dftb calculations on the structures and properties of … calculations on the structures and...

DFTB calculations on the structures and DFTB calculations on the structures and properties of MOFsproperties of MOFs

Introduction to computational exercisesIntroduction to computational exercises

Agnieszka Kuc1, Andrey Enyashin2, Gotthard Seifert3

1 School of Engineering and Science, Jacobs University, Bremen2 Inst Solid State Chem., Russian Acad Sci., Ekaterinburg3 Physikalische Chemie, Technische Universität, Dresden

WorkshopDresden, 24-25.02.2009

What are Computer Simulations?

● Computer simulations use computers to 'solve' numerically the

equations that govern a certain property or process

● Simulations present in everyday life (e.g. weather forecasts,

simulations in finances, etc.)

● Simulations of materials tell us in which way the building blocks

interact with one another and with environment, determine the

internal structure, the dynamic processes and the response to

external factors (T, p, electric and magnetic field, etc.)

Why are Simulations Interesting?

● Complement and alternative to experimental research

● Increasing scope and power with improving computer and codes

● Scale up with increase of computer power

● Experiments sometimes limited (control of conditions,

interpretation) and generally expensive

Step of a Simulation

● A model of the interactions between the 'blocks' that build the

material (covalent bonds, van der Waals interactions, etc.) and the

theory level (DFT, DFTB, etc.)

● A simulation algorithm - numerical solutions to the equations that

describe the model (geometry optimization, molecular dynamics,etc.)

● A set of tools for the analysis of the results of the simulation (e.g.

visualization programs)

Simulation - Questions

● What interactions model should I use (level of theory)?

● How do I begin the simulation?

● How many molecules or how big fragment do I need to consider?

● What is the size of my simulation box?

Simulation - Questions

● What interactions model should I use (level of theory)?

● How do I begin the simulation?

● How many molecules or how big fragment do I need to consider?

● What is the size of my simulation box?

Plan your simulation

well!

Simulation Starts

Let us start

Metal-Organic Frameworks

> 500 various MOFs already synthesized

Panella et al., Adv. Funct. Mater. 16 (2006) 520

IRMOF-1Li et al., Nature 402 (1999) 276

Cu-BTCChui et al., Science 283 (1999) 1148

Krawiec et al., Adv. Eng. Mater. 8 (2006) 293

Panella et al., Adv. Funct. Mater. 16 (2006) 520

Yaghi et al., Nature. 378 (1995) 703

Isoreticular chemistry

MOF Structures

metal-oxide-based connector

+

+

polycyclic-hydrocarbon-based linker

MOF Structures


IRMOF-13


+

+


MOF Structures


We can tune pore size and create

isoreticular series of MOFs!IRMOF-13


+

+


Step I

Build the model structure

Building Model Structures

These can often be created directly from a crystallographic data: *.cif files or fractional coordinates

or built using a 3D viewer program such as Molden or HyperChem

Structure Representation

Crystal structure (3D periodic object) represented with a unit cell

(simulation box)

bulk

Structure Representation

Crystal structure (3D periodic object) represented with a unit cell

(simulation box)

unit cellbulk

Periodic Boundary Conditions


Bloch's theorem

The eigenstates of the one-electron

Hamiltonian in a periodic potential has the

form of a plane wave times a function with

the periodicity of the Bravais lattice

Ψnk(r) = eik·runk(r)


Bloch's theorem

The eigenstates of the one-electron

Hamiltonian in a periodic potential has the

form of a plane wave times a function with

the periodicity of the Bravais lattice

Ψnk(r) = eik·runk(r)

● εn(k)= ε

n(k + K), periodic with periodicity K of a reciprocal lattice vector

● εn vary continuously with wavevector k - an energy band with band index n

● εn are periodic in k, all distinct values of ε

n(k) occur for k-values within the

first Brillouin zone of the reciprocal lattice

k-point Sampling

Essential for small cells

Real space Reciprocal space

Large cells can be represented with only one k-point

Γ point k = (0,0,0)

• Sampling all permitted electronic states requires convergence with respect to k-point sampling

• A Monkhorst-Pack - regular grid in reciprocal space, reduced by symmetry operations to special 'k-points'

• Insulators - convergence often achieved using Γ-point only

• Conductors - band structures much more complex, multiple k-points required

Step II

Choose the level of theory

Level of Theory

For solid state we can do DFT or DFTB calculations

Here we will use previously described DFTB method(much faster, results in good agreement with DFT)

Step III

Choose the simulation algorithm

Geometry Optimization

● Minimum energy configuration (global or local)

● T = 0K (no account for temperature)

● No information on real dynamics

Global minimum

Local minimum


• Minimize forces on atoms - same algorithms as for gas phase model optimization

• Optimize a, b, c parameters of unit cell according to Bravais lattices – minimize stress

(force/area) on the cell

• Optimize atom positions simultaneously with cell parameters




Global minimum

Local minimum


• Minimize forces on atoms - same algorithms as for gas phase model optimization

• Optimize a, b, c parameters of unit cell according to Bravais lattices – minimize stress

