Chapter 1. Introduction, perspectives, and aims. On the science
of simulation and modelling. Modelling at bulk, meso, and nano
scale. (2 hours).
Chapter 2. Experimental Techniques in Nanotechnology. Theory
and Experiment: “Two faces of the same coin” (2 hours).
Chapter 3. Introduction to Methods of the Classic and Quantum
Mechanics.
Force
Fields,
Semiempirical,
Plane-Wave
pseudopotential calculations. (2 hours)
Chapter 4. Introduction to Methods and Techniques of Quantum
Chemistry, Ab initio methods, and Methods based on Density
Functional Theory (DFT). (4 hours)
Chapter 5. Visualization codes, algorithms and programs.
GAUSSIAN; CRYSTAL, and VASP. (6 hours)
.
Chapter 6. Calculation of physical and chemical properties of
nanomaterials. (2 hours).
Chapter 7. Calculation of optical properties. Photoluminescence.
(3 hours).
Chapter
8.
Modelization
of
the
growth
mechanism
of
nanomaterials. Surface Energy and Wullf architecture (3 hours)
Chapter
9. Heterostructures Modeling. Simple and complex
metal oxides. (2 hours)
Chapter 10. Modelization of chemical reaction at surfaces.
Heterogeneous catalysis. Towards an undertanding of the
Nanocatalysis. (4 hours)
Chapter 5. Visualization codes,
algorithms and programs
Lourdes Gracia y Juan Andrés
Departamento de Química-Física y Analítica
Universitat Jaume I
Spain
&
CMDCM, Sao Carlos
Brazil
Sao Carlos, Novembro 2010
PROGRAM
- CRYSTAL
Graphical Interface
XCrysDen, Jmol
Ana-Band-DOS
- VASP
Molden
- GAUSSIAN
GaussView
CRYSTAL
CRYSTAL performs ab initio calculations on periodic systems within the
linear combination of atomic orbitals (LCAO) approximation. That is, the
crystalline orbitals (CO) are treated as linear combinations of Bloch
functions (BF),
defined in terms of local functions, hereafter indicated as atomic
orbitals (AO). Those local functions are expressed as linear
combination of a certain number of Gaussian type functions (GTF).
The "CRYSTAL tutorial project" :
http://www.theochem.unito.it/crystal_tuto/mssc2008_cd/tutorials/index.html
GEOMETRY
- The bulk structure (conventional cell, primitive cell)
- Creating a super cell
- Symmetry and geometry editing
- Removal, addition, substitution, displacement of atoms
- 2D input; the slab model
input keywords
ATOMSUBS substitution of atoms
ATOMREMO removal of atoms
ATOMINSE addition of atoms
ATOMDISP displacement of atoms
ATOMROT rotation of groups of atoms
SUPERCEL generation of super cell
SLABCUT generation of a slab parallel to a given plane
SLABCUT generation of a slab parallel to a given plane
Planes with different Miller indices in cubic crystals
(ℓmn) denotes a plane that intercepts the three points a1/ℓ, a2/m, and a3/n
Basis set
http://www.crystal.unito.it/Basis_Sets/Ptable.html
ECPs
The idea behind pseudopotentials is to treat the core electrons as effective
averaged potentials rather than actual particles. Thus, pseudopotentials are
modifications to the Hamiltonian.
ECP
Keyword
Hay and Wadt large core
HAYWLC
Hay and Wadt small core
HAYWSC
Durand and Barthelat
BARTHE or DURAND
Density Functional Theory (DFT) methods
The keyword DFT
selects a DFT Hamiltonian. Exchange-correlation
functionals are separated in an exchange component
(keyword
EXCHANGE) and a correlation component (keyword CORRELAT).
