Inverse quantum confinement in luminescent Silicon ... - TDDFT.org

Inverse quantum confinement in luminescentSilicon quantum dotsThomas NiehausBremen Center for Computational Materials ScienceElectron Dynamics in Complex Systems4th Time-Dependent Density-Functional Theory:Prospects and ApplicationsBenasque – January 2010Thomas Niehaus, TDDFT Benasque 2010 1 of 46

Outline1 Motivation2 Theoretical MethodsApproximate TDDFT ⇒ TD-DFTB3 Application to Silicon NanostructuresHydrogenated SiQD - Simple and boring?Functionalization of Dots - Making them usefulSurface reconstruction - Energetics and SpectraSilicon nanowires - Confinement in radial and axial dimensions4 Summary and OutlookThomas Niehaus, TDDFT Benasque 2010 3 of 46

What is a Quantum Dot?CharacteristicsAn object that confineselectrons in all three dimensionsMade mostly fromsemiconductor material.Extension of nanometersElectronic level spacingcomparable k B TThomas Niehaus, TDDFT Benasque 2010 4 of 46

Properties of Quantum Dots – Size dependent EmissionFluorescence of Cadmium SelenideQuantum Dotswww.chemie.uni-hamburg.de/pc/weller/Range of Emission WavelengthMichalet Science (2005)Thomas Niehaus, TDDFT Benasque 2010 5 of 46

Properties of Quantum Dots – Advantages over DyesDyeMouse intestinal sectionhttp://probes.invitrogen.comDotMulti-color labellingpossibleThomas Niehaus, TDDFT Benasque 2010 6 of 46

Properties of Quantum Dots – Advantages over DyesMedintz, Nature Materials (2005)top: Nucleus labelled with Dot, microtubules with Dyebottom: vice versaQuantum dots show only limitedphotobleachingThomas Niehaus, TDDFT Benasque 2010 7 of 46

Quantum dots as markers in biologyIn vitroIn vivoFunctionalization withantibodies or aptamersMouse cerebellumhttp://probes.invitrogen.comRat tumorGao, Science (2004)Thomas Niehaus, TDDFT Benasque 2010 8 of 46

Biocompatibility of Cadmium based DotsThomas Niehaus, TDDFT Benasque 2010 9 of 46

Silicon ???Bulk Siliconphonon assisted emissionfund. band gap of 1.13 eVThomas Niehaus, TDDFT Benasque 2010 10 of 46

Silicon ???Bulk Siliconphonon assisted emissionfund. band gap of 1.13 eVSilicon Quantum DotsDipole-allowed transitionBright emission in the visibleEmission from SiH nanoparticles underUV excitation, Belomin APL (2002)Thomas Niehaus, TDDFT Benasque 2010 10 of 46

Outline1 Motivation2 Theoretical MethodsApproximate TDDFT ⇒ TD-DFTB3 Application to Silicon NanostructuresHydrogenated SiQD - Simple and boring?Functionalization of Dots - Making them usefulSurface reconstruction - Energetics and SpectraSilicon nanowires - Confinement in radial and axial dimensions4 Summary and OutlookThomas Niehaus, TDDFT Benasque 2010 11 of 46

Solution strategies in TDDFTTime domainFrequency domaini ∂ ∂t φ i(r, t) =[− 1 ]2 ∇2 + v KS [ρ](r) φ i (r, t)∑Ω ijkl Fkl I = ωI2klF Iij◮ Valid for strong fields◮ Full spectrum at once◮ Only occ. orbitals required◮ No information on darkstatesFormally N 3 , can be made N◮ Detailed information on theexcited state◮ Singlet and Triplet◮ Both occ. and virt. orbitalsrequiredFormally N 6 , can be made N 3Thomas Niehaus, TDDFT Benasque 2010 12 of 46

Density Functional based Tight Binding (DFTB)Main approximations - Ground stateMinimal AO basis: ψ i (r) = ∑ µ c µiφ µ (r − R A )Density divided into reference density (ρ 0 ) and fluctuation:ρ = ρ 0 + δρTotal energy:E tot = ∑ i∑c µi Hµν[ρ 0 0 ]c νi + ∑ γ AB ∆q A ∆q B + U repµνABTwo-center approximation, matrix elements calculated not fittedγ AB represents e-e interaction (including xc)Variational principle yields molecular orbitals and energiesSeifert et al. Z. Phys. Chem. 267 529 (1986) Elstner et al. PRB 58 7260 (1998)Thomas Niehaus, TDDFT Benasque 2010 13 of 46

