The Helium Atom - Quantum Mechanics for Chemistry

The Helium Hamiltonian¶

The Schrodinger equation for the helium atom is already extremely complicated from a mathematical point of view. No analytic solution to this equation has been found. However, with certain approximations, useful results can be obtained. The Hamiltonian for the He atom can be written as:

{\hat{H} = \underbrace{-\frac{\hbar^2}{2m_e}\left(\Delta_1 + \Delta_2\right)}_{\textnormal{Kinetic energy}} \underbrace{- \frac{1}{4\pi\epsilon_0}\left(\frac{Ze^2}{r_1} + \frac{Ze^2}{r_2} \overbrace{- \frac{e^2}{r_{12}}}^{\textnormal{Tough!!}}\right)}_{\textnormal{Potential energy}}}

(1)

where $\Delta_1$ is the Laplacian for the coordinates of electron 1, $\Delta_2$ for electron 2, $r_1$ is the distance of electron 1 from the nucleus, $r_2$ is the distance of electron 2 from the nucleus, and $r_{12}$ is the distance between electrons 1 and 2. For the He atom $Z = 2$ .

The term that makes this problem unsolvable is the electron, electron repulsion $e^2/r_{12}$ . It couples the two electron coordinates so that the Hamiltonian no longer separates into independent one-electron pieces.

The Independent-Electron Approximation¶

A first approximation is to ignore the “tough” term containing $r_{12}$ . In this case the Hamiltonian becomes a sum of two hydrogenlike atoms:

{\hat{H} = \hat{H}_1 + \hat{H}_2}

(2)

{\hat{H}_1 = -\frac{\hbar^2}{2m_e}\Delta_1 - \frac{Ze^2}{4\pi\epsilon_0r_1}}

(3)

{\hat{H}_2 = -\frac{\hbar^2}{2m_e}\Delta_2 - \frac{Ze^2}{4\pi\epsilon_0r_2}}

(4)

Because the Hamiltonian is a sum of two independent parts, the Schrodinger equation separates into two equations, each a hydrogenlike atom problem:

{\hat{H}_1\psi(r_1) = E_1\psi(r_1)}

(5)

{\hat{H}_2\psi(r_2) = E_2\psi(r_2)}

(6)

The total energy is the sum of $E_1$ and $E_2$ , and the total wavefunction is a product of $\psi(r_1)$ and $\psi(r_2)$ . Based on our previous wavefunction table for hydrogenlike atoms, we have:

For a ground-state He atom both electrons reside in the lowest-energy orbital, so the total wavefunction is

\psi(r_1,r_2) = \psi(r_1)\psi(r_2) = \psi(1)\psi(2) = 1s(1)1s(2).

(9)

The energy obtained from this approximation is not sufficiently accurate (it misses electron, electron repulsion) but the wavefunction can be used for qualitative analysis. The variational principle gives a systematic way to assess how good our approximation is.

A Better Approximation: Variational Nuclear Charge¶

We can take the wavefunction from the previous step and use the nuclear charge $Z$ as a variational parameter. The variational principle states that minimization of the energy expectation value with respect to $Z$ should approach the true value from above (but obviously will not reach it).

By judging the energy, we can say that this new wavefunction is better than the previous one. The obtained value of $Z$ is less than the true $Z$ ( $=2$ ). This can be understood in terms of electrons shielding the nucleus from each other and hence giving a reduced effective nuclear charge.

If this trial wavefunction is used in calculating the energy expectation value, we get:

{E = \langle\psi |\hat{H}|\psi\rangle = \cdots = \left[ Z^2 - \frac{27Z}{8}\right]\frac{e^2}{4\pi\epsilon_0a_0}}

(10)

In order to minimize the energy we differentiate it with respect to $Z$ and set the result to zero (an extremum point; here it is clear that this point is a minimum):

{\frac{dE}{dZ} = \left(2Z - \frac{27}{8}\right)\frac{e^2}{4\pi\epsilon_0a_0} = 0}

(11)

This result could be improved by adding more terms and variables to the trial wavefunction. For example, higher hydrogenlike orbitals with appropriate variational coefficients would yield a much better result.

Another type of approximate method is based on perturbation theory, which would typically treat the electron, electron repulsion as an additional (small) perturbation to the independent-electron case above.