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The variational principle states that for any trial wavefunction φ, the expectation value of the energy ⟨φ|Ĥ|φ⟩/⟨φ|φ⟩ is always greater than or equal to the true ground-state energy E₀. This gives a systematic way to improve approximate wavefunctions by minimizing the energy with respect to variational parameters. The secular determinant |H − ES| = 0, obtained by expanding the trial function in a basis set, reduces the problem to matrix diagonalization. This principle underpins Hartree-Fock theory, DFT, and all basis-set electronic structure methods.
Apply the variational method to helium as a first example, using a screened hydrogenic wavefunction with the screening constant as the variational parameter. Observe how minimizing energy gives the optimal screening constant.
From solving the Schrödinger equation for the hydrogen atom, you know that exact analytical solutions exist for one-electron systems. But the moment you add a second electron — even for helium, the simplest multi-electron atom — the electron-electron repulsion term makes the equation analytically unsolvable. The variational principle provides a way forward: it turns the problem of solving a differential equation into the more tractable problem of *minimizing a function*. The principle states that for any normalized trial wavefunction φ you can write down, the expectation value of the energy ⟨φ|Ĥ|φ⟩ is guaranteed to be greater than or equal to the true ground-state energy E₀. This means the exact ground-state energy is a lower bound that no trial function can beat — every guess overshoots.
This guarantee converts quantum mechanics into an optimization problem. You propose a trial wavefunction with adjustable parameters — say, a hydrogen-like wavefunction for helium but with the nuclear charge Z replaced by an effective charge Z_eff that accounts for electron shielding. Then you compute the energy as a function of Z_eff and minimize. The resulting Z_eff ≈ 1.69 (rather than the bare nuclear charge of 2) captures the physical reality that each electron partially screens the nucleus from the other. The energy you obtain is remarkably close to the experimental value — within about 2% — from a single-parameter optimization. More parameters and more flexible trial functions systematically push the energy closer to the true value, and the variational principle guarantees you are always approaching from above.
The method becomes even more powerful when you expand the trial wavefunction as a linear combination of basis functions: φ = c₁χ₁ + c₂χ₂ + ... + c_Nχ_N. Now the variational parameters are the coefficients c_i. Minimizing the energy with respect to all coefficients leads to the secular equation |H − ES| = 0, where H is the matrix of Hamiltonian integrals H_ij = ⟨χᵢ|Ĥ|χⱼ⟩ and S is the overlap matrix S_ij = ⟨χᵢ|χⱼ⟩. This is an eigenvalue problem — a connection to the linear algebra you have studied. Diagonalizing the matrix gives you the best approximation to the ground-state energy (the lowest eigenvalue) and its wavefunction (the corresponding eigenvector), plus approximations to excited states as the higher eigenvalues.
This framework is the foundation of essentially all modern computational chemistry. Hartree-Fock theory uses the variational principle to find the best single-determinant wavefunction. Density functional theory, basis-set methods, and configuration interaction all rest on the same idea: propose a flexible functional form, compute the energy, and minimize. The variational principle is what makes these methods trustworthy — because the energy can only go down as you improve the trial function, you always know which direction "better" is. It transforms the intractable many-body Schrödinger equation into a systematic, improvable approximation scheme.