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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