introduction to quantum & computational chemistry for...

Introduction to Quantum & Computational Chemistry

for Electronically Excited StatesLecture 4

(visit http://www.lcpp.bgsu.edu)

Target: Computer simulations are employed to study the structure and reactivity of single molecules and molecular systems (molecule in solution on in a macromolecular cavity).

Tools: We need a series of software technologies to describe the electronic and geometrical structure of molecules and their time evolution.

http://www.lcpp.bgsu.edu

http://www.lcpp.bgsu.edu

The (3N-6 Dimensional) Potential Energy Surfaceof a Chemical Reaction

[T (R) + Vnn(R) + Eel(R)]

hν B

A

Conical Intersection (CI) or“Photochemical Funnel”A*

Excited State

Ground State

Photochemical Reaction Path

Minimum Energy Paths

Energy

Reaction Coordinate

CI

Ground StateExcited State

hν

A*

BA

1966 Zimmerman, 1972 Michl

TS

B

A

CI A*

B

A

Transition Vector (X 1) Branching (or g-h) plane (X1, X2)Stationary Point Singularity

One Product One or More Product

X1

X1X2

Photochemical Reaction Path

νh

254 nmliquid stateunder N2

benzvalene

Benzene Photochemistry

(q.y. 0.02)(e.g. Turro 1986)

(primary) (secondary)(excited state)

prefulvene

Ground state diradical intermediate

N of absorbed photonsN of photoproduct molecules

This conical intersection defines the "prefulvene" path

2.0 Å

1.4 Å

1.4 Å

ground stateallyl radical

Benzene Conical Intersection: Structure

half-broken bond

unpaired electrons

The wavefunction (electronic structure) does not change when passing through the CI.

diradical

Energy

Reaction Coordinate

CI


ΨAΨB

Kekule

Benzene Conical Intersection:Branching Space

X2= δ

δqψ1 ψ2X1 = δ( E − E )

δ q12

Benzene Conical Intersection:Branching Space

Gradient Difference

(fastest escape from energy degeneracy)

Derivative Coupling

(fastest change in the electronic structure)

Benzene Conical Intersection: Wavefunction

x1 x2

A consequence of the Geometric Phase theorem: the wavefunction (and bonding) changes sign along a loop that contains the intersection !

coupled electrons

coupled electrons

coupled electrons

Benzene Conical Intersection: the branching space

χ

3

1

x1

x2-x1-x2

x1

x2-x1

-x2

χ

1

3

31

Branching space diagram

unstable

unstable

π12

π2π32π4 S1

M*π12

π21π3

1π4 S2

π12

π22π3π4 S0 S0 σ1

2 π1

2π2σ2

S1 σ12

π1π22σ2

the first computations: 1969 Van der Lugt and Oosteroff and 1975 Devaquet et al. found that the point of return to the ground state (M*) is an energy minimum.

A bit of History

Avoded crossing

Cs symmetry

Suggests that the non-crossing rule applies not only to diatomic but also to polyatomic molecules

interpolated and symmetric reaction coordinate

State correlationdiagram

Slow decay (Fermi Golden Rule - coupling of vibrational states)

2A1

1B2

1A1

νh

E. Teller Isr. J. Chem. 7, 227, 1969

“…in a polyatomic molecule the non-crossing rule, which is rigorously valid for diatomics, fails and two electronic states, even if they have the same symmetry, are allowed to cross at a conical intersection..”.

“…radiationless decay from the upper to the lower intersecting state occurs within a single vibrational period when the system “travels” in the vicinity of such intersection points…”

A bit of History

H.C. Longuet-Higgins, “The Intersection of Potential Energy Surfaces in Polyatomic Molecules”, Proc. R. Soc. Lond. Ser. A., 344, 147-156, 1975

“…thereby disposing of a recent claim that the non-crossing rule for diatomic molecules applies also to polyatomic molecules...”.

