Adiabatic and Adiabatic and nonadiabaticnonadiabatic molecular dynamics molecular dynamics withwith multireferencemultireference abab initioinitio methodsmethods
Mario BarbattiMario Barbatti
Institute for Theoretical ChemistryUniversity of Vienna
Columbus in Rio: univie.ac.at/columbus/rio
Rio de Janeiro, Nov. 27- Dec. 02, 2005
OutlineOutline
First Lecture: An introduction to molecular dynamicsFirst Lecture: An introduction to molecular dynamics1. Dynamics, why?2. Overview of the available approaches
Second Lecture: Towards an implementation of surface hopping dynSecond Lecture: Towards an implementation of surface hopping dynamicsamics1. Practical aspects to be addressed2. The NEWTON-X program3. Some applications: theory and experiment
Part IIPart IITowards an implementation of Towards an implementation of
surface hopping dynamicssurface hopping dynamics
Practical aspects to be Practical aspects to be addressedaddressed
Initial conditionsInitial conditionsInitial conditions
R,P
E
E0 ± ∆E
Wigner distribution
S0
Sn
acceptdon’t accept
( )
2
2exp2exp1),(
22)0(
HO
eW
m
PRRPRP
ωα
αα
π
=
⎥⎦
⎤⎢⎣
⎡−⎥⎦
⎤⎢⎣
⎡ −−=
hhh
A good one is the Velocity Verlet (Swope et al. JCP 76, 637 (1982)):
For each nucleus I
Classical dynamics: integrator Classical dynamics: integrator Classical dynamics: integrator
[ ]
ttttttt
ttEM
t
ttttt
ttttttt
III
IRI
I
III
IIII
∆∆++⎟⎠⎞
⎜⎝⎛ ∆
+=∆+
∆+∇−=
∆+=⎟⎠⎞
⎜⎝⎛ ∆
+
∆+∆+=∆+
)(21
2)(
)(1)(
)(21)(
2
)(21)()()( 2
avv
Ra
avv
avRR
Quantum chemistry calculation
Any standard method can be used in the integration of the Newton equations.
Schlick, Barth and Mandziuk, Annu. Rev. Biophys. Struct. 26, 181 (1997).
Time-step for the classical equationsTimeTime--step for the step for the classicalclassical equationsequations
Time-step for the classical equationsTimeTime--step for the step for the classicalclassical equationsequations
Time step should not be larger than 1 fs (1/10v).
∆t = 0.5 fs assures a good level of conservation of energy most of the time.
Exception: dynamics close to the conical intersection may require 0.25 fs.
TDSE: integratorTDSE: integratorTDSE: integrator
• Fourth-order Runge-Kutta (RK4)
• Bulirsh-Stoer
BS works better than RK4
Numerical Recipes in Fortran
h(t) h(t+∆t)
∆t/ms
)11( ))(-)((+)(=1 -..mn=ttt+mnt
mn-tt+ s
ss
hhhh ∆⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛∆
. . .
0 10 20 30 40 50 60 70 80 90 100
1.00
1.01
1.02
1.03
1.04
2
3
ms = 20
ms = 10
Σi|ci|2
Time (fs)
ms = 1
∆t = 0.5 fs
Time-step for the quantum equationsTimeTime--step for the step for the quantumquantum equationsequations
)()( 2222 ttata ∆+→
NaN 222 =
( )0
)()()()(
21
2222
2212
=
∆+−=∆+−=
→
→
hop
hop
n
NttatattNtNn
Fewest switches: two statesFewest switches: two statesFewest switches: two states
Population in S2:
Trajectories in S2:
Minimum number of hoppings that keeps the correct number of trajectories:
⎪⎪⎩
⎪⎪⎨
⎧
≤
>⎟⎠⎞
⎜⎝⎛∆
≈= →→
0,0
0,)(
)( 11
1111
11
2
1212
dtdadt
dadt
datat
tNnP
hophop
Probability of hopping
0
1
P2→1
Fewest switches: several statesFewest switches: several statesFewest switches: several states
( ) ( )∑ ∑≠ ≠
⎥⎦⎤
⎢⎣⎡ −==
kl klklklklklklkk daHaba ** Re2Im2
h&
Tully, JCP 93, 1061 (1990)tabP
kk
klhoplk ∆=→
Example: Three states 313233 bba +=&
tabPhop ∆=→
33
3223
Only the fraction of derivative connected to the particular transition
0
1
P3→2
P3→2 +P3→1
R
E
Total energy
Forbidden hop
Forbidden hop makes the classical statistical distributions deviate from the quantum populations.
