Trial Wavefunctions

A variational calculation is only as good as its trial wavefunction. The variational principle guarantees an upper bound on the ground-state energy, but it does not guarantee that a poorly designed trial family will be close to the true state.

The art is to build a trial state that is simple enough to optimize and rich enough to contain the important physics.

For the relation between ansatz error, optimization error, upper bounds, and state diagnostics, see Variational and Bound Methods.

When expectation values are evaluated by sampling $\lvert\Psi_\theta\rvert^2$, Variational Monte Carlo adds practical requirements: stable amplitude ratios, finite local-energy variance, and adequate support for the sampler. Variational Many-Body States owns the structural comparison of Slater, Jastrow, paired, projected, matrix-product, and neural ansätze.

Respect Boundary Conditions

The trial wavefunction must live in the domain of the Hamiltonian. It should satisfy the same boundary conditions as the physical state:

• vanish at infinite walls;
• be continuous where the potential is finite;
• have the correct derivative behavior at finite jumps;
• be normalizable on the relevant domain;
• obey regularity conditions at the origin in radial problems.

Violating boundary conditions can produce a misleading energy expectation or a state that is not admissible at all.

Respect Symmetries

If the Hamiltonian commutes with a symmetry, choose trial states in the appropriate symmetry sector. For a parity-invariant one-dimensional Hamiltonian, a ground-state trial function should usually be even. For a central potential, use states with definite angular momentum when targeting a fixed angular-momentum sector.

Symmetry-adapted trial states have two advantages:

• forbidden mixing is avoided from the start;
• the number of variational parameters is reduced.

If symmetry is deliberately broken in a trial state, state why. Symmetry breaking may be useful as an intermediate approximation in many-body physics, but it should not happen accidentally.

Use Correct Limiting Behavior

Good trial functions build in known limits. Examples include:

• Gaussian tails for harmonic confinement;
• exponential tails for Coulombic bound states;
• oscillatory behavior in classically allowed regions;
• evanescent decay in forbidden regions;
• regular power-law behavior near the origin;
• cusp behavior when a singular interaction imposes it.

The asymptotic tail can dominate errors in weakly bound states. Short-distance behavior can dominate kinetic or interaction-energy errors.

Include Meaningful Parameters

A useful variational parameter usually controls a physical scale:

• width of a localized wavefunction;
• effective charge or screening length;
• displacement of a packet center;
• mixing angle between two configurations;
• correlation length between particles;
• basis cutoff or oscillator frequency in a finite-basis calculation.

Parameters should not merely decorate a trial function. They should allow the state to respond to the main physical competition in the Hamiltonian.

Once those coordinates have physical meaning, Variational Parameters explains their gradients, constraints, scaling, Hessian conditioning, and numerical optimization.

Balance Flexibility and Discipline

More parameters can lower the variational energy, but they also create risks:

• local minima;
• nearly redundant directions;
• numerical instability;
• loss of interpretability;
• fitting the energy while missing important observables.

Energy is often second-order sensitive to wavefunction errors, while other observables can be first-order sensitive. A very good energy does not automatically mean a very good wavefunction.

Compare Multiple Trial Families

A single trial family can hide bias. Compare at least two physically motivated families when possible. For example:

• a Gaussian and an exponential tail for a bound state;
• an uncorrelated product state and a correlated product state;
• a small analytic family and a finite-basis Rayleigh-Ritz result;
• a perturbative estimate and a variational estimate in their common regime.

Agreement between independent approximations is not proof, but disagreement is valuable information.

Examples

For the harmonic oscillator, a Gaussian trial state with adjustable width contains the exact ground state:

$$\psi_a(x) = \left(\frac{a}{\pi}\right)^{1/4} e^{-a x^2/2}.$$

The full normalization, energy expectation, width optimization, and comparison with a deliberately mismatched exponential family are worked through in Variational Estimate for the Harmonic Oscillator.

For Coulombic atoms, exponential orbitals are natural because hydrogenic bound states decay exponentially:

$$\phi_\zeta(r) = \left(\frac{\zeta^3}{\pi}\right)^{1/2} e^{-\zeta r}.$$

For two-particle systems, a product state may miss correlation. A simple correlated improvement can multiply by a factor depending on the separation:

$$\Psi(\mathbf r_1,\mathbf r_2) = \phi(\mathbf r_1)\phi(\mathbf r_2) F(|\mathbf r_1-\mathbf r_2|).$$

The factor $F$ can encode short-distance avoidance or long-distance correlation.

Common Mistakes

• Optimizing a trial function that violates boundary conditions.
• Forgetting to normalize or to include the normalization denominator.
• Choosing a trial family with the wrong symmetry.
• Adding parameters without checking whether they are identifiable or stable.
• Judging the state only by energy and not by observables or limiting behavior.
• Comparing two variational estimates as if the lower one is automatically accurate.

For tests that distinguish inadmissibility, ansatz bias, optimization failure, and numerical error, continue to Common Variational Pitfalls.

Exercises

1. Why is a Gaussian trial function natural for a harmonic oscillator but not automatically natural for a Coulomb tail?

Solution

The harmonic oscillator ground state is Gaussian because the potential is quadratic and the exact solution has a Gaussian tail. Coulomb bound states decay exponentially at large radius. A Gaussian may still be used as an approximation, but it builds in the wrong asymptotic tail for a Coulomb problem.

2. A Hamiltonian has parity symmetry and a nondegenerate ground state. Why should the ground-state trial wavefunction usually be chosen even?

Solution

For a parity-invariant Hamiltonian, eigenstates can be chosen with definite parity. A nondegenerate ground state in a typical one-dimensional bound problem has no node and is even. An odd trial state would be orthogonal to the even ground state and would instead target an excited sector.

3. What is the variational risk of adding a correlation factor $F(r_{12})$ with many free parameters?

Solution

The energy may decrease, but the optimization can develop local minima, redundant directions, or numerical instability. The parameters may also fit energy while giving unreliable observables. Additional flexibility should be accompanied by convergence checks and physical interpretation.

References

• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• A. Szabo and N. S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, Dover, 1996.