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Photochemistry: Photochemistry: adiabatic and nonadiabatic adiabatic and nonadiabatic molecular dynamics with molecular dynamics with
multireference ab initio methods multireference ab initio methods
Mario BarbattiMario Barbatti
Institute for Theoretical ChemistryUniversity of Vienna
COLUMBUS in BANGKOK (3-TSCOLUMBUS in BANGKOK (3-TS22CC22))Apr. 2 - 5, 2006Apr. 2 - 5, 2006
Burapha University, Bang Saen, ThailandBurapha University, Bang Saen, Thailand
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 dynamicsSecond Lecture: Towards an implementation of surface hopping dynamics1. The NEWTON-X program 2. Practical aspects to be adressed
Third Lecture: Some applications: theory and experiment• On the ambiguity of the experimental raw data • On how the initial surface can make difference• Intersection? Which of them?• Readressing the DNA/RNA bases problem
Part IIPart IITowards an Towards an
implementation of implementation of surface hopping surface hopping
dynamicsdynamics
Cândido Portinari, O lavrador de café, 1934
Quick overview of the Quick overview of the methodmethod
Surface hopping approach: adiabatic populationSurface hopping approach: adiabatic populationSurface hopping approach: adiabatic populationSurface hopping approach: adiabatic population
k
kti
k tetct k ),;(),(),,( )( rRRRr eH
ti
*ikki cca Population:
iki
i
ijij
i
kikjkjkj eaeaa hvhv
• Two electronic states are coupled only by the nonadiabatic coupling vector hij (adiabatic representation).
)()( )()( tdetccki
kitti
ikik
where kiriRkrikki td hvv
Time derivative Nonadiabatic coupling vector
Tully, JCP 93, 1061 (1990); Ferretti et al. JCP 104, 5517 (1996)
Surface hopping approach: fewest switchesSurface hopping approach: fewest switchesSurface hopping approach: fewest switchesSurface hopping approach: fewest switches
jk
i
kjjkjkeab hv
Re2
ta
bg
jj
jkjk Tully, JCP 93, 1061 (1990)
´´)(1
ttba
gtt
t jkjj
jk
Hammes-Schiffer and Tully, JCP 101, 4657 (1994)
15 16 17 180.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Pro
ba
bili
ty /
fs
Time (fs)
t = 0.1 fs, ms = 1, int = Butcher Tully Hammer-Schiffer and Tully
Surface hopping approach: semiclassical trajectoriesSurface hopping approach: semiclassical trajectoriesSurface hopping approach: semiclassical trajectoriesSurface hopping approach: semiclassical trajectories
2
2
dt
dME I
IiI
RR • Nuclear motion is obtained by integrating the Newton eqs.
• At each time, the dynamics is performed on one unique adiabatic state, Ei = Hii.
• In the adiabatic representation, Ei(R), Ei, and hji are obtained with traditional quantum chemistry methods.
• aji is obtained by integrating
iki
i
ijij
i
kikjkjkj eaeaa hvhv
• The transition probability gkj between two electronic states is calculated at each time step of the classical trajectory.
• A random event decides whether the system hops to other adiabatic state.
t
E
(Dis)advantages of the on-the-fly approach(Dis)advantages of the on-the-fly approach(Dis)advantages of the on-the-fly approach(Dis)advantages of the on-the-fly approach
Advantages:Advantages:• It is not need to get the complete surface. Only that regions spanned during the dynamics• It dispenses interpolation, extrapolation and fitting schemes
Disadvantages:Disadvantages:• Time-expensive dynamics• No non-local effects (tunneling)
Preparation of surface dynamicsConventionalConventional
dynamicsOn-the-flyOn-the-fly
Total timeTotal time
Practical aspects to be Practical aspects to be addressedaddressed
How to prepare initial conditionsHow to prepare initial conditionsHow to prepare initial conditionsHow to prepare initial conditions
R,P
E
S0
Sn
Microcanonical distributions:• Classical harmonic distribution• R-P uncorrelated quantum harmonic distrib. (Wigner)• R-P correlated quantum harmonic ditrib.
Canonical distribution:• Boltzmann
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 Classical dynamics: integrator
tttt
ttt
ttEM
t
tttt
t
ttttttt
III
IRI
I
III
IIII
)(2
1
2)(
)(1
)(
)(2
1)(
2
)(2
1)()()( 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 Time-step for the classicalclassical equations equationsTime-step for the Time-step for the classicalclassical equations equations
Time-step for the Time-step for the classicalclassical equations equationsTime-step for the Time-step for the classicalclassical equations equations
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 time.
