In this note, we derive (to third order in derivatives of the fluid velocity) a 2+1
dimensional theory of fluid dynamics that governs the evolution of generic long-
wavelength perturbations of a black brane or large black hole in four-dimensional
gravity with negative cosmological constant, applying a systematic procedure de-
veloped recently by Bhattacharyya, Hubeny, Minwalla, and Rangamani. In the
regime of validity of the fluid-dynamical description, the black-brane evolution
will generically correspond to a turbulent flow. Turbulence in 2+1 dimensions
has been well studied analytically, numerically, experimentally, and observation-
ally as it provides a first approximation to the large scale dynamics of planetary
atmospheres. These studies reveal dramatic differences between fluid flows in
2+1 and 3+1 dimensions, suggesting that the dynamics of perturbed four and
five dimensional large AdS black holes may be qualitatively different. However,
further investigation is required to understand whether these qualitative differ-
ences exist in the regime of fluid dynamics relevant to black hole dynamics.
Aurelian Isar - Decoherence And Transition From Quantum To Classical In Open ...SEENET-MTP
Lecture by Prof. Dr. Aurelian Isar (Department of Theoretical Physics, National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, Romania) on October 20, 2011 at the Faculty of Science and Mathematics, Nis, Serbia.
PROGRAMMA ATTIVITA’ DIDATTICA A.A. 2016/17
DOTTORATO DI RICERCA IN INGEGNERIA STRUTTURALE E GEOTECNICA
____________________________________________________________
STOCHASTIC DYNAMICS AND MONTE CARLO SIMULATION IN EARTHQUAKE ENGINEERING APPLICATIONS
Lecture Series by
Agathoklis Giaralis, Ph.D., M.ASCE., P.E. City, University of London
Visiting Professor Sapienza University of Rome
My introduction to electron correlation is based on multideterminant methods. I introduce the electron-electron cusp condition, configuration interaction, complete active space self consistent field (CASSCF), and just a little information about perturbation theories. These slides were part of a workshop I organized in 2014 at the University of Pittsburgh and for a guest lecture in a Chemical Engineering course at Pitt.
The Einstein field equation in terms of the Schrödinger equationVasil Penchev
The Einstein field equation (EFE) can be directly linked to the Schrödinger equation (SE) by meditation of the quantity of quantum information and its units: qubits
•
One qubit is an “atom” both of Hilbert space and Minkovski space underlying correspondingly quantum mechanics and special relativity
•
Pseudo-Riemannian space of general relativity being “deformed” Minkowski space therefore consists of “deformed” qubits directly referring to the eventual “deformation” of Hilbert space
Lecture by prof. dr Neven Bilic from the Ruđer Bošković Institute (Zagreb, Croatia) at the Faculty of Science and Mathematics (Niš, Serbia) on October 29, 2014.
The visit took place in the frame of the ICTP – SEENET-MTP project PRJ-09 “Cosmology and Strings”.
Understanding the experimental and mathematical derivation of Heisenberg's Uncertainty Principle. Simple application for estimating single degree of freedom particle in a potential free environment is also discussed.
In this note, we derive (to third order in derivatives of the fluid velocity) a 2+1
dimensional theory of fluid dynamics that governs the evolution of generic long-
wavelength perturbations of a black brane or large black hole in four-dimensional
gravity with negative cosmological constant, applying a systematic procedure de-
veloped recently by Bhattacharyya, Hubeny, Minwalla, and Rangamani. In the
regime of validity of the fluid-dynamical description, the black-brane evolution
will generically correspond to a turbulent flow. Turbulence in 2+1 dimensions
has been well studied analytically, numerically, experimentally, and observation-
ally as it provides a first approximation to the large scale dynamics of planetary
atmospheres. These studies reveal dramatic differences between fluid flows in
2+1 and 3+1 dimensions, suggesting that the dynamics of perturbed four and
five dimensional large AdS black holes may be qualitatively different. However,
further investigation is required to understand whether these qualitative differ-
ences exist in the regime of fluid dynamics relevant to black hole dynamics.
Aurelian Isar - Decoherence And Transition From Quantum To Classical In Open ...SEENET-MTP
Lecture by Prof. Dr. Aurelian Isar (Department of Theoretical Physics, National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, Romania) on October 20, 2011 at the Faculty of Science and Mathematics, Nis, Serbia.
PROGRAMMA ATTIVITA’ DIDATTICA A.A. 2016/17
DOTTORATO DI RICERCA IN INGEGNERIA STRUTTURALE E GEOTECNICA
____________________________________________________________
STOCHASTIC DYNAMICS AND MONTE CARLO SIMULATION IN EARTHQUAKE ENGINEERING APPLICATIONS
Lecture Series by
Agathoklis Giaralis, Ph.D., M.ASCE., P.E. City, University of London
Visiting Professor Sapienza University of Rome
My introduction to electron correlation is based on multideterminant methods. I introduce the electron-electron cusp condition, configuration interaction, complete active space self consistent field (CASSCF), and just a little information about perturbation theories. These slides were part of a workshop I organized in 2014 at the University of Pittsburgh and for a guest lecture in a Chemical Engineering course at Pitt.
