Introduction to Quantum & Computational Chemistry For

Introduction to Quantum & Computational Chemistry for Electronically Excited States Lecture 4

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.

(visit http://www.lcpp.bgsu.edu) The (3N-6 Dimensional) Potential Energy Surface of a Chemical Reaction

[T (R) + Vnn(R) + Eel(R)] Photochemical Reaction Path

Excited State Conical Intersection (CI) or A* “Photochemical Funnel” 1966 Zimmerman, 1972 Michl

B hν A*

Ground State CI hν B Energy A A

Excited State Ground State Minimum Energy Paths Reaction Coordinate Photochemical Reaction Path

Stationary Point Singularity

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

TS CI A*

A X X1 2

X1 B B A One Product One or More Product Benzene Photochemistry

(excited state) (primary) (secondary)

h ν

254 nm liquid state prefulvene benzvalene under N2 (q.y. 0.02) (e.g. Turro 1986)

N of photoproduct molecules N of absorbed photons

Ground state diradical intermediate Benzene Conical Intersection: Structure

unpaired electrons

ground state 1.4 Å allyl radical 2.0 Å half-broken bond

1.4 Å

This conical intersection defines the "prefulvene" path Benzene Conical Intersection:Branching Space

ΨB ΨA

diradical CI

Energy Kekule

Excited State Ground State

Reaction Coordinate

The wavefunction (electronic structure) does not change when passing through the CI. Benzene Conical Intersection:Branching Space

δ( E 2 − E1 ) δ X1 = X2 = ψ1 ψ2 δ q δq

Gradient Difference Derivative Coupling

(fastest escape from energy degeneracy) (fastest change in the electronic structure) Benzene Conical Intersection: Wavefunction

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

coupled electrons

x 1 x2 Benzene Conical Intersection: the branching space

χ unstable 3

x2 χ -x1 x1 3

-x2

3 1 unstable x1 -x2 -x1 x2 Branching space diagram A bit of History

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.

2 2 S1 σ1 π1π2 σ2 State correlation π 2 π π 2π S diagram 2A1 1 2 3 4 1

2 1 1 1B2 π1 π2 π3 π4 S2 M* Slow decay (Fermi Golden Avoded crossing h ν Rule - coupling of Suggests that the non- vibrational states) crossing rule applies not only to diatomic but also 2 2 1A π1 π2 π3π4 S0 S σ 2 π 2π σ to polyatomic molecules 1 0 1 1 2 2

Cs symmetry

interpolated and symmetric reaction coordinate A bit of History

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…”

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...”. Photophysics of octatetraene

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

hν’ CI hν B Energy A

Excited State Ground State

Reaction Coordinate 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.

1966, Howard Zimmerman

1970, Josef Michl

1974, Lionel Salem A bit of History

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

allow the use optimization methods (eg pseudo Newton-Raphson) allow to draw guess structures (eg for pericyclic reactions) !

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

2.17 Å

1.47 Å 2.08 Å

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

“statistical” demonstration using quantum chemistry 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

2A1

40 S2/S1 1B 2A 2 1 S1 S2/S1

1B 2 * S2 S2 M 20

h ν S1/S0

0 hν S1/S0

1A S 1 0 S0

Cs symmetry

1A 1 real crossing between 0.0 states of the same (A1) symmetry A bit of History

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

2 2 S1 σ1 π1π2 σ2

π 2 π π 2π S 1 2 3 4 1 S1 FC 2 1 1 S2 σ1 π1 π2 σ2

2 1 1 π1 π2 π3 π4 S2 M* CI

h ν S0 Gradient Based Coordinate

π 2 π 2π π S 2 2 1 2 3 4 0 S0 σ1 π1 π2σ2

Symmetry Based Coordinate Computational Tools

Intrinsic Reaction Coordinate (IRC)

A* Conical Interersection Optimization (CIO)

Initial Relaxation Direction (IRD) B

Trajectory (Classical or Semi-classical) Energy Minimum and Transition State A Optimization Photochemical Reaction Path in Textbooks 2001 Photochemical Reaction Path in Textbooks

