3.320: Lecture 5 (Feb 15 2005) THE MANY-BODY PROBLEM 3.320: Lecture 5 (Feb 15 2005) THE MANY-BODY...

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Transcript of 3.320: Lecture 5 (Feb 15 2005) THE MANY-BODY PROBLEM 3.320: Lecture 5 (Feb 15 2005) THE MANY-BODY...

  • 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