Ultrafast deactivation channels are not consistent with stable M* intermediate.

Energy

Reaction Coordinate

CI


hν

A*

B

A

Photophysics of octatetraene

hν’

1966, Howard Zimmerman

1970, Josef Michl

1974, Lionel Salem

A bit of History

Zimmerman, Michl and Salem were the first to suggest that, in photochemical organic reactions, the point of return M* may correspond to a conical intersection. Zimmerman and Michl call it photochemical funnel.

1982-1988 CASSCF Gradients of the Excited State Energy (Robb, Bernardi, Schlegel and Olivucci). Structure Predicted from Valence Bond Theory

1990 First Conical Intersection “Detected” for the Ethylene Dimerization (Bernardi, Olivucci, Robb). Computation is carried out on the CRAY-XMP in London.

1.47 Å

2.17 Å

2.08 Å

1990-2000: 25 different organic chromophores undergoing 16 different reactions

A bit of History

“statistical” demonstration using quantum chemistry

allow to draw guess structures (eg for pericyclic reactions) !

allow the use optimization methods (eg pseudo Newton-Raphson)

0.0

40

20

0

2A1

1B2

1A1

hν

real crossing between states of the same (A1) symmetry

A bit of History

First application of ab initio CASPT2//CASSCF: s-cis buta-1,3-diene J. Chem. Phys. 1995

excited state minimum energy path

S1

M*S2

S0 S0

S2

Cs symmetry

2A1

1B2 S2

1A1

S1/S0

S2/S1

S1/S0

S2/S1

νh

S1

S0

CI

π12

π2π32π4 S1

M*

Symmetry Based Coordinate

π12

π21π3

1π4 S2

π12

π22π3π4 S0 S0 σ1

2 π1

2π2σ2

S2 σ12

π11π2

1σ2

S1 σ12

π1π22σ2

Gradient Based Coordinate

the van der Lugt and Oosteroff result is consistent with the existence of a conical intersection at the bottom of the S1 energy surfaces

A bit of History

FC

νh

B

A

A*

Computational Tools

Conical InterersectionOptimization (CIO)

Intrinsic Reaction Coordinate (IRC)

Initial Relaxation Direction (IRD)

Energy Minimumand Transition StateOptimization

Trajectory (Classicalor Semi-classical)

Photochemical Reaction Path in Textbooks

2001

2008

“…the use of computational methods to elucidate reaction mechanisms has not really made a major impact on the way in which organic photochemist think about such mechanisms …”

6.13 Some Important and Unique Properties of Conical Intersections

6.12 The Non-Crossing Rule and Its Violations: Conical Intersections and their Visualization

6.30 Concerted Photochemical Pericyclic Reactions and Conical Intersections

5.6 Conical Intersections near Zero-Order Surface Crossings

1990

Photochemical Reaction Path in Textbooks

Turro, N. J. (1990). J. Photochem. Photobiol., A: Chemistry 51 63.

MOLECULAR AND ELECTRONIC STRUCTURE OF THE CROSSING: NATURE OF THE PHOTOCHEMICAL FUNNEL

EXCITED STATE REACTION PATHS: EXCITED STATE DECAY

GROUND STATE RELAXATION PATHS: PHOTOPRODUCT SELECTIVITY

TRAJECTORIES: REACTION TIME SCALES AND QUANTUM YIELDS

Computational Photochemistry

almost routine

feasible !

still unpractical or impossible

wavefunction/density (orbital occupancies)

branching plane

equilibrium geometries,transition states and minimum energy paths

Newton equations of motion

optimization of a singularity

S1

M*

S0 S0

S2

2A1

1B2 S2

1A1

S1/S0

S2/S1S1

M*

S0 S0

S2

2A1

1B2

S2

1A1

S1/S0

Avoided Crossing rule valid !

