Comparison of quantum Monte Carlo with time-dependent and static density-functional theory calculations of diamondoid excitation energies and Stokes shifts

We compute the absorption and emission energies and hence Stokes shifts of small diamondoids as a function of size using different theoretical approaches, including density-functional theory (DFT) and quantum Monte Carlo (QMC) calculations. The absorption spectra of these molecules are also investigated by time-dependent DFT and compared with experiment. We analyze the structural distortion and formation of a self-trapped exciton in the excited state, and we study the effects of these on the Stokes shift as a function of size. Compared to recent experiments, QMC overestimates the excitation energies by about 0.8(1) eV on average. Benefiting from a cancellation of errors, the optical gaps obtained in DFT calculations with the B3LYP functional are in better agreement with experiment. It is also shown that time-dependent B3LYP calculations can reproduce most of the features found in the experimental spectra. According to our calculations, the structures of diamondoids in the excited state show a distortion which is hardly noticeable compared to that found for methane. As the number of diamond cages is increased, the distortion mechanism abruptly changes character. We have shown that the Stokes shift is size dependent and decreases with the number of diamond cages. If we neglect orbital symmetry effects on the optical excitations, the rate of decrease in the Stokes shift is, on average, 0.1 eV per cage for small diamondoids.

Figure 1

(a) Schematic diagram of the ground- and excited-state singlet and triplet potential energy surfaces of a semiconductor NC in terms of the atomic coordinates ($Q$). Light is absorbed by exciting the NC from the atomic configuration that minimizes the ground-state energy, point A, to the optically active singlet state B${}_s$, which has the same geometry as point A. Emission occurs from the relaxed triplet state, point C, leading to a red-shift of the emission line. $E_A$, $E_B$, $E_C$, and $E_D$ are the energies corresponding to points A, B, C, and D, respectively, in the energy surfaces shown. (b) Schematic diagram in terms of $Q$ for delocalized excitons (DEs) and STEs in NCs, showing the corresponding absorption (solid lines) and emission (dashed lines) transitions. If the chemical bond is stretched beyond a critical value $Q$${}_C$, an STE state will be created.

Figure 2

TD-B3LYP excitation spectra along with experimental spectra taken from Ref. 10 for (a) adamantane, (b) diamantane, (c) triamantane, and (d) [123]tetramantane. The calculated lines are Gaussian broadened by about 0.03 eV. The first experimental and theoretical excitation energies are indicated by arrows in each panel. The structure of each molecule is given above the corresponding spectrum.

Figure 3

LDA energy diagram of up- and down-spin electron and hole levels of adamantane in a triplet excited state. The hole is indicated by a small circle. For each level, the energy (in a.u.) and the symmetry of the orbital are shown. (a),(b) Energy levels of down- and up-spin electrons at point B. Before Jahn-Teller relaxation, the hole and two T2-symmetric occupied down-spin levels are degenerate. (c),(d) Energy levels of, respectively, up- and down-spin electrons after the symmetry relaxation. The energy levels of the down-spin electrons split, with a reduction in the symmetry of the levels. $\Delta E_{\mathit{JT}}$ is the reduction in the energy of the degenerate levels after the Jahn-Teller relaxation.

Figure 4

Optimized geometries of methane and adamantane in (a) the ground-state and (b) the geometry- and symmetry-relaxed excited state, from LDA calculations using plane-wave basis sets. All lengths are in angstroms, and angles in degrees.

Figure 5

Distribution of the C-C (left) and C-H (right) bond lengths in the relaxed triplet excited states relative to their ground-state values: [$L{\left(\mathrm{C}-\mathrm{C}\right)}_{\mathit{ES}}-L{\left(\mathrm{C}-\mathrm{C}\right)}_{\mathit{GS}}{]/L\left(\mathrm{C}-\mathrm{C}\right)}_{\mathit{GS}}$ and [$L{\left(\mathrm{C}-\mathrm{H}\right)}_{\mathit{ES}}-L{\left(\mathrm{C}-\mathrm{H}\right)}_{\mathit{GS}}{]/L\left(\mathrm{C}-\mathrm{H}\right)}_{\mathit{GS}}$. $L{(\mathrm{C}-\mathrm{C})}_{\mathit{ES}}$ and $L{(\mathrm{C}-\mathrm{C})}_{\mathit{GS}}$ denote the C-C bond lengths in the excited and ground states, respectively. $L{\left(\mathrm{C}-\mathrm{H}\right)}_{\mathit{ES}}$ and $L{\left(\mathrm{C}-\mathrm{H}\right)}_{\mathit{GS}}$ denote the C-H bond lengths in the excited and ground states, respectively. (a) Adamantane, (b) diamantane, (c) triamantane, and (d) tetramantane. The lines are artificially Gaussian broadened by between 0.5 and 0.05. According to (d), nearly no change in C-H bond length is observed in the excited state in comparison with the ground state.

Figure 6

Isosurfaces of charge densities of the electron and hole orbitals with an isodensity value of 0.02 a.u. in the triplet excited-state configuration. The arrow indicates the most-stretched C-H bond of adamantane, on which the electron state localizes. (a) Adamantane, (b) diamantane, (c) triamantane, and (d) tetramantane.