Hybrid: the exchange functional is a linear combination of Hartree-Fock,
local and gradient-corrected exchange term
B3PW
EXCHANGE
BECKE
CORRELAT
PWGGA
HYBRID
20
NONLOCAL
0.9 0.81
B3LYP
EXCHANGE
BECKE
CORRELAT
LYP
HYBRID
20
NONLOCAL
0.9 0.81
% of Hartree-Fock exchange
weight of non local exchange and correlation
EXAMPLE
A PZT ten-layer slab model of the PT(100) surface. Substitution:40%Zr and 60% Ti
PZT-4060
CRYSTAL
000
P4MM
99
4.017 4.14
4
282 0.0 0.0 -0.029665
8 -0.5 -0.5 0.126996
8 0.0 -0.5 -0.373995
22 -0.5 -0.5 -0.479824
SLABCUT
slab model
100
1 10
ten layers
ATOMSUBS
2
substitution
3 40
23 40
OPTGEOM
ENDOPT
ENDG
pseudopotential
282 2
DURAND
0 1 3 2.0 1.0
0 1 1 2.0 1.0
22 7
0 0 8 2. 1.
0 1 6 8. 1.
0 1 4 8. 1.
0 1 1 2.0 1.0
0 1 1 0.0 1.0
0 3 4 2.0 0.972
0 3 1 0.0 1.0
84
0 0 6 2.0 1.0
0 1 3 6.0 1.0
0 1 1 0.0 1.0
0 3 1 0.0 1.0
40 8
0 0 9 2. 1.
0 1 7 8. 1.
0 1 6 8. 1.
0 1 3 8. 1.
0 3 6 10. 1.
0 1 1 2. 1.
0 3 2 2. 1.
0 3 1 0. 1.
99 0
END
DFT
B3LYP
END
SCFDIR
TOLINTEG
8 8 8 8 14
SHRINK
44
LEVSHIFT
31
MAXCYCLE
100
FMIXING
30
END
Hybrid functional
Coulomb and Exchange series tolerances
Pack-Monkhorst and Gilat shrinking factors
level shifter used to help convergence
% of hamiltonian matrix mixing to help convergence
OUTPUT
************************************************************************************
LATTICE PARAMETERS (ANGSTROMS AND DEGREES) - BOHR = 0.5291772083 ANGSTROM
PRIMITIVE CELL
A
B
C
ALPHA
BETA
GAMMA
4.01700000 4.14000000 500.00000000 90.000000 90.000000 90.000000
*******************************************************************************
ATOMS IN THE ASYMMETRIC UNIT 25 - ATOMS IN THE UNIT CELL: 25
ATOM
X/A
Y/B
Z(ANGSTROM)
*******************************************************************************
1T 8O
-5.000000000000E-01 1.269960000000E-01 9.038250000000E+00
2T 8O
0.000000000000E+00 -3.739950000000E-01 9.038250000000E+00
ZrO2
3 T 40 ZR -5.000000000000E-01 -4.798240000000E-01 9.038250000000E+00
4 T 282 PB 0.000000000000E+00 -2.966500000000E-02 7.029750000000E+00
PbO
5T 8O
-5.000000000000E-01 -3.739950000000E-01 7.029750000000E+00
6T 8O
-5.000000000000E-01 1.269960000000E-01 5.021250000000E+00
TiO2
7T 8O
0.000000000000E+00 -3.739950000000E-01 5.021250000000E+00
8 T 22 TI -5.000000000000E-01 -4.798240000000E-01 5.021250000000E+00
9 T 282 PB 0.000000000000E+00 -2.966500000000E-02 3.012750000000E+00
PbO
10 T 8 O -5.000000000000E-01 -3.739950000000E-01 3.012750000000E+00
11 T 8 O -5.000000000000E-01 1.269960000000E-01 1.004250000000E+00
TiO2
12 T 8 O
0.000000000000E+00 -3.739950000000E-01 1.004250000000E+00
13 T 22 TI -5.000000000000E-01 -4.798240000000E-01 1.004250000000E+00
14 T 282 PB 0.000000000000E+00 -2.966500000000E-02 -1.004250000000E+00
PbO
15 T 8 O -5.000000000000E-01 -3.739950000000E-01 -1.004250000000E+00
16 T 8 O -5.000000000000E-01 1.269960000000E-01 -3.012750000000E+00
TiO2
17 T 8 O
0.000000000000E+00 -3.739950000000E-01 -3.012750000000E+00
18 T 22 T I -5.000000000000E-01 -4.798240000000E-01 -3.012750000000E+00
19 T 282 PB 0.000000000000E+00 -2.966500000000E-02 -5.021250000000E+00
PbO
20 T 8 O -5.000000000000E-01 -3.739950000000E-01 -5.021250000000E+00
21 T 8 O -5.000000000000E-01 1.269960000000E-01 -7.029750000000E+00
ZrO2
22 T 8 O
0.000000000000E+00 -3.739950000000E-01 -7.029750000000E+00
23 T 40 ZR -5.000000000000E-01 -4.798240000000E-01 -7.029750000000E+00
24 T 282 PB 0.000000000000E+00 -2.966500000000E-02 -9.038250000000E+00
PbO
25 T 8 O -5.000000000000E-01 -3.739950000000E-01 -9.038250000000E+00
Spin-polarized systems
an unrestricted calculation must be performed
- UHF in input block 3 for HF hamiltonian
- SPIN in DFT input block for DFT hamiltonian
preparing an SCF guess driving the system to the desired spin state.