Avoided Crossing rule invalid

Avoided Crossingrule invalid

Different Electronic States =Different Conical Intersection Structure =

Different Chemistry

- +

+(π π*)2 π π*

Hydrocarbons Schiff bases

S1

νh νh

NH2 NH2+

σ-Bond Making

σ-Bond Breaking

C

Group (or σ-Bond) Exchange

The Chemistry of Conical Intersections:Bond-Making, Bond Breaking and Group Transfer

1

3

6

1

6

3

Polyenes (and polyene radicals)

Benzene Cyclohexadienes

The Chemistry of Conical Intersections:Conjugated Hydrocarbons

1.4 1.42.0

J. Am. Chem. Soc. 1995, 117, 11584-11585

Crossing between the ground state and a (π-π*) doubly excited state

S1

S0 ππ∗

ππ∗

E / k

cal m

ol-1

3 5 7 9

0

10

20

30

40

cyclizations

Z/E isomerization

90°

Selectivity may be due to differences in energy

S1

S0


NH2 (+)1

23

45

NH2 (+)

trans

1

23

45

The Chemistry of Conical Intersections:Protonated Schiff Bases

cis

Cis Form

Light

Retinal Rhodopsin

Trans Form

NH+1111NH+

(Appears in ca. 200 fs)

- +

1.46 Å

1.40 Å

1.38 Å

1.33 Å

1.38 Å

90°

e-+

Crossing between the ground state and a (π-π*) singly excited state

S1

S0

ππ∗

ππ∗

hν

N+

90°

Newman projection


X1

X2

NH2+

NH2+

+NH2

+

x1

x2

-x1

-x2

χ

NH2

NH2

N

N

Unstable (TS)

Unstable (TS)

Stretching

Motion Coupled to the Torsion


χ

x1

x2-x1-x2

NH2

+NH2

breaks thedouble bond homolitically

+

breaks thedouble bond hetherolitically

NH2

NH2

+

NH2

+

+


1.35

1.461.36

1.43

1.29

MEP co-ordinate (a. u.)

0.0 5.0 10.0 15.0 20.00

20

40

60

80

100

hν

1.39

1.391.46

1.42 1.3076.8

(291 nm)

91.2

1.36

1.431.42

1.43 1.30

FC

120

1.42

1.37

1.53

1.37

1.35

24.7

12.5

1.51.37 1.37

1.351.41

S1

S0

0.0

1.35

1.451.36

1.431.29

trans

180.0

S2

CI

Ener

gy (k

cal m

ol-1

)ππ∗

ππ∗


HN

Cα

Cα

O

O

W2

W1

Structure of Bovine (rod) Rhodopsin

Lys296

Glu113

11 12

S. T. Menon, M. Han, T. P. Sakmar, Physiological reviews 2001, 81, 1659.

Lys296

Try265

Glu113

Low Temperature Photochromism in Bovine Rhodopsin

Rh (λmax 498 nm) bathoRh (λmax 543 nm)

580 nm irradiation at 77 K

OpsinHN

Spalink, J. D.; Reynolds, A. H.; Rentzepis, P. M.; Sperling, W.; Applebury, M. L. Proc. Natl. Acad. Sci. U. S. A. 1983, 80, 1887-1891.

-144° (all-trans)

Kukura, P.; Mc Camant, D. W.; Yoon, S.; Wandschneider, D. B.; Mathies, R. A. Science. 2005, 310, 1006-1009.

-8° (11-cis)

Verdegem, P. J. E. et al. Biochemistry 1999, 38, 11316.

HN

Opsin

0.67 q.y.

hνB

A

The Photochemical Funnel is aConical Intersection (CI)

A*

Excited State

Ground State

Zimmerman, Michl, Salem

Bernardi, F., M. Olivucci, M. A. Robb, Chem. Soc. Rev. 1996, 25, 321 328.