DFTB approximation of TDDFT = TD-DFTB..jiEquation to solve (Singlets):∑kl[ω2ij δ ik δ jl + 2 √ ω ij K ij,kl√ωkl]FIkl = ω 2 I F IijCouplingmatrix K leads to correction of single-particle picture∫∫DFT: K ij,kl = 2( 1ψ i (r) ψ j (r)|r − r ′ | + δv )xc(r)δρ(r ′ ψ k (r ′ ) ψ l (r ′ ) drdr ′)Niehaus, PRB 63 085108 (2001)Thomas Niehaus, TDDFT Benasque 2010 14 of 46

DFTB approximation of TDDFT = TD-DFTB.jiEquation to solve (Singlets):∑kl[ω2ij δ ik δ jl + 2 √ ω ij K ij,kl√ωkl]FIkl = ω 2 I F Iij.Get single particle energies {ɛ i } from DFTBCouplingmatrix K leads to correction of single-particle picture∫∫DFT: K ij,kl = 2DFTB: K ij,kl = 2 ∑ AB( 1ψ i (r) ψ j (r)|r − r ′ | + δv )xc(r)δρ(r ′ ψ k (r ′ ) ψ l (r ′ ) drdr ′)q ij Aγ AB (|R A − R B |, U A , U B ) q klBNiehaus, PRB 63 085108 (2001)Thomas Niehaus, TDDFT Benasque 2010 14 of 46

TD-DFTB performanceExcitation energies (eV)Molecule State Exp.TD-DFTB TDDFT 1ω I ω ij ω I ω ijEthylen 3 B 1u (π → π ∗ ) 4.40 5.47 6.30 4.16 5.661 B 1u (π → π ∗ ) 7.65 7.81 6.30 7.44 5.66Propynal 3 A ′′ (n → π ∗ ) 2.99 4.04 4.04 2.74 3.151 A ′′ (n → π ∗ ) 3.56 4.04 4.04 3.37 3.15[1] BPW91//6-311+G**Thomas Niehaus, TDDFT Benasque 2010 15 of 46

TD-DFTB performanceExcitation energies (eV)Molecule State Exp.TD-DFTB TDDFT 1ω I ω ij ω I ω ijEthylen 3 B 1u (π → π ∗ ) 4.40 5.47 6.30 4.16 5.661 B 1u (π → π ∗ ) 7.65 7.81 6.30 7.44 5.66Propynal 3 A ′′ (n → π ∗ ) 2.99 4.04 4.04 2.74 3.151 A ′′ (n → π ∗ ) 3.56 4.04 4.04 3.37 3.15[1] BPW91//6-311+G**Mean absolute error (eV)BPW91TD-DFTB 6-311+G** STO-3GINDO/SSing. (16 Com.) 0.38 0.36 1.19 0.56Trip. (13/11 Com.) 0.64 0.37 0.49 1.38Thomas Niehaus, TDDFT Benasque 2010 15 of 46

Numerical issuesSum of CPU times (ground + excited state) for test set of 10 organicmolecules (< 10 atoms). TDDFT results are for the BPW91 functional.1e+041e+03CPU time [s]1e+02101TD-DFTB INDO/S STO-3G 6-311+G**Thomas Niehaus, TDDFT Benasque 2010 16 of 46

Numerical issuesScalingIterative subspace methods for diag.Formal scaling of N 6 reduced to N 3∑Ω ij,kl v kl =klωijv 2 ij + 4 √ ∑ω ij qµ( ∑ ijµ νγ µν( ∑ klq klν√ωkl v kl ))CPU time [s]1010101054323.293.30EXC. STATEEXC. ST. GRADIENTGROUND STATE2.92Analytical excited state gradients from auxiliary Lagrangianapproach (Furche and Ahlrichs)⇒ Heringer et al. JCC 28 2589 (2007)500 1000# basis functionsThomas Niehaus, TDDFT Benasque 2010 17 of 46

