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J. Phys. Chem. C 2007, 111, 12588-12593
Hydrogenated Silicon Nanoparticles Relaxed in Excited States X. Wang,† R. Q. Zhang,*,† T. A. Niehaus,‡,§ Th. Frauenheim,‡ and S. T. Lee† Centre of Super-Diamond and AdVanced Films (COSDAF) and Department of Physics and Materials Science, City UniVersity of Hong Kong, Hong Kong SAR, China, Bremen Center for Computational Material Science, UniVersity Bremen, 28334 Bremen, Germany, and Department of Molecular Biophysics, German Cancer Research Center, D-69120 Heidelberg, Germany ReceiVed: February 18, 2007; In Final Form: May 11, 2007
Small hydrogenated silicon particles with diameters smaller than 2 nm were geometrically relaxed in excited states using density functional tight-binding methods. The relaxation caused Stokes shifts and optical-absorption spectrum red shifts with the particle size increases, consistent with experimental observation. Interestingly, the emission energies were fluctuated around 1.51-3.25 eV for small-size particles with diameter smaller than 1.5 nm (eSi87H76), and became inversely proportional to the size at larger diameters. The electronic property changes due to quantum confinement effects and significant structural deformations were shown to be responsible for the above-described features of the small silicon nanoparticles.
I. Introduction Although bulk silicon is an indirect band gap material and is optically inactive, porous silicon was found to emit visible light at room temperature.1 The discovery has attracted extensive interest and lead to numerous experimental2-10 and theoretical studies on silicon nanostructures such as hydrogenated nanoparticles.11-17 For developing new optical nanodevices such as nanosensors, a wealth of experimental efforts has been made to measure and characterize the structural, optical, and electronic properties of various semiconductor nanoparticles. The experimental observations of photoluminescence (PL), Fourier transform infrared spectroscopy (FTIR) transmission, and cathodoluminescence (CL) spectra indicated that the quantum confinement effect is a significant factor for them to occur.2-4,9 Moreover, the surface structure and chemical adsorption also determine the electronic states of nanosize Si and further affect their optical emission.10,19-21 Contrary to the abundant experimental work of hydrogenated silicon nanostructures, theoretical efforts devoted to revealing the PL mechanism are rather limited. Attempts were made with various theoretical methods including time-dependent local density approximation (TD-LDA) and generalized gradient approximation (GGA) of density functional theory (DFT), diffusion quantum Monte Carlo (QMC) technique, and GW correction, etc.13,22-24 Consistent with experiments, the band gaps of silicon nanostructures were found blue-shifted from the infrared to the visible region as the size was reduced. Still, numerous controversies exist between theoretical and experimental results due to the use of different methodologies. The diversities in the results are mainly due to (i) the uncertain particle size in the measurement, (ii) the surface impurity of the hydrogenated silicon particles, and (iii) the approximations of the contemporary approaches for studying the excited-state properties. Puzder et al.13 computed the absorption and emission energies of small silicon nanocrystals, using LDA and QMC * To whom correspondence should be addressed. E-mail: aprqz@ cityu.edu.hk. † City University of Hong Kong. ‡ University Bremen. § German Cancer Research Center.
