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 solidsSolving Schrodinger’s equation for solids

2( ( ) ) ( ) 0kj kjV r E r Solution of differential equation is required

Properties of the potential ( )V r

ˆ( ) ( , ) ( )ll nlm lmr r E i Y r

Atomic potential(spherically symmetric)

nlmE

Properties of Solution for a Single AtomProperties of Solution for a Single Atom

2( ( ) ) ( ) 0sphV r E r

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 electronsE1s2,E2s2E2p6 and so on until we accomodate all N electrons within various slots.Thus we obtain a periodic table.

•For atom with given N we need to find discrete levels E1s,E2s,E3s,E2p,E3p,... •We need to order them from lowest to highest. •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 spinFor 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

2

( )ext

ZeV r

r

22 ( )

| |i ext ii i i j i j

eH V r

r r

1 1( ... ) ( ... )N NH r r E r r

where En is the set of many body energy levels. Hunds rules follow from this Hamiltonian

Bringing atoms together:Bringing atoms together:what to expect?what to expect?

Properties of the solutions: , ( )kj kjE r

2Ze

r

Core Levels

Energy Bands

Illustration: Periodic array of potential wells placed at distance L between the wells.

Solution for a Linear Atomic Chain

Basic property of the potential V(x+l)=V(x), and of the Hamiltonian H(x+l)=H(x)

Solving variationally:

1 2 3 4

( ( ) ) ( ) 0

( ) ( ) ( ) ( 2 ) ( 3 ) ...

( ) ( )i i ii i

H x E x

x A x A x l A x l A x l

A x il A x

We obtain infinite system of equations

1 12 2 13 3

21 1 2 23 3

1 1

( ) ... 0

( ) ... 0

...

... ( ) ... 0i i

E A H A H A

H A E A H A

H A E A

If overlap is only between nearest wells, we simplify it to be

1 0 1

1 1

( ) 0

...

( ) 0i i i

VA E A VA

VA E A VA

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.

2 2| ( ) | | ( ) |x x l

That is possible if ( ) ( )i xx e u x

where u(x) is a periodical function: ( ) ( )u x l u x

( )( ) ( ) ( ) ( )i x l i l i x i lx l e u x l e e u x e x

introduce periodic boundary condition

( ) ( ) ( )x L x Nl x ( ) ( ) ( )

1

2

2

i Nl i x i Nl

i Nl i L

n n

x Nl e e u x e x

e e

L n

nk

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

1 2 3 4( ) ( ) ( ) ( 2 ) ( 3 ) ...x A x A x l A x l A x l

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

( ) ( )mx ml x

with some coefficients c_n so that the combination

( ) ( )m m mm m

c r c r ml

obeys the periodicity condition automatically

1 1

( ) ( ) ( ( 1) )

( ) ( )

m m m mm m m

m m mm m

c x l c x ml l c x m l

c x ml c x

For this to happen

1

( ) ( )

i lm m

i lm m m m

c e c

c x l e c x

or, the coefficient is equal to

2

n

ni ml ik mlLe e

where n can be any integer.So we can label those combinations with index k_n

( ) ( )n nk ik ml

m

x e x ml

which automatically obey the periodicity condition for wave function (Bloch theorem)

( ) ( )n n nk ik l kx l e x

What we want is to is to use those linear combinations in finding the solutions

1 1 1 1 2 2

2 1 1 2 2 2

( ) ( )

| | | | ... 0

| | | | ... 0

...

n nk kn

k k k k k k

k k k k k k

x A x

H E A H E A

H E A H E A

The advantage of this formulation is seen because all off-diagonal matrix elements disappear. This is seen because

2 1

1 2

( )

0| | ( )

L i k k xk kH E e F x dx

F(x) is periodical function, therefore

2 1 2 1 2 1 2 1

2 1

( ) ( ) ( ) ( )

0 0 0( ) ( ) ( )

L l li k k x i k k x i k k ml i k k xk k

m

e F x dx e F x dx e e F x dx

Therefore, we automatically obtain the diagonalized Hamiltonian

1 1 1

2 2 2

| | 0

| | 0

...

k k k

k k k

H E A

H E A

That mean that the spectrum is known to us

| |

( ) ( )k k k

k k

E H

x x

In other words, we are able to solve the problem completely!

Let us analyze the solutions

| | ( ) ( ) ( )

~ ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) 2 cos( )

ikmlk k k

m

ikl

ikl

E H e dx x H x x ml

dx x H x x e dx x H x x l

e dx x H x x l V kl

0 2 cos( )kE E V 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

2 2nk 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

( ) ( ) ( ) ~ ( ) ( ) ( )iklm ikl iklk k

m

x x e x lm x e x l e x l

Summary: Bloch Property for a Local BasisSummary: Bloch Property for a Local Basis

Differential equation using expansion

( ) ( )kj kkj r A r

( )k rwhere is a basis set satisfying Bloch theorem

( ) ( )k ikR kr R e r

To force the Bloch property we now use instead of plane waves:

Linear combinations of local orbitals

( )( )k i k G rr e G

( ) ( )k ikR

R

r e r R

2( ( ) ) ( ) 0kj kjV r E r

Apply Variational PrincipleApply Variational Principle

Variational principle leads us to solve matrix eigenvalueproblem

2| |k k kjkjV E A

where

2| |k k kH V

( ) 0k k kjkjH E O A

|k k kO

is hamiltonian matrix

is overlap matrix

Linear Combination of Atomic Orbitals (LCAO)

ˆ( ) ( , ) ( )ll nlm lmr r E i Y r

( ) ( )k ikRnlm nlm

R

r e r R to be used in variational principle

Tailored to atomic potential

nlmE

LCAO MethodLCAO Method

Hamiltonian of LCAO Method: Hamiltonian of LCAO Method: Hoppings between the orbitalsHoppings between the orbitals

*

*

'0

( ) ( )

( ) ( )

( )

( ) ~ 2 cos( )

k k k

ikR

R

ikR

R

ikR

R

H r H r dV

e r H r R dV

e H R

E e t R E t kR

Tight-Binding ParametrizationTight-Binding Parametrization

In LCAO Method, the Hamitonain is parameterizedvia 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 HarmonicsThinking of Cubic Harmonics*

' ' ' ' ' '( ) (| |) ( ) (| |) ( )

( )

l m lm l l m l lm

lm l m

t R r Y r H r R Y r R dV

Y r Y A

- unitary transformation

Hoppings between various orbitalsHoppings between various orbitals

ss,sp,sd hoppingsss,sp,sd hoppings

pp,pd hoppingspp,pd hoppings

dd hoppingsdd 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 SoftwareMINDLab Software

http://www.physics.ucdavis.edu/~mindlab

Understanding s-electron band structure. Tight-binding parameterization for Na. s level position and hopping ratefor 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