Variational Method in Quantum Mechanics

In quantum mechanics, finding exact solutions to the Schrödinger equation is only possible for a handful of simple systems like the hydrogen atom. For anything more complex—from a simple helium atom to a large molecule—we must rely on approximation methods. The variational method is one of the most powerful and widely used tools for this task, providing a systematic way to calculate upper bounds on ground state energies using intelligent guesses. Its principle is beautifully simple: any reasonable guess for the wavefunction will yield an energy expectation value that is greater than or equal to the true ground state energy. This method forms the computational backbone for much of modern quantum chemistry and condensed matter physics, allowing us to probe systems where exact solutions are forever out of reach.

The Variational Principle

The foundation of the method is the variational principle. It states that for any system with a time-independent Hamiltonian $\hat{H}$, the energy expectation value calculated with any normalized trial wavefunction ${\psi }_{\text{trial}}$ is always greater than or equal to the true ground state energy $E_0$.

$$⟨E⟩=\frac{⟨{\psi }_{\text{trial}}|\hat{H}|{\psi }_{\text{trial}}⟩}{⟨{\psi }_{\text{trial}}|{\psi }_{\text{trial}}⟩}\ge E_0$$

The angled brackets $⟨⟩$ denote the integral over all coordinates. The power of this statement is its generality: your trial function does not need to be close to the true ground state wavefunction. If it is normalized, the calculated energy $⟨E⟩$ is guaranteed to be an upper bound to $E_0$. A lower $⟨E⟩$ means a better approximation. The equality holds only if ${\psi }_{\text{trial}}$ is identical to the true ground state wavefunction ${\psi }_0$. Think of it like trying to find the lowest point in a bowl by dropping marbles (trial functions) onto its surface; no marble can ever settle below the true bottom.

Constructing and Optimizing Trial Wavefunctions

The practical application of the method involves constructing a trial wavefunction with adjustable variational parameters. These are symbols, often denoted by Greek letters like $\alpha$ or $\beta$, built into the functional form of ${\psi }_{\text{trial}}$. Once constructed, you compute the energy expectation value $⟨E(\alpha ,\beta ,...)⟩$, which becomes a function of these parameters.

The next step is optimization. You minimize the expectation value with respect to each parameter by taking derivatives and setting them to zero: $$\frac{\partial ⟨E⟩}{\partial \alpha }=0,\frac{\partial ⟨E⟩}{\partial \beta }=0,...$$ The parameter values that satisfy these equations yield the lowest possible energy upper bound for that specific form of trial wavefunction. The quality of your final result depends entirely on the insight used to choose this functional form. A good trial wavefunction incorporates the essential physics of the problem, such as correct symmetry and asymptotic behavior.

Application to the Helium Atom

The helium atom is the classic textbook example. Its Hamiltonian includes kinetic energy for two electrons and three potential terms: the attraction of each electron to the nucleus (charge $Z=2$) and the repulsion between the electrons. The electron-electron repulsion term makes the Schrödinger equation unsolvable exactly.

A simple but effective trial wavefunction models each electron as being in a hydrogen-like 1s orbital, but with an effective nuclear charge $\zeta$ (zeta) that acts as the variational parameter. This accounts for the fact that each electron "shields" the other from the full charge of the nucleus. The trial function is: $${\psi }_{\text{trial}}({\mathbf{r}}_1,{\mathbf{r}}_2)=\frac{{\zeta }^3}{\pi a_0^3}{e}^{-\zeta (r_1+r_2)/a_0}$$ where $a_0$ is the Bohr radius.

You then compute $⟨E(\zeta )⟩$, which includes terms for kinetic energy, electron-nucleus attraction, and electron-electron repulsion. Minimizing this function with respect to $\zeta$ yields an optimal value ${\zeta }_{\text{opt}}\approx 1.69$ (less than the true nuclear charge of 2 due to shielding). The corresponding energy is approximately -77.5 eV, which is above the true ground state energy (-79.0 eV) but a significant improvement over a calculation that ignores electron correlation entirely. More sophisticated trial functions that include explicit dependence on the inter-electron distance $r_{12}$ can model the Coulomb hole—the reduced probability of electrons being close together due to repulsion—and yield results within 0.1% of the experimental value.

