From Ordinary to Blue Emission in Peralkylated n ... - ACS Publications

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From Ordinary to Blue Emission in Peralkylated n‑Oligosilanes: The Calculated Structure of Delocalized and Localized Singlet Excitons Matthew K. MacLeod† and Josef Michl*,†,‡ †

Department of Chemistry and Biochemistry, University of Colorado, Boulder, Colorado 80309-0215, United States Institute of Organic Chemistry and Biochemistry, Academy of Sciences of the Czech Republic, Flemingovo nám., 2, 16610 Praha 6, Czech Republic

‡

S Supporting Information *

ABSTRACT: Excited singlet state structures believed to be responsible for the Franck−Condon-allowed and the strongly Stokes-shifted (blue) emissions in linear permethylated oligosilanes (SinMe2n+2) have been found and characterized with time-dependent density functional (TD-DFT) methods for chain lengths 4 ≤ n ≤ 16. For chain lengths with n > 7, the S1 relaxed structures closely resemble the S0 equilibrium structures where all valence angles are tetrahedral and all backbone dihedral angles are transoid. At chain lengths with n < 8 more strongly distorted structures with one long Si−Si bond built from silicon 3p orbitals are encountered. The large Stokes shift is due more to a large destabilization of the ground state than the relaxation in the S1 excited state. For n = 7, both types of minima were located, exactly reproducing the borderline between the large-radius and the small-radius self-trapped excitons known from experiments.

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INTRODUCTION Linear persubstituted polysilanes (−RR′Si−)n, where R and R′ usually are alkyls or aryls, are very stable high polymers and have been long recognized as σ-conjugated analogues of πconjugated polyacetylenes.1,2 Among the many striking optical properties of polysilanes are an intense first absorption peak and an associated Franck−Condon (FC)-allowed fluorescence, remarkable for a small Stokes shift and a high quantum yield. Both are characteristic of chain segments in an extended nearly all-anti conformation, and it has been estimated3,4 that the emitting “large-radius” singlet exciton spans two or three dozen Si atoms. When R and R′ are alkyls, the absorption and emission both peak near 330 nm, a strikingly long wavelength for a structure devoid of unsaturation and lone pairs. When the substituents are aryls, the absorption and emission are shifted to even longer wavelengths. Steric interactions introduced by the R and R′ substituents5 usually make the strictly anti geometry with an Si−Si−Si−Si dihedral angle of 180° unfavorable, and dihedral angles of ∼165° (transoid) and ∼155° (deviant) are more common.6 They both still support σ conjugation almost as efficiently, whereas small dihedral angles such as ∼90° (ortho), ∼55° (gauche), or ∼30° (cisoid) interrupt it for reasons understandable in simple intuitive terms.7−10 The presence of such sharp turns in a long chain therefore causes its segmentation into relatively weakly communicating separate chromophores, and has drawn attention to the optical properties of short peralkylated chains in their extended (transoid or deviant) forms, which are dominant at low temperatures. © 2014 American Chemical Society

The absorption spectra of extended conformations of shorter oligosilanes display the anticipated effects of quantum confinement imposed on the normally longer exciton present in long polymer chains. They start with an intense peak, analogous to the first absorption peak of the polymer and attributable to a HOMO−LUMO transition of the σ(SiSi)σ*(SiSi) type. Its excitation energy increases steadily as the chain becomes shorter, in very good agreement with results of time-dependent density functional (TD-DFT) calculations.11 The shortest members of the series, the disilanes Si2R6, are exceptional in that their strong first transition corresponds to a σ(SiSi)π*(SiC) excitation, while their first σσ* excited state lies at higher energies and has not been well characterized.12,13 The terminating π* orbital of the first transition in disilanes is composed of antibonding orbitals associated with each lateral SiC bond and therefore consists primarily of 3p orbitals on the Si atoms. In oligosilanes with more than two silicon atoms, such a σπ* transition is also present, but a simple consideration shows that it must be forbidden or at best extremely weak and difficult to detect. According to TD-DFT calculations,11 the σπ* transition energy decreases with growing chain length, but more slowly than that of the σσ* transition. The σπ* transition has been detected in Si3Me8, where it still lies slightly below the Special Issue: Current Topics in Photochemistry Received: May 15, 2014 Revised: July 7, 2014 Published: July 11, 2014 10538

dx.doi.org/10.1021/jp504805y | J. Phys. Chem. A 2014, 118, 10538−10553

The Journal of Physical Chemistry A

Article

Table 1. PBE0/SVP Optimized Geometries of Ground State (S0) Equilibrium Structure (All-Transoid Si Backbone) and the Corresponding S1 Relaxed Structure 7* structure

n

7

1

ωSiSiSiSia/deg

∠SiSiSib/deg

2

163.3

111.3

3

164.0

110.5

4

164.0

111.3

5

163.3

110.5

6

111.3

∠CSiSic/deg

∠CSiCd/deg

SiSie/Å

SiCf/Å

109.4 110.1 111.3 108.6 111.1 110.5 108.0 110.3 108.1 108.1 110.3 108.6 110.5

108.5 109.0 108.5 108.5

2.365

108.6

2.367

109.0

2.367

108.6

2.367

108.5

2.365

108.5 109.0 108.5 108.3 109.6 108.3 109.2

2.389

109.7

2.428

109.7

2.428

109.7

2.396

109.2

2.389

1.895 1.893 1.893 1.902 1.903 1.902 1.902 1.902 1.902 1.902 1.902 1.903 1.902 1.893 1.895 1.893 1.899 1.890 1.891 1.895 1.895 1.893 1.892 1.892 1.892 1.893 1.892 1.895 1.895 1.890 1.899 1.891

7

7*

1

2

167.6

112.7

3

168.0

113.4

4

168.0

115.5

5

167.6

113.4

6

111.5 108.7 110.0 107.5 109.6 110.8 108.1 108.6 107.2 107.2 108.6 107.5 110.8

112.7

7

108.3 109.6 108.3

2.367

2.396

a

Dihedral angle for rotation about the Si(n)−Si(n+1) bond. The Si atoms are numbered consecutively starting at a chain end. bSiSiSi valence angle at Si(n). cCSiSi valence angle at Si(n). dCSiC valence angle at Si(n). eLength of the Si(n)−Si(n+1) bond. fLength of the Si(n)−C bond.

