Femtochemistry: A theoretical overviewFemtochemistry: A theoretical overview

Mario Barbattimario.barbatti@univie.ac.at

IV – Non-crossing rule and conical intersections

This lecture can be downloaded athttp://homepage.univie.ac.at/mario.barbatti/femtochem.html lecture4.ppt

2

The non-crossing ruleThe non-crossing rule

von Neumann and Wigner, Z. Phyzik 30, 467 (1929)Teller, JCP 41, 109 (1937)

“For diatomics, the potential energy curves of the electronic states of the same symmetry species cannot cross as the

internuclear distance is varied.”

3

Simple argumentSimple argument

Suppose a two-level molecule whose electronic Hamiltonian is H(R), where R is the internuclear coordinate.

Given a basis of unknown orthogonal functions 1 and 2, we want to solve the Schrödinger equation

Check it!

2/12

122

221122111 421

21

HHHHHE

2/12

122

221122112 421

21

HHHHHE

0 iEH 2,12211 icc iii

0det2221

1211

EHH

HEH *jijiij HRHH

The energies are given by

4

2/12

122

221122111 421

21

HHHHHE

2/12

122

221122112 421

21

HHHHHE

Simple argumentSimple argument

5

2/12

122

221122111 421

21

HHHHHE

2/12

122

221122112 421

21

HHHHHE

21 EE

0

0

12

2211

H

HH (i)

(ii)

It is unlikely (but not impossible) that by varying the unique parameter R conditions (i) and (ii) will

be simultaneously satisfied.

Simple argumentSimple argument

6

Rigorous proofRigorous proof

Naqvi and Brown, IJQC 6, 271 (1972)

Suppose that the degeneracy occurs at R0:

0201 RERE

R0

E

R

R+R

E1 = E0 + E1

E2 = E0 + E2

21 EE

To have a “crossing” not only the degeneracy condition is necessary. It is also need:

7

Rigorous proofRigorous proof

011 EH e

Schrödinger equation for the first state:

Small displacement from R0 to R0 + R:

1001

10101

00

EERRE

RR

HHRRH e

Neglecting terms in 2:

0011100 EHEH

Multiply by to the left and integrate over the nuclear coordinates: 0*2 R

01

021

01

02 EH Prove it!

8

Rigorous proofRigorous proof

After repeating the same steps for the second state:

01

021

01

02 EH

01

022

01

02 EH

Because of the second condition

01

022

01

021 EE

21 EE

Therefore 001

02 H

Expanding in Taylor to the first order

RRH

H 00

102 RH

9

Rigorous proofRigorous proof

Having a crossing between two states requires two conditions:

001

02 RH

0201 RERE (i) degeneracy

(ii) crossing

It is unlikely (but not impossible) that by varying the unique parameter R conditions (i) and (ii) will

be simultaneously satisfied.(a) If only (i) is satisfied it is not a crossing, but a complete state degeneracy

for any R.

012 (for any R)

001

021

01

02 EH

(b) If the states have different symmetries, (ii) is trivially satisfied because:

10

Conical intersectionsConical intersections

• In diatomics the unique parameter R is not enough to satisfy the two conditions for crossing.

• In polyatomics there are 3N-6 internal coordinates!

What does happen if the molecule has more than one degree of freedom?

11

Conical intersectionsConical intersections

Suppose a two-level molecule whose electronic Hamiltonian is H(R), where R are the nuclear coordinates.

Given a basis of unknown orthogonal functions 1 and 2, we want to solve the Schrödinger equation

2/12

122

221122111 421

21

HHHHHE

2/12

122

221122112 421

21

HHHHHE

0 ie EH 2,12211 icc iii

0det2221

1211

EHH

HEH *jijeiij HHH R

The energies are given by

12

Conical intersectionsConical intersections

In a more compact way:

2/1212

22,1 HE

where

221121

HH 221121

HH and

A degeneracy at Rx will happen if

0 XR

012 XH R

In general, two independent coordinates are necessary to tune these conditions.

13

Conical intersectionsConical intersections

In a more compact way:

2/1212

22,1 HE

where

221121

HH 221121

HH and

Expansion in first order around Rx for :

Rs

RGRG

RRRR

X

XX

XX HH

12

21

2211

2121

21

XiiXi H RG

14

Conical intersectionsConical intersections

In a more compact way:

2/1212

22,1 HE

where

221121

HH 221121

HH and

In first order around Rx each of these terms are:

RsRGG XXX1221 Xii

Xi H RG

RgRGG XXX1221

RfR XXHH 121212

And the energies in a point RX + R are in first order:

2/12

12

2

12122,1 RfRgRs XXXE

15

Conical intersectionsConical intersections

2/12

12

2

12122,1 RfRgRs XXXE

Writting

then

sincos

sin

cos

RsRs

fyfR

gxgR

yx

Rs

Rf

Rg

yxsyfxg ˆˆ;ˆ;ˆ srfg

2/122222,1 yfxgysxsE yx

16

Conical intersectionsConical intersections

2/12

12

2

12122,1 RfRgRs XXXE

2/122222,1 yfxgysxsE yx

Atchity, Xantheas, and Ruedenberg, J. Chem. Phys. 95, 1862 (1991)

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Conical intersectionsConical intersections

2/12

12

2

12122,1 RfRgRs XXXE

What does happen if the molecule is distorted along a direction that is perpendicular to g and f?