Extension to Molecular Systems: The Hydrogen Molecule Ion

The variational method also shines in molecular quantum mechanics. Consider the hydrogen molecule ion, $H_2^{+}$, a one-electron system where the electron is shared between two protons. The exact solution is possible but cumbersome; the variational approach provides clear physical insight.

A logical trial wavefunction is a linear combination of atomic orbitals (LCAO). For the ground state, we combine two hydrogen 1s orbitals, one centered on each proton (A and B): $${\psi }_{\text{trial}}=c_A{\psi }_{1s}(r_A)+c_B{\psi }_{1s}(r_B)$$ By symmetry, the electron density should be equal about each proton, so $c_A=\pm c_B$. This leads to two molecular orbitals: a bonding orbital ($c_A=c_B$) and an antibonding orbital ($c_A=-c_B$).

Using the trial function and the full Hamiltonian (including electron kinetic energy and attraction to both protons), you compute the energy expectation value $⟨E⟩$ as a function of the internuclear separation $R$. The variational principle not only gives an upper bound for the energy at each $R$ but also allows you to find the equilibrium bond length (the $R$ that minimizes $⟨E⟩$) and the bond dissociation energy. This simple LCAO model successfully predicts that $H_2^{+}$ is stable (has a bonding ground state), demonstrating the quantum mechanical origin of the chemical bond.

Common Pitfalls

Poor Choice of Trial Function: The most critical error is selecting a trial wavefunction that lacks the necessary symmetry or boundary conditions of the true system. For example, a trial function for a particle in a box that does not go to zero at the walls is fundamentally flawed and will yield poor results, regardless of parameter optimization. Always ensure your trial function respects the physical constraints of the problem.

Overlooking Normalization: The variational principle is strictly proven for normalized wavefunctions. If you forget to normalize ${\psi }_{\text{trial}}$ before computing $⟨E⟩$, or if your parameter optimization changes the normalization constant, your energy result is not guaranteed to be an upper bound. Always either use normalized functions or keep the explicit normalization denominator in the expectation value formula.

Misinterpreting the Result: The variational method provides an upper bound for the ground state energy. It does not provide a lower bound, nor does it guarantee that your optimized wavefunction is a good approximation for excited states. Furthermore, a low energy from a sophisticated trial function is strong evidence of its accuracy, but the only way to know the exact $E_0$ is to have a method that gives a lower bound to complement the variational upper bound.

Ignoring More Powerful Parameterizations: In the helium example, using a fixed hydrogen-like orbital without the effective charge $\zeta$ is a valid but poor trial function. Failing to include parameters that capture key correlations (like the $r_{12}$ term for electron repulsion) limits the accuracy you can achieve. The art of the method lies in choosing a parameterization that balances computational complexity with physical insight.

Summary

• The variational principle guarantees that the energy expectation value calculated from any normalized trial wavefunction is an upper bound to the true ground state energy $E_0$, with equality only if the trial function is the exact ground state.
• Practical application involves constructing a trial wavefunction with built-in variational parameters, computing the energy $⟨E⟩$ as a function of these parameters, and then minimizing it to find the best approximation possible within that chosen functional form.
• For the helium atom, a simple trial function using hydrogen-like orbitals with an effective nuclear charge $\zeta$ as a parameter accounts for electron shielding and yields a good first approximation to the ground state energy, which can be improved by adding electron correlation terms.
• For molecular systems like the hydrogen molecule ion ($H_2^{+}$), the variational method applied to a linear combination of atomic orbitals (LCAO) successfully predicts bond stability, equilibrium bond length, and provides a quantum mechanical model of chemical bonding.
• The power and accuracy of the method are entirely dependent on the physical insight used to select the trial wavefunction; a well-chosen function with few parameters can often yield better results than a poorly chosen one with many.