Trisilanes and disilanes do not fluoresce even at 10 K. A computational exploration of the S1 potential energy surfaces of permethylated trisilane19 and disilane20 identified minima analogous to those disscussed below for tetrasilane and longer chains, but they were extremely shallow. Their location very close to S1/S0 funnels (regions of near degeneracy) and the absence of a significant barrier on the way from the minima to the funnels account for the efficient return to the ground state S0 surface and the absence of fluorescence. This behavior is in perfect agreement with the anticipated effects of quantum confinement,21 which reduces the energy saving available at the vertically excited geometry by electron delocalization in the large-radius exciton. As the chain is shortened to seven silicons or fewer, the energy saving becomes smaller than the site distortion energy. As a result, it becomes preferable for one of the sites to distort, and the exciton collapses onto it. The actual nature of the geometrical distortion associated with the localization has remained unknown, as has the possible involvement of a mixing of the σσ* state with the nearby σπ* state as local symmetry is lowered. It was suggested that one of the Si−Si bonds stretches and serves as the excitation site, but no experimental and little computational22 evidence for this is available. In the present paper, we provide computational

σσ* transition,14 and in tetrasilanes, where the two transitions are essentially degenerate.15 It has not been detected in pentasilanes and longer oligosilanes, but is calculated to be present at energies only slightly higher than those of the intense σσ* transition. The emission spectra of extended conformations of short oligosilanes vary in a quite striking fashion as a function of the chain length.16,17 For permethylated decasilane and longer chains, the FC-allowed shape, very small Stokes shift, and high quantum yield are the same as in the high polymers. As the chain is shortened, the Stokes shift increases, and the emission band broadens, very slowly at first. When heptasilane is reached, an abrupt change occurs. While at least one conformer continues the steady trend of gradual change in fluorescence properties, others display a very different strongly Stokesshifted FC-forbidden broad blue emission with a lower roomtemperature quantum yield.18 The yield, however, increases to almost unity upon sufficient cooling.17 Permethylated hexasilane, pentasilane, and tetrasilane exhibit the same FC-forbidden blue fluorescence. It shifts to slightly longer wavelengths as the chain is shortened, and cooling to still lower temperatures is then necessary for the quantum yield to approach unity. 10539



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Table 2. B3LYP/def2-TZVP Calculated Vertical S0−S1 Absorption and S1−S0 Emission Energies and Oscillator Strengths at Optimized Geometries of SinMe2n+2 (n = 16−7), Stokes Shifts, and S1 Distortion Energiesa n 16 14 12 10 9 8 7

EVAb calcd 31 600 32 200 32 900 34 000 34 800 35 800 37 000

f VAc calcd 2.63 2.25 1.86 1.47 1.28 1.07 0.87

EVEd calcd 30 900 31 400 32 000 32 900 33 500 34 300 35 200

f VEe calcd

ESSf calcd

2.64 2.28 1.91 1.55 1.37 1.19 1.00

700 800 900 1100 1300 1500 1800

ESDg calcd 600 700 700 800 900 900 1000

EVAh obsd k

31 700 31 900l 32 700l 34 100 34 600l 35 500 36 600

f VAi obsd k

1.21 1.00m 0.79m 0.53 0.51m 0.39 0.31

ESSj obsd 400k 500m 500m 600 600m 800 1500

Energies are given in cm−1. bVertical S0−S1 absorption energy at the optimized S0 all-transoid geometry. cOscillator strength at the optimized S0 alltransoid geometry. dVertical S1−S0 emission energy at the optimized S1 all-transoid geometry. eOscillator strength at the optimized S1 all-transoid geometry. fStokes shift. gS1 state distortion energy. hExperimental absorption band maximum at 77 K in cyclopentane−isopentane (3:7 by volume) from ref 16. iExperimental S1−S0 oscillator strenth calculated in ref 11 from experimental data in ref 16. jExperimental Stokes shift from ref 16. k Measured at 77 K in 3-methylpentane glass.46 lExtrapolated from experimental data in ref 16. mExtrapolated from experimental data.16 a

Natural Bond Orbital (NBO) Analysis. NBOs were used to analyze excited states38 obtained from the S1 density matrix produced in a B3LYP/def2-TZVP calculation using the NBO 5.939 program linked to the GAMESS program suite,40 employing the 3-center 4-electron bond search parameter. S0−S1 Transition Densities. These were calculated using the configuration interaction singles (CIS) method41,42 and the 6-311G* basis set43,44 as implemented in QChem 3.2.45 This method yielded S1 states that were qualitatively similar to those obtained with TD-DFT (Supporting Information).

results for the geometries associated with the delocalized and the localized excitons in permethylated n-oligosilanes with 4− 16 silicon atoms. We find that all of the excitons are of σσ* as opposed to σπ* nature. The localized excitons responsible for the blue emission are of the bond-stretch type and can be located on one or another Si−Si bond in the chain, providing an example of bond-stretch isomerism.

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METHODS

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Geometries of Potential Energy Minima. Geometry optimizations for the ground state S0 (density functional theory, DFT)23,24 and first singlet excited state S1 (timedependent density functional theory in the Tamm−Dancoff approximation),25,26 were carried out with the PBE027 functional and the SVP28 basis set using Turbomole 6.229 and large integration grids (size 5).23 All minima found were verified by vibrational analysis. The use of S0 optimized geometries of the all-transoid conformers as starting structures for S1 optimizations produced delocalized exciton minima for SinMe2n+2 with 7 ≤ n ≤ 16. For Si6Me14 and Si5Me12 it yielded the minima 6*γ and 5*β (see the Results section for a description of these minima), respectively, and for Si4Me10, a structure characterized by narrow Si(1)Si(2)Si(3) and Si(2)Si(3)Si(4) valence angles and wide C(4)Si(2)Si(3) and C(7)Si(3)Si(2) valence angles (see the Results section for the atom numbering convention) similar to that found earlier.22 S1 optimization of Si7Me16 structures built from the S1 minima of 6 led to 7*α, 7*β, and 7*γ. The remaining S1 minima were located by optimizing starting structures built from analogous minima previously located by excited state stochastic search methods in shorter oligosilane chains.19,20 Absorption and Emission. Vertical absorption energies at each S0 minimum and vertical emission energies at each S1 minimum were calculated with the TD-DFT/RPA (random phase approximation) method and the B3LYP functional30,31 with the VWN 5 correlation functional,32 using the def2TZVP33 basis set in Turbomole 6.2. Oscillator strengths were obtained in the dipole length formulation. The site distortion energy was computed as the difference between the vertical FC and relaxed S1 energies, and the Stokes shift was obtained as the difference between the vertical absorption and emission electronic energies. Emission energies and S0−S1 oscillator strengths for the minima 4*α and 4*β were also calculated with the MSCASPT2/ANO-L method34−36 using MOLCAS 7.637 and agreed well with the TD-DFT results (Supporting Information).

RESULTS First excited singlet state geometries were optimized for SinMe2n+2 chain lengths of 6 < n ≤ 16 (delocalized excitons) and for chain lengths of 4 < n < 8, as well as some highly twisted conformers of longer chains (localized excitons), using TD-DFT methods (Figures 1−4, 7, 9−11, S1−S5, and Tables 1−7 and S1−S14). The computational results reproduce the absorption and emission energies and also the FC-allowed nature of the emission observed in the longer chains (small geometry relaxation) and FC-forbidden nature of the emission observed in the shorter chains (large geometry relaxation). In the shorter chains, the exciton can locate in one or another of the Si−Si bonds, which is then stretched, and its SiSiSi valence angles are distorted relative to the tetrahedral optimal ground state structure. Ordinary Emission: Delocalized Excitation and Relaxation. The PBE0/SVP optimized ground state geometries are essentially the same for chains of all lengths. Trends in the optimized geometry of the delocalized S1 state as a function of chain length interpolate smoothly between Si16Me34 and Si7Me16 (Figure 1 and Table 1, and Tables S1−S12 in the Supporting Information). The geometries of the optimized S0 and S1 states are nearly identical in 16 but differ noticeably in 7. The largest difference upon going from 16 to 16* is the increase of the central Si−Si− Si−Si dihedral angle by 3.5° to a value of 166.8°, while the terminal backbone dihedral angles only increase by 0.5° to 164.0°. In 16* the central Si−Si bond length is 2.390 Å, only 0.020 Å more than in its ground state 16, and the terminal Si− Si bond lengths are 2.369 Å, only 0.003 Å more than in 16. At 111.7°, the central Si−Si−Si valence angles in 16* are 1.0° larger than in 16, while terminal Si−Si−Si valence angles are 111.5°, only 0.2° larger. These minuscule differences are associated with a calculated Stokes shift of only 700 cm−1 for the FC-allowed emission, calculated at 30 900 cm−1, and with a 10540