“Photochemical” Trajectories

reactive trajectory

non reactive trajectory

Q. Wang, R. W. Schoenlein, L. A. Peteanu, R. A. Mathies, C. V. Shank, Science 1994, 266, 422.

Evolution of the reactive moiety (198 vibrational degrees of freedom) embedded in the protein cavity.

V0 (R)

V1 (R)

Lys296

δε

γ

α

11

913

QM/MMSpecifically

ParametrizedMM Frontier

multiconfigurational QM

Rhodopsin (x-ray structure)

Nicolas Ferré and Massimo Olivucci Theochem, 2003 632, 71-82.

Ferré, N., Cembran, A., Garavelli, M. and Olivucci, M. Theo. Chem. Acc. 2004 112 335-341.

Ferré, N. and Olivucci, M. J. Am. Chem. Soc., 2003 125, 6868-6869.

MM

Retinal chromophore

A QM/MM Protocol for Excited States

Fixed

Relaxed(< 4 Å)

HTot = HMM+HQM+HQM/MM

HQM = Te(r) + Vee(r) + Ven(r,R) + Vnn(R)

HMM = Vmm(rmm)

HQM/MM = Ve/mm(r, rmm) + Vn/mm(rmm,R)

A. Warshel, M. Levitt // Theoretical Studies of Enzymic Reactions: Dielectric, Electrostatic and steric stabilization of the carbonium ion in the reaction of lysozyme. // J.Mol.Biol. 103(1976), 227-49

o

Rrmm

y

x

z

r

A QM/MM Protocol for Excited States

QM

MM

Frontier

rmm r

R

0 30 60 90 120 S0

S1

0

2 0

4 0

6 0

Frutos, L.-M., Ferré, N. Andruniow, T., Santoro, F. and Olivucci, M. Proc. Nat. Acad. Sci. USA 2007 104 7764.

104 fs

Ene

rgy

(kca

l mol

-1)

Time (fs)

hν (702 nm)

Scaled-CASSCF/Amber Semi-classical Trajectories

Schapiro, I., Ryazantsev, M. N., Frutos, L. M., Ferré, N., R. Lindh, R., Olivucci, M. J. Am. Chem. Soc. 2011 133 3354.

50 fs

15 0 180 21 0

-144° (all-trans)HN

Opsin

OpsinHN

-8° (11-cis)

hν

bathoRh

Rh

Kakitani et al. J. Phys. Chem. B 1998 (120-170 fs)Kandori et al. Chem. Phys. Lett. 2001 (above 100 fs)Polli et al. Nature 2010 (above 70 fs)

D. Polli, P. Altoè, O. Weingart, K. M. Spillane, C. Manzoni, D. Brida, G. Tomasello, Garavelli, M. et al., Nature 2010, 467, 440.

OpsinHN

CASSCF/Amber Semi-classical Trajectory of Rh (200 fs)

Frutos, L.-M., Ferré, N. Andruniow, T., Santoro, F. and Olivucci, M. Proc. Nat. Acad. Sci. USA 2007 7764-7769.


CW CCW

bicycle-pedal

H H

HOOP

Evolution of the π-Electron Density of Rh

(reactive trajectory)

(Bond reconstitution)

Ener

gy

Reaction Coordinate

CI


hν

A*

BA

broken π-bond


Evolution of the π-Electron Density of Rh

(non reactive trajectory)

(Bond reconstitution)

Ener

gyReaction Coordinate

CI


hν

A*

BA

broken π-bond

HN

Cε

restrained QM atomrestrained


The Mulliken’s dream

“…In conclusion, I would like to emphasize strongly my belief that the era of computing chemists, when hundreds if not thousands of chemists will go to the

computing machine instead of the laboratory for increasingly many facets of chemical information, is

already at hand…”

Robert Mulliken, Nobel Lecture, 1966

Software: Molecular Orbitals rather than Atomic Orbitals for the description of the electronic structure of molecules

Faster computations Slower computations

introduction to quantum & computational chemistry for...

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