Quantum Mechanics Course

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Part VI, Chapter 3 | Page 2 of 3

Worked Example: Helium Ground State

Step-by-step variational calculation of the simplest two-electron atom

The helium atom -- with two electrons orbiting a nucleus of charge $Z=2$ -- is the simplest system that cannot be solved exactly. It provides the ideal testing ground for the variational method and demonstrates how a single clever parameter can capture the essential physics of electron-electron interactions.

The Helium Hamiltonian

The full Hamiltonian for helium (ignoring relativistic and spin effects) involves the kinetic energies of both electrons, their attraction to the nucleus, and their mutual repulsion:

$$\hat{H} = \underbrace{-\frac{\hbar^2}{2m_e}\nabla_1^2 - \frac{Ze^2}{4\pi\epsilon_0 r_1}}_{\text{electron 1}} + \underbrace{-\frac{\hbar^2}{2m_e}\nabla_2^2 - \frac{Ze^2}{4\pi\epsilon_0 r_2}}_{\text{electron 2}} + \underbrace{\frac{e^2}{4\pi\epsilon_0|\vec{r}_1 - \vec{r}_2|}}_{\text{repulsion}}$$

where $Z=2$ for helium. The electron-electron repulsion term $e^2/r_{12}$ makes this problem analytically unsolvable. Without it, the two electrons would be independent hydrogen-like atoms.

The Trial Wave Function

We use the simplest physically motivated trial function: each electron occupies a hydrogen-like 1s orbital, but with an effective nuclear charge $Z_{\text{eff}}$ as the variational parameter:

$$\psi(\vec{r}_1, \vec{r}_2) = \phi(\vec{r}_1)\phi(\vec{r}_2) = \frac{Z_{\text{eff}}^3}{\pi a_0^3}\, e^{-Z_{\text{eff}}(r_1 + r_2)/a_0}$$

The physical idea: each electron partially shields the nucleus from the other electron, so the effective nuclear charge "seen" by each electron is less than $Z=2$. We expect $1 < Z_{\text{eff}} < 2$.

Note: This product ansatz ignores electron correlation (the electrons don't "know" about each other's positions). More sophisticated trial functions include explicit $r_{12}$-dependent terms.

Step 1: Rewrite the Hamiltonian

It is useful to rewrite $\hat{H}$ in terms of $Z_{\text{eff}}$ by adding and subtracting terms:

$$\hat{H} = \underbrace{\left(-\frac{\hbar^2}{2m_e}\nabla_1^2 - \frac{Z_{\text{eff}}e^2}{4\pi\epsilon_0 r_1}\right)}_{\hat{h}_1(Z_{\text{eff}})} + \underbrace{\left(-\frac{\hbar^2}{2m_e}\nabla_2^2 - \frac{Z_{\text{eff}}e^2}{4\pi\epsilon_0 r_2}\right)}_{\hat{h}_2(Z_{\text{eff}})} + \frac{(Z_{\text{eff}} - Z)e^2}{4\pi\epsilon_0}\left(\frac{1}{r_1} + \frac{1}{r_2}\right) + \frac{e^2}{4\pi\epsilon_0 r_{12}}$$

The first two terms are hydrogen-like Hamiltonians with nuclear charge $Z_{\text{eff}}$, for which our trial function is an exact eigenstate.

Step 2: Evaluate Each Energy Contribution

Kinetic + Nuclear Attraction (with $Z_{\text{eff}}$)

Since $\phi$ is the exact ground state of a hydrogen-like atom with charge $Z_{\text{eff}}$:

$$\langle\hat{h}_1\rangle + \langle\hat{h}_2\rangle = 2 \times \left(-\frac{Z_{\text{eff}}^2 e^2}{8\pi\epsilon_0 a_0}\right) = -Z_{\text{eff}}^2 \times 13.6\text{ eV}$$

Nuclear Charge Correction

The correction for using $Z_{\text{eff}}$ instead of $Z$:

$$\left\langle\frac{(Z_{\text{eff}}-Z)e^2}{4\pi\epsilon_0}\left(\frac{1}{r_1}+\frac{1}{r_2}\right)\right\rangle = 2(Z_{\text{eff}}-Z)\frac{e^2}{4\pi\epsilon_0}\left\langle\frac{1}{r}\right\rangle$$

