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)

ttritrtrVtr

m ∂Ψ∂

=Ψ+Ψ∇−),(),(),(),(

22

2 r

hrrrh

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)

ttritrtrVtr

m ∂Ψ∂

=Ψ+Ψ∇−),(),(),(),(

22

2 r

hrrrh *

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

Stationary Schrödinger’s Equation (II)

)()( tfEtfdtdi =h

)()()(2

22

rErrVm

rrrh ϕϕ =⎥⎦

⎤⎢⎣

⎡+∇−

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

)()( tfEtfdtdi =h ⎟

⎠⎞

⎜⎝⎛−= tEitf

hexp)(

Free particle Ψ(x,t)=φ(x)f(t)

)()(2

22

xExm

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

22( ) 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 22 2 2

ˆ ( )2

H V rm x y z

∂ ∂ ∂= − ∇ + ∇ = + +

∂ ∂ ∂h

2 22

2 2 2 2 2

1 1 1ˆ sin ( )2 sin sin

H r V rm r r r r r

ϑϑ 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 rm 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

____________________________________________________________

Orthogonality, Expectation Values, and Dirac’s <bra|kets>

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

nkn

nnc ϕϕψ ∑=

=

ψϕψϕ mm EH =ˆ

mnkn

mn EcHc =∑=

ϕϕ ˆ,1

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

Matrix Formulation (II)

mnmkn

n EcHc =∑=

ϕϕ ˆ,1

mkn

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

HE α αα

α α

Ψ Ψ=

Ψ Ψ

( )expC rα αΨ = −

2 2 22

3 2

1 1,2 2

C C Crα α α α α απ π π

α α αΨ Ψ = Ψ − ∇ Ψ = Ψ − Ψ = −

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

Two-electron atom

),(),(||

121

21

21212121

22

21 rrErr

rrrZ

rZ

elrrrr

rr ψψ =⎥⎦

⎤⎢⎣

⎡−

+−−∇−∇−

Many-electron atom

21 1

1 1 ( ,..., ) ( ,..., )2 | |i n el n

i i i j ii i j

Z r r E r rr r r

ψ ψ>

⎡ ⎤− ∇ − + =⎢ ⎥

−⎢ ⎥⎣ ⎦∑ ∑ ∑∑ r r r r

r 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 jiee

i i IiINeie rr

VrRVVT||

1ˆˆ21ˆ 2

rrrr

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

Electrons and Nuclei

),...,,,...,(),...,,,...,(ˆ1111 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 threevariables a library; but the full specification of a single wavefunction 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

• This average repulsion is the electrostatic repulsion of the average charge density of all other electrons

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

Hartree EquationsThe Hartree equations can be obtained directly from the variationalprinciple, once the search is restricted to the many-body wavefunctions that are written as the product of single orbitals (i.e. we are working with independent electrons)

)()()(),...,( 22111 nnn rrrrr rL

rrrr ϕϕϕψ =

)()(||

1|)(|)(21 22

iiiijIij ij

jjiIi rrrdrr

rrRV rrrrr

rrrϕεϕϕ =

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−+−+∇− ∑ ∑∫

≠

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

The self-consistent field• The single-particle Hartree operator is self-

consistent ! I.e., it depends in itself on the orbitals that are the solution of all other Hartree equations

• We have n simultaneous integro-differential equations for the n orbitals

• Solution is achieved iteratively

)()(||

1|)(|)(21 22

iiiijIij ij

jjiIi rrrdrr

rrRV rrrrr

rrrϕεϕϕ =

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−+−+∇− ∑ ∑∫

≠

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

Iterations to self-consistency

• Initial guess at the orbitals• Construction of all the operators• Solution of the single-particle pseudo-

Schrodinger equations• With this new set of orbitals, construct the

Hartree operators again• Iterate the procedure until it (hopefully)

converges

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

Differential Analyzer

Vannevar Bush and the Differential Analyzer.Courtesy of the MIT Museum. Used with permission.

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

What’s missing

• It does not include correlation• The wavefunction is not antisymmetric• It does remove nl accidental degeneracy of

the hydrogenoid atoms

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

Spin-Statistics

• All elementary particles are either fermions(half-integer spins) or bosons (integer)

• A set of identical (indistinguishable) fermions has a wavefunction that is antisymmetric by exchange

• For bosons it is symmetric

),...,,...,,...,,(),...,,...,,...,,( 2121 njknkj rrrrrrrrrr rrrrrrrrr ψψ −=

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

Slater determinant

• An antisymmetric wavefunction is constructed via a Slater determinant of the individual orbitals (instead of just a product, as in the Hartree approach)

)()()(

)()()()()()(

!1),...,,( 222

111

21

nnn

n

rrr

rrrrrr

nrrr

rL

rrMOMM

rL

rr

rL

rr

rrr

νβα

νβα

νβα

ϕϕϕ

ϕϕϕϕϕϕ

ψ =

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

Pauli principle

• If two states are identical, the determinant vanishes (i.e. we can’t have two electrons in the same quantum state)

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

Hartree-Fock EquationsThe Hartree-Fock equations are, again, obtained from the variational principle: we look for the minimum of the many-electron Schroedinger equation in the class of all wavefunctions that are written as a single Slater determinant

)()()(||

1)(

)()(||

1)(

)()(21

*

*

2

iijjij

j

ijjij

j

iiI

Ii

rrrdrrr

r

rrdrrr

r

rrRV

rrrrrr

r

rrrrr

r

rrr

λµ

µλµ

λµµµ

λ

ϕεϕϕϕ

ϕϕϕ

ϕ

=⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

+⎥⎦

⎤⎢⎣

⎡−+∇−

∑ ∫

∫∑

∑

Slaterrr n =),...,( 1rrψ

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

Shell structure of atoms

• Self-interaction free• Good for atomic properties• Start higher-order perturbation theory• Exchange is in, correlation still out

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

Faster, or better

• The exchange integrals are the hidden cost (fourth power). Linear-scaling efforts underway

• Semi-empirical methods (ZDO, NDDO, INDO, CNDO, MINDO): neglect certain multi-center integrals

• Configuration interaction, Mǿller-Plesset

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

Restricted vs. Unrestricted

• Spinorbitals in the Slater determinant: spatial orbital times a spin function

• Unrestricted: different orbitals for different spins

• Restricted: same orbital part

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

Koopmans’ Theorems

• Total energy is invariant under unitary transformations

• It is not the sum of the canonical MO orbital energies

• Ionization energy, electron affinity are given by the eigenvalue of the respective MO, in the frozen orbitals approximation

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

Atomic Units and Conversion Factors(see handout)

1 a.u. = 2 Ry = 1 Ha

1 Ry = 13.6057 eV

1 eV = 23.05 kcal/mol

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

Software• Gaussian (http://www.gaussian.com)• Crystal (http://www.cse.clrc.ac.uk/cmg/CRYSTAL/,

http://www.theochem.unito.it/)

References• F. Jensen, Introduction to Computational Chemistry• J. M. Thijssen, Computational Physics• B. H. Bransden and C. J. Joachim, Quantum

Mechanics, and also Physics of Atoms and MoleculesFeb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari