1.021 , 3.021, 10.333, 22.00 : Introduction to Modeling and Simulation : Spring 2012 Part II – Quantum Mechanical Methods : Lecture 3

From Many-Body to Single-Particle: Quantum Modeling of Molecules

Jeffrey C. Grossman

Department of Materials Science and Engineering

Massac husetts Institute of T ec hnology

Par t II T opics

1. It ’ s a Quantum W orld: The Theor y of Quantum Mechanics

2. Quantum Mechanics: Practice Mak es P erf ect

3. Fr om Man y-Body to Single-Par ticle; Quantum Modeling of Molecules

4. Application of Quantum Modeling of Molecules: Solar Thermal Fuels

5. Application of Quantum Modeling of Molecules: Hydr ogen Storage

6. Fr om Atoms to Solids

7. Quantum Modeling of Solids: Basic Pr oper ties

8. Advanced Pr op . of Materials: What else can w e do?

9. Application of Quantum Modeling of Solids: Solar Cells Par t I

10. Application of Quantum Modeling of Solids: Solar Cells Par t II

11. Application of Quantum Modeling of Solids: Nanotechnolog y

Motivation

?

Last time: 1-electr on quantum mechanics to describe spectral lines

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

Image of NGC 604 nebula is in the public domain. Source: Hubble Space Telescope Institute (NASA). Via Wikimedia Commons .

.

?

T od a y: man y electr ons to describe materials.

Lesson outline

• Re vie w

• The Man y-body P r oblem

• Har tr ee and Har tr ee-F ock

• Density Functional Theor y

• Computational App r oaches

• Modeling Softwar e

• PWscf

R e vi e w: Schrödinger equation

H time independent: 1 ( ⌃ r , t ) = 1 ( ⌃ r ) · f ( t )

f ˙ ( t ) H 1 ( r r )

i n = = const. = E f ( t ) 1 ( r r )

H G ( → r ) = E G ( → r )

/ ( ⌃ r , t ) = / ( ⌃ r ) · e -

i E t

time independent Schrödinger equation stationar y Schrödinger equation

R e vi e w: The h yd r ogen atom

stationar y

Schrödinger equation H l = E l

T + V ⇥ G = E G

k 2 2

— 2 m D

k 2 2

+ V ⇥ G ( → r ) = E G ( → r )

e 2 ⇥

— 2 m D

— 4 v s r

G ( → r ) = E G ( → r )

Radial W a v efunctions f or a Coulomb V(r)

Z

r

Thickness dr

y

X

Angular Par ts

Image by MIT OpenCourseWare.

R e vi e w: The h yd r ogen atom

q u a n t u m n u m b e r s

R e vi e w: The h yd r ogen atom

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R e vi e w: The h yd r ogen atom

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R e vi e w: Next? Helium

e -

r 1 r 12

-

H 1 = E 1

+ e

r 2 H 1

+ H 2

+ W ⇥ G ( → r 1

, → r 2 ) = E G ( → r 1

, → r 2 )

T 1 + V 1 + T 2 + V 2 + W ⇥ G ( → r 1 , → r 2 ) = E G ( → r 1 , → r 2 )

k 2 2

e 2 k 2 2 e 2

e 2 ⇥

— 2 m D 1 —

4 v s 0 r 1

— 2 m D 2 —

4 v s 0 r 2

+ 4 v s

r 12

G ( r 1 , r 2 ) = E G ( r 1 , r 2 )

cannot be solv ed anal yticall y p r oblem!

R e vi e w: Spin

ne w quantum n umber : spin quantum n umber f or elect r ons: spin quantum n umber can ON L Y be

up d o wn

E v er ything is spinning ...

Stern–Gerlach experiment (1922)

F ⇧ = — D E

= D m ⇧ · B ⇧

Image courtesy of Teresa Knott.

E v er ything is spinning ...