How to treat them:• Reject all classically forbidden hop and keep the momentum.• Reject all classically forbidden hop and invert the momentum.• Use the time incertainty to search for a point in which the hop is allowed (Jasper et al. 116 5424 (2002)).
Forbidden hopsForbidden hopsForbidden hops
After hop, what are the new nuclear velocities?
• Adjust the velocities components in the direction of the nonadiabatic coupling vector h12.
Adjustment of momentum after hoppingAdjustment of momentum after hoppingAdjustment of momentum after hopping
R
E
KN(t)KN(t+∆t)
Total energy
Phase controlPhase controlPhase control
0 2 4 6 8 10 12
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15h1
x
Time (fs)
Component h1x Before the phase correction
CNH4+: MRCI/CAS(4,3)/6-31G*
Phase controlPhase controlPhase control
0 2 4 6 8 10 12
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15h1
x
Time (fs)
Component h1x Before the phase correction After the phase correction
CNH4+: MRCI/CAS(4,3)/6-31G*
Abrupt changes controlAbrupt changes controlAbrupt changes control
11.75 fs
01h
12.00 fs
OrthogonalizationOrthogonalizationOrthogonalization
ββ
ββ
cossin~sincos~
ijijij
ijijij
hgh
hgg
+=
+−=
ijijijij
ijij
gghhhg
⋅−⋅⋅
=2
2tan β
• The routine also gives the linear parameters:
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠
⎞⎜⎝
⎛−
∆++±+=
2/12222 )(
2)(
21 yxyxyxdE gh
yxgh σσ
F
S1
S0
F
F
S1
S0
F
g-h space orthogonalization [Yarkony, JCP 112, 2111 (2000)]
Abrupt changes controlAbrupt changes controlAbrupt changes control
01h 01~h
11.75 fs
12.00 fs
The NewtonThe Newton--X programX program
Adiabatic dynamicsAdiabatic dynamicsAdiabatic dynamics
Classical motion of the nuclei: Velocity Verlet [Swope et al. JCP 76, 637 (1982)]
R(t), v(t)
t+∆t, R(t+∆t), v(t+∆t/2)
COLUMBUS:COLUMBUS:E(t+∆t), ∇E(t+∆t)
v(t+∆t)
• Fortran 90 routines• Perl controler
Nonadiabatic dynamicsNonadiabaticNonadiabatic dynamicsdynamics
• Surface hopping: fewest switches
R(t), v(t)
t+∆t, R(t+∆t), v(t+∆t/2)
COLUMBUS:COLUMBUS:E(t+∆t), ∇E(t+∆t), hkl(t+∆t)
v(t+∆t)
ak, Pk→l(t+∆t)
∑ −=k
kti
kketct )()()( )( Rφψ γ
Integration:RK4 (with Granucci, Pisa)
Previous implementation:Bulirsh-Stoer (Pittner, Prague)
∑≠
− ⋅−=kl
klti
lkletctc hv)()()( γ&
)()( ttt
i el ψψ H=∂∂
h
)()()( * tctcta lkkl =
⎟⎟⎠
⎞⎜⎜⎝
⎛=→
kk
lklk a
taMaxtP δ&,0)(
Extensions to other methodsExtensions to other methodsExtensions to other methods
R(t), v(t)
t+∆t, R(t+∆t), v(t+∆t/2)
Any program:Any program:E(t+∆t), ∇E(t+∆t)
v(t+∆t)
Presently:
• COLUMBUS (nonadiabatic dynamics)• MCSCF• MRCI
• TURBOMOLE (adiabadic dynamics) • TD-DFT• RI-CC2
Features Options Dynamics On-the-fly adiabatic dynamics On-the-fly nonadiabatic dynamics with surface hopping (fewest switches) Adjustment of momentum along the nonadiabatic coupling vector Neglecting of forbidden hoppings Phase control of the CI vector (overlapping / inner-product) Yarkony orthogonalization of the nonadiabatic coupling vectors Topographic analysis of the conical intersections Damped dynamics (kinetic energy always null) TDSE integrated in substeps of the classical equations Portability COLUMBUS (CASSCF, MRCI)
TURBOMOLE (TD-DFT, RI-CC2) Analytical models
Initial conditions Wigner distribution / classical harmonic oscillator File management Friendly input via nxinp. Automatic management of files and directories in multiple
trajectories Documentation All options and routines are documented Output and analysis Statistical analysis of results (internal coordinates and forces, energies, wave function
properties) Recomputation of internal coordinates On-the-fly graphical outputs (MOLDEN, GNUPLOT)
Current capabilitiesCurrent capabilitiesCurrent capabilities
$NX/nxinp$NX/$NX/nxinpnxinp
------------------------------------------NEWTON-X
Newton dynamics close to the crossing seam------------------------------------------
MAIN MENU
1. GENERATE INITIAL CONDITIONS
2. SET BASIC INPUT
3. SET GENERAL OPTIONS
4. SET NONADIABATIC DYNAMICS
5. GENERATE TRAJECTORIES
6. SET STATISTICAL ANALYSIS
7. EXIT
Select one option (1-7):
$NX/nxinp$NX/$NX/nxinpnxinp
------------------------------------------NEWTON-X
Newton dynamics close to the crossing seam------------------------------------------
SET BASIC OPTIONS
nat: Number of atoms.There is no value attributed to natEnter the value of nat : 6Setting nat = 6
nstat: Number of states.The current value of nstat is: 2Enter the new value of nstat : 3Setting nstat = 3
nstatdyn: Initial state (1 - ground state).The current value of nstatdyn is: 2Enter the new value of nstatdyn : 2Setting nstatdyn = 2
prog: Quantum chemistry program and method
0 - ANALYTICAL MODEL1 - COLUMBUS2.0 - TURBOMOLE RI-CC22.1 - TURBOMOLE TD-DFT
The current value of prog is: 1Enter the new value of prog : 1
So many choicesSo many choices……What method should I use?What method should I use?
