Time-Dependent Density-Functional Theory (TD-DFT)...p4 Haibin Su ( 蘇海斌 ) NORDITA, 4 Oct. 2016...

NORDITA, 4 Oct. 2016 1

Mark E. Casida ( 樫田マーク , 卡西达马克 ) Laboratoire de Chimie Théorique (LCT)Département de Chimie Moléculaire (DCM)Institut de Chimie Moléculaire de Grenoble (ICMG)Université Grenoble Alpes (UGA)F-38041 GrenobleFrancee-mail: [email protected]

Time-Dependent Density-Functional Theory(TD-DFT)

NorditaStockholm, Sweden

4 October 20161 hour 45 min


VERI

TAS

LIBE

RANT

“The truth will set you free”(Liberating Truth)

Université de Grenoble

Established: 1339Recreated: 1809 Napolean appointed Joseph Fourier rector of the university.

Dissolved: during the reorganizationof 1970 to create Grenoble I, II, III,and the INPG.Currently: Université Grenoble-Alpes

“Home”


“A mountain at the end of every street.” –- Henri Beyl STENDHAL*

* Famous French author, originally from Grenoble.


CHEMISTRY: “Better things for better living through chemistry”Dupont Chemical Motto: 1935-1982

Publish(but don't

mention the10,000 failures)

An idea fora synthesis.Try it out.

Characterizethe product.

Did you get the right molecule?

Why not?New idea for the synthesis.

YesNo

* Physical chemistry and theoretical chemistry start here.

**

******

** *


EXPERIMENTAL PHOTOCHEMISTRY

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).]


LET'S MODEL IT!


p4

Haibin Su( 蘇海斌 )


HOPPING PROBABILITY

Helecr ; R t elec

r , t =i ℏd

dt

elecr ,t

elecr , t =∑m

melecr ; R t C

mt

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

Helecr ; R t

melecr ; R t =E

mR t

melecr ; R t

Probability of finding the system on surface m is

Pmt =∣C

mt ∣2

1st order equation.

Cmt =−i E

mt C

mt /ℏ−∑n

⟨m∣d n

dt⟩C

nt

IT IS A

LL GREEK TO M

E!


p4

Haibin Su( 蘇海斌 )


1 2 3 4

5 6 7

Enrico Tapavicza, Ivano Tavernelli and Ursula Röthlisberger, École Polytéchnique Fédérale de Lauzanne, 2007

TDPBE/TDA with Landau-Zener surface hopping.

Hot reactions (~4000K): cyclic-CH

2OCH

2 -> CH

2CH

2O -> CH

3 + CHO or CH

4 + CO


Goals :

Keep this presentation as simple as possible (Masters level).

But never be afraid to do something hard if it is also useful.

THIS IS A THEORY TALK

I want to give you an understanding of the mathematics and the physicsbehind the programs that do TD-DFT.


OUTLINE

I. Ab Initio Quantum Chemistry

Pause

II. (TD-)DFT


OUTLINE


A. Born-Oppenheimer ApproximationB. Hartree-Fock ApproximationC. Configuration InteractionD. Second QuantizationE. Equation-of-Motion/Superoperator Approach


OUTLINE




CONSIDER A MOLECULE ...

[Fe(H2O)

6]2+ [Fe(bpy)

3]2+[Fe(NH

3)]2+

M nuclei

R=( R1 , R2 ,⋯, RM )

N electrons

r=( r 1 , r 2 ,⋯, rM )

R=(XYZ )

x=( x1 , x2 ,⋯, xM )r=(

xyz ) x=(

xyzσ)

spin

Erwin SCHRÖDINGER

H ne(r ,R)Ψne(x ,R , t )=i ∂∂ tΨne(x ,R , t)

Hartree atomic units:

ℏ=me=e=1


BORN-OPPENHEIMER SEPARATION

H ne(r ,R)Ψne(x ,R , t )=i ∂∂ tΨne(x ,R , t)

H ne(r ,R)=T n(R)+V nn(R)+ H e(r ;R)

(1)

(2)

A separation of time scales approximation:1) The electrons “see” fixed nuclei.2) The nuclei “see” the time-averaged field of the electrons.

H e(r ;R)Ψ Ie (x ;R)=E I

e(R)Ψ Ie (x ;R)

Time-independent orbital Schrödinger equation:

(3)

Born-Oppenheimer expansion:

Ψne(x ,R , t)=∑IΨ I

e(r ;R)Ψ In(R , t ) (5)

(4)H e(r ;R)=T e (r )+V ne(r ;R)+V ee(r )


BORN-OPPENHEIMER GROUP EQUATION

−∑J {∑M

12mM

[∑K(δI , K ∇M+F I , K

(M ) (R))⋅(δK ,J ∇M+FK , J(M ) (R) ) ] }ΨJ

n(R , t )

+V I (R)Ψ In(R , t )=i ∂

∂ tΨ I

n(R , t ) (1)

(2)

(3)

Potential energy surface (PES)

V I (R)=E Ie(R)+V nn(R)

Derivative compling matrix (DCM)

F I , J (R)=⟨Ψ I

e(R)∣[ ∇M H e(R) ]∣Ψ I

e(R)⟩−δI , J ∇M E I

e(R)

EJe (R)−E I

e(R)

We may neglect DCM ifi. The force on the nuclei is sufficiently small.ii. The energy difference between different PESs is sufficiently big..