In quantum mechanics par ticles can h a v e a magnetic moment and a ”spin”

magnetic moment

m ⇥

spinning charge

E v er ything is spinning ...

conclusion f r om the Stern-Gerlach experiment

f or elect r ons: spin can ON L Y be

up d o wn

Spin Histor y

Disco v er ed in 1926 b y Goudsmit and Uhlenbeck

Part of a letter by L.H. Thomas to Goudsmit on Mar ch 25 1926 (sour ce: W ikipedia).

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Pauli ’ s exclusions principle

T w o elect r ons in a system cannot h a v e the same quantum n umbers!

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

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

magnetic m: -l,...,l spin: up , do wn

h ydr ogen

... ... ... ...

3s 3p 3d

2s 2p

1s

Pauli Exclusion Principle

“Alr eady in m y original pa per I str essed the ci r cumstance that I was unable to giv e a logical r eason f or the exclusion principle or to deduce it f r om mor e general assumptions. I had al w a ys the f eeling, and I still h a v e it tod a y , that this is a deficienc y . ”

W . Pauli, Exclusion Principle and Quantum Mechanics, Nobel prize acceptance lectur e , Stockholm (1946).

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P eriodic table

This image is in the public domain. Source: Wikimedia Commons .

The ma n y-body p r oblem

helium: 2e - i r on: 26e -

e -

r 1 r 12

+ e -

r 2

G = G ( → r 1 , . . . , → r n )

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Dirac Quotes

Y ear 1929…

The underl ying p h ysical l a ws necessar y f or the mathematical theor y of a large par t of p h ysics and the whole of chemistr y ar e thus completel y kno wn, and the difficulty is onl y that the exact a pplication of these l a ws leads to equations m uch too complicated to be soluble .

P .A.M. Dirac , Pr oc . Ro y . Soc . 123, 714 (1929)

...and in 1963

If ther e is no complete agr eement […] betw een the r esults of one ’ s w ork and the experiment, one should not allo w oneself to be too discouraged [...]

P .A.M. Dirac , Scientific American, M a y 1963

The Multi-Elect r on Hamiltonian

H l = E l

Remember

the g ood old da ys of the

— k D 2 — e ⇥ G ( r ) = E G ( r )

The y ’ r e o v er!

1- electr on H-atom??

2 m 4 v s 0 r

N h 2

1 N N

Z i Z j e 2 h 2

n N n

Z e 2 1 n n e 2

H     2

      2

   i

  

i  1

2 M i

2 i  1

j  1 i  j

R i  R j

r i

i  1 i  1 j  1

R i  r j

2 i  1 j  1

i  j

kinetic energ y of ions kinetic energ y of electr ons electr on-ion interaction potential energ y of ions electr on-electr on interaction

Born-Oppenheimer App r o ximation

Sour ce: Disco v er Magazine Online

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25

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Born-Oppenheimer App r o ximation (skinless v ersion)

• mass of n uclei exceeds that of the electr ons b y a factor of 1000 or mor e

• w e can neglect the kinetic energ y

of the n uclei

• tr eat the ion-ion interaction

classicall y

• significantl y simplifies the Hamiltonian f or the electr ons:

Images in public domain.

Born Oppenheimer

This term is just an external potential V(r j )

h 2 n 2

N n Z e 2

1 n n e 2

H  

2 m   r i

 i

R i  r j

 2  

i  1 i  1 j  1 i  1 j  1

Solutions

quantum chemistr y

density functional theor y

Molle r -Plesset per turbation theor y MP2

coupled cluster theor y CCSD(T)

Har t r ee App r oach

Write w a v efunction as a simple p r oduct of single par ticle states:

  r 1 , ..., r n    1  r 1   2  r 2  .. .  n

 r n 

Har d P r oduct of Easy

Leads to an equation w e can solv e on a computer!



 h 2 2

  2 m 

n

 V ext  r    



d r   r      r 

 j  1 

28  j  i 

Har t r ee App r oach



 h 2 2



n 

  2 m 

 V ext  r    

j  1

d r   i  r    i  i  r 



 j  i 

The solution f or each state depends on all the other states (th r ough the Coulomb term).

