Our Experience with TD-DFT and TD-DFTB in Biological, Bio-inspired, - - PowerPoint PPT Presentation

CECAM SISSA ICTP 22/09-2020 1

Mark E. Casida ( 樫田 マーグ

マーク , 卡西达 マーグ 马克 )

Professeur, chimie théorique Laboratoire de Spectrométrie, Interactions et Chimie Théorique (SITh) Département de Chimie Moléculaire (DCM, UMR CNRS/UGA 5250) Institut de Chimie Moléculaire de Grenoble (ICMG, FR-2607) Université Grenoble Alpes (UGA) 301 rue de la Chimie CS 40700 38058 Grenoble cedex 9 France e-mail: mark.casida@univ-grenoble-alpes.fr

Our Experience with TD-DFT and TD-DFTB in Biological, Bio-inspired, and Other Photoprocesses

SISSA Zoom Workshop Hosted by ICTP, Trieste, Italy 30 minutes

CECAM SISSA ICTP 22/09-2020 2

Lester Earl CASIDA 1904–1986

• Prof. Univ. Wisconsin, Madison

American Society of Animal Science has named a scholarship for doctoral students in Reproductive Physiology in his honor.

Lester Earl CASIDA Jr. 1928-2017

• Prof. Penn. State Univ.

Industrial and soil microbiology Discovered bacterium: Ensifer adherens Casida 1982

John Edward CASIDA 1929-2018

• Prof. Univ. Calif., Berkeley

Entomology and toxicology Very distinguished (member USA NAS, UK Royal Society, Wolf Prize in Agriculture, ...)

Mark Earl CASIDA 1957-?

• Prof. Univ. Grenoble Alpes

Theoretical Chemistry Casida equations for TD-DFT

CECAM SISSA ICTP 22/09-2020 3

CECAM SISSA ICTP 22/09-2020 4

OUTLINE

• I. PHOTOCHEMICAL THINKING
• II. ELECTRONIC EXCITED-STATE PROCESSES
• III. TD-DFT(B)
• IV. STATIC APPLICATIONS
• A. Luminescence indices
• B. Retinal
• C. Excitonic effects
• V. TD-DFT(B) FSSH
• VI. DYNAMICS APPLICATIONS
• A. Easy case: oxirane
• B. Hard case: pentacene/buckyball
• VII. CONCLUSION

CECAM SISSA ICTP 22/09-2020 5

O X I R A N E P H O T O C H E MI S T R Y T . I b u k i , M. I n a s a k i e t Y . T a k e s a k i , J . C h e m . P h y s . 5 9 , 2 7 6 ( 1 9 7 3 ).

CECAM SISSA ICTP 22/09-2020 6

LEWIS STRUCTURES

Gomer-Noyes Mechanism [E. Gomer et W.A. Noyes, Jr., J. Am. Chem. Soc. 72, 101 (1950);

• T. Ibuki, M. Inasaki et Y. Takesaki, J. Chem. Phys. 59, 2076 (1973).]

CECAM SISSA ICTP 22/09-2020 7

ORBITAL THINKING: WOODWARD-HOFFMANN RULES

CECAM SISSA ICTP 22/09-2020 8

S T A T E S : T WO

• O

R B I T A L T WO

• E

L E C T R O N MO D E L ( T O T E M) i a i a i a i a i a

a

 i 

a

 i 

a

 i 

a

 i 

∣i  i ∣ ∣a  i ∣ ∣i  a ∣ ∣i a ∣ ∣ a  i ∣

Singlet Triplets

 1

,

=∣i a ∣

 S

, MS

 1

, −1

=∣ a  i ∣  0

,

= 1

2

∣a

 i ∣∣i  a ∣  1

,

= 1

2

∣a

 i ∣−∣i  a ∣

S =a −i 2

a

i

∣f

H∣i

a

−a

a

∣f

H∣i

i



T =a −i −a a

∣f

H∣i

i



CECAM SISSA ICTP 22/09-2020 9

POTENTIAL ENERGY SURFACES (PESs) Original image: J. Michl and V. Bonacic-Koutecky, Electronic Aspects of Organic Photochemistry (Wiley: New York, 1990), p. 71. Embellishment: E. Tapvicza. REACTANT PRODUCT PRODUCT

CECAM SISSA ICTP 22/09-2020 10

WAYS WE TRY TO UNDERSTAND PHOTOCHEMICAL MECHANISMS

Lewis structures

Orbital models Potential energy surfaces* Pathway approach Minimum energy pathways Funnels Dynamics Ehrenfest Surface hopping

