girse 2009 2nd part: examples relaxation processes, molecular motions and electron spin resonance...

Girse 20092nd part: examples

Relaxation processes, molecular motions and Relaxation processes, molecular motions and electron spin resonanceelectron spin resonance

Antonino PolimenoAntonino PolimenoUniversità degli Studi di PadovaUniversità degli Studi di Padova

http://www.chimica.unipd.it/antonino.polimenohttp://www.chimica.unipd.it/antonino.polimeno

2. Applications

http://www.chimica.unipd.it/antonino.polimeno


T4 lysozyme

Biological Macromolecules3. Liang, Z.; Lou, Y.; Freed, J.H.; Columbus, L.; Hubbel, W. L. J. Phys. Chem. B 2004, 108, 17649

Labelled Peptides1. Carlotto, S.; Cimino, P.; Zerbetto, M.; Franco, L.; Corvaja, C.; Crisma, M.; Formaggio, F.; Toniolo, C.; Polimeno, A.; Barone, V. J. Am. Chem. Soc. 2007, 129, 11248

Materials2. Barone, V.; Brustolon, M.; Cimino, P.; Polimeno, A.; Zerbetto, M.; Zoleo, A. J. Am. Chem. Soc. 2006, 128, 1586

ApplicationsApplications


Interpretation based on Stochastic Liouville Equation (SLE) defined by the direct inclusion of motional dynamics via stochastic operators plus super Hamiltonian H

DIPOLE-DIPOLE

TENSORg, A TENSORS

DIFFUSION

TENSOR

ELECTRONIC STRUCTURE

Dissipative parameters, e.g. rotational diffusion tensors, can be determined via hydrodynamic modelling

Principal values and orientation of electron Zeeman tensor and hyperfine coupling tensors

Additional interactions; e.g. in double labeled systems, dipolar interaction based on the molecular structures beyond the point approximation

MOLECULAR GEOMETRY

Quantum Mechanical calculation pursued by Density Functional Theory (DFT) via adoption of mixed quantum-mechanical / molecular mechanical (QM/MM) methods

Barone, V.; Polimeno, A. Phys. Chem. Chem. Phys. 2006, 8, 4609

Integrated approachIntegrated approach


The SLE describes the variation in time of the system density matrix.

The density matrix / probability depends upon quantum pseudo-coordinates and classical (stochastic) variables1.

Q, t t

i H Q , Q, t Q Q,t L Q Q, t

L Q H Q , Q H Q Q

The cw-ESR spectrum is obtained from the spectral density

1

0 01 ˆ ˆˆˆ

X I X I eqI S i iH S P

1 1 1

1. Schneider, D. J.; Freed, J. H. Adv. Chem. Phys. 1989, 73, 387

SLESLE


Numerical implementation (1)Numerical implementation (1)

• The cw-ESR spectrum given is now identified as the real part of the spectral density for the auto-correlation function for the observable, usually named ‘starting’ vector corresponding to the x-component of the magnetization

1

0 0

1/ 2 1/ 2

1

ˆ

ˆX I eq

I v i v

iH

v I S P

1

1

L

L

• The spectrum is obtained by numerically evaluating the spectral density and this is usually attained via iterative algorithms, like Lanczos or conjugate gradients


Numerical implementation (2)Numerical implementation (2)

• The spectrum can be written in the form a continued fraction

I 0 1

i 0 1

22

i 0 2

32

i 0 3

n1

n 1 (i L n) n

nn 1

n n i L n ,

n n i L n 1


Numerical Implementation (3)Numerical Implementation (3)• The computational task is

carried on in finite arithmetic, by projecting the symmetrized time evolution operator and the starting vector on a basis set

• Symmetry arguments can be employed to significantly reduce the number of basis function sets required to achieve convergence,

• A numerical selection of a reduced basis set of functions based on the ‘pruning’ of basis elements with negligible contribution to the spectrum can be also used

pSqS p Iq I l ,l

• The matrix-vector expression for the c.f. coefficients is

n1

vn1(L

n1)v

n

nv

n 1

nv

nv

n,

nv

nv

n 1


Numerical implementation (4)Numerical implementation (4)• To evaluate matrix elements, one needs to make explicit the

dependence of the Liouvillean from magnetic and orientational parameters. We adopt a spherical irreducible tensorial representation

( , ')* ( , )' , ,

0,2 , '

ˆˆ ( )l

l l m l mmm LF MF MF LF

l m m l

H F A

D

LF

GF

AnF

MFGF

MFAnF

B0

MF

LFMF


Liouvillean symmetrizationLiouvillean symmetrizationThe Liouville operator is spanned in the set of basis functions