Using $\langle 1/r\rangle = Z_{\text{eff}}/a_0$ for a hydrogen-like 1s orbital:

$$= 2(Z_{\text{eff}}-Z) \times Z_{\text{eff}} \times 13.6\text{ eV} / (-1) = -2Z_{\text{eff}}(Z_{\text{eff}}-Z) \times 13.6\text{ eV}$$

Wait -- let us be more careful. We have $\langle 1/r \rangle = Z_{\text{eff}}/a_0$ and $e^2/(4\pi\epsilon_0 a_0) = 2 \times 13.6$ eV, so:

$$= 2(Z_{\text{eff}}-Z)\frac{Z_{\text{eff}}}{a_0}\frac{e^2}{4\pi\epsilon_0} = 2(Z_{\text{eff}}-Z)Z_{\text{eff}} \times 27.2\text{ eV}$$

Electron-Electron Repulsion

This is the most challenging integral. Using the multipole expansion and standard techniques:

$$\left\langle\frac{e^2}{4\pi\epsilon_0 r_{12}}\right\rangle = \frac{5}{8}\frac{Z_{\text{eff}} e^2}{4\pi\epsilon_0 a_0} = \frac{5}{8}Z_{\text{eff}} \times 27.2\text{ eV}$$

The factor $5/8$ is a famous result. It represents the average electron-electron repulsion energy for two electrons in identical 1s orbitals.

Step 3: Total Energy Functional

Combining all contributions (in units of $e^2/(4\pi\epsilon_0 a_0) = 27.2$ eV):

$$E(Z_{\text{eff}}) = \left[Z_{\text{eff}}^2 - 2Z \cdot Z_{\text{eff}} + \frac{5}{8}Z_{\text{eff}}\right] \times 13.6\text{ eV}$$

For helium ($Z = 2$):

$$E(Z_{\text{eff}}) = \left[Z_{\text{eff}}^2 - 4Z_{\text{eff}} + \frac{5}{4}Z_{\text{eff}}\right] \times 13.6\text{ eV} = \left[Z_{\text{eff}}^2 - \frac{27}{16}Z_{\text{eff}} \cdot \frac{16}{8}\right] \times 13.6\text{ eV}$$

More simply:

$$E(Z_{\text{eff}}) = \left(Z_{\text{eff}}^2 - \frac{27}{8}Z_{\text{eff}}\right) \times 27.2\text{ eV}$$

Step 4: Minimization

Setting $dE/dZ_{\text{eff}} = 0$:

$$\frac{dE}{dZ_{\text{eff}}} = \left(2Z_{\text{eff}} - \frac{27}{8}\right) \times 27.2\text{ eV} = 0$$
$$\boxed{Z_{\text{eff}}^* = \frac{27}{16} = 1.6875}$$

As expected, $Z_{\text{eff}} < 2$: each electron partially screens the nucleus from the other. The screening reduces the effective charge by $5/16 \approx 0.31$.

Substituting back:

$$\boxed{E_{\min} = -\left(\frac{27}{16}\right)^2 \times 27.2\text{ eV} = -77.5\text{ eV}}$$

Comparison with Experiment

How good is our result?

$$\begin{aligned} \text{Variational (1 parameter):} \quad & E = -77.5\text{ eV} \\ \text{Experimental:} \quad & E = -79.0\text{ eV} \\ \text{Error:} \quad & \approx 2\% \end{aligned}$$

Only 2% error with a single variational parameter! For comparison:

  • No interaction ($Z_{\text{eff}} = 2$): $E = -108.8$ eV (38% error)
  • First-order perturbation theory (treating $e^2/r_{12}$ as perturbation): $E = -74.8$ eV (5% error)
  • Variational ($Z_{\text{eff}} = 27/16$): $E = -77.5$ eV (2% error)
  • Hylleraas (6 parameters including $r_{12}$): $E = -79.0$ eV (0.01% error)
  • Modern computation (1000+ parameters): $E = -79.0051$ eV (matching experiment to ppm)

Physical Interpretation of Z_eff

The value $Z_{\text{eff}} = 27/16 \approx 1.69$ has a clear physical meaning: each electron "sees" the nuclear charge reduced by about $5/16 \approx 0.31$ due to the screening by the other electron. The inner electron partially blocks the nuclear attraction from the outer electron. This is the simplest version of the screening concept that underlies all of atomic structure theory.

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