* J. Michl and V. Bonacic-Koutecky, Electronic Aspects of Organic

Photochemistry (Wiley: New York, 1990)

(Star Trek 3D chess)

CECAM SISSA ICTP 22/09-2020 11

OUTLINE

• I. PHOTOCHEMICAL THINKING
• II. ELECTRONIC EXCITED-STATE PROCESSES
• III. TD-DFT(B)
• IV. STATIC APPLICATIONS
• A. Luminescence indices
• B. Retinal
• C. Excitonic effects
• V. TD-DFT(B) FSSH
• VI. DYNAMICS APPLICATIONS
• A. Easy case: oxirane
• B. Hard case: pentacene/buckyball
• VII. CONCLUSION

CECAM SISSA ICTP 22/09-2020 12

SOME EXAMPLES OF EXCITED-STATE PROCESSES IN BIOLOGY (all fall under a generalized notion of “photochemistry”)

CECAM SISSA ICTP 22/09-2020 13

PHOTOPHYSICS: Radiationless relaxation Luminescence Fluoresence Phosphorescence

Example: green fluorescent protein

A+h ν→A

*→A http://zeiss-campus.magnet.fsu.edu/print/probes/fpintroduction-print.html

CECAM SISSA ICTP 22/09-2020 14

PHOTOCHEMISTRY

Example: vision (rhodopsin)

A+h ν→A

*→B+C https://www.ncbi.nlm.nih.gov/books/NBK52768/

CECAM SISSA ICTP 22/09-2020 15

CHEMILUMINESCENCE

Example: firefly luciferon

B+C→A

*→A+h ν

CECAM SISSA ICTP 22/09-2020 16

E.H. White, J. Wiecko, and D.F. Roswell, “Photochemistry without light”,

• J. Am. Chem. Soc. 91, 5194 (1969).

A+h ν→A

*→B+C

A+Δ→A

*→B+C

CECAM SISSA ICTP 22/09-2020 17

https://www.slideshare.net/cdtpv/thursday-42325335 From a presentation by Neil Greenham

A

*+B→A+B *

EXCITONS

A

++B

• →A
• +B

+ Example: photosynthesis Excitons may be: Real --- energy and charge transfer Fictitious --- needed to understand complex systems

CECAM SISSA ICTP 22/09-2020 18

OUTLINE

• I. PHOTOCHEMICAL THINKING
• II. ELECTRONIC EXCITED-STATE PROCESSES
• III. TD-DFT(B)
• IV. STATIC APPLICATIONS
• A. Luminescence indices
• B. Retinal
• C. Excitonic effects
• V. TD-DFT(B) FSSH
• VI. DYNAMICS APPLICATIONS
• A. Easy case: oxirane
• B. Hard case: pentacene/buckyball
• VII. CONCLUSION

CECAM SISSA ICTP 22/09-2020 19

Cartoon given to me by Jean-Paul Malrieu 1995

CECAM SISSA ICTP 22/09-2020 20

Presentation uploaded by Deddy Tedjo https://slideplayer.com/slide/17033333/

CECAM SISSA ICTP 22/09-2020 21

F

• r

a s y s t e m , i n t i a l l y i n i t s g r

• u

n d s t a t e , e x p

• s

e d t

• t

i m e

• d

e p e n d e n t p e r t u r b a t i

• n

: R u n g e

• G

r

• s

s T h e

• r

e m : v

e x t

( r t ) i s d e t e r m i n e d b y ( r t ) u p t

• a

n a d d i t i v e f u n c t i

• n
• f

t i m e C

• r
• l

l a r y :

 rt N ,vext r tC t  H tC t te

−i∫t0

t Ct 'dt '

T I ME

• D

E P E N D E N T D E N S I T Y

• F

U N C T I O N A L T H E O R Y ( T D D F T ) [ E . R u n g e a n d E . K . U . G r

• s

s , P h y s . R e v . L e t t . 5 2 , 9 9 7 ( 1 9 8 4 ) ]

( R G 1 a s s u m e s f u n c t i

• n

s w i t h T a y l

• r

s e r i e s . )

CECAM SISSA ICTP 22/09-2020 22

TIME-DEPENDENT KOHN-SHAM EQUATION

[−1

2 ∇

2

v

e x t

 r t ∫  r ' t  ∣ r − r ' ∣ d  r ' v

x c

 r t ]i  r t =i ∂ ∂t i  r t   r t =∑i

 n i ∣i 

r t ∣

2

where and

v

x c

 r t =  A

x c

[]   r t 

(1) (2) (3) [E. Runge and E. K. U. Gross, Phys. Rev. Lett. 52, 997 (1984)]