BORN-OPPENHEIMER APPROXIMATION

We neglect the DCM.

( T n+V I (R) )Ψ I , vn (R , t )=i ∂

∂ tΨ I , v

n (R , t ) (1)

(2)

(3)

(4)



e(R)Ψ Ie (x ;R)

H e(r ;R)=T e (r )+V ne(r ;R)+V ee(r )

Ψ I , vne (x ,R , t )=Ψ I

e (x ;R)Ψ I , vn (R , t) (5)


BUT PHOTOCHEMISTRY RELIES ON THE BREAK DOWN OF THE BORN-OPPENHEIMER APPROXIMATION!


Let us concentrate on the electronic problem!


e(R)Ψ Ie(x ;R)

H Ψ I (x)=E IΨ I (x)

(1)

(2)


OUTLINE




THE VARIATIONAL PRINCIPLE*

We want to solve

H I=E I I ; E0≤E1≤E2≤⋯ (1)

We will use the variational principle

E0≤W [Ψ]=⟨Ψ trial∣H∣Ψ trial⟩

⟨Ψ trial∣Ψ trial⟩(2)

Valid for every test function test

, satisfying the same boundary conditions as I.

* Chimie Quantique 101 ?


THE PAULI PRINCIPLE

Ψ(1,2,⋯, j ,⋯, i ,⋯, N )=−Ψ(1,2,⋯, i ,⋯, j ,⋯, N )

Electrons are fermions:

(1)

where i= xi (2)

The simplest function satisfying equation (1) is aSlater determinant:

=∣1 ,2 ,⋯, N∣=1

N !det [

11 21 ⋯ N 112 22 ⋯ N 2⋮ ⋮ ⋱ ⋮

1N 2N ⋯ N N ] (3)

⟨ψi∣ψ j⟩=δi , j (4)where


FOR A (FICTICIOUS) SYSTEM OF NONINTERACTING ELECTRONS

h0ψi=ϵi

0ψi ; ϵ1

0≤ϵ2

0≤ϵ3

0≤⋯ (1)

H 0=∑i=1,N

h0(i)

H 0Ψ0=E0Ψ0

(2)

Ψ0=∣ψ1 ,ψ2 ,⋯,ψN∣

(3)

E00=∑i=1,N

ϵi0

(4) (5)


ROGUES GALLERY

Douglas RaynerHARTREE

VladimirFOCKWolfgang

PAULIJohn C. SLATER

Charlotte FROESE FISCHER*(applied mathematician and computer scientistwho worked on multiconfigurational Hartree-Fock)

* Lest we forget the heroinesamong our heroes!


THE HARTREE-FOCK (HF) APPROXIMATION

The test function is a Slater determinant.

We minimize the variational energy subject to the condition that the orbitalsare orthonormal. The result is

f ψi=ϵiψi (1)

f=h+ v SCF (2)

The Fock operator:

Self-consistent field (SCF) :

v SCF=vH+Σx=J−K (3)

Physics notation Chemistry notation


THE SELF-CONSISTENT FIELD

Density matrix γ(1,2)=∑iniψi(1)ψi

*(2) (1)

Density ρ(1)=∑ini∣ψi(1)∣

2 (2)

The Hartree (or Coulomb) potential

vH (1)=∫ρ(2)r12

d 2 (3)

The exchange self-energy (or exchange operator)

Σxϕ(1)=−∫γ(1,2)r12

ϕ(2)d 2 (4)


HARTREE-FOCK (HF) ENERGY

E=∑ini ⟨ψi∣h∣ψi⟩+ESCF=∑i

ni ϵi−ESCF (1)

ESCF=EH+Ex (2)

EH=∫∫ρ(1)ρ(2)

r12

d 1d 2 (3) (4)Ex=−∫∫∣γ(1,2)∣2

r12

d 1d 2


KOOPMANS' THEOREM*

* T. Koopmans, “Über die Zuordnung von Wellenfunktionen und Eigenwerten zu den EinzelnenElecktronen Eines Atomes'', Physica 1, 104 (1934).

In the HF approximation, in the absence of orbital relaxation:

−IPi=EN−EN−1i−1=i

Ionization potential (IP)(1)

The occupied orbitals “see” N-1 electrons.

−EAa=EN1a1−EN=a

Electron affinity (EA)

The unoccupied orbitals “see” N electrons.

M+IP→M++e-(v=0)

(2)

M+e-(v=0)→M-+EA (3)

(4)

Tjalling CharlesKOOPMANS


f ψi=ϵiψi

Linear Combination of Atomic Orbitals (LCAO) Theory

Want to solve the molecular orbital (MO) equation,

(1)

Expand each MO in a basis of atomic orbitals (AOs),

ψi=∑νχνc ν , i (2)

Either (i) rigorously by finding the MO coefficients c,i by doing a variational minimization of the energy or (ii) heuristically by inserting Eq. (2) in Eq. (1) and projecting:

∑νf χνc ν ,i≈ϵi∑ν

χν cν ,i (3)

∑ν⟨χμ∣f∣χν⟩cν , i=ϵi∑ν

⟨χμ∣χν⟩cν , i (4a) N equations inN unknowns

f c i=ϵi s c i (4b)


Per-Olov Löwdin's Solution

Solve s v i=λi v i (1) with v i+ v j=δi , j (2)

Construct s−1/2=∑iv i+ λi

−1/2 v i (3) and~f=s−1/2 f s−1/2

(4)

~f d i=ϵi di

may be obtained by solving(5)Then c i=s−1/2 di (6)

because ( s−1/2 f s−1/2 )⏟~f

( s+1/2 c i )⏟d i

=ϵi ( s+1/2 c i )⏟d i

(7)

Upsala,Sweden

Gainsville,Florida, USA


OUTLINE




We may finally start talking about excited states!!


