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 AnalíticaUniversitat Jaume I

Spain&

CMDCM, Sao CarlosBrazil

Sao Carlos, Novembro 2010

- GAUSSIAN

- CRYSTAL

- VASP

Graphical Interface

GaussView

PROGRAM

XCrysDen, JmolAna-Band-DOS

Molden

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).

CRYSTAL

The "CRYSTAL tutorial project" :http://www.theochem.unito.it/crystal_tuto/mssc2008_cd/tutorials/index.html

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

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

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

ECPs

ECP    Keyword

Hay and Wadt large  core  HAYWLC

Hay and Wadt small core   

HAYWSC

Durand and Barthelat    BARTHE or DURAND

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.

Basis set http://www.crystal.unito.it/Basis_Sets/Ptable.html

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

Density Functional Theory  (DFT) methods

B3PW B3LYP EXCHANGE EXCHANGE

BECKE BECKE CORRELAT CORRELAT

PWGGA LYP HYBRID HYBRID

20 20 NONLOCAL NONLOCAL

0.9 0.81 0.9 0.81

% of Hartree-Fock exchange

weight of non local exchange and correlation

EXAMPLEA PZT ten-layer slab model of the PT(100) surface. Substitution:40%Zr and 60% TiPZT-4060CRYSTAL0 0 0994.017 4.144282 0.0 0.0 -0.0296658 -0.5 -0.5 0.1269968 0.0 -0.5 -0.37399522 -0.5 -0.5 -0.479824SLABCUT1 0 01 10ATOMSUBS23 4023 40OPTGEOMENDOPTENDG282 2DURAND0 1 3 2.0 1.00 1 1 2.0 1.022 70 0 8 2. 1.0 1 6 8. 1.0 1 4 8. 1.0 1 1 2.0 1.00 1 1 0.0 1.00 3 4 2.0 0.9720 3 1 0.0 1.0

8 40 0 6 2.0 1.00 1 3 6.0 1.00 1 1 0.0 1.00 3 1 0.0 1.040 80 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 DFTB3LYPEND SCFDIRTOLINTEG8 8 8 8 14SHRINK4 4LEVSHIFT3 1MAXCYCLE100FMIXING30END

P 4 M M

Hybrid functional

pseudopotential

substitution

slab model

ten layers

Coulomb and Exchange series tolerances

level shifter used to help convergence

% of hamiltonian matrix mixing to help convergence

Pack-Monkhorst and Gilat shrinking factors

************************************************************************************ 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) ******************************************************************************* 1 T 8 O -5.000000000000E-01 1.269960000000E-01 9.038250000000E+00 2 T 8 O 0.000000000000E+00 -3.739950000000E-01 9.038250000000E+00 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 5 T 8 O -5.000000000000E-01 -3.739950000000E-01 7.029750000000E+00 6 T 8 O -5.000000000000E-01 1.269960000000E-01 5.021250000000E+00 7 T 8 O 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 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 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 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 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 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 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 25 T 8 O -5.000000000000E-01 -3.739950000000E-01 -9.038250000000E+00

OUTPUT

ZrO2

ZrO2

PbO

PbO

PbO

PbO

PbO

TiO2

TiO2

TiO2

input keywords

DFT B3LYPSPINEND…SPINLOCK2 2000

ATOMSPIN2 1 1 2 1

an unrestricted calculation must be performed- UHF in input block 3 for HF hamiltonian- SPIN in DFT input block for DFT hamiltonian 

Spin-polarized systems

preparing an SCF guess driving the system to the desired spin state.

lock in a given spin state

alpha-beta electrons locked to 2 for 2000 scf cycles

atom 1 and 2 have formally the same spin in the atomic wave function

NiO – CRYSTAL0 1 1 225 4.164 2 28 0. 0. 0. 8 .5 .5 .5 SUPERCEL 0 1 1 1 0 1 1 1 0 END basis set input END UHF TOLINTEG 7 7 7 7 14 END 8 8 TOLENE 7 LEVSHIFT 3 1 SPINLOCK 0 50 ATOMSPIN 21 1 2 -1 MAXCYCLE 90 END

2 Nickel atoms with antiparallel spins. Total spin 0

atom 1 and 2 have formally opposite spin in the atomic wavefunction

Anti ferromagnetic phase - 2 electrons up, 2 electrons down, total spin 0

Pack-Monkhorst and Gilat shrinking factors

Example: antiferromagnetic phase (NiO)

Frequency calculation

input

output

Vibrational frequencies Jmol interface

http://www.theochem.unito.it/crystal_tuto/mssc2008_cd/jmoledit/index.html

Frequency calculation output

input

Electron properties

XCrysDen

•Atoms and bond populations (Mulliken scheme) •Electron Charge Density •Band Structure •Density of States

  HF LDA GGA B3LYP

Mg 10.021 10.123 10.104 10.091

O 9.979 9.877 9.896 9.909

Here are the total atomic charges obtained at different level of theory:  

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.

