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Potential-functional embedding theory for molecules and materials
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10.1063/1.3659293
/content/aip/journal/jcp/135/19/10.1063/1.3659293
http://aip.metastore.ingenta.com/content/aip/journal/jcp/135/19/10.1063/1.3659293

Figures

Image of FIG. 1.
FIG. 1.

Flow chart for total energy minimization with respect to the embedding potential and subsystem electron densities{N K }. After initializing , we solve for the subsystem electron numbers {N K } by minimizing with fixed (step 2). In step 3, the gradient is calculated according to Eq. (20). With the gradient, we employ the quasi-Newton method (Ref. 41) to minimize E tot [u] (step 4). Once E tot is minimized, we update {N K } with the new (step 5). If E tot and {N K } are both converged, the code exits, otherwise we go back to step 3.

Image of FIG. 2.
FIG. 2.

(Upper plot) Comparison of electron densities along the H2 molecular axis: KS-DFT density (benchmark) is the black curve. The emb-OEP, emb-vW, emb-HC10, and emb-TF densities are shown with black crosses, black open-squares, blue dash-dots, and pink dashes, respectively. The electron density of each H atom from the emb-OEP calculations is shown by the red solid curves. (Lower plot) Contour plot for the embedding potential (in Hartree·bohr3) in a plane containing the H2 molecule, with the coordinates in Å.

Image of FIG. 3.
FIG. 3.

(Upper plot) Comparison of electron densities along the P2 molecular axis: KS-DFT density (benchmark) is the black curve. The emb-OEP, emb-HC10, and emb-TF results are given by the red circles, blue curve, and green curve, respectively. Electron densities for each P atom from the emb-OEP calculations are given by red solid curves. (Lower plot) Contour plot for the embedding potential (in Hartree·bohr3) in a plane containing the P2 molecule, with the coordinates in Å.

Image of FIG. 4.
FIG. 4.

(Upper plot) Comparison of electron densities along AlP bond axis: KS-DFT density (benchmark) is the black curve. The emb-OEP, emb-HC10, and emb-TF results are shown by red circles, blue dashes, and green dash-dots, respectively. (Middle plot) Electron densities of Al (red-dash dots) and P (red dashes) atoms from emb-OEP calculations. (Lower plot) Contour plot for the embedding potential (in Hartree·bohr3) in a plane containing the AlP molecule, with the coordinates in Å.

Image of FIG. 5.
FIG. 5.

(Upper plot) Comparison of electron densities along the LiH bond axis: KS-DFT density (benchmark) is given by the black curve and the emb-OEP density is shown in open circles. The inset shows the details at the peak of the density. Electron densities of H and Li atoms from emb-OEP calculations are shown by the red and blue curves. (Lower plot) Contour plot of the embedding potential (in Hartree·bohr3) in a plane containing the diatomic molecule, with the coordinates in Å. The same convention is used in Figs. 6 and 7.

Image of FIG. 6.
FIG. 6.

(Upper plot) Comparison of electron densities along the NaH bond axis. (Lower plot) Contour plot of the embedding potential (in Hartree·bohr3), with the coordinates in Å. See Fig. 5's caption for details.

Image of FIG. 7.
FIG. 7.

(Upper plot) Comparison of electron densities along the KH bond axis. (Lower plot) Contour plot of the embedding potential (in Hartree·bohr3), with the coordinates in Å. See Fig. 5's caption for details.

Image of FIG. 8.
FIG. 8.

(a) Contour plot of the embedding potential (Hartree·bohr3) from emb-OEP theory in a plane containing the H6 chain, with coordinates in Å. The six hydrogen atoms are marked with black dots. (b) Comparison of the electron density along the H6 chain for the KS-DFT benchmark (black solid curve) and the emb-OEP density sum (open circles). Subsystem electron densities are shown with dashes for each of the six hydrogen atoms. (c) Contour plot of the electron density difference between the KS-DFT benchmark and the emb-OEP scheme, with coordinates in Å.

Image of FIG. 9.
FIG. 9.

Total energies (in eV) per cubic unit cell of bulk NaCl versus the lattice constant (in Å). The unit cell contains four Na and four Cl atoms. The KS-DFT results (the benchmark) are shown by open squares. The emb-TF (triangles) and emb-OEP (crosses) results are also shown. Inset: detailed comparison between the benchmark and emb-OEP results. The emb-OEP data almost coincide with the benchmark.

Image of FIG. 10.
FIG. 10.

Total energy versus the distance d between the water molecule and the MgO(001) surface. Comparison between the benchmark (black squares), non-embedded results (labeled “shift-bare-ML,” blue triangles), shifted emb-TF results (labeled “shift-emb-TF,” green circles), and emb-OEP results (red circles). Both shift-bare-ML (shifted downwards by 1819.879 eV) and shift-emb-TF (shifted downwards by 2.539 eV) data are shifted to match the benchmark at d = 4.0 Å. The emb-OEP results are not shifted and are consistently higher than the benchmark, but with a small absolute error of less than 30 meV.

Image of FIG. 11.
FIG. 11.

Two contour plots of the embedding potential (in Hartree·bohr3) for the H2O molecule 2.3 Å above the MgO (100) surface. The O atom in the water molecule is directly above the labeled Mg atom in plane (A). Due to the periodicity, we only show the Mg and O atoms in the unit cell. The upper part of the cell is vacuum. The boxes with red edges are the unit cells used in our calculations. O, H, and Mg atoms are shown in red, yellow, and purple, respectively. (a) The contour plot is on the plane cut through the labeled Mg and O atoms, and is perpendicular to the MgO surface. (b) The contour plot is on a plane bisecting the top two layers (A) and (B), and is parallel to the MgO surface.

Tables

Generic image for table
Table I.

Comparison of the total energies (eV) of the KS-DFT benchmark and the emb-OEP theory for H2, P2, AlP, LiH, NaH, and KH diatomic molecules.

Generic image for table
Table II.

Upper table: Fractional electron numbers on the metallic atoms (Al, Li, Na, and K) in the heteronuclear diatomic molecules studied here. Lower table: fractional electron numbers on H atoms from left to right (labeled from 1 to 6, see Fig. 8) in the H6 chain. All results are from emb-OEP calculations.

Generic image for table
Table III.

Comparison of bulk NaCl bulk moduli (B0 in GPa), equilibrium lattice constants (a 0 in Å), and equilibrium energy per unit cell (E0 in eV) calculated via KS-DFT (the benchmark), emb-OEP theory, and emb-TF theory.

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/content/aip/journal/jcp/135/19/10.1063/1.3659293
2011-11-17
2014-04-23
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Potential-functional embedding theory for molecules and materials
http://aip.metastore.ingenta.com/content/aip/journal/jcp/135/19/10.1063/1.3659293
10.1063/1.3659293
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