CONFIGURATION INTERACTION (CI)

All of the orthonormal orbitals, taken together, form a complete (infinite) basis set that canbe used to expand any one-electron wave function.

Similarly, all of the (infinitely many) N-electron Slater determinants, taken together, Constitute a complete basis set that may be used to expand any N-electron wave function.

This type of N-electron expansion is called “complete CI” and it is just a nice dream!

Ψ=C0Φ+∑i , aCaiΦi

a+∑i , j , a , bCabjiΦij

ab+⋯ (1)

An expansion in all of possible determinants made from a finite orbital basis is called “full CI”and is only possible for small basis sets (and hence for small molecules).

For normal sized orbital basis sets, we can only do “truncated CI”.


TRUNCATED CI

test = truncated CI

Using this trial function in a variation calculation is a linear variation and specialtheorems apply.

H C I=E I C I

H=[H 0,0 H 0,1+ ⋯

H 0,1 H1,1 ⋯⋮ ⋮ ⋱

] C I=(C0

C2

⋮)

(C1 )ai=CaiH 0,0=⟨Φ∣H∣Φ⟩=EHF

( H 0,1 )ai=⟨Φia∣H∣Φ⟩=f a ,i=0

(1)

(2)

(3a)(4a)

(4b)

(H1,1 )ai , bj=δi , jδa , bEHF+Aai , bj

(3b)

(3c)

(3d)

(Brillouin's theorem)


Only for linear variations (such as CI):

THE HYLLERAAS-UNDHEIM-MACDONALD or CAUCHY'S INTERLEAVING THEOREM

1 2 3 4 5 6 7 8 9 ...

The nth level is an upper bound to the true energyof the nth state.


CONFIGURATION INTERACTION SINGLES (CIS)

[H 0,0 H 0,1+

H 0,1 H1,1] (C0

C1)=E (C0

C1) (1)

[EHF 0+

0 A+EHF] (C0

C1)=E (C0

C1) (2)

A C1=(E−EHF ) C1=ω C1 (3)

How do we solve such a big eigenvalue equation?


Exact up to numerical convergence. Not an approximation!


CONICAL (or diabolic) INTERSECTIONS


If we only count internal degrees of freedom, then the PES is

F=3N−6 *

* F=3N−5 for a linear molecule.

dimensional in a 3N-5 dimensional space (3N-4 for a linear molecule).

What dimension is the intersection space of two PESs?

Conditions restricting the dimensionality:

E Ie(R)=EJ

e (R)i) ⇒ dimension F=3N−7

ii) H I , Je (R)=0 ⇒ dimension F=3N−8

(3N-6)

(3N-7)**

** Exception: This is not a restrictionwhen it is automatically guaranteedeither for symmetry reasons or becauseof Brillouin's theorem!

This is a diabolo.


PHOTOCHEMICAL FUNNELS

Avoided crossing Conical intersection

Diatomics never have conicalintersections. However avoided crossings are possible betweenPESs corresponding to electronicstates with the same symmetry.


OUTLINE




(H1,1 )ai , bj=δi , jδa , bEHF+Aai , bj

HOW TO FIND A FORMULA FOR THE A MATRIX?

It is “easy” with Second Quantization.

Aai , bj=δi , jδa ,b (ϵa−ϵi )+[ai∣ jb ]−[ab∣ ji]

(1)

(2)

Answer:


His car from the 1970s.


For us, this is just a way to make it easier to do thealgebra needed when working with antisymmetricwave functions.

Only for Shaolin masters!?

(I don't think so.)


FUNDAMENTAL IDEAS

The physical vacuum Φ=∣ψ1 ,ψ2,⋯,ψN∣ (1)

Creation operator ar+Φ=∣ψr ,ψ1 ,ψ2,⋯,ψN∣ (2)

Annihilation operator

ar ar+Φ=ar∣ψr ,ψ1 ,ψ2,⋯,ψN∣=∣ψ1 ,ψ2,⋯,ψN∣=Φ (3)

Anticommutation rules:

[r , s ]+=rs+sr=0 (6) [r+ , s+ ]+=r+ s++s+ r+

=0 (7)

[r , s+ ]=r s++s+ r=δr , s (8)

Lazy notation:

r=ar r+=ar+

(4) (5)


SECOND-QUANTIZED OPERATORS

H=∑r , shr , sr

+ s+12∑p ,q , r , s

[ pr∣qs] p+q+ s r (1)

hr , s=⟨ψr∣h∣ψs ⟩

[ pq∣r s ]=∫∫ψp*(1)ψq(1)

1r12

ψr*(2)ψs(2)d 1d 2

(2)

Mullikan “charge cloud” notation:

(3)


WICK'S THEOREM (I)

ab c⋯gh⏟unoccupied

i j k lmn⏟occupied

o pq⋯ y z⏟free

“FORTRAN index” notation

Normal order: A product of creation and annihilation operators is in normal order when(1) For occupied orbitals, all the creation operators are to the right of all of the annihilation

operators ( ijk … m+ n+ …)(2) For unoccupied orbitals, all the annihilation operators are to the right of all the creation

operators (a+ b+ c+ … d e ...)

