Symmetry-adapted modeling for molecules and crystals

We have developed a symmetry-adapted modeling procedure for molecules and crystals. By using the completeness of multipoles to express spatial and time-reversal parity-specific anisotropic distributions, we can generate systematically the complete symmetry-adapted multipole basis set to describe any of electronic degrees of freedom in isolated cluster systems and periodic crystals. The symmetry-adapted modeling is then achieved by expressing the Hamiltonian in terms of the linear combination of these bases belonging to the identity irreducible representation, and the model parameters (linear coefficients) in the Hamiltonian can be determined so as to reproduce the electronic structures given by the density-functional computation. We demonstrate our method for the modeling of graphene and emphasize usefulness of the symmetry-adapted basis to analyze and predict physical phenomena and spontaneous symmetry breaking in a phase transition. The present method is complementary to de facto standard Wannier tight-binding modeling, and it provides us with a fundamental basis to develop a symmetry-based analysis for materials science.

Figure 1

Separation of the symmetry operation, $C_3$; (a) without separation and (b) operations separately to atomic sites/bonds and atomic degrees of freedom, e.g., spins.

Figure 1

Separation of the symmetry operation, $C_3$; (a) without separation and (b) operations separately to atomic sites/bonds and atomic degrees of freedom, e.g., spins.

Figure 2

Example of site clusters (different colored spheres) and bond clusters (different colored bonds) in the ${\mathrm{C}}_{3\mathrm{v}}$ (31m) point group (vertical mirrors in $xz$ and equivalent planes). The representative site and bond are denoted by a yellow circle and red arrow, respectively. The trigonal units are ${\mathit{a}}_1=(1,0,0)$, ${\mathit{a}}_2=(-1/2,\sqrt{3}/2,0)$, and ${\mathit{a}}_3=(0,0,1)$.

Figure 2

Example of site clusters (different colored spheres) and bond clusters (different colored bonds) in the ${\mathrm{C}}_{3\mathrm{v}}$ (31m) point group (vertical mirrors in $xz$ and equivalent planes). The representative site and bond are denoted by a yellow circle and red arrow, respectively. The trigonal units are ${\mathit{a}}_1=(1,0,0)$, ${\mathit{a}}_2=(-1/2,\sqrt{3}/2,0)$, and ${\mathit{a}}_3=(0,0,1)$.

Figure 3

(a) Full matrix form of SAMB ${\widehat{\mathbb{Z}}}_{l\xi ,sk}$, (b) atomic SAMB ${\mathbb{X}}_{l_1{\xi }_1,sk}$ with respect to atomic orbitals in each site or bond, and (c) cluster SAMB ${\mathbb{Y}}_{l_2{\xi }_2}$ for each site/bond clusters.

Figure 3

(a) Full matrix form of SAMB ${\widehat{\mathbb{Z}}}_{l\xi ,sk}$, (b) atomic SAMB ${\mathbb{X}}_{l_1{\xi }_1,sk}$ with respect to atomic orbitals in each site or bond, and (c) cluster SAMB ${\mathbb{Y}}_{l_2{\xi }_2}$ for each site/bond clusters.

Figure 4

Site cluster and nearest-neighbor and second-neighbor bond clusters in graphene. The representative site and bonds are indicated by the yellow circle and red arrows, respectively.

Figure 4

Site cluster and nearest-neighbor and second-neighbor bond clusters in graphene. The representative site and bonds are indicated by the yellow circle and red arrows, respectively.

Figure 5

Site cluster and the nearest-neighbor bond cluster SAMBs for graphene. The color and size (width) represent the weight of the components in each SAMB.

Figure 5

Site cluster and the nearest-neighbor bond cluster SAMBs for graphene. The color and size (width) represent the weight of the components in each SAMB.

Figure 6

Symmetry-adapted structure factors for the nearest-neighbor bonds in graphene. $k_x$ and $k_y$ are in unit of $2\pi /a$.

Figure 6

Symmetry-adapted structure factors for the nearest-neighbor bonds in graphene. $k_x$ and $k_y$ are in unit of $2\pi /a$.

Figure 7

The comparisons of the band dispersion between the DF Wannier and SAMB TB models. (a) The Brillouin zone and high-symmetry points, (b) the comparison of energy dispersions between DF (gray solid lines) and Wanner TB model (red dashed lines), and [(c)–(h)] the comparison of energy dispersion between the DF Wannier TB model (gray solid lines) and our SAMB TB model up to $N_{\max }^{\left(\mathrm{b}\right)}$-neighbor bonds (red dashed lines). The Fermi energy is set as the origin.

Figure 7

The comparisons of the band dispersion between the DF Wannier and SAMB TB models. (a) The Brillouin zone and high-symmetry points, (b) the comparison of energy dispersions between DF (gray solid lines) and Wanner TB model (red dashed lines), and [(c)–(h)] the comparison of energy dispersion between the DF Wannier TB model (gray solid lines) and our SAMB TB model up to $N_{\max }^{\left(\mathrm{b}\right)}$-neighbor bonds (red dashed lines). The Fermi energy is set as the origin.

Figure 8

(a) Comparison of the energy dispersion and DOS for the optimized SAMB TB with $N_{\max }^{\left(\mathrm{b}\right)}=6$. (b) The isoenergy surface at $\mu =-1.5$ eV. (c) The bond length dependence of the maximum strength of hoppings. The gray solid (red dashed) lines represent the results of the DF Wannier (SAMB) TB model.