This basis set generally leads to a Hermitean matrix representation of the Liouvillean, but Lanczos algorithm is written for symmetric matrices

Symmetrization of the Liouvillean is achieved by the basis transformation

The matrix elements in the new basis set are expressed in function of the matrix elements in the old basis


Pruning

Basis functions are kept or discharged in function of their relative importance on determining the line shape Weights are calculated with the criterion [7]

[7] D. J. Schneider, J. H. Freed, Adv. Chem. Phys. 73, 387 – 527 (1989)


int main (void) { int i; long double dw, Iw, omega0; FILE *ouf, *ouf2; phisics(); basi(); matrix(); stvec();

omega0=(g[0][0]+g[1][0]+g[2][0])/3.0; […] printf("\n\nnstep = "); scanf("%d",&nstep); lanczos();

[…] for (i=-10000;i<=10000;i++) { dw=(long double)i/250000.0; Iw=fracinf(dw-omega0); fprintf(ouf,"%Lf %Le\n",dw,Iw); }}

Main

Input physical

parameters

1

Build Liouville operator matrix

3

Project basis functions on

v

4

Lanczos tridiagonaliza

tion

5

Generate basis

functions indexes2

ESR line calculation

6


• Calculation of the spectrum is based on stochastic methods and in particular on the solution of the stochastic Liouville equation.

• In this equation dynamics is added as stochastic operators in the Liouville operator that describes the time evolution of the density matrix of the system.

• Full diffusion tensor is calculated via a hydrodynamic model that describes the molecule as an ensemble of interacting spherical beads surrounded by a locally isotropic continuous fluid.• Diffusion tensor depends on the geometry of the molecule and on the viscosity of the solvent.• Non-rigid molecules can be described with internal degrees of freedom represented by torsional angles.

• Quantum Mechanical calculations are employed to calculate: (i) structural properties like geometry and torsional potentials; (ii) magnetic properties, i.e. the magnetic tensors.

• E-SpiReS automatically generates an input file for Gaussian that can be edited by the user and submitted directly from the GUI.

• A number of parameters can be chosen to refine via a non linear least squares minimization routine based on the Levenberg - Marquardt method.

• Also an experimental spectrum can be loaded as reference.

Quantum MechanicsQuantum Mechanics

Hydro Dynamics

Hydro Dynamics

Stochastic Methods

Stochastic Methods

CW-ESR spectra

prediction

CW-ESR spectra

prediction

• Molecular geometry is given in Z-matrix or PDB format• Graphical definition of the form of the stochastic Liouville operator• All physical and calculation parameters are set here

Electron Spin Resonance Simulation is a multiscale software for the ab-initio calculation of cw-ESR spectra.

The core of E-SpiReS is written in C and parallelized under the MPI paradigm.

The graphic user interface (GUI) is written in JAVA to ensure good portability in every operating system.

Polimeno, A.; Zerbetto, M.; Barone V. Comp. Phys. Comm. – submitted


Electron Spin Resonance Simulation is a multiscale software for the ab-initio calculation of cw-ESR spectra.

The core of E-SpiReS is written in C and parallelized under the MPI paradigm.

The graphic user interface (GUI) is written in JAVA to ensure good portability in every operating system.

http://www.chimica.unipd.it/licc

The web interface, accessible from any browser, communicates with the cluster sending to it the requests of calculation and giving back to the user the spectrum




Polimeno, A.; Zerbetto, M.; Franco, L.; Maggini, M.; Corvaja, C. J. Am. Chem. Soc. 2006, 128, 4737

Isomer 2

Isomer 1

•Stochastic model: free Brownian rotator•Geometry & shape: MM level calculation•Diffusion tensor: hydrodynamic model• Interaction energy J obtained via fitting

• Diffusive operator

• Spin Hamiltonian

˜ DXXˆ J X

2 DYYˆ J Y

2 DZZˆ J Z

2

ˆ H e

hB0g1

ˆ S 1 e

hB0g2

ˆ S 2 eˆ I 1A1

ˆ S 1 eˆ I 2A 2

ˆ S 2

2eJˆ S 1ˆ S 2 0

4ge

2e2

hr3ˆ S 1ˆ S 2

3

r2ˆ S 1r ˆ S 2r


Barone, V.; Brustolon, M.; Cimino, P.; Polimeno, A.; Zerbetto, M.; Zoleo, A. J. Am. Chem. Soc. 2006, 128, 1586

N NO O

S

12

34

5

6

7

8

18

13

17 18

14 15

19

31

27

35

23

N NO O

S

12

34

5

6

7

8

18

13

17 18

14 15

19

31

27

35

23

p-(methyl thio) phenyl nitroxyl nitroxide (MTPNN)