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

2008

5.6 Conical Intersections near Zero-Order Surface Crossings

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

6.13 Some Important and Unique Properties of Conical Intersections

6.30 Concerted Photochemical Pericyclic Reactions and Conical Intersections Computational Photochemistry optimization of a singularity wavefunction/density (orbital occupancies) MOLECULAR AND ELECTRONIC STRUCTURE OF THE CROSSING: NATURE OF THE PHOTOCHEMICAL FUNNEL

equilibrium geometries,transition states and minimum energy paths

EXCITED STATE REACTION PATHS: EXCITED STATE DECAY

almost routine branching plane

GROUND STATE RELAXATION PATHS: PHOTOPRODUCT SELECTIVITY feasible ! Newton equations of motion

TRAJECTORIES: REACTION TIME SCALES AND QUANTUM YIELDS still unpractical or impossible Different Electronic States = Different Conical Intersection Structure = Different Chemistry

Hydrocarbons Schiff bases

S2 Avoided Crossing rule valid ! S 2 2A 1 S2 S2/S1 2A S1 1B 1 2 S1 S1

1B * 2 S2 M * Avoided Crossing rule invalid M

- + S1/S0 S1/S0 h ν Avoided Crossing h ν rule invalid

1A S 1 0 S0 1A S 1 0 S0

+ NH2 + NH2 (π π*)2 π π* The Chemistry of Conical Intersections: Bond-Making, Bond Breaking and Group Transfer

6 6

1 3

1 σ-Bond Making C

Group (or σ-Bond) Exchange σ-Bond Breaking The Chemistry of Conical Intersections: Conjugated Hydrocarbons

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

Crossing between the ground state and a (π-π*) doubly excited state 2.0 1.4 1.4 π∗ S Polyenes (and polyene radicals) 1 π

π∗ S0 π

Benzene Cyclohexadienes The Chemistry of Conical Intersections: Conjugated Hydrocarbons The Chemistry of Conical Intersections: Conjugated Hydrocarbons

Selectivity may be due to differences in energy

30 S1 -1 S0 20

E / mol kcal 10

90° 0 cyclizations

3 5 7 9 Z/E isomerization The Chemistry of Conical Intersections: Protonated Schiff Bases

- + 1 NH2 (+) 3 2 3 2 NH (+) 5 4 1 2 5 4 cis trans

Retinal Rhodopsin

11 Light 11 + + NH NH

Trans Form Cis Form (Appears in ca. 200 fs) The Chemistry of Conical Intersections: Protonated Schiff Bases

Crossing between the ground state and a (π-π*) singly 1.46 Å excited state 1.38 Å

π∗ + 1.38 Å 1.40 Å π N 1.33 Å 90° S1 hν + e- π∗ 90° π

S0 Newman projection The Chemistry of Conical Intersections: Protonated Schiff Bases

Motion Coupled to Unstable (TS) the Torsion +

NH2

N x2

X1 χ -x1 x1 + NH2 + NH2 NH2

-x2 N Stretching X + 2 NH2

Unstable (TS) The Chemistry of Conical Intersections: Protonated Schiff Bases

+ NH2 breaks the double bond hetherolitically + + NH2

NH2 breaks the double bond homolitically + + NH2 NH2

x1 -x2 -x1 x2 The Chemistry of Conical Intersections: Protonated Schiff Bases

0.0 1.46 1.43 12.5 π∗ 1.36 1.35 1.29 1.37 1.37 FC π 1.5 1.41 1.35 24.7 120 1.37 S2 1.37 1.35 π∗ 1.53 1.42 π 100 76.8

) 1.42 1.30 -1 80 1.39 1.46 S1 1.39 CI 60 91.2 1.43 1.30 1.43 hν (291 nm) 40 1.42 1.36

Energy (kcal mol S0 180.0 1.43 20 1.45 1.29 1.36 1.35 trans 0 0.0 5.0 10.0 15.0 20.0