CECAM SISSA ICTP 22/09-2020 23

E l e c t r i c

• F

i e l d I n d u c e d E l e c t r

• n

i c P

• l

a r i z a t i

• n

Classical model of a photon Induced dipole moment   t=−e 0∣ r∣ 0t 0t∣ r∣ 0

  t =  cos0t v r t=e  t ⋅ r

H H H H O H H H H

ℏ0

photon

 t

CECAM SISSA ICTP 22/09-2020 24

THE DYNAMIC POLARIZABILITY

it=i∑j i , j j cost⋯

r i ,r j=∑I≠0 2I 0∣r i∣ I I∣r j∣ 0 I

2− 2

=∑I≠0 f I I

2− 2

f I=2 3 I ∣ 0∣x∣ I∣

2∣ 0∣y∣ I∣ 2∣ 0∣z∣ I ∣ 2

Sum-over-states (SOS) theorem

f

I

I

How to make computationally convenient?

CECAM SISSA ICTP 22/09-2020 25

COMPUTATIONALLY CONVENIENT FORMULATION

[

A I  B I  B I  A I ]  X

I

 Y

I=I

[

1 0 0 −1

]

 X

I

 Y

I

Ma r k E . C a s i d a i n R e c e n t A d v a n c e s i n D e n s i t y F u n c t i

• n

a l Me t h

• d

s , P a r t I , e d i t e d b y D . P . C h

• n

g ( S i n g a p

• r

e , Wo r l d S c i e n t i fi c , 1 9 9 5 ) , p . 1 5 5 . " T i m e

• d

e p e n d e n t d e n s i t y

• f

u n c t i

• n

a l r e s p

• n

s e t h e

• r

y f

• r

m

• l

e c u l e s ' '

A

i j  , k l = , i , k

 j

, l

 j

− j K i j  , k l 

K

i j  , k l =∫∫ ∗i  

r  j



r  f

H x c  , 

r ,  r ' ; k



r ' ∗l



r ' d  r d  r ' B

i j  , k l =K i j  , l k 

“RPA” equation (1) where (2) (3) Coupling matrix (4)

CECAM SISSA ICTP 22/09-2020 26

Wh e r e C

• n

v e n t i

• n

a l T D

• D

F T Wo r k s B e s t

When the un-symmetry-broken ground-state DFT answer is good Low energy excitations of dominant single excitation character Not too much “charge-transfer” (really density relaxation) character Reasonably localized excitations “safe place” “reasonable risk” : where most applications are actually done “danger zone” : what I would like to do and where we are most “in danger” of learning something interesting

CECAM SISSA ICTP 22/09-2020 27

Despite its simplicity, people really like TD-DFT because it allows them to treat problems that they could not treat with

• ther methods!

CECAM SISSA ICTP 22/09-2020 28

DENSITY-FUNCTIONAL TIGHT BINDING (DFTB)

• Resembles DFT
• Minimal basis set
• Monopole expansion of ERIs (g integrals)
• Only 2-center integrals thanks to the potential or density superposition approximation
• Expansion of repulsion potential in terms of atomic pair potentials
• Self-consistent (response) correction
• Extended to TD-DFTB

I will never do semi-empirical theory! Sure you will! (A historic conversation From Montreal days) Thomas HEINE, Technische Universität Dresden

CECAM SISSA ICTP 22/09-2020 29

Murphy's Law: “If anything can possibly go wrong, it will.” (A good law for air craft engineers* and developers of theoretical methods.)

* Edward Murphy was an aeronautics engineer. The

• riginal meaning of the “law” was that you need to

be ultracareful when you want to design something safe and reliable.

CECAM SISSA ICTP 22/09-2020 30

WHERE TD-DFT WORKS BEST: Single electron (particle-hole) excitations Calculations without symmetry breaking Low-energy excitations Localized excitations Excitations without too much charge transfer Away from conical intersections Much progress has been made on overcoming these

• limitations. Only a little can presented here about

these problems and their solutions.

And if you want to look at charge transfer excitations?

CECAM SISSA ICTP 22/09-2020 31

RELATED PROBLEM: UNDERESTIMATED CHARGE TRANSFER EXCITATIONS

S =a −i a i

∣2

f

H f x c ,  f x c  , ∣i

a



T =a −i i a

∣f

x c , − f x c  , ∣a

i



(1) (2) Long-range charge transfer =>

C

T

=a −i E A −I P

(3) Should have

C

T

=E A −I P − 1 R

(4)

• A. Dreuw, J.L. Weisman, and M. Head-Gordon, “Long-range charge-transfer

excited-states in time-dependent density-functional theory require non-local exchange”,

• J. Chem. Phys. 119, 2943 (2003).

