Download Quantum Dynamics I - University of Warwick

Introduction
Scattering
Femtochemistry
Potential Energy Surfaces
Adiabatic / Diabatic
Kinetic Energy
Introduction to Quantum Dynamics:
Solving the Time-Dependent Schrödinger Equation
Graham Worth
Dept. of Chemistry, University College London, U.K.
1 / 30
Introduction
Scattering
Femtochemistry
Potential Energy Surfaces
Adiabatic / Diabatic
Kinetic Energy
Dynamical phenomena are described by the
Time-Dependent Schrödinger Equation
i~
∂
Ψ(R, r, t) = ĤΨ(R, r, t)
∂t
A wavepacket evolves in time driven by the Hamiltonian
X
i
Ψ(q, t) =
ci ψi e− ~ Ei t
(1)
(2)
i
where ψi are the eigenfunctions of the Hamiltonian
• D.J. Tannor “Introduction to Quantum Mechanics: A Time-Dependent
Perspective” (2007) University Science Books
http://www.weizmann.ac.il/chemphys/tannor/Book/
• G. C. Schatz and M.A. Ratner “Quantum mechanics in chemistry” (2002) Dover
• P.W. Atkins and R.S. Friedman “Molecular Quantum Mechanics” (2004) Oxford
• K.C. Kulander “Time-dependent methods for quantum dynamics” (1991) Elsevier
2 / 30
Introduction
Scattering
Femtochemistry
Potential Energy Surfaces
Adiabatic / Diabatic
Kinetic Energy
Aim of lectures:
• Introduce Chemical Dynamics
• Molecular Beams (scattering)
• Time-resolved spectroscopy (femtochemistry)
• The Time-dependent Schr"odinger Equation (TDSE)
• Born-Oppenheimer Approximation.
• Adiabatic and Diabatic Pictures
• Techniques used to solve TDSE numerically
• What is possible (bottlenecks / restrictions)
3 / 30
Introduction
Scattering
Femtochemistry
Potential Energy Surfaces
Adiabatic / Diabatic
Kinetic Energy
Molecular Beams and Scattering
Collimated beams of reactants intersect at right angles in high
vacuum (> 10−7 Torr)
Angular Distribution
Crossed
Molecular
Beams
Source B

Velocity
Distribution
Source A
Single collision (if any) occurs in crossing zone.
4 / 30
Introduction
Scattering
Femtochemistry
Potential Energy Surfaces
Adiabatic / Diabatic
Kinetic Energy
Collisions may result in 3 types of scattering:
• Elastic – Translational ∆E
A + BC(ν, J) −→ A + BC(ν, J)
• Inelastic – Rotational / vibrational ∆E
A + BC(ν, J) −→ A + BC(ν 0 , J 0 )
• Reactive – New chemical products
A + BC(ν, J) −→ AB(ν 0 , J 0 )
+C
Must be able to distinguish new products from the background of
elastic / inelastic scattered reactants. Implies sensitive and selective
detector
• Time-of-flight mass spectrometer (TOF)
• “universal detector”
• velocity and product identification
• specific rotational / vibrational states probed by laser
5 / 30
Introduction
Scattering
Femtochemistry
Potential Energy Surfaces
Adiabatic / Diabatic
Kinetic Energy
The Cross-section
b – impact parameter
R, θ – coordinates
Collision cross-section, σc , is
effective target size.
c
Differential cross-section, dσ
dω , is
effective target size as a function Expect a minimum translation energy for reaction
of scattering angle.
Z
2π
Z
π
dθ
σc =
0
dφ
0
dσc
dω
Not every collision results in reaction Reaction cross-section
σr < σc
6 / 30
Introduction
Scattering
Femtochemistry
Potential Energy Surfaces
Adiabatic / Diabatic
Kinetic Energy
Example: F + D2 −→ DF + D
Differential cross section at a relative
energy of 1.82 kcal mol−1 shows probability of DF appearing at angle Θ
and velocities (distance from scattering
centre).
• Contour map inhomogenous:
Preferential orientations.
• Mostly back scattered
⇒ head-on.
• All collisions have same
relative velocities (kinetic
energies). Each reaction
releases same energy,
distributed between
translational and internal
(vib-rot)
• Higher vibration ⇒
slower recoil
Θ = 180◦ initial direction of F beam
7 / 30
Introduction
Scattering
Femtochemistry
Potential Energy Surfaces
Adiabatic / Diabatic
Kinetic Energy
F + H2 Potential Surfaces
Product is hot with populated high vibrational states.