The Einstein field equation in terms of the Schrödinger equationVasil Penchev
The Einstein field equation (EFE) can be directly linked to the Schrödinger equation (SE) by meditation of the quantity of quantum information and its units: qubits
•
One qubit is an “atom” both of Hilbert space and Minkovski space underlying correspondingly quantum mechanics and special relativity
•
Pseudo-Riemannian space of general relativity being “deformed” Minkowski space therefore consists of “deformed” qubits directly referring to the eventual “deformation” of Hilbert space
Lecture by prof. dr Neven Bilic from the Ruđer Bošković Institute (Zagreb, Croatia) at the Faculty of Science and Mathematics (Niš, Serbia) on October 29, 2014.
The visit took place in the frame of the ICTP – SEENET-MTP project PRJ-09 “Cosmology and Strings”.
Understanding the experimental and mathematical derivation of Heisenberg's Uncertainty Principle. Simple application for estimating single degree of freedom particle in a potential free environment is also discussed.
Neutral Electronic Excitations: a Many-body approach to the optical absorptio...Claudio Attaccalite
Neutral Electronic Excitations: a Many-body approach to the optical absorption spectra.
Introduction to Bethe-Salpeter equation and linear response theory.
In this second lecture, I will discuss how to calculate polarization in terms of Berry phase, how to include GW correction in the real-time dynamics and electron-hole interaction.
UCSD NANO 266 Quantum Mechanical Modelling of Materials and Nanostructures is a graduate class that provides students with a highly practical introduction to the application of first principles quantum mechanical simulations to model, understand and predict the properties of materials and nano-structures. The syllabus includes: a brief introduction to quantum mechanics and the Hartree-Fock and density functional theory (DFT) formulations; practical simulation considerations such as convergence, selection of the appropriate functional and parameters; interpretation of the results from simulations, including the limits of accuracy of each method. Several lab sessions provide students with hands-on experience in the conduct of simulations. A key aspect of the course is in the use of programming to facilitate calculations and analysis.
In this talk I will present real-time spectroscopy and different code to perform this kind of calculations.
This presentation can be download here:
http://www.attaccalite.com/wp-content/uploads/2022/03/RealTime_Lausanne_2022.odp
Exact Sum Rules for Vector Channel at Finite Temperature and its Applications...Daisuke Satow
Slides used in presentation at:
“International School of Nuclear Physics 38th Course Nuclear matter under extreme conditions -Relativistic heavy-ion collisions”, in September, 2016 @ Erice, Italy
Dumitru Vulcanov - Numerical simulations with Ricci flow, an overview and cos...SEENET-MTP
Lecture by prof. dr Dumitru Vulcanov (dean of the Faculty of Physics, West University of Timisoara, Romania) on October 21, 2010 at the Faculty of Science and Mathematics, Nis, Serbia.
(May 29th, 2024) Advancements in Intravital Microscopy- Insights for Preclini...Scintica Instrumentation
Intravital microscopy (IVM) is a powerful tool utilized to study cellular behavior over time and space in vivo. Much of our understanding of cell biology has been accomplished using various in vitro and ex vivo methods; however, these studies do not necessarily reflect the natural dynamics of biological processes. Unlike traditional cell culture or fixed tissue imaging, IVM allows for the ultra-fast high-resolution imaging of cellular processes over time and space and were studied in its natural environment. Real-time visualization of biological processes in the context of an intact organism helps maintain physiological relevance and provide insights into the progression of disease, response to treatments or developmental processes.
In this webinar we give an overview of advanced applications of the IVM system in preclinical research. IVIM technology is a provider of all-in-one intravital microscopy systems and solutions optimized for in vivo imaging of live animal models at sub-micron resolution. The system’s unique features and user-friendly software enables researchers to probe fast dynamic biological processes such as immune cell tracking, cell-cell interaction as well as vascularization and tumor metastasis with exceptional detail. This webinar will also give an overview of IVM being utilized in drug development, offering a view into the intricate interaction between drugs/nanoparticles and tissues in vivo and allows for the evaluation of therapeutic intervention in a variety of tissues and organs. This interdisciplinary collaboration continues to drive the advancements of novel therapeutic strategies.
The increased availability of biomedical data, particularly in the public domain, offers the opportunity to better understand human health and to develop effective therapeutics for a wide range of unmet medical needs. However, data scientists remain stymied by the fact that data remain hard to find and to productively reuse because data and their metadata i) are wholly inaccessible, ii) are in non-standard or incompatible representations, iii) do not conform to community standards, and iv) have unclear or highly restricted terms and conditions that preclude legitimate reuse. These limitations require a rethink on data can be made machine and AI-ready - the key motivation behind the FAIR Guiding Principles. Concurrently, while recent efforts have explored the use of deep learning to fuse disparate data into predictive models for a wide range of biomedical applications, these models often fail even when the correct answer is already known, and fail to explain individual predictions in terms that data scientists can appreciate. These limitations suggest that new methods to produce practical artificial intelligence are still needed.
