Highly localized quasiatomic minimal basis orbitals for Mo from ab initio calculations

A minimal basis set of localized quasiatomic orbitals for Mo is constructed using the fully converged eigenstates from first-principles calculations with a large basis set. The orbitals, although similar in shape to those of a free atom, are slightly deformed such that it can reproduce all the occupied-state electronic properties of the system. They are very useful for analyzing chemical bonding by calculating the Mulliken overlap population and bond order index between atoms. In addition, the transferability of tight-binding parametrizations can be evaluated, for example, the effect of the two-center approximation.

Figure 1

Three-dimensional plots for the six QUAMBOs of Mo in bcc structure. A cubic unit cell of the bcc structure is also shown in the same scale as in the QUAMBO plot.

Figure 2

Contour plot of the QUAMBOs of Mo in bcc structure shown in Fig. 1: (a) $s$-like, (b) $d_{yz}$-like, (c) $d_{x^2-y^2}$-like, and (d) $d_{3z^2-r^2}$-like QUAMBOs with their corresponding QUAMBOs of Mo in fcc structure and free-atom orbitals for comparison. (a), (b), and (d) are plotted in the (100) plane and (c) is plotted in the (001) plane. The solid and dotted contour lines have opposite sign. The outermost contour line has the value of $0.02{\mathrm{\AA }}^{-3∕2}$. The contour step is $0.02{\mathrm{\AA }}^{-3∕2}$ for (a) and $0.04{\mathrm{\AA }}^{-3∕2}$ for (b)–(d). The two perpendicular lines passing through the center of the QUAMBOs indicate the directions of the nearest-neighbor atoms. If the nearest neighbors are out of the plane, the lines are drawn as dashed. The corners of the square indicate the positions of the second nearest neighbors.

Figure 3

Electronic band structure for bcc-Mo along symmetry directions calculated using a mixed-basis first-principles LDA method (solid line) and using QUAMBOs as a basis set (dotted line).

Figure 4

(Color online) (a) Electronic density of states of bcc-Mo obtained by using QUAMBOs as a basis set, compared with those from the original mixed-basis first-principles LDA calculations. (b) Projected density of states into $s$ and $d$ components by using QUAMBOs as a basis set and by simple projection (projecting the eigenstates onto spherical harmonics in atomic spheres). (c) Projected density of states using QUAMBOs illustrating the $s$ and $d$ contribution add up to the total density of states.

Figure 5

The matrix elements $M_{i0,jn}$ (top) and bond order index ${\mathrm{BO}}_{i0,jn}$ (bottom) between the origin atom and its various neighbors for Mo in bcc and fcc structures.

Figure 6

The matrix elements $M_{i0,jn}$ between the origin atom and its various neighbors for diamond-Si, fcc-Al, and bcc-Mo.

Figure 7

Overlap integrals as a function of interatomic distance for Mo in the bcc, fcc, and sc structures decomposed under the Slater-Koster tight-binding scheme.

Figure 8

Nonorthogonal hopping integrals for Mo as a function of interatomic distance in the bcc, fcc, and sc structures decomposed under the Slater-Koster tight-binding scheme. The black (gray) data points correspond to hopping integrals extracted from nearest (second nearest) neighbors, while the unshaded points correspond to third and further neighbors.

Figure 9

Contour plot of the (a) $s$-like, (b) $d_{yz}$-like, (c) $d_{x^2-y^2}$-like, and (d) $d_{3z^2-r^2}$-like orthogonal QUAMBOs in bcc structure at equilibrium lattice constant. The values of the contour lines and indications of the neighbors’ atomic positions are the same as in Fig. 2.

Figure 10

Contour plot of the (a) $s$-like, (b) $d_{yz}$-like, (c) $d_{x^2-y^2}$-like, and (d) $d_{3z^2-r^2}$-like orthogonal QUAMBOs in fcc structure at equilibrium lattice constant. The values of the contour lines and indications of the neighbors’ atomic positions are the same as in Fig. 2.

Figure 11

Orthogonal hopping integrals for Mo as a function of interatomic distance in the bcc, fcc, and sc structures decomposed under the Slater-Koster tight-binding scheme. The black (gray) data points correspond to hopping integrals extracted from nearest (second nearest) neighbors, while the unshaded points correspond to third and further neighbors.