Nonzero contractions:

i+ j=δi , j ab+=δa ,b

or

r+ s=nsδr , s

(occupied) (unoccupied)

s r+=(1−nr)δr , s=nrδr , s


WICK'S THEOREM (II)

Wick's Theorem (simplified) :The expectation value with respect to the physical vacuum is just the sum of all possiblefully-contracted products multiplied by +1 (-1) if the number of intersections of thecontraction lines is even (odd).

Example :


HARTREE-FOCK (HF)

EHF=∑ihi , i+

12∑i , j

([ii∣jj ]−[ ij∣ ji ] ) (1)

f=∑r , sf r , s r

+ s (2) f r , s=hr , s+∑i([rs∣ii ]−[ri∣is ]) (3)

If the orbital basis consists of HF orbitals, then

f r , s=ϵrδr , s f=∑rϵr r

+r(4) (5)


EVALUATION OF Hia,jb

One-electron part

hai , bj=⟨Φia∣h∣Φ j

b⟩=∑p ,q

h p, q ⟨Φ∣i+ a p+ q b+ j∣Φ⟩ (1)

=∑p ,qhp , q (i

+ a p+ q b+ j+i+ a p+ q b+ j+i+ a p+ q b+ j ) (2)

=∑p , qhp ,q (δi , jδa , pδb ,q−δa ,bδi ,q δ j , p+δi , jδa ,bδp , qnp ) (3)

=δi , jha, b−δa , bh j , i+δi , jδa , b∑khk , k (4)

=δi , jha ,b−δa ,bh j , i+δi , jδa ,b ⟨Φ∣h∣Φ⟩ (5)



Two-electron part

v ai , bj=⟨Φia∣vee∣Φ j

b⟩=∑p ,q , r , s[ pr∣qs ]⟨Φ∣i+ a p+ q+ s r b+ j∣Φ⟩ (1)

=∑p ,q ,r , s[ pr∣qs] i+ a p+ q+ s r b+ j+∑p ,q , r , s

[ pr∣qs ] i+ a p+ q+ s r b+ j(Term A) (Term B)

(2)



Two-electron part (continued)

+∑p , q , r , s[ pr∣qs] i+ a p+ q+ s r b+ j+∑p ,q , r , s

[ pr∣qs ]i+ a p+ q+ s r b+ j(Term C) (Term D)

+∑p ,q , r , s[ pr∣qs] i+ a p+ q+ s r b+ j+∑p ,q , r , s

[ pr∣qs ]i+ a p+ q+ s r b+ j(Term E) (Term F)

(1)

(2)





[ pr∣qs ]i+ a p+ q+ s r b+ j(Term G) (Term H)


[ pr∣qs ]i+ a p+ q+ s r b+ j(Term I) (Term J)

(1)

(2)





[ pr∣qs ]i+ a p+ q+ s r b+ j(Term K) (Term L)


[ pr∣qs ]i+ a p+ q+ s r b+ j(Term M) (Term N)

(1)

(2)



Two-electron part

v ai , bj=⟨Φia∣vee∣Φ j

b⟩=∑p ,q , r , s[ pr∣qs ]⟨Φ∣i+ a p+ q+ s r b+ j∣Φ⟩ (1)

=δi , jδa ,b∑k ,l[kk∣l l ]−δi , jδa, b [kl∣lk ]

(Term A) (Term B)

which is simply δi , jδa ,b ⟨Φ∣vee∣Φ⟩

(2)




+δi , j∑k[ab∣k k ]−δi , j∑k

[ak∣k b ]

(Terms C+E) (Terms D+F)

which is simply +δi , j va, bSCF

−δa ,b∑k[ ji∣k k ]+δi , j∑k

[ j k∣k i ]

(Terms H+I) (Terms G+J)

which is simply −δa ,b v j ,iSCF

(1)

(2)




+[a i∣ j b ]−[ab∣ j i ]

(Terms L+M) (Terms K+N)

Putting it all together gives:

H ai , bj=Aai ,bj+δi , jδa ,bEHF

where

Aai , bj=δi , j f a ,b−δa ,b f j ,i+[ai∣ jb ]−[ab∣ji ]

orAai , bj=δi , jδa , b (ϵa−ϵi )+[ai∣ jb ]−[ab∣ji]

in terms of HF MOs.

(1)

(2)

(3)

(4)


TWOORBITAL TWOELECTRON MODEL (TOTEM)*

ia

ia

ia

ia

ia

a i

a i

a i

a i ∣ii∣

∣ai∣

∣i a∣

∣i a∣

∣ai∣

Singlet

Triplets 1,0=∣i a∣

S , M

S

1,−1=∣ai∣

0,0=

1

2∣ai∣∣i a∣

1,0=

1

2∣ai∣−∣ia∣

* two-orbital two-electronmodel.


TOTEM CIS

[ϵa−ϵi+[ai∣ia ]−[aa∣ii ] [ai∣ia ][ai∣ia ] ϵa−ϵi+[ai∣ia ]−[aa∣ii ] ] (

X aiα

X aiβ)=ω (X aiα

X aiβ)

(X aiα

X aiβ)=(11 ) ⇒ ω=ϵa−ϵi+2 [ai∣ia ]−[aa∣ii]

(X aiα

X aiβ)=( 1−1 ) ⇒ ω=ϵa−ϵi−[aa∣ii]

Singlet type excitation

Triplet type excitation


A consequence of Koopmans' theorem is that the TOTEM formulae are almost never acccurate enough to be useful.