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Figure 1. PBE0/SVP optimized geometries of 7, 7*, 16, and 16*. Valence angles (under atoms), Si−Si bond lengths (above bonds), and SiSiSiSi dihedral angles (under central Si−Si bond). Angles in deg and lengths in Å (not shown if identical by symmetry or if all values are the same across a structure, as indicated by an arrow).

Figure 2. B3LYP/def2-TZVP HOMO (σ) and LUMO (σ*) at ground state equilibrium S0 (all-transoid conformer) 7 and relaxed S1 geometry 7*. All molecular orbitals are shown with the isodensity surface value of 0.05.

minimal difference of oscillator strengths at the two geometries (Table 2). Among permethylated oligosilanes, the heptasilane Si7Me16 was found to be the shortest chain capable of accommodating the delocalized exciton. The optimized S0 geometry 7 and the optimized S1 geometry 7* differ noticeably throughout the entire molecule (Figure 1), significantly more so than in the

longer chains. All backbone dihedral angles are increased toward the anti limit. The central angle is increased by 4.0° to 168.0° and the terminal angle by 4.3° to 167.6°. The changes are the largest at the center of the chain, where the SiSiSi valence angle is increased by 4.0° to 115.5°, whereas at the terminus it only grew by 1.4° to 112.7°, and where the central SiSi bond lengths increased by 0.061 Å to 2.428 Å, while the 10541



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Table 3. Optimized Geometries (PBE0/SVP) of S1 Bond-Stretch Minima in 4 (4*α and 4*β) and 5 (5*α and 5*β) minimum

n

4*α

1

2

ωSiSiSiSia/deg

180.0

3

∠SiSiSib/deg

∠CSiSic/deg

∠CSiCd/deg

SiSie/Å

SiCf/Å

110.1 110.0 110.1 108.3

2.477

146.8

121.7 102.1 102.1 108.8 108.8 107.7 107.7

107.4

2.378

107.4 107.4 108.2 107.9 108.9 107.5 109.2

2.394

109.2

2.393

107.9 108.9 107.5 110.9 110.9 111.0 108.4

2.493

107.4

2.391

107.5

2.372

107.5 108.4 107.5 108.0 109.3 108.6 108.0

2.361

109.3

2.423

108.6

2.376

1.889 1.890 1.890 1.927 1.927 1.910 1.910 1.902 1.902 1.905 1.894 1.904 1.895 1.900 1.951 1.900 1.951 1.904 1.895 1.894 1.884 1.890 1.891 1.909 1.910 1.908 1.908 1.906 1.906 1.884 1.890 1.891 1.896 1.903 1.896 1.905 1.905 1.915 1.914 1.900 1.898 1.898 1.892 1.891

127.8

4

4*β

1

2

180.0

3

117.7 106.4 106.4 89.0 101.5 108.2 107.8

138.2 138.2

4

5*α

1

2

179.7

139.5

3

177.4

126.4

4

119.9 101.8 101.7 109.3 109.3 108.0 108.3 107.3 107.6

119.4

5

5*β

1

2

175.2

125.0

3

179.0

140.0

4

112.4 109.5 109.3 98.7 96.8 108.7 108.3 107.9 107.7

122.5

5

107.5 108.5 107.5

2.473

2.473

2.445

2.523

a

Dihedral angle for rotation about the Si(n)-Si(n+1) bond. The Si atoms are numbered consecutively starting at a chain end. bSiSiSi valence angle at Si(n). cCSiSi valence angle at Si(n). dCSiC valence angle at Si(n). eLength of the Si(n)−Si(n+1) bond. fLength of the Si(n)−C bond.

terminal bond only grew by 0.024 Å to 2.389 Å. The calculated geometry relaxation in S1 released a site-distortion energy (ESD) of 1000 cm−1 and led to an emission energy for 7* of 35 200 cm−1 and a Stokes shift of 1800 cm−1. The nature of the S0−S1 transition is σσ* for both 7 and 7* (Figure 2). Blue Emission: Localized Excitation and Geometry Relaxation. At chain lengths of n < 8, the calculated smallradius singlet exciton of extended permethylated oligosilanes is largely localized at one of the single Si−Si bonds. For simplicity we shall refer to these optimized S1 geometries as Si−Si bondstretch minima, although they are strongly distorted from the optimized S0 geometry in valence and dihedral angles at the stretched bond as well.

To refer to the individual minima, we introduce the following composite symbols. Each one starts with the number of Si atoms in the chain (n), followed by a string of letter symbols (c, g, o, ...6) that describe a sequence of SiSiSiSi dihedral angles starting with Si(1) and carry a subscript (+ or −) indicating the handedness of the chain twist. This is followed by an asterisk (*) if the minimum is in the S1 surface. The absence of an asterisk implies that the miminum is in the S0 surface. Finally, we add a lower case Greek letter (α, β, ...) that specifies the first Si atom of the stretched bond. Since we presently deal almost exclusively with fully extended conformers, with all dihedral angles either anti or transoid, instead of full labels such as 10542



Article

Figure 3. Selected geometry parameters (PBE0/SVP) for 4*α, 4*β, 5*α, and 5*β.

Figure 4. Selected geometry parameters (PBE0/SVP) for minima 6*α, 6*β, and 6*γ.

6at+t+*α we use abbreviations such as 6*α, where dihedral angles that are all close to 180° will be implied. The situation is especially complicated in permethylated heptasilane, which represents a borderline case capable of supporting either a localized or a delocalized exciton depending on its conformation. We shall therefore first describe the results of calculations for the shorter chains, n = 4−6.