Outline1 Motivation2 Theoretical MethodsApproximate TDDFT ⇒ TD-DFTB3 Application to Silicon NanostructuresHydrogenated SiQD - Simple and boring?Functionalization of Dots - Making them usefulSurface reconstruction - Energetics and SpectraSilicon nanowires - Confinement in radial and axial dimensions4 Summary and OutlookThomas Niehaus, TDDFT Benasque 2010 18 of 46

Generation of modelsSpherical fragments Si 1−199from bulk SiPassivation with Hydrogen andrelaxationSi-Si bonds 2.33-2.38 ÅT d symmetryd(Si 5 ) = 0.45 nm to d(Si 199 ) =2.0 nmThomas Niehaus, TDDFT Benasque 2010 19 of 46

Validation of TD-DFTBS 1 − S 0 energy [eV]DFTB PBE B3LYP MR-MP2 expSiH 4 10.3 8.76 9.25 8.8Si 5 H 12 6.40 6.09 6.66 6.56 6.5Si 29 H 36 4.42 3.65 4.52 4.45Si 35 H 36 4.37 3.56 4.42 4.33Thomas Niehaus, TDDFT Benasque 2010 20 of 46

Validation of TD-DFTBS 1 − S 0 energy [eV]DFTB PBE B3LYP MR-MP2 expSiH 4 10.3 8.76 9.25 8.8Si 5 H 12 6.40 6.09 6.66 6.56 6.5Si 29 H 36 4.42 3.65 4.52 4.45Si 35 H 36 4.37 3.56 4.42 4.33B3LYP functional of choice for SiQDDFTB is fine for larger dotsThomas Niehaus, TDDFT Benasque 2010 20 of 46

UV-Vis absorption energies of H-SiQDAnticipated Particle-in-a-box behaviourWang, JPC C 111 12588 (2007), APL 90 123116 (2007)Thomas Niehaus, TDDFT Benasque 2010 21 of 46

Experimental reports on SiQD photoluminescencePeak PL versus Si nanocrystal sizeWilcoxon et al. PRB 60 2704 (1999)Thomas Niehaus, TDDFT Benasque 2010 22 of 46

Luminescence and excited state processesThomas Niehaus, TDDFT Benasque 2010 23 of 46

Luminescence properties of H-SiQDHuge Stokes shift for small particlesWang, JPC C 111 12588 (2007), APL 90 123116 (2007)Thomas Niehaus, TDDFT Benasque 2010 24 of 46

Structure relaxation in the excited stateHOMO LUMO LUMO HOMOS 0 opt. S 1 opt.S 1 optimum of Si 35 H 36Formation of a self-trapped excitonThomas Niehaus, TDDFT Benasque 2010 25 of 46

Functionalization of dotsWarner, Angew. Chem. (2005)Functionalization with allylamine leads toIncreased solubilityDecreased aggregationDecreased oxidationDoes coating destroy the optical properties?Thomas Niehaus, TDDFT Benasque 2010 26 of 46

Generation of modelsNo preference in adsorption siteSite Binding energy [eV]Si 35 H 36 1 2.442 2.433 2.44Si 59 H 60 1 2.362 2.443 2.43Complete coverage impossibleWang, JPC C 111 2394 (2007)Thomas Niehaus, TDDFT Benasque 2010 27 of 46

Simulated and experimental IR spectrumPartial oxidation?Wang, JPC C 111 2394 (2007)Warner, Angew. Chem. (2005)Thomas Niehaus, TDDFT Benasque 2010 28 of 46

TD-DFTB absorption energiesStrong quantum size effect – Littledependence on passivationThomas Niehaus, TDDFT Benasque 2010 29 of 46

Maximum absorption and emission peaks# Si # allylamine Absorption [nm] Emission [nm] Shift [nm]10 0 205 421 21629 0 281 405 12435 0 284 429 1454 284 425 1418 297 462 16559 0 334 390 56Exp. 320 480 160ResultsGood agreement between exp. and theory for Si 35Red light emission requires larger dots as Si 59Thomas Niehaus, TDDFT Benasque 2010 30 of 46