methods. Their results demonstrated that the optical characteristics, especially their absorption and emission energies, are sensitive to the size, surface structure, and chemistry of the semiconductor nanoparticles. It is widely recognized that different calculation methods may cause differences in the determined values of excited states. The diversity is mainly due to the different approximations adopted for spectrum calculations that include the treatment of Coulomb or excitonic interactions. In addition to the small silicon nanoparticles with diameters less than 1 nm (such as Si29H24, Si29H36, and Si35H36), the largesize particles are being paid increasing attention, due to the new advancements of computational method and capability. Chelikowsky and his co-workers11 have examined the TD-DFT, in particular, the TD-LDA, for the optical properties of silicon nanoparticles with diameters of 0-2 nm. It was considered that the TD-LDA method that utilizes a real-space description of the electronic structure problem is better than QMC in offering an efficient approach for large systems. Although silicon nanoparticles have become a subject of many computational studies at ab initio and DFT levels, the origin and the physical or chemical mechanisms of their outstanding optical properties are still unresolved. In this paper, we present a detailed analysis on the structural and optical properties of hydrogenated silicon nanoparticles ranging from Si5H12 to Si199H140 (diameter < 2 nm) relaxed in excited states. We aim at a systematic investigation of the structural, electronic, and optical properties as a function of the size of hydrogenated silicon particles, in particular those relaxed in excited states, in order to provide insight into the development of the silicon-based optical nanodevices. II. Models and Computational Details The silicon nanoparticles studied in this work were approximately spherical, centered on a Si atom or a Si10 core. All Si atoms were assumed to take their ideal bulk positions with bond lengths equal to those of bulk silicon, prior to geometry optimizations. We eliminated surface Si atoms that involve more than two dangling bonds, while the remaining dangling bonds were passivated with hydrogen atoms. Since every Si atom was in sp3 hybridization, the ground states of these particles are
10.1021/jp071384j CCC: $37.00 © 2007 American Chemical Society Published on Web 08/09/2007
Silicon Nanoparticles Relaxed in Excited States
J. Phys. Chem. C, Vol. 111, No. 34, 2007 12589 where.25,26 Furthermore, the time-dependent linear response extension of the DFTB scheme (TD-DFTB) has been used for the calculations of the excitation spectra.27 The excitation energies were obtained by building up the coupling matrix, which gives the response of the potential with respect to a change in the electron density. Several approximations such as a γ-approximation were adopted in the TD-DFTB method, offering a reasonably high computational efficiency with moderate accuracies which surpass those of the configuration interaction singles (CIS) and the random-phase approximation (RPA) and are close to that of the TD-DFT.28 To validate the TD-DFTB approach for the present study, several small-size particles were further calculated at the TDDFT/B3LYP level29 with basis sets 6-311G* using the GAUSSIAN 03 package.30 As shown in Table 1, the absorption energies at TD-DFT/B3LYP level are in good agreement with that of previous TD-DFT/B3LYP results.22,31 However, in comparison with experimental32 and other computational values,17 our TDDFTB method overestimates the first transition energies of the three smallest particles (e.g., SiH4, Si2H6, and Si3H8). This is because these excited states mainly behave with Rydberg character which cannot be captured in our TD-DFTB minimal basis. As the contribution of Rydberg states diminishes with the increase of particle size, our TD-DFTB results tend to be similar to our TD-DFT/B3LYP results when the number of Si atoms is larger than 5 and to even be close to other calculated results using higher accuracy computational methods such as the multireference second-order perturbation theory MRMP2.31,33 In the case of Si5H12, the absorption energy at TDDFTB (6.40 eV) is close to the experimental value (6.5 eV34) and the other high-level ab initio results.33 For Si29H36, the absorption energy of TD-DFTB (4.42 eV) is in excellent agreement with the results of MR-MP2 (4.45 eV 31) and TDDFT (4.53 eV 31). Similarly, TD-DFTB also reproduced accurately the absorption energy of Si35H36 (4.37 eV) compared to MR-MP2 (4.33 eV 33) and TD-DFT results (4.42 eV 31). Furthermore, for the light emission properties, these available theoretical methods present even larger discrepancies than that in the absorption property predictions of small clusters. Our TDDFTB and TD-DFT/B3LYP results show that, from Si5H12 to Si17H36, the two different methods lead to more similar emission energies with the size increase. The TD-DFTB (TD-DFT/ B3LYP) emission energies of Si5H12, Si10H16, and Si17H36 are 2.29 eV (2.96 eV), 1.56 eV (1.18 eV), and 2.40 eV (2.51 eV), respectively. Obviously, the TD-DFTB results tend to be close to the TD-DFT/B3LYP results as the size increases. Although the excited-state optimizations of Si29H36 and Si35H36 using TDDFT method exceed our computational capability, their corresponding emission energies from TD-DFTB (2.57 and 2.89 eV) are still in reasonable accordance with the TD-DFT/GGA-PBE calculations (2.29 and 2.64 eV), respectively.35,36 These comparisons confirm that the TD-DFTB method predicts reasonably accurate optical properties of hydrogenated silicon particles except several small particles (e.g., SiH4, Si2H6, and Si3H8).
