Emil J Zak Department of Physics and Astronomy University College London 10 London UK June 20 2017 quotSince in practice we normally cannot solve the full electronnuclear problem ID: 798187
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Slide1
The molecular geometric phase and light-induced conical intersections
Emil J. Zak
Department of Physics and AstronomyUniversity College London, London, UK
June 20,
2017
Slide2"Since, in practice, we normally cannot solve the full electron-nuclear problem,
it will be extremely important to find simple mathematical criteria by which one can know (without doing the full calculation) if a nonvanishing
BO-Berry phase survives in the exact electron-nuclear treatment. This interesting—but difficult—question will be the subject of future research."
Slide3Questions
1. Do natural conical intersections
(NCI) and light-induced conical intersections (LICI) have identical nature?2. What is the role of the vibronic coupling in the mechanism of generating LICIs?
3. Is the geometric phase an artifact of the Born-Oppenheimer approximation?
4. What are the possible applications of geometric phase effects near LICIs?
Slide4Light-induced conical intersection
NaH:
Born-Oppenheimer, diabatic Hamiltonian
Slide5Light-induced conical intersection
NaH:
AC electric field:
Slide6Light-induced conical intersection
NaH:
AC electric field:
Molecule-field Interaction energy:
Slide7Light-induced conical intersection
NaH:
Molecule-field Interaction energy:
Dynamical variables:
Slide8Light-induced conical intersection
Moving into adiabatic picture
Slide9Light-induced conical intersection
Moving into adiabatic picture
Slide10LICI vs laser frequency
- chirp
Slide11LICI vs laser intensity
- chirp
Slide12Effect of vibronic coupling on the LICI
Slide13Effect of vibronic coupling on the LICI
The vibronic coupling breaks the symmetry generated by the direction of the AC electric field
The effect of the vibronic coupling is expected to be subtle, yet measurable
Conditions for LICI
Slide14Questions
1. Do natural conical intersections
(NCI) and light-induced conical intersections (LICI) have identical nature?2. What is the role of the vibronic coupling in the mechanism of generating LICIs?
3. Is the geometric phase an artifact of the Born-Oppenheimer approximation?
4. What are the possible applications of geometric phase effects near LICIs?
Slide15Questions
1. Do natural conical intersections
(NCI) and light-induced conical intersections (LICI) have identical nature?2. What is the role of the vibronic coupling in the mechanism of generating LICIs?3. Is the geometric phase an artifact of the Born-Oppenheimer approximation?
4. What are the possible applications of geometric phase effects near LICIs?
Slide16Natural Conical Intersection
Slide17Are LICIs and NCIs 'identical in nature‘?
Non-adiabatic coupling terms (NACT)
Diagonal Born-Oppenheimer correction (DBOC)
Vector potential
Real and double valued
Adiabatic ↔ Diabatic
Slide18Discontinuous mixing angle
NCI and LICI have such discontinuity
Slide19Derivative couplings
The Berry phase
A tool for identifying conical intersections
Light-induced non-adiabatic coupling
Slide20Derivative couplings
Slide21Derivative couplings
a
Slide22Derivative couplings
and the geometric phase
a
Slide23Questions
1. Do natural conical intersections
(NCI) and light-induced conical intersections (LICI) have identical nature?2. What is the role of the vibronic coupling in the mechanism of generating LICIs?3. Is the geometric phase an artifact of the Born-Oppenheimer approximation?
4. What are the possible applications of geometric phase effects near LICIs?
Slide24G. J Halász et al 2011 J. Phys. B: At. Mol. Opt. Phys. 44 175102
G. J Halász et al, J. Phys. Chem. Lett. 2015, 6, 348−354
LICI and NCI are 'identical in nature'
Slide25Questions
1. Do natural conical intersections
(NCI) and light-induced conical intersections (LICI) have identical nature?2. What is the role of the vibronic coupling in the mechanism of generating LICIs?3. Is the geometric phase an artifact of the Born-Oppenheimer approximation?
4. What are the possible applications of geometric phase effects near LICIs?
Slide26Questions
1. Do natural conical intersections (NCI) and light-induced conical intersections (LICI) have identical nature?
2. What is the role of the vibronic coupling in the mechanism of generating LICIs?3. Is the geometric phase an artifact of the Born-Oppenheimer approximation?4. What are the possible applications of geometric phase effects near LICIs?
Slide27The geometric phase as a tool for detection of LICIs and NCIs
Figure. The geometric phase as
a function of the center of integration for a circular loop with radius equal to 1.0
Slide28Does the Berry Phase survive in the LVC model?
Slide29Does the Berry Phase survive in the LVC model?
YES
Slide30Control of non-adiabatic effects
with strong fields?
- chirp
and c - chirp
Slide31Nuclear dynamics: coming soon...
Slide32Two CIs – quadratic vibronic coupling
Quadratic vibronic coupling does not change the qualitative picture
Slide33Does the Berry Phase survive in the exact picture (full non-adiabatic coupling)?
Type of Conical intersection
Berry phaseBorn-Oppenheimer: natural CIpBorn-Oppenheimer: Natural CI +
Light-induced potentialp
Born-Oppenheimer: Light-induced
conical intersection
p
Linear vibronic coupling: Light-induced conical intersection
p
diabatic
0
Slide34Non-adiabatic coupling term in the adiabatic representation is singular at the conical intersection:
Resolvent diabatisation
Slide35Non-adiabatic coupling term in the adiabatic representation can we rewritten in the following form:
Resolvent expansion method – adapted from Quantum Field Theory
Resolvent diabatisation
Slide36Non-adiabatic coupling term in the adiabatic representation can we rewritten in the following form:
Resolvent expansion method – adapted from Quantum Field Theory
Resolvent diabatisation
...
