Quantum mechanics/Many electron systems

Subject classification: this is a chemistry resource.

Type classification: this is a lesson resource.

Orbitals

The Hamiltonian of a system made by N particles (with coordinates r₁, r₂, ..., r_N) that do not interact is:

${\hat {H}}(r_{1},r_{2},...,r_{N})={\hat {h}}(r_{1})+{\hat {h}}(r_{2})+...+{\hat {h}}(r_{N})$ (Eq. 1)

According to the rules of separable Hamiltonians, the eigenfunctions Ψ(r₁, r₂, ..., r_N) of ${\hat {H}}$ should be the product of the eigenfunctions of the individual Hamiltonians ${\hat {h}}(r_{i})$.

$\Psi (r_{1},r_{2},...,r_{N})=\varphi _{1}(r_{1})\varphi _{2}(r_{2})...\varphi _{N}(r_{N})$ (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.

${\hat {H}}_{\mbox{He}}(r_{1},r_{2})=-{\frac {\hbar ^{2}}{2m_{e}}}\nabla _{1}^{2}-{\frac {2e^{2}}{r_{1}}}-{\frac {\hbar ^{2}}{2m_{e}}}\nabla _{2}^{2}-{\frac {2e^{2}}{r_{2}}}+{\frac {e^{2}}{r_{12}}}$ (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 (r₁₂ is their distance). The repulsion between electrons makes the Hamiltonian non-separable. 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.

${\hat {H}}_{\mbox{He}}(r_{1},r_{2})=-{\frac {\hbar ^{2}}{2m_{e}}}\nabla _{1}^{2}-{\frac {2e^{2}}{r_{1}}}+U(r_{1})-{\frac {\hbar ^{2}}{2m_{e}}}\nabla _{2}^{2}-{\frac {2e^{2}}{r_{1}}}+U(r_{2})={\hat {h}}(r_{1})+{\hat {h}}(r_{2})$ (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 ${\hat {h}}(r_{i})$ 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 electron-electron repulsion and considering only an average repulsion with the other electrons.

Multi electron wavefunction

Spin must be included

If we make the Hamiltonian separable as the sum of single one-electron Hamiltonians (like Eq. 1) the wavefunction cannot be just Ψ(r₁, r₂, ..., r_N) = φ₁(r₁)φ₂(r₂)...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(r₁)α(1)φ_B(r₂)β(2) but this is not correct yet.

Electrons must be indistinguishable

The probability |Ψ(r₁, r₂)|² should be identical to the probability |Ψ(r₂, r₁)|² because the electrons have no label and they cannot be told apart because of Heisenberg principle. You can naively think that Ψ(r₁, r₂)=±Ψ(r₂, r₁) 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. Multi-electron wavefunctions are antisymmetric.

Pauli principle

The consequences of postulate 6 are the following:

φ_A(r₁)α(1)φ_B(r₂)β(2) is not a good wavefunction because if the coordinates are interchanged you get φ_A(r₂)α(2)φ_B(r₁)β(1) which is not -φ_A(r₁)α(1)φ_B(r₂)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(r₁)φ_B(r₂) + φ_B(r₁)φ_A(r₂)][α(1)β(2) - β(1)α(2)] or (Eq. 6)
• [φ_A(r₁)φ_B(r₂) - φ_B(r₁)φ_A(r₂)][α(1)α(2)] or
• [φ_A(r₁)φ_B(r₂) - φ_B(r₁)φ_A(r₂)][β(1)β(2)] or
• [φ_A(r₁)φ_B(r₂) - φ_B(r₁)φ_A(r₂)][α(1)β(2) + β(1)α(2)]

The only possibility to describe two electrons in the same orbital φ_A is

• [φ_A(r₁)φ_A(r₂)][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

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 hydrogen-like 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 1s¹2s¹ leads to 4 different possible antisymmetric wavefunctions:

$\Psi _{S1}=[\varphi _{A}(r_{1})\varphi _{B}(r_{2})+\varphi _{B}(r_{1})\varphi _{A}(r_{2})][\alpha (1)\beta (2)-\beta (1)\alpha (2)]$ (Eq. 6)

$\Psi _{S3}=[\varphi _{A}(r_{1})\varphi _{B}(r_{2})-\varphi _{B}(r_{1})\varphi _{A}(r_{2})]{\begin{cases}\beta (1)\beta (2)\\\alpha (1)\beta (2)+\beta (1)\alpha (2)\\\alpha (1)\alpha (2)\end{cases}}$

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 Explain why an electron in a 2p orbital is more screened (smaller Zeff) with respect to an electron in a 2s orbital. The orbital energy can be identified with the ionization energy (removal of an electron from that orbital). The first ionization energy of Na is 5.14 eV (the electron is removed from the 3s orbital). What is Zeff for the electron in 3s? What is an orbital? Solutions The density is closer to the nucleus for orbital 2s which feels a greater attraction. $E_{n}({\mbox{eV}})=-{\frac {(Z_{\mbox{eff}})^{2}\cdot 13.6}{n^{2}}}$ Since n=3 and E3=-5.14 eV, therefore $Z_{\mbox{eff}}=\left({\frac {3^{2}\cdot 5.14}{13.6}}\right)^{\begin{matrix}{\frac {1}{2}}\end{matrix}}$ Any answer other than "a single electron wavefunction" is wrong.

Next: Lesson 12 - Atomic spectra