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
Intro Electronic Structure Dynamics SHARC
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
F. Plasser Nonadiabatic dynamics 2 / 30
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
F. Plasser Nonadiabatic dynamics 3 / 30
Intro Electronic Structure Dynamics SHARC Nonadiabatic coupling Spin-orbit coupling
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
F. Plasser Nonadiabatic dynamics 4 / 30
Intro Electronic Structure Dynamics SHARC Nonadiabatic coupling Spin-orbit coupling
Transitions between PES
Different coupling termsI Nonadiabatic couplingI Spin-orbit couplingI Coupling to an external fieldI ...→ Surface hopping with arbitrary couplings (SHARC)
F. Plasser Nonadiabatic dynamics 5 / 30
Intro Electronic Structure Dynamics SHARC Nonadiabatic coupling Spin-orbit coupling
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
Intro Electronic Structure Dynamics SHARC Nonadiabatic coupling Spin-orbit coupling
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
F. Plasser Nonadiabatic dynamics 7 / 30
Intro Electronic Structure Dynamics SHARC Nonadiabatic coupling Spin-orbit coupling
Nonadiabatic coupling
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◦
F. Plasser Nonadiabatic dynamics 8 / 30
Intro Electronic Structure Dynamics SHARC Nonadiabatic coupling Spin-orbit coupling
Nonadiabatic coupling
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◦
F. Plasser Nonadiabatic dynamics 9 / 30
Intro Electronic Structure Dynamics SHARC Nonadiabatic coupling Spin-orbit coupling
Nonadiabatic coupling
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
F. Plasser Nonadiabatic dynamics 10 / 30
Intro Electronic Structure Dynamics SHARC Nonadiabatic coupling Spin-orbit coupling
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
F. Plasser Nonadiabatic dynamics 11 / 30
Intro Electronic Structure Dynamics SHARC Nonadiabatic coupling Spin-orbit coupling
Overlaps
Wave function overlaps
SIJ = 〈ΨI(R)|ΨJ(R′)〉
Many-electron wave functionsI Expansion into Slater determinants
|ΨI〉 =
nCI∑k=1
dkI |Φk〉
I Expansion into MOs- α and β spin
|Φk〉 = |ϕ1 . . . ϕnα ϕnα+1 . . . ϕn|
F. Plasser Nonadiabatic dynamics 12 / 30
Intro Electronic Structure Dynamics SHARC Nonadiabatic coupling Spin-orbit coupling
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?
F. Plasser Nonadiabatic dynamics 13 / 30
Intro Electronic Structure Dynamics SHARC Nonadiabatic coupling Spin-orbit coupling
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
F. Plasser Nonadiabatic dynamics 14 / 30
Intro Electronic Structure Dynamics SHARC Nonadiabatic coupling Spin-orbit coupling
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
⟩
F. Plasser Nonadiabatic dynamics 15 / 30
Intro Electronic Structure Dynamics SHARC Nonadiabatic coupling Spin-orbit coupling
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
Intro Electronic Structure Dynamics SHARC Nonadiabatic coupling Spin-orbit coupling
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)
F. Plasser Nonadiabatic dynamics 17 / 30
Intro Electronic Structure Dynamics SHARC Nonadiabatic coupling Spin-orbit coupling
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.
F. Plasser Nonadiabatic dynamics 18 / 30
Intro Electronic Structure Dynamics SHARC Nonadiabatic coupling Spin-orbit coupling
Nonadiabatic coupling
I Transitions determined by
Wave function overlaps
SIJ = 〈ΨI(R)|ΨJ(R′)〉
, Transferable to any method / program package, Computationally efficient
F. Plasser Nonadiabatic dynamics 19 / 30
Intro Electronic Structure Dynamics SHARC Nonadiabatic coupling Spin-orbit coupling
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
F. Plasser Nonadiabatic dynamics 20 / 30
Intro Electronic Structure Dynamics SHARC Nonadiabatic coupling Spin-orbit coupling
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.
F. Plasser Nonadiabatic dynamics 21 / 30
Intro Electronic Structure Dynamics SHARC Nonadiabatic coupling Spin-orbit coupling
Spin-orbit coupling
I Spin-orbit coupling/ Complicated underlying theoryI Determined in mean-field approximation→ One-electron operator, Transferable, Low computational cost
F. Plasser Nonadiabatic dynamics 22 / 30
Intro Electronic Structure Dynamics SHARC Nonadiabatic coupling Spin-orbit coupling
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
F. Plasser Nonadiabatic dynamics 23 / 30
Intro Electronic Structure Dynamics SHARC Nonadiabatic coupling Spin-orbit coupling
Spin-orbit coupling
Diagonalization
UTHSOCU =
ε0 0 0 . . .0 ε1 0 . . .0 0 ε2 . . ....
. . .
εα Diagonal energiesU Diagonal/MCH transformation matrix
F. Plasser Nonadiabatic dynamics 24 / 30
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
F. Plasser Nonadiabatic dynamics 25 / 30
Intro Electronic Structure Dynamics SHARC Example
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
Intro Electronic Structure Dynamics SHARC Example
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?
F. Plasser Nonadiabatic dynamics 27 / 30
Intro Electronic Structure Dynamics SHARC Example
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)
F. Plasser Nonadiabatic dynamics 28 / 30
Intro Electronic Structure Dynamics SHARC Example
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
F. Plasser Nonadiabatic dynamics 29 / 30
Intro Electronic Structure Dynamics SHARC
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 ...
F. Plasser Nonadiabatic dynamics 30 / 30