Metodo PMM applicato allo studio delle trasizioni di spin

1. Metodo PMM applicatoallo studio delletrasizionidi spin Ing. Paolo Marracino DIET E-mail: marracino@die.uniroma1.it

3. BirdsCompass

4. Meccanismo ipotizzato a livello microscopico: effetto del campo magnetico sul processo di rilassamento tra stato elettronico eccitato (dal fotone) e ground state In a system with two spin 1/2 particles - for example the proton and electron in the ground state of hydrogen, measured on a given axis, each particle can be either spin up or spin down so the system has four basis states …un po’ di teoria Es: atomo con più orbitali Principio di esclusione di Pauli OK Principio di esclusione di Pauli OK hv TRIPLETTO

5. Stati elettronici e vibrazionali Energia potenziale della molecola di idrogeno in funzione della distanza tra i due nuclei. Le linee orizzontali indicano i livelli vibrazionali, De è l'energia di dissociazione della molecola.

6. Schema del passaggio da uno stato fondamentale (S0) ad uno stato eccitato (S1) e da questo di nuovo allo stato fondamentale. I segmenti terminanti con una freccia rappresentano le transizioni, quelli ondulati rappresentano le transizioni che non producono emissione diluce (Cantor and Schimmel, Biophysical Chemistry).

7. Benchmark reaction indole riboflavin • Simulazioni MD di: • Fl+In (in acqua) • Fl-+In+ (in acqua) 20 MD simulationsofthe neutral complex and the 20 MD simulation of the ionic complex

8. In order to model chemical reactions in complex atomic-molecular systems: use of a mixed quantum mechanics/molecular dynamics (QM/MD) theoretical computational methodology based on the perturbed matrix method (PMM) 1) Classical degrees of freedom (e.g. roto­translational coordinates) are treated by classical atomistic Molecular Dynamics (MD) simulations; 2) Quantum degrees of freedom (e.g. electronic coordinates) of the relevant subpart of the system, the quantum center (QC), are treated by high level quantum chemical calculations; 3) Coupling between quantum and classical degrees of freedom is obtained by the Perturbed Matrix Method (PMM) providing at each MD frame the perturbed quantum eigenstates and properties of the QC Centri quantististici considerati: Flavina e Indolo separatamente

9. PMM: theoreticalvalidation Classical simulations affect the unperturbed Hamiltonian via electrostatic or magnetostatic perturbations The environmentalEenvor Benv take into account bothendogenous or exogenous EM fields Hp=H0 - Eenv·μE Electricperturbation Hp=H0 - Benv·μB Magneticperturbation Example: modeling a 3-state chemicalreaction in protein Eenv=0: electronic transfer doesnotoccour Difference in free energy isessentially due to the electrostaticfieldproducedby the proteinsurrounding the active site Eenvconsidered: electronic transfer actuallyoccours

10. Effetti del campo magnetico? Con il PMM si tratta in modo dinamico: Spin-Spinmagneticinteractions are treatedby the dipolarapproximationwith the correctionsto the perturbedSpinstatesgivenby first orderperturbationtheory.

11. Adding Spin-Spin Interaction Spin-Spin magneticinteractions are treated by the dipolarapproximation with the corrections to the perturbed Spin statesgiven by secondorderperturbationtheory. FromLiterature Perturbedtripletstates Tc Energy Tb S Ta B Erice 2012

12. Limiti Attualmente non sono considerati nel termine perturbativo (endogeno) • Interazioni spin-nucleo (iperfini) • Interazioni spin-orbita* (momento magnetico intrinseco e orbitale) • Interazioni nucleo-nucleo (deboli)

13. Interazione spin-orbita le due curve di energia calcolate con l'accoppiamento spin-orbita seguano sostanzialmente l'andamento della curva corrispondente al tripletto fino al punto di incrocio tra le due curve e della curva corrispondente al singoletto dopo il punto di incrocio

14. Schema reazione

15. Schema cinetico

16. Passi dello schema (1-2) • The first step of the reaction (from 1 to 2 in) is the photon excitation ofFlavin from its ground state to a triplet excited state

17. Passi dello schema (2-3) • A series of following processes, experimentally characterized by the quenching constant Kq, drives the reaction from step 2 to 3. At first, the formation of the complex occurs, characterized by an association constant (not explicitly calculated) which is likely to determine the overall quenching constant. Then, with a series of consecutive electron transfer (ET) processes (shown in inset A) and a vibrational relaxation, the triplet ground state is reached. Another ET drives the reaction to step 3

18. Nota • Note that the reactionpath will pass through the singlet state of the ionic couple as no reaction event is possible between the vibrationally excited state of the triplet ground state of the neutral complex and the singlet state of the ionic couple

19. Come calcolare gli ET several abscissa crossings, each corresponding to an electron transfer transition

20. Passi dello schema (3-4) • From the step 3, two reaction channels are possible: the ionic couple can dissociate or a spin transition between the triplet and singlet state can occur ionic couple dissociation: calculated in water

21. Spin transition • the spin interconversion between the Ta triplet state and the singlet state results very slow (10 ns-1 µs from literature data ) BUT….. • Up to now we can only calculate the fast interconversion between Tb and the singlet state and the magnetic field effect on such trasition Some perturbationterms are missing!!!!

