introduction to the computational laboratory
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INTRODUCTIONto the Computational Laboratory
Giuseppe [email protected]
Imperial College London - Chemistry Department
Thomas Young Centre:
the London Centre for Theory and Simulation of Materials
1 February 2010 – 8 March 2010
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Outline
1 TIMETABLE and DEADLINE
2 AIMS and SYSTEMS
3 VIBRATIONS
4 HOW
5 SIMULATION TYPES
6 CLASSICAL SIMULATION
7 FIRST STEP
8 HOW TO WRITE A REPORT
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TIMETABLE and DEADLINE → GROUP A
Statistical Mechanics: The Thermal Expansion of MgOMon Tue Thur Fri
2:00-5:00 08/03/10 09/03/10 11/03/10 12/03/10
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TIMETABLE and DEADLINE → GROUP A
Statistical Mechanics: The Thermal Expansion of MgOMon Tue Thur Fri
2:00-5:00 08/03/10 09/03/10 11/03/10 12/03/10
DEADLINE:
when? 13:00pm on Wed the 17th of Marchwhere? ROOM 442 to Ms Lisa Benbow
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TIMETABLE and DEADLINE → GROUP B
Statistical Mechanics: The Thermal Expansion of MgOMon Tue Thur Fri
2:00-5:00 01/02/10 02/02/10 04/02/10 05/02/10
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TIMETABLE and DEADLINE → GROUP B
Statistical Mechanics: The Thermal Expansion of MgOMon Tue Thur Fri
2:00-5:00 01/02/10 02/02/10 04/02/10 05/02/10
DEADLINE:
when? 13:00pm on Wed the 10th of Februarywhere? ROOM 442 to Ms Lisa Benbow
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AIM:
The Thermal Expansion of MgO
to predict how the material expands when heated;
to calculate the thermal expansion coefficient:α = 1
V0
(
∂V∂T
)
P
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AIM:
The Thermal Expansion of MgO
to predict how the material expands when heated;
to calculate the thermal expansion coefficient:α = 1
V0
(
∂V∂T
)
P
QUASI-HARMONIC APPROXIMATION (LD)to compute vibrational energy levels of MgO;to understand the phonon dispersion of a material andthe vibrational density of state;
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AIM:
The Thermal Expansion of MgO
to predict how the material expands when heated;
to calculate the thermal expansion coefficient:α = 1
V0
(
∂V∂T
)
P
QUASI-HARMONIC APPROXIMATION (LD)to compute vibrational energy levels of MgO;to understand the phonon dispersion of a material andthe vibrational density of state;
MOLECULAR DYNAMICS (MD)to simulate the vibrations as random motions of atomsinside a cell;
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AIM:
The Thermal Expansion of MgO
to predict how the material expands when heated;
to calculate the thermal expansion coefficient:α = 1
V0
(
∂V∂T
)
P
QUASI-HARMONIC APPROXIMATION (LD)to compute vibrational energy levels of MgO;to understand the phonon dispersion of a material andthe vibrational density of state;
MOLECULAR DYNAMICS (MD)to simulate the vibrations as random motions of atomsinside a cell;
to compare QUASI-HARMONIC APPROXIMATION withMOLECULAR DYNAMICS results.
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SYSTEMS: MgO crystal (fcc)
IDEAL, NON DEFECTIVE, PERIODIC SYSTEM IN 3D
CONVENTIONAL CELLac=bc=cc ; αc , βc , γc
Nc : number of atomsVc : volume
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SYSTEMS: MgO crystal (fcc)
IDEAL, NON DEFECTIVE, PERIODIC SYSTEM IN 3D
CONVENTIONAL CELLac=bc=cc ; αc , βc , γc
Nc : number of atomsVc : volume
PRIMITIVE CELLap=bp=cp; αp, βp, γp
Nc : number of atomsVp: volume
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SYSTEMS: MgO crystal (fcc)
IDEAL, NON DEFECTIVE, PERIODIC SYSTEM IN 3D
CONVENTIONAL CELLac=bc=cc ; αc , βc , γc
Nc : number of atomsVc : volume
PRIMITIVE CELLap=bp=cp; αp, βp, γp
Nc : number of atomsVp: volume
SUPERCELLas=bs=cs= 2 × ac
αs , βs , γs
Ns : number of atomsVs : volume
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SYSTEMS II: lattice parameter of MgO
What is the lattice parameter of MgO?