input keywords
DFT
B3LYP
SPIN
END
…
lock in a given spin state
SPINLOCK
2 2000
alpha-beta electrons locked to 2 for 2000 scf cycles
ATOMSPIN
21121
atom 1 and 2 have formally the same spin in the atomic wave function
NiO – Anti ferromagnetic phase - 2 electrons up, 2 electrons down, total spin 0
CRYSTAL
011
225
4.164
2
28 0. 0. 0.
8 .5 .5 .5
SUPERCEL
011 101 110
END
basis set input
END
UHF
TOLINTEG
7 7 7 7 14
END
Pack-Monkhorst and Gilat shrinking factors
8 8
TOLENE
7
LEVSHIFT
31
SPINLOCK
0 50
2 Nickel atoms with antiparallel spins. Total spin 0
ATOMSPIN
2
atom 1 and 2 have formally opposite spin in the atomic wavefunction
1 1 2 -1
MAXCYCLE
90
END
Example: antiferromagnetic phase (NiO)
Frequency calculation
output
input
Vibrational frequencies
Jmol interface
http://www.theochem.unito.it/crystal_tuto/mssc2008_cd/jmoledit/index.html
Frequency calculation
input
output
Electron properties
•Atoms and bond populations (Mulliken scheme)
PPAN
•Electron Charge Density
ECHG
XCrysDen
•Band Structure
BAND
ANA-BAND-DOS
DOSS
•Density of States
Here are the total atomic charges obtained at different level of theory:
HF
LDA
GGA
B3LYP
Mg
10.021
10.123
10.104
10.091
O
9.979
9.877
9.896
9.909
Note that Mulliken population analysis is an arbitrary scheme for partitioning total
electron charge in atom and bond contributions. Atomic charges, unlike the
electron density, are not a quantum mechanical observable, and are not
unambiguously predictable from first principles.
ECHG → XCrysDen
CRYSTAL computes the charge density in a grid of points defined in input.
-total electron density maps
-difference maps: difference between the crystal electron density
and a "reference" electron density.
