Why Approximation Methods Are Needed

Introduction

The particle in a box, the harmonic oscillator, the rigid rotor and the hydrogen atom can all be solved exactly with pen and paper. It is tempting to think that with enough effort any atom or molecule could be treated the same way. It cannot. As soon as a system contains two electrons, the repulsion between them couples their motions so tightly that no exact analytical solution exists. The rest of quantum chemistry is, in large part, the art of approximating well. This page explains why that art is necessary.

Core explanation

The helium Hamiltonian. Helium has a nucleus of charge +2 and two electrons. Treating the nucleus as fixed, the Hamiltonian in atomic units is

Ĥ = −½∇₁² − ½∇₂² − 2/r₁ − 2/r₂ + 1/r₁₂.

The first four terms are two copies of the hydrogen-like Hamiltonian for Z = 2. The last term, 1/r₁₂, is the repulsion between the electrons, where r₁₂ = r ₁ − r ₂ .

Why separation fails. If 1/r₁₂ were absent, Ĥ would be a sum of a part depending only on electron 1 and a part depending only on electron 2. The wavefunction would then factorise as ψ(1)ψ(2), and the energy would be the sum of two hydrogen-like energies. The repulsion term depends on the coordinates of both electrons together, including the angle between them. It cannot be written as a sum of one-electron pieces, so the six-dimensional equation cannot be split into two three-dimensional ones. This is the quantum version of the classical three-body problem, which also has no general closed-form solution.

How large is the repulsion? Ignoring 1/r₁₂ gives E = 2 × (−Z²/2) = −4 Eₕ = −108.8 eV for helium. The experimental energy (the sum of both ionisation energies, 24.6 + 54.4 eV) is −79.0 eV. The neglected repulsion is therefore worth about 30 eV, roughly seven times the bond energy of H₂. It is far too large to be treated as a tiny correction.

What is exactly solvable. Exact solutions exist for any one-electron system with simple potentials: hydrogen-like ions and, within the Born–Oppenheimer approximation, the H₂⁺ molecule-ion (in special elliptical coordinates). Everything chemically interesting beyond that — helium, lithium, H₂, water, proteins — requires approximation.

The main strategies.

- Variation method: guess a trial wavefunction with adjustable parameters and minimise the energy. The variation principle guarantees the result lies at or above the true ground-state energy. - Perturbation theory: start from a solvable problem and treat the difficult term as a small addition, correcting the energy order by order. - Self-consistent field (Hartree–Fock): each electron moves in the averaged field of the others, and the orbitals are refined until they reproduce the field that generated them. - Electron correlation methods and density functional theory: recover the effects of electrons instantaneously avoiding one another, which the averaged picture misses.

Each method trades accuracy against computational cost, and choosing the right one is a key skill.

Step-by-step reasoning

To see whether a problem needs approximation:

1. Write the full Hamiltonian, including every pairwise Coulomb interaction. 2. Ask whether it separates into independent one-particle parts. 3. If any term links two electrons (1/rᵢⱼ), exact separation fails. 4. Estimate the size of that term; if it is large, simple perturbative neglect will not suffice. 5. Choose a systematic approximation, such as the variation method.

Visual explanation

Picture the six-dimensional space of two electron positions as a vast landscape. Without repulsion, it is a product of two identical valleys and can be mapped one axis at a time. Repulsion adds a ridge along every point where the two electrons coincide, carving a shape that cannot be described one electron at a time.

Real-world analogy

Predicting the path of one ball rolling in a bowl is easy. Add a second ball that collides with the first and the paths become tangled: each ball's motion depends at every moment on where the other is. You can still predict the average behaviour well, but not with a single tidy formula.

Real-world example

Modern drug design relies on computed binding energies between molecules and protein pockets. None of these systems can be solved exactly, yet approximate methods such as density functional theory routinely predict geometries to within about 0.01 Å and relative energies to within a few kJ mol⁻¹, good enough to guide which compounds a laboratory makes next.

Why?

Why not just solve the equation numerically on a grid? The wavefunction of N electrons lives in 3N dimensions. With only ten grid points per dimension, a ten-electron molecule would need 10³⁰ values — far beyond any computer. Clever approximations that exploit chemical structure are essential, not merely convenient.

Common misconception

"Approximate methods give unreliable answers." Approximation is not guesswork. Systematic methods can be improved step by step, and for helium variational calculations agree with experiment to more than ten significant figures. The approximations are controlled and their errors can be estimated.

Worked example

Question: First-order perturbation theory adds the average repulsion ⟨1/r₁₂⟩ = 5Z/8 Eₕ to the independent-electron energy of helium. Calculate the resulting energy and compare it with experiment.

Reasoning: With Z = 2, the correction is 5 × 2/8 = 1.25 Eₕ = 34.0 eV. The zeroth-order energy is −108.8 eV, so E ≈ −108.8 + 34.0 = −74.8 eV. Experiment gives −79.0 eV.

Answer: About −74.8 eV, an error of roughly 4 eV (5 %). Including the average repulsion recovers most of the effect, but not all of it.

Quick check

1. Which term in the helium Hamiltonian prevents the Schrödinger equation from being separated into one-electron equations? Answer: The electron–electron repulsion term 1/r₁₂, because it depends on the positions of both electrons at once.

Exam focus

Be able to write the helium Hamiltonian, identify 1/r₁₂ as the non-separable term, and quote the independent-electron energy (−108.8 eV) against experiment (−79.0 eV). Examiners often ask you to name and briefly contrast the variation and perturbation approaches.

Advanced insight

The helium ground state has become a benchmark of precision. In 1929 Hylleraas introduced trial functions depending explicitly on r₁₂, and modern versions with thousands of terms give −2.903 724 377 Eₕ, with agreement to experiment limited mainly by relativistic and quantum electrodynamic corrections rather than by the approximation itself.

Summary

Only one-electron systems have exact analytical solutions. For two or more electrons the repulsion 1/rᵢⱼ couples their coordinates, making the Schrödinger equation non-separable. In helium this term is worth about 30 eV, far too large to ignore. Quantum chemistry therefore relies on systematic approximations: the variation method, perturbation theory, self-consistent field methods and correlation or density functional methods.

Practice questions

1. What would the ground-state energy of helium be if electron repulsion were ignored? Give the answer in eV. Answer: 2 × (−13.6 × 2²) = −108.8 eV. 2. Why is the H₂⁺ ion exactly solvable within the Born–Oppenheimer approximation while H₂ is not? Answer: H₂⁺ has only one electron, so there is no electron–electron repulsion term; H₂ has two electrons and a non-separable 1/r₁₂ term. 3. Name two general approximation methods and state the key idea of each. Answer: The variation method minimises the energy of a trial wavefunction, giving an upper bound; perturbation theory corrects a solvable model by treating the difficult term as a small addition. 4. Explain why storing a wavefunction on a grid becomes impossible for large molecules. Answer: The number of grid points grows exponentially with the number of electrons, since the wavefunction depends on 3N coordinates, so memory requirements explode.