Download Second Lecture: Towards an implementation of surface hopping

Photochemistry:
adiabatic and nonadiabatic molecular dynamics
with multireference ab initio methods
Mario Barbatti
Institute for Theoretical Chemistry
University of Vienna
COLUMBUS in BANGKOK (3-TS2C2)
Apr. 2 - 5, 2006
Burapha University, Bang Saen, Thailand
Outline
First Lecture: An introduction to molecular dynamics
1. Dynamics, why?
2. Overview of the available approaches
Second Lecture: Towards an implementation of surface hopping dynamics
1. 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 II
Towards an implementation of
surface hopping dynamics
Cândido Portinari, O lavrador de café, 1934
Quick overview of the method
Surface hopping approach: adiabatic population
 (r, R, t )   ck (R, t )ei k (t )k (R; r, t )
i
k
ck   ci (t )e
ik
i   k ( t )  i ( t ) 
d ki (t ) where d ki  k

i
t
r

 H e
t
 v  k  R i
Time derivative
r
 v  h ki
Nonadiabatic coupling vector
Tully, JCP 93, 1061 (1990); Ferretti et al. JCP 104, 5517 (1996)
*
Population: aki  ck ci

akj   akie
i kj
i
v  hij  aij e kj v  h ki

i
• Two electronic states are coupled only by the nonadiabatic coupling vector hij
(adiabatic representation).
Surface hopping approach: fewest switches
gk  j 
b jk
t
a jj
gk  j 
1
a jj

t  t
t
i jk
v  h
jk
Tully, JCP 93, 1061 (1990)
b jk (t´)t´
Hammes-Schiffer and Tully, JCP 101, 4657 (1994)
t = 0.1 fs, ms = 1, int = Butcher
1.4
Tully
Hammer-Schiffer and Tully
1.2
Probability / fs

b jk  2 Re akje
1.0
0.8
0.6
0.4
0.2
0.0
15
16
17
Time (fs)
18
Surface hopping approach: semiclassical trajectories
d 2R I
• Nuclear motion is obtained by integrating the Newton eqs.   R I Ei  M I
dt 2
• 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

akj   akie
i kj
E
i
v  hij  aij e kj v  h ki

i
• 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
(Dis)advantages of the on-the-fly approach
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:
• Time-expensive dynamics
• No non-local effects (tunneling)
Conventional
Preparation of surface
On-the-fly
dynamics
Total time
dynamics
Practical aspects to be
addressed
How to prepare initial conditions
E
Sn
S0
R,P
Microcanonical distributions:
• Classical harmonic distribution
• R-P uncorrelated quantum harmonic distrib. (Wigner)
• R-P correlated quantum harmonic ditrib.
Canonical distribution:
• Boltzmann
Classical dynamics: integrator
Any standard method can be used in the integration of the Newton equations.
A good one is the Velocity Verlet (Swope et al. JCP 76, 637 (1982)):
For each nucleus I
1
R I (t  t )  R I (t )  v I (t )t  a I (t )t 2
2
1
 t 
v I  t    v I (t )  a I (t )t
2
2

1
a I (t )  
 R E R I (t  t )
MI
 t  1
v I (t  t )  v I  t    a I (t  t )t
2 2

Quantum chemistry calculation
Time-step for the classical equations
Schlick, Barth and Mandziuk, Annu. Rev. Biophys. Struct. 26, 181 (1997).
Time-step for the classical 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: integrator
Numerical Recipes in Fortran
Constant time step:
• Fourth-order Runge-Kutta (RK4)
Adaptive time step:
• Bulirsh-Stoer
Adaptive works better than
constant time step
TDSE: integrators
0.7
Some step-constant integrators available in
NEWTON-X:
t = 0.5 fs; ms = 1
Runge-Kutta
Adams Moulton (6)
Propagator
Butcher
0.6
• Polynomial, 3rd order
• Runge-Kutta, 4th order
• Adams Moulton predictor-corrector, 5th order
• Adams Moulton predictor-corrector, 6th order
• Unitary propagator
• Butcher, 5th order
a00
0.5
0.4
0.3
0.2
0.1
0.0
14
16
18
20
22
24
Time (fs)
26
28
30
Time-step for the quantum equations
t/ms
...
h(t)
h(t+t)

 n 
n
h t+t 1-   = h(t ) + (h(t+t ) - h(t ))
ms
 ms  

 |c |
2
i
(n=1..ms -1)
t = 0.5 fs
i
3
ms = 1
2
1.04
1.03
ms = 10
1.02
1.01
ms = 20
1.00
0
10
20
30
40
50
Time (fs)
60
70
80
90
100
Fewest switches: two states
Population in S2: a22 (t )  a22 (t  t )
Trajectories in S2: N2  a22 N
Minimum number of hoppings that keeps the correct number of trajectories:
n2hop
1  N 2 (t )  N 2 (t  t )
 a22 (t )  a22 (t  t ) N
n1hop
2  0
1
Probability of hopping
 t  da11  da11
0