MEP co-ordinate (a. u.) Structure of Bovine (rod) Rhodopsin

Cα W1 11 12 O

HN Glu113 O W2

Lys296

Cα

Glu113

Lys296

Try265 S. T. Menon, M. Han, T. P. Sakmar, Physiological reviews 2001, 81, 1659. Low Temperature Photochromism in Bovine Rhodopsin

580 nm irradiation at 77 K

-8° (11-cis) Opsin -144° (all-trans)

HN 0.67 q.y. Opsin

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

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. Kukura, P.; Mc Camant, D. W.; Yoon, S.; Wandschneider, D. B.; Mathies, R. A. Science. 2005, 310, 1006-1009. Verdegem, P. J. E. et al. Biochemistry 1999, 38, 11316. “Photochemical” Trajectories

V1 (R) Excited State The Photochemical Funnel is a Conical Intersection (CI) A* Zimmerman, Michl, Salem

reactive trajectory B hν

V0 (R) Ground State

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

Bernardi, F., M. Olivucci, M. A. Robb, Chem. Soc. Rev. 1996, 25, 321 328. Q. Wang, R. W. Schoenlein, L. A. Peteanu, R. A. Mathies, C. V. Shank, Science 1994, 266, 422. A QM/MM Protocol for Excited States

Fixed

Lys296 MM α Relaxed γ (< 4 Å) 9 δ 13 ε 11 QM/MM Speciﬁcally Retinal Parametrized MM Frontier chromophore multiconﬁgurational QM

Rhodopsin (x-ray structure)

Ferré, N. and Olivucci, M. J. Am. Chem. Soc., 2003 125, 6868-6869. 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. A QM/MM Protocol for Excited States

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)

z rmm r R r MM rmm o x QM y Frontier 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 Scaled-CASSCF/Amber Semi-classical Trajectories

-8° (11-cis) 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)

HN Opsin Rh

50 fs 104 fs Opsin 6 0 -144° (all-trans)

) HN

-1 4 0 hν hν (702 nm)

2 0 bathoRh

Energy (kcal mol Time (fs) 0 S 0 30 60 90 20 1 1 15 0 180 21 0 S 0

Frutos, L.-M., Ferré, N. Andruniow, T., Santoro, F. and Olivucci, M. Proc. Nat. Acad. Sci. USA 2007 104 7764. D. Polli, P. Altoè, O. Weingart, K. M. Spillane, C. Manzoni, D. Brida, G. Tomasello, Garavelli, M. et al., Nature 2010, 467, 440. Schapiro, I., Ryazantsev, M. N., Frutos, L. M., Ferré, N., R. Lindh, R., Olivucci, M. J. Am. Chem. Soc. 2011 133 3354. CASSCF/Amber Semi-classical Trajectory of Rh (200 fs)

bicycle-pedal HOOP CW CCW

H H

HN Opsin

Schapiro, I., Ryazantsev, M. N., Frutos, L. M., Ferré, N., R. Lindh, R., Olivucci, M. J. Am. Chem. Soc. 2011 133 3354. Frutos, L.-M., Ferré, N. Andruniow, T., Santoro, F. and Olivucci, M. Proc. Nat. Acad. Sci. USA 2007 7764-7769. Evolution of the π-Electron Density of Rh (reactive trajectory)

broken π-bond

hν

CI Energy A B

Excited State Ground State (Bond reconstitution) Reaction Coordinate

Schapiro, I., Ryazantsev, M. N., Frutos, L. M., Ferré, N., R. Lindh, R., Olivucci, M. J. Am. Chem. Soc. 2011 133 3354. Evolution of the π-Electron Density of Rh (non reactive trajectory)

Cε restrained restrained QM atom

broken π-bond

hν

A CI

Energy (Bond reconstitution) B

Excited State Ground State

Reaction Coordinate

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

Robert Mulliken, Nobel Lecture, 1966

“…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…”

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

Faster computations Slower computations