CECAM SISSA ICTP 22/09-2020 32

L-Criterion*

* M.J.G. Peach, P. Benfield, T. Helgaker, and D.J. Tozer, J. Chem. Phys. 128, 044118 (2008). “Excitation energies in density functional theory: An evaluation and a diagnostic test”

L=∑i,a κia

2 Oia

∑i ,a κia

2

(1)

κia=Xia+ Y ia

(2)

Oia=∫∣ψi(⃗ r)∣∣ψa(⃗ r)∣d ⃗ r

(3)

Small values (< 0.3) of L indicate a high likelihood of a “charge-transfer” underestimation.

CECAM SISSA ICTP 22/09-2020 33

R A N G E

• S

E P A R A T E D H Y B R I D S ( R S H )

1 r12 =erfc r12 r12



SHORT RANGE

erf  r12 r12



LONG RANGE

Molecules: SR <-> DFT LR <-> WF (e.g. HF) Solids: SR <-> WF LR <-> DFT

Applications in TDDFT:

• Y. Tawada, T. Tsuneda, S. Yanagisawa, T. Yanai, and K. Hirao, J. Chem. Phys. 120, 8425 (2004).
• S. Tokura, T. Tsuneda, and K. Hirao, J. Theoretical and Computational Chem. 5, 925 (2006).

O.A. Vydrov and G.E. Scuseria, J. Chem. Phys. 125, 234109 (2006). M.J.G. Peach, E.I. Tellgrent, P. Salek, T. Helgaker, and D.J. Tozer, J. Phys. Chem. A 111, 11930 (2007).

• E. Livshits and R. Baer, Phys. Chem. Chem. Phys. 9, 2932 (2007).

CECAM SISSA ICTP 22/09-2020 34

OUTLINE

• I. PHOTOCHEMICAL THINKING
• II. ELECTRONIC EXCITED-STATE PROCESSES
• III. TD-DFT(B)
• IV. STATIC APPLICATIONS
• A. Luminescence indices
• B. Retinal
• C. Excitonic effects
• V. TD-DFT(B) FSSH
• VI. DYNAMICS APPLICATIONS
• A. Easy case: oxirane
• B. Hard case: pentacene/buckyball
• VII. CONCLUSION

CECAM SISSA ICTP 22/09-2020 35

PHOTOSYSTEM II (GREEN PLANTS) From http://en.wikipedia.org/wiki/Photosystem_II Creates H+ gradient which is used in the process of making ATP.

CECAM SISSA ICTP 22/09-2020 36

PHOTOSYSTEM II: Cyanobacteria photosystem II, Monomer, PDB 2AXT. From http://en.wikipedia.org/wiki/Photosystem_II

CECAM SISSA ICTP 22/09-2020 37

PHOTOSYSTEM II (GREEN PLANTS) Image from Sébastien Liatard's oral defense. (1) (2) (3) OEC =

• xygen

evolving complex

CECAM SISSA ICTP 22/09-2020 38

BIO-INSPIRED CHEMISTRY

Donner – Pigment – Acceptor D-P*-A

1) Excitation 2) Charge separation

D+-P-A-

CECAM SISSA ICTP 22/09-2020 39

MAGIC OF [Ru(bpy)3]2+ Wavelength (nm)

Solar spectrum

Image from Sébastien Liatard's oral defense. Long-lived excited state Phosphorescence Easy charge transfer

CECAM SISSA ICTP 22/09-2020 40

CAN WE ADD “WIRES”? CAN WE MAKE IT LOOK MORE LIKE PS II? Image from Sébastien Liatard's oral defense.

CECAM SISSA ICTP 22/09-2020 41

TIME RESOLVED PHOTOLUMINESCENCE https://www.picoquant.com/applications/category/life-science/time-resolved-fluorescence The experiment measures the rate

• f disappearance of the excited state,

either by luminescence or by other mechanisms such as radiationless relaxation.

CECAM SISSA ICTP 22/09-2020 42

LUMINESCENCE LIFETIME DEPENDS UPON THE HEIGHT OF A BARRIER

1MLCT 3MLCT 3MLCT 3MC 3MC 3MC 1MC 1GS 1GS

REACTION COORDINATE

3MC 3MC/1GS 1GS 3MLCT

ENERGY

Ea

CECAM SISSA ICTP 22/09-2020 43

Section “E. Collection of spectroscopic, redox, photochemical, and photophysical data”

• pp. 145-243 is a gold mine of data on many different Ru(II) polypyridine complexes!!

Allows us to extract an empirical barrier height: Eave

CECAM SISSA ICTP 22/09-2020 44

FINAL LUMINESCENCE INDEX?