Nº of point along the
B-A segment
Input ECHG
ECHG
0
65
COORDINA
-4. -4. 0.0
4. -4. 0.0
4. 4. 0.0
END
ECHG → XCrysDen
Example: Slab PZT
ECHG → XCrysDen
Example: Bulk ST
Supercell 2x2x2
Band Structure: ANA-BAND
by Nélio H. Nicoleti, POSMAT, Campus Bauru
Input Band
Band Structure: Plot with Origin
.dat
-points
Fermi energy (eV)
Fermi energy (-4.04 eV) scaled at 0
8
6
E (eV)
4
2
0
-2
-4
-6

k1
k2
k3
k4
k5

Density of States: ANA-DOS
Only T of .out
R points of .out
Density of States: ANA-DOS
Input: DOS total
projection onto all the AOs
calculation of eigenvectors
shrinking factor for reciprocal space Pack-Monkhorst net
1: evaluation of the Fermi level with the new k-points net
0: no print options
keyword
3:
number of projections
80: number of points along the energy axis in which the
DOSS is calculated;
20: first band
30: last band
1:
plot option (if 1, the program stores the data in fort.25);
15: degree of the polynomial used for the DOSS expansion;
0:
printing option
Output: DOS
.inf from ANA-DOS
Input: DOS atomico
.inf
dxy
dxz
dy 2
dz 2
dx 2 - y 2
.dat plotted with Origin
B
C
D
E
Pb
O
Ti
tot
DOS
total DOS
DOS projected on atoms
-8
-6
-4
-2
0
2
4
dxy
dy2
dz2
dx2-y2
E (eV)
DOS projected on
atomic orbitals of Ti
-8
-6
-4
-2
E (eV)
0
2
4
VASP
complex package for performing ab-initio quantum-mechanical molecular
dynamics (MD) simulations using pseudopotentials or the projector-augmented
wave method and a plane wave basis set.
• The approach is based on the (finite-temperature) local-density approximation
with the free energy as variational quantity and an exact evaluation of the
instantaneous electronic ground state at each MD time step.
• VASP uses efficient matrix diagonalisation schemes and an efficient
Pulay/Broyden charge density mixing.
• The interaction between ions and electrons is described by ultra-soft
Vanderbilt pseudopotentials (US-PP) or by the projector-augmented wave
(PAW) method. They allow for a considerable reduction of the number of planewaves per atom for transition metals and first row elements.
• Forces and the full stress tensor can be calculated with VASP and used to
relax atoms into their instantaneous ground-state.
Input files for VASP
• POSCAR: contains the lattice geometry and the
ionic positions
• POTCAR: contains the pseudopotential for each
atomic species used in the calculation
• INCAR: central input file of VASP. It determines
'what to do and how to do it'
• KPOINTS: contain the k-point coordinates and
weights or the mesh size for creating the k-point
grid
POSCAR
Cubic STO
3.904
1.0 0.0 0.0
0.0 1.0 0.0
0.0 0.0 1.0
113
Direct
0.5 0.5 0.5
0.0 0.0 0.0
0.5 0.5 0.0
0.0 0.5 0.5
0.5 0.0 0.5
Cubic STO
3.904
1.0 0.0 0.0
0.0 1.0 0.0
0.0 0.0 1.0
113
Selective dynamics
Cartesian
1.952 1.952 1.952 T T F
0.0 0.0
0.0 T F F
1.952 1.952 0.0 T T T
0.0 1.952 1.952 F F F
1.952 0.0
1.952 F F F
scaling factor (lattice constant)
the three lattice vectors defining the unit cell
number of atoms per atomic species
fractional coordinates
crystal
0.5 0.5 0.5 Sr
0.0 0.0 0.0 Ti
0.5 0.0 0.0 O
Only some coordinates of the atom will be
allowed to change during the ionic relaxation
lattice vectors (lattice.f)
POTCAR
Plane waves (PW’s) pseudopotentials
• Natural choice for system with periodic boundary conditions
• It is easy to pass from real- to reciprocal space representation
• No Pulay correction to forces on atoms
• Basis set convergence easy to control
• Electron-ion interaction must be represented by pseudopotentials (US) or
projector-augmented wave (PAW) potentials
Reconstruction of exact wavefunction in the core region
Contains:
- the pseudopotential for each atomic species
- information about the atoms
their mass
their valence electrons
the energy of the reference configuration for which the pseudopotential
was created.
- a default energy cutoff (ENMAX and ENMIN line)
On a UNIX machine, con-cat three POTCAR files:
> cat ~/pot/Al/POTCAR ~/pot/C/POTCAR ~/pot/H/POTCAR >POTCAR
POTCAR
Sr
PAW_PBE
KPOINTS
Automatic mesh
0
Monkhorst-Pack
666
0. 0. 0.