Figure 1. Schematic energy diagram of the ground state and the first excited singlet state of silicon nanoparticles.
favored to be in a high symmetry belonging to the Td point group. Therefore, the anisotropic effects in these particles were expected to be small. A schematic diagram of the vertical excitation and deexcitation between the ground and the first excited states is shown in Figure 1. The absorption energy (Eabs) is the difference in the energy required to excite the electron from the point 1 to point 2 in Figure 1. The emission energy (Eemi) is the energy released during the electronic relaxation from point 3 to point 4. The structure relaxations in excited states lead to an energy decrease from point 2 to point 3. The total energies of the two ground-state points 1 and 4 are determined from closed shell electronic configurations, while the electronic configurations of the excited-states 2 and 3 correspond to two single-occupation electrons with contrary spins (see Figure 1). The Stokes shift was calculated according to EStokes-shift ) Eabs - Eemi ) (E2 - E1) - (E3 - E4), where E1, E2, E3, and E4 correspond to the total energies at points 1-4, respectively. The structural optimization and ground-state properties of the particles were determined at first by a self-consistent charge density functional tight-binding method (SCC-DFTB) with the basis of numerically described s, p and d atomic orbitals for silicon and s orbital for hydrogen. This method is a secondorder perturbation approach to DFT. The Hamiltonian and overlap matrix elements were evaluated by a two-center approximation. Charge transfers were taken into account through the incorporation of a self-consistency scheme for Mulliken charges on the basis of the second-order expansion of the Kohn-Sham energy in terms of charge density fluctuations. The details of the SCC-DFTB method could be found else-
TABLE 1: Comparison of Calculated Absorption Energy Eabs (eV) of Several Small Hydrogenated Silicon Particles with Previous Calculational and Experimental Data cluster SiH4 Si2H6 Si3H8 Si5H12 Si10H16 Si29H36 Si35H36
TD-DFTB (our work) 10.30 8.21 7.38 6.40 6.04 4.42 4.37
TD-DFT/B3LYP (our work) 9.28 7.55 6.92 6.58 5.82 4.55 4.42
∆SCF/QMC 23
9.1 7.4 23 6.8 23 6.1 23 5.3 13 5.0 13
TD-DFT/BP 22
TD-DFT/B3LYP
MR-MP2
22
8.98 7.25 22 6.17 22 6.10 33
9.25 7.51 22 6.36 22 6.66 31
6.56 33
3.84 22 3.70 22
4.53 31 4.42 31
4.45 31 4.33 33
GGA-PBE 8.76
24
6.09 24 4.81 24 3.65 24 3.56 24
expt 8.8 32 7.6 32 6.6 32 6.5 34
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Figure 2. (a) Comparison of Si-Si bond lengths in ground states (black lines) and excited states (red lines). The solid lines correspond to the maximal bond lengths, while the dashed lines correspond to the minimal bond lengths. (b) Schematic geometrical relaxation diagram of the Si66H64 in excited state.
Although previous reports15,16,31,33 claimed accurate calculations of some small silicon particles, most of these particles are quite small in size, due to the need of a great computational cost in excited-state study. Therefore, the TD-DFTB is a moderate theoretical method offering acceptable accuracy to study the excited states of the hydrogenated silicon particles and enables our study to be extended from several atoms to nanometer-size systems. III. Results and Discussion Structural Distortions. A series of spherical hydrogenated silicon nanoparticles ranging from 5 to 199 silicon atoms were examined in this work. The geometrical optimizations show that, in ground states, Si-Si and Si-H bond lengths are about 2.342.38 and 1.50 Å, respectively, and bond angles remain at ∼109°. They are in good agreement with the experimental values of the bulk silicon and the silane