Infrared chemiluminescence results – emission due to excited states
generated in chemical reaction
8 / 30
Introduction
Scattering
Femtochemistry
Potential Energy Surfaces
Adiabatic / Diabatic
Kinetic Energy
H + H2 −→ H2 + H
1
Simplest “Reaction”
Reaction Probability
0.8
2
ν=0→ν=0
0.6
~ω = 0.27eV
0.4
0.2
ν=1
0
0.5
1.5
0.75
1
1.25
1.5
1.75
T =300K
=0;1
[
A2]
Energy [eV]
1
ν=0
0.5
0
0.5
1
1.5
Etrans
2
2.5
Reaction Probability
0.4
0.5
0.3
ν=1→ν=1
0.2
~ω = 0.79eV
0.1
Reaction Cross-section
(probability) for H + D2
0
0.8
1
1.2
1.4
1.6
1.8
2
Energy [eV]
State-to-state cross-sections
H + H2
9 / 30
Introduction
Scattering
Femtochemistry
Potential Energy Surfaces
Adiabatic / Diabatic
Kinetic Energy
Pump-Probe Experiments: Femtochemistry
10 / 30
Introduction
Scattering
Femtochemistry
Potential Energy Surfaces
Adiabatic / Diabatic
Kinetic Energy
Ultrafast molecular vibrations are the fundamental motions that
characterize chemical bonding and determine molecular dynamics at
the molecular level.
Typical periods of motion: Vibrational ∼ 100 fs
Rotational ∼ 100 ps
(1 fs = 10−15 s)
(1 ps = 10−12 s)
Short (femtosecond) laser pulses allow us to “watch” the molecular
motion
Basic scheme:
1. pump laser pulse starts reaction
2. probe laser pulse probes molecules as reaction proceeds
3. Detection of probe signal
11 / 30
Introduction
Scattering
Femtochemistry
Potential Energy Surfaces
Adiabatic / Diabatic
Kinetic Energy
Transient Spectra for NaI dissociation
NaI∗ −→ [Na · · · I]‡∗ −→ Na + I
Pump constant, change probe
• (c) is resonant with Na D-lines
• step-wise escape of Na
• non-resonant same frequency
• trapped portion of
wavepacket
• T = 1.2 ps
12 / 30
Introduction
Scattering
Femtochemistry
Potential Energy Surfaces
Adiabatic / Diabatic
Kinetic Energy
Energetics described by the covalent (NaI) and ionic (Na+ I− ) potential
energy curves which cross at an internuclear distance RC
Non-adiabatic (2 interacting
states).
• In adiabatic picture
curves do not cross
• If system is
adiabatic,
bound-state
• In diabatic picture curves
cross
• If system is diabatic,
dissociation
Which it is depends on coupling between states.
13 / 30
Introduction
Scattering
Femtochemistry
Potential Energy Surfaces
Adiabatic / Diabatic
Kinetic Energy
Time-resolved study - Rhodopsin
• Initial excitation - HOOP
mode
• after 50 fs S1 −→ S2
• energy −→ HT
Kukura et al Science 310: 1006 (2005)
14 / 30
Introduction
Scattering
Femtochemistry
Potential Energy Surfaces
Adiabatic / Diabatic
Kinetic Energy
The Time-Dependent Schrödinger Equation
i~
∂
Ψ(R, r, t) = ĤΨ(R, r, t)
∂t
(3)
If the Hamiltonian is time-independent, formal solution
Ψ(t) = exp −i Ĥt Ψ(0)
Further, if we can write
then
(4)
Phase factor
Ψ(x, t) = Ψi (x)e−iωi t
(5)
'$
∂
Ψ(x, t) = ~ωi Ψi (x)e−iωi t
∂t
&%
by comparison with the TDSE, Ψi are solutions
to the
time-independent Schrödinger equation
i~
(6)
ĤΨi = Ei Ψi = ~ωi Ψi
(7)
15 / 30
Introduction
Scattering
Femtochemistry
Potential Energy Surfaces
Adiabatic / Diabatic
Kinetic Energy
Ψi is a Stationary State as expectation values (properties) are
time-independent
hÔi = hΨi |Ô|Ψi ieiωi t e−iωi t = hΨi |Ô|Ψi i
(8)
If wavefunction is a superposition of stationary states,
X
χ(x, t) =
ci Ψi (x)e−iωi t
(9)
i
now,
hÔi(t) = −i~
XX
i
ci∗ cj hΨi |O|Ψj iei(ωi −ωj )t
(10)
j
An expectation value changes with time and depends on the initial
function (ci coefficients).