In this talk, I will discuss our work in (1) building an integrative knowledge infrastructure to prepare FAIR and "AI-ready" data and services along with (2) neurosymbolic AI methods to improve the quality of predictions and to generate plausible explanations. Attention is given to standards, platforms, and methods to wrangle knowledge into simple, but effective semantic and latent representations, and to make these available into standards-compliant and discoverable interfaces that can be used in model building, validation, and explanation. Our work, and those of others in the field, creates a baseline for building trustworthy and easy to deploy AI models in biomedicine.
Bio
Dr. Michel Dumontier is the Distinguished Professor of Data Science at Maastricht University, founder and executive director of the Institute of Data Science, and co-founder of the FAIR (Findable, Accessible, Interoperable and Reusable) data principles. His research explores socio-technological approaches for responsible discovery science, which includes collaborative multi-modal knowledge graphs, privacy-preserving distributed data mining, and AI methods for drug discovery and personalized medicine. His work is supported through the Dutch National Research Agenda, the Netherlands Organisation for Scientific Research, Horizon Europe, the European Open Science Cloud, the US National Institutes of Health, and a Marie-Curie Innovative Training Network. He is the editor-in-chief for the journal Data Science and is internationally recognized for his contributions in bioinformatics, biomedical informatics, and semantic technologies including ontologies and linked data.
Professional air quality monitoring systems provide immediate, on-site data for analysis, compliance, and decision-making.
Monitor common gases, weather parameters, particulates.
Introduction:
RNA interference (RNAi) or Post-Transcriptional Gene Silencing (PTGS) is an important biological process for modulating eukaryotic gene expression.
It is highly conserved process of posttranscriptional gene silencing by which double stranded RNA (dsRNA) causes sequence-specific degradation of mRNA sequences.
dsRNA-induced gene silencing (RNAi) is reported in a wide range of eukaryotes ranging from worms, insects, mammals and plants.
This process mediates resistance to both endogenous parasitic and exogenous pathogenic nucleic acids, and regulates the expression of protein-coding genes.
What are small ncRNAs?
micro RNA (miRNA)
short interfering RNA (siRNA)
Properties of small non-coding RNA:
Involved in silencing mRNA transcripts.
Called “small” because they are usually only about 21-24 nucleotides long.
Synthesized by first cutting up longer precursor sequences (like the 61nt one that Lee discovered).
Silence an mRNA by base pairing with some sequence on the mRNA.
Discovery of siRNA?
The first small RNA:
In 1993 Rosalind Lee (Victor Ambros lab) was studying a non- coding gene in C. elegans, lin-4, that was involved in silencing of another gene, lin-14, at the appropriate time in the
development of the worm C. elegans.
Two small transcripts of lin-4 (22nt and 61nt) were found to be complementary to a sequence in the 3' UTR of lin-14.
Because lin-4 encoded no protein, she deduced that it must be these transcripts that are causing the silencing by RNA-RNA interactions.
Types of RNAi ( non coding RNA)
MiRNA
Length (23-25 nt)
Trans acting
Binds with target MRNA in mismatch
Translation inhibition
Si RNA
Length 21 nt.
Cis acting
Bind with target Mrna in perfect complementary sequence
Piwi-RNA
Length ; 25 to 36 nt.
Expressed in Germ Cells
Regulates trnasposomes activity
MECHANISM OF RNAI:
First the double-stranded RNA teams up with a protein complex named Dicer, which cuts the long RNA into short pieces.
Then another protein complex called RISC (RNA-induced silencing complex) discards one of the two RNA strands.
The RISC-docked, single-stranded RNA then pairs with the homologous mRNA and destroys it.
THE RISC COMPLEX:
RISC is large(>500kD) RNA multi- protein Binding complex which triggers MRNA degradation in response to MRNA
Unwinding of double stranded Si RNA by ATP independent Helicase
Active component of RISC is Ago proteins( ENDONUCLEASE) which cleave target MRNA.
DICER: endonuclease (RNase Family III)
Argonaute: Central Component of the RNA-Induced Silencing Complex (RISC)
One strand of the dsRNA produced by Dicer is retained in the RISC complex in association with Argonaute
ARGONAUTE PROTEIN :
1.PAZ(PIWI/Argonaute/ Zwille)- Recognition of target MRNA
2.PIWI (p-element induced wimpy Testis)- breaks Phosphodiester bond of mRNA.)RNAse H activity.
MiRNA:
The Double-stranded RNAs are naturally produced in eukaryotic cells during development, and they have a key role in regulating gene expression .
THE IMPORTANCE OF MARTIAN ATMOSPHERE SAMPLE RETURN.Sérgio Sacani
The return of a sample of near-surface atmosphere from Mars would facilitate answers to several first-order science questions surrounding the formation and evolution of the planet. One of the important aspects of terrestrial planet formation in general is the role that primary atmospheres played in influencing the chemistry and structure of the planets and their antecedents. Studies of the martian atmosphere can be used to investigate the role of a primary atmosphere in its history. Atmosphere samples would also inform our understanding of the near-surface chemistry of the planet, and ultimately the prospects for life. High-precision isotopic analyses of constituent gases are needed to address these questions, requiring that the analyses are made on returned samples rather than in situ.
