Rare event techniques are used to sample rare thermal activation processes that cannot be directly observed in simulation or experiment. Common methods include constrained minimization, nudged elastic band, dimer method, and Monte Carlo. Nudged elastic band constructs a chain of images describing the minimum energy pathway between reactants and products on a potential energy surface. Monte Carlo uses random sampling to infer properties from many probabilistic experiments weighted by the Boltzmann factor. These methods help characterize transition states that are challenging to describe with a single reaction coordinate.
BIOS203 Lecture 1: Introduction to potentials and minimizationbios203
Lecture 1 for BIOS 203 Mini-course at Stanford University taught by Heather J. Kulik. http://bios203.stanford.edu for more info or email bios203.course@gmail.com
Lecture 6: Introduction to Quantum Chemical Simulation graduate course taught at MIT in Fall 2014 by Heather Kulik. This course covers: wavefunction theory, density functional theory, force fields and molecular dynamics and sampling.
Lecture 5: Introduction to Quantum Chemical Simulation graduate course taught at MIT in Fall 2014 by Heather Kulik. This course covers: wavefunction theory, density functional theory, force fields and molecular dynamics and sampling.
BIOS203 Lecture 1: Introduction to potentials and minimizationbios203
Lecture 1 for BIOS 203 Mini-course at Stanford University taught by Heather J. Kulik. http://bios203.stanford.edu for more info or email bios203.course@gmail.com
Lecture 6: Introduction to Quantum Chemical Simulation graduate course taught at MIT in Fall 2014 by Heather Kulik. This course covers: wavefunction theory, density functional theory, force fields and molecular dynamics and sampling.
Lecture 5: Introduction to Quantum Chemical Simulation graduate course taught at MIT in Fall 2014 by Heather Kulik. This course covers: wavefunction theory, density functional theory, force fields and molecular dynamics and sampling.
In computational physics and Quantum chemistry, the Hartree–Fock (HF) method also known as self consistent method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system or many electron system in a stationary state
In this talk I will discuss different approximations in DFT: pseduo-potentials, exchange correlation functions.
The presentation can be downloaded here:
http://www.attaccalite.com/wp-content/uploads/2022/03/dft_approximations.odp
Physics dictionary for CBSE, ISCE, Class X Students by Arun Umraossuserd6b1fd
Dictionaries are very important. Without definitions of scientific words you can not understand the theories or theorems. This dictionary explains nearly all the terms used in CBSE Class X science book.
Lecture 8: Introduction to Quantum Chemical Simulation graduate course taught at MIT in Fall 2014 by Heather Kulik. This course covers: wavefunction theory, density functional theory, force fields and molecular dynamics and sampling.
Lecture 2: Introduction to Quantum Chemical Simulation graduate course taught at MIT in Fall 2014 by Heather Kulik. This course covers: wavefunction theory, density functional theory, force fields and molecular dynamics and sampling.
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.
Lecture 4: Introduction to Quantum Chemical Simulation graduate course taught at MIT in Fall 2014 by Heather Kulik. This course covers: wavefunction theory, density functional theory, force fields and molecular dynamics and sampling.
Study on the dependency of steady state response on the ratio of larmor and r...eSAT Journals
Abstract In this project we simulate with very high accuracy specially to study the dependency of the steady state power and dispersion output on the ratio (r) between Larmor and Rabi frequency for the electron spin resonance experiment by the matlab software (version 7.9.0.529(R2009b)). Where the sample material (DPPH) has been kept in a strong static magnetic field (B0) and in orthogonal direction a high frequency electromagnetic field (B1(t)) has been applied. We divide our simulation into two parts. In the first part we ignore the terms and observe the dependency of the power maximum on the amplitude of the oscillating e.m. field B1 (for fixed (ωL) Larmor frequency) and on ωL (for fixed B1). Also observe a clear shift (Δω) of the power maxima (Pmax) from ωL. In our second part we consider the term and the ratio (r) between Larmor and Rabi frequency and observe the shift (Δω) of the power maxima (Pmax) from ωL and change in peak to peak line width (ΔBPP) with B1 both depends upon the ratio r. we consider various range of r ([0.83,5], [16,100], [88.3,500], [1000,2000], [833.3,5000]) and observe these dependency. We observe as the ratio of r increases the output i.e. shift (Δω) and the change in ΔBPP with B1 decreases and converges to the case of neglecting terms. We also observe the shift (Δω) follows some non linear relationship with B1. Keywords: E.S.R., Larmor, Rabi, Ratio r, Spin
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.
