Ultrafast processes in molecules

Ultrafast processes in molecules

Mario [email protected]

III – Adiabatic approximation and non-adiabatic corrections

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2

Diabatic x adiabatic

From Greek diabatos: to be crossed or passed, fordable

diabatic = with crossing

a-diabatic = without crossing

non-a-diabatic = with crossing!?

3


In thermodynamics

without exchanging (cross) heat or energy with environment

4


In quantum mechanics

“A physical system remains in its instantaneous eigenstate if a given perturbation is acting on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian's spectrum.” Adiabatic theorem (Born and Fock, 1928).

E

x

In this example (adiabatic process), the spring constant k of a harmonic oscillator is slowly (adiabatically) changed. The system remains in the ground state, which is adjusted also smoothly to the new potential shape. Its state is always an eingenstate of the Hamiltonian at each time (“no crossing”).

k

5


In quantum mechanics

“A physical system remains in its instantaneous eigenstate if a given perturbation is acting on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian's spectrum.” Adiabatic theorem (Born and Fock, 1928).

In this example (diabatic process), the spring constant k of a harmonic oscillator is suddenly (diabatically) changed. The system remains in the original state, which is not a eingenstate of the new Hamiltonian. It is a superposition (“crossing”) of several eingenstates of the new Hamiltonian.

E

x

k

6


In quantum chemistry

“The nuclear vibration in a molecule is a slowly acting perturbation to the electronic Hamiltonian. Therefore, the electronic system remains in its instantaneous eigenstate if there is a gap between the eigenvalue and the rest of the Hamiltonian's spectrum.”

This is another way to say that:

The electrons see the nuclei instantaneously frozen

7

Beyond adiabatic approximation I: Time independent

Time-independent formulation

Since i is a complete basis, any function in the Hilbert space can be exactly written as a linear combination of i.

0 UH

eN HT HTN – Kinetic energy nucleiHe – potential energy terms

Rr;ii depends on the electronic coordinates r and parametrically on the nuclear coordinates R.

i which solves: 0 iie EH (adiabatic basis)

ikki

8

sN

kkk

1

;, RrRRr kk nuclear wave function

Multiply by i at left and integrate in the electronic coordinates

sN

kkkniii TUE

1

0

22

1

22

22 M

N

I I

IN

at

MT

BABABAAB 222 2

Non-adiabatic coupling terms

sN

kkNikkMikMiiN TETU

1

2 0

Prove it!

eN HT H 0 UH

9


sN

kkNikkMikMiiN TETU

1

2 0

If non-adiabatic coupling terms = 0

0 iiN ETU

Nuclear vibrational problem.

If Ei is expanded to the second order around the equilibrium position:

Nat

k

Nat

llk

eqlk

ieqii qq

qqEEE

3

1

3

121R eqkkkk xxMq ,

2/1

it can be treated by normal mode analysis.

10

Beyond adiabatic approximation II: Time dependent (adiabatic representation)

Time-dependent formulation

eN HT HTN – Kinetic energy nucleiHe – potential energy terms

i which solves: 0 ie EH

Rr;ii depends on the electronic coordinates r and parametrically on the nuclear coordinates R.

Since i is a complete basis, any function in the Hilbert space can be exactly written as a linear combination of i.

0),,(

tt

i RrH

(adiabatic basis)

11

sN

kkk

1

;, RrRRr kk nuclear wave function

Prove it!

Multiply by i at left and integrate in the electronic coordinates

Time dependent Schrödinger equation for the nuclei

sN

kkkNikiiiN T

tiET

ti

1

0

eN HT H

0),,(

tt

i RrH

12


sN

kkkNikiiiN T

tiET

ti

1

0

First suppose the couplings are null (adiabatic approximation):

0

iiN ETt

i

Independent equations for each surface.

E0

time0(t)

0(t0)

0(t)

13

Classical limit of the nuclear motion

),(exp),(),( tSitAti RRR

Write nuclear wave function in polar form

tLdttS

0'),(R The phase (action) is the integral

of the Lagrangian

0

iiNi ETt

i Adiabatic approximation

I IIi

I AA

ME

MS

tS 22

22

02

2

Ii

I

EMS

tS Classical limit 0

Tully, Faraday Discuss. 110, 407 (1998)

14

02

2

Ii

I

EMS

tS

Hamilton-Jacobi Equation

To solve the Hamilton-Jacobi equation for the action is totally equivalente to solve the Newton`s equations for the coordinates!

dtdME I

IiR2

Newton equations

In the classical limit, the solutions of the time dependent Schrödinger equation for the nuclei in the adiabatic approximation

are equivalent to the solutions of the Newton`s equations.

In which cases does this classical limit lose validity?

15

I IIi

I AA

ME

MS

tS 22

22

adiabatic quantum terms ≠ 01

non-adiabatic coupling terms ≠ 0

sN

kkkNikiiiN T

tiET

ti

1

0

2

In which cases does this classical limit lose validity?