A non-stationary wavefunction is called a WAVEPACKET.
16 / 30
Introduction
Scattering
Femtochemistry
Potential Energy Surfaces
Adiabatic / Diabatic
Kinetic Energy
Free Particle
The functions
E
Ψk = eikx e−i ~ t
represent a particle with an exact momentum
p̂Ψk = −i~
d
Ψk = k ~Ψk
dx
But, particle is not localised. Take a superposition
Z ∞
χ(x, t) =
dk C(k )Ψk (x, t)
−∞
where C(k ) is a suitable function
17 / 30
Introduction
Scattering
Femtochemistry
Potential Energy Surfaces
Adiabatic / Diabatic
Kinetic Energy
E.g. Form a Gaussian wavepacket
2
−a (k − k0 )2
C(k ) = N exp
2
x − x0 (t)x0 (t)
iγ
χ(x, t) = N0 e exp −
+ ik0 x
2a2 δ
where
x0 (t) =
~k0 t
m
so wavepacket moves to right with velocity
~k0
m .
The functions Ψk form a basis sutiable to describe free motion.
18 / 30
Introduction
Scattering
Femtochemistry
Potential Energy Surfaces
Adiabatic / Diabatic
Kinetic Energy
Further, width of density, < x 2 > − < x >2 , is
2 2
∆(t) = a (ln 2) 1 +
~ t
m 2 a4
21
and as time increases. packet spreads out.
k0 ~
t0
k0 ~
t0 + ∆t
19 / 30
Introduction
Scattering
Femtochemistry
Potential Energy Surfaces
Adiabatic / Diabatic
Kinetic Energy
Bound Motion
Ĥ = −
E
Ψ0
Ψ1
Ψ2
5
~ω
2
3
~ω
2
1
~ω
2
~2 ∂ 2
+ 12 mω 2 x 2
2
2m ∂x
1 mω 2
2
= N0 e − 2 ~ x
r
mω 2 − 1 mω2 x 2
= N1
xe 2 ~
~
1 mω 2 2
mω 2 2
= N2 4
x − 2 e− 2 ~ x
~
The functions Ψk form a basis sutiable to describe bound motion.
X
χ(x, t) =
ci (t)Ψi (x, t)
i
20 / 30
Introduction
Scattering
Femtochemistry
Potential Energy Surfaces
Adiabatic / Diabatic
Kinetic Energy
The Born-Oppenheimer Approximation
Start using Born representation
X
Ψ(q, r) =
χi (q)Φi (r; q) ,
(11)
i
where electronic functions are solutions to clamped nucleus
Hamiltonian
Ĥel Φi (r; q) = Vi (R)Φi (r; q) .
(12)
The full Hamiltonian is
Ĥ(q, r) = T̂n (q) + Ĥel (q, r) ,
(13)
Integrate out electronic degrees of freedom to obtain
1
∂χ
−
(∇1 + F)2 + V χ = i~
,
2M
∂t
(14)
21 / 30
Introduction
Scattering
Femtochemistry
Potential Energy Surfaces
Adiabatic / Diabatic
Kinetic Energy
The Adiabatic Picture
where
Fij = hΦi | ∇Φj i
(15)
is the derivative coupling vector
Assuming
F
M
≈0
∂χ
(16)
∂t
and nuclei move over a single adiabatic potential energy surface, V ,
which can be obtained from quantum chemistry calculations.
h
i
T̂n + V χ = i~
Unfortunately,
Fij =
hΦi | ∇Ĥel | Φj i
Vj − Vi
for i 6= j
.
(17)
22 / 30
Introduction
Scattering
Femtochemistry
Potential Energy Surfaces
Adiabatic / Diabatic
Kinetic Energy
The Diabatic Picture
First we separate out a group of coupled states from the rest
h
i
∂χ(g)
(g)
(g) 2
(g)
(T̂n 1 + F ) + V
χ(g) = i~
∂t
,
(18)
To remove singularities, find a suitable unitary transformation
Φ̃ = S(q)Φ
(19)
such that the Hamiltonian can be written
[TN 1 + W] χ = i~
∂χ
∂t
(20)
,
where all elements of W are potential-like terms
Worth and Cederbaum Ann. Rev. Phys. Chem. (2004) 55: 127
23 / 30
Introduction
Scattering
Femtochemistry
Potential Energy Surfaces
Adiabatic / Diabatic
Kinetic Energy
• Result 1: Electronic motion contained in potential energy
surfaces which can be calculated using quantum chemistry
• Problem 1: Potential surfaces are calculated in the adiabatic
picture. Dynamics run in the diabatic picture
Solution is to diabatise adiabatic surfaces for the dynamics.