1. Path integral representation and quantum-classical
correspondence for nonadiabatic systems
1
Mikiya Fujii, Yamashita-Ushiyama Lab,
Dept. of Chemical System Engineering, The Unviersity of Tokyo
1. Introduction to nonadiabatic transitions
2. Nonadiabatic path integral based on overlap integrals
3. Nonadiabatic partition functions: nonadiabatic beads model
4. Semiclassical nonadiabatic kernel: rigorous surface hopping
5. Semiclassical quantization of nonadiabatic systems:
quantum-classical correspondence in nonadiabatic steady states
2. Transitions of nuclear wavepackets between
electronic eigenstates (adiabatic surfaces)
Femtosecond time-resolved spectroscopy of the dynamics at conical intersections, G.
Stock and W. Domcke, in: Conical Intersections, eds: W. Domcke, D. R. Yarkony, and H.
Koppel, (World Scientific, Singapore, 2003) , figure from http://www.moldyn.uni-
freiburg.de/research/ultrafast_nonadiabatic_photoreactions.html
NonAdiabatic Transitions (NATs)
G.-J. Kroes, Science 321, 794 (2008).
⃝surface reactions
G. Cerullo et.al, Nature 467, 440 (2010)
⃝vision
X.-Y. Zhu et.al, Nature Materials 12, 66 (2012)
⃝organic solar cells
⃝transition probability(’30∼)
• Landau–Zener
• Stueckelberg
• Zhu-Nakamura
⃝photo reactions
R. J. Sension et.al,
PCCP 16, 4439(2014)
applicationsbasics
⃝Theortical methods
T. Kubar and M. Elstner, J. R. Soc. Int.
2013 10, 20130415
Ehrenfest
Surface hopping
3. Notations
Electronic Hamiltonian
ˆHe( ˆR) =
ˆp2
2me
+ Vee(ˆr) + VNe(ˆr, ˆR) + VNN ( ˆR)
ˆHe(R)| n; Ri = ✏n(R)| n; Ri
Time independent electronic Schrödinger equation
Arbitrary state ket of a molecule
ni-th adiabatic surface
nuclear wavepacket
on ni-th adiabatic surface
3
| (t)i =
Z
dR
X
n
n(R, t)|Ri| n; Ri
Total Hamiltonian for molecules
ˆH = ˆTN + ˆHe( ˆR)
4. Schrödinger equation for NATs
Total wave function
is substituted to the time-dependent Schrödinger eq.
(r, R, t) =
X
n
n(R, t) n(r; R)
4
i~ ˙ (r, R, t) =
~2
2M
@2
@R2
~2
2me
@2
@r2
+ Vee(r) + VNe(r, R) + VNN (R) (r, R, t)
Multiplying
⇤
n(r; R) from left and integration r leads to
Nonadiabatic coupling between n-th and n’-th
adiabatic surfaces (derivative couplings)
i~ ˙n(R, t) =
~2
2M
@2
@R2
+ Vn(R) n(R, t)
X
m
~2
M
Xnm(R) 0
m(R, t) +
~2
2M
Ynm(R) m(R, t)
Xnm(R) =
Z
dr ⇤
n(r; R)
@
@R
m(r; R)
Ynm(R) =
Z
dr ⇤
n(r; R)
@2
@R2 m(r; R)
5. 5
1. Introduction to nonadiabatic transitions
2. Nonadiabatic path integral based on overlap integrals:
derivative couplings vs. overlap integrals
3. Nonadiabatic partition functions:
nonadiabatic beads model
4. Semiclassical nonadiabatic kernel:
rigorous surface hopping
5. Semiclassical quantization of nonadiabatic systems:
quantum-classical correspondence in nonadiabatic steady states
CONTENTS
6. NATs via overlap integrals
Total state ket of molecules is substituted to the time-dependent Schrödinger eq.:
| (t)i =
Z
dR
X
n
n(R, t)|Ri| n; Ri i~ ˙| (t)i = [ ˆTN + ˆHe]| (t)i
i~
Z
dR0
X
n0
˙n0 (R0
, t)|R0
i| n0 ; R0
i = [ ˆTN + ˆHe( ˆR)]
Z
dR0
X
n0
n0 (R0
, t)|R0
i| n0 ; R0
i
6
h n; R|hR|Multiplying from left leads to
i~
Z
dR0
X
n0
˙n0 (R0
, t)hR|R0
ih n; R| n0 ; R0
i = i~ ˙n(R, t)
(R R0
) nn0
Left=
2nd term
of right
= h n; R|hR| ˆHe(R0
)
Z
dR0
X
n0
n0 (R0
, t)|R0
i| n0 ; R0
i = ✏n(R) n(R, t)
7. NATs via overlap integrals
= h n; R|hR| ˆTN
Z
dR0
X
n0
n0 (R0
, t)|R0
i| n0 ; R0
i
=
Z
dR0
X
n0
n0 (R0
, t)h n; R|hR| ˆTN |R0
i| n0 ; R0
i
=
Z
dR0
X
n0
n0 (R0
