Quantum espresso

Quantum espresso

(Parte 6 de 7)

As a consequence of symmetry, roto-translated KS orbitals are KS orbitals with the rotated Bloch vector: Sψi,k(r) ≡ ψi,k(R−1r − f) = ψi,Rk(r). Where, strictly speaking, the resulting wave-function at Rk does not necessarily have the same band index as the original one but could be some unitary transformation of states at Rk that share with it the same singleparticle eigenvalue. Since quantities of physical interest are invariant for unitary rotations among degenerate states this additional complication has no effect on the final result.

This is the basis for the symmetrization procedure used in PWscf. One introduces a non-symmetrized charge density (labeled by superscript (ns)) calculated on the irreducible BZ (IBZ):

The factors wk (“weights”) are proportional to the number of vectors in the star (i.e. inequivalent k vectors among all the {Rk} vectors generated by the point-group rotations) and are normalized to 1: ∑ k∈IBZ wk = 1. Weights can either be calculated or deduced from the literature on the special-point technique[89, 90]. The charge density is then symmetrized as:

where the sum runs over all Ns symmetry operations. The symmetrization technique can be extended to all quantities that are expressed as sums over the BZ. Hellmann-Feynman forces Fs on atom s are thus calculated as follows:

QUANTUM ESPRESSO 29 where S−1(s) labels the atom into which the s−th atom transforms (modulo a lattice translation vector) after application of S−1, the symmetry operation inverse of S. In a similar way one determines the symmetrized stress, using the rule for matrix transformation under a rotation:

RαγRβδ σ (ns) γδ . (A.28)

The PHonon package supplements the above technique with a further strategy. Given the phonon wave-vector q, the small group of q (the subgroup Sq of crystal symmetry operations that leave q invariant) is identified and the reducible representation defined by the 3Nat atomic displacements along cartesian axis is decomposed into nirr irreducible representations

j , j = 1,, nirr. The dimensions of the irreducible representations are small,

with νj ≤ 3 in most cases, up to 6 in some special cases (zone-boundary wave-vectors q in nonsymmorphic groups). Each irrep, j, is therefore defined by a set of νj linear combinations of atomic displacements that transform into each other under the symmetry operations of the small group of q. In the self-consistent solution of the linear response equations, only perturbations associated to a given irrep need to be treated together and different irreps can be solved independently. This feature is exploited to reduce the amount of memory required by the calculation and is suitable for coarse-grained parallelization and for execution on a Grid infrastructure [192].

The wavefunction response, ∆ψ (j,α)

α = 1,, νj labels different partners of the given irrep), is then calculated. The lattice-

periodic unsymmetrized charge response, ∆ρ (ns) q,j,α(r), has the form:

wkψ∗ k,i(r)∆ψ where the notation IBZ(q) indicates the IBZ calculated assuming the small group of q as symmetry group, and the weights wk are calculated accordingly. The symmetrized charge response is calculated as e−iqf νj∑

where D(Sq) is the matrix representation of the action of the symmetry operation Sq ≡ {R|f} for the j−th irrep γ (q)j . At the end of the self-consistent procedure, the force constant matrix

Csα,tβ(q) (where s, t label atoms, α, β cartesian coordinates) is calculated. Force constants at all vectors in the star of q are then obtained using symmetry:

where S ≡ {R|f} is a symmetry operation of the crystal group but not of the small group of q.

QUANTUM ESPRESSO 30

Appendix A.5. Fock exchange

Hybrid functionals are characterized by the inclusion of a fraction of exact (i.e. non-local) Fock exchange in the definition of the exchange-correlation functional. For a periodic system, the Fock exchange energy per unit cell is given by:

where an insulating and non magnetic system is assumed for simplicity. Integrals and wavefunction normalizations are defined over the whole crystal volume, V = NΩ (Ω being the unit cell volume), and the summations run over all occupied bands and all N k-points defined in the BZ by Born-von Kármán boundary conditions. The calculation of this term is performed exploiting the dual-space formalism: auxiliary codensities, ρk′,v′

computed in real space and transformed to reciprocal space by FFT, where the associated electrostatic energies are accumulated. The application of the Fock exchange operator to a wavefunction involves additional FFTs and real-space array multiplications. These basic operations need to be repeated for all the occupied bands and all the points in the BZ grid. For this reason the computational cost of the exact exchange calculation is very high, at least an order of magnitude larger than for non-hybrid functional calculations.

In order to limit the computational cost, an auxiliary grid of q-points in the BZ, centered at the Γ point, can be introduced and the summation over k′ be limited to the subset k′ = k+q. Of course convergence with respect to this additional parameter needs to be checked, but often a grid coarser than the one used for computing densities and potentials is sufficient.

The direct evaluation of the Fock energy on regular grids in the BZ is however problematic due to an integrable divergence that appears in the q → 0 limit. This problem is addressed resorting to a procedure, first proposed by Gygi and Baldereschi [193], where an integrable term that displays the same divergence is subtracted from the expression for the exchange energy and its analytic integral over the BZ is separately added back to it. Some care must still be paid [177] in order to estimate the contribution of the q = 0 term in the sum, which contains a 0/0 limit that cannot be calculated from information at q = 0 only. This term is estimated [177] assuming that the grid of q-points used for evaluating the exchange integrals is dense enough that a coarser grid, including only every second point in each direction, would also be equally accurate. Since the limiting term contributes to the integral with different weights in the two grids, one can extract its value from the condition that the two integral give the same result. This procedure removes an error proportional to the inverse of the unit cell volume Ω that would otherwise appear if this term were simply neglected.

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(Parte 6 de 7)

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