talk 3: nonadiabatic dynamics including triplet states · 2018-01-04 · f. plasser nonadiabatic...

Intro Electronic Structure Dynamics SHARC

Talk 3:Nonadiabatic dynamics including triplet states

Felix Plasser

Institute for Theoretical Chemistry, University of Vienna

Helsinki, 19 December 2017

F. Plasser Nonadiabatic dynamics 1 / 30


Introduction

Dynamics on coupled potentialenergy surfaces (PES)

I Photon absorptionI Motion on the PESI Transitions between different PES- Internal conversion(same multiplicity)

- Intersystem crossing(different multiplicity)

GoalI Learn how to simulate these

processes→ Electronic structure ingredients→ Dynamics simulations


Intro Electronic Structure Dynamics SHARC Nonadiabatic coupling Spin-orbit coupling

Photon absorption

I Absorption intensity determined by

Transition dipole moment

µ0I = 〈Ψ0| µ |ΨI〉

, Readily available, Computationally cheap



Motion on the PES

I Motion on the PES determined by

Electronic energy gradient

∇EI = ∇〈ΨI | H |ΨI〉

, Readily available/ Cost per gradient ≈ energy evaluation



Transitions between PES

Different coupling termsI Nonadiabatic couplingI Spin-orbit couplingI Coupling to an external fieldI ...→ Surface hopping with arbitrary couplings (SHARC)



Nonadiabatic coupling

Potential curveI Selenocroleine- Twist around double bond- T2/T1

I T1 - Two minima:nπ∗ and ππ∗ character

I States cross around 55◦

I T1 and T2 exchange character

y z

x

Se

C

C

C

H

H

HH

Se

C

C

Cθ

1 F. Plasser et al. J. Chem. Theory Comput. 2016, 12, 1207.F. Plasser Nonadiabatic dynamics 6 / 30


Avoided Crossing

ZoomI Avoided crossing at 58◦

- Diabatic states (nπ∗, ππ∗)follow straight lines

- Adiabatic states changecharacter

- No crossing

y z

x

Se

C

C

C

H

H

HH

Se

C

C

Cθ




State overlapI Orthogonal states〈Ψ1(R0)|Ψ2(R0)〉 = 0

I State character changes〈Ψ1(R0)|Ψ2(R1)〉 ≈ 1

I Difference quotient⟨Ψ1(R0)

∣∣∣Ψ2(R1)−Ψ2(R0)R1−R0

⟩≈ 1

R1−R0

I Nonadiabatic coupling⟨Ψ1(R0)

∣∣ ∂∂RΨ2(R0)

⟩≈ 1

R1−R0

R0 = 50◦, R1 = 65◦




Trajectory going through the crossing

I Same adiabatic surfaceT2 → T2

→ Different diabatic surfaceππ∗ → nπ∗

I Same diabatic surfaceππ∗ → ππ∗

→ Different adiabatic surfaceT2 → T1

- Surface hop

R0 = 50◦, R1 = 65◦




I Transitions determined by

Nonadiabatic coupling vectors

hIJ(R) = 〈ΨI(R)|∇ΨJ(R)〉

/ Only implemented for some quantum chemistry methods / programpackages

/ Computationally expensive- One computation per pair of states



Nonadiabatic Interactions

Nonadiabatic coupling vectors

hIJ(R) = 〈ΨI(R)|∇ΨJ(R)〉

R Vector of nuclear coordinates

I Expressed in terms of wavefunction overlaps

SIJ(R,R′) = 〈ΨI(R)|ΨJ(R′)〉hIJ(R) = ∇′ 〈ΨI(R)|ΨJ(R′)〉 |R′=R = ∇′SIJ(R,R′)|R′=R

I Discrete

hIJ(R) ·∆R ≈ SIJ(R,R + ∆R)

I Applicable to trajectory dynamics simulations



Overlaps

I Overlap as double sum over Slater determinant overlaps

SIJ = 〈ΨI |Ψ′J〉 =

nCI∑k=1

n′CI∑l=1

dkId′lJ 〈Φk|Φ′l〉

I Computed as determinant over MO overlaps〈Φk|Φ′l〉 =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

⟨ϕ1

∣∣ϕ′1⟩ . . .⟨ϕ1

∣∣∣ϕ′nα⟩...

. . .... 0⟨

ϕnα∣∣ϕ′1⟩ . . .

⟨ϕnα

∣∣∣ϕ′nα⟩ ⟨ϕnα+1

∣∣∣ϕ′nα+1

⟩. . .

⟨ϕnα+1

∣∣∣ϕ′l(n)

⟩0

.

.

.. . .

.

.

.⟨ϕn

∣∣∣ϕ′nα+1

⟩. . .

