Slide Lecture 13 · Slide 2 Reminder: Electrons As Waves The wave is an excitation (a vibration):...
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Slide 1 Quantum Mechanics: the Practice
Transcript of Slide Lecture 13 · Slide 2 Reminder: Electrons As Waves The wave is an excitation (a vibration):...
Slide 1
Quantum Mechanics: the Practice
Slide 2
Reminder: Electrons As Waves
The wave is an excitation (a vibration): We need to know the amplitude of the excitation at every point and at every instant
),( trrΨ=Ψ
Wavelength • momentum = Planck?
λ • p = h ( h = 6.6 x 10-34 J s )
Slide 3
Wave Mechanics
)()(),( tfrtr rr ψ=Ψ
ttritrtrVtr
m ∂Ψ∂=Ψ+Ψ∇− ),(),(),(),(
22
2 r
hrrrh
ttf
firrVr
mr ∂∂=
+∇− )(1)()()(
2)(1 2
2
hrrrh
r ψψψ
)()()(2
22
rErrVm
rrrh ψψ =
+∇−
Slide 4
Stationary Schroedinger’s equation
)()()(2
)( 22
rErrVm
a rrrh ψψ =
+∇−
Vm
VTH ˆ2
ˆˆˆ 22
+∇−=+= h
)()(ˆ)( rErHb rr ψψ =
•It’s not proven – it’s postulated, and it is confirmed experimentally
•It’s an eigenvalue equation
•Boundary conditions (and regularity) must be specified
Slide 5
Interpretation of the Quantum Wavefunction
Remember the free particle, and the principle of indetermination:if the momentum is perfectly known, the position is perfectly unknown
2),( trrΨ is the probability of finding an electron in r and t
)](exp[),( trkiAtr ω−•=Ψ rrr
)exp()()()(),( Etirtfrtrh
rrr −==Ψ ψψIf V=V(r) , it’s separable:
Slide 6
Infinite Square Well
Slide 7
Finite Square Well
Slide 8
A Central Potential (e.g. the Nucleus)
2
2
2
2
2
222
2
)(2
ˆzyx
rVH∂∂+
∂∂+
∂∂=∇+∇−= rh
µ
)(sin1sin
sin11
2ˆ
2
2
2222
2
2
rVrrr
rrr
H +
∂∂+
∂∂
∂∂+
∂∂
∂∂−=
ϕϑϑϑ
ϑϑµh
),()()( ϕϑψ lmElmElm YrRr =r
)()()(2
)1(22 2
2
2
22
rRErRrVr
lldrd
rdrd
ElEl =
+++
+−
µµhh
Slide 9
Solutions in a Coulomb
Potential: the Radial
Wavefunctions
Slide 10
Solutions in a Coulomb Potential: the Periodic Table
http://www.orbitals.com/orb/orbtable.htm
5 d orbital
Slide 11
<Bra|kets>
>== ψψψ |)(rr
ijjiji rdrr δψψψψ =>=< ∫rrr )()(| *
iiiii ErdrrVm
rH =
+−>=< ∫
rrrhr )()(2
)(|ˆ|2
* ψψψψ
Slide 12
Variational Principle
[ ]><
><=φφ
φφφ|
|ˆ| HE
[ ] 0EE ≥φ
Slide 13
Electrons and Nuclei
NeNNeee VVVTH −−− +++= ˆˆˆˆˆ
•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
),...,(),...,(ˆ11 ntotn rrErrH rrrr ψψ =
Slide 14
Two-electron atom
),(),(||
121
21
21212121
22
21 rrErr
rrrZ
rZ rrrr
rr ψψ =
−
+−−∇−∇−
Slide 15
Energy of a collection of atoms
• VN-N: electrostatic nucleus-nucleus repulsion• Te: quantum kinetic energy of the electrons• Ve-N: electrostatic electron-nucleus attraction
(electrons in the field of all the nuclei)• Ve-e: electron-electron interactions
( ) ∑∑∑ ∑ ∑>
−− −=
−=∇−=i ij ji
eei i I
iINeie rrVrRVVT
||1ˆˆ
21ˆ 2
rrrr
Slide 16
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
Slide 17
Hartree Equations
)()(||
1|)(|)(21 22
iiiijIij ij
jjiIi rrrdrr
rrRV rrrrr
rrrϕεϕϕ =
−+−+∇− ∑ ∑∫
≠
)()()(),...,( 22111 nnn rrrrr rL
rrrr ϕϕϕψ =
•The Hartree equations can be obtained directly from the variational principle, once the search is restricted to the many-body wavefunctions that are written – as above – as the product of single orbitals (i.e. we are working with independent electrons)
Slide 18
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
Slide 19
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
Slide 20
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 −=
Slide 21
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 =
Slide 22
Pauli principle
• If two states are identical, the determinant vanishes (I.e. we can’t have two electrons in the same quantum state)
Slide 23
Hartree-Fock Equations
)()()(||
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
•The 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
Slide 24
Density-functional Theory
• Conceptually very different from Hartree-Fock –variational principle on the charge density
• In practice, equations have the same form, but for the exchange energy – obtained from the density, not the wavefunctions
• It’s exact in principle, but approximate in practice: different forms for the exchange-correlation density: LDA, GGA, hybrids (Hartree-Fock exchange + density-functional correlations)
