Three-photon excitation in fluorescence microscopy

Abstract

In density-functional calculations of total energies and forces, the fractional occupation numbers used for the discrete sampling of the Brillouin zone are often generated by “finite-temperature” broadening schemes with a finite broadening parameter σ. It is shown that if the broadening parameters are chosen properly, the results obtained with different broadening schemes, e.g., with Fermi-Dirac broadening or with Gaussian broadening, will differ only by terms of the order σ4. As a further result, it is demonstrated that if the ground-state energy for σ=0 is approximated by 1/2[F(σ)+E(σ)], where E(σ) and F(σ) denote the energy and a suitably defined “free energy,” the correction terms are of order σ4 rather than of order σ3 as is commonly assumed.