Metodo PMM applicato allo studio delle trasizioni di spin

Ing. Paolo Marracino

DIET

E-mail: marracino@die.uniroma1.it

Birds Compass

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

Meccanismo ipotizzato a livello microscopico: effetto del campo magnetico sul processo di rilassamento tra stato elettronico eccitato (dal fotone) e ground state…un po’ di teoria

Principio di esclusione di Pauli OK

hvPrincipio di esclusione di Pauli OK

TRIPLETTO

Es: atomo con più orbitali

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.

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

indole

riboflavinSimulazioni MD di:•Fl+In (in acqua)•Fl-+In+ (in acqua)

20 MD simulations of the neutral complex and the 20 MD simulation of the ionic complex

Benchmark reaction

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. roto translational 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

Classical simulations affect the unperturbed Hamiltonian via electrostatic or magnetostatic perturbations

Hp=H0 - Eenv·μE

P M M : th eo reti cal valid ati o n

Difference in free energy is essentially due to the electrostatic field produced by the protein surrounding the active site

Example: modeling a 3-state chemical reaction in protein

Eenv=0: electronic transfer does not occour

Eenv considered: electronic transfer actually occours

0( )ln pE EA KT e

Hp=H0 - Benv·μB

The environmental Eenv or Benv take into account both endogenous or exogenous EM fields

Electric perturbation

Magnetic perturbation

Effetti del campo magnetico?

Spin-Spin magnetic interactions are treated by the dipolar approximation with the corrections to the perturbed Spin states given by first order perturbation theory.

Con il PMM si tratta in modo dinamico:

Erice 2012

Adding Spin-Spin InteractionSpin-Spin magnetic interactions are treated by the dipolar approximation with the corrections to the perturbed Spin states given by second order perturbation theory.

From Literature

B

Energy Tc

Tb

Ta

S

Perturbed triplet states

Limiti

• Interazioni spin-nucleo (iperfini)• Interazioni spin-orbita* (momento magnetico

intrinseco e orbitale)• Interazioni nucleo-nucleo (deboli)

Attualmente non sono considerati nel termine perturbativo (endogeno)

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

Schema reazione

Schema cinetico

Passi dello schema (1-2)

• The first step of the reaction (from 1 to 2 in) is the photon excitation of Flavin from its ground state to a triplet excited state

• 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

Passi dello schema (2-3)

Nota• Note that the reaction path 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

Come calcolare gli ET

several abscissa crossings, each correspondingto an electron transfer transition

• 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

Passi dello schema (3-4)

ionic couple dissociation:calculated in water

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 perturbation terms are missing!!!!

Theory 1/4We 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.

Theory 2/4

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 .

Theory 3/4•The dipolar perturbation operator used to construct the perturbed (spin) Hamiltonian matrix is:

•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

Theory 4/4

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.

Spin transition

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 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

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.

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

Passi dello schema (4-5)

Confronto con dati sperimentaliPro: dati sperimentali sulla stessa reazioneContro: la reazione avviene in micellaI 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