Conclusions on allylamine functionalizationAllylamine is a promising candidate for protection andfunctionalization of Silicon quantum dots withoutperturbation of their favorable optical properties.Wang, Zhang, Niehaus, Frauenheim JPC C 111 2394 (2007)Thomas Niehaus, TDDFT Benasque 2010 31 of 46

Reconstructed Silicon Quantum DotsDimerformation byH 2abstractionSi 29 H 36Si 29 H 24Thomas Niehaus, TDDFT Benasque 2010 32 of 46

Reconstructed Silicon Quantum DotsDimerformation byH 2abstractionSi 29 H 36Si 29 H 24Formation energy:E f = E re +(∆n/2)E H2 −E blThomas Niehaus, TDDFT Benasque 2010 32 of 46

Optical propertiesInverse quantum confinement effectWang, Zhang, Lee, Frauenheim, Niehaus APL 93 243120 (2008)Thomas Niehaus, TDDFT Benasque 2010 33 of 46

Effects of confinement in radial and axial dimensions30Absorption strength [arb. units]252015105Si 112H98Si 128H110Si 144H122Si 160H134Si 176H146Finite models of SiNW with [110]growth direction02.9 3 3.1 3.2 3.3Energy[eV]Absorption and emission spectra forSiNW with d = 0.84 nmExcited state relaxation does not quenchluminescenceThomas Niehaus, TDDFT Benasque 2010 34 of 46

Size dependence of exciton localizationHOMO and LUMOd=0.84 nm and l=1.17 nmEnergy[eV]654321d=0.84nmEnergy[eV]654321d=0.98nmEnergy[eV]00 0.5 1 1.5 2 2.5 3Length[nm]654321d=1.08nm00 0.5 1 1.5 2 2.5 3Length[nm]Energy[eV]00 0.5 1 1.5 2 2.5 3Length[nm]654321d=1.24nm00 0.5 1 1.5 2 2.5 3Length[nm]d=0.84 nm and l=2.32 nmExciton delocalizes for rods with l > 2 nmWang, Zhang, Frauenheim, Niehaus JPC C 113 12935 (2009)Thomas Niehaus, TDDFT Benasque 2010 35 of 46

Outline1 Motivation2 Theoretical MethodsApproximate TDDFT ⇒ TD-DFTB3 Application to Silicon NanostructuresHydrogenated SiQD - Simple and boring?Functionalization of Dots - Making them usefulSurface reconstruction - Energetics and SpectraSilicon nanowires - Confinement in radial and axial dimensions4 Summary and OutlookThomas Niehaus, TDDFT Benasque 2010 36 of 46

Summary on Silicon nanostructuresMain findingsExcited state relaxation is crucialSelf-trapped excitons form for nanostructures < 2 nmOxidation is not the only source of redshifted emissionReconstruction has to be consideredThomas Niehaus, TDDFT Benasque 2010 37 of 46

Other TD-DFTB implementationsLinear response O(N)TD-DFTB for open systemsSi 717 H 300F. Wang et al. PRB 76 045114 (2007)Wang et al., to be published (2010)Nonadiabatic Molecular dynamicsNA-MD: FemtochemistryNiehaus et al. EPJD 35 467 (2005)Niehaus et al. EPJD 35 467 (2005)i ∂ ∂t σ = [HDFTB , σ]Thomas Niehaus, TDDFT Benasque 2010 38 of 46

Other TD-DFTB implementationsLinear response O(N)TD-DFTB for open systemsSi 717 H 300F. Wang et al. PRB 76 045114 (2007)Wang et al., to be published (2010)Nonadiabatic Molecular dynamicsNA-MD: FemtochemistryNiehaus et al. EPJD 35 467 (2005)i ∂ ∂t σ D = [H DFTBD, σ D ] + i∑α=L,RNiehaus et al. EPJD 35 467 (2005)Thomas Niehaus, TDDFT Benasque 2010 38 of 46Q α

