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We define a set G of (Equation presented)-dimensional vectors (Equation presented) with (Equation presented) and the (Equation presented) component (Equation presented) of (Equation presented) as (Equation presented). All elements of G are orthogonal and eigenvectors to (Equation presented) with the eigenvalue (Equation presented).
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We define a set G of (Equation presented)-dimensional vectors (Equation presented) with (Equation presented) and the (Equation presented) component (Equation presented) of (Equation presented) as (Equation presented). All elements of G are orthogonal and eigenvectors to (Equation presented) with the eigenvalue (Equation presented). The component vectors of the set G will also act as a basis to a (Equation presented)-dimensional subspace (Equation presented) of the (Equation presented). We define a second set U of (Equation presented)-dimensional vectors (Equation presented) with (Equation presented) and the (Equation presented) components (Equation presented) of (Equation presented) as (Equation presented). Like the elements of G, all elements of U are orthogonal and eigenvectors of (Equation presented), but with the eigenvalue (Equation presented). Furthermore, all elements of U are orthogonal to all elements of G. As before for G, the elements of U also act as a basis to an (Equation presented)-dimensional subspace (Equation presented) of the (Equation presented). As all elements of U are orthogonal to G, (Equation presented) and (Equation presented) are orthogonal subspaces of (Equation presented). From this fact together with (Equation presented)+(Equation presented)=(Equation presented) follows now (Equation presented). Obviously, all normal modes with (Equation presented) ((Equation presented)) must belong to the subspace (Equation presented) ((Equation presented)), simultaneously they must also be a basis to the subspaces as the set of all vectors describing the normal modes must be a basis for the (Equation presented). The total number of normal modes belonging to one distinct eigenvalue of (Equation presented) is therefore preserved, independently of the actual choice of the normal modes.
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Theory on the level of the PBEsol exchange-correlation functional as implemented into the quantum espresso suite.
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Control calculations for slabs of 1-5 layers of (Equation presented) and (Equation presented) were performed in the frame of density functional (perturbation) theory on the level of the PBEsol exchange-correlation functional as implemented into the quantum espresso suite. The electrostatic potential between valence electrons and chemically inert atomic core of Mo, W, and S was modeled by the recently proposed GBRV high-throughput pseudopotentials [51] with a cutoff of 80 Ry. All reciprocal space integrations were performed by a discrete (Equation presented)-point sampling of (Equation presented) points in the Brillouin zone. We fully optimized the atomic positions and cell parameters of the considered systems until the residual forces between atoms was smaller than 0.001 eV/Å and the pressure on the cell was lower than 0.001 GPa. Interactions of the slabs with residual periodic images due to the 3D boundary conditions were minimized by maintaining a vacuum layer between slabs of at least 25 Å. After obtaining a converged electronic ground state, the frequencies and atomic displacements of the (Equation presented)-point phonons was calculated through density functional perturbation theory (DFPT).
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Control calculations for slabs of 1-5 layers of (Equation presented) and (Equation presented) were performed in the frame of density functional (perturbation)
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J.-U. Lee, J. Park, Y.-W. Son, and H. Cheong, Nanoscale 7, 3229 (2015). 2040-3364 10.1039/C4NR05785F
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(2015)
Nanoscale
, vol.7
, pp. 3229
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Lee, J.-U.1
Park, J.2
Son, Y.-W.3
Cheong, H.4
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