The variational principle

Result

The variational principle

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Statement

An upper bound on the ground-state energy.

Why it matters

particle-in-a-box and hydrogen-atom-solution are among the very rare quantum systems solvable exactly in closed form; essentially every molecule of real chemical interest has a Schrödinger equation with no exact analytical solution at all, a direct consequence of the many-electron problem, even after born-oppenheimer's separation of nuclear and electronic motion. The variational principle is the foundational theorem that makes approximate, yet rigorously bounded and systematically improvable, solutions possible in this overwhelmingly common situation, and it underlies both huckel-theory's treatment of conjugated \(\pi\) systems and lcao-molecular-orbitals's construction of molecular orbitals from atomic ones.

Its particular value is that it converts an otherwise unconstrained approximation problem into a well-posed optimisation: any trial wavefunction gives a computable number that is guaranteed never to fall below the true answer, turning "find a good approximate wavefunction" into "minimise a computable energy," a strategy that underlies essentially all of modern computational quantum chemistry.

Hypotheses

The trial wavefunction \(\phi\) is normalisable and satisfies the same boundary conditions as the true ground-state wavefunction.Without this, the relevant integrals in the proof need not converge, and \(\phi\) would not correspond to a physically admissible quantum state to compare in the first place. The comparison is made specifically against the true ground-state energy \(E_0\) of the exact Hamiltonian.No such universal lower-bound guarantee exists for excited states in general; restricted variants of the theorem for excited states require additional symmetry constraints, outside this page's scope. The theorem guarantees only that the trial energy is an upper bound on \(E_0\); it does not guarantee that a trial function giving a low energy is a good approximation to the true wavefunction in every other respect, such as for computing other observables.

Proof

\phi=\sum_n c_n\psi_n, \qquad \hat H\psi_n=E_n\psi_n,\ E_0\le E_1\le E_2\le\cdots

Expand an arbitrary, normalised trial function \(\phi\) in the complete set of true (generally unknown) eigenfunctions of \(\hat H\), ordered by increasing energy starting from the ground state \(E_0\). A

\langle\phi|\hat H|\phi\rangle=\sum_n|c_n|^2E_n

Using orthonormality of the true eigenfunctions to eliminate all cross terms, the expectation value of the energy for this trial function reduces to a weighted sum of the true eigenvalues, weighted by the squared expansion coefficients. A

E_n\ge E_0\ \forall n \;\Rightarrow\; \langle\phi|\hat H|\phi\rangle \ge E_0\sum_n|c_n|^2

Since every eigenvalue is at least as large as the ground-state energy by definition, replacing each \(E_n\) in Step 2's sum by \(E_0\) can only decrease or leave unchanged the total, giving this inequality directly. A

Normalisation of the trial function fixes the sum of squared coefficients to exactly \(1\), so Step 3's inequality reduces directly to the variational bound: any admissible trial function's energy expectation value is at least \(E_0\), regardless of how \(\phi\) was chosen. A

\text{Linear variation method: expand } \phi \text{ in a finite basis with adjustable coefficients, minimise the trial energy.}

Operationalising the bound: expanding \(\phi\) as a linear combination of a fixed, finite basis set with adjustable coefficients and minimising the resulting trial energy with respect to those coefficients is a calculus problem yielding the secular equations that lcao-molecular-orbitals and huckel-theory both solve in their respective, more specialised contexts. B

Result

\langle\phi|\hat H|\phi\rangle \;\ge\; E_0 \quad \text{for any normalised, admissible trial function } \phi

Reading. Any trial wavefunction's computed energy expectation value can only overestimate, or at best exactly equal, the true ground-state energy, never underestimate it — which is what makes minimising the trial energy over adjustable parameters a rigorously meaningful strategy for approaching the true ground state from above.

Scope. Holds for any normalised, admissible trial function (Hypotheses); the bound applies specifically to the ground state, not, without further qualification, to excited states.

Corollaries & converses

Because the bound holds for any admissible trial function, adding more flexibility (more adjustable parameters, or a larger basis set) can only lower or leave unchanged the resulting variational energy, never raise it — a systematic route to improvement exploited by huckel-theory and lcao-molecular-orbitals alike.
The linear variation method's secular equations (Step 5) reduce the variational problem to a matrix eigenvalue problem, the same computational structure underlying essentially all modern quantum chemistry methods built on this foundation.
born-oppenheimer's separation of nuclear and electronic motion is what makes an electronic-only trial wavefunction, at fixed nuclear positions, a meaningful, well-posed variational object in the first place.

Fails without

Drop normalisation or use a trial function violating the correct boundary conditions (Hypotheses): the integrals defining the energy expectation value may not converge, or may not correspond to a physically admissible state at all, and the guaranteed upper-bound relationship of the Result no longer follows from Steps 1–4 as derived.
Apply the plain ground-state bound directly to an excited state without the required additional constraints: a trial function aimed at an excited state can, without those extra constraints, variationally collapse toward the ground state instead, since nothing in the bare theorem as proven here prevents this from happening.