SWCNT Growth from Chiral and Achiral Carbon Nanorings: Prediction of Chiralit...Stephan Irle
Catalyst-free, chirality-controlled growth of chiral and achiral single-walled carbon nanotubes (SWCNTs) from organic precursors is demonstrated using quantum chemical simulations [1]. Growth of (4,3), (6,5), (6,1), (10,1), (6,6) and (8,0) SWCNTs was induced by ethynyl radical (C2H) addition to organic precursors. These simulations show a strong dependence of the SWCNT growth rate on the chiral angle, θ. The SWCNT diameter however does not influence the SWCNT growth rate under these conditions. This agreement with a previously proposed screw-dislocation-like model of transition metal-catalyzed SWCNT growth rates [2] indicates that the SWCNT growth rate is an intrinsic property of the SWCNT edge itself. Conversely, we predict that the rate of local SWCNT growth via Diels-Alder cycloaddition of C2H2 is strongly influenced by the diameter of the SWCNT. We therefore predict the existence of a maximum local growth rate for an optimum diameter/chirality combination at a given C2H/C2H2 ratio. We also find that the ability of a SWCNT to avoid defect formation during growth is an intrinsic quality of the SWCNT edge.
References:
[1] Li, H.-B.; Page, A. J.; Irle, S.; Morokuma, K. J. Am. Chem. Soc. 2012, 134, 15887-15896.
[2] Ding, F.; Harutyunyan, A. R.; Yakobson, B. I. Proc. Natl. Acad. Sci. 2009, 106, 2506-2509.
In computational physics and Quantum chemistry, the Hartree–Fock (HF) method also known as self consistent method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system or many electron system in a stationary state
In this talk I will discuss different approximations in DFT: pseduo-potentials, exchange correlation functions.
The presentation can be downloaded here:
http://www.attaccalite.com/wp-content/uploads/2022/03/dft_approximations.odp
Physics dictionary for CBSE, ISCE, Class X Students by Arun Umraossuserd6b1fd
Dictionaries are very important. Without definitions of scientific words you can not understand the theories or theorems. This dictionary explains nearly all the terms used in CBSE Class X science book.
Lecture 8: Introduction to Quantum Chemical Simulation graduate course taught at MIT in Fall 2014 by Heather Kulik. This course covers: wavefunction theory, density functional theory, force fields and molecular dynamics and sampling.
Lecture 2: Introduction to Quantum Chemical Simulation graduate course taught at MIT in Fall 2014 by Heather Kulik. This course covers: wavefunction theory, density functional theory, force fields and molecular dynamics and sampling.
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.
Lecture 4: Introduction to Quantum Chemical Simulation graduate course taught at MIT in Fall 2014 by Heather Kulik. This course covers: wavefunction theory, density functional theory, force fields and molecular dynamics and sampling.
Study on the dependency of steady state response on the ratio of larmor and r...eSAT Journals
Abstract In this project we simulate with very high accuracy specially to study the dependency of the steady state power and dispersion output on the ratio (r) between Larmor and Rabi frequency for the electron spin resonance experiment by the matlab software (version 7.9.0.529(R2009b)). Where the sample material (DPPH) has been kept in a strong static magnetic field (B0) and in orthogonal direction a high frequency electromagnetic field (B1(t)) has been applied. We divide our simulation into two parts. In the first part we ignore the terms and observe the dependency of the power maximum on the amplitude of the oscillating e.m. field B1 (for fixed (ωL) Larmor frequency) and on ωL (for fixed B1). Also observe a clear shift (Δω) of the power maxima (Pmax) from ωL. In our second part we consider the term and the ratio (r) between Larmor and Rabi frequency and observe the shift (Δω) of the power maxima (Pmax) from ωL and change in peak to peak line width (ΔBPP) with B1 both depends upon the ratio r. we consider various range of r ([0.83,5], [16,100], [88.3,500], [1000,2000], [833.3,5000]) and observe these dependency. We observe as the ratio of r increases the output i.e. shift (Δω) and the change in ΔBPP with B1 decreases and converges to the case of neglecting terms. We also observe the shift (Δω) follows some non linear relationship with B1. Keywords: E.S.R., Larmor, Rabi, Ratio r, Spin
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.