22. Theory 1/4 We have explicitly calculated the dipolar interaction between two radicals by means of a perturbing approach in the flavin-indole electron transfer reaction. This electron transfer process involves, as a second reaction step,a triplet to singlet spin state relaxation of the ionic complex. Simulation details:: • TD-DFT calculations were used to provide a reliable unperturbed basis set for both the donor and acceptor quantum centers, i.e. the flavin and indole molecules. • Classical molecular dynamics simulations were performed on a system consisted of a dodecahedral box in which riboavin (Fl) and indole (In) molecules were placed surrounded by Single Point Charge (SPC) water molecules resulting in a typical density of 55.32 mol/l. • Randomly selected structures obtained by the extended trajectory were then used as starting conformations for 20 shorter trajectories (2 ns) guaranteeing a proper sampling.

23. Theory 2/4 Simulation details:: In order to characterize the spin transitions, the eigenvalues and eigenvectors of the perturbed spin Hamiltonian matrix were calculated: one of the four eigenvalues corresponds to the singlet state being always equal to zero, while the other three eigenvalues correspond to three perturbed triplet states resulting from combinations of the T0, T+1 and T-1 spin eigenstates used as basis set . • TD-DFT calculations were used to provide a reliable unperturbed basis set for both the donor and acceptor quantum centers, i.e. the flavin and indole molecules. • Classical molecular dynamics simulations were performed on a system consisted of a dodecahedral box in which riboavin (Fl) and indole (In) molecules were placed surrounded by Single Point Charge (SPC) water molecules resulting in a typical density of 55.32 mol/l. • Randomly selected structures obtained by the extended trajectory were then used as starting conformations for 20 shorter trajectories (2 ns) guaranteeing a proper sampling.

24. Theory 3/4 • The dipolar perturbation operator used to construct the perturbed (spin) Hamiltonian matrix is: Simulation details:: • TD-DFT calculations were used to provide a reliable unperturbed basis set for both the donor and acceptor quantum centers, i.e. the flavin and indole molecules. • Classical molecular dynamics simulations were performed on a system consisted of a dodecahedral box in which riboavin (Fl) and indole (In) molecules were placed surrounded by Single Point Charge (SPC) water molecules resulting in a typical density of 55.32 mol/l. • Randomly selected structures obtained by the extended trajectory were then used as starting conformations for 20 shorter trajectories (2 ns) guaranteeing a proper sampling. • The magnetic dipole operators associated to the two radical pairs are generally equal to: • The effect of an exogenous magnetic field can be considered as a further additive perturbing term to V given by: These operators dinamically affect the reaction considered

25. Theory 4/4 Simulation details:: The true spin state (adiabatic state) is virtually identical to either the triplet or singlet spin state except at each crossing, where it becomes a triplet-singlet combination. Therefore, the true spin relaxation process is given by the spin transition provided along the adiabatic energy surface that we approximate via the triplet-singlet energy surfaces. In this conceptual scheme each crossing then becomes a signature for the spin transition within the adiabatic state. • TD-DFT calculations were used to provide a reliable unperturbed basis set for both the donor and acceptor quantum centers, i.e. the flavin and indole molecules. • Classical molecular dynamics simulations were performed on a system consisted of a dodecahedral box in which riboavin (Fl) and indole (In) molecules were placed surrounded by Single Point Charge (SPC) water molecules resulting in a typical density of 55.32 mol/l. • Randomly selected structures obtained by the extended trajectory were then used as starting conformations for 20 shorter trajectories (2 ns) guaranteeing a proper sampling.

26. Spintransition

27. RP are always in the lowest triplet energy state (black line): spin transitions from this state to the singlet state (red line) are thus crucial in the overall process. The dipolar interaction seems to modulate magnetic spin transitions. In fact, when the dipolar coupling is considerably stronger than hyperfine coupling [4], a significant energy separation of the spin states is present and hence no spin transition can occur. The two radicals have to be far enough apart in order to have a vanishing dipolar interaction comparable with the hyperfine one, as clearly apparent considering both panel A and B where a direct comparison between interradical dipolar energy and interradical distance is shown

28. Passi dello schema (4-5) • Once in the singlet state (step 4) another ET drives the system to the singlet ground state of the neutral complex (step 5).

29. Confronto con dati sperimentali Pro: dati sperimentali sulla stessa reazione Contro: la reazione avviene in micella I parametri cinetici che non sono influenzati (o poco) dall’ambiente sono la Kq, le K di interconversione magnetica e tutte le ET I parametri cinetici fortemente influenzati (o poco) dall’ambiente sono le K di dissociazione Stima indiretta della K di dissociazione (da inserire nel nostro modello) da dati sperimentali (essenzialmente la micella stabilizza le specie coinvolte nella reazione sfavorendo la dissociazione) fattore riduttivo utilizzato=1000