,
hop

n21  a11 (t )  dt 
dt
P2hop



1
N 2 (t ) 
da11
0
,
0

dt
P2→1
0
Fewest switches: several states
2

akk   bkl    Imakl* H kl   2 Reakl* d kl 

l k
l k  
Pkhop
l 
bkl
t
akk
Tully, JCP 93, 1061 (1990)
Pkhop
l 
1 t  t
blk (t´)t´
akk t
Hammer-Schiffer and Tully, JCP 101, 4657 (1994)
1
Example: Three states a33  b32  b31
P3hop
2 
b32
t
a33
P3→2 +P3→1
Only the fraction of derivative
connected to the particular transition
P3→2
0
Forbidden hops
E
Forbidden hop
Total energy
R
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)).
Adjustment of momentum after hopping
E
Total energy
KN(t)
KN(t+t)
R
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
 

e  h kl sin   g kl cos
Phase control
CNH4+: MRCI/CAS(4,3)/6-31G*
Component h1x
Before the phase correction
0.15
0.10
h1x
0.05
0.00
-0.05
-0.10
-0.15
0
2
4
6
Time (fs)
8
10
12
Compare h(t) and h(t+ t)
Phase control
CNH4+: MRCI/CAS(4,3)/6-31G*
Component h1x
Before the phase correction
After the phase correction
0.15
0.10
h1x
0.05
0.00
-0.05
-0.10
-0.15
0
2
4
6
Time (fs)
8
10
12
Compare h(t) and h(t+ t)
Abrupt changes control
h 01
11.75 fs
12.00 fs
Orthogonalization
g-h space orthogonalization [Yarkony, JCP 112, 2111 (2000)]
~
gij  g ij cos   hij sin 
~
hij  g ij sin   hij cos 
tan 2  
2g ij  hij
hij  h ij  g ij  g ij
• The routine also gives the linear parameters:
1/ 2



1 2

gh
2
2
2
E  d gh  x x   y y   ( x  y ) 
(x  y ) 
2
2

 

When surface hopping fails
• SH is supposed to reproduce quantum distributions, in the sense that
fraction of trajectories (t) = adiabatic population (t)
Eq. (1)
This statement should be true for:
Number of forbidden hops → 0
Number 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.
NEWTON-X: a package for Newtonian
dynamics close to the crossing seam
M. Barbatti, G. Granucci, H. Lischka
and M. Ruckenbauer (2005-2006)
NX 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: 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: 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 design
• Fortran 90 routines
• Perl controller
Initial condition generation
R(t), v(t)
t+t, R(t+t), v(t+t/2)
Tables and graphics
provide:
Ek(t+t), hkl(t+t)
akk, Pkl(t+t)
v(t+t)
Statistical analysis
Adiabatic dynamics
R(t), v(t)
t+t, R(t+t), v(t+t/2)
provide:
Ek(t+t)
v(t+t)
Methods available
R(t), v(t)
t+t, R(t+t), v(t+t/2)
provide:
Ek(t+t), hkl(t+t)
akk, Pkl(t+t)
v(t+t)
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]
So many choices…
What method should I use?
Present situation of quantum chemistry methods
Method
MR-CISD
EOM-CC
RI-CC2
CASPT2
MR-MP2
CISD/QCISD
CASSCF
TD-DFT
FOMO/AM1
Single/Multi
Reference
MR
SR
SR
MR
MR
SR
MR
SR
MR
Analytical
gradients









Coupling
vectors









Computational
effort
Typical implementation
Columbus
Aces II
Turbomole
Molcas / Molpro
Gamess
Molpro / Gaussian
Columbus / Molpro
Turbomole
Mopac (Pisa)
Comparison among methods
CNH4+: MRCI/CAS(4,3)/6-31G*
TD-DFT
CASSCF
Energy
RI-CC2
0
5
10
15
Time (fs)
20
25
30 0
5
10
15
Time (fs)
20
25
30 0
5
10
15
20
25
30
Time (fs)
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).
A 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.
Conclusions
Correlation Method
…
Full-CI
MRCI
CASSCF
HF
DZ
TZ …
Basis set
BS limit
Conclusions
Correlation Method
…
Full-CI
MRCI
< 6 heavy atoms
CASSCF
6-10 heavy atoms
HF
DZ
Dynamics method
TZ …
Basis set
BS limit
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
Next lecture:
• Adiabatic and nonadiabatic dynamics methods will be used in the
investigation of some examples of photoexcited systems