Structure 4*α has an elongated terminal Si−Si bond at 2.477 Å, but its central Si−Si bond is nearly equally extended at 2.473 Å (other computational methods show a larger difference between these bond lengths47). This optimized geometry has Cs symmetry with a backbone dihedral angle of 180° (4a*α). In 4*β, the central bond is stretched to 2.473 Å and the terminal bonds are only slightly elongated, to 2.394 Å. The 10543



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Table 4. PBE0/SVP Optimized Geometries of S1 Bond-Stretch Minima 6*α, 6*β, and 6*γ Minimum

n

6*α

1

ωSiSiSiSia/deg

∠SiSiSib/deg

2

179.7

132.2

3

173.0

124.4

4

167.6

119.7

5

116.2

∠CSiSic/deg

∠CSiSid/deg

SiSie/Å

SiCf/Å

117.9 101.5 101.7 111.3 111.4 108.4 108.0 108.8 107.0 107.4 108.1

111.7 111.7 111.7 107.9

2.512

108.5

2.400

107.7

2.380

111.7

2.369

107.7 107.8 108.5 108.2 109.6 108.9 109.9

2.361

108.9

2.410

108.2

2.387

107.7

2.372

107.7 107.8 108.6 108.3 109.0 108.0 110.4

2.375

109.0

2.536

109.0

2.373

110.4

2.375

1.881 1.886 1.901 1.899 1.898 1.907 1.907 1.906 1.905 1.905 1.906 1.896 1.896 1.896 1.901 1.895 1.895 1.895 1.894 1.903 1.902 1.904 1.905 1.904 1.904 1.890 1.890 1.897 1.899 1.893 1.894 1.902 1.901 1.915 1.896 1.915 1.896 1.901 1.902 1.899 1.894 1.893

6

6*β

1

2

174.7

122.1

3

177.5

134.2

4

172.3

121.1

5

112.6 110.1 108.7 98.6 97.4 110.0 110.5 107.8 108.6 107.5 108.2

117.1

6

6*γ

1

2

176.2

115.6

3

180.0

131.1

4

−176.2

131.1

5

112.4 109.9 109.2 107.8 107.5 92.4 99.2 110.8 111.0 108.1 108.8

115.6

6

108.0 108.3 109.0

2.415

2.544

2.373

a


contracted C−Si−Si valence angles. For example, in 4*α the C(4)−Si(2)−Si(1) valence angle is 90.8° and in 4*β the C(4)−Si(2)−Si(3) valence angle is 89.0°. C−Si−C valence angles widen to 120° at the TBP limit. This structural change is not as pronounced as previous distortions; in 4*α the C(4)− Si(2)−C(5) valence angle is 114.2° and in 4*β the internal C− Si−C valence angles are essentially exactly tetrahedral at 109.2°. Structures 5*α and 6*α are similar to 4*α. The internal bond-stretch minima 5*β and 6*β have no direct analogue in permethylated tetrasilane, but the local structure around the most stretched Si−Si bond is similar to that in 4*α. Structure 6*γ is similar to 4*β as both minima possess Ci symmetry (Figures 3 and 4). As the chain length grows in terminal and internal bond-stretch minima, the symmetry is lowered to C1, since the dihedral angles farther from the stretched Si−Si bond

backbone dihedral angle is again 180° (4a*β). The point symmetry group is not the anticipated C2h, but only Ci, due to a slight twist of the internal methyl groups and differences in Ci− Si−Si bond angles (Table 3, Figure 3). Besides localized Si−Si bond-stretching, 4*α and 4*β exhibit other drastic deviations from the ground state tetrahedral structure 4 (Table 3). These include some much wider and some much narrower valence angles, as if the two Si atoms of the stretched bond tried to move from a tetrahedral toward a trigonal bipyramidal (TBP) geometry. In the TBP limit the Si− Si−Si valence angles are 180°. In 4*α the Si(2)−Si(3)−Si(4) valence angle is increased more toward the TBP limit at 146.8° than the adjacent Si(1)−Si(2)−Si(3) valence angle (127.8°). In 4*β both Si−Si−Si valence angles are 138.2°. At the TBP limit the C−Si−Si valence angles are 90°. Both 4*α and 4*β have 10544



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Table 5. B3LYP/def2-TZVP Vertical Absorption EVA and S0−S1 Oscillator Strengths f for Ground State Equilibrium Structures and Emission Energies EVE, S0−S1 Oscillator Strengths f, Stokes shifts ESS, and S0−S1 Distortion Energies ESD for the BondStretch Minimaa S1 Species 7*α 7*β 7*γ 6*α 6*β 6*γ 5*α 5*β 4*α 4*β

EVAb calcd 37 37 37 38 38 38 40 40 43 43

000 000 000 500 500 500 600 600 600 600

fc calcd 0.67 0.67 0.67 0.67 0.67 0.67 0.47 0.47 0.30 0.30

EVEd calcd 29 28 30 28 27 27 26 27 26 25

900 500 100 900 800 500 600 600 800 700

fe calcd

ESSf calcd

ESDg calcd

0.42 0.44 0.62 0.32 0.36 0.40 0.22 0.20 0.12 0.12

7 100 8 500 6 900 9 600 10 700 11 000 14 000 13 000 16 800 17 900

400 1 100 1 500 700 1 400 1 600 2 900 2 500 4 200 4 400

EVAh obsd 36 36 36 38 38 38 40 40 43 43

600 600 600 100 100 100 100 100 100 100

f VAi obsd

ESSj obsd

0.31 0.31 0.31 0.23 0.23 0.23 0.18 0.18 0.16 0.16

6 100k 6 100k 6 100k 11 300l 11 300l 11 300l 14 100l 14 100l 16 800m 16 800m

In cm−1. bVertical S0-S1 absorption energy at the optimized S0 7t+t+t+o+, 6t+t+t+, 5t+t+, and 4t+ structures. cOscillator strengths at the same geometries. dVertical S1−S0 emission energy at the optimized S1 geometries of the same conformations. eOscillator strengths at the same geometries. f Stokes shift. gS1 state distortion energy. hExperimental absorption band maximum at 77 K in cyclopentane-isopentane (3:7 by volume) from ref 16. i Experimental S1−S0 oscillator strenth calculated in ref 11 from experimental data in ref 16. jExperimental Stokes shift.16 k130 K.16 l120 K.16 mNeat at 17 K.16 a

Figure 5. S0−S1 transition densities (CIS/6-311G*) for the bond-stretch minima 4*α, 4*β, 5*α, and 5*β, and the ground state equilibrium geometries 4 and 5 (the isodensity surface value is 0.005). The S1 state is of σσ* nature.

are no longer anti. Simultaneously, the extent to which the distorted Si−Si bond is stretched increases, e.g., from 2.477 Å in 4*α to 2.493 Å in 5*α and to 2.512 Å in 6*α. In 6*γ, the central Si−Si bond length is 2.536 Å. The adjacent Si(2)−Si(3) bond lengths are calculated to be 2.523, 2.544, and 2.373 Å for 5*β, 6*β, and 6*γ, respectively (Table 4). As the chain length increases, a larger number of the adjacent Si−Si bonds are elongated relative to the ground state, but to a lesser degree, and the Si−Si−Si valence angle distortions become less drastic; e.g., the terminal Si(1)−Si(2)−Si(3) valence angle is 139.5° and 132.2° in 5*α and 6*α, respectively, compared to 146.8° in 4*α. Thus, the Si−Si−Si valence angle distortions toward the TBP limit (180°) are reduced as the chain length is increased. The Si−Si−C valence angle distortions toward the TBP geometry are also are less pronounced at longer chain lengths, (92.4° in 6*γ) than the acute CSiSi valence angles at shorter chain lengths (89.0° in 4*β).

