Femtochemistry: A theoretical overview

1
Femtochemistry A theoretical overview
III Adiabatic approximation and non-adiabatic
corrections
Mario Barbatti mario.barbatti_at_univie.ac.at
This lecture can be downloaded at http//homepage.
univie.ac.at/mario.barbatti/femtochem.html
lecture3.ppt
2
Diabatic x adiabatic
From Greek diabatos to be crossed or passed,
fordable
diabatic with crossing a-diabatic
without crossing non-a-diabatic with
crossing!?
3
Diabatic x adiabatic
In thermodynamics
without exchanging (cross) heat or energy with
environment
4
Diabatic x adiabatic
In quantum mechanics
A physical system remains in its instantaneous
eigenstate if a given perturbation is acting on
it slowly enough and if there is a gap between
the eigenvalue and the rest of the Hamiltonian's
spectrum. Adiabatic theorem (Born and Fock,
1928).
In this example (adiabatic process), the spring
constant k of a harmonic oscillator is slowly
(adiabatically) changed. The system remains in
the ground state, which is adjusted also smoothly
to the new potential shape. Its state is always
an eingenstate of the Hamiltonian at each time
(no crossing).
5
Diabatic x adiabatic
In quantum mechanics
A physical system remains in its instantaneous
eigenstate if a given perturbation is acting on
it slowly enough and if there is a gap between
the eigenvalue and the rest of the Hamiltonian's
spectrum. Adiabatic theorem (Born and Fock,
1928).
E
x
In this example (diabatic process), the spring
constant k of a harmonic oscillator is suddenly
(diabatically) changed. The system remains in the
original state, which is not a eingenstate of the
new Hamiltonian. It is a superposition
(crossing) of several eingenstates of the new
Hamiltonian.
6
Diabatic x adiabatic
In quantum chemistry
The nuclear vibration in a molecule is a slowly
acting perturbation to the electronic
Hamiltonian. Therefore, the electronic system
remains in its instantaneous eigenstate if there
is a gap between the eigenvalue and the rest of
the Hamiltonian's spectrum.
This is another way to say that The electrons
see the nuclei instantaneously frozen
7
Beyond adiabatic approximation I Time independent
Time-independent formulation
8
nuclear wave function
9
(No Transcript)
10
Beyond adiabatic approximation II Time dependent
(adiabatic representation)
Time-dependent formulation
TN Kinetic energy nuclei He potential energy
terms
(adiabatic basis)
which solves
depends on the electronic coordinates r and
parametrically on the nuclear coordinates R.
is a complete basis, any function in the Hilbert
space can be exactly written as a linear
combination of yi.
Since
11
nuclear wave function
Prove it!
12
Non-adiabatic coupling terms
13
Classical limit of the nuclear motion
Adiabatic approximation
Write nuclear wave function in polar form
The phase (action) is the integral of the
Lagrangian
Classical limit
Tully, Faraday Discuss. 110, 407 (1998)
14
Hamilton-Jacobi Equation
To solve the Hamilton-Jacobi equation for the
action is totally equivalente to solve the
Newtons equations for the coordinates!
Newtons equations
In the classical limit, the solutions of the time
dependent Schrödinger equation for the nuclei in
the adiabatic approximation are equivalent to the
solutions of the Newtons equations.
In which cases does this classical limit lose
validity?
15
In which cases does this classical limit lose
validity?
16
Non-adiabatic coupling terms
E1
x2 (a0)
E0
x1 (a0)
17
Beyond adiabatic approximation III Time
dependent (general representation)
18
A very important result
Nuclear kinetic energy operator
19
For example, Surface Hopping
20
(No Transcript)
21
I. Time-independent
II. Time-dependent (adiabatic basis)
III. Time-dependent (general basis)
22
Coupling terms
This term is often neglected in local
approximations
23
non-adiabatic coupling terms
In both cases, the derivatives are in nuclear
coordinates.
24
Present situation of quantum chemistry methods
Methods allowing for excited-state calculations
25
Why are non-adiabatic coupling vectors important?

• They define the limit of validity of the
  adiabatic approximation and of the breakup of the
  Hilbert space into uncoupled subspaces, which is
  important for reducing the dimensionality of the
  problem to be treated.
• The coupling vectors allow to connect the
  Hilbert space from one set of nuclear coordinates
  R to another R DR nearby, which is important
  for the time-dependent formulation of the
  problem.
• The coupling vectors define one of the
  directions of the branching space around the
  conical intersections, which is important for the
  localization of these points of degeneracy.

26
Why are non-adiabatic coupling vectors important?

• They define the limit of validity of the
  adiabatic approximation and of the breakup of the
  Hilbert space into uncoupled subspaces, which is
  important for reducing the dimensionality of the
  problem to be treated.

If h2k 0 (k ? 2), then state 2 can be treated
alone (adiabatically).
27
Why are non-adiabatic coupling vectors important?

• The coupling vectors allow to connect the
  Hilbert space from one set of nuclear coordinates
  R to another R DR nearby, which is important
  for the time-dependent formulation of the problem.

R
R
RDR
t
t
tDt
28
Why are non-adiabatic coupling vectors important?

• The coupling vectors define one of the
  directions of the branching space around the
  conical intersections, which is important for the
  localization of these points of degeneracy.

29
Relation between time derivative and spatial
derivative couplings
0
30
Pittner, Lischka, and Barbatti, Chem. Phys. 356,
147 (2009)
31

• Implemented for MRCI, MCSCF, and TD-DFT
• Computational saving may be one-order higher
• For TD-DFT it is restricted to excited state
  crossings
• It can (in principle) be implemented for any
  method

32
Next lecture

• Non-crossing rule
• Conical intersections

Contact mario.barbatti_at_univie.ac.at This lecture
can be downloaded at http//homepage.univie.ac.at/
mario.barbatti/femtochem.html lecture3.ppt