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Physics 250-06 “Advanced Electronic Structure”
Earlier Electronic Structure Methods
Contents:
1. Solution for a Single Atom
2. Solution for Linear Atomic Chain
3. LCAO Method and Tight-Binding Representation
4. Tight-Binding with BandLab
Solving Schrodinger’s equation for solids
Solution of differential equation is required
(2  V (r)  Ekj ) kj (r)  0
Properties of the potential
V (r)
Properties of Solution for a Single Atom
(  Vsph (r)  E )  (r)  0
2
  ( r )   l ( r, Enlm )i Ylm ( rˆ)
l
Atomic potential
(spherically symmetric)
Enlm
Property of solution for symmetric potential
Discrete set of levels is obtained. While
degeneracy with respect to m remains,
degeneracy with respect to l is now lifted since V
V(r) is different from –Ze2/r . We can label the
levels by main quantum number and orbital
quantum number. For given l, n=l+1,l+2,...
which is the property of the solution.
For l=0 we have the states E1s, E2s,E3s,E4s,...
For l=1 we have the states E2p, E3p,E4p,E5p,...
For l=2 we have the states E3d, E4d,E5d,E6d,...
No artificial degeneracy as in the hydrogen atom
case:
E2s=E2p,E3s=E3p=E3d,E4s=E4p=E4d=E4f…
At some point, level Enl becomes above zero, i.e
the spectrum changes from discrete to
continuous.
Note also that this spectrum of levels is a
functional of V(r), i.e. it is different for a
given atom with a given number of electrons N.
We can interpret different solutions as wave functions with different numbers of nodes.
For example, for l=2, there are solutions of the equations with no nodes, with one node, with two
nodes, etc. Number of nodes = n-l-1. If n runs from l+1, l+2, ..., nodes =0,1,2,3...
Periodic table of elements
The one-electron approximation is very useful as it allows to understand what happens
if we have many electrons accommodated over different levels.
Let us take atom with N electrons. Lets us find all discrete levels E1s<E2s<E2p<...
Since electrons are the fermions they obey Fermi-Dirac statistics, i.e. they cannot be
more than two electrons (with opposite spins) occupying given non-degenerate level.
If level is N-fold degenerate (for example, p level is 3 fold degenerate) then it can accomondate 2N
electrons.
So we can now fill various atomic shells with electrons
E1s2,E2s2E2p6 and so on until we accomodate all N electrons within various slots.
Thus we obtain a periodic table.
i.For atom with given N we need to find discrete levels E1s,E2s,E3s,E2p,E3p,...
ii.We need to order them from lowest to highest.
iii.We need to fill the levels with electrons.
In many cases, for a given number N ordering the levels is simple E1s<E2s<E2p<E3s...
Therefore atoms of the periodic table have configurations which is easy to obtain.
In some cases (since levels depend on Vscf(r) which depends on N) this rule is violated. This for
example happens for later 3d elements.
Hunds rules
For a given l shell, the electrons will occupy the slots to maximize the total spin
For a given l shell, the electrons will maximize the total spin and total angular momentum
Hunds rules cannot be understood on the basis of one-electron self-consistent approximation.
However, for atomic system, the many body problem for N electrons moving in the
Coulomb potential can be solved
e2
H    i  Vext ( ri )  
i
i
i  j | ri  r j |
2
Ze2
Vext ( r )  
r
H  (r1...rN )  E  (r1...rN )
where En is the set of many body energy levels. Hunds rules follow from this Hamiltonian
Bringing atoms together:
what to expect?
Properties of the solutions:
Ekj , kj ( r )
Energy Bands
Core Levels
Ze2

r
Solution for a Linear Atomic Chain
Illustration: Periodic array of potential wells placed at distance L between the wells.
Basic property of the potential V(x+l)=V(x), and of the Hamiltonian H(x+l)=H(x)
Solving variationally:
( H ( x )  E ) ( x )  0
 ( x )  A1 ( x )  A2  ( x  l )  A3  ( x  2l )  A4  ( x  3l )  ...



 A  ( x  il )   A  ( x )
i 
i
i 
i
i
We obtain infinite system of equations
(  E ) A1  H12 A2  H13 A3  ...  0
H 21 A1  (  E ) A2  H 23 A3  ...  0
...
H i1 A1  ...  (  E ) Ai  ...  0
If overlap is only between nearest wells, we simplify it to be
VA1  (   E ) A0  VA1  0
...
VAi 1  (  E ) Ai  VAi 1  0
This is infinite set of equations. At first glance it seems that we cannot solve it. But we can do it indeed.
Since Hamitonian is periodical function, all wells are equivalent for the electron.
That would mean that the probability to find the electron in each of the well should be the same.
| ( x) |2 | ( x  l ) |2
That is possible if
 ( x )  ei xu( x )
where u(x) is a periodical function:
u( x  l )  u( x )
 ( x  l )  ei ( x l )u( x  l )  ei l ei xu( x )  eil ( x )
introduce periodic boundary condition
 ( x  L)   ( x  Nl )   ( x )
 ( x  Nl )  ei Nl ei x u( x )  ei Nl ( x )
ei Nl  ei L  1
 L  2 n
2 n
  n 
 kn
L
where n is any integer number and phase is frequently called the wave vector k_n
Once we understand the form for the wave function, we already see one problem with the representation
 ( x)  A1 ( x)  A2  ( x  l )  A3 ( x  2l )  A4  ( x  3l )  ...
Each term in this expansion is not a periodical function, only the entire sum is periodical. The question
is if we can construct another linear combinations of those basis functions so that each of the terms
obeys the periodicity condition. In other words, we would like to have such combinations of
 ( x  ml )   m ( x )
with some coefficients c_n so that the combination
c
m
 m ( r )   cm  ( r  ml )
m
m
obeys the periodicity condition automatically
c
m
 m ( x  l )   cm  ( x  ml  l )   cm  ( x  (m  1)l ) 
m
m
c
m 1
m
m
 ( x  ml )   cm1  m ( x )
m
For this to happen
cm1  ei l cm
c
m
 m ( x  l )  ei l  cm  m ( x )
or, the coefficient is equal to
e
i
2 n
ml
L
 eiknml
where n can be any integer.
So we can label those combinations with index k_n
 k ( x )   eik ml  ( x  ml )
n
n
m
which automatically obey the periodicity condition for wave function (Bloch theorem)
 ( x  l )  e  ( x)
kn
iknl
kn
What we want is to is to use those linear combinations in finding the solutions
 ( x )   Ak  k ( x )
n
n
n
  k1 | H  E |  k1  Ak1    k1 | H  E |  k2  Ak2  ...  0
  k2 | H  E |  k1  Ak1    k2 | H  E |  k2  Ak2  ...  0
...
The advantage of this formulation is seen because all off-diagonal matrix elements disappear.
This is seen because
L
  k1 | H  E |  k2   ei ( k 2 k1 ) x F ( x)dx
0
F(x) is periodical function, therefore