The pattern of the most acute SiSiC internal angles is different in the different bond-stretch minima. In the central bond-stretch minima the two most acute SiSiC valence angles are related by inversion symmetry and there is one on each Si atom of the stretched bond, e.g., C(4)Si(2)Si(3) and C(7)Si(3)Si(2) in 4*β. This is similar for the next two most acute SiSiC valence angles. This pattern is seen in 5*β, 6*β, and 7*β (Tables 3, 4, and 6). An additional internal bondstretch minimum with local Ci symmetry around the Si(2)− Si(3) bond and an energy equal to that of 5*β was found for Si5Me12 (5*β′, Supporting Information). The terminal bondstretch minima that have Cs symmetry (4*α) or at least local Cs symmetry around the Si(1)−Si(2) bond (5*α, 6*α) display the two most acute CSiSi valence angles centered on Si(2), followed by two larger angles centered on Si(1). The most acute SiSiC valence angles in the internal bond-stretch minima 5*β, 6*β, and 7*β follow the pattern observed in the terminal 10545



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Figure 6. S0−S1 transition densities (CIS/6-311G*) for bond-stretch minima 6*α, 6*β, and 6*γ. The isodensity surface value is 0.005. The S1 state is of σσ* nature.

Figure 7. Selected geometry parameters (PBE0/SVP) for bond-stretch minima 7*α, 7*β, and 7*γ.

The S0−S1 oscillator strength ranges from 0.12 (4*β and 4*α) to 0.40 (6*γ). At each chain length, it is approximately half of that computed at the S0 equilibrium geometry. Terminal bond-stretch minima n*α have larger EVE values and lower oscillator strengths, Stokes shifts, and site distortion energies than internal minima, n*β and n*γ. The electronic nature of the S0−S1 transition is best described as HOMO−LUMO (σσ*) promotion, as was the case for the delocalized excitons, but now the HOMO is largely localized on the longest Si−Si bond (Figure 5). This localizes the transition density on this Si−Si bond.47

bond-stretch minima, in that the two most acute CSiSi valence angles are both centered on Si(3) and the next pair of most acute angles is located on Si(2). The calculated vertical emission energies for 4*−6* increase as the chain grows longer, from 25 700 cm−1 for 4*β to 28 900 cm−1 for 6*α (Table 5). The Stokes shift drops from 17 900 cm−1 in 4*β to 9600 cm−1 in 6*α, and the site distortion energy from 4400 cm−1 in 4*β to 700 cm−1 in 6*α. The largest variation in vertical emission energies at one chain length is 1400 cm−1 (6*β−6*γ) and the average is 1200 cm−1. The vertical emission energy of 27 800 cm−1 for 6*β is closer to that of 6*γ at 27 500 cm−1 than to that of 6*α at 28 900 cm−1. 10546



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Table 6. PBE0/SVP Optimized Geometries of S1 Bond-Stretch Minima 7*α, 7*β, and 7*γ structure

n

7*α

1

ωSiSiSiSia/deg

∠SiSiSib/deg

2

179.2

127.9

3

174.6

122.2

4

172.9

120.4

5

91.0

118.1

6

113.0

∠CSiSic/deg

∠CSiCd/deg

SiSie/Å

SiCf/Ås

116.3 101.7 102.0 112.2 112.3 108.2 108.7 106.8 109.0 106.4 108.7 107.5 108.7

112.0 112.2 112.0 107.2

2.521

106.3

2.404

108.0

2.387

112.0

2.385

112.2

2.364

108.6 107.9 108.0 108.2 109.6 109.0 112.1

2.361

110.0

2.401

109.0

2.393

108.1

2.387

108.8

2.364

108.0 108.8 108.1 108.5 109.6 108.7 110.2

2.381

109.3

2.576

109.6

2.379

108.1

2.399

109.1

2.368

1.894 1.895 1.898 1.893 1.894 1.905 1.906 1.905 1.906 1.904 1.908 1.905 1.911 1.884 1.884 1.879 1.897 1.894 1.894 1.892 1.892 1.899 1.898 1.904 1.903 1.906 1.903 1.905 1.911 1.897 1.889 1.889 1.897 1.892 1.894 1.898 1.898 1.890 1.893 1.898 1.897 1.904 1.899 1.904 1.910 1.891 1.898 1.891

7

7*β

1

2

176.8

120.4

3

177.9

131.1

4

173.9

120.5

5

92.2

118.1

6

111.2 110.2 108.7 98.2 97.6 110.1 110.7 108.5 108.9 108.6 105.6 108.2 108.3

112.8

7

7*γ

1

111.1 109.6 109.4 108.3 110.2 99.6 98.1 112.0 110.4 109.7 106.3 109.2 108.4

111.9 2

175.5 120.6

3 173.8

127.9

168.9

117.4

95.6

111.6

4 5 6 7

109.1 108.1 108.2

2.402

2.552

2.353

a


The Borderline Case (n = 7). Permethylated heptasilane is able to support both delocalized and localized excitation in S1, depending on its conformation. The delocalized minimum is found in the fully extended 7t+t+t+t+ conformer and was discussed above. Minima of the latter type are found in other conformers and a few of them have been derived from analogous minima in permethylated hexasilane by replacement of a terminal methyl group with a SiMe3 group in a way that yields a terminal dihedral angle of ∼90° (Figure 7), followed by

geometry optimization in S1. The structures of the resulting S1 minima are similar to those in the parent hexasilanes. Notable differences are a slight extension of the most stretched Si−Si bond and a partial relaxation of the associated valence angles toward the tetrahedral value. For 7*γ, the Si(3)−Si(4) bond length is 2.576 Å instead of 2.536 Å in 6*γ, in 7*β the Si(2)−Si(3) bond length is 2.552 Å as opposed to 2.544 Å in 6*β, and in 7*α the Si(1)−Si(2) bond length is 2.521 Å in contrast to 2.512 Å in 6*α. The 10547



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terminal Si(1)Si(2)Si(3) valence angle is reduced to 127.9° in 7*α from 132.2° in 6*α, the Si(2)Si(3)Si(4) valence angle decreases to 131.1° in 7*β from 134.2° in 6*β, and the Si(3)Si(4)Si(5) valence angle is relaxed to 127.9° in 7*γ from 131.1° in 6*γ. The terminal backbone dihedral angles are close to 90° at 95.6°, 92.2°, and 91.0° for 7*γ, 7*β, and 7*α, respectively (Table 6, Figure 7). The calculated emission energies for the various S1 isomers of permethylated heptasilane with a localized exciton range from 28 500 to 30 100 cm−1. On average, they are slightly higher than those of similar isomers of permethylated hexasilanes, but they still are much lower than that of the delocalized exciton 7*, which yielded an only slightly Stokesshifted emission at 35 200 cm−1. The average Stokes shift of 7*α, 7*β, and 7*γ is 7500 cm−1, about four times that of 7* (1 800 cm−1). The S0−S1 oscillator strength for 7*γ is 0.62, larger than for the analogue 6*γ (0.40), but lower than for the delocalized excitation in 7* (1.00). S0−S1 transition densities are localized on the most stretched Si−Si bond (Figure 8 and Supporting Information). Figure 9. Plot of the S0 (blue) and S1 (red) energy (cm−1/103) versus the terminal SiSiSiSi dihedral angle (with all other geometrical parameters PBE0/SVP optimized), indicating the crossover between delocalized excitation in 7t+t+t+t+* and the self-trapped exciton in 7t+t+t+o+*β. The HOMO (below) and the LUMO (above) at 125 and 126° are shown.