Common errors

Assuming a lower computed variational energy always means a uniformly better wavefunction in every respect, when it strictly only guarantees a value closer to (or, at best, equal to) the true ground-state energy.
Applying the plain ground-state bound uncritically to claims about an excited-state energy, without the additional constraints excited-state variational treatments actually require.
Forgetting to normalise the trial function, or to include the normalisation factor explicitly, before comparing the computed expectation value against \(E_0\).
Believing that because the true exact energy is unknown, the quality of a variational result cannot be assessed at all; in practice, systematically enlarging the trial function's flexibility and checking that the computed energy keeps decreasing toward a stable value is exactly the standard convergence check used in computational chemistry.

Discussion

The variational principle's specific application to bound the Schrödinger equation's ground-state energy became a central working tool as quantum chemistry developed through the 1920s and 1930s, essentially from the moment the equation was recognised as unsolvable in closed form for anything beyond the very simplest one-electron systems, hydrogen-atom-solution's result being essentially the last of these to yield fully to exact solution for a real chemical system.

The linear variation method's secular equations, in the special case of the Hückel approximation applied to conjugated \(\pi\) systems (huckel-theory), reduce to an especially simple form because most off-diagonal matrix elements are set to zero by assumption, illustrating how the fully general variational machinery developed here specialises cleanly into the much more restricted, hand-calculable models built on top of it elsewhere in the unit.

Common misconception: that the variational principle proves any specific trial function is a good approximation. It proves only the one-directional inequality (Result) — a poor trial function still gives a valid, but possibly very loose, upper bound, and only comparison between multiple trial functions, or against experiment, can establish which is actually a good approximation.

Worked examples

\text{Particle in a box, } 0\le x\le L\text{: trial function } \phi(x)=x(L-x)

Applying the variational method with a deliberately simple, non-eigenfunction trial function satisfying the correct boundary conditions (\(\phi=0\) at \(x=0,L\)), \(\phi''=-2\), so \(\hat H\phi=-\tfrac{\hbar^2}{2m}(-2)=\tfrac{\hbar^2}{m}\) (a constant), giving \(\langle\phi|\hat H\phi\rangle=\tfrac{\hbar^2}{m}\int_0^Lx(L-x)\,dx=\tfrac{\hbar^2}{m}\cdot\tfrac{L^3}{6}\), and \(\langle\phi|\phi\rangle=\int_0^Lx^2(L-x)^2dx=\tfrac{L^5}{30}\). B

E_{\text{trial}}=\frac{\langle\phi|\hat H\phi\rangle}{\langle\phi|\phi\rangle}=\frac{(\hbar^2/m)(L^3/6)}{L^5/30}=\frac{5\hbar^2}{mL^2}

Dividing the two integrals from Step 1 gives a variational energy of exactly \(5\hbar^2/(mL^2)\), to be compared against particle-in-a-box's exact ground-state result. B

E_{\text{trial}}=5\,\frac{\hbar^2}{mL^2}\approx5.000\,\frac{\hbar^2}{mL^2}\quad\text{vs}\quad E_0^{\text{exact}}=\frac{\pi^2}{2}\frac{\hbar^2}{mL^2}\approx4.935\,\frac{\hbar^2}{mL^2}

Reading. The simple parabolic trial function, which is not the true sine-shaped ground-state wavefunction, gives an energy only about \(1.3\%\) above the exact particle-in-a-box ground state, and, exactly as the Result requires, never falls below it.

Scope. This numerical check, applying the variational method to a system whose exact solution is independently known, is a standard way of confirming the theorem's guaranteed-upper-bound behaviour directly.

Problems

Explain why the computed variational energy \(5\hbar^2/(mL^2)\) in the Worked example must be greater than or equal to, not less than, the true ground-state energy, referencing the Result directly.
Solution
The trial function \(\phi(x)=x(L-x)\) is normalisable and satisfies the correct boundary conditions (Hypotheses), so the Result's guarantee, \(\langle\phi|\hat H|\phi\rangle\ge E_0\), applies directly; since it is not the true ground-state eigenfunction, the inequality is expected to be strict, exactly as observed (\(5>\pi^2/2\)).
A student adds a second adjustable parameter to a one-parameter trial function and finds the resulting variational energy is lower than before. Explain why this outcome is guaranteed never to reverse (i.e. the energy can never rise) as more parameters are added.
Solution
By Corollaries, a trial function with more adjustable parameters strictly contains every trial function reachable with fewer parameters as a special case (setting the new parameter's optimal value to reduce to the old function is always an available option); since the variational method always finds the lowest energy achievable within the trial function's full flexibility, adding parameters can only find an equal or lower minimum, never a higher one.
Explain, using the variational principle, why a Hartree-Fock calculation's computed electronic energy is always greater than or equal to the true, exact non-relativistic ground-state energy of the same system.
Solution
The Hartree-Fock wavefunction (a single Slater determinant) is a specific, admissible trial function for the true electronic Hamiltonian; by the Result, its computed energy expectation value must be greater than or equal to the true ground-state energy \(E_0\), with equality only in the (essentially never realised) case that the true ground state is itself exactly a single Slater determinant.