Efficient calculation of electronic absorption spectra by means of intensity selected TD-DFTB 1,2 1 1 2 Robert R¨ uger , You Lu , Erik van Lenthe and Lucas Visscher 1 Scientific 2 Computing & Modelling NV Division of Theoretical Chemistry, VU University Amsterdam Summary of ground state SCC-DFTB i 1 hφi|H |φii + 2 ˆ0 N atom X AB 1 ∆qAγAB ∆qB + 2 N atom X I UAB AB Orbital energies: P P I LCAO basis: φi(~ r) = A µ∈A cµiχµ(~r) I Hamiltonian matrix elements are parametrized from DFT: free atom ε Dµ E for µ = ν ˆ 0 0 0 ˆ hχµ|H |χν i = χµ T + vKS [ρA + ρB ] χν for µ ∈ A, ν ∈ B, A 6= B 0 otherwise Self-consistent charge correction: P 0 P 0 I Total density: ρ = ρ + δρ = A ρA + A δρA ~ A|) I Density fluctuations: δρA(~ r) ≈ ∆qAξA(|~r − R free atom I ∆qA = qA − q are calculated by Mulliken charge A analysis from the molecular orbitals: Nel X X X 1 qA = cµiSµν cνi + cνiSνµcµi 2 i ν γAB = d3~r 2 1 δ Exc 3 0 d ~r ξA(~r) + 0 |~r − ~r | δρ(~r)δρ(~r 0) Z 3 ρ Example: Fullerene C60 [2] = δij δab (εa − εi) + 4 εa − Z Z S Kia,jb = d3~r d3~r 0φi(~r)φa(~r) 6 I large number of low-lying single orbital transitions with almost no oscillator strength: Number of single orbital transitions all 18% (fia > 0.005) Absorption [arb. units] { 800 all 15% (fia > 0.01) 500 400 300 3 4 200 all fia > 0.001 fia > 0.02 fia > 0.1 1000 100 10 1 1 2 5 6 (εa − εi ) [eV] I intensity all 7% (fia > 0.05) S/T p εiKia,jb εb − εj 2 δ Exc 1 0 0 + φ (~ r )φ (~ r ) j b 0 0 GS |~r − ~r | δρ(~r)δρ(~r ) ρ Eigenvalue equation that gives excitation energies and excited states in the basis of the single orbital transitions I Ω too large to be diagonalized directly ⇒ iterative methods selection reduces the number of excitations in the target window and makes the calculation of interesting quantities possible, e.g. spectral shift upon embedding of the chromophore into the protein environment: I 2 3 4 5 6 ∆I [eV] Tyrosine Ubiquitin Goal: Find an approximation for the coupling matrices K that is easy to calculate within the SCC-DFTB framework! lot of single orbital transitions have zero oscillator strength due to symmetry I essentially the same absorption spectrum can be obtained with only 25% of the basis I quality of the approximation decreases as more transitions are removed Original derivation by Niehaus: I Atomic contributions to the transition density: X pij (~r) = φi(~r)φj (~r) = pij,A(~r) Results: Carefully applied intensity selection preserves the resulting absorption spectra within the general accuracy of the TD-DFTB method, while the reduced basis size together with the smaller number of excitations per energy interval reduce the computational effort considerably. Linear response TD-DFTB [3] S/T Ia 7 Absorption [arb. units] √ 2 5 λ [nm] Linear response excitations from Casida’s eq.: Ω F~I = ∆I F~I S/T Ωia,jb 4 Energy[eV] Example: Ubiquitin (1231 atoms) all 25% (fia > 0.001) { Single orbital transitions: I excited states and excitation energies are straightforward to calculate I in itself not a very good approximation for the true excitations I can be used as the basis of a Nocc × Nvirt dimensional space in which other approximations of the excited states can be expressed DFTB (Quasinano) Question: Can one remove the single orbital transitions with small oscillator strengths from the basis without changing the resulting linear response excitations too much? 0 ξB (~r 0) Summary of linear response TD-DFT DFT (PBE-D3(BJ)+TZP) Observations for non-intensity selected TD-DFTB: I The dimension of the single orbital transition space grows quadratically with system size. I Only a limited number of excitations can be calculated when using iterative eigensolvers. ~ia contribute little to the I Single orbital transitions with small transition dipole moments d q P 2(εa−εi) ~ ~ transition dipole moments of the linear response excitations: dI = ia F d ia,I ia ∆I Precalculate γ for all elements and distances using DFT: Z experimental [5] Intensity selected TD-DFTB µ∈A I TD-DFTB is an approximation of TD-DFT ⇒ same problems and limitations I TD-DFTB results are on average not much worse than TD-DFT, see [3] and [4] I TD-DFTB is several orders of magnitude faster than TD-DFT; possible applications: I structures which are too large to be treated with regular TD-DFT I pre-screening of a large number of different structures Example: fac-Ir(ppy)3 Absorption [arb. units] EDFTB = Nel X The pros and cons of TD-DFTB [1] 240 250 260 270 280 290 300 λ [nm] I easier to use than QM/MM with model systems! A I Multipole expansion and monopolar approximation: ~ A|) pij,A(~r) ≈ qij,AξA(|~r − R X X 1 qij,A = cµiSµν cνj + cνiSνµcµj 2 ν µ∈A I Insert expansion into the (singlet) coupling matrix: X S Kia,jb = qia,Aγ˜AB qjb,B Z γ˜AB = I 3 d ~r Z AB 2 1 δ Exc d ~r ξA(~r) + 0 |~r − ~r | δρ(~r)δρ(~r 0) 3 0 ρGS Ignore charge fluctuations and assume γ˜AB ≈ γAB . Conclusions & Outlook TD-DFTB is a useful tool to quickly calculate the absorption spectrum of large structures. I Intensity selection can be used to further reduce the computational cost of TD-DFTB. I An implementation of TD-DFTB with intensity selection will be available in ADF2014. I References ξB (~r 0) ⇒ Straightforward way to build the coupling matrix from ground state results and parameters! [1] Augusto F. Oliveira et al., J. Braz. Chem. Soc. 20, 1193-1205 (2009) [2] M. E. Casida and D. P. Chong, Recent Advances in Density Functional Methods, 1, p. 155, 1995 [3] Thomas Niehaus et al., Phys. Rev. B 63, 085108 (2001) [4] Fabio Trani et al., J. Chem. Theory Comput. 2011, 7, 3304–3313 [5] J. Fine et al., Molecular Physics 110:15-16, 1849-1862 (2012) mail: [email protected]

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