Variational approaches

More accurate results for the bound state energy levels of muonic molecules have been obtained with variational calculations. Szalewicz reviewed recent progress on the variational approaches [85]. In these approaches, the exact wave function $\Psi$ for the Hamiltonian ${\cal H}$ is approximated by an expansion with a finite set of suitable basis functions

$$\Psi ( \mbox{\boldmath$r_{1},r_{2}$ } ) \sim \tilde{\Psi } ... ...1}^{N} c_{i} \varphi _{i} ( \mbox{\boldmath$r_{1},r_{2}$ } ).$$ (32)

The approximate energy levels $\tilde{E}_{n}$, and approximate eigenfunctions $c_{i} \varphi _{i}$ are found by diagonalizing an $N \times N$matrix. The variational principle states

$$\frac{ < \tilde{\Psi} \vert {\cal H} \tilde{\Psi} > } {<\tilde{\Psi} \vert \tilde{\Psi} >} \geq E_{0},$$ (33)

where E₀ is the ground state energy, hence, all the approximate energies $\tilde{E}_{n}$ are upper bounds to the exact energies. The basis functions, also called trial functions, contain some non-linear parameters, which are to be optimized by repeating the diagonalization and finding $\tilde{E}_{n}$ with a systematic variation of the parameters.

The choice of the basis functions is a major factor which determines the accuracy and convergence of the calculations. Physical intuition and computational experience in choosing the functions can be rewarded with faster convergence and better accuracy.