Multi-electron atoms#

What you need to know

  • We approximate many-electron wavefunctions using hydrogenic orbitals as building blocks.

  • Unlike the hydrogen atom, multi-electron problems do not separate and therefore do not admit exact analytical solutions.

  • Spin and antisymmetry impose strict constraints on allowed electronic wavefunctions.

Orbital Approximation#

  • For multi-electron atoms, the Hamiltonian depends on the coordinates of all electrons. The coordinates cannot be separated, so the Schrödinger equation is not analytically solvable.

  • A practical approximation is to represent the wavefunction as a product of single-electron orbitals:

    • These orbitals are obtained computationally (e.g., variationally).

Orbital Approximation

\[ \Psi(r_1, r_2, \ldots, r_n) \approx \phi_1(r_1)\phi_2(r_2)\cdots\phi_n(r_n) \]

  • \(\Psi\) is the multi-electron wavefunction.

  • Each \(\phi_i(r)\) is an atomic orbital occupied by one electron.

Helium Wavefunction#

For helium, the simplest guess would be

\[ \Psi(r_1, r_2) \approx \phi_1(r_1)\phi_2(r_2). \]

This form, however, suffers from three fundamental issues:

  1. Indistinguishability Electrons are identical; the wavefunction cannot assign “electron 1” to “orbital 1” and “electron 2” to “orbital 2.”

  2. Missing spin information The spatial wavefunction must combine with a spin function so that the total wavefunction is antisymmetric.

  3. Neglect of electron–electron correlation The product form assumes electrons move independently, which leads to quantitative errors in energies.

Issue 1: Indistinguishability and Antisymmetry#

Particles such as electrons are indistinguishable. Therefore the probability density must remain unchanged upon interchange:

\[ |\Psi(r_1,r_2)|^2 = |\Psi(r_2,r_1)|^2, \]

which implies

\[ \Psi(r_1,r_2) = \pm \Psi(r_2,r_1). \]

Electrons are fermions, and Pauli’s principle dictates that their total wavefunction must be antisymmetric:

Antisymmetry Requirement

\[ \Psi(\ldots, r_m, \ldots, r_n, \ldots) = -\Psi(\ldots, r_n, \ldots, r_m, \ldots). \]

A simple way to generate symmetric/antisymmetric forms for a two-variable function \(f(x,y)\):

  • Symmetric:

    \[ g_+(x,y) = f(x,y) + f(y,x) \]

  • Antisymmetric:

    \[ g_-(x,y) = f(x,y) - f(y,x) \]

For helium, the antisymmetrized spatial wavefunction is

\[ \Psi(r_1,r_2) \propto \phi_1(r_1)\phi_2(r_2) - \phi_1(r_2)\phi_2(r_1). \]

Issue 2: Spin Requirement#

Electrons have two spin states, \(\alpha\) and \(\beta\). The total wavefunction (spatial × spin) must be antisymmetric:

  • If spatial part is symmetric, spin part must be antisymmetric.

  • If spatial part is antisymmetric, spin part must be symmetric.

Possible two-electron spin combinations:

  • Symmetric:

    \[ \alpha(1)\alpha(2) \]
    \[ \beta(1)\beta(2) \]
    \[ \frac{1}{\sqrt{2}}\left[\alpha(1)\beta(2) + \alpha(2)\beta(1)\right] \]

  • Antisymmetric:

    \[ \frac{1}{\sqrt{2}}\left[\alpha(1)\beta(2) - \alpha(2)\beta(1)\right] \]

Pauli Exclusion Principle#

A key result of antisymmetry is the Pauli Exclusion Principle:

No two electrons can possess identical sets of quantum numbers.

