Variational Method

Chem 3240 · Lecture 6.1

Davit Potoyan

Why We Need Approximations

  • The Schrodinger equation is solvable exactly only for a handful of systems.
  • The hydrogen atom is the most complex atom with a closed form.
  • Helium and any multi-electron system are intractable.
  • We need a systematic way to approximate and to rank approximations.

The Core Idea

  • Begin with an educated guess: a trial wavefunction \(|\phi\rangle\).
  • The guess carries adjustable parameters.
  • Tune the parameters to minimize the energy, driving the answer toward the exact value.

The Variational Theorem

For any trial function, the computed energy is an upper bound.

\[E_{\phi} = \frac{\langle \phi \mid \hat{H} \mid \phi\rangle}{\langle \phi \mid \phi\rangle} \geq E_0\]

  • \(\hat{H}\) is the Hamiltonian of the problem.
  • \(|\phi\rangle\) is the trial function with unknown parameters.
  • \(E_0\) is the true ground-state energy (usually unknown).

What the Theorem Buys Us

  • The ground state is the lowest possible energy.
  • Minimizing the energy functional gives the best answer for a given trial form.
  • More parameters mean more handles, hence more accuracy.
  • Equality holds only when \(\phi\) equals the exact ground state.

Example: Particle in a Box

Trial function obeying \(\psi_t(0) = \psi_t(a) = 0\):

\[\psi_t(x) = \frac{\sqrt{30}}{a^{5/2}}\,x(a - x)\]

  • Plug into \(\hat{H} = -\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2}\) and integrate.

\[E_\phi = \frac{5\hbar^2}{a^2 m} \;\geq\; E_1\]

  • A clean upper bound from a crude parabola guess.

Example: Hydrogen with a Gaussian

  • Trial function \(\phi = e^{-\alpha r^2}\) for the 1s ground state.

\[E_{trial}(\alpha) = \frac{3\hbar^2 \alpha}{2m_e} - \frac{e^2 \alpha^{1/2}}{\sqrt{2}\,\epsilon_0 \pi^{3/2}}\]

  • Minimize: \(\dfrac{\partial E_{trial}}{\partial \alpha} = 0\).

\[E_0 = -0.5\,h, \qquad E_{trial}(\alpha_{min}) = -0.424\,h\]

  • About 15% error. A Gaussian lacks the exponential’s cusp.

The Helium Atom Is Tough

\[\hat{H} = -\frac{\hbar^2}{2m_e}(\Delta_1 + \Delta_2) - \frac{1}{4\pi\epsilon_0}\left(\frac{Ze^2}{r_1} + \frac{Ze^2}{r_2} - \frac{e^2}{r_{12}}\right)\]

  • The electron-electron term \(\dfrac{e^2}{r_{12}}\) blocks any analytic solution.
  • Drop it and \(\hat{H}\) separates into two hydrogen-like atoms.

Helium: From Crude to Variational

  • Independent electrons (\(1s(1)\,1s(2)\)): \(E = -74.8\) eV.
  • Now treat the nuclear charge \(Z\) as a variational parameter: \[E(Z) = \left[Z^2 - \frac{27Z}{8}\right]\frac{e^2}{4\pi\epsilon_0 a_0}\]
  • Minimizing gives \(Z = \dfrac{27}{16} \approx 1.7\) and \(E \approx -77.5\) eV.
  • \(Z < 2\): each electron shields the nucleus. Exact is \(-79.0\) eV.

The Linear Variational Method

  • Expand the trial function in a basis set \(f_n\):

\[\phi(r) = \sum_n^N c_n f_n(r)\]

  • Minimize over the coefficients \(c_n\), not over a functional form.
  • Turns a hard differential equation into linear algebra.
  • This idea underlies Hartree-Fock and modern electronic structure.

The Generalized Eigenvalue Problem

\[\mathbf{H}\mathbf{c} = E\,\mathbf{S}\mathbf{c}\]

\[\begin{bmatrix} H_{11} & H_{12} \\ H_{12} & H_{22} \end{bmatrix} \begin{bmatrix} c_1 \\ c_2 \end{bmatrix} = E \begin{bmatrix} S_{11} & S_{12} \\ S_{12} & S_{22} \end{bmatrix} \begin{bmatrix} c_1 \\ c_2 \end{bmatrix}\]

  • \(H_{ij} = \langle f_i|\hat{H}|f_j\rangle\), \(\quad S_{ij} = \langle f_i|f_j\rangle\).
  • Lowest eigenvalue is the energy estimate; its eigenvector gives the best \(\mathbf{c}\).

The Secular Determinant

A nontrivial solution exists only when

\[\det(\mathbf{H} - E\,\mathbf{S}) = 0\]

  • Solving this determinant yields the variational energies.
  • For an orthonormal basis (\(\mathbf{S} = \mathbf{I}\)) it reduces to \[\mathbf{H}\mathbf{c} = E\mathbf{c}\]
  • Box example with two functions: \(E_\phi = 4.9349\,\tfrac{\hbar^2}{m}\) vs exact \(4.9348\,\tfrac{\hbar^2}{m}\).

Takeaway

Any trial wavefunction gives an energy that is an upper bound on the true ground state, \(E_\phi \geq E_0\). Minimizing over parameters sharpens the estimate, and expanding in a basis set turns the search into a generalized eigenvalue problem \(\mathbf{H}\mathbf{c} = E\mathbf{S}\mathbf{c}\), the foundation of electronic-structure theory.