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We present a new optimization method and one of its applications in quantum chemistry. The problem is to find the most stable configurations of sodium clusters. We have to minimize a weighted sum of eigenvalues of the Hamiltonian operator which corresponds to the energy of the molecules. According to the Rellich theorem, the eigenvalues/eigenvectors are analytic when the matrix follows an analytic path. ; Using automatic differentiation, we compute higher-order derivatives of the matrix. The eigenvalue/eigenvector derivatives are calculated by solving several factorized linear systems. The higher-order derivative method is used to obtain an explicit solution of the cost function (Taylor's expansion). ; This approach leads to a new global optimization algorithm whose idea is as follows: if we apply the optimality condition to the approximated cost function, we obtain a polynomial equation. We find its roots. The "best" zeros are used to restart the algorithm with a new search direction. ; This algorithm presents a natural way of parallelization: computation of Taylor's expansion and its minima in many directions can be performed independently. This method applied to our chemical problem gives very promising results. All the lowest energies of small sodium clusters have been found. This optimization algorithm can also be applied in dynamics of mechanical structures.