Exceptions: • Dynamics close to the conical intersection may require 0.25 fs• Dissociation processes may require even smaller time steps
TDSE: integratorTDSE: integratorTDSE: integratorTDSE: integrator
• Fourth-order Runge-Kutta (RK4)
• Bulirsh-Stoer
Adaptive works better than Adaptive works better than constant time stepconstant time step
Numerical Recipes in Fortran
Constant time step:Constant time step:
Adaptive time step:Adaptive time step:
TDSE: integratorsTDSE: integratorsTDSE: integratorsTDSE: integrators
Some step-constant integrators available in NEWTON-X: • Polynomial, 3rd order • Runge-Kutta, 4th order • Adams Moulton predictor-corrector, 5th order• Adams Moulton predictor-corrector, 6th order• Unitary propagator• Butcher, 5th order 14 16 18 20 22 24 26 28 30
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7t = 0.5 fs; ms = 1
Runge-Kutta Adams Moulton (6) Propagator Butcher
a00
Time (fs)
h(t) h(t+t)
t/ms
)11( ))(-)((+)(=1 -..mn=ttt+m
nt
m
n-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
i|c
i|2
Time (fs)
ms = 1
t = 0.5 fs
Time-step for the Time-step for the quantumquantum equations equationsTime-step for the Time-step for the quantumquantum equations equations
)()( 2222 ttata
NaN 222
0
)()(
)()(
21
2222
2212
hop
hop
n
Nttata
ttNtNn
Fewest switches: two statesFewest 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
dt
da
dt
da
dt
da
ta
t
tN
nP
hophop
Probability of hopping
0
1
P2→1
Fewest switches: several statesFewest switches: several statesFewest switches: several statesFewest switches: several states
kl klklklklklklkk daHaba ** Re2Im
2
Tully, JCP 93, 1061 (1990)ta
bP
kk
klhoplk
Example: Three states 313233 bba
ta
bPhop
33
3223
Only the fraction of derivative connected to the particular transition
0
1
P3→2
P3→2 +P3→1
´´)(1
ttba
Ptt
t lkkk
hoplk
Hammer-Schiffer and Tully, JCP 101, 4657 (1994)
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 uncertainty to search for a point in which the hop is allowed (Jasper et al. 116 5424 (2002)).
Forbidden hopsForbidden hopsForbidden hopsForbidden hops
Adjustment of momentum after hoppingAdjustment of momentum after hoppingAdjustment of momentum after hoppingAdjustment of momentum after hopping
R
E
KN(t)KN(t+t)
Total energy
After hop, what are the new nuclear velocities?
• Redistribute the energy excess equally among all degrees• Adjust velocities components in the direction of the nonadiabatic coupling vector h12
• Adjust velocities components in the direction of the difference gradient vector g12
• Adjust velocities in the direction cossin klkl ghe
Phase controlPhase controlPhase controlPhase control
0 2 4 6 8 10 12
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
h1
x
Time (fs)
Component h1x Before the phase correction
CNH4+: MRCI/CAS(4,3)/6-31G*
Compare h(t) and h(t+ t)
Phase controlPhase controlPhase controlPhase control
0 2 4 6 8 10 12
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
h1
x
Time (fs)
Component h1x Before the phase correction After the phase correction
CNH4+: MRCI/CAS(4,3)/6-31G*
Compare h(t) and h(t+ t)
Abrupt changes controlAbrupt changes controlAbrupt changes controlAbrupt changes control
11.75 fs
01h
12.00 fs
OrthogonalizationOrthogonalizationOrthogonalizationOrthogonalization
cossin~
sincos~
ijijij
ijijij
hgh
hgg
ijijijij
ijij
gghh
hg
2
2tan
• The routine also gives the linear parameters:
2/1
2222 )(2
)(2
1yxyxyxdE gh
yxgh
F
S1
S0
F
F
S1
S0
F
g-h space orthogonalization [Yarkony, JCP 112, 2111 (2000)]
When surface hopping failsWhen surface hopping failsWhen surface hopping failsWhen surface hopping fails
• SH is supposed to reproduce quantum distributions, in the sense thatfraction of trajectories (t) = adiabatic population (t) Eq. (1)
This statement should be true for:Number of forbidden hops → 0Number of trajectories → infinity
Granucci and Persico have shown that for some cases, even if these conditions are satisfied, Eq. (1) may be not true.