• Stochastic model: Brownian rotator +

conformational dynamics• Geometry & shape: QM level calculation

(Gaussian, DFT-B3LYP)•Diffusion tensor: hydrodynamic model• Diffusive operator


˜ DXXˆ J X

2 DYYˆ J Y

2 DZZˆ J Z

2

ˆ H e

hB0g1

ˆ S 1 eˆ I 1A1

ˆ S 1 eˆ I 2A 2

ˆ S 1


The oligopeptide is labeled with two nitroxide radicals, in the form of -amino acid TOAC (2,2,6,6-tetrametyl-1oxyl-4amino-4-carboxylic acid)

T2F

MF

LF

B0

Zerbetto, M.; Carlotto, S.; Polimeno, A.; Corvaja, C.; Franco, L.; Toniolo, C.; Formaggio, F.; Barone, V.; Cimino, P. J. Phys. Chem. B 2007, 111, 2668

Carlotto, S.; Cimino, P.; Zerbetto, M.; Franco, L.; Corvaja, C.; Crisma, M.; Formaggio, F.; Toniolo, C.; Polimeno, A.; Barone, V. J. Am. Chem. Soc. 2007, 129, 11248

˜ DXXˆ J X

2 DYYˆ J Y

2 DZZˆ J Z

2

ˆ H e

hB0g1

ˆ S 1 e

hB0g2

ˆ S 2 eˆ I 1A1

ˆ S 1 eˆ I 2A 2

ˆ S 2 2eJˆ S 1ˆ S 2 ˆ S 1T ˆ S 2


2

2 220

33 22

3

4

x x y x z

e ex y y y z

x z y z z

r r r r rg

r r r r rr r

r r r r r

T 1

2 ' '' ' '' ' '' ' ''

212 , 12 12

, 512

ˆ1 2 2 1 1 2 2 1

3ˆ

rT

r

T N T

r r

At short distances between the two spin probes the calculation of the dipolar interaction tensor must consider the distribution of the unpaired electrons over the anti-bonding orbitals.

R

2,0T versus distance calculated via approximated expression (dashed line) and exact treatment (solid line).


Simulations of cw-ESR spectra of the peptide in four different solvents. Red solid line are experimental spectra, black dashed line are theoretical spectra

Acetonytrile

Chloroform

Methanol Toluene


Polimeno, A.; Carlotto, S.; Zerbetto, M.; Huber, M.; Toniolo, C. – in preparation

This preliminary study is based on

•Molecular structure from molecular mechanics calculations

•Magnetic tensors from literature

•Diffusion tensor calculated via a hydrodynamic method

X-band cw-EPR spectrum in acetonitrile at 293 K

W-band cw-EPR spectrum in acetonitrile at 293 K

Structure of the peptide:

Fmoc-Aib-TOAC-(Aib)5-TOAC-Aib-OMe


Zerbetto, M.; Polimeno, A.; Cimino, P.; Barone, V. J. Chem. Phys. 2008, 128, 24501

ˆ ˆ M

/

trDRR DRI

DIR DII

Peq ,

ˆ M

/

Peq

1 ,

Peq , exp V , /kT exp Vext /kT exp Vint /kT

Vext D0,02

Vint ne in

n

• Diffusive operator


ˆ H e

hB0g1

ˆ S 1 eˆ I 1A1

ˆ S 1 eˆ I 2A 2

ˆ S 1

Isotropic phaseIsotropic phase

Nematic phaseNematic phase


We investigate the temperature dependence of cw-EPR line shape measured at the early stages of the methyl methacrilic polymerization.

Hermosilla, L.; Sieiro, C.; Calle, P.; Zerbetto, M.; Polimeno, A. J. Phys. Chem. B 2008, 112, 11202

We assume the following model for the dynamics:

with a diffusive part describing rotation about the C – C bond:

and a random-walk component that we interpret as the effect of the propagation reaction that imposes random changes to the internal angle

W RW Peq

P ,t t

ˆ P ,t ˆ DP , t ˆ RW P ,t

ˆ D

DII Peq

Peq 1

ˆ RW P ,t d0

2 P ,t W P ,t W


Quantum mechanical calculations conducted ad DFT level, B3LYP functional, 6-31G* basis set

Dependence of internal potential on

Internal torsional potential:

U ne in

n

n n

*

Dependence of and ’ hyperfine constants on

Spin Hamiltonian:

ˆ H e

hgB0

ˆ S e an I n ˆ S n1

5

girse 2009 2nd part: examples relaxation processes, molecular motions and electron spin resonance...

Documents