Wavefunctions, Observables and Variational Bounds

Introduction

An electronic-structure calculation reports energies, orbitals and properties, but those numbers come from an approximate quantum state. The wavefunction encodes amplitudes from which probabilities and average observables are computed. A central quality check is normalization: the state must represent total probability one. The variational principle then gives a rigorous direction for ground-state energy error when a legitimate trial wavefunction is used with a fixed Hamiltonian. It explains why adding flexibility to a basis can lower a calculated ground-state energy, while also warning that a lower energy is not automatically a better prediction of every chemical observable.

Core explanation

For a state ψ depending on coordinates x, normalization is ∫ ψ(x) ²dx = 1 over the allowed domain. The probability of finding a particle in a region is the integral of ψ ² over that region, not ψ itself. If a trial function is not normalized, an expectation value can still be calculated as ⟨A⟩ = ⟨ψ Â ψ⟩/⟨ψ ψ⟩. For a normalized state, the denominator is one. Observables are represented by Hermitian operators so their measured eigenvalues and expectation values are real. Energy is the expectation of the Hamiltonian, while other operators can represent position, momentum or a molecular electric dipole within a chosen model.

The variational theorem states that for any acceptable normalized trial state φ of a fixed Hamiltonian with ground energy E₀, E trial = ⟨φ Ĥ φ⟩ ≥ E₀. Expand φ in exact normalized energy eigenstates ψ n: φ = Σ n c nψ n. Then E trial = Σ n c n ²E n, with Σ n c n ² = 1. Since every E n ≥ E₀, the weighted average cannot be below E₀. Equality holds if the trial state lies entirely in the ground-state subspace. A university quantum-chemistry derivation explains this proof and how it supports linear variational methods in quantum chemistry. An ACS educational application lets students test the method on a particle-in-a-box system with known exact levels.

In a linear variational method, choose basis functions χ i and write φ = Σ i a iχ i. Optimize coefficients a i to minimize the energy ratio. If the basis is enlarged by adding functions while retaining all earlier choices, the old optimized state is still available as a possible state. The minimum in the enlarged space therefore cannot be higher than the previous minimum, assuming numerical solution and Hamiltonian are unchanged. It may stay the same or fall. This is a statement about variational ground-state energy for one model, not a universal claim that every reported excitation energy, dipole or reaction energy improves monotonically with basis size. Differences between two approximate total energies can move in either direction because each state has its own error.

The theorem is often invoked in Hartree–Fock theory, where orbitals are optimized within a determinant ansatz. Hartree–Fock ground-state energy is an upper bound to the exact nonrelativistic ground energy for the same fixed-nuclei Hamiltonian, provided the calculation is genuinely variational and the state space is appropriate. Some common correlated methods, perturbation treatments and approximate density functionals do not give the same simple upper-bound guarantee. A computed energy lower than a reference benchmark is not automatically a violation of quantum mechanics; the two calculations may use different Hamiltonians, approximations or energy conventions, or the method may not be variational. The Rayleigh–Ritz discussion in current primary research states the upper bound for a trial state of the same Hamiltonian.

Variational energy can be deceptively insensitive to small wavefunction errors. A trial state close to the true ground state may give a very good energy while another observable that depends linearly on a particular wavefunction feature remains noticeably wrong. Conversely, fitting a flexible wavefunction to minimize energy does not correct an omitted physical effect such as solvent, relativistic behavior or an incorrect spin state. Comparing calculations requires keeping nuclear geometry, electron number, Hamiltonian and reference convention fixed for an upper-bound statement. If a basis improvement changes the optimized geometry, the problem being compared also changes, so the simple fixed-geometry inequality needs care.

Normalization and phase are worth separating. Multiplying ψ by a constant changes its norm until it is renormalized; multiplying by a global complex phase does not change ψ ² or ordinary observable expectations. A molecular orbital diagram is one representation of a many-electron state, not the complete measured object. Wavefunction-based interpretation is strongest when tied to calculated observables and convergence tests, rather than the visual appeal of an orbital plot.

Step-by-step reasoning

1. Define the Hamiltonian, geometry, charge and spin state to which the calculation applies. 2. Check that a trial wavefunction is mathematically acceptable and normalized, or retain the norm denominator. 3. Compute energy as ⟨φ Ĥ φ⟩/⟨φ φ⟩ and optimize only within the allowed trial family. 4. Use the variational theorem to compare with the exact ground energy of that same Hamiltonian. 5. When enlarging a nested basis, verify that previous trial states remain available and numerical convergence is sound. 6. Evaluate other observables and missing physics separately instead of inferring them from a lower energy alone.

Visual explanation

Draw an energy axis with the exact ground energy E₀ as a bottom line. Place several trial-state energies above it, with arrows downward as basis flexibility increases. Label one arrow “same Hamiltonian, nested basis.” Draw a second axis for a dipole prediction with points moving nonmonotonically as basis size changes to show that a better variational energy does not ensure every property improves smoothly. Beside it, draw two probability curves with unit areas but different shapes, emphasizing normalization versus accuracy of distribution.