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SYSTEMS II: lattice parameter of MgO
What is the lattice parameter of MgO?
CONVENTIONAL CELL
ac=bc=cc ; αc , βc , γc
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SYSTEMS II: lattice parameter of MgO
What is the lattice parameter of MgO?
CONVENTIONAL CELL
ac=bc=cc ; αc , βc , γc
ac
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SYSTEMS III
Is it possible to move
from the CONVENTIONAL CELL
to the PRIMITIVE CELL?
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SYSTEMS III
Is it possible to move
from the CONVENTIONAL CELL
to the PRIMITIVE CELL?
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VIBRATIONS
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VIBRATIONS:
WHY ARE THEY IMPORTANT? 1
Atoms vibrate around their equilibrium positions.
thermal properties: heat capacity, expansion
phase transitions, including melting
transport: thermal conductivity, sound
electrical properties, e.g., superconductivity
dielectric phenomena at low frequencies
1From Prof N. M. Harrison’s Lectrure Notes: Vibrations in crystals
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VIBRATIONS: FROM A FINITE SYSTEM
TO AN INFINITE SYSTEM 1
1From Prof N. M. Harrison’s Lectrure Notes: Vibrations in crystals
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VIBRATIONS:
WAVELENGTH - WAVEVECTOR 1
k =2π
λ
In this case, as k increases, the energy of vibration increasesand the frequency too.
1From Prof N. M. Harrison’s Lectrure Notes: Vibrations in crystals
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VIBRATIONS:
PHONON DISPERSION 1
VIBRATIONAL FREQUENCY ω AS A FUNCTION OF k
1From Prof N. M. Harrison’s Lectrure Notes: Vibrations in crystals
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VIBRATIONS:
PHONON DISPERSION 1
VIBRATIONAL FREQUENCY ω AS A FUNCTION OF k
SIMILARLY, ELECTRONIC BAND STRUCTURE OF THEHYDROGEN POLYMER
1From Prof N. M. Harrison’s Lectrure Notes: Vibrations in crystals
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VIBRATIONS:
PHONON?
electron: WAVE or PARTICLE
radiation: WAVE or PARTICLE
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VIBRATIONS:
PHONON?
electron: WAVE or PARTICLE
radiation: WAVE or PARTICLE (photon)
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VIBRATIONS:
PHONON?
electron: WAVE or PARTICLE
radiation: WAVE or PARTICLE (photon)
vibration: WAVE or PARTICLE
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VIBRATIONS:
PHONON?
electron: WAVE or PARTICLE
radiation: WAVE or PARTICLE (photon)
vibration: WAVE or PARTICLE (phonon)
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VIBRATIONS:
1D MONOATOMIC CHAIN (OR POLYMER) 1
INFINITE NUMBER OF VIBRATIONS → BRANCH
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VIBRATIONS:
1D MONOATOMIC CHAIN (OR POLYMER) 1
INFINITE NUMBER OF VIBRATIONS → BRANCH
ωk =
√
4J
M|sin(ka/2)|
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VIBRATIONS:
1D MONOATOMIC CHAIN (OR POLYMER) 1
INFINITE NUMBER OF VIBRATIONS → BRANCH
ωk =
√
4J
M|sin(ka/2)|
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VIBRATIONS:
DIRECT AND RECIPROCAL SPACE
1 PERIODIC DIRECTION IN THE
DIRECT SPACE!
a
1 PERIODIC DIRECTION IN THE
RECIPROCAL SPACE!
a∗ = 2πa
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VIBRATIONS:
1D DIATOMIC CHAIN
(SIMILARLY TO THE POLYMER WITH H2 PER CELL)
FOLDING PROCESS!!!