[MCA+17] Denis Magero, Mark E. Casida, George Amolo, Nicholas Makau, and Lusweti Kituyi,

• J. Photochem. Photobiol. A 348, 305 (2017). Preprint: https://arxiv.org/abs/1707.03665

"Partial Density of States Ligand Field Theory (PDOS-LFT): Recovering a LFT-Like Picture and Application to Photoproperties of Ruthenium(II) Polypyridine Complexes" Not perfect, but very good considering all the approximations made! (eV)

Δ E=ϵeg

*−ϵπ *

¯ E= ϵeg

*+ϵπ *

2

CECAM SISSA ICTP 22/09-2020 45 5 10 15 20 25 30 35 100 200 300 400 500 600 700 800 900 [Ru(X)3]2+ [Ru(Y)2]2+ [Ru(bpy)2(X)]2+ [Ru(bpy)(Y)2]2+ [Ru(X)2(Y)]2+

(bar E)^2/(Delta E) (eV) Delta Eave (cm-1)

BREAKDOWN BY LIGAND FAMILIES

??? [Ru(bpy)2(DIAFO)]2+ [Ru(bpy)2(4,4'-DTB-bpy))]2+ [Ru(phen)2(4,7-dhy-phen))]2+

CECAM SISSA ICTP 22/09-2020 46

BREAKDOWN BY LIGAND FAMILIES

CECAM SISSA ICTP 22/09-2020 47

OUTLINE

• I. PHOTOCHEMICAL THINKING
• II. ELECTRONIC EXCITED-STATE PROCESSES
• III. TD-DFT(B)
• IV. STATIC APPLICATIONS
• A. Luminescence indices
• B. Retinal
• C. Excitonic effects
• V. TD-DFT(B) FSSH
• VI. DYNAMICS APPLICATIONS
• A. Easy case: oxirane
• B. Hard case: pentacene/buckyball
• VII. CONCLUSION

CECAM SISSA ICTP 22/09-2020 48

• O. Valsson, C. Filippi, and M.E. Casida, J. Chem. Phys. 142, 144104 (2015)

Regarding the use and misuse of retinal protonated Schiff base photochemistry as a test case for time-dependent density-functional theory

CECAM SISSA ICTP 22/09-2020 49

http://www.chm.bris.ac.uk/motm/retinal/retinalv.htm

SOME BIOCHEMISTRY

Elizabeth Taylor 11Z 11Z

CECAM SISSA ICTP 22/09-2020 50

RETINAL PROTONATED SCHIFF BASE (PSB)

5 6 7 8 9 10 11 13 12 14 15

h ν 11Z → 11 E

1 2 3 4 16 11 12 11 12 11 12 11 12

CECAM SISSA ICTP 22/09-2020 51

d d d d d d d d d d d d d d d d d = double bond in ground state

θ=93

• g=0
• θ=−10
• g=43.6
• CASPT2

CASSCF

θ=8.1

• g=112.7
• CASPT2

B3LYP

• CAM-B3LYP
• θ=−27.4
• g=17.7
• LC-BLYP

B3LYP CAM-B3LYP

θ=−0.2

• g=86.8
• LC-BLYP

θ=−2.4

• g=85.4
• θ=−1.2
• g=85.2

CECAM SISSA ICTP 22/09-2020 52

Scan around single bond

CECAM SISSA ICTP 22/09-2020 53

OUTLINE

• I. PHOTOCHEMICAL THINKING
• II. ELECTRONIC EXCITED-STATE PROCESSES
• III. TD-DFT(B)
• IV. STATIC APPLICATIONS
• A. Luminescence indices
• B. Retinal
• C. Excitonic effects
• V. TD-DFT(B) FSSH
• VI. DYNAMICS APPLICATIONS
• A. Easy case: oxirane
• B. Hard case: pentacene/buckyball
• VII. CONCLUSION

CECAM SISSA ICTP 22/09-2020 54

A.A.M.H.M. Darghouth, G.C. Correa, S. Juillard, M.E. Casida, A. Humeniuk, and R. Mitrić, "Davydov-Type Excitonic Effects on the Absorption Spectra of Parallel-Stacked and Herringbone Aggregates of Pentacene: Time-Dependent Density-Functional Theory and Time-Dependent Density-Functional Tight Binding”, J. Chem. Phys. 149, 134111 (2018)

CECAM SISSA ICTP 22/09-2020 55

Spectral effects seen in dye aggregates: J-aggregates (Jelly/Scheibe) spectrum shifts to lower energies and new narrow peaks appear. H-aggregates (hypsochromic) spectrum shifts to higher energies. Image from

• Chem. Soc. Rev.

