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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
for the Hamiltonian
is
approximated by an expansion with a finite set of suitable basis functions

(32) 
The approximate energy levels
,
and approximate
eigenfunctions
are found by diagonalizing an
matrix.
The variational principle states

(33) 
where E_{0} is the ground state energy,
hence, all the approximate energies
are upper bounds
to the exact energies. The basis functions, also called trial functions,
contain some nonlinear parameters, which are to be optimized by repeating
the diagonalization and finding
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: Hylleraas type basis expansion
Up: Muonic molecule and the
Previous: Improved Adiabatic methods