Other TD-DFTB implementationsLinear response O(N)TD-DFTB for open systemsSi 717 H 300F. Wang et al. PRB 76 045114 (2007)Wang et al., to be published (2010)Nonadiabatic Molecular dynamicsNA-MD: FemtochemistryNiehaus et al. EPJD 35 467 (2005)Niehaus et al. EPJD 35 467 (2005)i ∂ ∂t |ψ ∑occi 〉 = H DFTB |ψ i 〉 ∧ M α ¨R α = −i〈ψ i | dHDFTBdR α|ψ i 〉 − dErepdR αThomas Niehaus, TDDFT Benasque 2010 38 of 46

Other TD-DFTB implementationsLinear response O(N)TD-DFTB for open systemsSi 717 H 300F. Wang et al. PRB 76 045114 (2007)Wang et al., to be published (2010)Nonadiabatic Molecular dynamicsNA-MD: FemtochemistryNiehaus et al. EPJD 35 467 (2005)Niehaus et al. EPJD 35 467 (2005)i ∂ ∂t |ψ ∑occi 〉 = H DFTB |ψ i 〉 ∧ M α ¨R α = −i〈ψ i | dHDFTBdR α|ψ i 〉 − dErepdR αThomas Niehaus, TDDFT Benasque 2010 38 of 46

www.dftb-plus.infoThomas Niehaus, TDDFT Benasque 2010 39 of 46

Thanks ∞Xian Wang, Quan-Song Li, Chensheng Lin andRuiqin ZhangThe EDiCS group at BCCMSBernhard GrundkötterChiYung YamSetiantoYong WangThomas Niehaus, TDDFT Benasque 2010 40 of 46

Density Functional based Tight-BindingAtomic DFT calculations ⇒ ρ A (r), φ µ(r), ɛ µ ⇒ ρ 0 = ∑ A ρ ASeifert et al. Z. Phys. Chem. 267 529 (1986) Elstner et al. PRB 58 7260 (1998)Thomas Niehaus, TDDFT Benasque 2010 41 of 46

Density Functional based Tight-BindingAtomic DFT calculations ⇒ ρ A (r), φ µ(r), ɛ µ ⇒ ρ 0 = ∑ A ρ AConstruction of molecular Hamiltonian H 0 µν[ρ 0]⎧⎨ ɛ µ : µ = νHµν 0 = 〈φ µ|H KS [ρ A + ρ B ]|φ ν〉 : µ ∈ A, ν ∈ B⎩0 : otherwiseSeifert et al. Z. Phys. Chem. 267 529 (1986) Elstner et al. PRB 58 7260 (1998)Thomas Niehaus, TDDFT Benasque 2010 41 of 46

Density Functional based Tight-BindingAtomic DFT calculations ⇒ ρ A (r), φ µ(r), ɛ µ ⇒ ρ 0 = ∑ A ρ AConstruction of molecular Hamiltonian H 0 µν[ρ 0] + H 1 µν[ρ − ρ 0]⎧⎨ ɛ µ : µ = νHµν 0 = 〈φ µ|H KS [ρ A + ρ B ]|φ ν〉 : µ ∈ A, ν ∈ B Hµν 1 = 1 ∑⎩2 Sµν (γ µα + γ να)∆q α0 : otherwiseα∫∫ ( )γ µν = |φ µ(r)| 2 1|r − r ′ | + fxc[ρ0] |φ ν(r ′ )| 2 drdr ′Seifert et al. Z. Phys. Chem. 267 529 (1986) Elstner et al. PRB 58 7260 (1998)Thomas Niehaus, TDDFT Benasque 2010 41 of 46

Density Functional based Tight-BindingAtomic DFT calculations ⇒ ρ A (r), φ µ(r), ɛ µ ⇒ ρ 0 = ∑ A ρ AConstruction of molecular Hamiltonian H 0 µν[ρ 0] + H 1 µν[ρ − ρ 0]⎧⎨ ɛ µ : µ = νHµν 0 = 〈φ µ|H KS [ρ A + ρ B ]|φ ν〉 : µ ∈ A, ν ∈ B⎩0 : otherwise∫∫γ µν =Hµν 1 = 1 ∑2 Sµν (γ µα + γ να)∆q α|φ µ(r)| 2 ( 1|r − r ′ | + fxc[ρ0] )|φ ν(r ′ )| 2 drdr ′ ≈ γ µν(U A , U B , |R A − R B |)αSeifert et al. Z. Phys. Chem. 267 529 (1986) Elstner et al. PRB 58 7260 (1998)Thomas Niehaus, TDDFT Benasque 2010 41 of 46

