the spectral method: time-dependent quantum dynamics of fhf - : potential energy surface,...
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The spectral method: time-dependent quantum dynamics of FHF-: Potential Energy Surface, Vibrational Eigenfunctions and Infrared Spectrum.
Guillermo Pérez Hernández
Friedrich Schiller Universität JenaInstitut für Physikalische Chemie
3 December 2007, Jena
2
OUTLINE
- FHF-
- The vibrational problem
- Theoretical approaches- Time dependent approach: the Spectral Method
- Results- Quantum Chemistry- Nuclear Dynamics- Spectra
- Conclusions and outloook
3
FHF-
- Bifluoride Ion:
- dF-F ~ 2.27-2.28 Å
- 20 e (closed shell ground state g.).
- Heavy-light-heavy system ( Zundel cation, CdH2)
Very strong HB ~ 35-40 Kcal / mol
Suitable for high-level ab-initio calculations!
4
The vibrational problem
- Normal modes of vibration around the equilibrium geometry
- Vibrational degrees of freedom: 4.
symmetric stretch
asymmetric stretch
bending (x2)
3
2
1
6
The vibrational problem
- Vibrations: INFRARED (IR) Spectrum
SOLID:B. S. Ault, JPC 83, 837 (1979)
GAS:Kawaguchi et al, JCP 87, 6838 (1987)
3 (as. stretch)
2 (bending)KFH2 in solid Ar matrix
7
vibrational coordinates (q)
What do theoretical approaches consist of?
Theoretical approaches
en
erg
y /
cm
-1
h2
h3
electronic ground state
(q)Ψ E nn ,
(q)Ψ E mm ,
(q)Ψ E kk ,
TIME INDEPENDENT
SCHRÖDINGER EQUATION
potential energy surface (PES)
9
- The many different approaches differ basically in these aspects:
- Quantum Chemistry for calculation of PES.- Methods: (HF, MP2, CISD..., CCSD....)
- Basis sets: (augmented, polarized....)
- Nuclear Dynamics for obtention of eigenfunctions and eigenvalues.- Selection of coordinates: (normal mode, bond-angle, spherical...)
- Representation of space: (on a grid, with analytical functions...)
- Representation of PES: (on a grid, fit to analytical eigenfunctions, force fields...)
- „Technical“ approximations
- Variational procedures mostly involved.
- They are time independent.
Theoretical approaches
10
- The Spectral Method (Feit et al., J. Comp. P, 41, 112 (1982)).
- Obtains time independent information in a time dependent fashion.
- Implies solving the time dependent Schrödinger Equation,
which has the solution:
)tΨ(qeΨ(q,t) ti0,H
Time dependent approach: the spectral method
Ψ(q,t)(q,t)Ψ(q,t)t
i H
initial statestate at time t
economic, flexible !
11
Time dependent approach: Split-Operator
)tΨ(qeΨ(q,t) ti0,H
How to apply the time-evolution operator?
12
- The time propagation provides
time dependent information about
the system: wavefunctions, mean
values:
1. Time dependent approach: the spectral method
time
vib
. coord t2
t3
ti
tf
t1
t0 )Ψ(q,t0
)Ψ(q,t1
)Ψ(q,t2
)Ψ(q,t3
)Ψ(q,ti
)Ψ(q,t f
)( 0tq
)( 1tq
)( 2tq
)( 3tq
)( itq
)( ftq
time wfs exp. values
15
5E
2.Time dependent approach: the spectral method
- The power spectrum displays the eigenenergies .
energy
sp
ectr
um
1E
2E
3E
En}{
- Eigenvalues are known, but they are only "numbers“.
- Still no information about the eigenfunctions
themselves.
- Which eigenvalue corresponds to which eigenfunction?
4E
16
5E
energy
sp
ectr
um
1E
2E
3E
4E
3. Time dependent approach: the spectral method
- Obtaining the eigenfunctions: the filtering procedure.
(q)Ψt)Ψ(qedt ktiE
t
t
k
f
,0
filtering operation
filtered eigenfunctionfilter value
(q)Ψ1
(q)Ψ 2
(q)Ψ3
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)(, nkn
nn EE(q)ΨCq)Ψ(t tiE
nnn
ne(q)ΨCq)Ψ(t , tEEi
nnn
nke(q)ΨCq)Ψ(t )(, (q)Ψq)Ψ(t k,
Time dependent approach: the spectral method
- Obtaining the eigenfunctions: the filtering procedure.
0
0
t
t
dt tiEke
filter value
filtering operation
filtered eigenfunction
18
RESULTS
Quantum Chemistry
19
- Coordinate system
- Molecule is assumed to have zero total angular momentum (J=0).
X
Y
Z
xy
0. Results: Quantum Chemistry
Grid size:
R 64 points
x 64 points
y 64 points
),,(222
2
2
2
,
2
2
2
,
2
22
yxRVRmymxm
HFFHFHF
643 points!R
20
- PES
- Global minimum at x=0, y=0 and R= 2.28 Å (Req exp. = 2.28 Å)
0. Results: Quantum Chemistry (CCSD(T) / aug-cc-pvtz)
R = 2.28 Å
X
Y
R
21
0. Results: Quantum Chemistry (CCSD(T) / aug-cc-pvtz)
R = 1.82 Å R = 3.2 Å
- PES
22
RESULTS
Nuclear Dynamics
23
- Initial wavefunction displaced from global minium (R≠2.28, x,y≠0).
- Total propagation time 32 ps.
- Expectation values for the first
200 fs
q)Ψ(tq)Ψ(t ,,0
1. Results: Nuclear dynamics. Propagation.
- Autocorrelation function for
the first 200 fs.
24
2. Results: Spectrum
resolution ~
propagation time
1
resolution = 2 cm-1
Ψ(q,t))Ψ(q,tdtet
t
iEt0
0
prop. time = 33 ps
25
Results
- Example of a filter operation:
- After relative short integration time, the filtered wavefunction shows its characteristic structure with nodes.
t = 0 fs t = 0 fs t = 0 fs
t = 150 fs t = 150 fs t = 150 fs
26
Movie time
27
3. Results: Eigenfunctions.
- Vibrational ground state and fundamental modes:R
y
x
1
2
3
28
2. Results:Transition frequencies.
- Comparison of the obtained transition frequencies with other available
theoretical and experimental data:
29
IR Spectra: Modulating intensities
Transition probability: permanent dipole moment:x and y
30
IR-Spectra: Intensities
0 0 1 0 0 1
1 0 1
1 0 1
2 0 1
3 0 12 0 1
0 1 0 0 1 0
1 0 1 1 0 1
)(qx
)(qy
31
Conclusions and outlook
- Time dependent approach provides very flexible, economic and reliable method.
- Good agreement with experimental and other theoretical data.
- Outlook:
- Take into account the degenerate bending mode
- Trigger that motion with a laser?
32
Acknowledgments
- Prof. Leticia González
- Dr. Jesús González-Vázquez
- GK #788
... and you!