HF DFT


OUTLINE




The idea here is to present a method which is relatively “easy”and “elegant” to understand but which gives us the same equations as response theory.


EQUATION-OF-MOTION (EOM)

The EOMH O= [ H , O ]= O

has excitation-type solutions

[ H ,∣I0∣ ]=E I−E0∣I0∣

and de-excitation-type solutions

[ H ,∣0I∣ ]=E0−E I ∣0I∣

(1)

(2)

(3)

(it also has other solutions, but these are all we care about here!)

Let us seek a solution of the form

O+=∑aia+ i X ai+∑ai

i+aY ai (4)


SUPEROPERATOR METRIC AND MATRIX EQUATION

H O= [ H , O ]= O (1)

O+=∑aia+ i X ai+∑ai

i+aY ai (2)

Inserting (2) into (1) and using the “metric”

( A∣B )=⟨Φ∣[ A+ , B ]∣Φ⟩ (3)

gives us

[ A BB∗ A∗ ] XY = [1 0

0 −1 ] XY (4)

with Aai , bj=δi , jδa ,b(ϵa−ϵi)+[ai∣ jb]−[ab∣ ji ]

Bai , bj=[ib∣ ja]−[ jb∣ia]

(5)

(6)


PAIRED SOLUTIONS

[ A BB∗ A∗ ] XY = [1 0

0 −1 ] XY (1)

[ A BB∗ A∗ ] Y

∗

X∗ =− [1 00 −1 ] Y

∗

X∗ (2)

excitation

corresponding de-excitation


THE TAMM-DANCOFF APPROXIMATION

[ A B=0B∗=0 A∗ ] XY = [1 0

0 −1 ] XY (1)

A X= X (2)

This is the same as CIS !!


TOTEM

singlet

triplet

T= [a−iia∣f H∣ai−ii∣f H∣aa ] [a−i−ia∣f H∣ai−ii∣f H∣aa ]

S= [a−iia∣f H∣ai−ii∣f H∣aa ] [a−i3ia∣f H∣ai−ii∣f H∣aa ]

T will be imaginary when

ia∣f H∣aiii∣f H∣aaa−i

Triplet-type instability!

(1a)

(2a)

(3)

≈a−i−ii∣f H∣aa (1b)

≈a−i2ia∣f H∣ai−ii∣f H∣aa (2b)


HF STABILITY ANALYSIS*

Given a spin-restricted solution (same orbitals for different spin), is it possible that there is alower energy solution with different orbitals for different spin?

rr =ei Ri I

rr

Look at an arbitrary unitary transformation of the orbitals:

E=E02 [ R A−B RI AB I ]O 3

We find

But another way to write the EOM matrix equation is

AB A−B ZI=

I2 Z

IConclusion : Symmetry breaking will occur if there are imaginary excitation energies.

* J. Cizek and J. Paldus, J. Chem. Phys. 47, 3976 (1967).

H H H H (1)

(2)

(3)

(4)


IME OUT

LET'S TAKE ABREAK


OUTLINE

II. (TD-)DFT

A. DFT i. Formalism ii. Fonctionals B. TD-DFT iii. Formalism iv. Response theory v. The “Casida equation” vi. Absorption spectra vii. Closing remarks C. Some Problems


OUTLINE

II. (TD-)DFT



Walter KOHN (9 March 1923 - 19 April 2016)

● Father of modern DFT● Theoretical physicist and a great friend to chemists● 1998 Nobel prize in Chemistry


OUTLINE

II. (TD-)DFT



1st HK Theorem: The external potential is determined up to an additive Constant by the groundstate charge density.

Corollary:

N , vextC HC

I,

I=E

I−E

0

2nd HK Theorem: The groundstate energy and charge density may be obtained by minimizing the variational expression

E=F []∫ vextr r d r

The functional F[] is "universel" in the sense of being independent of vext

.

The Two HohenbergKohn (HK) Theorems[Phys. Rev. 136, B864 (1964)]

(1)

(2)


The LevyLieb Constrained Search Formalism

Avoids the vrepresentablity problem.It assumes that the denisty is Nrepresentable (and it really is!)The exact functional is actually known! (but it is not practical!)

F []=min⟨∣TV ee∣ ⟩

⟨∣ ⟩

Ensemble formulation:

F []=minD tr [ TV ee D ]

D=∑IpI∣ I I∣

pI is the probability of observing the state I.

(1)

(2)

(3)


Introducing N orthonormal KohnSham orbitals allows us to give an exactdescription of an important part of the total energy.

E=∑ini

i∣−

1

2∇ 2v

ext∣

i

1

2∫∫

r1r

2

∣r 1−r 2

∣d r 1

d r 2E

xc[]

wherer =∑i

ni∣

ir ∣2

Variational minimization subject to the orbital orthonormality contraint gives the KohnSham equation.

