William A. Goddard, III, [email protected]
description
Transcript of William A. Goddard, III, [email protected]
EEWS-90.502-Goddard-L04 1© copyright 2009 William A. Goddard III, all rights reserved
Nature of the Chemical Bond with applications to catalysis, materials
science, nanotechnology, surface science, bioinorganic chemistry, and energy
William A. Goddard, III, [email protected] Professor at EEWS-KAIST and
Charles and Mary Ferkel Professor of Chemistry, Materials Science, and Applied Physics,
California Institute of Technology
Course number: KAIST EEWS 80.502 Room E11-101Hours: 0900-1030 Tuesday and Thursday
Senior Assistant: Dr. Hyungjun Kim: [email protected] Assistant: Ms. Ga In Lee: [email protected]
Lecture 4, September 10, 2009
EEWS-90.502-Goddard-L04 2© copyright 2009 William A. Goddard III, all rights reserved
Last time: The nodal Theorem
E00 < E10 < E20< E21 and E00 < E01 < E11 < E21
but nodal argument does not indicate the relative energies of E10 and E20 versus E01
+
Φ00 Φ10
Φ01 Φ20+-
+-
+-+
- Φ11+-
++-+
+- -Φ21
The nodal Theorem sometimes orders excited states in 2D, 3D
The ground state of a system is nodeless (more properly, the ground state never changes sign). Useful in reasoning about wavefunctions. Implies that ground state wavefunctions for H2
+ are g not u)
For one dimensional finite systems, we can order all eigenstates by the number of nodes E0 < E1 < E2 .... En < En+1
EEWS-90.502-Goddard-L04 3© copyright 2009 William A. Goddard III, all rights reserved
4th postulate of QM
Consider the exact eigenstate of a system
HΦ = EΦ
and multiply the Schrödinger equation by some CONSTANT phase factor (independent of position and time)
exp(i) = ei
ei HΦ = H (ei Φ) = E (ei Φ)
Thus Φ and (ei Φ) lead to identical properties and we consider them to describe exactly the same state.
EEWS-90.502-Goddard-L04 4© copyright 2009 William A. Goddard III, all rights reserved
The Hamiltonian for H2+
For H atom the Hamiltonian is
Ĥ = - (Ћ2/2me)– e2/r or
Ĥ = - ½ – 1/r (in atomic units)
r
Coordinates of H atom
Coordinates of H2
+
For H2+ molecule the Hamiltonian (in atomic units) is
Ĥ = - ½ + V(r) where 1 1 1
We will rewrite this as
EEWS-90.502-Goddard-L04 5© copyright 2009 William A. Goddard III, all rights reserved
The Schrödinger Equation for H2+
The exact (electronic wavefunction of H2+ is obtained by solving
Here we can ignore the 1/R term (not depend on electron coordinates) to write
where is the electronic energy Then the total energy E becomes
E= + 1/RR
Since v(r) depends on R, the wavefunction φ depends on R.
Thus for each R we solve for φ and and add to 1/R to get the total energy E(R)
E(R)
EEWS-90.502-Goddard-L04 6© copyright 2009 William A. Goddard III, all rights reserved
Inversion Symmetry
The operation of inversion (denoted as ) through the origin of a coordinate system changes the coordinates as
x -x
y -y
z -z
Taking the origin of the coordinate system as the bond midpoint, inversion changes the electronic coordinates as illustrated.
I
The identity or do-nothing operator, is called einheit. e
Applying the inversion twice, leads to the identity
x -x x; y -y y z -z z Nothing changes!
