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Plane-wave pseudopotential implementation of explicit integrators for time-dependent Kohn-Sham equations in large-scale simulations
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10.1063/1.4758792
/content/aip/journal/jcp/137/22/10.1063/1.4758792
http://aip.metastore.ingenta.com/content/aip/journal/jcp/137/22/10.1063/1.4758792
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Figures

Image of FIG. 1.
FIG. 1.

Cubic unit cell containing a single Na atom (small circle). To prepare a non-equilibrium initial condition, the Na 3s wave function (represented by the yellow isosurface) has been shifted by (0.32, 0.32, 0.32) Å, see (a). In (b) and (c), two snapshots of the RK4 simulation are shown.

Image of FIG. 2.
FIG. 2.

Total energy E tot (in eV) of the Na atom (at t = 0 fs, the 3s wave function was shifted by (0.32, 0.32, 0.32) Å from its equilibrium position in real space) as a function of time t (in fs). In (a), the Euler scheme (black solid line, Δt = 0.069 as) and the sc100-SOD second-order finite-difference scheme (red solid line, Δt = 0.069 as) are compared to the Runge-Kutta propagators (blue solid line). The second-order (Δt = 0.069 as) and the fourth-order (Δt = 0.691 as) Runge-Kutta scheme yield the same trajectory for the times shown in (a). In (b), the fully self-consistent second-order finite-difference method (green solid line) is compared to the sc100-SOD (red solid line) and the non-self-consistent (red dotted line) one for Δt = 0.069 as.

Image of FIG. 3.
FIG. 3.

Total energy E tot (in eV) of the Na atom as a function of time t (in fs). The curves in (a) result from the sc100-SOD second-order finite-difference scheme and the ones in (b) from the second-order Runge-Kutta scheme. Time steps of Δt = 0.069 as (black solid lines), Δt = 0.104 as (red solid lines), and Δt = 0.138 as (green solid lines) were used.

Image of FIG. 4.
FIG. 4.

Time steps Δt (in as) for which the non-self-consistent second-order finite-difference scheme (a) or the fourth-order Runge-Kutta scheme (b) are stable (green triangles pointing up) or unstable (red triangles pointing down), depending on the respective plane-wave cutoff energy E cut (in Ry) used.

Image of FIG. 5.
FIG. 5.

Sum of the expectation values (in eV) of all valence wave functions i of the Na atom (at t = 0 fs the 3s wave function was shifted by (0.32, 0.32, 0.32) Å from its equilibrium position in real space) as a function of time t (in fs). In (a) the Euler scheme (black solid line, Δt = 0.069 as) and the sc100-SOD second-order finite-difference scheme (red solid line, Δt = 0.069 as) are compared to the Runge-Kutta propagators (blue solid line). The second-order (Δt = 0.069 as) and the fourth-order (Δt = 0.691 as) Runge-Kutta scheme yield the same trajectory for the times shown in (a). In (b) the fully self-consistent second-order finite-difference method (green solid line) is compared to the sc100-SOD (red solid line) and the non-self-consistent (red dotted line) one for Δt = 0.069 as.

Image of FIG. 6.
FIG. 6.

Total energy E tot (in eV) of the Na atom (at t = 0 fs the 3s wave function was shifted by (0.32, 0.32, 0.32) Å from its equilibrium position in real space) as a function of time t (in fs). The results have been obtained using the RK4 propagator and Δt = 0.691 as.

Image of FIG. 7.
FIG. 7.

The 64-atom unit cell of MgO (Mg atoms red circles, O atoms blue circles) containing a single Na atom (black circle) on an oxygen lattice position.

Image of FIG. 8.
FIG. 8.

Sum of the expectation values (in eV) of all valence wave functions in the Na:MgO 64-atom supercell (at t = 0 fs the Na-induced level within the MgO gap was shifted by (0.032, 0.032, 0.032) Å from its equilibrium position in real space) as a function of time t (in fs). The sc100-SOD second-order finite-difference scheme (red solid line, Δt = 0.069 as) is compared to the second-order (green solid line, Δt = 0.069 as) and the fourth-order (blue solid line, Δt = 0.691 as) Runge-Kutta scheme.

Image of FIG. 9.
FIG. 9.

Total energy E tot (in eV) of the Na:MgO 64-atom supercell (at t = 0 fs the Na-induced level within the MgO gap was shifted by (0.032, 0.032, 0.032) Å from its equilibrium position in real space) as a function of time t (in fs). The results have been obtained using the fourth-order Runge-Kutta propagator and Δt = 0.691 as.

Image of FIG. 10.
FIG. 10.

The number of steps that can be performed within 1 s wall time is plotted versus the number of processing cores used in the calculation for the Na:MgO 64-atom supercell. The steepest descent algorithm (black curve) is compared to the RK4 propagation (red curve).

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/content/aip/journal/jcp/137/22/10.1063/1.4758792
2012-10-22
2014-04-24
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Plane-wave pseudopotential implementation of explicit integrators for time-dependent Kohn-Sham equations in large-scale simulations
http://aip.metastore.ingenta.com/content/aip/journal/jcp/137/22/10.1063/1.4758792
10.1063/1.4758792
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