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3.021J / 1.021J / 10.333J / 18.361J / 22.00J Introduction to Modeling and Simulation Spring 2008

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1.021/3.021/10.333/18.361/22.00 Introduction to Modeling and Simulation

Timo Thonhauser

Department of Materials Science and Engineering Massachusetts Institute of Technology

From Many-Body to Single-Particle: Quantum

Modeling of Molecules

3. From Many-Body toSingle-Particle; Quantum Modeling of Molecules

5.

Class outline

2

theory practice

1. It’s A Quantum World: The Theory of Quantum Mechanics

2. Quantum Mechanics: Practice Makes Perfect

4. From Atoms to Solids

Quantum Modeling of Solids: Basic Properties

6. Advanced Prop. of Materials; What else can we do?

7. Review session

3. From Many-Body toSingle-Particle; Quantum Modeling of Molecules

Motivation!

3

? Image of hydrogen orbitals removed due to copyright restrictions. Please see: http://www.kfunigraz.ac.at/imawww/vqm/images/qm/es421.jpg

http://www.kfunigraz.ac.at/imawww/vqm/images/qm/es421.jpg

Lesson outline

4

Review

The Many-body Problem

Density Functional Theory

Computational Approaches

Modeling Software

PWscf Please see: http://www.quantum-espresso.org/

http://www.quantum-espresso.org/

Review: Schrödinger equation

5

H time independent: ψ(�r, t) = ψ(�r) · f(t)

i� ḟ(t)

f(t) =

Hψ(�r)

ψ(�r) = const. = E

ψ(�r, t) = ψ(�r) · e − i � EtHψ(�r) = Eψ(�r)

time independent Schrödinger equation stationary Schrödinger equation

Review: The hydrogen atom

6

Please see: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydwf.html#c3

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydwf.html#c3

Review: The hydrogen atom

7

Please see: http://hyperphysics.phy-astr.gsu.edu/hbase/imgmod/hydspe.gif

http://hyperphysics.phy-astr.gsu.edu/hbase/imgmod/hydspe.gif

r1

Review: Next? Helium!

8

problem!cannot be solved analytically

+

e -

e -

r2

r12

� H1 + H2 + W

� ψ(�r1, �r2) = Eψ(�r1, �r2)

� T1 + V1 + T2 + V2 + W

� ψ(�r1, �r2) = Eψ(�r1, �r2)

� − �

2

2m ∇2 1 −

e2

4π�0r1 − �

2

2m ∇2 2 −

e2

4π�0r2 +

e2

4π�0r12

� ψ(�r1, �r2) = Eψ(�r1, �r2)

Hψ = Eψ

Review: Spin

9

for electrons: spin quantum number can ONLY be

up down

new quantum number: spin quantum number

Review: exclusions principle

10

Two electrons in a system cannot have the same quantum numbers!

hydrogen

1s

2p 3p 3d3s

2s

... ... ... ...quantum numbers: main n: 1,2,3 ...

orbital l: 0,1,...,n-1 magnetic m: -l,...,l

spin: up, down

Review: Periodic table

11

Image from Wikimedia Commons, http://commons.wikimedia.org

http://commons.wikimedia.org

The many-body problem

12

helium: 2e-

iron: 26e-

Image removed due to copyright restrictions. Please see: http://environmentalchemistry.com/yogi/periodic/Fe.html

+

e-

r 1

e -

r 2

r 12

http://environmentalchemistry.com/yogi/periodic/Fe.html

Solutions

13

Moller-Plesset perturbation theory

MP2

coupled cluster theory CCSD(T)

quantum chemistry density

functional theory

computational expense for system size N:

O(N5-N7)

O(N3); O(N)

Why DFT?

14

Quantum Chemistry methods; MP2, CCSD(T)

Density Functional Theory

Examplesilicon

2 atoms/unitcell

DFT: 0.1 sec

MP2: 0.1 sec

CCSD(T): 0.1 sec

silicon 100 atoms/unitcell

DFT: 5 hours

MP2: 1 year

CCSD(T): 2000 years!!!

Why DFT?

15 Figure by MIT OpenCourseWare.

100,000

10,000

1000

100

10

2003 2007 2011 20151

Year

Num

ber o

f Ato

ms

Linear

scali

ng DF

T

DFT

QMC

CCSD(T)

Exact treatment

MP2

wave func

tion:

complicate

d!

electrondensity:easy!

Density functional theory

ψ = ψ(�r1, �r2, . . . , �rN )

n = n(�r)

16

Density functional theory

17

Total energy is a functional of the electron density!

kinetic

ion-ion ion-electron

electron-electron

E[n] = T [n] + Vii + Vie[n] + Vee[n]

Image removed due to copyright restrictions.Please see: http://www.nd.edu/~ed/Images/BmimPF6chrg.jpg

http://www.nd.edu/~ed/Images/BmimPF6chrg.jpg

�

Density functional theory

E[n] = T [n] + Vii + Vie[n] + Vee[n] kinetic ion-ion ion-electron electron-electron

electron density n(�r) = |φi(�r)|2 i

Eground state = min E[n]

φ

Find the wave functions that minimize the

energy using a functional derivative!