43, 5211 (2014)

CECAM SISSA ICTP 22/09-2020 56

Davydov Splitting in Pentacene

BSE/GW results* Shifted BSE/GW results J-aggregate * P. Cudazzo, F. Sottile, A. Rubio, and M. Gatti, “Topical Review: Exciton dispersion in molecular Solids,” J. Phys. Condens. Matter 27, 113204 (2015).

CECAM SISSA ICTP 22/09-2020 57

K a s h a ' s E x c i t

• n

Mo d e l *

* [KRE65] M. Kasha, H.R. Rawls, and A. El Bayoumi, “The exciton model in molecular Spectroscopy”, Pure Appl. Chem. 11, 371 (1965).

• 1. Historical theory (1960s). Necessarily very approximate!
• 2. Assumes a van der Waals (vdW) dimer.
• 3. Perturbative. Only zero and first order.
• 4. Point-dipole point-dipole approximation.

Ψ1

0→Ψ1 I

Ψ2

0→Ψ2 I

=E1

I−E1

=E1

I−E1

(1) (2) (3) (4)

^ H= ^ H 1+ ^ H 2+ ^ V 12 Ψ0=Ψ1

0 Ψ2

ΨI=C1 Ψ1

I Ψ2 0+C2 Ψ1 0 Ψ2 I

(5) (6) (7) Michael Kasha (1920-2013)

CECAM SISSA ICTP 22/09-2020 58

Ground-State Energy

E0=⟨Ψ1

0 Ψ2 0∣ ^

H∣Ψ1

0 Ψ2 0⟩

(1)

E0=⟨Ψ1

0 Ψ2 0∣ ^

H 1∣Ψ1

0 Ψ2 0⟩+⟨Ψ1 0 Ψ2 0∣ ^

H 2∣Ψ1

0 Ψ2 0⟩+⟨Ψ1 0 Ψ2 0∣^

V 12∣Ψ1

0 Ψ2 0⟩

(3)

E0=⟨Ψ1

0∣ ^

H 1∣Ψ1

0⟩⟨Ψ2 0∣Ψ2 0⟩+⟨Ψ1 0∣Ψ1 0⟩⟨Ψ2 0∣ ^

H 2∣Ψ2

0⟩+⟨Ψ1 0 Ψ2 0∣^

V 12∣Ψ1

0 Ψ2 0⟩ (4)

E0=E1

0+E2 0+⟨Ψ1 0 Ψ2 0∣^

V 12∣Ψ1

0 Ψ2 0⟩

(5)

E0=E1

0+E2 0+EvdW

(6)

^ H= ^ H 1+ ^ H 2+ ^ V 12

(2)

CECAM SISSA ICTP 22/09-2020 59

Excited-State Energy

[

A B B A]( C1 C2)=E

I

(

C1 C2)

(1)

ΨI=C1 Ψ1

I Ψ2 0+C2 Ψ1 0 Ψ2 I

(2)

A=⟨Ψ1

I Ψ2 0∣ ^

H∣Ψ1

I Ψ2 0⟩=⟨Ψ1 0 Ψ2 I∣ ^

H∣Ψ1

0 Ψ2 I⟩

B=⟨Ψ1

0 Ψ2 I∣ ^

H∣Ψ1

I Ψ2 0⟩=⟨Ψ1 I Ψ2 0∣ ^

H∣Ψ1

0 Ψ2 I⟩

(3) (4)

Ψ±

I = 1

√2

(Ψ1

I Ψ2 0±Ψ1 0 Ψ2 I )

E ±

I =A±B

(5) (6)

CECAM SISSA ICTP 22/09-2020 60

Exciton Splitting

(1) (2)

A=⟨Ψ1

I Ψ2 0∣ ^

H∣Ψ1

I Ψ2 0⟩

B=⟨Ψ1

0 Ψ2 I∣ ^

H∣Ψ1

I Ψ2 0⟩

(3) (4)

±

I =E ± I −E 0=1 0+(EvdW I

−EvdW )±Eexciton splitting

(5) (6)

^ H= ^ H 1+ ^ H 2+ ^ V 12 A=E1

I+E2 0+⟨Ψ1 I Ψ2 0∣^

V 12∣Ψ1

I Ψ2 0⟩=E1 I+E2 0+EvdW I

B=⟨Ψ1

0 Ψ2 I∣^

V 12∣Ψ1

I Ψ2 0⟩=Eexciton splitting

E ±

I =E1 I+E2 0+EvdW I

±Eexciton splitting

(7)