L
0
e
i ( k 2  k1 ) x
l
F ( x )dx   e
0
i ( k 2  k1 ) x
F ( x )dx  e
i ( k 2  k1 ) ml
m
l
  k2k1  ei ( k 2 k1 ) x F ( x )dx
Therefore, we automatically obtain the diagonalized Hamiltonian
  k1 | H  E |  k1  Ak1  0
  k2 | H  E |  k2  Ak2  0
...
That mean that the spectrum is known to us
Ek   k | H |  k 
 k ( x)   k ( x)
In other words, we are able to solve the problem completely!
0
Let us analyze the solutions
Ek   k | H |  k   eikml  dx  ( x ) H ( x )  ( x  ml ) 
m
~  dx  ( x ) H ( x )  ( x ) eikl  dx  ( x ) H ( x )  ( x  l ) 
e  ikl  dx  ( x ) H ( x )  ( x  l )    2V cos(kl )
Ek  E0  2V cos(kl )
We see that we can draw the solutions as a cosine centered around h as a function of wave number k.
That is a band. Due to periodicity of cos, it is sufficient to draw it within –pi<kl<pi.
We do not forget of course that k are discrete set of numbers
kn 
2
2
n
n
L
Nl
However since we consider N to be very large number, we can deal with k as with continuous argument.
The wave functions corresponding to each solutions are simply
 k ( x )   k ( x )   eiklm  ( x  lm) ~  ( x )  eikl  ( x  l )  e ikl  ( x  l )
m
Summary: Bloch Property for a Local Basis
(2  V (r)  Ekj ) kj (r)  0
Differential equation using expansion
 kj ( r )   A  ( r )
kj
where
 ( r )
k
k

is a basis set satisfying Bloch theorem
 ( r  R )  e  ( r )
k
ikR
k
To force the Bloch property we now use instead of plane waves:
 ( r )  e
k
i ( k G ) r
 G
Linear combinations of local orbitals
k ( r )   eikR  ( r  R)
R
Apply Variational Principle
Variational principle leads us to solve matrix eigenvalue
problem
k
2
k
kj


|


V

E
|


A
 
kj

 

k
k
kj
(
H

E
O
)
A
  kj    0

where
H    |   V |   
k
k
2
O    |   
k
k
k
k
is hamiltonian matrix
is overlap matrix
LCAO Method
Linear Combination of Atomic Orbitals
(LCAO)
 ( r )   l ( r, Enlm )i Ylm ( rˆ)
l
Enlm
Tailored to atomic potential
k
 nlm
( r )   eikR  nlm ( r  R)
R
to be used in variational
principle
Hamiltonian of LCAO Method:
Hoppings between the orbitals
H    
k
k*

( r ) H   ( r )dV 
k
ikR
*
e

   (r ) H   (r  R)dV
R
  eikR H  ( R ) 
R
E    e t ( R ) ~ E   2t cos(kR0 )
' ikR
R
Tight-Binding Parametrization
In LCAO Method, the Hamitonain is parameterized
via on site energies of the orbitals and
nearest-neighbor hopping integrals between
the orbitals.
In many situations symmetry plays an important role since
for many orbitals hoppings intergrals between them are
automatically equal zero.
Thinking of Cubic Harmonics
tl ' m ' lm ( R )   l ' (| r |)Y *l ' m ' ( r ) Hl ' (| r  R |)Ylm ( r  R )dV
Ylm ( r )  Yl A m

- unitary transformation
Hoppings between various orbitals
ss,sp,sd hoppings
pp,pd hoppings
dd hoppings
Illustration, CuO2 plane
It is clear that
levels are non-bonding and are all occupied.
The same is true for
level of copper, and for
orbitals. The active degrees of freedom here are
orbitals which have hopping rate
.
Using the active degrees of freedom as the basis
the hamiltonian has the form
Let us introduce another basis set
of bonding (b),
antibonding (a), and non-bonding (n) orbitals
Within this basis set, the Hamiltonian becomes
The bands are now seen as the bonding band below and the
antibonding band above. The non--bonding band is also present.
MINDLab Software
http://www.physics.ucdavis.edu/~mindlab
Understanding s-electron band structure.
Tight-binding parameterization for Na. s level position and hopping rate
for s electrons.
Understanding s-d electron band structure.
Tight-binding parameterization for Cu.
s level position and hopping rate for s electrons.
d level position and narrow d-bands. s-d hybridization