, t)hR| ˆTN |R0
ih n; R| n0 ; R0
i
1st term
of right
Overlap integral between
different nuclear coordinates
i~ ˙n(R, t) =
Z
dR0
X
n0
hR| ˆTN |R0
ih n; R| n0 ; R0
i n0 (R0
, t) + ✏n(R) n(R, t)
Namely,
[
Nonadiabatic interaction between
n-th and n’-th adiabatic surfaces
via overlap integrals
7
commutable
8. Differential form vs. integral form of
Schrödinger equation
i~ ˙n(R, t) =
Z
dR0
X
n0
hR| ˆTN |R0
ih n; R| n0 ; R0
i n0 (R0
, t) + ✏n(R) n(R, t)
⃝differential form: NATS via derivative couplings
⃝Integral form: NATs via overlap integrals
They are Mathematically equivalent
Nonlocal propagation from R’ to R
↓
Suitable for the path integral
representation8
i~ ˙n(R, t) =
~2
2M
@2
@R2
✏n(R) n(R, t)
X
n0
~2
M
⌧
n(r; R)
@
@R
n0 (r; R)
@
@R
n0 (R, t)
X
n0
~2
2M
⌧
n(r; R)
@2
@R2 n0 (r; R) n0 (R, t)
9. Introduction of Nonadiabatic Kernel
Considering the infinitesimal time kernel of a molecule
h nf
; Rf |hRf |e
i
~
ˆH t
|Rii| ni
; Rii
Trotter decmp.
' h nf
; Rf |hRf |e
i
~
ˆTN t
e
i
h
ˆHe( ˆR) t
|Rii| ni
; Rii
9
= h nf
; Rf | ni
; RiihRf |e
i
~
ˆTN t
|Riie
i
~ ✏ni
(Ri) t
adiabatic propagation
on ni-th adiabatic surface
overlap integral
between ni@Ri and nf@Rf
, representing nonadiabatic transition
Repeating this infinitesimal time kernel gives a finite time kernel
= h nf
; Rf |hRf |e
i
~
ˆTN t
|Riie
i
h
ˆHe(Ri) t
| ni
; Rii
10. 10
K =
Z
D [R(⌧), n(⌧)] ⇠ exp
i
~
S
②Infinite product of the overlap integrals
(phase weighted probability of each path)
①Nuclear paths that are evolving through arbitrary
positions and electronic eigenstates
{R(⌧), n(⌧)}
NonAdiabatic Path Integral (NAPI)
This nonadiabatic kernel holds 2 differences from adiabatic kernel
⇠ ⌘ lim
!1
Y
k=0
h n(tk+1); R(tk+1)| n(tk); R(tk)i
J. R. Schmidt and J. C. Tully, J. Chem. Phys. 127, 094103 (2007)
M. Fujii, J. Chem. Phys. 135, 114102 (2011)
11. NA Schrödinger eq. is revisited from the NAPI
n(x, t + ✏) =
X
m
Z 1
1
d⌘hn; x|m; x + ⌘i exp
i
~
⌘2
2✏
i
~
Vm(x + ⌘)✏ m(x + ⌘, t)
Time propagation with infinitesimal time-width in NAPI:✏
p
2~✏ < ⌘ <
p
2~✏
The main contribution is from the range:
⌘2
2~✏
' 1
i.e.,
Then, we expand the NAPI up to .✏ or ⌘2
12. n(R, t + ✏) =
X
m
Z 1
1
d⌘A exp
M⌘2
2i~✏
⇢
hn; R|m; Ri m(R, t) +
1
i~
hn; R|m; RiVm(R) m(R, t)✏
+hn; R|m; Ri
@ m
@R
⌘ + Xnm(R) m(R, t)⌘
+hn; R|m; Ri
@2
m
@R2
⌘2
2
+ Xnm(R)
@ m
@R
⌘2
+ Ynm(R) m(R, t)
⌘2
2
nm
By solving the Gaussian integrals,
the nonadiabatic Schrödinger eq. is revisited:
i~ ˙n(R, t) =
~2
2M
@2
@R2
+ Vn(R) n(R, t)
X
m
~2
M
Xnm(R) 0
m(R, t) +
~2
2M
Ynm(R) m(R, t)
Xnm(R) =
Z
dr ⇤
n(r; R)
@
@R
m(r; R)
Ynm(R) =
Z
dr ⇤
n(r; R)
@2
@R2 m(r; R)
13. K =
Z
D [R(⌧), n(⌧)] ⇠ exp
i
~
S
i~ ˙n(R, t) =
~2
2M
@2
@R2
+ Vn(R) n(R, t)
X
m
~2
M
Xnm(R) 0
m(R, t) +
~2
2M
Ynm(R) m(R, t)
Xnm(R) =
Z
dr ⇤
n(r; R)
@
@R
m(r; R)
Ynm(R) =
Z
dr ⇤
n(r; R)
@2
@R2 m(r; R)
⇠ ⌘ lim
!1
Y
k=0
h n(tk+1); R(tk+1)| n(tk); R(tk)i
Nonadiabatic path integral with overlap integrals
Nonadiabatic Schrödinger eq. with derivative couplings
Mathematically equivalent
14. 14
1. Introduction to nonadiabatic transitions
2. Nonadiabatic path integral based on overlap integrals:
derivative couplings vs. overlap integrals
3. Nonadiabatic partition functions:
nonadiabatic beads model
4. Semiclassical nonadiabatic kernel:
rigorous surface hopping
5. Semiclassical quantization of nonadiabatic systems:
quantum-classical correspondence in nonadiabatic steady states
