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.

Required background. The Variational Principle gives the energy bound, and Rayleigh–Ritz explains linear trial spaces. Symmetry Constraints identify allowed sectors.

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 compares Slater, Jastrow, paired, projected, matrix-product, and neural ansätze.

Admissible trial wavefunctions

A finite variational energy requires membership in the Hamiltonian’s quadratic-form domain, which is often larger than its operator domain. For a one-dimensional Schrödinger operator with Dirichlet walls, use continuous functions that vanish at the walls, have a square-integrable weak derivative, and a well-defined potential form. Evaluate the kinetic energy as $\hbar^2\int|\psi'|^2/(2m)$.

A kink is therefore not automatically forbidden: $e^{-a|x|}$ has finite kinetic energy on the line even though its second derivative includes a delta distribution. A discontinuous step, in contrast, is not in this kinetic form domain. Do not integrate a classical second derivative piecewise and discard the singular contribution.

Derivative matching and cusp conditions of an exact eigenfunction are valuable design information, but not always admissibility conditions for a variational trial function. For example, smooth Gaussians are admissible for a Coulomb Hamiltonian despite failing its exact nuclear cusp. For Robin boundaries or point interactions, include the corresponding boundary or contact terms in the energy form.

Respect Symmetries

If the Hamiltonian commutes with a symmetry, choose trial states in the appropriate symmetry sector. For a real scalar Schrödinger operator on a connected, parity-symmetric one-dimensional domain, a nondegenerate ground state is nodeless and even. Parity symmetry alone does not establish this for arbitrary spinor or nonlocal Hamiltonians. 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 decay times appropriate powers 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. With $r$ a length and $\zeta>0$ an inverse length, a normalized example is

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

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.

Exercises

A cusp can be admissible

For $\psi_a(x)=\sqrt a\,e^{-a|x|}$, compute $\int|\psi_a'|^2dx$ and $\langle X^2\rangle$. Is this an admissible oscillator trial state?

Solution

The integrals are $a^2$ and $1/(2a^2)$. Hence $E(a)=\hbar^2a^2/(2m)+m\omega^2/(4a^2)$ is finite. The kink excludes this state from the oscillator’s operator domain, but not its form domain; the variational bound applies to this energy form. The oscillator example compares its optimized energy with the Gaussian result.

Project a broken-parity trial state

For $[H,P]=0$ with $P^2=I$, split a normalized trial state into its even and odd components. Show that its energy is a weighted average of the two normalized sector energies.

Solution

Let $\psi_\pm=(I\pm P)\psi/2$ and $w_\pm=\|\psi_\pm\|^2$. Orthogonality and $[H,P]=0$ give $E=w_+E_++w_-E_-$ with $w_++w_-=1$. At least one nonzero component has energy no higher than the original state. The theorem alone does not identify which sector; knowledge of the target ground-state symmetry does.

References

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