Density Functional based Tight-BindingAtomic DFT calculations ⇒ ρ A (r), φ µ(r), ɛ µ ⇒ ρ 0 = ∑ A ρ AConstruction of molecular Hamiltonian H 0 µν[ρ 0] + H 1 µν[ρ − ρ 0]⎧⎨ ɛ µ : µ = νHµν 0 = 〈φ µ|H KS [ρ A + ρ B ]|φ ν〉 : µ ∈ A, ν ∈ B⎩0 : otherwise∫∫γ µν =Hµν 1 = 1 ∑2 Sµν (γ µα + γ να)∆q α|φ µ(r)| 2 ( 1|r − r ′ | + fxc[ρ0] )|φ ν(r ′ )| 2 drdr ′ ≈ γ µν(U A , U B , |R A − R B |)αTotal energyE tot = ∑ µνP µνHµν 0 + 1 ∑γ µν∆q µ∆q ν + E rep2µνSeifert et al. Z. Phys. Chem. 267 529 (1986) Elstner et al. PRB 58 7260 (1998)Thomas Niehaus, TDDFT Benasque 2010 41 of 46

Linear response in DFTB: TD-DFTBEquation to solve (singlets):K Sij,kl =∑kl∫∫[ωijδ 2 ik δ jl + 4 √ ω ij Kij,klS √ ]ωkl Fkl I = ωI2F Iij( )1ψ i (r) ψ j (r)|r − r ′ | + fxc[ρ] ψ k (r ′ ) ψ l (r ′ ) drdr ′Thomas Niehaus, TDDFT Benasque 2010 42 of 46

Linear response in DFTB: TD-DFTBEquation to solve (singlets):K Sij,kl =∑kl∫∫[ωijδ 2 ik δ jl + 4 √ ω ij Kij,klS √ ]ωkl Fkl I = ωI2F Iij( )1ψ i (r) ψ j (r)|r − r ′ | + fxc[ρ] ψ k (r ′ ) ψ l (r ′ ) drdr ′Approximation:K Sij,kl = ∑µναβc µi c νj c αk c βl∫∫( )1φ µ(r) φ ν(r)|r − r ′ | + fxc[ρ] φ α(r ′ ) φ β (r ′ ) drdr ′Thomas Niehaus, TDDFT Benasque 2010 42 of 46

Linear response in DFTB: TD-DFTBEquation to solve (singlets):K Sij,kl =∑kl∫∫[ωijδ 2 ik δ jl + 4 √ ω ij Kij,klS √ ]ωkl Fkl I = ωI2F Iij( )1ψ i (r) ψ j (r)|r − r ′ | + fxc[ρ] ψ k (r ′ ) ψ l (r ′ ) drdr ′Approximation:K Sij,kl = ∑µναβc µi c νj c αk c βl∫∫( )1φ µ(r) φ ν(r)|r − r ′ | + fxc[ρ] φ α(r ′ ) φ β (r ′ ) drdr ′Mulliken approximation: φ µ(r) φ ν(r) ≈ 1 2 Sµν (|φ µ(r)| 2 + |φ ν(r)| 2)Thomas Niehaus, TDDFT Benasque 2010 42 of 46

Linear response in DFTB: TD-DFTBEquation to solve (singlets):K Sij,kl =∑kl∫∫[ωijδ 2 ik δ jl + 4 √ ω ij Kij,klS √ ]ωkl Fkl I = ωI2F Iij( )1ψ i (r) ψ j (r)|r − r ′ | + fxc[ρ] ψ k (r ′ ) ψ l (r ′ ) drdr ′Approximation:K Sij,kl = ∑µναβc µi c νj c αk c βl∫∫( )1φ µ(r) φ ν(r)|r − r ′ | + fxc[ρ] φ α(r ′ ) φ β (r ′ ) drdr ′Mulliken approximation: φ µ(r) φ ν(r) ≈ 1 (2 Sµν |φ µ(r)| 2 + |φ ν(r)| 2)Kij,kl S ≈ ∑ ∫∫ ( )qµq ij νkl |φ µ(r)| 2 1|r − r ′ | + fxc[ρ] |φ ν(r ′ )| 2 drdr ′µνThomas Niehaus, TDDFT Benasque 2010 42 of 46