! number of k-points = 0 ->automatic generation scheme
! select Monkhorst-Pack
! size of mesh (6x6x6 points along b1, b2, b3)
! shift of the k-mesh
The number of k-points depends critically on the necessary precision and whether the
system is metallic. Metallic systems require an order of magnitude more k-points than
semiconducting and insulating systems
- For semiconductors or insulators use always tetrahedron method with Blöch
corrections (ISMEAR=-5)
- For relaxations in metals always use ISMEAR=1 (defect). The method of
Methfessel-Paxton (MP) also results in a very accurate description of the total
energy for large super cells
INCAR
GGA = PW | PB| 91 | B3LYP
1 a RMM-DIIS quasi-Newton algorithm is used to relax the ions
IBRION
2 a conjugate-gradient algorithm 1. forces calculated for the initial positions
2. trial (or predictor step)
3. corrector step.
controls whether the stress tensor is calculated
ISIF
ISIF
opt
calculate
calculate
relax
change
change
force
stress tensor
ions
cell shape
cell volume
0
yes
no
yes
no
no
1
yes
trace only
yes
no
no
2
yes
yes
yes
no
no
3
yes
yes
yes
yes
yes
4
yes
yes
yes
yes
no
5
yes
yes
no
yes
no
6
yes
yes
no
yes
yes
7
yes
yes
no
no
yes
OUTCAR
INCAR
freq
calculate the Hessian matrix, finite differences
time-step for ionic-motion
ion is displaced in each direction by a small positive and negative displacement
OUTCAR
spin polarized calculations
ISPIN=2
NUPDOWN
Molden
Visualization of CONTCAR
GAUSSIAN
An electronic structure package capable of predicting many properties
of atoms, molecules, and reactive systems, e.g.
• Energies
• Structures
• Vibrational frequencies
utilizing ab initio, density functional theory (DFT), semi-empirical,
molecular mechanics, and hybrid methods.
Types of Calculations
• single point energy and properties (electron density, dipole moment, …)
• geometry optimization
• frequency
• reaction path following
Modelling chemical reactivity
Gas phase PES
Energy
TS
products
ΔE‡
ΔE
reactants
Potential Energy Surface (PES)
Transition State
Adiabatic surface
Products
Reaction pathway
Reactants
Levels of Theory Available:
– semi-empirical
AM1, PM3, MNDO, …
– density functional theory
B3LYP, MPW1PW91, …
– ab initio
HF, MP2, CCSD, CCSD(T), …
Basis set
The set of underlying approximations
used to describe the chemical system.
Higher levels of theory are often more
accurate however they come at much
greater computational cost.
https://bse.pnl.gov/bse/portal
Input
V=O vanadyl bond
V-O-Ti sites
VTi3O10H3 cluster
The default optimization algorithm included in Gaussian is the "Berny
algorithm" developed by Bernhard Schlegel.
This algorithm uses the forces acting on the atoms of a given structure
together with the second derivative matrix (called the Hessian matrix) to
predict energetically more favorable structures and thus optimize the
molecular structure towards the next local minimum on the potential
energy surface.
For each step of the geometry optimization, Gaussian will write to the output:
a) the current structure of the system,
b) the energy for this structure,
c) the derivative of the energy with respect to the geometric variables (the
gradients),
d) a summary of the convergence criteria.
After each iteration of the geometry optimization, the output files contain a
summary of the current stage of the optimization:
remaining force on an atom
structural change of one
coordinate
RMS (root mean square)= average
Polarizable Continuum Model (PCM)
The solvation model based on the partition of the system into two subsytems, the
molecule under scrutiny (“the solute”) and the "environment". This latter is treated
as a macroscopic and continuous medium characterized by some specific
macroscopic physical properties, its dielectric permittivity.
Tomasi et al.
Decomposition of the PCM free energy:
G = Gel + Gdis + Grep + Gcav
electrostatic
dispersion
repulsion
cavitation
- cavity defined through interlocking van der Waals-spheres centered at atomic
positions. The reaction field is represented through point charges located on the
surface of the molecular cavity
Input options
b3lyp/6-31g* opt scrf=(iefpcm,solvent=benzene)