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Solutions for LCAO model exercises

Question 1

  1. See lecture notes.
  2. The atomic number of Tungsten is 74: 1s22s22p63s23p64s23d104p65s24d105p66s24f145d4
  3. Cu=[Ar]4s23d9Pd=[Kr]5s24d8Ag=[Kr]5s24d9Au=[Xe]6s24f145d9

Question 2

  1. $$ \psi(x) = \begin{cases} &\sqrt{κ}e^{κ(x-x_1)}, xx_1 \end{cases} $$ Where κ=2mE2=mV02. The energy is given by ϵ1=ϵ2=mV0222 The wave function of a single delta peak is given by ψ1(x)=mV0emV02|xx1| ψ2(x) can be found by replacing x1 by x2

  2. H=mV022(1/2+exp(2mV02|x2x1|)exp(mV02|x2x1|)exp(mV02|x2x1|)1/2+exp(2mV02|x2x1|))
  3. ϵ±=β(1/2+exp(2α)±exp(α)) Where β=mV022 and α=mV02|x2x1|

Question 3

1.

HE=exE,

2. H^=(E0t tE0)+(1|exE|11|exE|2 2|exE|12|exE|2)=(E0γt tE0+γ), where γ=edE/2 and have used \(1|exE|1=edE/21|1=edE/2\)

3.

The eigenstates of the Hamiltonian are given by: E±=E0±t2+γ2 The ground state wave function is: |ψ=t(γ+γ2+t2)2+t2(γ+t2+γ2t 1) |ψ=γ+t2+γ2(γ+γ2+t2)2+t2|1+t(γ+γ2+t2)2+t2|2

4. P=2γ2E(1γ2+t2)