Method Single/Multi Reference
Analytical gradients
Coupling vectors
Computational effort
Typical implementation
MR-CISD MR √ √ Columbus RI-CC2 SR √ × Turbomole CASPT2 MR × × Molcas / Molpro MR-MP2 MR × × Gamess CISD/QCISD SR √ × Molpro / Gaussian CASSCF MR √ √ Columbus / Molpro TD-DFT SR √ × Turbomole FOMO/AM1 MR √ √ Mopac (Pisa)
Present situation of quantum chemistry methodsPresent situation of quantum chemistry methodsPresent situation of quantum chemistry methods
Comparison among methodsComparison among methodsComparison among methods
TD-DFT MCSCF
Time0 tc tfinal
0 5 10 15 20 25 30
Ene
rgy
Time (fs)
RI-CC2
0 5 10 15 20 25 30
Time (fs)
TD-DFT
0 5 10 15 20 25 30
Time (fs)
CASSCF
A basic protocolA basic protocolA basic protocol
• Use TD-DFT for large systems (> 10 heavy atoms) with one single configuration dominating the region of the phase space spanned by the dynamics. Test against CASSCF and RI-CC2.
• Use TD-DFT for large systems until a multireference region of the phase space be reached. Switch to CASSCF to continue the dynamics.
• Use RI-CC2 for medium systems (6-10 heavy atoms) under the same conditions as in the previous two points.
• Use CASSCF for medium systems with strong multireference character in all phase space. Test against MRCI.
• Use MRCI for small systems (< 6 heavy atoms).
• In all cases, when the number of relevant internal coordinates is small (2-4) and they can easily be determined, test against wave-packet dynamics.
Some applications:Some applications:theory and experimenttheory and experiment
On the ambiguity of the On the ambiguity of the experimental raw dataexperimental raw data
Bonačić-Koutecký and Mitrić, Chem. Rev. 105, 11 (2005).
Experimental: pump-probeExperimental: pumpExperimental: pump--probeprobe
Analysis of the experimental resultsAnalysis of the experimental resultsAnalysis of the experimental results
Fuß et al., Faraday Discuss. 127, 23 (2004).
0 100 200 300 4000.0
0.2
0.4
0.6
0.8
1.0
Ave
rage
occ
upat
ion
Time (fs)
SS11
SS22
SS00
Barbatti, Granucci, Persico and Lischka, CPL 401, 276 (2005)
The lifetime of ethyleneThe lifetime of ethyleneThe lifetime of ethylene
Method Lifetime (fs) DTSH [1] S1→S2,S0 139±1a / 105±1b DTSH [1] S1,S2→S0 188±2a / 137±2b DTSH [2] S1→S2,S0 50 AIMS [3] > 250 AIMS [4] 180 MCDTH [5] >> 100 Experimental [6] 30±15 Experimental [7] 20±10 Experimental (τ1 + τ 2) [8] 25±5
[1] Barbatti, Granucci, Persico and Lischka, CPL 401, 276 (2005).[2] Granucci, Persico, and Toniolo, JCP 114 (2001) 10608.[3] Ben-Nun and Martínez, CPL 298 (1998) 57.[4] Ben-Nun, Quenneville, and Martínez, JPCA 104 (2000) 5161.[5] Viel, Krawczyk, Manthe, Domcke, JCP 120 (2004) 11000. [6] Farmanara, Stert, and Radloff, CPL 288 (1998) 518.[7] Mestagh, Visticot, Elhanine, Soep, JCP 113 (2000) 237.[8] Stert, Lippert, Ritze, and Radloff, CPL 388 (2004) 144.
a Evb = 6.2 ± 0.3 eV.
b Evc = 7.4 ± 0.15 eV.