Slide37A trivial example:
Resolvent diabatisation
DBOC
Singular at the conical intersection
Slide38Singularities problem
DBOC
Second order expansion
Slide39Singularities problem
DBOC
Third order expansion
Slide40Singularities problem
DBOC
Increasing expansion order
Slide41Singularities problem
DBOC
Increasing expansion order
The berry phase is zero in any finite order of approximation.
Slide42Start with the exact factorization and expand in terms of adiabatic wavefunctions
Find a quasi-diabatic representation for which
Use resolvent expansion for a pair of representations:
Not all hope is lost...
Slide43Type of Conical intersectionBerry phase
Born-Oppenheimer: natural CIp
Born-Oppenheimer: Natural CI + Light-induced potentialpBorn-Oppenheimer: Light-induced
conical intersection
p
Linear vibronic coupling: Light-induced conical intersection
p
diabatic
0
Resolvent (trivial expansion)
0 for any finite order
Exact factorisation
p?!
Exact
?
The Berry Phase in different approaches
Slide44Non-adiabatic Berry Phase
The Berry Phase around a CI is non-zero in the Born-Oppenheimer approximationThe Berry Phase around a CI is
non-zero when
some non-adiabatic couplings
are included
The Berry Phase around a CI is
zero
in the
Diabatic representation
,
which is unitarily connected with the adiabatic representation
The Berry Phase is an artifact of the adiabatic representation.
But does it mean it is not necessary?
Kendrick et. al, Nature comm., 6:7918 (2015)
Natan, et. all, PRL 116, 143004 (2016)
A.Izmaylov et. al, J. Chem. Theory Comput., 2016, 12 (11), pp 5278–5283
Dynamics of nuclei sometimes strongly depends on geometric phase effects
Slide45Exact factorization
Theorem 1. The exact solution of Schrodinger equation can be written as a single product
Where Satisfies the partial normalization condition
For any
A. Abedi, N. T. Maitra and E. K. U. Gross, J. Chem. Phys., 137, 22A530, (2012).
A. Abedi, F. Agostini, Y. Suzuki and E. K. U. Gross, Phys. Rev. Lett., 110, 263001, (2013)
Slide46Fortunately we have an expression for the
exact geometric phase:
Gauge invariant difference between the ‘total probability current density’ and the ‘nuclear probability current density’:
Non-adiabatic Berry Phase
Gauge invariant
Slide47Born-Huang expansion
Looking for terms with non-zero curl.Take a B-O electronic state. Expand it in some complete basis
For two electronic states:
Slide48Outlook
Polyatomics: Stark induced CI‘s
Optical-Topological Cooling (Stark decelerator)
Slide49Berry phase
M. Berry: what about closed loops in parameter space?
Additional phase, independent on basis used: geometric property of the parameters space
Slide50Berry phase
M. Berry: what about closed loops in parameter space?
Additional phase, independent on basis used: geometric property of the parameters space
Berry connection
Slide51Exact factorization
Theorem. The exact solution of Schrodinger equation can be written as a single product
Where Satisfies the partial normalization condition
For any
Slide52Exact factorization
Theorem 1. The exact solution of Schrodinger equation can be written as a single product
General form:
Slide53Exact factorization
Theorem 2. The wavefunctions from theorem 1 satisfy following equations
Slide54Exact factorization
Theorem 2. The wavefunctions from theorem 1 satisfy following equations
Slide55Exact factorization
Theorem 2. The wavefunctions from theorem 1 satisfy following equations
Generalized, exact PESElectron-nuclear coupling
Slide56Gauge invariance
Can be gauged away
Gauge invariant
Slide57Gauge invariance
Gauge invariant
Gauge invariantCurrent densityUsing Stokes theorem (in 3D)
Slide58Na_3 molecule – theorerical model
Independently of the model, eigenfunctions take the following form:
Double-valued wavefunction!But...no spin involved. What's going on?
Recall MTPC course, spin and 'belt experiment'.
Total wavefunction must be single valued.
Slide59Na_3 molecule - experiments
Simple model:
Slide60Na_3 molecule - experiments
In more familiar form:
In linear approximation (compare to Thursday's class result)
Slide61Na_3 molecule – theorerical model
Independently of the model, eigenfunctions take the following form:
Theta depends on vibrational coordinates!Linear model
quadratic model
Slide62Na_3 molecule – theorerical model
Independently of the model, eigenfunctions take the following form:
Double-valued wavefunction!But...no spin involved. What's going on?
Recall MTPC course, spin and 'belt experiment'.
Slide63Na_3 molecule – theorerical model
Independently of the model, eigenfunctions take the following form:
Double-valued wavefunction!But...no spin involved. What's going on?
Recall MTPC course, spin and 'belt experiment'.
Total wavefunction must be single valued.
Slide64Na_3 molecule – Berry phase
Real & double valued
complex & single valued
Slide65Na_3 molecule – Berry phase
Real & double valued
complex & single valued
Slide66Na_3 molecule – Berry phase
Real & double valued
complex & single valued
Not allowed to put in Berry Phase formula
allowed to put in Berry Phase formula
Slide67Na_3 molecule – Berry phase
1-vector
1-form
Slide68Na_3 molecule – Berry phase
From adiabatic theorem:
If a hamiltonian (system) is dependen on a set of n slowly varying* parameters, then if the system is initially in its I-th eigenstate, after time t, it will remain in the same state. The only thing that can change is the phase of the wavefunction:
Set of molecular parameters:
Slide69Na_3 molecule – Berry phase
Berry: what about closed loops in parameter space?
Additional phase, independent on basis used: geometric property of parameters space
Slide70Berry phase
Any 2x2 hermitian matrix can be represented in the form:
Solid angle defined by closed path C