Linear approximation fails

Crossing seam

XE 122,1 sE

RperpendRx

E1

E2

18

Conical intersectionsConical intersections

What does happen if the molecule is distorted along a direction that is parallel to g or f?

Linear approximation fails

Crossing seam

E

RparallelRx

E1

E2

2/12

12

2

12122,1 RfRgRs XXXE

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Branching spaceBranching space

• Starting at the conical intersection, geometrical displacement in the „branching space“ lifts the degeneracy linearly.

• The branching space is the plane defined by the vectors g and f.

• Geometrical displacements along the other 3N-8 internal coordinates keeps the degeneracy (in first order). These coordinate space is called „seam“ or „intersection“ space.

Note that XXX EEH 12211212 hf

2112 h Non-adiabatic coupling vector

For this reason the branching space is also referred as g-h space.

See the proof, e.g., in Hu at al. J. Chem. Phys. 127, 064103 2007 (Eqs. 2 and 3)

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• The coupling vectors define one of the directions of the branching space around the conical intersections, which is important for the localization of these points of degeneracy.

Why are non-adiabatic coupling vectors important?Why are non-adiabatic coupling vectors important?

21

Conical intersections are not rareConical intersections are not rare

“When one encounters a local minimum (along a path) of the gap between two potential energy surfaces, almost always it is the shoulder of a

conical intersection. Conical intersections are not rare; true avoided intersections are much less

likely.”

Truhlar and Mead, Phys. Rev. A 68, 032501 (2003)

R

E

<< 1 a.u.

R

E

0~

2/432

N

tot

tot

CIV

minV ~ O(1) is the density of zeros in the Hel matrix.

22

Conical intersections are connectedConical intersections are connected

Crossing seam

Minimum on the crossing seam (MXS)

Energy

R

4

5

6

7

8

-100

-50

0

50

100

4060

80100

120

Ene

rgy

(eV

)

(degre

es)

degrees)

EthylideneEthylidene

PyramidalizedPyramidalized

H-migrationH-migration

Barbatti, Paier and Lischka, J. Chem. Phys. 121, 11614 (2004)

CC3V3V

Crossing seam in ethylene

Example of dynamics results

Torsion Torsion + Pyramid.+ Pyramid.

H-migrationH-migration

Pyram. MXSPyram. MXS

Ethylidene MXSEthylidene MXS

~7.6 eV~7.6 eV

~ 100-140 fs~ 100-140 fs

60%60%

11%11%23%23%

26

Conical intersections are distortedConical intersections are distorted

2/1

22222,1 22

1yxyxyxdE gh

yxgh

It can be rewritten as a general cone equation (Yarkony, JCP 114, 2601 (2001)):

2

22

ghgh d

hg asymmetry parameter

2/122hg ghd pitch parameter

ggs

ghx d

hhs

ghy d

tilt parameters

2/12

12

2

12122,1 RfRgRs XXXE

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h01 g01

Energy

Example: pyrrole

28

h01 g01

Energy

Photoproduct depends on the direction that the molecule leaves the intersection

29Burghardt, Cederbaum, and Hynes, Faraday. Discuss. 127, 395 (2004)

Example: protonated Schiff Base

In gas phase In water

NH2+

r

30

0 20 40 60 80 100 120 140 160 180 200-2

0

2

4

6

8

10

0 20 40 60 80 100 120 140 160 180 200-2

0

2

4

6

8

10

energ

y (

eV

)

S0 S1 S2

time (fs)

gashpase

aqua

Ruckenbauer, Barbatti, Niller, and Lischka, JPCA 2009

31

32

0 50 100 150 200 250 300 350 400 450 500

0

30

60

90

120

150

180

210

240

270

300

330

360

cent

ral t

orsi

on (

°)

time (fs)0 50 100 150 200 250 300 350 400 450 500

0

30

60

90

120

150

180

210

240

270

300

330

360

cent

ral t

orsi

on (

°)

time (fs)

gasphase water

33

Intersections don’t need to be conical!Intersections don’t need to be conical!

Bersuker, The Jahn Teller effect, 2006

Yarkony, Rev. Mod. Phys. 68, 985 (1996)

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Next lectureNext lecture

• Finding conical intersections

Contactmario.barbatti@univie.ac.at

This lecture can be downloaded athttp://homepage.univie.ac.at/mario.barbatti/femtochem.html lecture4.ppt