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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

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

 \begin{displaymath}
\frac{ < \tilde{\Psi} \vert {\cal H} \tilde{\Psi} > } {<\tilde{\Psi} \vert
\tilde{\Psi} >} \geq E_{0},
\end{displaymath} (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.



 
next up previous contents
Next: Hylleraas type basis expansion Up: Muonic molecule and the Previous: Improved Adiabatic methods