[−1

2∇

2v

extr ∫

r '

∣r−r '∣d r 'v

xcr ] i

r = i ir

where the exchangecorrelation (xc) potential is vxc[]r =

Exc[]

r

The KohnSham Formulation[Phys. Rev. 140, A1133 (1965)]

(1)

(2)

(3)

(4)


We need ...v

xc[]r =

Exc[]

r

Definition E xc[]−E xc []=∫

E xc []

r r d r

Functional Derivatives

f xc[]r 1 ,r 2=

2 E xc[]

r 1 r 2

gxc []r 1 ,r 2 ,r 3=

3 E xc []

r 1 r 2 r 3

and sometimes also

(1)

(2)

(3)

(4)


« Le but n'est pas toujours placé pour être atteint, mais pour servirde point de mire ou de direction. »* --- Joseph Joubert (p. 221 of Receuil des pensées de M. Joubert by F.-R. De Chateaubriand, 1838)

* The target is not always meant to be hit, but rather to show where we should aim.

?


OUTLINE

II. (TD-)DFT



● Up to this point, we have only discussed pure DFT (depends only on .)● Today pure DFT is pure spin-DFT (depends on

and on

.)

● Today we really see pure DFT (pure Kohn-Sham). Most people use some type of generalized Kohn-Sham (GKS) that includes an orbital dependence.

Taking Stock


“Jacob's Ladder”William Blake

watercolor1799-1800


Size of the molecule

ab initio DFT

hybrid/OEP

mGGA*

GGA

LDA

“No computer land.”

HARTREE WORLD

THEORETICALCHEMISTRYHEAVEN**

Jacob's Ladder

r

x r =∣∇ r ∣

r 4/3

r =∑in

i∣∇

i∣

2

KxHF

MBPT

* or include ∇2

pure DFT

** 1 kcal/mol or better precision


RangeSeparated Hybrids (RSH*)

1r12

=erfc (γr12)

r12⏟SHORT RANGE

+erf (γ r12)

r12⏟LONGRANGE

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

Solids: SR <-> WF LR <-> DFT

TD-DFT applications: 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).

The favorite type of GKS is lessAnd less B3LYP and more andMore a RSH.

* because Nature is not always short-sighted.

The RSH idea came originally from Andreas Savin (Université Pierre et Marie Curie, Paris.)


SEMIEMPIRICAL DISPERSION CORRECTION*

* S. Grimme, J. Antony, S. Ehrlich, and H. Krieg, J. Chem. Phys. 132, 154104 (2010).“A consistent and accurate ab initio parameterization of density functional dispersioncorrection (DFT-D) for the 94 elements H-Pu”

Semi-empirical (molecular-modeling-type) formula designed to interpolate between twophysical limits: ''the asymptotic part that is described very accurately by [TD-DFT] (for atomic or molecular fragments) and the short-range regime for which standard [DFAs] often yield a rather accurate description of the exchange-correlation problem''.

Edisp=∑A ,B∑n=6,8,10,. ..sn

CnA , B

RA ,Bn [1+6 (

RAB

sr ,nR0A , B )

−αn

]−∑A ,B ,C

√C6ABC6

ACC6BC(3cosθacosθbcosθc+1)

(RABRBC RCA )3 [1+6 (

RABC

sr ,3R0A ,B )

−α3

]


OUTLINE

II. (TD-)DFT



Given a system, initially in its ground state, exposed to a timedependentperturbation:

1st Theorem (RG1): vext

(rt) is determined by (rt) up to an arbitrary additive function of time.

Corollary: r t N , vext r t C t H t C t t e−i∫t0

tC t ' dt '

2nd Theorem (RG2): The timedependent chargedensity is a stationary pointof the FrenkelDirac action

A[]=∫t0

t t ' ∣i ∂

∂ t '− H t ' ∣ t ' dt '

TIMEDEPENDENT DENSITYFUNCTIONAL THEORY (TDDFT)[E. Runge and E.K.U. Gross, Phys. Rev. Lett. 52, 997 (1984)]

(1)

(2)

[RG2 suffers from a “causality paradox”. But this difficulty now seems to have been solved by G. Vignale, Phys. Rev. A 77, 062511 (2008). “Realtime solution of the causality paradox of timedependent densityfunctional theory”]

(RG1 has only been proven for functions whose time dependence can be expanded in a Taylor series.)


TIME-DEPENDENT KOHN-SHAM EQUATION

[−12∇

2vext r t ∫

r ' t ∣r−r '∣

d r 'vxc r t ] i r t =i ∂∂ t i r t

r t =∑ini∣ i r t ∣2where

and vxc r t = Axc []

r t

(1)

(2)

(3)

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

Remark: If the initial state is not the ground state, then

v xc r , t =v xc [ , 0,0]r , t (4)


TD-DFT ADIABATIC APPROXIMATION (AA)

t r =r t

This defines “conventional TD-DFT”.

v xc r t = A xc []

r t v xc r t =

Exc [t ]

t r

Assume that the exchange-correlation (xc) potential responds instantaneously and withoutmemory to any change in the time-dependent potential.


OUTLINE

II. (TD-)DFT



ELECTRONIC POLARIZATION INDUCED BY THE APPLICATION OFA TIME-DEPENDENT ELECTRIC FIELD

Classical model of a photon

Induced dipole moment

t =−e 0∣r∣

0t

0 t ∣r∣

0

t = cos0t

v r t =e t ⋅r

HH

O

HH Hℏ

0

photon

t

(1)

(2)

(3)


The linear response for a property a at

to the perturbation b b t = b cos0t

HI=E

I

I

[ Hb t ]0t =i ℏ ∂

∂ t

0t

0 t =[

0

0t ]e

−iE0t / ℏ

which allows us to deduce the 1st order equation

b t 0=i ℏ ∂

∂ t− HE

0

0t

REPONSE THEORY FOR CHEMISTS …

(1)

(2)

(3)

(4)

(5)


ENERGY REPRESENTATION

Fourier transforms

f =∫−∞∞

ei t f t dt

f t =1

2∫−∞

∞

e−i t f d

b t 0=i ℏ ∂

∂ t− HE

0

0t

Transform the 1st order equation

b 0=ℏ− HE

0

0

Solve by Rayleigh-Schrödinger perturbation theory!