molecule. However, the Si-Si bonds in the core region of the particle are a little longer than the others due to the traction of the outer atoms. In the excited-state configurations, the Td symmetry of particles is distorted due to the electronic relaxation. The Si-H bond lengths remain practically unchanged for both the ground and excited states, in contrast with the Si-Si distances. It has also been reported24 that it is the Si shell that is excited rather than the H atoms which are simply localized on the surface. To qualitatively evaluate the structural changes, the calculated
longest (solid curves) and shortest (dash curves) Si-Si bond lengths of the considered particles in both the ground- and excited-state configurations are shown in Figure 2a. The shortest Si-Si bond lengths saturate roughly at 2.33 Å no matter if the particle is in the ground state or excited state. However, the longest Si-Si bond lengths of ground states and excited states are quite different. The former is nearly saturated at 2.37 Å, while the latter changes remarkably with the particle size change. For example, in the Si10H16 particle, one Si-Si bond length extends from 2.34 to 2.86 Å under excitation. Strong distortions can also be found in other small particles such as Si26H32, Si29H36, Si35H36, and Si66H64. Figure 2 shows a rapid decrease of the longest Si-Si bond lengths of excited-state configurations. The longest Si-Si bond lengths tend to become less than 2.41 Å when the sizes increase to Si87H76 and saturate at a constant (2.39 Å) at Si123H100. Furthermore, most of the structure distortions take place in the first-neighbor silicon atoms of the particle core, which consists of a Si atom or a Si10 in the center. The first-neighbor atoms favor departure a little from the particle core due to their repulsion, while the second- or third-neighbor atoms tend to restrict the motion of the first-neighbor atoms by forming a rigid network. Contrary to the small increases of SiSi bonds in Si59H60 and Si75H76, the Si-Si distance of Si66H64 shows a significant extension from 2.37 to 2.74 Å, due to the different growth bases. The latter is grown from a Si10 core, while the former two start with a central Si atom. For Si66H64,
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J. Phys. Chem. C, Vol. 111, No. 34, 2007 12591
TABLE 2: Energy Levels of HOMO and LUMO (eV), Energy Gaps of HOMO and LUMO (∆E) (eV), Absorption and Emission Energies (eV) of Different Size Silicon Particles (Diameters Denoted by d0 (nm)) in Their Ground and First Excited Singlet Statesa ground state
excited state
cluster
d0
HOMO
LUMO
∆E
Eabs
HOMO
LUMO
∆E
Eemi
Si5H12 Si10H16 Si26H32 Si29H36 Si35H36 Si59H60 Si66H64 Si75H76 Si78H64 Si84H64 Si87H76 Si123H100 Si147H100 Si172H120 Si199H140
0.45 0.55 0.91 1.03 1.10 1.36 1.38 1.41 1.45 1.47 1.48 1.74 1.76 1.82 2.00
-6.68 -6.60 -6.12 -6.17 -6.05 -5.90 -5.82 -5.76 -5.76 -5.74 -5.66 -5.63 -5.55 -5.48 -5.50
-0.50 -0.61 -1.55 -1.84 -1.74 -2.24 -2.29 -2.30 -2.22 -1.92 -2.24 -2.56 -2.50 -2.56 -2.73
6.18 5.99 4.57 4.33 4.31 3.66 3.53 3.45 3.54 3.82 3.43 3.07 3.04 2.92 2.77
6.40 6.04 4.63 4.41 4.37 3.72 3.59 3.51 3.58 3.58 3.47 3.11 3.08 2.96 2.81
-5.30 -5.85 -5.71 -5.91 -5.91 -5.73 -5.72 -5.61 -5.65 -5.64 -5.59 -5.57 -5.52 -5.45 -5.47
-4.14 -4.29 -4.21 -3.36 -3.32 -2.84 -3.34 -2.75 -2.71 -2.69 -2.54 -2.70 -2.51 -2.64 -2.79
1.16 1.55 1.50 2.55 2.58 2.89 2.38 2.87 2.95 2.95 3.05 2.87 3.00 2.81 2.68
2.29 (17f15) 1.56 (29f28) 1.51 (69f68) 2.57 (77f76) 2.89 (89f86) 3.18 (149f146) 2.55 (165f162) 3.12 (189f186) 3.13 (189f187) 3.12 (201f198) 3.25 (213f210) 3.04 (297f294) 2.94 (345f343) 2.90 (405f403) 2.76 (469f467)
a
The numbers in parentheses denote the number of electronic transition orbitals in excited states.