SWCNT Growth from Chiral and Achiral Carbon Nanorings: Prediction of Chiralit...Stephan Irle
Catalyst-free, chirality-controlled growth of chiral and achiral single-walled carbon nanotubes (SWCNTs) from organic precursors is demonstrated using quantum chemical simulations [1]. Growth of (4,3), (6,5), (6,1), (10,1), (6,6) and (8,0) SWCNTs was induced by ethynyl radical (C2H) addition to organic precursors. These simulations show a strong dependence of the SWCNT growth rate on the chiral angle, θ. The SWCNT diameter however does not influence the SWCNT growth rate under these conditions. This agreement with a previously proposed screw-dislocation-like model of transition metal-catalyzed SWCNT growth rates [2] indicates that the SWCNT growth rate is an intrinsic property of the SWCNT edge itself. Conversely, we predict that the rate of local SWCNT growth via Diels-Alder cycloaddition of C2H2 is strongly influenced by the diameter of the SWCNT. We therefore predict the existence of a maximum local growth rate for an optimum diameter/chirality combination at a given C2H/C2H2 ratio. We also find that the ability of a SWCNT to avoid defect formation during growth is an intrinsic quality of the SWCNT edge.
References:
[1] Li, H.-B.; Page, A. J.; Irle, S.; Morokuma, K. J. Am. Chem. Soc. 2012, 134, 15887-15896.
[2] Ding, F.; Harutyunyan, A. R.; Yakobson, B. I. Proc. Natl. Acad. Sci. 2009, 106, 2506-2509.
Il Sottoscala9 è uno spazio dedicato alla musica, all'arte, allo spettacolo e alla cultura in ogni sua forma, in più ospita la sede provinciale del circolo Arci Latina. In vista del nuovo anno, il direttivo del Sottoscala9 ha deciso di compiere un’azione di rinnovamento globale partendo dalla sistemazione logistica del contenitore che ospiterà la nuova programmazione alla ridefinizione della propria identità visiva.
Erica Zigelman, a New York City school principal, graduated from New York University with a bachelor’s degree in education, and continued at the school to obtain a master’s degree. Since 1979, Erica Zigelman has served as principal in the New York City Department of Education.
This briefing describes the applications for a private, off-the-grid, mesh connected IRC server that can be deployed in support of emergency communications. This system would be particulary useful in shelter, emegerency operation center, and campus triage support scenarios.
Optical fiber communication Part 2 Sources and DetectorsMadhumita Tamhane
For optical fiber communication, major light sources are hetero-junction-structured semiconductor laser diode and light emitting diodes. Heterojunction consists of two adjoining semiconductor materials with different bandgap energies. They have adequate power for wide range of applications. Detectors used are PiN diode and Avalanche Photodiode. Being very small in size and feeding to small core optical fiber, it is very important to study emission characteristics of sources and their coupling to fiber. As it can operate for low power over a long distance, received power is very small, hence study of noise characteristics of detectors is very essential...
Synchronverters: Inverters that Mimic Synchronous GeneratorsQing-Chang Zhong
Inverters are made mathematically equivalent to conventional synchronous generators, which considerably facilitates the integration of renewable energy and distributed generation into smart grids.
Similar to BIOS203 Lecture 7: Rare event techniques (20)
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
2. The sampling bottleneck
Transition
Fast: oscillations around each minimum. State
S l o w: the jump over the barrier from
one minimum to the other.
Energy
E
van’t Hoff-Arrhenius relationship: a
-Ea kBT
t jump ~ t vibe kbT
Reactants
Products
Example:
Reaction coordinate
kcal
Ea » 17 ; T = 300 K
mol For a thermally activated process,
t vib » 10-8 s timescale to observe event in real time
is so slow. It would take 1015 1 fs MD
t jump » 1s steps to observe the event directly!