⟨ϕn∣∣ϕ′n⟩

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣I Formal scaling: O(nCIn

′CIn

3el)

I Simplifications?



Overlaps

I Two independent factors for α and β spin

〈Φk|Φ′l〉 =∣∣∣∣∣∣∣⟨ϕ1

∣∣ϕ′1⟩ . . .⟨ϕ1

∣∣∣ϕ′nα⟩...

. . ....⟨

ϕnα∣∣ϕ′1⟩ . . .

⟨ϕnα

∣∣∣ϕ′nα⟩∣∣∣∣∣∣∣×∣∣∣∣∣∣∣⟨ϕnα+1

∣∣∣ϕ′nα+1

⟩. . .

⟨ϕnα+1

∣∣∣ϕ′l(n)

⟩...

. . ....⟨

ϕn

∣∣∣ϕ′nα+1

⟩. . .

⟨ϕn∣∣ϕ′n⟩

∣∣∣∣∣∣∣= SklSkl

I Spin-factors reappearI Strategy: Precompute and store these factors



Overlaps

SIJ

ContractUnique factorsSkl, Skl

Precompute

Sort

Slater De-terminants|Φk〉 ,

∣∣Φ′l⟩CI-coefficientsdkI , d

′lJ

MO overlaps⟨ϕp|ϕ′q

⟩

MO coefficientsCpµ, C

′qν

Double moleculeAO overlaps⟨χµ|χ′ν

⟩



Verification

Y

C

C

C

H

H

HH

Se

C

C

Cθ

Verification for selenoacroleine torsionI New code1 against existing state-of-the-art code2

Implem. Method 〈T1(50◦)|T1(55◦)〉〈T1(50◦)|T2(55◦)〉 tCPU (s)current CASSCF(6,5) 0.6873547950 0.7107005295 0Ref. 2 CASSCF(6,5) 0.6873547949 0.7107005297 0current MR-CIS(4,3) 0.9839833569 0.1084043350 33Ref. 2 MR-CIS(4,3) 0.9839833570 0.1084043349 43769

I Quantitative agreementI 1000 times faster

1 FP, M. Ruckenbauer, S. Mai, M. Oppel, P. Marquetand, L. González JCTC 2016, 12,1207.

2 J. Pittner et al. Chem. Phys. 2009, 356, 147-152.F. Plasser Nonadiabatic dynamics 16 / 30


Performance

Tim

e(core

seco

nds)

0.1

1

10

100

1,000

10,000

100,000

npair

1e+05 1e+06 1e+07 1e+08 1e+09 1e+10 1e+11 1e+12

I Uniform performanceI Over 7 orders of magnitude in problem sizeI For various wave function models

I 2-3 orders of magnitude faster than previous code (X)



Overlaps

I Integration into the SHARC dynamics code1

I Interface to various other electronic structure codes

Multireference methods Columbus, MolcasCorrelated single-reference methods Turbomole2

Time-dependent DFT ADF, GaussianI Photoelectron spectra / Dyson orbitals3

I Wavefunction analysis4

1 S. Mai, P. Marquetand, L. González IJQC 2015, 115, 1215, https://sharc-md.org/.2 S. Mai, FP, M. Pabst, A. Köhn, L. González JCP 2017, 147, 184109.3 M. Ruckenbauer, S. Mai, P. Marquetand, L. González Sci. Rep. 2016, 6, 35522.4 FP, L. L. González JCP 2016, 145, 021103.




I Transitions determined by

Wave function overlaps

SIJ = 〈ΨI(R)|ΨJ(R′)〉

, Transferable to any method / program package, Computationally efficient



Spin-orbit coupling

I Relativistic effect- Exact equation not known (combination of quantum mechanics andrelativity)

- Good approximations exist

Breit-Pauli Hamiltonian

HSO,BP =1

2c2

nel∑i=1

nnuc∑K=1

ZK(riK × pi) · sir3iK

−

1

2c2

nel∑i,j 6=i

(rij × pi) · sir3ij

+1

2c2

nel∑i,j 6=i

(rij × pi) · sjr3ij

I Spin-orbit coupling

(r× p) · s = L · s



Spin-orbit coupling

Breit-Pauli Hamiltonian

HSO =1

2c2

nel∑i=1

nnuc∑K=1

ZK(riK × pi) · sir3iK

−

1

2c2

nel∑i,j 6=i

(rij × pi) · sir3ij

+1

2c2

nel∑i,j 6=i

(rij × pi) · sjr3ij

I One- and two-electron terms→ Cost reduced through mean-field approximation- SOMF (Spin-orbit mean field)- AMFI (Atomic mean field integrals)

1 A. Berning, M. Schweizer, H. Werner, et al. Mol. Phys. 2000, 98, 1823.2 F. Neese J. Chem. Phys. 2005, 122, 034107.



Spin-orbit coupling

I Spin-orbit coupling/ Complicated underlying theoryI Determined in mean-field approximation→ One-electron operator, Transferable, Low computational cost



Spin-orbit coupling

I Quasi-degenerate perturbation theory

Spin-orbit Hamiltonian matrix

HSOC =

E0 〈Ψ0| HSO |Ψ1〉 . . .

