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107 lines
5.2 KiB
Plaintext
107 lines
5.2 KiB
Plaintext
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A simplified rotational invariant DFT+U method is presently implemented in
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the pw.x code of Quantum ESPRESSO. The implemented functional is the
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one proposed, among others, by S.L.Dudarev et al. in PRB, 57, 1505 (1998).
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A discussion of the method, more details about the current implementation
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and a description of a method to compute the Hubbard U parameter
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are contained in Matteo Cococcioni's PhD thesis at SISSA and in the
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paper "Linear-response approach to the calculation of the effective"
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interaction parameters in the LDA+U method" by M. Cococcioni and
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S. de Gironcoli, PRB 71, 035105 (2005). See also how to compute
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the Hubbard parameters using the HP code whcih is part of Quantum ESPRESSO:
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I. Timrov, N. Marzari, M. Cococcioni, PRB 98, 085127 (2018) and
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PRB 103, 045141 (2021).
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A classical example for DFT+U method is FeO that is incorrectly predicted
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to be a metal by LDA and GGA while it is an insulating antiferromagnetic
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material in real world.
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In this example we use FeO in order to illustrate some of the input
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variables involved in DFT+U calculations.
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Computational parameters (as wfc and density cutoff, k-points grid etc.)
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are set so as to make the example reasonably fast and the results are
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NOT meant to be converged in any sense.
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The first run is just plain LDA calculation for FeO in the rhombohedral
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(antiferromagnetic) cell. There are 2 types of Fe atoms in the input
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because the desired magnetic structure is antiferromagnetic and opposite
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starting_magnetization for the two types is suggested.
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The lda_plus_u flag is enabled in the input and a tiny value is set
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for the Hubbard U of the two Iron types in order to force the code to
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write out the occupation matrices for the localized Fe-3d states without
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affecting the LDA result.
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Looking at the output it is clear that the resulting solution is metallic:
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the "correction for metal" energy term is clearly non zero and the Fermi
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energy falls in the middle of the bands.
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Coming to the occupation of the localized 3d states one can see that
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they are completely filled for the majority spin [spin 1(up) for atom
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3 and spin 2(down) for atom 4] while minority-spin components only are
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partially filled and with FRACTIONAL occupations.
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In the second run of the example a test value for the Hubbard U
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parameter is adopted (for demonstration purposes) and the calculation
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is repeated. Note that for production calculations we suggest to compute
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Hubbard U from first principles using the HP code of Quantum ESPRESSO.
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The DFT+U method is now active and disfavors fractional occupations.
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In spite of that the system still, painfully, converges to a metallic
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solution with similar fractional occupations as the LDA solution.
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This is due to the fact that DFT+U calculations can exhibit---even more
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than spin polarized calculations do---several solutions and one is not
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guaranteed to fall in the desired global minimum automatically.
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We have to live with that and manage to explore several possibilities by
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suggesting to the system different starting points.
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This can be done by setting the starting occupation matrices of the
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system in a user defined way.
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This is done by exploiting the starting_ns_eigenvalue input variable as
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in the third calculation of this example.
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From literature or simple electron counting, one knows that in the
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minority spin component one would like to occupy completely a single state
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leaving the other as empty as possible.
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So, in the third run, by means of the starting_ns_eigenvalue variable,
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one enforces the complete occupation of the fifth eigenvalue of the
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minority spin components of each Fe atomic type.
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Why the fifth eigenvalue? Because from the "standard DFT+U" run we know
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that at the first iteration this is the one that is non-degenerate and
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if occupied completely could lead to an insulating result.
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This calculation converges rather easily to the desired insulating solution.
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In the output we can see that the "correction for metal" energy term is
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essentially zero and Fermi energy falls in a gap.
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A comment about energetics:
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Plain LDA calculation has the lowest energy, as expected, since the +U
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term is a positive defined penalty function added to it and energy can
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only go up.
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Notice however that the "standard DFT+U" calculation, the one with
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fractional occupation of minority-spin levels, has a higher energy than
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the "user defined ns" one, where one manages to completely fill the
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desired level. This shows that this later one is indeed the ground state,
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or at least, a better solution of the problem (still higher than plain
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LDA, of course).
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Looking at the output of these calculations one can notice that even in
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the insulating solution obtained starting with user-defined ns matrices,
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many of the minority spin occupations are still fractional while the DFT+U
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method would like them to be either 0 or 1.
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This is because the projector on localized 3d states used in the calculation
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are based on atomic orbitals that are somehow different from the crystal
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wavefunctions. So some "spurious" 3d level occupation comes from Oxygen
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2s and 2p states that protrude toward Iron sites.
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This is not wrong in general, the important thing is to be consistent
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and use of the U parameter appropriate for the chosen projector.
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See PP/examples/example06 for a calculation using Wannier functions.
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