Linear response in DFTB: TD-DFTBEquation to solve (singlets):K Sij,kl =∑kl∫∫[ωijδ 2 ik δ jl + 4 √ ω ij Kij,klS √ ]ωkl Fkl I = ωI2F Iij( )1ψ i (r) ψ j (r)|r − r ′ | + fxc[ρ] ψ k (r ′ ) ψ l (r ′ ) drdr ′Approximation:K Sij,kl = ∑µναβc µi c νj c αk c βl∫∫( )1φ µ(r) φ ν(r)|r − r ′ | + fxc[ρ] φ α(r ′ ) φ β (r ′ ) drdr ′Mulliken approximation: φ µ(r) φ ν(r) ≈ 1 2 Sµν (|φ µ(r)| 2 + |φ ν(r)| 2)Kij,kl S ≈ ∑ qµq ij ν kl γ µν(U A , U B , |R A − R B |)µνThomas Niehaus, TDDFT Benasque 2010 42 of 46

Linear response in DFTB: TD-DFTBEquation to solve (triplets):∑klK Tij,kl =[ωijδ 2 ik δ jl + 4 √ ω ij Kij,klT √ ]ωkl Fkl I = ωI2∫∫ψ i (r) ψ j (r)F Iij∂ 2 E xc∂m(r)∂m(r ′ ) ψ k(r ′ ) ψ l (r ′ ) drdr ′Niehaus et al. PRB 63 085108 (2001)Thomas Niehaus, TDDFT Benasque 2010 43 of 46

Linear response in DFTB: TD-DFTBEquation to solve (triplets):∑klK Tij,kl =[ωijδ 2 ik δ jl + 4 √ ω ij Kij,klT √ ]ωkl Fkl I = ωI2∫∫ψ i (r) ψ j (r)F Iij∂ 2 E xc∂m(r)∂m(r ′ ) ψ k(r ′ ) ψ l (r ′ ) drdr ′Approximation:Kij,kl T ≈ ∑ ∫∫qµq ij νklµν|φ µ(r)| 2 ∂ 2 E xc∂m(r)∂m(r ′ ) |φν(r′ )| 2 drdr ′Niehaus et al. PRB 63 085108 (2001)Thomas Niehaus, TDDFT Benasque 2010 43 of 46

Linear response in DFTB: TD-DFTBEquation to solve (triplets):∑klK Tij,kl =[ωijδ 2 ik δ jl + 4 √ ω ij Kij,klT √ ]ωkl Fkl I = ωI2∫∫ψ i (r) ψ j (r)F Iij∂ 2 E xc∂m(r)∂m(r ′ ) ψ k(r ′ ) ψ l (r ′ ) drdr ′Approximation:Kij,kl T ≈ ∑ ∫∫qµq ij νklµν|φ µ(r)| 2 ∂ 2 E xc∂m(r)∂m(r ′ ) |φν(r′ )| 2 drdr ′≈ ∑ µq ij µq klµ M µ Onsite only, M µ from atomic DFTNiehaus et al. PRB 63 085108 (2001)Thomas Niehaus, TDDFT Benasque 2010 43 of 46

Curse and Blessing of Oxygen – The CurseOxidation leads toSize-independent emissionWeak luminescenceThomas Niehaus, TDDFT Benasque 2010 44 of 46

Curse and Blessing of Oxygen – The BlessingLi APL 91 043106 (2007)SiQD with Si=O bond optimized in S 0 and S 1Si 5 H 12 photodissociatesLUMO strongly localized around OxygenOxygen stabilizes Silicon coreThomas Niehaus, TDDFT Benasque 2010 45 of 46

Curse and Blessing of Oxygen – The Blessing but . . .HOMO-LUMO gap of hydrogenated and oxidized SiQDNo size dependence, emission in the IRThomas Niehaus, TDDFT Benasque 2010 46 of 46