I I I e= ( )2 = Thus The inversion operator is of order 2
EEWS-90.502-Goddard-L04 7© copyright 2009 William A. Goddard III, all rights reserved
Symmetry Theorem
If h(r) is invariant under inversion: h(-r) = h(r)
Then for all eigenfunctions φ(r) of h
h(r)φ(r) = φ(r)
I
I
φg(r) = + φg(r)
φu(r) = - φu(r)
g for gerade or even
u for ungerade or odd
φg(r) φu(r)
EEWS-90.502-Goddard-L04 8© copyright 2009 William A. Goddard III, all rights reserved
Now consider symmetry for H2 molecule
For H2 the Hamiltonian is
1/r12 interaction between 2 electrons
all terms depending only on electron i
For multielectron systems, inversion, , inverts all electron coordinates simultaneously
I
H(1,2)Φ(1,2) = E Φ(1,2) If H(1,2) is invariant under inversion: H(-r1,-r2) = H(r1,r2) Then for all eigenfunctionsΦ(1,2) of H
EEWS-90.502-Goddard-L04 9© copyright 2009 William A. Goddard III, all rights reserved
inversion symmetry for H2 wavefunctions
g
g
u
u
EEWS-90.502-Goddard-L04 10© copyright 2009 William A. Goddard III, all rights reserved
Permutation Symmetry
Transposing the two electrons in H(1,2) must leave the Hamiltonian invariant since the electrons are identical
H(2,1) = h(2) + h(1) + 1/r12 + 1/R = H(1,2)
We denote transposition as where Φ(1,2) = Φ(2,1)
Applying twice leads to the identity e,
Φ(1,2) = Φ(1,2)
Thus the previous arguments on inversion apply equally to transposition
EEWS-90.502-Goddard-L04 11© copyright 2009 William A. Goddard III, all rights reserved
permutational symmetry for H2 wavefunctions
symmetric
antisymmetric
EEWS-90.502-Goddard-L04 12© copyright 2009 William A. Goddard III, all rights reserved
Electron spin
Consider application of a magnetic field
Our hamiltonian has not no terms dependent on the magnetic field.
Hence no effect.
But experimentally there is a huge effect. Namely
The ground state of H atom splits into two states
This leads to the 5th postulate of QM
In addition to the 3 spatial coordinates x,y,z each electron has internal or spin coordinates which leads to a magnetic dipole that is either aligned with the external magnetic field or it is opposite.
We label these as for spin up and for spin down. Thus the ground states of H atom are φ(xyz)(spin) and φ(xyz)(spin)
EEWS-90.502-Goddard-L04 13© copyright 2009 William A. Goddard III, all rights reserved
Spin states for 1 electron systems
Our Hamiltonian does not involve any terms dependent on the spin, so without a magnetic field we have 2 degenerate states for H atom.
φ(r) with up-spin, ms = +1/2
φ(r)with down-spin, ms = -1/2
The electron is said to have a spin anglular momentum of S=1/2 with projections along a polar axis (say the external magnetic moment) of +1/2 (spin up) or -1/2 (down spin). This explains the observed splitting of the H atom into two states in a magnetic fieldSimilarly for H2
+ the ground state becomes φg(r) and φg(r)
While the excited state becomesφu(r) and φu(r)
EEWS-90.502-Goddard-L04 14© copyright 2009 William A. Goddard III, all rights reserved
Electron spin
+½ or or up-spin and -½ or or down-spin
But the only external manifestation is that this spin leads to a magnetic moment that interacts with an external magnetic field to splt into two states, one more stable and the other less stable by an equal amount.
B=0 Increasing B
Now the wavefunction of an atom is written as ψ(r,)
where r refers to the vector of 3 spatial coordinates, x,y,z
while to the internal spin coordinates
So far we have considered the electron as a point particle with mass, me, and charge, -e.
In fact the electron has internal coordinates, that we refer to as spin, with two possible angular momenta.
E = -Bzsz
EEWS-90.502-Goddard-L04 15© copyright 2009 William A. Goddard III, all rights reserved
Spinorbitals
The Hamiltonian does not depend on spin the spatial and spin coordinates are independent. Hence the wavefunction can be written as a product of
a spatial wavefunction, φ(called an orbital, and
a spin function, х( or .
ψ(r,) = φ(х(
where r refers to the vector of 3 spatial coordinates, x,y,z
while to the internal spin coordinates.