• w e don ’ t kno w these solutions a priori

• m ust be solv ed iterativ el y:

- guess f orm f or { Ψ i in ( r ) }

- compute single par ticle Hamiltonians

- generate { Ψ i out ( r ) }

- compar e with old

- if diff er ent set { Ψ i in ( r ) } = { Ψ i out ( r ) } and r epeat

- if same , y ou ar e done

Simple Pictu r e ...But...

Interacting Non-Interacting

-

After all this w ork, ther e is still one major p r oblem: the solution is fundamentall y w r ong

The fix brings us back to spin!

Symmetr y Holds the K e y

Speculation: e v er ything w e kno w with scientific cer tainty is someho w dictated b y symmetr y .

The r elationship betw een symmetr y and quantum mechanics is par ticularl y striking.

Exchange Symmetr y

• all electrons are indistinguishable

• electrons that made da V inci, Newton, and Einstein who

they were, are identical to those within our molecules

a bit humbling...

• so if

• I show you a system containing electrons

• you look away

• I exchange two electrons in the system

• you resume looking at system

• there is no experiment that you can conduct that will

indicate that I have switched the two electrons

Mathematicall y

• define the exchange operator :

 12  1  r 1   2  r 2 

  1  r 2   2  r 1 

• exchange operator eigen values ar e ±1:

suppose  ^    

12 12    2   

 2  1 , or    1.

Empiricall y

• all quantum mechanical states ar e also eigenfunctions of exchange operators

- those with eigen value 1 (symmetric) ar e kno wn as Bosons

- those with eigen value -1 (antisymmetric) ar e kno wn

as F ermions

• p r of ound implications f or materials p r oper ties

- w a v efunctions f or our man y elect r on p r oblem m ust be anti-symmetric under exchange

- implies Pauli exclusion principle

Har t r ee- F ock

• Emplo ying Har tr ee's a pp r oach, but

- enf o r cing the anti-symmetr y condition

- accounting f or spin

• Leads to a r emarkable r esult:



 h 2

 2  V

n

 r     d r



  r    

 d r 

e  *  r     r     r    

 r 

  2 m

ext

  i

s i , s j

r   r j i j i i

 j  1  j  1

 j  i  j  i

• Har tr ee-F ock theor y is the f oundation of molecular orbital theor y .

• It is based upon a choice of w a v efunction that guarantees antisymmetr y betw een elect r ons.

But...Har t r ee- F ock

• neglects impor tant contribution to elect r on energ y (called “cor r elation” energ y)

• difficult to deal with: integral operator ma k es solution complicated

• supe r ceded b y another a pp r oach: density functional theor y

Solutions

quantum chemistr y

density functional theor y

Molle r -Plesset per turbation theor y MP2

coupled cluster theor y CCSD(T)

Solving the Sch r odinger Equation

No matter ho w y ou slice it, the w a v efunction is a beast of an entity to h a v e to deal with.

F or example: consider that w e h a v e n elect r ons populating a 3D space .

Let ’ s divide 3D space into NxNxN=2x2x2 grid points.

T o r econstruct Ψ (r), ho w man y points m ust w e k eep track of?

Solving the Sch r odinger Eq.

divide 3D space into NxNxN=2x2x2 grid points

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Ψ = Ψ (r 1 , . . . , r n ) # points = N 3n

W orking with the Density

The elect r on density seems to be a mor e manageable quantit y .

W ouldn ’ t it be nice if w e could r ef orm ulate our p r oblem in terms of the densit y , rather than the w a v efunction?

Energ y Elect r on Density

E 0 =E[ n 0 ]

W alter K ohn (left), r eceiving the Nobel prize in chemistr y

in 1998.

W h y DFT?

computational expense f or system size N:

Quantum Chemistr y O(N 5 -N 7 )

methods; MP2, CCSD(T)

Density Functional Theor y

Example

O(N 3 ); O(N)

silicon silicon

2 atoms/unit cell 100 atoms/unit cell

DFT : 0.1 sec

CCSD(T): 0.1 sec DFT : 5 hours

CCSD(T): 2000

y ears!!!