Real-world analogy

An architect compares designs under one fixed set of constraints and costs. Allowing more design options cannot make the best achievable cost within that set worse, because the old option remains available. This resembles the nested variational basis. The analogy is limited: an energy minimum is determined by a quantum operator, and changing the physical Hamiltonian is like changing the cost rules entirely, so the old bound no longer applies directly.

Real-world example

A researcher calculates an iron complex using two nested orbital basis sets with the same approximate Hamiltonian and geometry. The larger basis lowers the variational Hartree–Fock total energy. That is expected. The researcher then compares a spin-state energy gap and finds it changes sign. The total-energy upper-bound property does not tell which spin state is correct or guarantee cancellation of errors between them. Magnetic and spectroscopic experiments, electron-correlation methods and checks of basis convergence are needed before assigning the complex's ground spin state.

Why?

Why can a trial ground-state energy not be lower than the true ground energy of the same Hamiltonian? Any valid trial state is a mixture of exact eigenstates, and its average energy is a weighted average of their eigenvalues. The ground eigenvalue is the smallest weightable value, so no weighted average can go below it. This reasoning needs the same Hamiltonian and a normalized, physically allowed state; it does not compare arbitrary outputs from different models.

Common misconception

“A lower calculated energy always means a more accurate chemical model.” Lower energy within a nested variational space can indicate better approximation to the same model's ground state, but an inappropriate model can still be wrong about solvent, reaction outcome or spectroscopy. Another error is dropping the norm denominator for an unnormalized trial function. A third is extending the ground-state bound automatically to excitation energies or nonvariational computational methods, where different conditions apply.

Worked example

Suppose a fixed Hamiltonian has exact orthonormal eigenstates ψ₀ and ψ₁ with energies E₀ = −10.0 and E₁ = −6.0 arbitrary energy units. Choose φ = √0.75 ψ₀ + √0.25 ψ₁. Its norm is 0.75+0.25 = 1, so E trial = 0.75(−10.0)+0.25(−6.0) = −9.0 units. This is 1.0 unit above E₀ despite containing 75% ground-state probability weight. If a second trial state has 90% ψ₀ and 10% ψ₁ weight, its energy is −9.6 units, lower and closer to E₀. The example illustrates the bound; a real computational basis may mix many exact states and does not reveal these weights directly.

Quick check

1. What denominator is required if a trial wavefunction has not been normalized? Answer: Divide ⟨ψ Â ψ⟩ by ⟨ψ ψ⟩ to obtain the expectation value. 2. Can adding functions to a truly nested variational basis raise its optimized ground-state energy for an unchanged Hamiltonian? Answer: No. The previous optimized state remains available, so the new minimum is no higher, barring numerical or setup changes.

Exam focus

Write the normalized expectation value and the variational inequality E trial ≥ E₀ for one fixed Hamiltonian. Derive the bound from expansion in exact eigenstates when asked. Explain why a nested larger basis can lower an optimized ground-state energy, but distinguish total energy from reaction-energy differences and other observables. State clearly that changing Hamiltonian, geometry, physical environment or method may invalidate a simple comparison of upper bounds.

Advanced insight

The variational principle is both a proof and an algorithmic strategy. Rayleigh–Ritz turns coefficient optimization into a matrix eigenvalue problem, while modern variational algorithms optimize more complicated state parameters. Near-degenerate states challenge a single-determinant ansatz even if its energy is minimized perfectly within that restricted form. Energy accuracy can conceal errors in density or response properties, so robust computational reporting includes basis convergence, state character and independent property checks. The useful claim is precise: the bound belongs to an acceptable trial state and a specified Hamiltonian's ground energy.

Summary

Wavefunctions encode quantum amplitudes, and observable expectations require correct normalization. The variational theorem guarantees that an acceptable trial state's energy is at least the exact ground energy for the same Hamiltonian. Enlarging a nested trial space cannot raise its optimized ground-state energy, but that monotonic statement does not automatically extend to every property, energy difference or changed model. Method choice and validation remain essential.

Practice questions

1. A normalized state has weights 0.6 and 0.4 on levels of 2 and 5 eV. What is its energy expectation? Answer: 0.6×2 + 0.4×5 = 3.2 eV, above the 2 eV ground level. 2. Why can two reaction-energy estimates change nonmonotonically as basis size increases even if both total energies decrease? Answer: Reactant and product total energies can fall by different amounts, so their difference can move either way. 3. If a trial function is multiplied by 3, does its normalized energy expectation change? Answer: No. The factor of nine in numerator and denominator cancels, provided the state remains otherwise the same. 4. Does the variational bound alone prove that an approximate solvent-free calculation matches a measured solution equilibrium? Answer: No. The bound concerns the chosen Hamiltonian's ground energy and does not supply missing solvent or thermal physics.