0=∑I≠0

I

I∣b ∣

0

−I

ℏI=E

I−E

0where ℏ=1

Joseph FourierPrefect of the Isère

(Grenoble)1802-1815

(1)

(2)

(3)

(4)

(5) (6)


TIME REPRESENTATION

∫−∞∞

ei t cos0t dt=[

0−

0]

Since

∫−∞∞

ei t sin0t dt=

i[

0−

0]

we find

0t =∑I≠0

I

I

I∣b∣

0

0

2−

I

2cos

0t

−i ∑I≠0

I

0

I∣b∣

0

0

2−

I

2sin

0t

(1)

(2)

(3)


RESPONSE OF AN ARBITRARY PROPERTY

at =0∣a∣

0t

0t ∣a∣

0

at =∑I≠0

2Iℜe

0∣a∣

I

I∣b∣

0

0

2−

I

2cos

0t

∑I≠02

0ℑm

0∣a∣

I

I∣b∣

0

I

2−

0

2sin

0t

Electronic polarizabilities, NMR chemical shifts

Circular dichroïsme

(1)

(2)


THE DYNAMIC POLARIZABILITY

it =

i∑ j

i , j

jcos t⋯

i , j =∑I≠0

2I 0∣r i∣ I I∣r j∣ 0

I2−

2

=∑I≠0

fI

I

2−

2

fI=

2

3

I∣

0∣x∣

I∣

2∣

0∣y∣

I∣

2∣

0∣z∣

I∣

2

Sum-over-states (SOS*)f

I

ωI

(1)

(2)

(3)

(4)


OUTLINE

II. (TD-)DFT



FREQUENCY/ENERGY FORMULATION

[ AI BI

BI AI ] X I

Y I=I [1 0

0 −1 ] X I

Y I

Mark E. Casida in Recent Advances in Density Functional Methods, Part I, edited by D.P. Chong (Singapore, World Scientific, 1995), p. 155."Timedependent densityfunctional response theory for molecules'' *

Aij ,kl = ,i ,k j , l i− jK ij , kl

K ij , kl=∫∫ ∗i r j r f Hxc ,r ,r ' ; k r ' ∗l r ' d r d r '

Bij ,kl =K ij , lk

“RPA” equation

(1)

or (2)

(3)

Coupling matrix

(4)

Remark: The general formulation does not make the adiabatic approximation.

* 1246 citations according to Google Scholar Citations, 1 Oct. 2016.


OUTLINE

II. (TD-)DFT



From Mark E. Casida, Bhaarathi Natarajan, and Thierry Deutsch, http://arxiv.org/abs/1102.1849.in Fundamentals of Time-Dependent Density-Functional Theory, edited by Miquel Marques, Neepa Maitra, Fernando Noguiera, E.K.U. Gross, and Angel Rubio, Lecture Notes in Physics, Vol. 837 (Springer Verlag: 2011), p. 279. "Non-Born-Oppenheimer dynamics and conical intersections"


BEER'S LAW*

I 0 I

cuvette

Absorbance (unitless)

A=C l

incident intensity transmitted intensity

A=log10

I 0

I

Beer's law

lwidth (cm)

Concentration (mol/L) C

Molar extinction coefficiient (L/mol.cm)

(1)

(2)

* August Beer (1825-1863). The term Beer's law appears to date from 1889.


=

[Co(bpy)3]2+

From A. Vargas et al., J. Chem. Theory Comput. 2, 1342 (2006)

cobalt(II)tris(2,2’-biypiridine) 2E t2g6 eg

1

t2g

eg

experiment

theory


STICK SPECTRUM

E1=h1

E2=h2

● Vertical transition (ground-state equilibrium geometry)

● Neglect vibrations (and rotations) ● Neglect other contributions to the

width of the peaks (e.g., solvent, spectrometer artifacts)

● Average over rotations● Dipole-dipole approximation

Spectral function (units: cm)

Oscillator strength (unitless)

S =∑If I − I

f I=2Ime

3ℏ∣⟨ 0∣r∣ I ⟩∣

2

=2

Angular frequency (rad/s)

h=ℏ

(1)

(2)

(3) (4)


EXPERIMENTAL OSCILLATOR STRENGTHS

We assume that the photon absorption process is sudden and use Fermi's (2nd) GoldenRule to obtain (in Gaussian units):

f I=me c ln 10

N A e2 ∫ d

You must integrate over all of the spectra coming from the electronic transition!

f I=me c 40 ln 10

N A e2 ∫ d

In SI units, this becomes

f I=(1.44e-19 mol.cm.s/L)∫ ϵ(ν)d ν

which makes

(1)

(2)

(3)

ICBST*

* It Can Be Shown That


THEORETICAL CALCULATION OF THE MOLAR EXTINCTION COEFFICIENT

f I=me c 40 ln 10

N A e2 ∫ d

For a single transition,

= N A e

2

mec 40 ln 10f I − I

For several transitions,

= N A e

2

mec 40 ln 10S

S =∑If I − I

(1)

(2)

(3)

(4)


GAUSSIAN CONVOLUTION

Normalized gaussian function

g = e−2

Full width at half maximum (FWHM) FWHM=2 ln 2

= N A e

2

mec 40 ln 10

×∑If I g− I

experiment

theoryOne program:

spectrum.py By Pablo Baudin and M.E.C.

