TDDFT as a tool in biophysics

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TDDFT as a tool in biophysics The primary event in vision Robert Send Universität Karlsruhe 09.09.08 Robert Send TDDFT as a tool in biophysics 09.09.08 1 / 28

Outline 1 Human vision 2 The methods 3 The photoreaction mechanism: Three methods two results Robert Send TDDFT as a tool in biophysics 09.09.08 2 / 28

Human vision Robert Send TDDFT as a tool in biophysics 09.09.08 3 / 28

Human vision Robert Send TDDFT as a tool in biophysics 09.09.08 4 / 28

Human vision Robert Send TDDFT as a tool in biophysics 09.09.08 5 / 28

Human vision The chromophor: 11-cis retinal Robert Send TDDFT as a tool in biophysics 09.09.08 6 / 28

Human vision The photoproduct: all-trans retinal Robert Send TDDFT as a tool in biophysics 09.09.08 7 / 28

Human vision Photoreaction to all-trans retinal 1 Extremely fast (200 fs) Very efficient (quantum yield of 0.67) Very selective Tunable absorption maximum 1 P.Kukura et al. Science 310 1006 (2005) and references therein Robert Send TDDFT as a tool in biophysics 09.09.08 8 / 28

Human vision The chromophor: 11-cis retinal Polyene chain Protonated Schiff base: NH + 2 β-ionone ring Charge-transfer upon excitation 1 1 R. Mathies and L. Stryer Proc. Natl. Acad. Sci. USA 73 2169 (1967) Robert Send TDDFT as a tool in biophysics 09.09.08 9 / 28

TDDFT Ansatz: γ α (x, x ) = ia ( X α ia φ a (x)φ i (x ) + Y α ia φ i (x)φ a (x ) ) Robert Send TDDFT as a tool in biophysics 09.09.08 10 / 28

TDDFT Ansatz: γ α (x, x ) = ia ( X α ia φ a (x)φ i (x ) + Y α ia φ i (x)φ a (x ) ) Casida s equation: [( A(ωn ) B(ω n ) B(ω n ) A(ω n ) ) ω n ( 1 0 0 1 )] ( Xn Y n ) = 0 Robert Send TDDFT as a tool in biophysics 09.09.08 10 / 28

TDDFT Ansatz: γ α (x, x ) = ia ( X α ia φ a (x)φ i (x ) + Y α ia φ i (x)φ a (x ) ) Casida s equation: [( A(ωn ) B(ω n ) B(ω n ) A(ω n ) ) ω n ( 1 0 0 1 )] ( Xn Y n ) = 0 with C iajb (ω) = (A(ω) + B(ω)) iajb = (ɛ a ɛ i )δ ij δ ab (A(ω) B(ω)) iajb = (A(ω) + B(ω)) iajb + 2C iajb (ω) ( ) 1 dxdx φ i (x)φ a (x) r r + f xc(ω, x, x ) φ j (x )φ b (x ) Robert Send TDDFT as a tool in biophysics 09.09.08 10 / 28

Coupled Cluster methods Ground state energy: E = E HF + Φ HF Ĥe ˆT Φ HF with exp( ˆT 1 + ˆT 2 + ˆT 3...) = 1 + ˆT 1 + ˆT 2 + 1 2 ˆT 1 2 + ˆT 1 ˆT 2 + 1 2 ˆT 2 2 +... Robert Send TDDFT as a tool in biophysics 09.09.08 11 / 28

Coupled Cluster methods Ground state energy: with E = E HF + Φ HF Ĥe ˆT Φ HF exp( ˆT 1 + ˆT 2 + ˆT 3...) = 1 + ˆT 1 + ˆT 2 + 1 2 ˆT 2 1 + ˆT 1 ˆT 2 + 1 2 ˆT 2 2 +... Excitation energy: with (A ω n I) X n = 0 A µν = Φ HF ˆτ + µ e ˆT [ Ĥ, ˆτ ν ] e ˆT Φ HF T I µν = Φ HF τ ˆ µ e ˆ ˆτ ν e T Φ HF = δ µν Robert Send TDDFT as a tool in biophysics 09.09.08 11 / 28

Complete Active Space Self-Consistent Field (CASSCF) Choose: m active orbitals, n α,β active electrons Form Φ CAS with ( ) ( m m n α n β ) determinants Optimize coefficients and orbitals E = min ω i Ψ SA c 0,cs r,...,{φ i Ĥ Ψ SA i i } (12,12): 853776 Slater determinants Multireference: important for bond twists i Robert Send TDDFT as a tool in biophysics 09.09.08 12 / 28