Non-trivial.
24 / 30
Introduction
Scattering
Femtochemistry
Potential Energy Surfaces
Adiabatic / Diabatic
Kinetic Energy
Conical Intersections
H
Butatriene Radical Cation
H
C
C
C
C
H
H
CoIn
11
V [eV]
10.5
Diabatic
FC
10
•
9.5
9
•
•
Xmin
TS
Amin
•
8.5
•
11
90
-2
-1
0
-30
1
2
Q14
3
30
0 θ (deg)
-60
4
-90
10.5
V [eV]
60
10
9.5
9
8.5
Adiabatic
-90
-60
-2
-30
-1
0
0
1
Q14
2
30
3
4
θ
60
90
25 / 30
Introduction
Scattering
Femtochemistry
Potential Energy Surfaces
Adiabatic / Diabatic
Kinetic Energy
Coordinates: The Kinetic Energy Operator
In Cartesian coordinates,
3
1 X ∂2
T =
−
2
2mi
∂xiα
i=1
α=1
N
X
(21)
This includes COM and ROT - continua. To remove these
contributions use, e.g. Jacobi coordinates
T
C
r
θ
Q
Q
Q
Q
Q
Q
B
Q
R QQQQ
Q
QQ
Sukiasyan and
JCP (02) : 116
A
Meyer
1
∂2
1 ∂2
= −
−
2µR R 2 ∂R 2
2µr r 2 ∂r 2
1
1
+(
+
)j 2
2
2
2µR R
2µr r
1
−
(J(J + 1) − 2K 2 )
2
2µR R
1 p
−
(J(J + 1) − K (K ± 1)j±
2µR R 2
(22)
26 / 30
Introduction
Scattering
Femtochemistry
Potential Energy Surfaces
Adiabatic / Diabatic
Kinetic Energy
6 Dimensional Jacobi Coordinates
2T̂
3
X
1
1
1 ∂2
~L†~L1 )BF
= −
R
+
(
+
)(
i
1
2
2
µi Ri ∂Ri2
µ
R
µ
R
1
3
1
3
i=1
+(
1
1
+
)(~L†2~L2 )BF
2
2
µ2 R2
µ3 R3
(~J 2 − 2~J(~L1 + ~L2 ) + 2~L1~L2 )BF
+
µ3 R32
.
(23)
Gatti et al JCP (05) 123: 174311
Other coordinates: Hyperspherical, Radau, ....
27 / 30
Introduction
Scattering
Femtochemistry
Potential Energy Surfaces
Adiabatic / Diabatic
Kinetic Energy
Normal modes
Final example, choose rectilinear coordinates so that force constant
matrix (Hessian) is diagonal,
Wij =
∂2V
∂xi ∂xj
(24)
then expanding around the minimum on the potential surface
V =
3N−6
X
i=1
ωi 2
Q + O(3)
2 i
(25)
COM and ROT removed and
T =
3N−6
X
i=1
ωi ∂ 2
−
2 ∂Qi2
(26)
Very simple, but PES only suitable for small displacements.
Wilson, Cross and Decius “Molecular Vibrations” (1980) Dover
28 / 30
Introduction
Scattering
Femtochemistry
Potential Energy Surfaces
Adiabatic / Diabatic
Kinetic Energy
• Result 2: Can select coordinates so that COM (and some ROT)
motion removed and KEO has a simple form.
• Problem 2: In general, simple KEO coordinates are not optimal
for PES representation.
In general, simple KEO coordinates are not optimal for PES
representation and vice versa
29 / 30
Introduction
Scattering
Femtochemistry
Potential Energy Surfaces
Adiabatic / Diabatic
Kinetic Energy
Summary
• Chemical physics is study of molecular interactions and resulting
dynamics
• Molecular beam scattering experiments provide details of
interactions on ground-state
• Cross-section relates to probability of process, e.g. reaction,
occuring
• Femtochemistry experiments probe dynamics on excited surface
• pump-probe experiments create and watch wavepacket
• Initialisation of a reaction creates a wavepacket, a solution of the
TDSE
• Starting point to solving the TDSE is the Born-Oppenheimer
Approximation
• Nuclear / electronic coupling leads to breakdown of BO
• Adiabatic and Diabatic Pictures
• To solve TDSE need Ĥ: PES + KEO
30 / 30