(1)

(2)

(3)

Normally experiment and properlynormalized theoretical spectra haveintensities which agree to within an order of magnitude: ● Imperfect spectrometers● Neglect of vibrational coupling● Neglect of solvent● ???


CALCULATION OF THE ABSORPTION SPECTRUM OF [Ru(trpy)3]2+

Example prepared by Denis MAGERO with Gaussian09.

Optimisation beginning with theX-ray structure.

B3LYP/6-31G(d)+LANL2DZ ECP on Ru


HOW GOOD IS THE GEOMETRY?

Gas phase B3LYP calculations give a geometry close to the one measuredin the crystal.

D. Magero


Exp.

CIS

TD-HFTD-B3LYP

Only the TD-DFT spectrumis close enough to the experiment to allow anassignment.

D. Magero

CALCULATION OF THE ABSORPTION SPECTRUM OF [Ru(trpy)3]2+


OUTLINE

II. (TD-)DFT



RECENT BOOKS ON TD-DFT

Book: Time-Dependent Density Functional Theory, Edited by M.A.L. Marques, C.A. Ullrich, F. Nogueira, A. Rubio, K. Burke, and E.K.U. Gross, Lecture Notes in Physics vol. 706 (Springer: Berlin, 2006).

Book: Fundamentals of Time-Dependent Density-Functional Theory, edited by M. Marques, N. Maitra, F. Noguiera, E.K.U. Gross, and A. Rubio, Lecture Notes in Physics, Vol. 837 (Springer: Berlin, 2011)

Book: Time-Dependent Density Functional Theory, C.A. Ullrich (Oxford: 2012)


OUTLINE

II. (TD-)DFT



Joseph Kosuth1987


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


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

SOME BIOCHEMISTRY

ElizabethTaylor

11Z11Z


CLASSIC SINGLET MECHANISM


RECENT AB INITIO CALCULATIONS AND EXPERIMENT SAYS THAT THE PHOTOREACTION IS A BIT MORE COMPLICATED.

[GML+13] S. Gozem, F. Melaccio, R. Lindh, A.I. Krylov, A.A. Granovsky, C. Angeli, and M. Olivucci, J. Chem. Theory Comput. 9, 4495 (2013). “Mapping the excited state potential energy surface of a retinal chromophore model with multireference and equation-of-motioncoupled cluster methods”[SKS11] R. Send, V.R.I. Kaila and D. Sundholm, J. Chem. Theory Comput. 7, 2473 (2011). “Benchmarking the approximate second-order coupled cluster method on biochromophores”[VF10] O. Valsson and C. Filippi, J. Chem. Theory Comput. 6, 1275 (2010). “Photoisomerization of model retinal chromophores: Insight with Quantum Monte Carlo and multiconfiguration perturbation theory”[ZHC09] G. Zgrablić, S. Haacke, and M. Chergui, J. Phys. B 113, 4384 (2009). “Heterogeneityand relaxation dynamics of the photoexcited retinal Schiff base cation in solution”Warning: trans → cis


RETINAL PROTONATED SCHIFF BASE (PSB)

56

7

8

9

10

1113

1214

15

hν11Z → 11E

12

3

416

11 12 11 12

11 12 11 12


Bond Length Alternation (BLA)

BLA = < R(C-C) > - < R(C=C) >where C-C and C=C refer to the ground state.


Classical mechanismonly accessed aftersurmounting anexcited-state activation barrier.

Dynamic correlation leads to a floppy excited state with more equal bond lengths and rotations around “double bonds”.

d

dd

d

d

dd

dd

d

dd

d

d

dd

d = double bond in ground state


d

dd

d

d

dd

dd

d

dd

d

d

dd

d = double bond in ground state

θ=93o

γ=0o

θ=−10o

γ=43.6o

CASPT2

CASSCF

θ=8.1o

γ=112.7o

CASPT2

B3LYP---

CAM-B3LYP---

θ=−27.4o

γ=17.7o

LC-BLYP

B3LYP CAM-B3LYP

θ=−0.2o

γ=86.8o

LC-BLYP

θ=−2.4o

γ=85.4oθ=−1.2o

γ=85.2o


-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”

Λ=∑i , a

κia2 Oia

∑i , aκia

2(1)

κia=X ia+ Y ia (2)

Oia=∫∣ψi( r )∣∣ψa( r )∣d r (3)

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


Scan aroundsingle bond


(TD-)DFT is advancing faster than many users can keep up!

The first events in photobiological systems appear to be within reach. Techniques are now being formulated which promise to take us even

further.


That's about it … … except for a few wise words.


Special thanks to Lars G.M. Pettersson, Patrick Norman,and to everyone else

TACK SÅ MYCKET

TACK SÅ MYCKET

Time-Dependent Density-Functional Theory (TD-DFT)...p4 Haibin Su ( 蘇海斌 ) NORDITA, 4 Oct. 2016...

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