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Generator Coordinate Method (GCM) in Molecules and Atomic Nuclei: Variational Principle and Diagonalization

The Generator Coordinate Method (GCM) in molecules and atomic nuclei employs a variational principle leading to the Hill-Wheeler equation and Hamiltonian diagonalization. The norm kernel, crucial for the GCM states, cannot be inverted due to zero eigenvalues. However, dealing with zero eigenvalue states leads to GCM states with zero norms, allowing a focus on states with positive norms. The symmetric orthogonalization of states and introduction of natural states form a complete set in the space of weight functions, enabling a coupling between intrinsic states and dynamics determined by the choice of collective coordinates. The GCM+HF+BCS approach, exemplified by Bender, Flocard, and Heenen, combines deformed mean fields, GCM eigenstates with HF+BCS for collective wave functions. The Gaussian Overlap Approximation, Collective Hamiltonian, and Collective Mass provide insights into zero-point energy corrections and collective levels.

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Generator Coordinate Method (GCM) in Molecules and Atomic Nuclei: Variational Principle and Diagonalization

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  1. Generator Coordinate Method (GCM) In molecules In atomic nuclei Variational principle leads to the Hill-Wheeler equation Hamiltonian kernel Norm kernel Diagonalization of the Hamiltonian in the nonorthogonal “basis” of the generator states. The norm kernel cannot be inverted since it has zero eigenvalues!

  2. Again, we have to deal with zero eigenvalue states • Fortunately, the corresponding states lead to GCM states f0 with zero norm! • Consequently, we can restrict ourselves to states with nk>0. Symmetric orthogonalization • Since N is a norm, its eigenvalues are non-negative (nk0) • The functions uk form a complete set in the space of weight functions f(Q) Let us introduce the basis of natural states: • Those states are orthogonal • They span a sub Hilbert space called “collective” subspace (the smallest Hilbert space containing all the generating states

  3. GCM eigenstates weight functions collective vawe functions • Microscopic • Based on variational principle • Allows for a coupling between intrinsic states • Dynamics determined by the choice of Q’s • A useful point for further approximations (e.g., GOA, CSE, etc.) • Transformation to the lab system still necessary, unless the Euler (or gauge) angles are taken as collective coordinates

  4. An example: projected GCM+HF+BCS M. Bender ,H. Flocard ,P.-H. Heenen nucl-th/0305021 Usual starting point: deformed mean fields PESs and GCM eigenstates HF+BCS Projected Collective wave functions ground state SD ND

  5. The Gaussian Overlap Approximation For small values of s, one obtains: Collective momentum operator If the fluctuation in collective momentum is large, one ends up with the Gaussian shape. (Well justified for heavy systems.) Collective Hamiltonian Collective mass (collective inertia) Collective potential Zero-point energy

  6. An example: GOA+HF(B) Gogny calculations J. Libert, M. Girod, and J.-P. Delaroche Phys. Rev. C 60 , 054301 (1999) zero-point energy corrections collective levels collective wave functions

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