EEWS-90.502-Goddard-L04 16© copyright 2009 William A. Goddard III, all rights reserved
spinorbitals for two-electron systems
Thus for a two-electron system with independent electrons, the wavefunction becomes
Ψ(1,2) = Ψ) = ψa() ψb()
= φa(хa(φb(хb(
φa(φb(хa(хb(
Where the last term factors the total wavefunction into space and spin parts
EEWS-90.502-Goddard-L04 17© copyright 2009 William A. Goddard III, all rights reserved
Spin states for 2-electron systems
Since each electron can have up or down spin, any two-electron system, such as H2 molecule will lead to 4 possible spin states each with the same energy
Φ(1,2)
Φ(1,2)
Φ(1,2)
Φ(1,2)
This immediately raises an issue with permutational symmetry
Since the Hamiltonian is invariant under interchange of the spin for electron 1 and the spin for electron 2, the two-electron spin functions must be symmetric or antisymmetric with respect to interchange of the spin coordinates, 1 and 2
Neither symmetric nor antisymmetric
Symmetric spin
Symmetric spin
EEWS-90.502-Goddard-L04 18© copyright 2009 William A. Goddard III, all rights reserved
Spin states for 2 electron systems
Combining the two-electron spin functions to form symmetric and antisymmetric combinations leads to
Φ(1,2)
Φ(1,2) [ +
Φ(1,2)
Φ(1,2) [ -
Adding the spin quantum numbers, ms, to obtain the total spin projection, MS = ms1 + ms2 leads to the numbers above.
The three symmetric spin states are considered to have spin S=1 with components +1.0,-1, which are referred to as a triplet state (since it leads to 3 levels in a magnetic field)
The antisymmetric state is considered to have spin S=0 with just one component, 0. It is called a singlet state.
Antisymmetric spin
Symmetric spin
MS
+1
0
-1
0
EEWS-90.502-Goddard-L04 19© copyright 2009 William A. Goddard III, all rights reserved
Permutational symmetry
Our Hamiltonian for H2,
H(1,2) =h(1) + h(2) + 1/r12 + 1/R
Does not involve spin
This it is invariant under 3 kinds of permutaions
Space only:
Spin only:
Space and spin simultaneously: )
Since doing any of these interchanges twice leads to the identity, we know from previous arguments that Ψ(2,1) = Ψ(1,2) symmetry for transposing spin and space coord
Φ(2,1) = Φ(1,2) symmetry for transposing space coord
Χ(2,1) = Χ(1,2) symmetry for transposing spin coord
EEWS-90.502-Goddard-L04 20© copyright 2009 William A. Goddard III, all rights reserved
Permutational symmetries for H2 and He
H2
He
the only states observed are
those for which the
wavefunction changes sign
upon transposing all coordinates of electron 1 and
2
Leads to the 6th postulate of
QM
EEWS-90.502-Goddard-L04 21© copyright 2009 William A. Goddard III, all rights reserved
The 6th postulate of QM: the Pauli Principle
For every state of an electronic system
H(1,2,…i…j…N)Ψ(1,2,…i…j…N) = EΨ(1,2,…i…j…N)
The electronic wavefunction Ψ(1,2,…i…j…N) changes sign upon transposing the total (space and spin) coordinates of any two electrons
Ψ(1,2,…j…i…N) = - Ψ(1,2,…i…j…N)
We can write this as
ij Ψ = - Ψ for all I and j
EEWS-90.502-Goddard-L04 22© copyright 2009 William A. Goddard III, all rights reserved
Implications of the Pauli Principle
Consider two independent electrons,
1 on the earth described by ψe(1)
and 2 on the moon described by ψm(2)
Ψ(1,2)= ψe(1) ψm(2)
And test whether this satisfies the Pauli Principle
Ψ(2,1)= ψm(1) ψe(2) ≠ - ψe(1) ψm(2)
Thus the Pauli Principle does NOT allow the simple product wavefunction for two independent electrons