PPAN

ECHG

BAND DOSS

ANA-BAND-DOS

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

ECHG 0

65 COORDINA -4. -4. 0.0 4. -4. 0.0 4. 4. 0.0

END

Input ECHG

ECHG → XCrysDen

ECHG → XCrysDen

Example: Slab PZT

ECHG → XCrysDen

Example: Bulk STSupercell 2x2x2

Band Structure: ANA-BAND

Input Band

by Nélio H. Nicoleti, POSMAT, Campus Bauru

Band Structure: Plot with Origin

.dat

Fermi energy (eV)-points

Fermi energy (-4.04 eV) scaled at 0

-6

-4

-2

0

2

4

6

8

E (

eV)

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 totalOutput: DOS

1: evaluation of the Fermi level with the new k-points net0: no print options

keyword

3:      number of projections 80: number of points along the energy axis in which the DOSS is calculated;20:    first band30:    last band1:      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

calculation of eigenvectors

shrinking factor for reciprocal space Pack-Monkhorst net

projection onto all the AOs

.inf

Input: DOS atomico

dxy

dxz

dy 2

dz 2

dx 2y

2-

.inf from ANA-DOS

PbOTitot

dxydy2

dz2

dx2-y2

DOS projected on atomic orbitals of Ti

total DOS

DOS projected on atoms

.dat plotted with Origin

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 plane-waves 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.

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

Input files for VASP

Cubic STO 3.904 1.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 1.0 1 1 3 Direct0.5 0.5 0.5 0.0 0.0 0.0 0.5 0.5 0.0 0.0 0.5 0.50.5 0.0 0.5

POSCAR

scaling factor (lattice constant)

the three lattice vectors defining the unit cell

number of atoms per atomic species

fractional coordinates

Only some coordinates of the atom will be allowed to change during the ionic relaxation

Cubic STO3.904 1.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 1.0 1 1 3 Selective dynamics Cartesian1.952 1.952 1.952 T T F0.0 0.0 0.0 T F F 1.952 1.952 0.0 T T T 0.0 1.952 1.952 F F F1.952 0.0 1.952 F F F

0.5 0.5 0.5 Sr0.0 0.0 0.0 Ti0.5 0.0 0.0 O

crystal

lattice vectors (lattice.f)

POTCAR

On a UNIX machine, con-cat three POTCAR files:

Contains: - the pseudopotential for each atomic species - information about the atoms

their masstheir valence electronsthe energy of the reference configuration for which the pseudopotential

was created.- a default energy cutoff (ENMAX and ENMIN line)

> cat ~/pot/Al/POTCAR ~/pot/C/POTCAR ~/pot/H/POTCAR >POTCAR

Reconstruction of exact wavefunction in the core region

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) orprojector-augmented wave (PAW) potentials

POTCAR

Sr PAW_PBE

Automatic mesh0 ! number of k-points = 0 ->automatic generation scheme Monkhorst-Pack ! select Monkhorst-Pack 6 6 6 ! size of mesh (6x6x6 points along b1, b2, b3) 0. 0. 0. ! shift of the k-mesh

KPOINTS

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

1 a RMM-DIIS quasi-Newton algorithm is used to relax the ionsIBRION2 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 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

GGA = PW | PB| 91 | B3LYP

opt

OUTCAR

INCAR

freq

time-step for ionic-motion

OUTCAR

calculate the Hessian matrix, finite differences

ion is displaced in each direction by a small positive and negative displacement

spin polarized calculations

ISPIN=2NUPDOWN

MoldenVisualization 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 reactivityGas phase PES

RC

En

erg

y

ΔE

ΔE‡

d1 / Å

d2 / Å

TS

reactants

products

Potential Energy Surface (PES)

Adiabatic surface

Reactants

Transition State

Products

Reaction pathway

Levels of Theory Available:

– semi-empiricalAM1, PM3, MNDO, …– density functional theoryB3LYP, MPW1PW91, …– ab initioHF, MP2, CCSD, CCSD(T), …

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.

Basis set https://bse.pnl.gov/bse/portal

Input

VTi3O10H3 cluster

V=O vanadyl bond V-O-Ti sites

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:

RMS (root mean square)= average

remaining force on an atom

structural change of onecoordinate

Tomasi et al.

- 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

Decomposition of the PCM free energy:

G = Gel + Gdis + Grep + Gcav

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.

Polarizable Continuum Model (PCM)

electrostatic dispersion repulsion cavitation

b3lyp/6-31g* opt scrf=(iefpcm,solvent=benzene)

Input options