(force/area) on the cell

• Optimize atom positions simultaneously with cell parameters

• Steepest Descent

• Conjugate Gradient Methods

• Newton-Raphson Methods

di = -g

i + β

id

i-1

mixture of current and previous

negative gradient search direction




Global minimum

Local minimum

Electronic Properties

Band structure

● εn(k) – electronic band structure - collection of energy eigenstates within the

first Brillouin zone

● Energy is plotted versus k for particular trajectories in k-space usually between

high-symmetry points in the Brillouin zone

● Resulting lines represent the allowed energy states – energy bands


band diagram of Si

(right) reciprocal lattice (bcc) and the first Brillouin zone of the fcc lattice (left)

Band structure

● εn(k) – electronic band structure - collection of energy eigenstates within the

first Brillouin zone

● Energy is plotted versus k for particular trajectories in k-space usually between

high-symmetry points in the Brillouin zone

● Resulting lines represent the allowed energy states – energy bands


Density of electron states

● Number of states in unit energy interval [Dn(ε)] obtained from the band

structure εn(k) via an integration in k-space over the Brillouin zone of a constant

energy surface [Sn(ε)] of the band structure

Dn(ε) = 2/(2π)3 ∫Sn(ε) [dS/∇εn(k)]

Dn(ε) has units per energy

per volume

Mechanical properties

● Estimation of bulk and/or shear moduli

● Elastic constants (stiness) cij calculated using the finite-difference

scheme (derivatives of the total energy with strain εi,j)

δ/δεi·(δE/δε

j) = c

ij

● The matrix of constants used to obtain the bulk modulus

B = 1/9[c11 + c22 + c33 + 2 (c12 + c13 + c23)] (orthorhombic lattices)

or

B = [Δc33 + 2c13]/[Δ + 2] (hexagonal lattices)

where Δ = [c11 + c12 − 2c13]/[c33 − c13]

● The shear modulus G can be calculated according to

G = 1/15 [(c11 + c22 + c33 − c12 − c13 − c23) + 3 (c44 + c55 + c66)]

Step IV

Analyze the results

Results and Discussion

Zn4O

IRMOF-1

+

PAH/PH

We kept connector unchanged and introduced various

linkers – IRMOF series

(based on the same cubic topology)

Structures of interest

Results and DiscussionKuc et al., J. Phys. Chem. B 111 (2007) 8179

Linkers: polycyclic hydrocarbons and carbon cages

More than 30 structures forming isoreticular series of IRMOFs studied!

Results and DiscussionKuc et al., J. Phys. Chem. B 111 (2007) 8179

IRMOF-M0(reference system) = 1.19 g cm-3

= 4.17eVB = 24.3 GPaE = -313.78 kJ mol-1

IRMOF-1 = 0.53 g cm-3

= 3.73 eVB = 8.70 GPaE = -309.51 kJ mol-1

IRMOF-M1b = 0.32 g cm-3

= 2.13 eVB = 3.20 GPaE = -276.78 kJ mol-1

IRMOF-8 = 0.42 g cm-3

= 2.83 eVB = 2.40 GPaE = -299.77 kJ mol-1

● Small bulk moduli: 0.5 – 24 GPa

● Semiconductors/insulators: band gaps 1 – 5 eV

● Same charge distribution in the crystal structure and in building blocks

Formation energy

Zn4O(OH)

6 + 3R(COOH)

2 Zn

4OR

3(COO)

6 + 6H

2O

Results and Discussion

● Bimodal pore size distribution

- Larger: 3D interconnected square

channels

- Smaller: triangular cavities

● Properties very similar to Zn-IRMOFs

Cu2 paddle-wheel View along [111] direction

View along [100] direction

Chui et al., Science 283 (1999) 1148Krawiec et al., Adv. Eng. Mater. 8 (2006) 293

Cu-BTC: Cu3(BTC)

2(H2O)

3 (BTC: 1,3,5-benzenetricarboxylate)

Conclusions

● Description of simulation steps was given

- choose and build a model (simulation box)

- choose the level of theory according to expected interactions

- choose the algorithm for calculations

- analyze results with additional tools (e.g. visualization programs)

● Crystal structures can be represented with elementary unit cell

● K-points or Γ-point approximation used in calculations

● Results of calculations on structure and properties of Metal-Organic

Frameworks discussed on exemplary structures

● DFTB is a fast method that gives results in agreement with experiment

and DFT calculations

Acknowledgments

Knut Vietze for computational assistance

Support of this research is acknowledged to DFG

Calculations partially performed in ZIH Dresden

Figures made with GTK Display Interface for Structures 0.89

Thank you for your attention!

Recommended Literature

'Computational Chemistry: A Practical Guide for Applying

Techniques to Real-World Problems'

D. C. Young, Wiley-VCH 2001

'Introduction to Computational Chemistry'

Frank Jensen, Wiley-VCH 1999

'Computational Chemistry Workbook – Basics'

T. Heine, J.-O. Joswig, A. Gelessus, Wiley-VCH, in press, scheduled

Juli 2009

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