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VIBRATIONS:
1D HETERO DIATOMIC CHAIN 1
1From Prof N. M. Harrison’s Lectrure Notes: Vibrations in crystals
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VIBRATIONS:
OPTIC AND ACOUSTIC MODES
OPTIC:1) has a frequency that is in the vicinity of the optical region ofthe electromagnentic spectrum2) the atomic motions associated are the same as the responseto an oscillating electromagnentic field
ACOUSTIOC:1) has acoustic frequency
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VIBRATIONS:
LONGITUDINAL AND TRANSVERSE MODES 1
1From Prof N. M. Harrison’s Lectrure Notes: Vibrations in crystals
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VIBRATIONS:
3D HETERO DIATOMIC CRYSTAL 1
Neutron data for GaAs
1From Prof N. M. Harrison’s Lectrure Notes: Vibrations in crystals
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HOW?
COMPUTATIONAL EXPERIMENT / SIMULATION
program
input
Environment:the choice of the Operating System → linux
Interface:DLV = package for the visualisation of materialsstructures and properties.
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SIMULATION TYPES
CLASSICAL SIMULATION
Newton law → GULPQUANTUM-MECHANICAL SIMULATION
Schroedinger equation → CRYSTAL
Systems under investigation
Properties
Accuracy
Computational time
Resources
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CLASSICAL SIMULATION
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INTERATOMIC POTENTIAL
coulombic interaction
short term repusilve contribution
Morse-like potential
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QUASI-HARMONIC APPROXIMATION 1
HELMHOLTZ FREEENERGY
1From Prof N. M. Harrison’s Lectrure Notes: Vibrations in crystals
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QUASI-HARMONIC APPROXIMATION I
HELMHOLTZ FREEENERGY
F = E − TS
F = F (T , V )
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MOLECULAR DYNAMICS 1
Initial configuration and initial velocities:the initial configuration will be that of ideal MgOthe velocities will be random but scaled to produceroughly the target temperature.
Compute the forces on the atoms (F).
Compute the accelerations a=F/m
Update the velocities: Vnew = Vold + a * dt
Update the positions of the atoms: Rnew = Rold + Vnew* dt
Repeat until average properties like E and T settle down
Once settled measure some properties.
1From Prof N. M. Harrison’s Lectrure Notes: Vibrations in crystals
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GEOMETRY OPTIMIZATION I
What does it mean?
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GEOMETRY OPTIMIZATION I
What does it mean?
Minimization of the energy as a function
of the atomic position
(xi , yi , zi , with i = 1, N ,
and N is the number of atoms)
and of the lattice parameters
(a,b,c,α,β,γ)
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GEOMETRY OPTIMIZATION II
How many variables for MgO?
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FIRST STEP
1 Restart the PCs that are running Windows;
2 Once you have a black screen with the following line:> boot:Type linux> boot: linuxPress Enter
3 Use your login and passwd as in Window
4 Open firefox: the web browser
5 http://www.ch.ic.ac.uk/harrison/Teaching/teaching.html
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HOW TO WRITE A REPORT I
by Giulia C. De Fusco
Introduction
the system
the methodology (theory in use)
the aims of the exercise
the tools in use (programs)
Body of the text
write it like a scientific paper (well-articulated sentences,NOT a list of two-word answers)
analyse critically obtained data and given answers
round numerical answers to a specific number of decimalplaces (i.e. 4)
add literature/web citations whenever a comparison withexperimental data is required
add explicitely every formula used one to obtain results
check spelling
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HOW TO WRITE A REPORT II
by Giulia C. De Fusco
Pictures
max 20
reasonably sized (NOT one-page sized pictures, but stillreadable)
white background (follow the instructions given on thewebsite clicking on the link ’How to save a picture for yourreport’)
described in caption or in the text
Graphs
add labels and units
add a critical comment whenever required (NOT a merelydescriptive comment)
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HOW TO WRITE A REPORT III
by Giulia C. De Fusco
Tables
add labels and units
round numerical answers to a specific number of decimalplaces (i.e. 4)
repeat heading if the table cannot fit in a single page
Conclusions
give a general description of your calculations and yourmain findings
outline the differences between the methods in use and theresults obtained
analyse critically these differences
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THANK YOU!!!