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Femtochemistry: A theoretical overview. V – Finding conical intersections. Mario Barbatti [email protected] This lecture can be downloaded at http://homepage.univie.ac.at/mario.barbatti/femtochem.html lecture5.ppt. Where are the conical intersections?. formamide. pyridone.

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Femtochemistry: A theoretical overview

V – Finding conical intersections

Mario Barbatti

[email protected]

This lecture can be downloaded at

http://homepage.univie.ac.at/mario.barbatti/femtochem.html

lecture5.ppt


Where are the conical intersections?

formamide

pyridone

Antol et al. JCP 127, 234303 (2007)



Conical intersections: Twisted-pyramidalized

Barbatti et al. PCCP 10, 482 (2008)


Conical intersections in rings: Stretched-bipyramidalized


The biradical character

CH2NH2+ MXS

The biradical character

Aminopyrimidine MXS


The biradical character1

p1*

p2

S0 ~ (p2)2

S1 ~ (p2)1(p1*)1

The biradical character


One step back single p bonds
One step back: single p-bonds

p*2

CH2NH2+

CH2CH2

pp*

pp*

sp*

p2

p2

p*2

CH2CHF

CH2SiH2

pp*

p*2

pp*

p2

p2

Barbatti et al. PCCP 10, 482 (2008)


One step back single p bonds1

p*2

C2H4

pp*

p2

One step back: single p-bonds

b


One step back single p bonds2
One step back: single p-bonds

The energy gap at 90° depends on the electronegativity difference (d) along the bond.

Michl and Bonačić-Koutecký,

Electronic Aspects of Organic Photochem. 1990


One step back single p bonds3
One step back: single p-bonds

  • d depends on:

  • substituents

  • solvation

  • other nuclear coordinates

For a large molecule is always possible to find an adequate geometric configuration that sets d to the intersection value.


Urocanic acid

  • Major UVB absorber in skin

  • Photoaging

  • UV-induced immunosuppression


Finding conical intersections

  • Conventional geometry optimization:

  • Minimize: f(R) = EJ

  • Conical intersection optimization:

  • Minimize: f(R) = EJ

  • Subject to: EJ – EI = 0

    • HIJ = 0

  • Three basic algorithms:

  • Penalty function (Ciminelli, Granucci, and Persico, 2004; MOPAC)

  • Gradient projection (Beapark, Robb, and Schlegel 1994; GAUSSIAN)

  • Lagrange-Newton (Manaa and Yarkony, 1993; COLUMBUS)

Keal et al., Theor. Chem. Acc. 118, 837 (2007)


This term minimizes

the energy average

This term (penalty) minimizes

the energy difference

Penalty function

Function to be optimized:

Recommended values for the constants:

c1 = 5 (kcal.mol-1)-1

c2 = 5 kcal.mol-1


Gradient DE2

E

E

E2

E2

E1

Projection of gradient of EJ

E1

Rparallel

Rx

Rperpend

Rx

Gradient projection method

Minimize in the branching space:

EJ - EI

Minimize in the intersection space:

EJ


Minimize energy difference

along the branching space

Minimize energy along the

intersection space

Gradient projection method

Gradient used in the optimization procedure:

Constants:

c1 > 0

0 < c2 1


Suppose that L was determined at x0 and l0. If L(x,l) is quadratic, it will have a minimum (or maximum) at [x1 = x0 + Dx, l1 = l0 + Dl], where Dx and Dl are given by:

Lagrange-Newton Method

A simple example:

Optimization of f(x)

Subject to r(x) = k

Lagrangian function:



Lagrange-Newton Method

Solving this system of equations for Dx and Dl will allow to find the extreme of L at (x1,l1). If L is not quadratic, repeat the procedure iteratively until converge the result.


allows for geometric

restrictions

restricts energy

difference to 0

minimizes energy

of one state

restricts non-diagonal

Hamiltonian terms to 0

Lagrange-Newton Method

In the case of conical intersections, Lagrangian function to be optimized:


Expanding the Lagrangian to the second order, the following set of equations

is obtained:

Compare with the simple one-dimensional example:

Lagrange-Newton Method

Lagrangian function to be optimized:


Expanding the Lagrangian to the second order, the following set of equations

is obtained:

Solve these equations for

Update

Repeat until converge.

Lagrange-Newton Method

Lagrangian function to be optimized:


Comparison of methods set of equations

LN is the most efficient in terms of optimization procedure.

GP is also a good method. Robb’s group is developing higher-order optimization based on this method.

PF is still worth using when h is not available.

Keal et al., Theor. Chem.

Acc. 118, 837 (2007)


Crossing of states with different multiplicities set of equations

Example: thymine

Serrano-Pérez et al., J. Phys. Chem. B 111, 11880 (2007)


Crossing of states with different multiplicities set of equations

Lagrangian function to be optimized:

Now the equations are:

Different from intersections between states with the same multiplicity, when different multiplicities are involved the branching space is one dimensional.


Three-states conical intersections set of equations

Example: cytosine

Kistler and Matsika, J. Chem. Phys. 128, 215102 (2008)


Conical intersections between three states set of equations

Lagrangian function to be optimized:

This leads to the following set of equations to be solved:

Matsika and Yarkony, J. Chem. Phys. 117, 6907 (2002)


Fast H elimination set of equations

Slow H elimination

Example of application: photochemistry of imidazole

Devine et al. J. Chem. Phys. 125, 184302 (2006)


Fast H elimination set of equations

Slow H elimination

Example of application: photochemistry of imidazole

Fast H elimination:ps* dissociative state

Slow H elimination: dissociation of the hot ground state formed by internal conversion

How are the conical intersections

in imidazole?

Devine et al. J. Chem. Phys. 125, 184302 (2006)



Barbatti et al., J. Chem. Phys. set of equations 130, 034305 (2009)


It is not a minimum on the set of equations

crossing seam, it is a maximum!

Crossing seam

Geometry-restricted optimization (dihedral angles kept constant)



At a certain excitation energy: set of equations

1. Which reaction path is the most important for the excited-state relaxation?

2. How long does this relaxation take?

3. Which products are formed?


Time evolution set of equations


Next lecture set of equations

  • Transition probabilities

Contact

[email protected]

This lecture can be downloaded at

http://homepage.univie.ac.at/mario.barbatti/femtochem.html

lecture5.ppt


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