Survey of theoretical and experimental predictionsSurvey of theoretical and experimental predictionsSurvey of theoretical and experimental predictions
Eion-Epot
S0
S1
C2H4+
MXS
0 20 40 60 80 100 120 140 160 180 2002
4
6
8
τbτc
172 nm (7.21 eV)
τnd
267 nm (4.6 eV)E ion-E
pot (
eV)
Time (fs)
Ev = 6.2 eV Ev = 7.4 eV
Radloff has a rather distinct explanation!CPL 388 (2004) 144.
The time-window questionThe timeThe time--window questionwindow question
On how the initial surface On how the initial surface can make a differencecan make a difference
0
2
4
6
8
10
12
14
1.2
1.3
1.4
1.5
1.6
015
3045
6075
90
Ener
gy (e
V)
CN Stre
tch (Å
)
Torsion (°)
Basic scenery expected:Torsion + decay at twisted MXS
CNH4+: MRCI/CAS(4,3)/6-31G*CNHCNH44
++: MRCI/CAS(4,3)/6: MRCI/CAS(4,3)/6--31G*31G*
This and other movies are available at:homepage.univie.ac.at/mario.barbatti
Barbatti, Aquino and Lischka, Mol. Phys. in press (2005)
0
2
4
6
8
10
12
14
1.2
1.3
1.4
1.5
1.6
015
3045
6075
90
Ener
gy (e
V)
CN Stre
tch (Å
)
Torsion (°)
Large momentum along CN stretch isacquired in the S2-S1 transition.
Stretching + pyramidalization dominate
CNH4+: MRCI/CAS(4,3)/6-31G*CNHCNH44
++: MRCI/CAS(4,3)/6: MRCI/CAS(4,3)/6--31G*31G*
0
2
4
6
8
10
12
14
1.2
1.3
1.4
1.5
1.6
015
3045
6075
90
Ener
gy (e
V)
CN Stre
tch (Å
)
Torsion (°)
If S1 (σπ*) is directly populated,the torsion should dominate thedynamics.
CNH4+: MRCI/CAS(4,3)/6-31G*CNHCNH44
++: MRCI/CAS(4,3)/6: MRCI/CAS(4,3)/6--31G*31G*
This and other movies are available at:homepage.univie.ac.at/mario.barbatti
Intersection? Intersection? Which of them?Which of them?
The S0/S1 crossing seamThe SThe S00/S/S11 crossing seamcrossing seam
4
5
6
7
8
-100
-50
0
50
100
4060
80100
120
Ene
rgy
(eV
)
β (degrees)
θ (degrees)
EthylideneEthylidene
PyramidalizedPyramidalized
HH--migrationmigration
Barbatti, Paier and Lischka, JCP 121, 11614 (2004).
θθββ
Ohmine 1985Freund and Klessinger 1998Ben-Nun and Martinez 1998Laino and Passerone 2004
CC3V3V
Evolution on the configuration spaceEvolution on the configuration spaceEvolution on the configuration space
20 40 60 80 100 120 1400
10
20
30
0
10
20
30
0
10
20
30
Hydrogen Migration (degrees)
Twisted
planarPy
ram
idal
izat
ion
(deg
rees
)
Ethylid. MXS
Pyramid. MXS
H-mig. c.i.
First S1→S0 hopping.
Torsion Torsion + Pyramid.+ Pyramid.
HH--migrationmigrationPyram. MXSPyram. MXS
Ethylidene MXSEthylidene MXS
~7.6 eV~7.6 eV
ττ ~ 100~ 100--140 fs140 fs
60%60%
11%11%23%23%
Barbatti, Ruckenbauer and Lischka, JCP 122, 174307 (2005)
Conical intersections in ethyleneConical intersections in ethyleneConical intersections in ethylene
Readressing the DNA bases problemReadressing the DNA bases problem
An example: photodynamics of DNA basesAn example: An example: photodynamicsphotodynamics of DNA of DNA basesbases
What has theory to say?
• The decay can be due to πσ*/S0 crossing Sobolewski and Domcke, Eur. Phys. J. D 20, 369 (2002).
• Or maybe due to the ππ*/S0 crossingMarian, JCP 122, 104314 (2005).