• SH, as any trajectory-independent semiclassical method, cannot account for quantum interference effects and quantization of vibrational and rotational motions.
• It is unclear how good the fewest switches approach in the proximity of conical intersections is.
NNEWTONEWTON-X: a package for Newtonian -X: a package for Newtonian dynamics close to the crossing seamdynamics close to the crossing seam
M. Barbatti, G. Granucci, H. LischkaM. Barbatti, G. Granucci, H. Lischka and M. Ruckenbauer (2005-2006) and M. Ruckenbauer (2005-2006)
NX aimsNX aimsNX aimsNX aims
• Easy and practical of using: just make the inputs and start the simulations; monitor partial results on-the-fly; get relevant summary of results at the end;
• Robust: if the input is right, the job will run: in case of error, messages must guide the user to fix the problem;
• Flexible: some different case to study or new method to implement? It should be easy to change the code;
• Open source: in the future, NX should be opened to the community.
NX input facility:NX input facility: nxinp nxinpNX input facility:NX input facility: nxinp nxinp
------------------------------------------ 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 input facility:NX input facility: nxinp nxinpNX input facility:NX input facility: nxinp nxinp
------------------------------------------ NEWTON-X Newton dynamics close to the crossing seam ------------------------------------------
SET BASIC OPTIONS
nat: Number of atoms. There is no value attributed to nat Enter the value of nat : 6 Setting nat = 6
nstat: Number of states. The current value of nstat is: 2 Enter the new value of nstat : 3 Setting nstat = 3
nstatdyn: Initial state (1 - ground state). The current value of nstatdyn is: 2 Enter the new value of nstatdyn : 2 Setting nstatdyn = 2
prog: Quantum chemistry program and method
0 - ANALYTICAL MODEL 1 - COLUMBUS 2.0 - TURBOMOLE RI-CC2 2.1 - TURBOMOLE TD-DFT The current value of prog is: 1 Enter the new value of prog : 1
NX modular designNX modular designNX modular designNX modular design
R(t), v(t)
t+t, R(t+t), v(t+t/2)
provide:provide:Ek(t+t), hkl(t+t)
v(t+t)
akk, Pkl(t+t)
Initial condition generation
Statistical analysis
Tables and graphics
• Fortran 90 routines• Perl controller
Adiabatic dynamicsAdiabatic dynamicsAdiabatic dynamicsAdiabatic dynamics
R(t), v(t)
t+t, R(t+t), v(t+t/2)
provide:provide:Ek(t+t)
v(t+t)
Methods availableMethods availableMethods availableMethods available
Presently:• COLUMBUS [(non)adiabatic dynamics]
• MCSCF• MRCI
• TURBOMOLE [adiabadic dynamics]• TD-DFT• RI-CC2
• Analytical models [user provided]
Being implemented:• COLUMBUS + TINKER
• QM/MM [(non)adiabatic dynamics]
To be implemented:• ACES II
• EOM-CC [(non)adiabatic dynamics]
R(t), v(t)
t+t, R(t+t), v(t+t/2)
provide:provide:Ek(t+t), hkl(t+t)
v(t+t)
akk, Pkl(t+t)
So many choices… So 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 EOM-CC SR Aces II 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 methodsPresent situation of quantum chemistry methods
Comparison among methodsComparison among methodsComparison among methodsComparison among methods
0 5 10 15 20 25 30
En
erg
y
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
Adiabatic dynamics can be used to find out the most relevant relaxation paths.
But be careful with the limitation of each method (CT states in TD-DFT for example).
CNH4+: MRCI/CAS(4,3)/6-31G*
A basic protocolA 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 RI-CC2 for medium systems (6-10 heavy atoms) under the same conditions as in the previous point.
• 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.
ConclusionsConclusionsConclusionsConclusions
Basis set
Correlation Method
DZ TZ BS limit
HF
CASSCF
MRCI
Full-CI
…
…
ConclusionsConclusionsConclusionsConclusions
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
Next lecture:Next lecture:• Adiabatic and nonadiabatic dynamics methods will be used in the investigation of some examples of photoexcited systems
This lecture:This lecture:• Surface hopping is one of the most popular methods available for nonadiabatic dynamics• Its implementation is direct and it can be used with any quantum chemistry method that can provide analytical excited-state gradients and analytical nonadiabatic coupling vectors• These requirements are fulfilled by only a few methods such as CASSCF, MRCI and (partially) EOM-CC