in the Hamiltonian, we should include it in the wavefunction. For example, the wavefunction of two electrons, the first one with spin α in orbital φ_{A} and the second with spin β in orbital φ_{B} could be written as φ_{A}(r_{1})α(1)φ_{B}(r_{2})β(2) but this is not correct yet.
Electrons must be indistinguishable[edit]
The probability Ψ(r_{1}, r_{2})² should be identical to the probability Ψ(r_{2}, r_{1})² because the electrons have no label and they cannot be told apart because of Heisenberg principle. You can naively think that Ψ(r_{1}, r_{2})=±Ψ(r_{2}, r_{1}) but it turns out that the sign must always be minus for the electrons. This is an additional postulate of quantum mechanics.
Postulate 6. Interchanging all the coordinates of two particles in a wavefunction, the resulting wavefunction must be either identical or the negative of the initial wavefunction. Particles whose wavefunction remains unchanged upon interchange of the coordinates are called bosons. Particles whose wavefunction changes sign upon interchange of the coordinates are called fermions. Electrons are fermions. Their wavefunctions always change sign when the coordinates of two electrons are interchanged. Multielectron wavefunctions are antisymmetric. 
Pauli principle[edit]
The consequences of postulate 6 are the following:
φ_{A}(r_{1})α(1)φ_{B}(r_{2})β(2) is not a good wavefunction because if the coordinates are interchanged you get φ_{A}(r_{2})α(2)φ_{B}(r_{1})β(1) which is not φ_{A}(r_{1})α(1)φ_{B}(r_{2})β(2).
The only possibility to describe two electrons in orbitals φ_{A} and φ_{B} satisfying the antisymmetry of the wavefunction is to write the wavefunction as
 [φ_{A}(r_{1})φ_{B}(r_{2}) + φ_{B}(r_{1})φ_{A}(r_{2})][α(1)β(2)  β(1)α(2)] or (Eq. 6)
 [φ_{A}(r_{1})φ_{B}(r_{2})  φ_{B}(r_{1})φ_{A}(r_{2})][α(1)α(2)] or
 [φ_{A}(r_{1})φ_{B}(r_{2})  φ_{B}(r_{1})φ_{A}(r_{2})][β(1)β(2)] or
 [φ_{A}(r_{1})φ_{B}(r_{2})  φ_{B}(r_{1})φ_{A}(r_{2})][α(1)β(2) + β(1)α(2)]
The only possibility to describe two electrons in the same orbital φ_{A} is
 [φ_{A}(r_{1})φ_{A}(r_{2})][α(1)β(2)  β(1)α(2)] (Eq. 7)
This implies that two electrons in the same orbital must have opposite spin.
There is no way to have two electrons in the same orbital with the same spin and still satisfy postulate 6. This is the Pauli exclusion principle. The qualitative idea that electrons fill the orbitals two electrons per orbital with opposite spin is a consequence of postulate 6.
Orbital energy in multi electron atoms[edit]
To describe qualitatively all atoms, we always start with the approximation of Equation 4, i.e. we assume that the electrons feel an 'average' repulsion of the other electrons and we consider the total wavefunction as the product of the individual orbitals with maximum two electrons per orbital (the wavefunction should be written to make it antisymmetric when we include spin). The calculation of the orbital energy is too advanced, but the following general points can be made:
 Atomic orbitals are still labelled as 1s, 2s, 2p, 3s, 3p, 3d, etc. The angular part of the orbital is again given by spherical harmonics (because the potential still has a spherical symmetry).
 The radial part is not identical to that of the hydrogen atom. You can assume that each electron is attracted to the nucleus by an effective positive charge Z_{eff} (smaller than the actual charge of the nucleus because the other electrons are partially screening this charge) and use the hydrogenlike radial wavefunction R(r).
 The orbitals with larger angular momentum quantum number l and same principal quantum number n are better screened (see Exercise) i.e. are less attracted by the nucleus and have therefore higher energy. Orbital 2s of the Li atom has lower energy than orbital 2p (in hydrogen they have the same energy).
The electronic configuration is the description of which orbitals are filled and by how many electrons. You find it for each atom in several periodic tables.
However, the configuration is not enough to describe the state of an atom. For example, the excited state configuration of helium 1s^{1}2s^{1} leads to 4 different possible antisymmetric wavefunctions:
(Eq. 6)
The energy depends on the radial part so Ψ_{S1} and Ψ_{S3} must have different energies. A description of the wavefunction more accurate than the configuration is the spectroscopic term which will be discussed in the next lesson.
Exercise

Next: Lesson 12  Atomic spectra