The Use of Density Functional Theory in the

Calculation of Thermodynamic Properties

Dr Andrew Scott, University of Leeds, UK

Hume-Rothery Seminar, Derby, 2016

Materials Chemistry Committee

Contents

Introduction

Density functional theory

- background

- accuracy – exchange correlation functional

- precision – k-point mesh, basis set expansion

- geometry optimisation

Case studies

- Ni, V

- Ni3V

- Ni,V sigma phase

Computational cost

Summary

- Accurate thermodynamic data and phase diagrams are vital in materials research

- Data historically provided by experimental determination of properties

- Accurate data determined directly from theory without any empirical input – the so-called ab-

initio methods – can been used as an important additional source of information

- Density functional theory suitable for studying all elements and compounds and shown to

provide accurate property data for metals, alloys, ionic, covalent and molecular systems.

- Properties such as heats of formation, phase stabilities, surface energies and elastic constants

can be obtained from known stable or metastable structures

- Information added to experimental data to improve phase diagram assessments using the

CALPHAD method

Exact – all terms known except the exchange correlation potential, Vxc

Each electron is surrounded by its own mutual exclusion zone or hole.

The exact shape of this hole is only known for a free electron gas and this is the

basis of the first approximation to this term, the local density approximation

(LDA) (Kohn, 1965)

)()()]()([)(2

22

rErrVrVrm

xcH

Ab-initio methods are based on quantum theory

Schrodinger equation (1926) – exact, essentially impossible to solve for

systems with more than one electron

Density functional theory (1964) – Hohenberg and Kohn

‘simplified’ – now based on electron density

Walter Kohn

Nobel Prize, 1998

1887-1961

Exchange-correlation potential – gives the accuracy of the calculation

Local density approximation (LDA)

- assumed to have limited applicability – metallic bonding

- proved to be successful for a whole range of ionic, covalent and metallic materials

- limitations

- tends to ‘over-bind’ – lattice parameters/ bonds too short cf expt

- underestimates band gaps in semi-conductors and insulators

- incorrectly predicts the ground state of structure of iron to be non-magnetic and fcc

Generalised gradient approximations (GGA)

- ‘improved’ exchange-correlation functionals developed from early 1990s

Dispersion corrected functionals – molecular systems

Hybrid functionals

You should always review the literature, talk to colleagues and test the validity of the

approximation for your system of interest.

Numerical precision

k-points

- DFT calculations run under periodic boundary

conditions i.e. a unit cell.

- Ideally, we would calculate the electron

density at all points in the unit cell.

- computationally expensive

- use a mesh of points – (k-points) – and

converge the property of interest (e.g.

energy) with the k-mesh density

fcc Pd: Energy convergence with k mesh density

Kinetic energy cut off = 200 eV

-3205.60

-3205.40

-3205.20

-3205.00

-3204.80

-3204.60

-3204.40

-3204.20

-3204.00

-3203.80

-3203.60

2 3 4 5 6 8 10 16 32

K point mesh

En

erg

y (

eV

)

-25.00

-20.00

-15.00

-10.00

-5.00

0.00

5.00

10.00

15.00

20.00

25.00

Err

or

(kJ/m

ol)

Convergence error

Energy

Numerical precision

basis set size

- expansion of the wavefunction term

- many codes this is an energy term, the

kinetic energy cut-off

- as with the k-points, this term is

increased until the property is converged

fcc Pd: Energy convergence with kinetic energy cut-off (k point mesh = 6x6x6)

-3207.00

-3206.90

-3206.80

-3206.70

-3206.60

-3206.50

-3206.40

-3206.30

-3206.20

-3206.10

-3206.00

250 300 350 400 450 500 550 600 650 700 750 800 850

Kinetic energy cut-off (eV)

En

erg

y (

eV

)

-5.00

-4.50

-4.00

-3.50

-3.00

-2.50

-2.00

-1.50

-1.00

-0.50

0.00

Err

or

(kJ/m

ol)

Energy

Convergence error

Our group typically converges to match experimental heats of formation errors of

approximately 0.5 kJ/mol of atoms.

BiPd geometry optimisation

-95.0

-94.0

-93.0

-92.0

-91.0

-90.0

-89.0

-88.0

-87.0

0 5 10 15 20 25 30 35

Step

En

thalp

y f

orm

ati

on

<B

iPd

> k

J/m

ol

Geometry/ structure optimisation

- should use energies from minimum energy

structures

- optimise lattice parameters/ internal atoms

positions to minimise forces

- optimised structures will be close (~+/- 2%) to

known parameters

- vital for unknown structures, defects, surface etc

Historical evolution of the predicted equilibrium lattice parameter for silicon Kurt Lejaeghere et al. Science 2016;351:aad3000

Codes using the same exchange-correlation functional now give the same answer

Early work may have been compromised by limitations of available computing power

Case studies

Elements – phase/ lattice stabilities

- vanadium

- nickel

Simple compound –

- Ni3V – heat of formation

Sigma phase, Ni,V

- 3 sub-lattice model

- 5 sub-lattice model

Structure databases

e.g. http://icsd.cds.rsc.org/

DFT code, parameters

- calculations were performed using the pseudopotential code, Castep

- GGA-PBE exchange-correlation functional

- k-point density 0.04 Å-1 and kinetic energy cut-off of 440 eV giving good numerical

precision

- ‘on the fly’ ultrasoft pseudopotentials were used for all the elements

- calculations were spin polarized and all structures were geometry optimized

Lattice

parameter

Energy Lattice

stability

(DFT)