Complete Active Space Self-Consistent Field (CASSCF) Choose: m active orbitals, n α,β active electrons Form Φ CAS with ( ) ( m m n α n β ) determinants Optimize coefficients and orbitals E = min ω i Ψ SA c 0,cs r,...,{φ i Ĥ Ψ SA i i } (12,12): 853776 Slater determinants Multireference: important for bond twists i Robert Send TDDFT as a tool in biophysics 09.09.08 12 / 28

CASPT2: CASSCF includes static correlation Dynamic correlation missing CASPT2: Perturbation theory for the inactive space CASPT2 energy: Second order estimate for E(FCI ) E(CASSCF ) Accurate excitation energies for small molecules PT: correction for correlation energy Robert Send TDDFT as a tool in biophysics 09.09.08 13 / 28

Method overview TDDFT CC2 CAS single reference single reference multireference linear response linear response state averaged CT: f 2 xc CT: lin. response CT: reliable 3 large molecules large molecules limited active space 2 A. Dreuw, J.L. Weisman, and M. Head-Gordon J. Chem. Phys. 119 2943 (2003) 3 M. Wanko et al. J. Chem. Phys. 120 1674 (2004) Robert Send TDDFT as a tool in biophysics 09.09.08 14 / 28

Three methods two results CAS as a reference method CASPT2 reliable for small molecules Reliable for charge-transfer states Multireference for strongly twisted bonds and at conical intersections TDDFT applicable? Charge-transfer issue TDDFT results contradict CASPT2 results Robert Send TDDFT as a tool in biophysics 09.09.08 15 / 28

Back to the chromophore Vertical excitation energies (in ev) Method 1a 2a Ref. BP86/TZVP 2.22 2.69 B3LYP/TZVP 2.34 3.10 [1] CC2/TZVPP 2.14 3.21 [2] CASPT2/6-31G* 2.28 3.49 [3] CASPT2/ANO 2.05 2.84 [4] Experiment 2.03 3.18 [5] [1] R. S. and D. Sundholm J. Phys. Chem. A 111 27 (2007) [2] R. S. and D. Sundholm Phys. Chem. Chem. Phys. 9 2862 (2007) [3] A. Cembran et al. J. Phys. Chem. A 109 6597 (2005) [4] S. Sekharan et al. Biophys. J. 91 L07 (2006) [5] I.B. Nielsen et al. Phys. Rev. Lett. 96 018304 (2006) Robert Send TDDFT as a tool in biophysics 09.09.08 16 / 28

Back to the chromophore Vertical excitation energies (in ev) Method 1a 2a Ref. BP86/TZVP 2.22 2.69 B3LYP/TZVP 2.34 3.10 [1] CC2/TZVPP 2.14 3.21 [2] CASPT2/6-31G* 2.28 3.49 [3] CASPT2/ANO 2.05 2.84 [4] Experiment 2.03 3.18 [5] Charge-transfer problems Coupled cluster-series convergence Basis sets Active space CAS as a reference? [1] R. S. and D. Sundholm J. Phys. Chem. A 111 27 (2007) [2] R. S. and D. Sundholm Phys. Chem. Chem. Phys. 9 2862 (2007) [3] A. Cembran et al. J. Phys. Chem. A 109 6597 (2005) [4] S. Sekharan et al. Biophys. J. 91 L07 (2006) [5] I.B. Nielsen et al. Phys. Rev. Lett. 96 018304 (2006) Robert Send TDDFT as a tool in biophysics 09.09.08 17 / 28

Applicability of TDDFT Do charge-transfer problems occur? Check and compare: Optimization with hybrid/non-hybrid functionals Stabilization with point charges Stabilization with polar molecules Optimization with CC2 Convergence of the CC series Robert Send TDDFT as a tool in biophysics 09.09.08 18 / 28

Applicability of TDDFT Do charge-transfer problems occur? Check and compare: Optimization with hybrid/non-hybrid functionals Stabilization with point charges Stabilization with polar molecules Optimization with CC2 Convergence of the CC series 11-cis retinal: all optimization results agree exception: angle of the β-ionone ring Robert Send TDDFT as a tool in biophysics 09.09.08 18 / 28

Optimizing the excited state Bond length alternation 4 5 150 148 ground state 1a TDDFT 1a CASSCF(12,12) 146 144 142 pm 140 138 136 134 132 C6-C7 C7-C8 C8-C9 C9-C10 C10-C11 C11-C12 C12-C13 C13-C14 C14-C15 C15-N16 4 A. Cembran et al. J. Phys. Chem. A 109 6597 (2005) 5 R. S. and D. Sundholm Phys. Chem. Chem. Phys. 9 2862 (2007) Robert Send TDDFT as a tool in biophysics 09.09.08 19 / 28

