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Femtochemistry: A theoretical overview

V – Finding conical intersections

Mario Barbatti

This lecture can be downloaded at

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

lecture5.ppt

Conical intersections: Twisted-pyramidalized

Barbatti et al. PCCP 10, 482 (2008)

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-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-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.

- Major UVB absorber in skin
- Photoaging
- UV-induced immunosuppression

- 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)

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

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

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:

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.

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:

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

Example: thymine

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

Crossing of states with different multiplicities

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

Example: cytosine

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

Conical intersections between three states

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)

Slow H elimination

Example of application: photochemistry of imidazole

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

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)

crossing seam, it is a maximum!

Crossing seam

Geometry-restricted optimization (dihedral angles kept constant)

At a certain excitation energy:

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?

- Transition probabilities

Contact

This lecture can be downloaded at

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

lecture5.ppt

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