3. How to sample rare events
• Map out the PES
• Constrained minimization
• (Guided) synchronous transit
• Nudged elastic band/string method
• Dimer method
• Monte Carlo
• Umbrella sampling
• Metadynamics
4. Mapping a complete PES
For very small
systems: we can
map out a fully ab
initio potential
energy surface.
e.g. CH4+Cl, CH5+,
etc. see work of
Bowman group.
But this PES mapping is impractical
for anything but very small systems.
5. Constrained minimizations
Works well for simple
reaction coordinates: Fails for complex
interpolate between
reaction pathways.
reactant and products,
using some geometric
constraints to define
PES
6. Synchronous transit
Quadratic synchronous
transit (QST): Search for a
maximum along an arc
R
between R and P, minimum on
perpendicular direction.
Guided QST:
Start with QST and then follow
TS
eigenvector to the saddle point.
Works well only for simple P
reaction coordinates and small
molecule systems.
7. Nudged elastic band
Chain-of-states method: string of images/geometries is
used to describe the minimum energy pathway.
A chain gang initial state final state guesses
Springs keep interpolated
images separated:
Our chain of states are propagated on the
potential energy surface until we find a
minimum energy path.
9. Nudged elastic band
Reactants,
intermediates,
products
Saddle point
Mueller
potential
10. Nudged elastic band
Reactants,
intermediates,
products
Saddle point
minimum
energy path
ÑE ( Ri ) ^ = 0
Mueller
potential
11. Nudged elastic band
Reactants,
intermediates,
products
Saddle point
minimum
energy path
NEB initial
guess from
interpolation
Mueller
potential
12. Nudged elastic band
Reactants,
intermediates,
Fi
products
^
i ti
ˆ
F
i Saddle point
S
F
i
FNEB minimum
i
energy path
NEB initial
guess from
interpolation
NEB image
Mueller
potential
13. Nudged elastic band
NEB image force:
^ S
F i
NEB
= F +F
i i
Fi^ = - ( ÑE ( Ri ) - ÑE ( Ri ) × t it i )
ˆˆ
Fi
True forces: ignore component that
minimizes energy parallel to path,
only minimize perpendicularly. ^
i ti
ˆ
F
i
F = k ( Ri+1 - Ri - Ri - Ri-1 ) t i
S
i
ˆ S
F
i
Spring forces: only want F i
NEB
component of this force that keeps
images separated (along path).
14. Nudged elastic band
Climbing image method: improved resolution of the saddle point
F = -ÑE ( R j ) + 2ÑE ( R j ) × t jt j
CI
j
ˆ ˆ
True forces: component
parallel to band is inverted,
image moves up the band. with CI
Spring forces: CI image
feels no spring forces. no CI
15. Nudged elastic band
Variable springs method: improved resolution of the saddle point
ì æ Emax - Ei ö
ï kmax - Dk ç
ï ÷
ki ' = í è Emax - Eref ø if Ei > Eref
ï
ï
î kmin if Ei < Eref
variable
springs
Eref
Spring forces: stiffer
springs for high energy
fixed
points to ensure resolution
springs
of the saddle point.
16. Nudged elastic band
Improved tangent method: for improved stability
When parallel forces are large and
LEPS
potential perpendicular are small, path can
get unstable, kinking.
Improved tangent:
ì Ri+1 - Ri
ï if Ei+1 > Ei > Ei-1
NEB
ï Ri+1 - Ri
ti = í
ï Ri - Ri-1
if Ei+1 < Ei < Ei-1
ï Ri - Ri-1
MEP
î
Resulting NEB path looks like MEP!
17. Nudged elastic band
Practical challenges:
1) Stability of the calculation depends on the number of images. If images are
too close together, we may be unstable.
2) Convergence to minimize forces may be slow and depends on the
minimizer used.
3) Initial estimates of MEP based on cartesian interpolation may be poor (vs.
internal coordinates). Initial estimate needs to be good to speed
convergence.
4) Rotations, translations may enter erroneously into path.
5) Our idea of what the MEP should be biases the solution that we can find.
