Variational principle

The simplest of the Quantum Monte Carlo methods is the variational method (VMC). The idea of this method is to use an approximate wave function $\psi _T$ (variational or trial wave function) and then by sampling the probability distribution

$$p({\bf R})~=~\vert\psi_T({\bf R})\vert^2$$ (2.1)

calculate averages of physical quantities. It is easy to show that the average

$$E_T = \frac{\langle\psi_T\vert\hat H\vert\psi_T\rangle}{\langle\psi_T\vert\psi_T\rangle} \ge E_0$$ (2.2)

gives an upper bound to the ground-state energy. By minimizing the variational energy with respect to the external parameters one can optimize the wave function within the given class of wave functions considered.

Importantly, the variational principle also applies to excited states. For a trial wave function $\psi _T$ with a given symmetry, the variational estimate provides an upper bound to the energy of the lowest excited state of the Hamiltonian $\hat H$ with that symmetry.