one of the first-neighbor surface Si atoms is repulsed by a Si10 core (see the Figure 2b), whereas the Si atoms of Si59H60 and Si75H76 are restricted by surface Si atoms such that their Si-Si bond lengths could only change slightly (below 0.06 Å). We conclude that the geometry relaxation in excited states is prevalent in the silicon particles, especially in the small-size ones. The distortions mainly take place in the first-neighbor silicon atoms of the particle centers. Optical and Electronic Properties. Table 2 lists our calculated HOMO (the highest occupied molecular orbital) and LUMO (the lowest unoccupied molecular orbital) energy levels, the energy gap ∆EH-L, and the optical gap in both the ground and first-excited states. In the ground states, the electron transitions from HOMO to LUMO lead to the lowest absorption energy and cause an obvious absorption peak. The HOMOs are three-degenerate with T symmetry, while the LUMOs are in A1 symmetry. Table 2 indeed shows that the HOMO-LUMO energy gaps and absorption energies decrease as the particle size increases. It means that the energy gap between HOMO and LUMO determines the change of optical absorption. In this paper, we define the orbital which is occupied by one excited electron to be the LUMO and the orbital with the LUMO-1 energy level to be the HOMO of the excited state. Compared with those in the ground states, the energy levels of HOMOs lift up slightly in their first-excited single states, but this change is reduced when the particle sizes increase. On the other hand, the LUMO energy levels acutely drop in small particles under excitation. For example, in the Si5H12, the LUMO energy level moves down from -0.50 to -4.14 eV, and the energy decrease of LUMO in Si10H16 is 3.68 eV. This decrease also becomes weak as the particle size increases and saturates at Si123H100. Thus, in the excited state, the energy gap between HOMO and LUMO is smaller than that of ground state. From Si5H12 to Si87H76, the energy gaps of excited states increase from 1.16 to 3.05 eV, and then slowly decrease as the size further increases (see the data in Table 2). The structure relaxation under excitation leads the energy level of degenerate orbitals to split. The three-degenerate HOMOs of the ground state correspond to HOMO, HOMO-1, and HOMO-2 orbitals in the excited state whose energy levels are close. Electrons prefer to transit from the LUMO to one of the HOMO-n (n ) 0∼2). Hence, the change of emission energy does not purely follow that of ∆EH-L (see the relative data in Table 2). When the number of Si atoms is smaller than 87, the emission energies
Figure 3. Absorption and emission energies and Stokes shift of silicon particles (ranging from Si26H32 to Si199H140) versus the number of silicon atoms.
fluctuate between 1.51 and 3.25 eV. The wave troughs often appear in the small particles with a Si10 core (e.g., Si26H32 and Si66H64). However, this fluctuation will slow down as the size further increases. Further Comparison and Analysis. Our results show that the optical properties of hydrogenated silicon nanoparticles are size-dependent. Figure 3 plots the absorption and emission energies from Table 2 and also the Stokes shift about which numerous experimental data have been reported in recent years.10,36-38 Both the absorption energy and Stokes shift are sensitive to the size of the nanoparticle, and they decrease with the size increase. The corresponding curves in Figure 3 prefigure that they will finally approach experimental values of silicon bulk if the size is extended infinitely. The size dependence of absorption gaps and Stokes shifts has also been confirmed in previous reports,13 though there appeared to be little consensus among experimental and computational results. The significant diversities of energy gaps and Stokes shifts have been a longstanding problem due to the difficulty in synthesizing monodisperse samples with well-characterized surfaces. The silicon surface oxidized easily at room temperature possibly leads to a dipole-forbidden yellow-red emission. Although the clean silicon surface could be simulated theoretically, inconsistent results were often obtained due to different approximations adopted for reducing the great computational cost of excited-state
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Figure 4. Electronic transition orbitals of silicon nanoparticles in ground and excited states. Wave function isosurfaces were plotted at the same isovalue.