16


sN

kkkNikiiiN T

tiET

ti

1

0

x1 (a0)

x 2 (a

0)

E0

E1

1(t)

0(t)

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Beyond adiabatic approximation III: Time dependent (general representation)

0),,(

tt

i RrH

i which solves: 0 ie EH (adiabatic basis)

Before:

We got:

sN

kkkNikiiiN T

tiET

ti

1

0

i general orthonormal complete basis

Now:

We get:

sN

kkkeikNikiiN HT

tiT

ti

1

0

another non-adiabatic coupling term

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A very important result:

N

II

IN MT

1

22

21

Nuclear kinetic energy operator

0NT

sN

kkkeiki

i Ht

it

i1

0

This equation is the basis for the mixed quantum-classical methods such as Surface Hopping and Mean Field

(Ehrenfest) dynamics.

sN

kkkeikNikiiN HT

tiT

ti

1

0

0NT

19

For example, Surface Hopping

0 iie EH For a fixed nuclear geometry, solve time-independent Schrödinger Eq. for electrons. Get the energy gradient.

1

dtdME I

IiR2

Use the energy gradient to update the nuclear geometry according to the Newton`s Eq.

2

sN

kkkeiki

i Ht

it

i1

0

For the new nuclear geometry (only!), solve the TDSE and correct classical solution by performing a hopping if necessary.

3

Go back to step 1 and repeat the procedure until the end of the trajectory.

4

Repeat procedure for a large number of trajectories to have the nuclear wave packet information.

5

20

sN

kkNikkikiiN T

tiET

ti

1

0

I. Time-independent

II. Time-dependent (adiabatic basis)

III. Time-dependent (general basis)

sN

kkNikkMikMiiN TETU

1

2 0

sN

kkeikkNikkikiN HT

tiT

ti

1

0

IV. Time-dependent (diabatic basis)

sN

kkeikkNikiN HTT

ti

1

0

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Coupling terms

kNi T This term is often neglected in local approximations

kei H This term is simplified to the Born-Oppenheimer energy if the electronic basis is adiabatic. Otherwise, it can be important.

ki t

Using the chain rule, this term can be written as:

v

kiki t v is the nuclear velocity.

ki The non-adiabatic coupling vector is by definition equal zero in an electronic diabatic basis. In adiabatic basis it is important close to degeneracies. It diverges at conical intersections.

i k Transition dipole moment. Coupling through an electric field.

SOi kH Spin-orbit couplig.

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non-adiabatic coupling terms

kNi 2kNi

In both cases, the derivatives are in nuclear coordinates.

N

ki

N

ki

N

ki

ki

ki

ki

kNi

zyx

zyx

111

sN

kkkeikNikiiN HT

tiT

ti

1

0

23

Present situation of quantum chemistry methods

Method Single/Multi Reference

Analytical gradients

Coupling vectors

Computational effort

Typical implementation

MR-CISD MR Columbus EOM-CC SR Aces2 SAC-CI SR Gaussian CC2 / ADC SR Turbomole CASPT2 MR Molpro MRPT2 MR Gamess CISD/QCISD SR Molpro / Gaussian MCSCF MR Columbus / Molpro DFT/MRCI MR S. Grimme (Münster) OM2 MR W. Thiel (Mülheim) TD-DFT SR Turbomole TD-DFTB SR M. Elstner (Braunschweig) FOMO/AM1 MR Mopac (Pisa)

Methods allowing for excited-state calculations:

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• They define the limit of validity of the adiabatic approximation and of the breakup of the Hilbert space into uncoupled subspaces, which is important for reducing the dimensionality of the problem to be treated.

• The coupling vectors allow to connect the Hilbert space from one set of nuclear coordinates R to another R + DR nearby, which is important for the time-dependent formulation of the problem.

• 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?

kNiik h

25

• They define the limit of validity of the adiabatic approximation and of the breakup of the Hilbert space into uncoupled subspaces, which is important for reducing the dimensionality of the problem to be treated.


E

1

2

345

6

If h2k ~ 0 (k ≠ 2), then state 2 can be treated alone (adiabatically).

E

1

2

345

6

If hik ≠ 0 for i = 1, 2, k = 1, 2 and hik ~ 0 for i = 1, 2 and k > 2, then states 1,2 for a uncoupled subspace and can be isolated treated.

26

• The coupling vectors allow to connect the Hilbert space from one set of nuclear coordinates R to another R + DR nearby, which is important for the time-dependent formulation of the problem.


R

t

2

1

;;; RN

kkikii DDD

RrhRRrRRr

Prove it!

Hint: Expand the electronic wave function till the firts order and use the fact that the first derivativeis a function of R and, therefore can be expanded in terms of

i

iR

R+DR

t t+Dt

27

• 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.


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kiki t

v

Relation between time derivative and spatial derivative couplings

ttttttt kikiki DD

1

tttt ki DD 1

= 0

Computation of the coupling can be reduced to the computation of overlaps!

01321

tt

t ki

ttttttttt jiji DDDDD

2221

0

ttttttt jiji DDD

21

1

Hammes-Schiffer and Tully, J. Chem. Phys. 101, 4657 (1994)

29

-0,025

0,000

0,000

0,001

0 2 4 6 8 10 12 14 16 18 20

0,000

0,002

v.h HST CDA

21 12

hopping

Cou

plin

g (a

u)

23

Time (fs)

13

Pittner, Lischka, and Barbatti, Chem. Phys. 356, 147 (2009)

30

Next lecture

• Non-crossing rule• Conical intersections

[email protected]

mailto:[email protected]

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Ultrafast processes in molecules