state of all the permethylated oligosilanes examined, and in those S1 states in which the excitation is delocalized and the emission ordinary. In Si−Si bonds, their occupancies are close to unity, whereas in Si−C bonds the Si NHO have occupancies of about 0.6 e, while the occupancies of their C(sp3) NHO partners are close to 1.4 e, reflecting the low electronegativity of silicon relative to carbon. Suggestive natural resonance structures (Chart 1) and significant deviations from sp3 hybridization at silicon atoms located at and in the vicinity of the stretched Si−Si bond are observed for S1 states that contain a localized exciton. In terminal bond-stretch minima n*α, the electronic structure at Si(2) resembles that in dimethylsilylene. Important resonance structures place a lone pair on Si(2), while Si(1) and Si(3) are coupled through a “long” covalent or ionic bond. In internal bond-stretch minima such as n*β, both silicon atoms of the stretched bond have partial silylene character. In 4*α, the Si(1) atom has nearly sp3 orbitals pointing toward methyl groups and an sp4 hybrid pointing toward Si(2). Three approximately sp2 hybridized orbitals on Si(2) are used to attach Si(3) and two methyl groups. The Si(2) NHO pointing to Si(1) is an almost pure 3p natural atomic orbital (sp10) and has a low occupancy of 0.75 e. The NHO pointing to Si(3) deviates by 31.5° from the line connecting the Si(2) and Si(3) nuclei and has an increased occupancy of 1.22 e (Figure 10). This deviation from the line of centers significantly exceeds that of other NHOs along the silicon backbone (see Supporting Information). The Si(3) atom has an sp4 orbital pointing to Si(2), and the remaining NHOs on Si(3) and Si(4) are almost exactly sp3 hybridized. In the internal bond-stretch minimum 4*β the terminal Si atoms have sp3 hybrid orbitals pointing toward carbon atoms and sp4 hybrid orbitals pointing to the internal Si atoms. The NHOs between Si(2) and Si(3) have decreased s character (sp4) and lower occupancy (0.92 e) while the adjacent Si NHOs

Figure 8. S0−S1 transition densities (CIS/6-311G*) for the central bond-stretch minimum 7*γ and the large radius exciton S1 geometry 7*, at the isodensity surface value of 0.005. The S1 state is of σσ* nature.

The site distortion energies ESD, computed relative to the 7t+t+t+o+* S1 energy, show that 7*γ is the most stable of the isomeric blue emitters (at 1500 cm−1) followed closely by 7*β (1100 cm−1). The terminal bond-stretch minimum 7*α has the smallest distortion value at 400 cm−1. These values follow the general trend that appears throughout this work: minima that contain their most stretched Si−Si bond closer to the chain terminus have lower ESD values. The delocalized exciton 7* is found to be the global minimum on the S1 surface, more stable by 1700, 1000, and 600 cm−1 than 7*α, 7*β, and 7*γ, respectively. The energy barrier in the S1 surface estimated by scanning the terminal all-Si dihedral angle from its value in 7t+t+t+t+* (delocalized excitation) to its value in 7t+t+t+o+*β (localized excitation) is ∼1400 cm−1 (Figure 9). The localization occurs at terminal Si backbone dihedral angle values 125−126°. If the same path is followed on the S0 surface, the barrier is much larger, 6700 cm−1. Natural Orbital Analysis. Weinhold analysis48 was performed on the density matrices obtained from DFT for the S0 state and from the TD-DFT response equations for the S1 state. The natural hybrid orbitals (NHOs) on all silicon atoms are found to be very close to tetrahedral (sp3) in the S0 10548



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Chart 1. Dominant Natural Resonance Theory (NRT) Structures (TD-B3LYP/def2-TZVP) in the S1 State at the Bond-Stretch Minima 4*α and 4*βa

a

Weights are given relative to the main resonance structure.

remains sp4 hybridized, the p character of the adjacent Si(2) NHO pointing to Si(1) is reduced from sp10 in 4*α to sp7 in 5*α and to sp5 in 6*α. The Si(2) NHO pointing to Si(3) remains sp2 in both 5*α and 6*α, while the Si(2) hybrids pointing to methyl groups gradually gain p character as the chain length increases, and are sp3 in 6*α and sp2.5 in 5*α. The Si(3) hybrid pointing to Si(2) remains sp4 in both 5*α and 6*α and all other Si hybrid orbitals remain close to sp3. The Si(2) NHO pointed toward Si(3) deviates less from the Si−Si line of centers as the chain length is increased; the deviation is 22.8° and 11.6° for 5*α and 6*α, respectively. NHO occupations are also similar. The results of the NHO analysis of 6*β and 5*β resemble those of the terminal bond-stretched minima. Since the most stretched Si−Si bond is no longer Si(1)−Si(2) but Si(2)− Si(3), the Si(3) atom now contains the high p character hybrid orbital pointed toward Si(2). This hybrid is sp5 and sp6 in 6*β and 5*β, respectively. In both of these minima the adjacent NHO on Si(3) pointing toward Si(4) has the same sp2 hybridization as the Si(2) hybrid orbital in 5*α and 6*α. The results of the NHO analysis for 6*γ are close to those of 4*β. The NHOs centered along the central bond, Si(3)−Si(4) in 6*γ, have less p character (sp3.5) than in 4*β (sp4). These NHOs deviate less from the line of centers in 6*γ (11.6°) than in 4*β (17.7°). The NHO occupancies of the central Si−Si NHOs decrease slightly to 0.88 e in 6*γ from 0.92 e in 4*β.

Figure 10. NHOs of Si(2) pointing to adjacent Si atoms in 4 (A), 4*β (B), and 4*α (C). All sp3 orbitals are shown in purple and orange, those with increased s character in green and yellow, and those with increased p character in blue and red. The deviation between the Si(2)−Si(3) line of centers and the Si(2) NHO direction (black arrow) is shown in degrees. The Si backbone NHOs located on Si(3) are identical by symmetry in 4 and 4*β; additional NHO line of centers deviations in 4*α are given in the Supporting Information.