CECAM SISSA ICTP 22/09-2020 61

Transition Moments

(1) (2)

⃗ μ

I 0=⟨Ψ I∣⃗

r∣Ψ

0⟩

⃗ μ±

I 0= 1

√2 ⟨Ψ1

I Ψ2 0±Ψ1 0 Ψ2 I∣⃗

r∣Ψ1

0 Ψ2 0⟩

(3) (4)

⃗ μ±

I 0= 1

√2

(⃗

μ1

I 0±⃗

μ2

I 0)

f ±

I =± I

3 ∣⃗ μ1

I 0±⃗

μ2

I 0∣ 2

• Fig. 1. Ref. [KRE65]

bright dark (5)

⃗ μ1/2⊥⃗ r

Example: Parallel stacking

(6)

f +

I =4

3 1

I∣⃗

μ1

I 0∣ 2

f −

I =0

(7)

CECAM SISSA ICTP 22/09-2020 62

We can apply this to stacked pentacene!!

DS DS

CECAM SISSA ICTP 22/09-2020 63

OUTLINE

• I. PHOTOCHEMICAL THINKING
• II. ELECTRONIC EXCITED-STATE PROCESSES
• III. TD-DFT(B)
• IV. STATIC APPLICATIONS
• A. Luminescence indices
• B. Retinal
• C. Excitonic effects
• V. TD-DFT(B) FSSH
• VI. DYNAMICS APPLICATIONS
• A. Easy case: oxirane
• B. Hard case: pentacene/buckyball
• VII. CONCLUSION

CECAM SISSA ICTP 22/09-2020 64

MIXED QUANTUM/CLASSICAL DYNAMICS

Electrons treated quantum mechanically. Nuclei are treated classically. Nuclei :

M ¨ R =−∇ V

mR



Electrons :

 H

e l e c

r ; R t 

e l e c

r , t =i ℏ d d t 

e l e c

r , t 

Ehrenfest dynamics:

M ¨ R =−∇ ⟨Ψ

e l e c

(t )∣V (R )∣Ψ

e l e c

(t )⟩

Surface hopping: Ehrenfest dynamics moves on an average PES which is physically incorrect and lacks microscopic reversability. Surface hopping has microscopic reversability and can, in principle, produce relative yields for competing products in a chemical reaction, but is much more difficult to implement and carry out.

CECAM SISSA ICTP 22/09-2020 65

SURFACE HOPPING

 H

e l e c

r ; R t 

e l e c

r , t =i ℏ d d t 

e l e c

r , t  

e l e c

r , t =∑m m

e l e c

r ; R t C

mt



Expand the time-dependent wave function in terms of the solutions of the time-independent Schrödinger equation.

 H

e l e c

r ; R t  m

e l e c

r ; R t =E

mR

t  m

e l e c

r ; R t 

Probability of finding the system on surface m is

P

mt

=∣C

mt

∣

2

1st order equation.

˙ C

mt

=−i E

mt

C

mt

/ℏ−∑n〈m ∣d n d t 〉C

n

t 

CECAM SISSA ICTP 22/09-2020 66

TULLY’S FEWEST SWITCHES SURFACE HOPPING (FSSH)

P

m, n

t =C

mt

C

n ∗t



Density matrix

g

m n

t ,  t = ˙ P

m, n

t  t P

m, m

= 2 ℜ ˙ C

mt

C

n ∗t

 t P

m, m

Probability of jumping from surface m to surface n in the interval (t,t+Δt) Generate a random number x in (0,1) and compare

g

m n

t ,  t 

If , then jump and set

C

n

t =1

Otherwise continue on surface m J.C. Tully, J. Chem. Phys. 93, 1061 (1990); S. Hammes-Schiffer and J.C. Tully, J. Chem.

• Phys. 101, 4657 (1994).

DETAILS

The hopping probability is 100% where two surfaces intersect. After a hop, nuclear kinetic energies need to be readjusted to conserve energy. The electronic numerical integration needs to be finer than the numerical integration for the nuclei.

NOT A DETAIL

It is important not to read too much meaning into a single FSSH trajectories. Only ensemble averages over swarms of trajectories are physical.

CECAM SISSA ICTP 22/09-2020 68

DECOHERENCE CORRECTION

Overcoherence in the Ehrenfest method: the system moves on an average PES. Roughly: The classical part of the calculation should collapse the wave function so that the nuclei are moving on only one PES once we are outside of the crossing region. Otherwise the method is overcoherent. Overcoherence in Tully’s method: the electronic wave function continues incorrectly to have components on both surfaces after leaving the crossing region. Introducing decoherence corrections makes the two methods increasingly similar.