CONTENTS
15. Z( ) = Tre
ˆH
Quantum MC
by Adiabatic beads
Quantum MC
by Nonadiabatic beads
Nonadaibatic Partition function
K = e
i
~
ˆHt
t = i~time propagator partition function
16. Z( ) = Tr
h
e
ˆH
· · · e
ˆH
i
Boltzmann operator is divided to Γ peaces:
ˆ1 =
Z
dR
X
n
|Ri| n; Rih n; R|hR|
Inserting identity operators
leads to
⇠ =
Y
k=1
h nk
; Rk| nk+1
; Rk+1i
Z( ) =
Z
dR1 · · · dR
X
n1···n
⇠hR1|e
ˆHn1 |R2i · · · hR |e
ˆHn
|R1i
ˆHn = ˆTN + ✏n( ˆR)
Infinite product of overlaps:
n-th adibatic Hamiltonian:
16
17. 17
The divided Boltzmann operators can be written as
hR|e
ˆHn
|R0
i = lim
!1
⇢0(R, R0
; )e ✏n(R0
)
⇢0(R, R0
; ) =
✓
M
2⇡~2
◆1
2
e
M
2~2 (R R0
)2
Boltzmann operator for free particles
After all, we obtained following representation:
Z( ) = lim
!1
✓
M
2⇡~2
◆ 2 X
n1,··· ,n
Z
dR1, · · · , dR
⇥⇠ exp
✓ X
k=1
M
2~2 2
(Rk Rk+1)2
+
✏nk
(Rk)
◆
nonadiabatic beads
18. quantum-classical mapping
under thermal equilibrium
Z( ) = Tre
ˆHTo calculate the partition function:
Hbeads =
X
k=1
M
2~2 2
(Rk Rk+1)2
+
✏nk
(Rk)
with weighting factor:
⇠ =
Y
k=1
h nk
; Rk| nk+1
; Rk+1iThis nonadiabatic beads model can be applied to
thermal average of physical quantities
quantum nonadiabatic particle classical nonadiabatic beads
18
classical mapping
J. R. Schmidt and et.al, JCP 127, 094103 (2007)
19. A simple model
, with m=1 [amu].
J. Morelli and S. Hammes-Schiffer, Chem. Phys. Lett. 269, 161 (1997)
22. quantum-classical mapping
under thermal equilibrium
To calculate the partition function and thermal average
Hbeads =
X
k=1
M
2~2 2
(Rk Rk+1)2
+
✏nk
(Rk)
with weighting factor:
⇠ =
Y
k=1
h nk
; Rk| nk+1
; Rk+1i
classical mapping
quantum nonadiabatic particle classical nonadiabatic beads
22 J. R. Schmidt and et.al, JCP 127, 094103 (2007)
23. 23
1. Introduction to nonadiabatic transitions
2. Nonadiabatic path integral based on overlap integrals:
derivative couplings vs. overlap integrals
3. Nonadiabatic partition functions:
nonadiabatic beads model
4. Semiclassical nonadiabatic kernel:
rigorous surface hopping
5. Semiclassical quantization of nonadiabatic systems:
quantum-classical correspondence in nonadiabatic steady states
CONTENTS
24. Semiclassical propagator (adiabatic)
Stationary phase approx. is applied to the time propagator
K = hRf |e i ˆH(tf ti)/~
|Rii =
Z
D[R(⌧)] exp
i
~
S[R(⌧)]
Ri
R1
R2
Rf
RN 1
RN
R0
t1
t2
tN 1
tf
ti
t
stationary phase condition:minimum action integral→classical trajectory:
S[R(⌧)]
R(⌧)
= 0
Rcl(⌧)
Rcl(⌧)
: Maslov index
S[Rcl(⌧)]: action integral
dRt
dPi
: Stability matrix
Formulated with quantities along classical trajectories
Quantum-Classical correspondence in dynamics
KSC =
X
Rcl
(2⇡i~)
1
2
dRt
dPi
1
2
exp
i
~
Scl[Rcl(⌧)]
i⇡
2
⌫
24
25. (a)→(b): Stationary approximation for summing up all trajectories on a surface
K /
Z
dth
Z
dRh⇠J exp
i
~
SnJ
cl (Rf , Rh) ⇠I exp
i
~
SnI
cl (Rh, R0)
25
Semiclassical approximation of the nonadiabatic kernel
(stationary phase approximation on the each surface)
26. K /
Z
dth
Z
dRh⇠J exp
i
~
SnJ
cl (Rf , Rh) ⇠I exp
i
~
SnI
cl (Rh, R0)
(b)→(c): Stationary approximation for the integral related to hopping points, Rh
d
dRh
[SnJ
cl (Rf , Rh) + SnI
cl (Rh, R0)] = PJ + PI = 0
Stationary phase condition:
momentum conservation
26
Semiclassical approximation of the nonadiabatic kernel
(stationary phase approximation for the hopping point)
27. 27
Nonadiabatic Semiclassical Kernel