Lattice

stability

(SGTE)

Å eV/atom kJ/(mole

atoms)

kJ/(mole

atoms)

bcc a=3.0016

(-0.9%)

-1951.77778 0

fcc a=3.8206 -1951.51756 +25.1 7.5

hcp a=2.8125

c=4.0737

-1951.44057 +32.5 4.0

DFT predicts vanadium to be body centred cubic and non-magnetic bcc V

fcc Ni

Lattice

parameter

Energy Lattice

stability

(DFT)

Lattice

stability

(SGTE)

Å eV kJ/(mole

atoms)

kJ/(mole

atoms)

fcc a=3.520

(-0.1%)

-1375.96867 0 0

hcp a=2.4893 -1375.94478 +2.3 +2.9

c=4.0823

bcc a=2.8020 -1375.87072 +9.5 +8.0

DFT predicts nickel to be face centred cubic and magnetic

Turchi, 2007, addresses the discrepancies between ab-initio and CALPHAD lattice

stabilities, with the conclusion that ab-initio data from dynamically unstable structures are

not valid. (can discuss further at coffee)

body centred tetragonal Ni3V

Heat of formation of Ni3V

Lattice

parameter

Heat of formation

Å kJ/(mole atoms)

a=3.5312

(-0.3%)

-23.9 CASTEP - DFT This work

c=7.2157

(-0.1%)

-21.7 Calorimetry Gao, 1995

-23.9 VASP - DFT Colinet, 2002

Ni-V sigma phase – 30 atom unit cell, 3 sub-lattice model

10AB 16AB 4A a (Å) c (Å) ΔHf,

0 K (kJ/mol)

V10 V16 V4 V1.00

Ni0.00

9.213 4.742 4.7

Ni10 V16 V4 V0.67

Ni0.33

9.081 4.600 -9.2

V10 Ni16 V4 V0.47

Ni0.53

8.762 4.716 -11.3

Ni10 Ni16 V4 V0.13

Ni0.87

8.609 4.607 -2.8

Ni10 Ni16 Ni4 V0.00

Ni30

8.575 4.555 10.7

Initial results

-15.0

-10.0

-5.0

0.0

5.0

10.0

15.0

0 20 40 60 80 100

%V (Ni,V sigma)

Heat

Fo

rmati

on

(kJ/m

ol)

Theory

Expt. (Watson)

T(K)

X(V)

700

1100

1500

1900

2300

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Ni V

Ni-V sigma phase – 5 sub-lattice model – on-going work

Computational cost

- calculation time scales with the cube of

the number of atoms (electrons) in the

unit cell

- time dependent on k-points, basis set

size, complexity of x-c functional

pseudopotential codes

- only consider the valence electrons in

the calculation, to reduce the compute

time – similar accuracy cf all-electron

- parallel calculations – multi-core (high

memory) desktop or high performance

computer

Scaling with number of atoms

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 20 40 60 80 100 120 140 160

Number of atoms

Tim

e (

s)

0

5

10

15

20

25

30

35

40

45

50

0 5 10 15 20 25 30 35 40

Number of atoms

Tim

e (

s)

1997 Silicon Graphics O2 (muffin)

1 core, memory 64Mb (£7,550)

2002 Leeds Grid Node 1 (maxima)

60 cores, total memory 168GB

2002 Leeds Grid Node 2 (Snowdon)

282 cores, total memory 282Gb, Myrinet B

2005 Leeds Grid Node 3 (Everest)

444 cores, total memory 1008Gb, Myrinet F

2008 National Grid Service (Leeds)

160 cores (Gb) total memory ~224Gb

2010- Leeds ARC1

2592 cores (1.5-2Gb), Infiniband,

2013- N8 (Leeds) Polaris

5056 cores (4Gb), 256 cores (16Gb),InfiniBand

2013 iDataplex - Blue Wonder

8192 cores; 64 TB memory, 158.7 TeraFlop/s

2014 - ? ARC2, ARC3, N8-2? + multicore, large memory pc

There are many DFT codes available for the researcher. What factors should be considered

when deciding on the code to use?

i) cost – free academic codes to commercial codes

ii) support – large user base (listserver)/ workshops

iii) ongoing development – latest X-C functionals, optimized to run in parallel on latest

computing architecture

iv) good interface to aid setting up the calculation and analyzing the results

Recommended: the all-electron code, WIEN2k (Blaha, 2001) and the pseudopotential codes,

Castep (Clark, 2005) and VASP (Kresse, 1996)

Summary

DFT can give ground state (0 K) properties:

- lattice parameters

- bulk energies

- lattice/phase stabilities

- elastic properties

- surface energies

Temperature dependent calculations possible but at computational cost

Excellent accuracy in most cases for stable or metastable structures. Systems of 10s of atoms

can be studied on a desktop pc and 100s of atoms on a parallel high performance computer.

Thermodynamic data used by CALPHAD community to improve existing phase diagram

assessments

All work in collaboration Dr Andy Watson