〈Ψ1| HSO |Ψ0〉 E1 . . .

〈Ψ2| HSO |Ψ0〉〈Ψ2| HSO |Ψ1〉 . . ....

. . .

EI Eigenvalues of molecular Coulomb Hamiltonian (MCH), “spin-free” energies



Spin-orbit coupling

Diagonalization

UTHSOCU =

ε0 0 0 . . .0 ε1 0 . . .0 0 ε2 . . ....

. . .

εα Diagonal energiesU Diagonal/MCH transformation matrix


Intro Electronic Structure Dynamics SHARC Example

Representations

Spectr.

1ππ∗ 1nπ∗

3nπ∗E

nerg

y MCH

S1

S2

T1

Coordinate

Diagonal

1

2 345

I Spectroscopic, state character 1nπ∗,1 ππ∗,3 ππ∗, . . .

- Potential couplings, spin-orbit couplings- Quasidiabatic → beneficial for quantum dynamicsI Molecular Coulomb Hamiltonian (MCH)- Nonadiabatic couplings, spin-orbit couplings- Usually used in quantum chemistry computationsI Diagonal- Nonadiabatic couplings→ Beneficial for surface hopping dynamics



Propagation

SHARC method1

I Nuclei propagated on diagonal surfaces→ Gradients of several MCH states needed

I Electronic coefficients propagated in the MCH representation

Three-step propagator

cdiag(t)U−→ cMCH(t)

Prop.−−−→ cMCH(t+ ∆t)UT

−−→ cdiag(t+ ∆t)

1 S. Mai, P. Marquetand, L. González IJQC 2015, 115, 1215.F. Plasser Nonadiabatic dynamics 26 / 30


Example

I H2CS

Exp

erim

ent

MRCISD+P(12,10)

MS-C

ASPT2(12,10)

ADC(2)

SCS-A

DC(2)

SOS-A

DC(2)

CC2

SCS-C

C2

SOS-C

C2

MS-C

ASPT2(10,6)

SA-C

ASSCF(10,6)

BP86

PBE

B3LY

P

mPW

1PW

BHHLY

P

1.5

2.0

2.5

3.0

3.5

4.0

Method

Energy(eV)

S1(1nπ∗) T1(

3nπ∗) T2(3ππ∗)

? Are the dynamics similar?



Dynamics

I Reference CASPT2→ No ISC

! ISC for for other methods- SA-CASSCF- B3LYP- BHLYP

MS-CASPT2(10,6)

0.01

0.02

0.03

0.04

0.05S0 S1 T1 T2

SA-CASSCF(10,6)

0.01

0.02

0.03

0.04

0.05

BP86

0.01

0.02

0.03

0.04

Population PBE

0.01

0.02

0.03

0.04

Population

B3LYP

0.01

0.02

0.03

0.04 BHHLYP

200 4000.00

0.05

0.10

0.15

Propagation Time (fs)ADC(2)

200 4000.00

0.01

0.02

0.03

0.04

Propagation Time (fs)



Potential Curves

I Different behavior afterdouble bond is broken

→ Multireference methodsneeded

- Pure DFT better thanhybrids

MRCISD+P(12,10)

1

2

3

4

5MS-CASPT2(12,10)

ADC(2)

1GS (1nπ∗) (3nπ∗) (3ππ∗)

SCS-ADC(2) SOS-ADC(2)

1

2

3

4

5

MS-CASPT2(10,6)

1

2

3

4

En

erg

y(e

V)

SA-CASSCF(10,6)

CC2 SCS-CC2 SOS-CC2

1

2

3

4

En

erg

y(e

V)

BP86

1.6 1.90

1

2

3

4

PBE

1.6 1.9

B3LYP

1.6 1.9

C-S Distance (A)

mPW1PW

1.6 1.9

BHHLYP

1.6 1.9 2.20

1

2

3

4



Software

SHARC - Surface hopping with arbitrary couplings

I Couplings- Nonadiabatic couplings- Spin-orbit couplings- Coupling to external field

I Various quantum chemistry methods- CASSCF - Molpro, MOLCAS- MRCI - Columbus- TDDFT - ADF, Gaussian- ADC(2) - Turbomole

I New release coming soon ...


talk 3: nonadiabatic dynamics including triplet states · 2018-01-04 · f. plasser nonadiabatic...

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