Linear Variational Method and Applications

Instead of optimizing a non-linear parameter, expand the trial function as a linear combination of fixed basis functions $\{|\phi_i\rangle\}$:

The coefficients $c_i$ are the variational parameters. The energy functional is:

where $H_{ij} = \langle\phi_i|\hat{H}|\phi_j\rangle$ is the Hamiltonian matrix and $S_{ij} = \langle\phi_i|\phi_j\rangle$ is the overlap matrix.

Minimizing $E$ with respect to $c_i^*$ (treating $c_i$ and $c_i^*$ as independent) leads to the generalized eigenvalue equation:

In component form:

Non-trivial solutions exist when:

This secular equation gives $N$ eigenvalues $E_1 \leq E_2 \leq \cdots \leq E_N$. If the basis functions are orthonormal ($S_{ij} = \delta_{ij}$), this reduces to the standard eigenvalue problem $\mathbf{H}\vec{c} = E\vec{c}$.

Important property: The lowest eigenvalue $E_1$ is an upper bound on the true ground state energy. The second eigenvalue $E_2$ is an upper bound on the first excited state, and so on. This is the Hylleraas-Undheim-MacDonald theorem.

The Huckel method applies the linear variational method to $\pi$-electron systems in conjugated molecules. For a molecule with $N$ carbon atoms contributing $p_z$ orbitals:

The Huckel approximations are:

Example: Benzene ($C_6H_6$). The $6 \times 6$ Huckel matrix gives eigenvalues:

This gives the energy levels $\alpha + 2\beta$ (1-fold), $\alpha + \beta$ (2-fold), $\alpha - \beta$ (2-fold), $\alpha - 2\beta$ (1-fold), reproducing the observed stability of benzene through delocalization energy.

The variational principle directly gives bounds on the ground state. For excited states, we use orthogonality constraints:

Method 1: Explicit Orthogonality

If we know (or have a good approximation to) the ground state $|\psi_0\rangle$, restrict the trial function to be orthogonal to it:

Method 2: Symmetry

If the ground state has even parity, use an odd trial function for the first excited state. Orthogonality is automatic by symmetry:

Method 3: Linear Variational Method

The Ritz method automatically gives upper bounds on multiple energy levels simultaneously. The $k$-th eigenvalue provides an upper bound on $E_{k-1}$.

The variational method underpins virtually all of modern computational quantum mechanics:

• Hartree-Fock method: Variational optimization of a single Slater determinant. Each electron moves in the average field of all others. Foundation of quantum chemistry.
• Configuration Interaction (CI): Linear variational method using multiple Slater determinants. Systematically improvable but computationally expensive.
• Density Functional Theory (DFT): Variational principle for the electron density rather than the wave function. Scales well to large systems.
• Variational Monte Carlo (VMC): Evaluate variational integrals using stochastic sampling. Can handle explicit electron-electron correlation.
• Tensor network methods: Variational optimization over matrix product states (DMRG) for strongly correlated systems.
• Variational quantum eigensolver (VQE): Hybrid quantum-classical algorithm where a quantum computer evaluates the energy and a classical optimizer adjusts parameters.