optimize_hubbard_time_step Subroutine

    subroutine optimize_hubbard_time_step()
        ! routine to set the optimal time-step for the hubbard model,
        ! where the pgens and matrix elements are set
        use SystemData, only: uhub, bhub, nel, omega, treal

        real(dp) :: p_elec, p_hole, time_step, mat_ele, death_prob

        ! first of all write down the notes here..

        ! the first implementation should just use the non-optimized values
        ! from the current implementation..
        ! afterwards i want to optimize especially the real-space hubbard
        ! implementation, since this is done really inefficient right now!
        ! although it is just wrong how it is done currently in the
        ! real-space hubbard model.. damn..
        ! are my results still valid??

        if (tReal) then
            ! in the real-space hubbard model the electron is picked
            ! uniformly
            p_elec = 1.0_dp / real(nel, dp)
            ! and the hole should be picked from the neighbors:
!             p_hole = 1.0_dp / (2.0_dp * real(dims, dp))

            p_hole = min(1.0_dp / real(nBasis / 2 - nOccAlpha, dp), &
                         1.0_dp / real(nBasis / 2 - nOccBeta, dp))

            ! and the matrix element is always just -t
            mat_ele = bhub

            ! so the off-diagonal time-step should be
            time_step = p_elec * p_hole / abs(mat_ele)

            ! but one has to also consider the diagonal part for the
            ! death probability (to limit it to < 1)
            ! there can be at most half the number of electrons double
            ! occupied sites!
            death_prob = Uhub * nel / 2.0_dp

            print *, "optimized time-step for real-space hubbard: ", time_step
            if (t_trans_corr_2body .or. t_trans_corr) then
                print *, "BUT: transcorrelated Hamiltonian used! "
                print *, " so matrix elements are not uniform anymore!"
            end if

        else
            ! in the momentum space hubbard the electrons fix the the holes
            ! to be picked! atleast if one is picked the 2nd hole is also
            ! chosen!

            ! for the 2-body transcorrelation we have to consider the
            ! different types of excitations and find the minimum tau
            ! for it..
            ! but also the matrix elements are not uniform.. so one
            ! would need to find the worst H_ij/pgen ratio, which is
            ! no feasible here.. thats actually what the tau-search is
            ! doing..

            p_elec = 2.0_dp / real(nOccAlpha * nOccBeta, dp)

            ! and the holes are all the remaining possible ones
            p_hole = 1.0_dp / real(nbasis - nel, dp)

            ! the matrix element is always U (or U/2 ?? ) check!
            mat_ele = uhub / omega

            time_step = p_elec * p_hole / abs(mat_ele)

            ! the diagonal element is -2t cos(k_vec)
            death_prob = 2.0_dp * abs(bhub)

            print *, "optimized time-step for the momentum space hubbard: ", time_step
            if (t_trans_corr_2body .or. t_trans_corr) then
                print *, "BUT: transcorrelated Hamiltonian used! "
                print *, " so matrix elements and pgens are not uniform anymore!"
                print *, " thus TAU is just a rough estimate! "
            end if

        end if

        ! with this stuff i can make the optimal time-step!
        ! but for the real-space hubbard i have to first implement the
        ! better choosing of only neighboring holes!
        if (tau > time_step) then
            print *, "initial guessed or input-provided time-step too large!"
        else
            print *, "initial guessed or input-provided time-step too small!"
        end if
        call assign_value_to_tau(clamp(0.1_dp * time_step, min_tau, max_tau), 'Initial guess hubbard')
        print *, "setting time-step to 0.1 * optimal: ", tau
        print *, "and turning tau-search OFF!"

        tau_search_method = possible_tau_search_methods%OFF
        t_fill_frequency_hists = .false.
        t_test_hist_tau = .false.

    end subroutine optimize_hubbard_time_step