calculations. Encouragingly, our results of absorption gaps and Stoke shifts are in good agreement with the previous results using QMC method13 and are close to the MR-MP2 results.31 It confirms again that the TD-DFTB is accurate enough to study the excited states of silicon nanoparticles. Compared with absorption energies and Stoke shift, the emission energies are relatively scarcely reported, not only due to the above reasons but also because of the difficulty in geometric optimizations of excited states by theoretical approaches and structural instability of the excited states in experimental synthesis. Here, the TD-DFTB method presumed the tendency of emission energy changes as shown by the solid curve in Figure 3. The 1.5 nm appears to be a critical size for the emission energies of silicon nanoparticles. The emission
energies show an anomalous fluctuation for those with the diameters smaller than 1.5 nm (eSi87H76) but decrease slowly like absorption energies at sizes larger than that. Hence, the Stoke shifts tend to be zero if the size increases illimitably, which is in reasonable agreement with previous theoretical results.13 The low values of emission energies appearing in the particles Si26H32 and Si66H64 can be explained by their geometrical structure features. Since both Si26H32 and Si66H64 involve a Si10 core and the bare first-neighbor atoms, their firstneighbor atoms are able to be distorted strongly without the restriction of second-neighbor atoms. The strong structure relaxation causes the rearrangement of energy levels in the excited-state and smaller transition gap, resulting in a remarkable reduction of the emission energy.
Silicon Nanoparticles Relaxed in Excited States To further understand the optical characters, the electronic orbitals were studied. Figure 4 displays the schematic diagrams of electronic orbital isosurfaces, where the red and blue areas denote the signs ((), respectively. We find that the HOMO in the ground state and one of the HOMO-n (n ) 0-2) in the excited state are similar. It indicates that the electronic excitation affects slightly the properties of HOMO despite causing the orbital split of three-degenerated HOMOs. However, from the ground state to the excited state, the structure relaxation causes an obvious transformation of LUMO. When the size is smaller than Si84H64, the LUMO is more localized in excited state than that in the ground state. The electronic orbital localizes in the area where the structure distortion occurs. In Si87H76, the LUMO distributes across more atoms than that in other smaller size particles. Furthermore, in Si123H100, the change of LUMO from the ground state to the excited state becomes quite small, indicating that the emission energy will be close to the absorption energy when the size is larger than Si123H100 whose structure relaxation becomes small. Figure 4 also shows the special distribution of electronic orbital in several atoms of Si66H64, which is at the structural distortion position under excitation. The exceptional structural relaxation which has been explained in a previous description leads to a motion of LUMO to lower energy, and hence the emission energy of Si66H64 is smaller than that of other particles such as Si59H60 and Si75H76. In general, the geometrical distortion can also be used to explain the changes of orbital forms. In the small-size particles (below the size of Si75H76), the distortion takes place in a direction along which the internal stress starts to be released first, giving more localized LUMOs. On the contrary, in larger particles (>Si87H76), the strong restriction from the rigid network formed by the numerous outer atoms results in a weak directional preference, so the LUMOs are delocalized in the whole nanoparticles. IV. Conclusions The absorption energy and Stokes shift are monotonically inversely proportional to the size of the nanoparticle. They are reduced with the size increase, and their values tend to saturate at the bulk value when the size increases further. However, for emission energy, there is no such size dependence observed until the size is larger than 1.5 nm. When the size is smaller than 1.5 nm, the structure relaxation plays an important role for the optical properties. Several atoms near the particle center favor being repulsed under excitation, which is the main reason for the energy level of LUMO decrease in excited states. However, when the size is larger than 1.5 nm, the size dependence dominates the optical properties. Acknowledgment. The work described in this paper is supported by grants from the Research Grants Council of Hong Kong SAR [Project Nos. CityU 103106 and CityU 3/04C]. References and Notes (1) Canham, L. T. Appl. Phys. Lett. 1990, 57, 1046. (2) Cullis, A. G.; Canham, L. T.; Calcott, P. D. J. J. Appl. Phys. 1997, 82, 909. (3) Wolkin, M. V.; Jorne, J.; Fauchet, P. M.; Allan, G.; Delerue, C. Phys. ReV. Lett. 1999, 82, 197. (4) Kim, K. Phys. ReV. B 1998, 57, 13072. (5) Kanemitsu, Y. Phys. ReV. B 1994, 49, 16845. (6) Tilley, R. D.; Warner, J. H.; Yamamoto, K.; Matsui, I.; Fujimori, H. Chem. Commun. (Cambridge) 2005, 14, 1836. (7) Wilcoxon, J. P.; Samara, G. A.; Provencio, P. N. Phys. ReV. B 1999, 60, 2704. (8) Garrido, B.; Lopez, M.; Gonzalez, O.; Perez-Rodriquez, A.; Morante, J. R.; Bonafos, C. Appl. Phys. Lett. 2000, 77, 3143.
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