22.3.2: ii. Exercises

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Q1

Show that the configuration (determinant) corresponding to the \(Li^+ 1s(\alpha )1s(\alpha )\) state vanishes.

Q2

Construct the 3 triplet and 1 singlet wavefunctions for the \(Li^+ 1s^12s^1\) configuration. Show that each state is a proper eigenfunction of \(S^2 \text{ and } S_z\) (use raising and lowering operators for \(S^2\))

Q3

Construct wavefunctions for each of the following states of \(CH_2:\)
\begin{align} &a.) & ^1B_1(1_{a1}^22_{a1}^21_{b2}^23_{a1}^11_{b1}^1) \\ &b.) & ^3B_1(1_{a1}^22_{a1}^21_{b2}^23_{a1}^11_{b1}^1) \\ & c.) & ^1A_1(1_{a1}^22_{a1}^21_{b2}^23_{a1}^2) \end{align}

Q4

Construct wavefunctions for each state of the \(1\sigma^22\sigma^23\sigma^21\pi^2\) configuration of NH.

Q5

Construct wavefunctions for each state of the \( 1s^12s^13s^1 \) configuration of Li.

Q6

Determine all term symbols that arise from the \( 1s^22s^22p^23d^1\) configuration of the excited N atom.

Q7

Calculate the energy (using Slater Condon rules) associated with the ep valence electrons for the following states of the C atom.

i. \(^3P(M_L=1,M_S=1),\)
ii. \(^3P(M_L=0,M_S=0),\)
iii. \(^1S(M_L=0, M_S=0), \text{ and } \)
iv. \(^1D(M_L=0, M_S=0)\).

Q8

Calculate the energy (using Slater Condon rules) associated with the \(\pi\) valence electrons for the following states of the NH molecule.
i. \(^1\Delta (M_L=2, M_S=0),\)
ii. \(^1\Sigma (M_L=0, M_S=0), \text{ and }\)
iii. \(^3\Sigma (M_L=0, M_S=0).\)