EEWS-90.502-Goddard-L04 23© copyright 2009 William A. Goddard III, all rights reserved
Quick fix to satisfy the Pauli Principle
Combine the product wavefunctions to form a symmetric combination
Ψs(1,2)= ψe(1) ψm(2) + ψm(1) ψe(2)
And an antisymmetric combination
Ψa(1,2)= ψe(1) ψm(2) - ψm(1) ψe(2)
We see that
12 Ψs(1,2) = Ψs(2,1) = Ψs(1,2) (boson symmetry)
12 Ψa(1,2) = Ψa(2,1) = -Ψa(1,2) (Fermion symmetry)
Thus the Pauli Principle only allows the antisymmetric combination for two independent electrons
EEWS-90.502-Goddard-L04 24© copyright 2009 William A. Goddard III, all rights reserved
Consider some simple cases: identical spinorbitals
Ψ(1,2)= ψe(1) ψm(2) - ψm(1) ψe(2)
Identical spinorbitals: assume that ψm = ψe
Then Ψ(1,2)= ψe(1) ψe(2) - ψe(1) ψe(2) = 0
Thus two electrons cannot be in identical spinorbitals
Note that if ψm = eiψe where is a constant phase factor, we still get zero
EEWS-90.502-Goddard-L04 25© copyright 2009 William A. Goddard III, all rights reserved
Consider some simple cases: orthogonality
Consider the wavefuntion
Ψold(1,2)= ψe(1) ψm(2) - ψm(1) ψe(2)
where the spinorbitals ψm and ψe are orthogonal
hence <ψm|ψe> = 0
Define a new spinorbital θm = ψm + ψe (ignore normalization)
That is not orthogonal to ψe. Then
Ψnew(1,2)= ψe(1) θm(2) - θm(1) ψe(2) =
ψe(1) θm(2) + ψe(1) ψe(2) - θm(1) ψe(2) - ψe(1) ψe(2)
= ψe(1) ψm(2) - ψm(1) ψe(2) =Ψold(1,2) Thus the Pauli Principle leads to orthogonality of spinorbitals for different electrons, <ψi|ψj> = ij = 1 if i=j
=0 if i≠j
EEWS-90.502-Goddard-L04 26© copyright 2009 William A. Goddard III, all rights reserved
Consider some simple cases: nonuniqueness
Starting with the wavefuntion
Ψold(1,2)= ψe(1) ψm(2) - ψm(1) ψe(2)
Consider the new spinorbitals θm and θe where
θm = (cos) ψm + (sin) ψe
θe = (cos) ψe - (sin) ψm Note that <θi|θj> = ij
Then Ψnew(1,2)= θe(1) θm(2) - θm(1) θe(2) =
+(cos)2 ψe(1)ψm(2) +(cos)(sin) ψe(1)ψe(2)
-(sin)(cos) ψm(1) ψm(2) - (sin)2 ψm(1) ψe(2)
-(cos)2 ψm(1) ψe(2) +(cos)(sin) ψm(1) ψm(2)
-(sin)(cos) ψe(1) ψe(2) +(sin)2 ψe(1) ψm(2)
[(cos)2+(sin)2] [ψe(1)ψm(2) - ψm(1) ψe(2)] =Ψold(1,2) Thus linear combinations of the spinorbitals do not change Ψ(1,2)
EEWS-90.502-Goddard-L04 27© copyright 2009 William A. Goddard III, all rights reserved
Determinants
The determinant of a matrix is defined as
The determinant is zero if any two columns (or rows) are identical
Adding some amount of any one column to any other column leaves the determinant unchanged.
Thus each column can be made orthogonal to all other columns.(and the same for rows)The above properities are just those of the Pauli PrincipleThus we will take determinants of our wavefunctions.
EEWS-90.502-Goddard-L04 28© copyright 2009 William A. Goddard III, all rights reserved
The antisymmetrized wavefunction
Where the antisymmetrizer can be thought of as the determinant operator.
Similarly starting with the 3!=6 product wavefunctions of the form
Now put the spinorbitals into the matrix and take the deteminant
The only combination satisfying the Pauil Principle is
EEWS-90.502-Goddard-L04 29© copyright 2009 William A. Goddard III, all rights reserved
Example:
From the properties of determinants we know that interchanging any two columns (or rows) that is interchanging any two spinorbitals, merely changes the sign of the wavefunction
Interchanging electrons 1 and 3 leads to
Guaranteeing that the Pauli Principle is always satisfied