The Variational Method

Perturbation theory needs a Hamiltonian split into a solvable piece and a small correction. Many problems have no such split: the two electrons of helium repel each other with a term as large as their attraction to the nucleus, and no piece is small. The variational method sidesteps the requirement entirely. It rests on a single inequality — the energy of any trial state overestimates the ground-state energy — and converts finding the ground state into minimizing a function of a few parameters. The method gives no error bar by itself, but it gives a rigorous ceiling, and a well-chosen trial family brings that ceiling within a percent of the truth.

The variational theorem

Let $\hat{H}$ have a discrete spectrum bounded below, with ground-state energy $E_{g s}$ . For any normalizable state $∣ ψ ⟩$ , the expectation of the energy cannot fall below $E_{g s}$ .

The bound is one-sided and cheap: it needs only an integral, not a solution. Its power comes from the weak dependence of $⟨ H ⟩$ on errors in $∣ ψ ⟩$ . If the trial state differs from the true ground state by an amount of order $ϵ$ , its energy differs by order $ϵ^{2}$ , because the ground state is a stationary point of $⟨ H ⟩$ . A crude trial wavefunction, wrong by 10% in shape, can give an energy wrong by only 1%.

Every trial state's energy lies at or above the true ground level; the closer the trial to the true ground state the tighter the ceiling, and the minimum over a family is the best available upper bound.

The recipe

The method is mechanical once a trial family is chosen.

Choose a trial family $ψ (r; a_{1}, \dots, a_{k})$ with adjustable parameters, guided by the qualitative shape expected of the ground state (nodeless, decaying, symmetric).
Compute the energy $E (a_{1}, \dots, a_{k}) = ⟨ ψ ∣ \hat{H} ∣ ψ ⟩ / ⟨ ψ ∣ ψ ⟩$ as a function of the parameters.
Minimize by solving $\partial E / \partial a_{i} = 0$ . The minimum value is the best upper bound the family can give, and the minimizing parameters give the best approximate ground state in that family.

When the family contains the true ground state, the method returns it exactly. A Gaussian trial for the harmonic oscillator, $ψ (x) = e^{- b x^{2}}$ , minimizes to $b = mω /2ℏ$ and $E_{min} = \frac{1}{2} ℏ ω$ , the exact ground state, because the true ground state is a Gaussian. The interesting cases are those where it is not.

The ground state of helium

Helium is the first problem with no exact solution and no small parameter. The Hamiltonian for two electrons about a charge- $2$ nucleus is

\hat{H} = - \frac{ℏ ^{2}}{2 m} (\nabla_{1}^{2} + \nabla_{2}^{2}) - \frac{e ^{2}}{4 π ϵ _{0}} (\frac{2}{r _{1}} + \frac{2}{r _{2}}) + \frac{e ^{2}}{4 π ϵ _{0}} \frac{1}{∣ r _{1} - r _{2} ∣},

where the last term, the electron–electron repulsion, is comparable in size to the attraction and cannot be treated as a perturbation. The physical picture suggests the trial: each electron partially screens the nucleus from the other, so each sees an effective charge $Z$ somewhat less than $2$ . Take a product of hydrogenic $1 s$ orbitals with $Z$ as the variational parameter,

ψ (r_{1}, r_{2}) = \frac{Z ^{3}}{π a _{0}^{3}} e^{- Z (r_{1} + r_{2}) / a_{0}} .

Evaluating each piece in Rydberg units gives the kinetic energy $2 Z^{2}$ , the nuclear attraction $- 8 Z$ (the actual charge is $2$ , not $Z$ ), and the electron–electron repulsion $+ \frac{5}{4} Z$ , so

E (Z) = (2 Z^{2} - \frac{27}{4} Z) Ry .

Minimizing, $d E / d Z = 4 Z - \frac{27}{4} = 0$ gives the effective charge and energy

Z = \frac{27}{16} = 1.69, E_{min} = - \frac{729}{128} Ry \approx - 77.5 eV .

Each helium electron sees a nucleus partially screened by the other, so the best hydrogenic trial uses an effective charge below the bare value; the variational minimum lands within two percent of the measured energy.

The hydrogen molecular ion

The variational method predicts chemical bonding. The ion $H_{2}^{+}$ is one electron shared by two protons a distance $R$ apart. Fix the protons (the Born–Oppenheimer approximation: nuclei move slowly) and use a linear combination of atomic orbitals, a sum of ground-state hydrogen orbitals centered on each proton,

ψ_{\pm} (r) = A_{\pm} [ψ_{0} (r - r_{1}) \pm ψ_{0} (r - r_{2})],

with no free scale parameter but the internuclear distance $R$ as the variable to minimize over. The symmetric (bonding) combination $ψ_{+}$ piles electron density between the protons, where it is attracted to both; the antisymmetric (antibonding) combination $ψ_{-}$ has a node between them and depletes that density.

Computing $⟨ H ⟩ (R)$ and adding the proton–proton repulsion, the bonding curve develops a minimum below the energy of a separated hydrogen atom plus a proton, while the antibonding curve is repulsive at every $R$ .

The prediction is qualitatively decisive (a bound state exists) and quantitatively fair: the measured bond length is $R = 2.0 a_{0} = 1.06 \overset{˚}{A}$ and the binding energy $2.8 eV$ . The gap is the price of a rigid trial built from undistorted atomic orbitals; allowing the orbitals to contract toward the bond tightens both numbers.

The bonding combination's energy dips below the separated-atom limit and produces a minimum at finite internuclear distance, a bound molecule; the antibonding combination rises monotonically and does not bind.

Excited states through orthogonality

The theorem bounds the ground state, but a refinement bounds excited states. If the trial state is orthogonal to the exact ground state, its energy bounds the first excited level.

The obstacle is that the exact ground state is usually unknown, so exact orthogonality is hard to impose. Symmetry rescues the common cases. When the ground state is even under a symmetry (parity, for a symmetric potential), any odd trial state is automatically orthogonal to it, so an odd trial family bounds the lowest odd state without knowing the ground state at all. More generally, a trial state built in a symmetry sector distinct from the ground state's is orthogonal to it by construction, and the variational minimum in that sector bounds the lowest state carrying those quantum numbers.

A trial state chosen odd under parity is orthogonal to an even ground state for free, so its variational minimum bounds the lowest odd state from above, extending the method up the spectrum sector by sector.

The variational method and perturbation theory are complementary. Perturbation theory gives a systematic series when a small parameter exists; the variational method gives a rigorous bound when none does, at the cost of a guess and a one-sided estimate. The final method of the module, the WKB approximation, takes a third route: it exploits a slowly varying potential to build the wavefunction semiclassically, recovering both bound-state quantization and tunneling rates.