Lecture 4, September 10, 2009. Nature of the Chemical Bond with applications to catalysis, materials science, nanotechnology, surface science, bioinorganic chemistry, and energy. Course number: KAIST EEWS 80.502 Room E11-101 Hours: 0900-1030 Tuesday and Thursday. - PowerPoint PPT Presentation
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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
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)
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
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.
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
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.
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
= 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
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.
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