3.320: Lecture 5 (Feb 15 2005)

THE MANYTHE MANY--BODY PROBLEMBODY PROBLEM

Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

When is a particle like a wave ?

Wavelength • momentum = Planck ↕

λ • p = h ( h = 6.6 x 10-34 J s )

),( trrΨ=Ψ

Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

Time-dependent Schrödinger’s equation (Newton’s 2nd law for quantum objects)

t tritrtrVtr

m ∂ Ψ∂

=Ψ+Ψ∇− ),(),(),(),(

2 2

2 r

h rrrh

1925-onwards: E. Schrödinger (wave equation), W. Heisenberg (matrix formulation), P.A.M. Dirac (relativistic)

Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

Stationary Schrödinger’s Equation (I)

t tritrtrVtr

m ∂ Ψ∂

=Ψ+Ψ∇− ),(),(),(),(

2 2

2 r

h rrrh *

Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

Stationary Schrödinger’s Equation (II)

)()( tfEtf dt di =h

)()()( 2

2 2

rErrV m

rrrh ϕϕ =⎥ ⎦

⎤ ⎢ ⎣

⎡ +∇−

Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

)()( tfEtf dt di =h ⎟

⎠ ⎞

⎜ ⎝ ⎛−= tEitf

h exp)(

Free particle Ψ(x,t)=φ(x)f(t)

)()( 2

2 2

xEx m

ϕϕ =∇− h ⎟⎟ ⎠

⎞ ⎜⎜ ⎝

⎛ = xmEix

h

2exp)(ϕ

Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

Interpretation of the Quantum Wavefunction (Copenhagen)

2( , )x tΨ is the probability of finding an electron in x and t

2 2( ) exp( ) ( )ix Et xϕ ϕ− =

h

Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

A Traveling “Plane” Wave

( , ) exp[ ( )]x t i kx tωΨ ∝ −

Diagram of plane wave removed for copyright reasons.

Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

Metal Surfaces (I)

Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

Metal Surfaces (II)

Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

Infinite Square Well

Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

16

14

12

10

8

6

4

2

0

n=1

n=2

n=3

n=4

-a 0 a x

ψ1

ψ2

ψ3

ψ4

ψ(x)8ma2

π2h2 E

Figure by MIT OCW.

Finite Square Well

Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

aψ2(x)

aψ3(x) aψ4(x)

aψ1(x)

1

-1

1

0.5

-0.5 -2 -1 2

x/a

0

1

-1

1

0.5

-2 -1 2

x/a

0

1

-1

1

0.5

-2 -1 20

1

-1

1-2 -1 2

x/a x/a

0

Figure by MIT OCW.

A Central Potential (e.g. the Nucleus) 2 2 2 2

2 2 2 2 2

ˆ ( ) 2

H V r m x y z

∂ ∂ ∂ = − ∇ + ∇ = + +

∂ ∂ ∂ h

2 2 2

2 2 2 2 2

1 1 1ˆ sin ( ) 2 sin sin

H r V r m r r r r r

ϑ ϑ ϑ ϑ ϑ ϕ

⎡ ⎤∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞= − + + +⎢ ⎜ ⎟ ⎜ ⎟ ⎥∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦

h

),()()( ϕϑψ lmElmElm YrRr = r

2 2 2

2 2

2 ( 1) ( ) ( ) ( ) 2 2 El El

d d l l V r R r E R r m dr r dr rµ

⎡ ⎤⎛ ⎞ + − + + + =⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦

h h

Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

Solutions in a Coulomb Potential: the Periodic Table

http://www.orbitals.com/orb/orbtable.htm

Courtesy of David Manthey. Used with permission.

Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

____________________________________________________________

http://www.orbitals.com/orb/orbtable.htm http://www.orbitals.com/orb/orbtable.htm

Orthogonality, Expectation Values, and Dirac’s

ψψψ == )(rr

ijjiji rdrr δψψψψ ==∫ rrr )()(*

iiiii EHrdrrVm r ==⎥

⎦

⎤ ⎢ ⎣

⎡ +−∫ ψψψψ ˆ)()(2)(

2 * rrrhr

Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

Matrix Formulation (I)

ψψψψ EHrErH =⇔= ˆ)()(ˆ rr

{ } functions orthogonalk ,1

n kn

nnc ϕϕψ ∑ =

=

ψϕψϕ mm EH =ˆ

mn kn

mn EcHc =∑ =

ϕϕ ˆ ,1

Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

Matrix Formulation (II)

mnm kn

n EcHc =∑ =

ϕϕ ˆ ,1

m kn

nmn EccH =∑ = ,1

⎟ ⎟ ⎟ ⎟ ⎟ ⎟

⎠

⎞

⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎝

⎛

=

⎟ ⎟ ⎟ ⎟ ⎟ ⎟

⎠

⎞

⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎝

⎛

⋅

⎟ ⎟ ⎟ ⎟ ⎟ ⎟

⎠

⎞

⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎝

⎛

kkkkk

k

c

c

E

c

c

HH

HH

.

.

.

.

.

.

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

...... 11

1

111

Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

Variational Principle

[ ] ˆ| | |

HE < Φ Φ >Φ = < Φ Φ >

If , then Φ is the ground state wavefunction, and viceversa…

[ ] 0E EΦ ≥ [ ] 0E EΦ =

Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

Energy of an Hydrogen Atom

Ĥ E α αα

α α

Ψ Ψ =

Ψ Ψ

( )expC rα αΨ = −

2 2 2 2

3 2

1 1, 2 2

C C C rα α α α α α

π π π α α α

Ψ Ψ = Ψ − ∇ Ψ = Ψ − Ψ = −

Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

Two-electron atom

),(),( ||

1 2 1

2 1

2121 2121

2 2

2 1 rrErrrrr

Z r Z

el rrrr

rr ψψ =⎥ ⎦

⎤ ⎢ ⎣

⎡ −

+−−∇−∇−

Many-electron atom

2 1 1

1 1 ( ,..., ) ( ,..., ) 2 | |i n el ni i i j ii i j

Z r r E r r r r r

ψ ψ >

⎡ ⎤ − ∇ − + =⎢ ⎥

−⎢ ⎥⎣ ⎦ ∑ ∑ ∑∑ r r r rr r

Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

Energy of a collection of atoms

NeNNeeNe VVVTTH −−− ++++= ˆˆˆˆˆˆ • Te: quantum kinetic energy of the electrons • Ve-e: electron-electron interactions • VN-N: electrostatic nucleus-nucleus repulsion • Ve-N: electrostatic electron-nucleus attraction

(electrons in the field of all the nuclei)

( ) ∑∑∑ ∑ ∑ >

−− − =⎥

⎦

⎤ ⎢ ⎣

⎡ −=∇−=

i ij ji ee

i i I iINeie rr

VrRVVT ||

1ˆˆ 2 1ˆ 2 rr

rr

Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

Electrons and Nuclei

),...,,,...,(),...,,,...,(ˆ 1111 NntotNn RRrrERRrrH rrrrrrrr ψψ =

•We treat only the electrons as quantum particles, in the field of the fixed (or slowly varying) nuclei

•This is generically called the adiabatic or Born- Oppenheimer approximation

•Adiabatic means that there is no coupling between different electronic surfaces; B-O no influence of the ionic motion on one electronic surface.

Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

Complexity of the many-body Ψ

“…Some form of approximation is essential, and this would mean the construction of tables. The tabulation function of one variable requires a page, of two variables a volume and of three variables a library; but the full specification of a single wave function of neutral iron is a function of 78 variables. It would be rather crude to restrict to 10 the number of values of each variable at which to tabulate this function, but even so, full tabulation would require 1078 entries.”

Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

Mean-field approach

• Independent particle model (Hartree): each electron moves in an effective potential, representing the attraction of the nuclei and the average effect of the repulsive interactions of the other electrons

• T