• Or nπ*/S0 crossing, who knows?Chen and Li, JPCA 109, 8443 (2005).
• Maybe, there’s no crossing at all!Langer and Doltsinis, PCCP 6, 2742 (2004)
• Our own dynamics simulations (TD-DFT(B3LYP)/SVP) do not show any crossing too…
• But the dynamics calculations are not reliable enough until now: or they miss the MR character, or the nonadiabatic character, or both!
Experimental lifetimes of the excited state of DNA/RNA basis: ~ 1 ps
An example: photodynamics of DNA basesAn example: An example: photodynamicsphotodynamics of DNA of DNA basesbases
Amino pyrimidine as a model for adenine
9H-adenine 4-amino pyrimidine
An example: photodynamics of DNA basesAn example: An example: photodynamicsphotodynamics of DNA of DNA basesbases
4-amino pyrimidineSA-3-CAS(5,6)/6-31G*
Time (fs)
The system moves toward the crossing seam with out-of-plane motions similar to that predicted by Marian for adenine (JCP 122, 104314 (2005))
An example: photodynamics of DNA basesAn example: An example: photodynamicsphotodynamics of DNA of DNA basesbases
4-amino pyrimidineSA-3-CAS(5,6)/6-31G*
Time (fs)
If the H-atoms that replace the ring have a big isotopic mass, the out-of-plane motion is inhibited and there is no crossing.
An example: photodynamics of DNA basesAn example: An example: photodynamicsphotodynamics of DNA of DNA basesbases
4-amino pyrimidineSA-3-CAS(5,6)/6-31G*
Average probability is 4×10-5
in each 0.05 fs.
In 1 ps, the probability is 0.8.
The system does not need a crossing to decay in 1 ps. The weak S1/S0 coupling is enough to induce the transfer.
0 100 200 300 400 500
0.0000
0.0002
0.0004
0.0006
0.0008
0.0010
0.0012
0.0014
0.008
Time (fs)
P(1→0) hopping probability in 0.05 fs
ConclusionsConclusions
ConclusionsConclusionsConclusions
Basis set
Correlation Method
DZ TZ BS limit
HF
CASSCF
MRCI
Full-CI
…
…
ConclusionsConclusionsConclusions
Basis set
Correlation Method
DZ TZ BS limit
HF
CASSCF
MRCI
Full-CI
…
…
Dynamics method
Surf. Hopping and Mean Field
Multiple Spawning
Wavepacket dynamics
Static calculations
Adiabatic dynamics
< 6 heavy atoms
6-10 heavy atoms
ConclusionsConclusionsConclusions
Dynamics, Why?• Ab initio dynamics can be used to characterize lifetimes, and relaxations paths in the ground and excited state, occurring in the time scale of sub-picosecond.• Dynamics reveal features that is not easily found by static methods.• Dynamics can be used to locate the important regions of the crossing seam.
Excited-state dynamics around the world (a very incomplete and biased inventory…)• Semiclassical dynamics:Tully; Jasper and Thrular; Robb, Olivucci and Buss; Barbatti and Lischka; Marx and Doltsinis (Car-Parrinelo); Granucci and Persico (local diabatization); Martinez-Nuñez; Pittner and Bonačić-Koutecký; Neufeld.
•Multiple spawningBen-Nun and Martínez
• Wave packetGonzalez and Manz; Manthe and Cederbaum; Viel and Domcke; Jacubtz; de Vivie-Riedle and Hofmann; Schinke; Köppel.
Our contributionOur contributionOur contribution
Towards an implementation of surface hopping dynamics• NEWTON-X: new implementation of on-the-fly surface hopping dynamics.• Energies, gradients and nonadiabatic coupling vectors can be read from any quantum chemical package.• For MRCI and CASSCF dynamics, COLUMBUS has been used.• For TD-DFT and RI-CC2, TURBOMOLE.
Where are we going now?• QM/MM: investigation of the influence of the environment on the relaxations paths, lifetimes and efficiency of the crossing seam.
MRCI, CAS
Force field
CollaboratorsCollaboratorsCollaborators
Vienna:Hans LischkaAdélia AquinoMatthias Ruckenbauer (Master student)Gunther Zechmann (Master student, now at Warwick)Daniela Raab
Pisa:Giovanni Granucci
Prague:Jiri Pittner
Zagreb:Mario Vazdar
ConclusionsConclusionsConclusions
Columbus in Rio: univie.ac.at/columbus/rio
Rio de Janeiro, Nov. 27- Dec. 02, 2005
• References and additional information: homepage.univie.ac.at/mario.barbatti
• Support: CNPq (Brazil), Austrian Science Fund (Special project F16)
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