MP2: 0.1 sec MP2: 1 y ear

100,000

10,000

1000

100

MP2

QMC

CCSD(T)

10

Exact treatment

1

2003

2007

201 1

2015

Y ear

W h y DFT?

Density functional theor y

G = G ( → r 1 , → r 2 , . . . , → r N )

n = n ( ⇤ r )

W alter DFT

K ohn 1964

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Density functional theor y

ion

electr on density n

T otal energ y is a functional of the elect r on densit y .

E [ n ] = T [ n ] + V ii + V ie [ n ] + V ee [ n

kinetic ion-electr on

ion-ion electr on-electr on

Density functional theor y

E [ n ] = T [ n ] + V ii + V ie [ n ] + V ee [ n

kinetic ion-ion ion-electr on electr on-electr on

elect r on density n ( ⌅ r ) = Σ | c i ( ⌅ r ) | 2

i

E gr ound stat e = mi n E [ n

c

Find the w a v e functions that minimize the energ y using a functional derivativ e .

Density Functional Theor y

Finding the minim um leads to K ohn-Sham equations

ion potential Har tr ee potential exchange-cor r elation

potential

equations f or non-interacting elect r ons

Density functional theor y

Onl y one p r oblem: v xc not kno wn!

a pp r o ximations necessar y

local density general gradient a pp r o ximation a pp r o ximation

L D A GGA

Self-consistent cycle

K ohn-Sham equations

n ( ⌅ r ) =

Σ | c i ( ⌅ r ) | 2

i

Modeling softwa r e

name license basis functions pr o/con

Basis functions

Matrix eigen value equation:

H 1 = E 1

H Σ c i c i = E Σ c i c i

⇥ = Σ c i c i

i

expansion in or thonormalized basis functions

∫ d → r ф H Σ Σ

i

c i ф i = E d → r ф c i ф i

j j

i i

Σ H j i c i = E c j

i

H ⇤ c = E ⇤ c

Plane w a v es as basis functions

plane w a v e expansion: G ( → r ) = Σ

j

c e i G ⇧ j · ⇧ r

plane wa v e

Cutoff f or a maxim um G is necessar y and r esults in a finite basis set.

Plane w a v es ar e periodic , thus the w a v e function is periodic!

periodic cr ystals: atoms, molecule s I m : age by MIT OpenCourseWare.

P erf ect!!! (next lectur e) be car eful!!!

Put molecule in a big b o x

unit cell

Mak e sur e the separation is big enough, so that w e do not include ar tificial interaction!

Images by MIT OpenCourseWare.

DFT calculations

total energ y = -84.80957141 Ry total energ y = -84.80938034 Ry total energ y = -84.81157880 Ry total energ y = -84.81278531 Ry

total energ y = -84.81312816 Ry

exiting loop;

total energ y = -84.81322862 Ry r esult pr ecise enough

total energ y = -84.81323129 Ry

At the end w e get:

1) elect r onic charge density

2) total energ y

• structur e

• bulk modulus binding energies

• shear modulus

r eaction paths

f or ces

• elastic constants pr essur e

• vibrational pr oper ties

• sound v elocity

str ess ...

?

Co n v ergence

PWscf input

wate r .input

What atoms ar e in v olv ed?

&control

Wher e ar e the atoms sitting?

/ pseudo_dir = ''”

Ho w big is the unit cell?

At what point do w e cut the basis off? When to exit the scf loop?

&system

ibrav = 1

celldm (1) = 15.0

nat = 3

ntyp = 2

occupations = 'fi xed'

/ ecutw fc = 60.0

&electrons

/ conv_thr = 1.0d-8

All possible parameters ar e described in INPUT_P W .

ATOMIC_SPECIES

H 1.00794 h ydrog en .U P F

O 15.9994 oxyg en .U P F

ATOMIC_POSITIONS {bohr} O 0.0 0.0 0.0

H 2.0 0.0 0.0

H 0.0 3.0 0.0