CASSCF: Ionic states vs. tetraradicals 6 Polyene isomerization via tetraradicals 6 M. Garavelli et al. J. Photochem. Photobiol. A 114 109 (1998) Robert Send TDDFT as a tool in biophysics 09.09.08 20 / 28

CASSCF: Ionic states vs. tetraradicals 6 Polyene isomerization via tetraradicals PSB isomerization via ionic states with bond length inversion 6 M. Garavelli et al. J. Photochem. Photobiol. A 114 109 (1998) Robert Send TDDFT as a tool in biophysics 09.09.08 20 / 28

CAS: Conical intersection region 7 8 C-C stretch vibrations Torsion twist around the 11-12-double bond 7 R. Gonzalez-Luque Proc. Nat. Acad. Sci. 97 9379 (2000) 8 A. Cembran et al. J. Phys. Chem. A 109 6597 (2005) Robert Send TDDFT as a tool in biophysics 09.09.08 21 / 28

TDDFT: The β-ionone ring twist 9 9 R. S. and D. Sundholm J. Phys. Chem. A 111 8766 (2007) Robert Send TDDFT as a tool in biophysics 09.09.08 22 / 28

TDDFT: The β-ionone ring twist 9 9 R. S. and D. Sundholm J. Phys. Chem. A 111 8766 (2007) Robert Send TDDFT as a tool in biophysics 09.09.08 22 / 28

TDDFT: The β-ionone ring twist 10 Three ground state isomers Excitation energy dependent on ring-torsion Single-bond barriers dependent on ring-torsion Not entirely supported at CC2 level 10 R. S. and D. Sundholm J. Phys. Chem. A 111 8766 (2007) Robert Send TDDFT as a tool in biophysics 09.09.08 23 / 28

TDDFT: The conical intersection 11 12 Twist around the 10-11-single bond Two methyl groups block each other Relaxation to the curl isomer 11 R. S. and D. Sundholm J. Phys. Chem. A 111 8766 (2007) 12 R. S. and D. Sundholm J. Mol. Model. 14 717 (2008) Robert Send TDDFT as a tool in biophysics 09.09.08 24 / 28

Soft experimental evidence Inside the protein pocket The double bond isomerization starts after 20 fs 13 The isomerization takes place in the ground state 14 There is an intermediate called photorhodopsin 15 Electron density at the ring changes upon isomerization 15 The isomerization does not happen when the ring is removed 16 13 T. Kakitani et al. J. Phys. Chem. A 102 1334 (1998) 14 P. Kukura et al. Science 310 1006 (2005) 15 H. Nakamichi and T. Okada Angew. Chem. 118 4376 (2006) 16 F. Bartl et al. J. Biol. Chem. 280 34259 (2005) Robert Send TDDFT as a tool in biophysics 09.09.08 25 / 28

Conclusions Applicability of TDDFT in retinal calculations New isomerization mechanism Robert Send TDDFT as a tool in biophysics 09.09.08 26 / 28

Conclusions Applicability of TDDFT in retinal calculations New isomerization mechanism Importance of single bond twists Importance of electrons in the β-ionone ring Robert Send TDDFT as a tool in biophysics 09.09.08 26 / 28

Conclusions Applicability of TDDFT in retinal calculations New isomerization mechanism Importance of single bond twists Importance of electrons in the β-ionone ring Doubts about CAS (12,12)-reliability Doubts about basis sets used in CAS calculations CAS might not be applicable in retinal calculations Robert Send TDDFT as a tool in biophysics 09.09.08 26 / 28

Problems Three methods with different weaknesses Two different results in agreement with experiment Robert Send TDDFT as a tool in biophysics 09.09.08 27 / 28

Problems Three methods with different weaknesses Two different results in agreement with experiment New evaluation of the methods needed New experiments to evaluate the methods needed Robert Send TDDFT as a tool in biophysics 09.09.08 27 / 28

Problems Three methods with different weaknesses Two different results in agreement with experiment New evaluation of the methods needed New experiments to evaluate the methods needed No reliability close to conical intersections Dynamical treatment needed Robert Send TDDFT as a tool in biophysics 09.09.08 27 / 28

Problems Three methods with different weaknesses Two different results in agreement with experiment New evaluation of the methods needed New experiments to evaluate the methods needed No reliability close to conical intersections Dynamical treatment needed Too many publications on retinal The major questions remain unanswered Robert Send TDDFT as a tool in biophysics 09.09.08 27 / 28

Acknowledgements Dage Sundholm (Helsinki) Filipp Furche (Irvine) Reinhart Ahlrichs (Karlsruhe) TURBOMOLE Robert Send TDDFT as a tool in biophysics 09.09.08 28 / 28

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