■

DISCUSSION We start by providing an overview of the experimental facts and analyzing them in terms of the classical physical model for onedimensional excitons.49,50 We will find that it accounts for the observations qualitatively and even semiquantitatively extremely well and provides considerable insight into the relation of large-radius to small-radius self-trapped excitons. We then use the results of numerical TD-DFT computations to take an independent look at the situation and to analyze those aspects that the classical model does not address, specifically the detailed geometrical structure of the two types of excitons and the existence of more than one possible location

pointed toward terminal Si atoms have slightly increased s character (sp2.8) and occupancy (1.12 e). Internal Si NHOs pointed toward methyl groups also have a slightly increased s character (sp2.8). The NHO located on Si(2) and pointing toward Si(3) deviates from the Si(2)−Si(3) line of centers by 17.7°, about half of what was found in 4*α (Figure 10). As the chain length is increased, the departure from sp3 hybridization becomes less pronounced at the terminal bondstretch minima. While the NHO on Si(1) pointing to Si(2) 10549



Article

energy of Si2Me6 is also unknown. The 1σσ* excitation energy was earlier believed16 to be 53 500 cm−1, but more recently, this absorption peak was reassigned as 1σπ* and the 1σσ* excitation was attributed to a higher energy peak at 61 500 cm−1.13 However, the terminating σ* orbital of this transition is calculated to have σ*(SiC) rather than σ*(SiSi) character, and in Si2Me6 no pure σ(SiSi)σ*(SiSi) state seems to exist. It does exist in Si2t−Bu6, whose SiSi bond is much longer, and where it is located at ∼52 000 cm −1 . Assuming that a true σ(SiSi)σ*(SiSi) excitation energy in Si2Me6 would lie somewhere near 60 000 cm−1, its Stokes shift can be very roughly estimated at 36 000 cm−1 and our best experimental estimate of S is ∼18 000 cm−1. B can be approximated for each n as the difference between the observed excitation energy and the vaguely known σ(SiSi)σ*(SiSi) excitation energy in Si2Me6, and g can then be evaluated (Table 7, Figure 11). The observed and the

of a small-radius exciton in a short oligosilane, which leads to isomerism. Classical Treatment. It has been known for some time that the permethylated linear oligosilane series SinMe2n+2 represents an exemplary realization of the classically anticipated response of singlet excitation in one dimension to quantum confinement.16,17 In very long chains electronic excitation is present as a large-radius exciton with an extrapolated excitation energy of 30 000 cm−1 (Table 7). The exciton extends over two or three Table 7. Experimental16 (exp.) and Calculated (calcd., B3LYP/16 def2-TZVP) Exciton Band Half-Width (B, in cm−1) and Exciton−Phonon Coupling Constant (g) for the Series SinMe2n+2, n = 3−16 and (extrapolated) n = ∞ n 3 4 5 6 7 8 9 10 12 14 16 ∞

B exp.a

B calcdb

12 900 16 900 19 900 21 900 23 400 24 500 25 400f 25 900 27 300f 28 100f 28 300 30 000f

e

13 100 16 400 19 400 21 500 23 000 24 200 25 200 26 000 27 100 27 800 28 400 28 800g

g exp.c

g calcdd

1.40 1.07 0.90 0.82 0.77 0.73 0.71f 0.69 0.66f 0.64f 0.63 0.60

1.32 1.05 0.89 0.80 0.75 0.71 0.68 0.66 0.64 0.62 0.61 0.60

a

From observed absorption band maxima16 and the latest13 Si2Me6 σ(SiSi)σ*(SiSi) EVA estimate of 60 000 cm−1. bDifference between the EVA = 60 000 cm−1 excitation energy estimated for Si2Me6 and the EVA value calculated for a alonger chain. cRatio S/B of the experimental site distortion energy S = 18 000 cm−1, taken as one-half of the estimated Stokes shift (ESS) in Si2Me6, to the observed exciton band half width B exp. dRatio S/B of the calculated site distortion energy S = 17 300 cm−1, obtained from the computed EVE value for Si2Me6,20 to the calculated exciton band half width B calcd. eThe Si3Me8 EVA value for the σσ* state was taken as 46 900 cm−1 from a B3LYP/def2-TZVP calculation.47 fThis value is extrapolated from experimental data.16 g Extrapolated from computed EVE values.

Figure 11. Experimental16 (red) and calculated (blue, B3LYP/def2TZVP) exciton phonon coupling constant g fitted from data for the series SinMe2n+2, n = 3−16.

calculated g values agree very well. The calculated g crossover value of 0.75 for n = 7 exactly matches theoretical theoretical result for the critical value for the collapse of the large radius exciton. In permethylated oligosilanes, the abrupt collapse of the large-radius to a small-radius self-trapped exciton is observed to occur at n = 7, and the degree of agreement with the simple classical analysis is astonishing. For n = 8, the Stokes shift is small, only 800 cm−1 (Table 2). In the fully extended alltransoid conformer of permethylheptasilane (n = 7), the Stokes shift is still only 1500 cm−1 (Table 5) and the excitation clearly extends over all six Si−Si bonds, whereas in another conformer or conformers the Stokes shift is large (6600 cm−1, Table 5), and the excitation is mostly localized on one Si−Si bond. For n = 6 and shorter chains, only the small-radius exciton is present and the Stokes shift increases even further, to 16 800 cm−1 (Table 5) in permethylated tetrasilane. Quantum Chemical Calculations. It is seen from Tables 2, 5, and 7 that the observed behavior of singlet excitation in permethylated oligosilane series agrees essentially perfectly with the results of our TD-DFT calculations. This is certainly true of state energies, but also the very large calculated S0−S1 oscillator strengths of the delocalized excitations fit observations qualitatively, although they are excessive.11 The transition is dominated by a HOMO to LUMO promotion and the transition dipole moment is directed along the Si backbone of the chain. The S0−S1 transition density, which is less

dozen Si−Si bonds3,4 and emits with a nearly vanishing Stokes shift (400 cm−1 in permethylated hexadecasilane and less in longer chains, cf. Table 2). In very short chains, excitation is present as a small-radius strongly localized exciton emitting with a huge Stokes shift (16 800 cm−1 in permethylated tetrasilane, Tables 5 and 7). The transition from the former to the latter is induced by a gradual change in the exciton− phonon coupling constant g as the chain is shortened and is expected to occur when g reaches a critical value of 0.7− 0.75.49,50 The value of g is defined as S/B, where S is the site distortion energy and B is the half-width of the exciton band. The exact value of S is experimentally unknown but it can be assumed to be nearly independent of n. It can be roughly estimated as equal to half of the Stokes shift for a single site, since in approximate theory the destabilization of the ground state is equal to the stabilization of the excited state at the optimized distorted geometry of the small-radius exciton localized to a single site.49,50 Unfortunately, only a rough estimate of the Stokes shift for hexamethyldisilane can be made, since disilanes do not fluoresce observably. Extrapolation from longer chains suggests that Si2Me6 would fluoresce close to 24 700 cm−1.16 A computational estimate is 25 500 cm−1 (B3LYP/def2-TZVP).20 The vertical σ(SiSi)σ*(SiSi) excitation 10550



