Errors in Hellmann-Feynman forces due to occupation-number broadening and how they can be corrected

Abstract

In ab initio calculations of electronic structures, total energies, and forces, it is convenient and often even necessary to employ a broadening of the occupation numbers. If done carefully, this improves the accuracy of the calculated electron densities and total energies and stabilizes the convergence of the iterative approach towards self-consistency. However, such a broadening may lead to an error in the calculation of the forces. Accurate forces are needed for an efficient geometry optimization of polyatomic systems and for ab initio molecular dynamics ͑MD͒ calculations. The relevance of this error and possible ways to correct it will be discussed in this paper. The first approach is computationally very simple and in fact exact for small MD time steps. This is demonstrated for the example of the vibration of a carbon dimer and for the relaxation of the top layer of the ͑111͒ surfaces of aluminum and platinum. The second, more general, scheme employs linearresponse theory and is applied to the calculation of the surface relaxation of Al͑111͒. We will show that the quadratic dependence of the forces on the broadening width enables an efficient extrapolation to the correct result. Finally the results of these correction methods will be compared to the forces obtained by using the smearing scheme, which has been proposed by Methfessel and Paxton. ͓S0163-1829͑97͒04647-X͔In ab initio electronic structure and total-energy calculations the integrals over the Brillouin zone are commonly replaced by the sum over a mesh of k points.1,2 This approach is very efficient for insulators, but for metallic systems convergence with respect to the number of k points becomes slow. Here the introduction of fractional occupation numbers is a convenient way to improve the k-space integration and in addition to stabilize the convergence in the iterative approach to self-consistency. In these broadening schemes the eigenstates are occupied according to a smooth function, e.g., a Gaussian 3 or the Fermi function 4,5,7,8 at a finite temperature.When a broadening scheme is employed in a density functional theory calculation, the computed electron density of the ground state n 0 (r) does not minimize the functional of the total energy E but the functional of the free energy A: A͓T el ;n͔ϭE͓T el ;n͔ϪT el S͓T el ;n͔, ͑1͒where S denotes the entropy associated with the occupation numbers of the Kohn-Sham orbitals and T el is the broadening parameter. In the case of Fermi broadening 6 we getSince the temperatures commonly used for the broadening are much higher than the physical ones ͑it is convenient to use k B T el ϳ0.1 eV), neither the total energies nor the free energies ͓Eq. ͑1͔͒ are directly meaningful.One way to obtain the ground-state energy at zero temperature is based on the well-known fact 6 that for the freeelectron gas, the quantities A and E depend quadratically on T el . Therefore one can write:

Ab initio Calculations of Phonon Dispersion Relations in Aluminium

“…The influence of the Fermi-smearing parameter on the results was also checked. In agreement with [18], no significant difference was observed by its changing from 0.1 eV to 0.2 eV and the latter value was employed in the calculations. The results appeared to be influenced strongly (in particular the convergence of acoustic branches at Γ point) by the accuracy in evaluation of the HF forces.…”

Nonthermal solid-to-solid phase transitions in tungsten

“…Below, we study the modification of interatomic forces at a given electronic temperature. One can show that, at a given T e , the force F I acting on the atom I at a position R I is given by the derivative of the Mermin functional [46][47][48]:…”

Generation and orientation of organoxenon molecule H–Xe–CCH in the gas phase

“…The irreducible wedge of the first Brillouin zone is sampled by 10 k-points. To improve convergence a Pulay mixing scheme 50,51 has been employed combined with a thermal Fermi broadening of 0.1 eV, extrapolating the energies to zero temperature 52 . To check on the dependence of the obtained results on the exchange-correlation functional, the full PESs have been calculated using two different functionals, the PBE 53 and the RPBE 54 functional.…”