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Solution of Hartree-Fock equations

Solution of Hartree-Fock equations. Take the initial single-particle hamiltonian h  Find eigenvectors of h : Take the A lowest-energy eigenvectors The largest s.p. energy is the Fermi level Find the density matrix Calculate  Go back to 2. Note that the HF equation

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Solution of Hartree-Fock equations

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  1. Solution of Hartree-Fock equations Take the initial single-particle hamiltonian h Find eigenvectors of h: Take the A lowest-energy eigenvectors The largest s.p. energy is the Fermi level Find the density matrix Calculate  Go back to 2. Note that the HF equation [h,]=0 is met by construction

  2. Hartree Fock stability condition HF minimum should correspond to the positive second derivative of the energy. This implies that independent variations are only in the ph channel (pp and hh matrix elements of r(1) and r(2) vanish) The HF stability matrix (must be positively defined)

  3. Constrained Hartree-Fock Often, we are interested not only in the local HF minima but also in the whole potential energy surface (PES) Routhian (cranked Hamiltonian) EHF q Collective coordinate • Several ways of solving the CHF equations • (linear constraint, quadratic constraint, gradient method) • Usually: many-dimensional surfaces • Static picture - fluctuations are not included!

  4. Self-consistent HF symmetries • If a certain symmetry for the solution is expected, one can start with an initial density which has this symmetry • If one starts with a certain symmetry, one will always stay within this symmetry (if the deepest minimum is deformed, one will never get to it starting with a spherically symmetric density matrix) • The above discussion can be generalized to density dependent interactions satisfying the condition:

  5. If we have a solution with a (spontaneously) broken symmetry S… …then every transformed (rotated) state is also a solution of the HF equations! Goldstone Theorem If a symmetry is broken, there appears a zero-energy mode (Goldstone boson!)

  6. Hartree-Fock-Bogoliubov A. Two-body (density-dependent) interaction: B. Variational principle: C. Trial wave functions: product states general Bogoliubov transformation HFB wave function is the quasuparticle vacuum • HFB - quasiparticles incorporated • into the HF formalism • HFB wave function - the most general • product wave function consisting of independently • moving quasiparticles (in HF: particles) • Selfconsistent description of coupling between p-h • an p-p channels

  7. HFB - density matrix and pairing tensor HFB density matrix HFB pairing tensor Generalized density matrix Eigenvalues are 0 or 1 (thus defining occupations of quasiparticle states) Independent quasiparticle Hamiltonian (HHFB) Quasiparticle interaction

  8. HFB equations HFB Hamiltonian HF Hamiltonian Selfconsistent HF field Selfconsistent pair field HFB equations Complicated eigenvalue problem • HFB equations treat p-h and p-p on the same footing • D is a state-dependent field. In general, it depends on • density and has a kinetic term • The generalized density matrix and the HFB Hamiltonian • can be diagonalized simultaneously • Fermi level determined from the particle number • equation • Often it is convenien to express the HFB equations • in the canonical basis

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