The Variation Principle

Introduction

When the Schrödinger equation cannot be solved exactly, we need a method that tells us not only an approximate answer but also how to judge whether one approximation is better than another. The variation principle does exactly that. It says that any reasonable guess at the ground-state wavefunction produces an energy that is too high, never too low. This turns the search for a good wavefunction into a straightforward competition: the lower the energy, the better the guess. It is the foundation of most practical quantum chemistry.

Core explanation

The statement. Let Ĥ be the Hamiltonian of a system whose true ground-state energy is E₀. Take any trial function ψ that is well behaved (single-valued, continuous, square-integrable and satisfying the same boundary conditions as the true wavefunction). Define the variational energy, also called the Rayleigh ratio,

E trial = ∫ψ Ĥψ dτ / ∫ψ ψ dτ.

Then

E trial ≥ E₀,

with equality if and only if ψ is the exact ground-state wavefunction (up to a constant factor). The denominator means ψ need not be normalised in advance; the ratio takes care of normalisation automatically.

What it means. The ground-state energy is the lowest possible expectation value of the Hamiltonian. Any imperfect trial function contains admixtures of excited states, which have higher energies and so pull the average upwards. The true ground state is the unique function that achieves the minimum.

How to use it. In practice the trial function contains one or more adjustable parameters, such as an orbital exponent. The energy is calculated as a function of these parameters and then minimised. Because every trial energy lies above E₀, the lowest value found is the best estimate available within that family of functions, and it is guaranteed to be an upper bound.

Quality of the energy. A useful property is that errors in the energy are second order in errors of the wavefunction. If the trial function differs from the exact one by a small amount of order ε, the energy error is of order ε². Even a rough wavefunction can therefore give a surprisingly good energy. The flip side is that a good energy does not guarantee that other properties, such as dipole moments, are equally accurate.

Scope. The principle applies to the ground state. It can be extended to excited states only if the trial function is constrained to be orthogonal to all lower exact states, or by the linear variation method, which yields upper bounds to several levels at once.

Formulae

E trial = ∫ψ Ĥψ dτ / ∫ψ ψ dτ ≥ E₀. For a trial function containing parameter α, the best estimate satisfies dE trial/dα = 0.

Step-by-step reasoning

To apply the variation principle:

1. Choose a trial function that obeys the boundary conditions and has the right general shape. 2. Include one or more adjustable parameters. 3. Evaluate the numerator ∫ψ Ĥψ dτ and the denominator ∫ψ ψ dτ. 4. Form the ratio to obtain E trial as a function of the parameters. 5. Minimise with respect to each parameter; the minimum is the best upper bound.

Visual explanation

Picture a landscape whose height at each point is the energy of a different trial function. The exact ground state sits at the bottom of the deepest valley. Wherever you stand, you are at or above that floor; walking downhill by adjusting parameters brings you closer to it, but you can never tunnel below.

Real-world analogy

Imagine finding the lowest point of a valley in fog by checking the altitude on a GPS at many spots. Every reading you take is at or above the valley floor. The lowest reading so far is your best estimate, and you know for certain the true floor is no higher than that.

Real-world example

Every Hartree–Fock calculation performed in chemistry software is a variational calculation. The program adjusts orbital coefficients to minimise the energy, and larger basis sets always give energies that are lower or equal, never higher. Chemists use this monotonic convergence to judge whether a basis set is adequate.

Why?

Why can't a trial function give an energy below the true ground state? Any trial function can be written as a mixture of the exact eigenfunctions of Ĥ. Its energy is a weighted average of their eigenvalues, with positive weights. An average of numbers that are all at least E₀ cannot fall below E₀.

Common misconception

"A lower variational energy means the wavefunction is more accurate for every property." The energy is the quantity being optimised, so it converges fastest. Properties that depend on regions the energy is insensitive to, such as the far tail of the wavefunction, can remain poor even when the energy is excellent.

Worked example

Question: For a particle of mass m in a box from x = 0 to x = L, use the trial function ψ = x(L − x) to estimate the ground-state energy and compare it with the exact value h²/(8mL²).

Reasoning: Inside the box Ĥ = −(ħ²/2m) d²/dx². Integration by parts gives ∫ψĤψ dx = (ħ²/2m)∫(dψ/dx)² dx = (ħ²/2m) × L³/3. The denominator is ∫x²(L − x)² dx = L⁵/30. The ratio is E trial = (ħ²/2m) × 10/L² = 5ħ²/(mL²) = 5h²/(4π²mL²) = 0.1267 h²/(mL²).

Answer: E trial = 0.1267 h²/(mL²), compared with the exact 0.1250 h²/(mL²). The estimate is only 1.3 % too high, and it is above the exact value, as the principle requires.

Quick check

1. A student's trial function for hydrogen gives an energy of −0.52 Eₕ. What can you conclude? Answer: There is an error in the calculation, because the exact ground-state energy is −0.5 Eₕ and no trial energy can lie below it.

Exam focus

State the principle precisely, including the conditions on the trial function and the equality case. Be able to write the Rayleigh ratio, explain why it is an upper bound, and carry out a one-parameter or parameter-free calculation such as the box example above.

Advanced insight

The variation principle can also be phrased as a statement that the exact eigenfunctions are the stationary points of the Rayleigh ratio. The ground state is the global minimum; excited states are saddle points. This viewpoint underlies modern methods such as coupled-cluster response theory and the variational quantum eigensolver used on quantum computers.

Summary

For any acceptable trial function, the variational energy ∫ψ Ĥψ dτ / ∫ψ ψ dτ is greater than or equal to the true ground-state energy, with equality only for the exact wavefunction. Minimising this energy with respect to adjustable parameters gives the best upper bound within the chosen family. Energy errors are second order in wavefunction errors, which makes the method efficient.

Practice questions

1. Write the expression for the variational energy of an unnormalised trial function. Answer: E trial = ∫ψ Ĥψ dτ / ∫ψ ψ dτ. 2. Under what condition does the variational energy equal the true ground-state energy? Answer: Only when the trial function is the exact ground-state wavefunction, possibly multiplied by a constant. 3. Two trial functions for the same molecule give energies of −1.128 Eₕ and −1.133 Eₕ. Which is better, and why? Answer: The second, because both are upper bounds and the lower energy is closer to the true ground-state value. 4. Why must the trial function for a particle in a box vanish at x = 0 and x = L? Answer: It must satisfy the same boundary conditions as the true wavefunction, which is zero at the walls where the potential is infinite. 5. Explain why the energy from a variational calculation is often more accurate than the wavefunction itself. Answer: The energy error is second order in the wavefunction error, so a wavefunction error of about 10 % gives an energy error of only about 1 %.