Optical excitations in organic molecules, clusters, and defects studied by first-principles Green’s function methods

Spectroscopic and optical properties of nanosystems and point defects are discussed within the framework of Green’s function methods. We use an approach based on evaluating the self-energy in the so-called GW approximation and solving the Bethe-Salpeter equation in the space of single-particle transitions. Plasmon-pole models or numerical energy integration, which have been used in most of the previous GW calculations, are not used. Fourier transforms of the dielectric function are also avoided. This approach is applied to benzene, naphthalene, passivated silicon clusters (containing more than one hundred atoms), and the $F$ center in LiCl. In the latter, excitonic effects and the $1s\to 2p$ defect line are identified in the energy-resolved dielectric function. We also compare optical spectra obtained by solving the Bethe-Salpeter equation and by using time-dependent density-functional theory in the local, adiabatic approximation. From this comparison, we conclude that both methods give similar predictions for optical excitations in benzene and naphthalene, but they differ in the spectra of small silicon clusters. As cluster size increases, both methods predict very low cross section for photoabsorption in the optical and near ultraviolet ranges. For the larger clusters, the computed cross section shows a slow increase as a function of photon frequency. Ionization potentials and electron affinities of molecules and clusters are also calculated.

Figure 1

Feynman diagrams for the polarizability operator $\Pi$ (a), (b) and the function $\Phi$ (c), (d). The local exchange-correlation kernel is represented by a crossed line. Solid oriented lines are Green’s functions. Dashed lines denote the bare Coulomb potential.

Figure 2

Ionization energies of benzene, associated to all occupied orbitals in the molecule, as predicted within ${\mathrm{GW}}_0$ and ${\mathrm{GW}}_f$. The theoretical ionization energy is the negative of a quasiparticle energy eigenvalue. Experimental data from Ref. 54.

Figure 3

Absorption spectrum of benzene, modified from numerical results published in Ref. 26. Results from both, TDLDA (upper panel) and BSE (lower panel) methods are shown. Absorption lines were broadened by Gaussian convolution with dispersion $0.15\mathrm{eV}$ (below $10\mathrm{eV}$) and $0.5\mathrm{eV}$ (above $10\mathrm{eV}$). The measured spectrum (Ref. 61) is shown by the dashed lines. In the lower panel, the BSE was solved using two approximations for the self-energy: ${\mathrm{GW}}_0$ (dotted line) and ${\mathrm{GW}}_f$ (solid line).

Figure 4

Ionization energies of naphthalene, associated to all occupied $\pi$ orbitals and the first $\sigma$ orbitals in the molecule. Experimental data from Ref. 67.

Figure 5

First ionization potential and electron affinity of passivated silicon clusters, calculated within the ${\mathrm{GW}}_f$ approximation (solid lines) and $\Delta \mathrm{SCF}$ (dotted lines). Experimental data from Ref. 71. $\Delta \mathrm{SCF}$ results include spin-polarization effects.

Figure 6

Diagonal matrix elements of the operator $\Sigma -V_{xc}$, c.f. Eq. (32), evaluated on the basis of Kohn-Sham eigenfunctions. Matrix elements for occupied (unoccupied) orbitals are represented by crosses (“plus” signs). Self-energy is evaluated within the ${\mathrm{GW}}_f$ approximation and at the extrapolated quasiparticle energy (see the text).

Figure 7

Cross section for photoabsorption in passivated silicon clusters, calculated within TDLDA (dashed lines) and BSE (solid lines). The self-energy operator used in BSE is calculated within the ${\mathrm{GW}}_f$ approximation. The cross section is normalized by the number of silicon atoms in each cluster. Absorption lines were broadened by a Gaussian convolution with dispersion $0.1\mathrm{eV}$.

Figure 8

Minimum gap (dashed lines and open symbols) and optical gap (solid lines and filled symbols) for passivated silicon clusters. Circles denote TDLDA gaps. Triangles denote ${\mathrm{GW}}_f+\mathrm{BSE}$ gaps. The optical gap is obtained by integrating the oscillator strength up to $p=1\times {10}^{-4}$. The minimum gap is defined as the lowest spin-singlet excitation energy $(p=0)$ in the system. Experimental data from Refs. 71, 82.

Figure 9

Imaginary part of the macroscopic dielectric function, ${ϵ}_2$ for a $F$ center embedded in LiCl. Solid and dashed lines correspond to $\mathrm{GW}+\mathrm{BSE}$ and TDLDA calculations, respectively. The self-energy operator used in BSE is calculated within the ${\mathrm{GW}}_f$ approximation. The position of the $1s\to 2p$ transition line and electronic bands gaps within LDA and ${\mathrm{GW}}_f$ are shown with arrows. A Gaussian convolution with dispersion $0.2\mathrm{eV}$ was used to smear out the absorption lines.