For helium’s ground state (\(1s^2\)), the properly antisymmetrized total wavefunction is

\[ \Psi = \frac{1}{\sqrt{2}} \left[1s(1)1s(2) + 1s(2)1s(1)\right] \left[\alpha(1)\beta(2) - \alpha(2)\beta(1)\right]. \]

Slater Determinants#

Slater introduced a compact representation of antisymmetric wavefunctions:

\[ \Psi(r_1,\ldots, r_n) = \frac{1}{\sqrt{n!}} \begin{vmatrix} \chi_1(1) & \chi_2(1) & \cdots \ \chi_1(2) & \chi_2(2) & \cdots \ \vdots & \vdots & \ddots \end{vmatrix}, \]

where each \(\chi_i\) is an orbital × spin function. For helium ground state:

\[ \Psi = \frac{1}{\sqrt{2}} \begin{vmatrix} 1s(1)\alpha(1) & 1s(1)\beta(1) \ 1s(2)\alpha(2) & 1s(2)\beta(2) \end{vmatrix}. \]

Singlet and Triplet States of Helium#

For an excited configuration (e.g., \(1s,2s\)), the symmetric and antisymmetric spatial combinations are:

\[ \Psi_{\text{sym}} = \frac{1}{\sqrt{2}}\left[1s(1)2s(2) + 1s(2)2s(1)\right], \]
\[ \Psi_{\text{antisym}} = \frac{1}{\sqrt{2}}\left[1s(1)2s(2) - 1s(2)2s(1)\right]. \]

Combining with spin symmetries:

  • Triplet (symmetric spin)

  • Singlet (antisymmetric spin)

Action of Spin Operators#

  • Triplet:

\[ \hat{S}_z|\psi_1\rangle = +\hbar|\psi_1\rangle, \qquad \hat{S}_z|\psi_2\rangle = 0, \qquad \hat{S}_z|\psi_3\rangle = -\hbar|\psi_3\rangle, \]
\[ \hat{S}^2|\psi_i\rangle = 2\hbar^2|\psi_i\rangle. \]

  • Singlet:

\[ \hat{S}_z|\psi_4\rangle = 0, \qquad \hat{S}^2|\psi_4\rangle = 0. \]

Energies of Multi-Electron States#

The Hamiltonian is

\[ \hat{H} = \hat{H}_1 + \hat{H}_2 + \hat{H}_{12}, \]

with \(\hat{H}_{12}\) corresponding to electron–electron repulsion.

Useful Integrals#

Single-electron energy

\[ I(a) = \int \phi_a^*(r) \left[ -\frac{\hbar^2}{2m}\nabla^2 - \frac{Ze^2}{4\pi\epsilon_0 r} \right] \phi_a(r)\, d\tau \]

Coulomb Integral

\[ J_{ij} = \int |\phi_i(r_1)|^2 \frac{e^2}{4\pi\epsilon_0 r_{12}} |\phi_j(r_2)|^2 , d^3r_1, d^3r_2, \]

always positive.

Exchange Integral

\[ K_{ij} = \int \phi_i^*(r_1)\phi_j^*(r_2) \frac{e^2}{4\pi\epsilon_0 r_{12}} \phi_j(r_1)\phi_i(r_2), d^3r_1, d^3r_2. \]

Positive, but leads to energy lowering for triplet states.

Energies of \(1s2s\) Singlet and Triplet#

Triplet (antisymmetric spatial):

\[ E_{\text{triplet}} = I(1s) + I(2s) + J(1s,2s) - K(1s,2s) \]

Singlet (symmetric spatial):

\[ E_{\text{singlet}} = I(1s) + I(2s) + J(1s,2s) + K(1s,2s) \]

Thus the triplet state is lower in energy due to exchange stabilization.

Hund’s Rule and the Aufbau Principle#

Aufbau Principle#

Electrons fill orbitals in order of increasing energy.

Pauli Exclusion Principle#

Each orbital holds at most two electrons with opposite spins.

Hund’s Rule#

Electrons occupy degenerate orbitals singly with parallel spins before pairing. This reflects exchange stabilization.