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The nature of bonding at the partly TBP Si atoms in the selftrapped excitons is revealed by the NHO analysis. This shows that excitation leads to a loss of electron density in the bond between the two silicon atom on which the exciton is localized, facilitated by their rehybridization from sp3 to sp2, with the long Si−Si bond that carries the “hole” formed by essentially pure 3p orbitals. If we view the distorted relaxed exciton structure as an effort by the molecule to accommodate a hole and a particle simultaneously, where, then, is the corresponding “particle”? We find that extra electron density is moved from bonds to the σ* antibonding region between Si atoms, and this explains why the NHOs along the Si backbone that show increased occupation numbers deviate from the line of centers. The bonding pattern in the terminal bond-stretch minimum resembles that of the wide Si−Si−Si valence angle minimum in octamethyltrisilane19 and could also have similarities to recently described σ̂ long bond species.53 The hybridization pattern at the minima also connects the shallow minima with dimethylsilylene extrusion conical intersections and helps us to understand the small quantum yields of room temperature fluorescence from the short chains and its complete absence from the shortest chains even at the lowest temperatures.17 Returning now to somewhat longer chains that still support localized excitons, it is noteworthy that there is distinct evidence for sigma conjugation of the distorted site with the rest of the chain. While the transition density is primarily localized on just two Si atoms, geometrical distortions are important for up to six Si atoms and their in-plane methyl substituents. If the excitation were fully localized at a single Si− Si bond we would not expect geometrical differences from the ground state equilibrium structure at the other Si atoms, nor any differences in the emission energies from chains of different length, and this is clearly not the case. This reflects the semidelocalized nature of the blue excitons and connects them to the delocalized excitation limit. While the main purpose of this study is to report new structures believed to be responsible for the blue emission in linear oligosilane chains, it also addresses a piece of the larger puzzle of how sigma bonds accommodate electronic energy due to excitation. At the longer chain limit Si−Si bonds stretch and the all-Si valence and dihedral angles increase. The latter two geometry distortions are also found in the relaxation of the radical anion (negative polaron), which houses an excess electron in the σ* orbital.54 Hence the blue self-trapped excitons differ from the delocalized excitons and the extended radical anions mainly in terms of more pronounced Si−Si bond stretching and angular distortion toward TBP geometries. Considering now the broader implications of the results obtained here, we note that structures analogous to the bondstretch minima described here have been found for the disilane and trisilane,19,20 albeit with extremely low barriers to funnels. It is thus conceivable that excitation could be localized in short segments of suitable conformers of longer chains, possibly giving rise to more minima than presented here. Blue emission could thus also be expected from chains longer than heptasilane if their structure is constrained and sigma conjugation is less effective.47 The reduction of the stabilizing effects of sigma delocalization can be accomplished by dihedral angle twisting, which permits an effective fragmentation of an oligosilane chromophoric unit to units for which exciton self-trapping and blue emission are favored. The effects of sigma conjugation are cut off at backbone dihedral angles near 90°.1,9,10 The situation can be illustrated on the case of Si7Me16. When the terminal

sensitive to the choice of molecular orbital basis (canonical or localized), confirms this result. The decreasing calculated difference between the absorption and emission oscillator strength as the chain length increases is compatible with the calculated decrease in structural differences between the optimized S1 and all-transoid ground state geometries and with the observed qualitative trend in the degree of FC allowedness of the observed absorption and emission bands. The delocalized excitons have equilibrium geometries very similar to those of the ground state. The calculated geometrical rearrangements occur throughout the silicon backbone, as one would expect for delocalized excitation. In the excited state, the Si−Si bonds are slightly stretched and the SiSiSi valence angles increased. These effects are most pronounced toward the center of the chain and thus correlate with the transition density, which is also maximized in the middle of the oligosilane chain. However, the calculations provide information that goes well beyond the classical model in that they predict the specific nature of the exciton−phonon coupling. Concrete structures for the geometrical distortions that occur upon relaxation of the excited singlet state into a localized self-trapped exciton are of particular interest, and relate to previous proposals for the assignment of the nature of the blue emission.1,5,22,51,52 For each chain length of seven or fewer silicon atoms, we find multiple S1 minima that can be described as self-trapped exciton minima and are probably collectively responsible for the observed blue emission. As expected, the degree of excitation localization increases as the chain is shortened, and the geometrical differences relative to the ground state become more pronounced. We find that the label “bond-stretch” minimum is somewhat of a misnomer, since the increase in the Si−Si bond length relative to the ground state is accompanied by very significant changes in valence angles, too. Indeed, during a computational geometry optimization on the S1 surface, starting with a ground state equilibrium geometry with a single Si−Si bond stretched is not enough to produce convergence to one of these minima. As the chain is shortened, an increasing fraction of the reorganization energy is associated with valence angle distortions as opposed to Si−Si bond length elongation. In a sense, the purest self-trapped excitons in the oligosilanes considered presently are therefore found in the shortest chain, tetrasilane. These structures have two adjacent silicon atoms with valence angles intermediate between tetrahedral and trigonal bipyramidal (TBP), as was found recently for octamethyltrisilane.19 The absence of fluorescence in trisilanes and disilanes is apparently related to the extreme valence angle distortions that lead very close to conical intersections that are ultimately responsible for return to the ground state with the extrusion of dimethylsilylene from the peralkyloligosilane, the signature photochemical event in this class of compounds.19,20 The simplest way to describe the geometries of these bond-stretch minima with their wide Si−Si−Si valence angles, contracted C− Si−Si valence angles, and elongated Si−Si bonds is to refer to TBP bonding between dimethylsilylene fragments. For a lack of a better brief label we shall continue to use the term “bondstretch minimum”, and this allows us to view the multiple minima in the S1 surface as describing “bond-stretch isomers”. Such isomerism is rare in the ground state, but surely common in the excited states of systems that support self-trapped excitons, since it is essential for exciton hopping. 10551



■

dihedral angle in an otherwise all-transoid backbone reaches approximately 125°, this chain begins to behave like Si6Me14 and no longer supports a delocalized exciton, and this underlies the ability of Si7Me16 to display both ordinary and blue fluorescence, depending on excitation wavelength.

CONCLUSIONS We find a remarkable agreement between experiment and both the classical simple models for exciton behavior and the TDDFT description of oligosilane excited states. Not only are the FC-allowed and forbidden emission energies reproduced, but the crossover between these two regimes is reproduced as well. The calculations show that the self-trapped (blue) exciton in permethylated oligosilanes is of σσ* nature and is localized primarily over one Si−Si bond, as best seen in images of the transition density. At equilibrium geometry, this bond is stretched and is formed by nearly pure 3p orbitals of the two sp2 hybridized silicon atoms, which adopt geometries intermediate between tetrahedral and trigonal bipyramidal. When the excitation is viewed as a particle-hole pair, the stretched Si− Si bond accommodates the “hole”, while the “particle” is accommodated in the antibonding orbitals of the sigma bonds present. In longer chains there is a strong preference to delocalize the excitation but dihedral angle twisting has the capacity to reduce or nearly completely break stabilization of excited states by sigma conjugation and can localize the excitation. Blue emission in longer chains containing distorted or conformationally confined Si backbones is to be expected. In summary, the results of the numerical computations presented here confirmed the qualitative concepts and understanding based on the simple classical exciton model, and have gone well beyond them by providing specific geometrical structures for the large-radius excitons and smallradius bond-stretch excitons in permethylated oligosilanes, responsible for their ordinary and blue emission, respectively. We did not address the origin of the green emission, occasionally also observed in constrained peralkylated oligosilanes, which we believe to be associated with a geometrically different kind of small-radius self-trapped exciton.19,20,47 ASSOCIATED CONTENT

S Supporting Information *

S0 and S1 (large radius exciton) equilibrium geometries for 16− 7. Additional 5*β′ conformer. MS-CASPT2 description of 4*α and 4*β. Additional molecular orbitals and S0−S1 transition densities. NHO line of centers deviation for 4, 4*α and 4*β. Fitting parameters for estimates of the exciton−phonon coupling constant g. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]; phone: 303-492-6519. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This material is based upon work supported by the National Science Foundation under Grant No. (CHE-1265922). 10552



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