c.f., Adiabatic semiclassical kernel
KSC =
X
Rcl
(2⇡i~)
1
2
dRt
dPi
1
2
exp
i
~
Scl[Rcl(⌧)]
i⇡
2
⌫
KSC =
X
Rhcl
(2⇡i~)
1
2 ⇠
dRt
dPi
1
2
exp
i
~
Scl[Rhcl(⌧)]
i⇡
2
⌫
①Hopping classical trajectories
Two differences from adiabatic semiclassical kernel
⇠ ⌘ lim
!1
Y
k=0
h n(tk+1); R(tk+1)| n(tk); R(tk)i
amplitude of each overlap means probability
of the hopping at each time step
②Infinite product of the overlap integrals
(phase weight probability of each hopping calssical traj.)
28. Nemerical example(Nonadiabatic SC-IVR, Herman-Kluk)
M. Fujii, J. Chem. Phys. 135, 114102 (2011)28
Black: Numerical exact
Blue&Green: present semi classical
107 trajectories
avoided crossing
Nonadiabatic wavepacket dynamics
including phase accompanied by
nonadiabatic transition is also reproduced.
Namely, rigorous surface hopping.
29. M. Fujii, J. Chem. Phys. 135, 114102 (2011)
29
Nemerical example(Nonadiabatic SC-IVR, Herman-Kluk)
30. Quantum-classical correspondence
in nonadiabatic dynamics
quantum wavepacket dynamics classical hopping dynamics
Classical hopping trajectories are taken out as dominant terms
of nonadiabatic propagation of quantum wavepackets
stationary phase
31. 31
1. Introduction to nonadiabatic transitions
2. Nonadiabatic path integral based on overlap integrals:
derivative couplings vs. overlap integrals
3. Nonadiabatic partition functions:
nonadiabatic beads model
4. Semiclassical nonadiabatic kernel:
rigorous surface hopping
5. Semiclassical quantization of nonadiabatic systems:
quantum-classical correspondence in nonadiabatic steady states
CONTENTS
32. Semiclassical Quantization
Revealing correspondence between time-invariant structures in
classical mechanics and steady states in quantum mechanics
e.g. Bohr’s model for Hydrogen, Bohr-Sommerfeld, Einstein–
Brillouin–Keller, etc
ˆH| i = E| i
steady states
in quantum mechanics
q
p
time-invariant structures
in phase space of
classical mechanics
periodic
orbits torus
big← →small~
32
33. Objective
Finding a quantum-classical correspondence for nonadiabatic steady states
i.e. How time-invariant structures in nuclear phase space should be quantized
Especially, the semiclassical concepts of the nonadiabatic transition (i.e. classical
dynamics on adiabatic surfaces and hopping) should be held.
!
The reason is that some pioneering studies that treat electrons and nuclei in equal-
footing-manner have been already presented for the semiclassical quantization.
e.g. Meyer-Miller (JCP 70, 3214 (1979)) and Stock-Thoss (PRL. 78, 578 (1997))
big← →small~
nonadiabatic
eigenstates
q
p
?nuclear phase space
34. Gutzwiller s trace formula
Semiclassical approximation to DOS, which has revealed correspondence between
quantum energy levels and classical periodic orbits through divergences of DOS.
classical action: Phase space volume
⌫ = 2
Scl
= 2⇡E/!
e.g. Harmonic oscillator
Maslov index: number of intersects between
trajectory and R-axis
geometric quantity
of a cycle of primitive
periodic orbit
number of cycle of primitive periodic orbit
Sum of k-cycle diverges at quantum energy levels
1 = exp
✓
i
~
2⇡E
!
i⇡
◆
) En =
✓
n +
1
2
◆
~!
}
⌦(E) /
1X
k=0
exp
✓
i
~
Scl i⇡
2
⌫
◆ k
=
1 exp
✓
i
~
Scl i⇡
2
⌫
◆ 1
34
35. ⌦(E) /
X
2PHPO
1 ⇠ exp
✓
i
~
Scl i⇡
2
⌫
◆ 1
①Sum of “Primitive Hopping Periodic Orbits (PHPO)”
Taking the summation of geometric series related to k, naively, leads to
⇠ < 1This term does not diverge because .