18. Dimer method
Two separated states, R1
and R2. R1
States are pushed up the
PES, inverting the force
along the lowest
frequency mode.
Also rotate the dimer to R2
sample PES. Dimer
force
Method scales well with
increasing complexity! Real
force
19. String method
• Similar to NEB in
construction/cost.
• Images propagated on path,
following force perpendicular to
path.
• No springs, images are adjusted
slightly following force
propagation to ensure spacing.
• Finite temperature extension.
• “Growing string” method allows
the path to change in time to allow
for variations in MEP.
20. Monte Carlo
If we are interested in macroscopic averages, these are thermodynamical
quantities that are difficult to extract from direct dynamics.
Origin of Monte Carlo (1940s Los Alamos- Ulam/von Neumann):
Rather than deterministic mathematical methods…
Infer instead the answer from the outcome of
many probabilistic, random experiments.
Today:
(1) scientific simulations
(2) used to simulate real events, e.g.
Stock market, etc.
21. Basic principle of Monte Carlo
p
1
Acircle = Asquare =1
4
y
For random selections, we are either a “hit”
inside the circle or a “miss” outside the circle.
nhits Acircle p
= =
0
nhits + nmiss Asquare 4 0 x 1
hit miss
We can estimate p in this way, but it will take thousands
of attempts to get a reasonable estimate!
22. Monte Carlo
Can we pick where we sample such that their weight is proportional to e-bE?
M
e- b H v M
A =å M
Av ?
A = å Av
v=1
å e- b Hv v=1
v=1
Random sample probability-weighted sample
Metropolis algorithm:
“Walk” through phase space, with proper P for infinite time limit
Random starting state
Pick a trial state j from I with some rate W0ij
Accept j with some probability Pij
23. A Monte Carlo algorithm
Compute
Initial Perturb the
energy for
configuration system
perturbation
Accept
Is DE<0?
Accept with
probability
P α e-ΔE/kT
24. Monte Carlo timescales
Monte Carlo timescale has no true meaning:
not a dynamical timescale but a measure of how much phase space has
been sampled.
We get an
property
average of our
property at long
MC timescales.
MC timescale
We can bias our dynamics: way perturbations are sampled can be
determined by the kind of phenomena we’re trying to describe: exchange
across long distances, nearest neighbor exchanges, etc.
25. Practical challenges for TSs
Characterization of the TS is only as good as the energetic
model being used.
Transition states often have open-shell, multi-reference
character:
CCSD(T), which is great for local minima, often fails for
transition states as a result of triples amplitudes, multi-
reference character.
Density functionals that yield 1-2 kcal/mol error in minima
often underestimate TSs by about 3 kcal/mol.
26. Beyond MEPs
Conical intersections: beyond the Born-oppenheimer approximation, coupling of
states and transfer between states govern key phenomena.
Also: electron transfer (Marcus theory), proton transfer (need quantum nuclear
effects), allosteric transitions that are difficult to describe by a single coordinate…
27. Follow-up reading
• Synchronous transit
– T. A. Halgren and W. N. Lipscomb “The synchronous-transit method for
determining reaction pathways and locating molecular transition states”
Chem. Phys. Lett. (1977).
– C. Peng and H. B. Schlegel “Combining synchronous transit and quasi-
Newton methods to find transition states” Israel J. of Chem. (1993).
• Chain of states techniques
– D. Sheppard, R. Terrell, and G. Henkelman “Optimization methods for
finding minimum energy paths” J. Chem. Phys. (2008).
– W. E., W. Ren, and E. Vanden-Eijnden “String method for the study of
rare events” Phys. Rev. B. (2002).
– G. Henkelman and H. Jonsson “A dimer method for finding saddle
points on high dimensional potential surfaces using only first
derivatives”J. Chem. Phys. (1999).
• Monte Carlo
– D. P. Landau and K. Binder A Guide to Monte Carlo Simulations in
Statistical Physics. Cambridge University Press 2nd Edition (2005).
– R. H. Swendsen and J.-S. Wang “Nonuniversal critical dynamics in
Monte Carlo simulations”. Phys. Rev. Lett. (1987).