CECAM SISSA ICTP 22/09-2020 69

OUTLINE

• I. PHOTOCHEMICAL THINKING
• II. ELECTRONIC EXCITED-STATE PROCESSES
• III. TD-DFT(B)
• IV. STATIC APPLICATIONS
• A. Luminescence indices
• B. Retinal
• C. Excitonic effects
• V. TD-DFT(B) FSSH
• VI. DYNAMICS APPLICATIONS
• A. Easy case: oxirane
• B. Hard case: pentacene/buckyball
• VII. CONCLUSION

CECAM SISSA ICTP 22/09-2020 70

1 2 3 4 5 6 7

• E. Tapavicza, I. Tavernelli, U. Röthlisberger, C. Filippi, and M.E. Casida,

“Mixed time-dependent density-functional theory/classical surface hopping study

• f oxirane photochemistry”, J. Chem. Phys. 129, 124108 (2008).

Gomer-Noyes mechanism

CECAM SISSA ICTP 22/09-2020 71

OUTLINE

• I. PHOTOCHEMICAL THINKING
• II. ELECTRONIC EXCITED-STATE PROCESSES
• III. TD-DFT(B)
• IV. STATIC APPLICATIONS
• A. Luminescence indices
• B. Retinal
• C. Excitonic effects
• V. TD-DFT(B) FSSH
• VI. DYNAMICS APPLICATIONS
• A. Easy case: oxirane
• B. Hard case: pentacene/buckyball
• VII. CONCLUSION

CECAM SISSA ICTP 22/09-2020 72

JCP, submitted

CECAM SISSA ICTP 22/09-2020 73

ELA*'S TUPPERWARE ANALOGY * Gabriela Calinao Correa, NSF iREU student with us 17 May – 31 July 2016.

CECAM SISSA ICTP 22/09-2020 74

CECAM SISSA ICTP 22/09-2020 75

PENTACENE/C60 TANG CELL (and spherical cow ) As a first approximation, neglect nuclear motion and just think in terms of electrons.

[10] C. W. T ang, Appl. Phys. Lett. 48 , 183 (1986), T wolayer organic photovoltaic cell . [8] S. Yoo, B. Domercq, and B. Kippelen, Appl. Phys. Lett. 85 , 5427 (2004), Effjcient thin-fjlm

• rganic solar cells based on pentacene/C60 heterojunctions .

bathocuproine bathocuproine ITO = Indium Tin Oxide

CECAM SISSA ICTP 22/09-2020 76

OUR SYSTEM

CECAM SISSA ICTP 22/09-2020 77

OUR OBSERVABLES P F

qp

P

qp

F

qh

P

qh

F Charge transfer (CT) and energy transfer (ET) in terms

• f particles in the “conduction band” (i.e., empty orbitals)

and holes in the “valence band”) (i.e., filled orbitals):

CT=qh

P−qp P=q p F−qh P

ET=1−(q p

P+qh P)=(qh F+q p F)−1

This definition is not unique (e.g., Kasha’s exciton model.)

Now, run a swarm of about 100 trajectories and ensemble average!

CECAM SISSA ICTP 22/09-2020 78

THE IMPORTANCE OF THE DECOHERENCE CORRECTION

CECAM SISSA ICTP 22/09-2020 79

RESULTS AS A FUNCTION OF RANGE SEPARATION PARAMETER

CECAM SISSA ICTP 22/09-2020 80

Reasonable ET and CT are for Rlc > 10 a0 .

CECAM SISSA ICTP 22/09-2020 81

OUR SYSTEM

Rlc≈10a0

We should not add HF exchange for smaller distances in this model.

CECAM SISSA ICTP 22/09-2020 82

OUTLINE

• I. PHOTOCHEMICAL THINKING
• II. ELECTRONIC EXCITED-STATE PROCESSES
• III. TD-DFT(B)
• IV. STATIC APPLICATIONS
• A. Luminescence indices
• B. Retinal
• C. Excitonic effects
• V. TD-DFT(B) FSSH
• VI. DYNAMICS APPLICATIONS
• A. Easy case: oxirane
• B. Hard case: pentacene/buckyball
• VII. CONCLUSION

CECAM SISSA ICTP 22/09-2020 83

JUST A GLIMPSE … … OF THE TIP OF THE ICEBERG

CECAM SISSA ICTP 22/09-2020 84

JUST A GLIMPSE … … OF THE TIP OF THE ICEBERG

CECAM SISSA ICTP 22/09-2020 85

SPECIAL THANKS TO You !

and to the organizers !!

https://www.charmingitaly.com/different-types-of-italian-coffee/