35
②Infinite product of the overlap integrals: ⇠ ⌘ lim
!1
Y
k=0
h n(tk+1); R(tk+1)| n(tk); R(tk)i
There are 2 differences from the Gutzwiller’s (adiabatic) trace formula
⌦(E) /
X
2PHPO
1X
k=0
⇠ exp
✓
i
~
Scl i⇡
2
⌫
◆ k
Nonadiabatic Trace formula
That is, individual PHPO cannot be quantized.
We must introduce another way to take the summation of
infinite number of the PHPOs
36. Bit sequence which represents PHPO
A concrete example: Two adiabatic harmonic oscillators which interact nonadiabatically at the origin only.
D12 = (R) sin(✓)
Ri Rj Rk Rl 0, 1, 1, and 0 are assigned when a
trajectory passes through Ri, Rj, Rk, and Rl,
respectively
Assignment of bit
e.g.,
adiabatic (no hopping) PO: 0000000…
!
!
diabatic (fully nonadiabatic) PO: 0101010…
!
!
Periodic bit sequences representing PHPOs can be expressed with dots on the fist and last bits
˙011˙1 ⌘ 011101110111 · · ·
˙0˙1 ⌘ 0101010101 · · ·
36
We can also confirm that the periodic and non-periodic orbits correspond to rational and irrational
numbers, respectively, because periodic bit sequences correspond to rational number in binary digits.
So, the number of periodic orbits is countable infinite while the number of arbitrary orbits is uncountable
infinite.
37. D12 = (R) sin(✓)
Ri Rj Rk Rl ˙01000111001˙1
0 in odd-numbered bits means “returning to Ri”.
Decomposition of each PHPO
01 + 00 + 0111 + 0011
At the 0 in odd-numbered bits, we can decompose this
PHPO to “more primitive (prime) bits (PHPOs)”.
Threfore, arbitrary hopping periodic orbits passing through Ri can be represented by combinations of
these prime PHPOs:
00, 01, 0110, 0111, 0010, 0011
,where 1 means combinations of 11 and 10.
Hereafter, this set of prime PHOPs are represented as
S0 ⌘
38. (Ⅰ) All prime PHPOs in Si pass through the same phase space point
(Ⅱ) Any pair of prime PHPOs (Γ, Γ’) in Si is coprime:
0
6⇢ _ r 0
62 S
38
S0 ⌘
00, 01, 0110, 0111, 0010, 0011
A set Si of prime PHPOs
D12 = (R) sin(✓)
Ri Rj Rk Rl
39. Sum of all HPOs as combination of coprime PHPOs
Sum of all HPOs, for example,
started from Ri
D12 = (R) sin(✓)
Ri Rj Rk Rl
00
01
+
+
0110
+
...
0000
0101
+
+
000110...
000000
010101...
k = 1
k = 2
k = 3
}
}
}
= Sum of geometric series
of sum of prime PHPOs
=
X
k2N
(00 + 01 + 0110)
k
=
X
k2N
X
2S0
!k
41. Quantum-classical correspondence
in nonadiabatic steady states
quantum nonadiabatic
eigenstates
Time-invariant structure in
classical nuclear phase space
big← →small~ S0 ⌘
1 =
X
2S
⇠ exp
✓
i
~
Scl i⇡
2
⌫
◆
Semiclassical Quantization condition
42. Summary of this talk
S0 ⌘
1. Nonadiabatic path integral with overlap integrals
K =
Z
D [R(⌧), n(⌧)] ⇠ exp
i
~
S
⇠ ⌘ lim
!1
Y
k=0
h n(tk+1); R(tk+1)| n(tk); R(tk)i
3. Nonadiabatic semiclassical kernel (“rigorous” surface hopping)
KSC =
X
Rhcl
(2⇡i~)
1
2 ⇠
dRt
dPi
1
2
exp
i
~
Scl[Rhcl(⌧)]
i⇡
2
⌫
4. Semiclassical quantization condition
⌦(E) /
X
2PHPO
1X
k=0
⇠ exp
✓
i
~
Scl i⇡
2
⌫
◆ k
Nonadiabatic trace formula
42
M. Fujii, JCP, 135, 114102 (2011)
M. Fujii and K. Yamashita, JCP, 142, 074104 (2015)
arXiv:1406.3769
J. R. Schmidt and et.al, JCP, 127, 094103 (2007)
M. Fujii, JCP, 135, 114102 (2011)
2. Nonadiabatic beads
J. R. Schmidt and et.al, JCP 127, 094103 (2007)
Classical mapping under
thermal equilibrium
Classical counterparts of
nonadiabatic wavepacket dynamics
Classical counterparts of nonadiabatic eigenstates
43. Acknowledgments
• I appreciate valuable discussions with
Prof. K. Yamashita
Pfof. K. Takatsuka
Prof. H. Ushiyama
Prof. O. Kühn
• This work was supported by
JSPS KAKENHI Grant No. 24750012
CREST, JST.
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