18

∑

Density functional theory

Finding the minimum leads to

Kohn-Sham equations

ion potential Hartree potential exchange-correlation potential

equations for non-interacting electrons

19

Density functional theory

20

Only one problem: vxc not known!!! approximations necessary

local density approximation

LDA

general gradient approximation

GGA

3.021j Introduction to Modeling and Simulation N. Marzari (MIT, May 2003)

selfconsistency

Self-consistent cycle

21

Image removed due to copyright restrictions. Please see: Fig. 7 in Payne, M. C., et al. "Iterative Minimization Techniques for ab Initio Total-Energy Calculations: Molecular Dynamics and Conjugate Gradients." Reviews of Modern Physics 64 (October 1992): 1045-1097.

Modeling software

22

name basis functionslicense pro/con

�

� �

�

� �

Basis functions

23

Matrix eigenvalue equation:

Hψ = Eψ

ψ = i

ciφi

H i

ciφi = E i

ciφi

i

Hjici = Ecj

H�c = E�c

d�r φ ∗ j H �

i

ciφi = E d�r φ ∗ j �

i

ciφi

expansion in orthonormalized basis functions

∑

∑ ∑

∫ ∫

∑

Plane waves as basis functions

24

Cutoff for a maximum G is necessary and results in a finite basis set.

Plane waves are periodic, thus the wave function is periodic!

plane wave expansion: plane wave

ψ(�r) = �

j

cj e i �Gj ·�r

periodic crystals:

Perfect!!! (lecture 4)

atoms, molecules: Image from Wikimedia Commons,http://commons.wikimedia.org.be careful!!!

Image from Open Clip Art Library, http://openclipart.org

http://commons.wikimedia.orghttp://openclipart.org

Put molecule in a big box!

25

unit cell

Make sure the separation is big enough, so that we do not include

artificial interaction!

Image from Wikimedia Commons, http://commons.wikimedia.org.

http://commons.wikimedia.org

DFT calculations

sc

f lo

op

total energy = -84.80957141 Ry

total energy = -84.80938034 Rytotal energy = -84.81157880 Rytotal energy = -84.81278531 Ry

exiting loop;total energy = -84.81312816 Ry

-84.81323129 Ry-84.81322862 Ry result precise enoughtotal energy =

total energy =

1) electronic charge densityAt the end we get:

2) total energy

26

Properties related to total energy

binding energiesstructure

reaction pathsbulk modulus

forcesshear modulus

pressureelastic constants

stressvibrational properties

...sound velocity

enough?

Convergence

28

Was my

box big

enough?

Was my basis big

Did I exit th

e scf

loop at the r

ight

point?

?

www.pwscf.org

29

http://www.pwscf.org/

PWscf users guide

User’s Guide for

Quantum-ESPRESSO

(version 3.2)

30

Please see: http://www.pwscf.org/ and http://www.democritos.it/scientific.php

http://www.pwscf.org/http://www.democritos.it/scientific.php

PWscf input

31

What atoms are involved? Where are the atoms sitting? How big is the unit cell? At what point do we cut the basis off? When to exit the scf loop?

All possible parameters are described in

INPUT_PW.

&control pseudo_dir = ””

/

&systemibrav = 1

celldm(1) = 15.0 nat = 3 ntyp = 2 occupations = 'fixed' ecutwfc = 60.0

/

&electrons conv_thr = 1.0d-8

/

ATOMIC_SPECIES H 1.00794 hydrogen.UPFO 15.9994 oxygen.UPF

ATOMIC_POSITIONS {bohr}O 0.0 0.0 0.0

H 2.0 0.0 0.0 H 0.0 3.0 0.0

K_POINTS {gamma}

water.input

GenePattern

32 Courtesy of Justin Riley. Used with permission. http://web.mit.edu/star/molsim/index.html.

http://web.mit.edu/star/molsim/index.html

GenePattern

Screenshot of the Genepattern PW module removed due to copyright restrictions. Please see: http://web.mit.edu/star/molsim/index.html

http://web.mit.edu/star/molsim/index.html

GenePattern

34

Screenshot of the Genepattern LoopPWscf module removed due to copyright restrictions. Please see: http://web.mit.edu/star/molsim/index.html

http://web.mit.edu/star/molsim/index.html

cytosin

e

adenin

eg

uan

ine

Current research ...

35

thym

ine

rise?

twist?

propeller twist?

backbone?

Current research ...

36

Review

37

Review

The Many-body Problem

Density Functional Theory

Computational Approaches

Modeling Software

PWscf Please see: http://www.quantum-espresso.org/

http://www.quantum-espresso.org/

Literature

Richard M. Martin, Electronic Structure

Kieron Burke, The ABC of DFT

chem.ps.uci.edu/~kieron/dft/

wikipedia, “many-body physics”, “density functional theory”, “pwscf”, “pseudopotentials”, ...

38

3. From Many-Body toSingle-Particle; Quantum Modeling of Molecules

5.

Class outline

39

theory practice

1. It’s A Quantum World: The Theory of Quantum Mechanics

2. Quantum Mechanics: Practice Makes Perfect

4. From Atoms to Solids

Quantum Modeling of Solids: Basic Properties

6. Advanced Prop. of Materials; What else can we do?

7. Review session

3. From Many-Body toSingle-Particle; Quantum Modeling of Molecules

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