Chem 113A HW7 - Solutions - Chem 113A Homework Assignment...

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Chem 113A Homework Assignment Week 9 Solutions 1) The ground state configuration for beryllium is 1s 2 2s 2 so the slater determinant is: 𝜓 = 1 √4! | 1?(1)?(1) 1?(1)?(1) 2?(1)?(1) 2?(1)?(1) 1?(2)?(2) 1?(2)?(2) 2?(2)?(2) 2?(2)?(2) 1?(3)?(3) 1?(3)?(3) 2?(3)?(3) 2?(3)?(3) 1?(4)?(4) 1?(4)?(4) 2?(4)?(4) 2?(4)?(4) | 2) The wavefunction for the ground state helium atom is given by: 𝜓 = 1 √2 (1??(1)1??(2)− 1??(1)1??(2)) And we know 𝑆 ̂ 𝑧 ? = 1 2 ? and 𝑆 ̂ 𝑧 ? = 1 2 ? therefore: 𝑆 ̂ 𝑧,𝑡?𝑡𝑎? 𝜓 = 1 √2 (𝑆 ̂ 𝑧,1 + 𝑆 ̂ 𝑧,2 )(1??(1)1??(2) − 1??(1)1??(2)) = 1 √2 (( 1 2 1 2 ) 1??(1)1??(2)− (− 1 2 + 1 2 ) 1??(1)1??(2)) = 0 So 𝜓 is an eigenfunction of the operator 𝑆 ̂ 𝑧,𝑡?𝑡𝑎? with eigenvalue 0. 3) First we need to find ? 𝜙 ? 𝜙 = ∫ 𝜙 𝐻 ̂ 𝜙?𝜏 ∫ 𝜙 𝜙 ?𝜏 𝐻 ̂ 𝜙 = (− 2 2? ? 2 ?𝑥 2 + ?𝑥 4 ) ? ?𝑥 2 2 = − 2 2? ? ?𝑥 (−?𝑥? ?𝑥 2 2 ) + ?𝑥 4 ? ?𝑥 2 2 = − 2 2? (−?? ?𝑥 2 2 + ? 2 𝑥 2 ? ?𝑥 2 2 ) + ?𝑥 4 ? ?𝑥 2 2 ∫ 𝜙 𝐻 ̂ 𝜙?𝜏 = ∫ ? ?𝑥 2 2 [− 2 2? (−?? ?𝑥 2 2 + ? 2 𝑥 2 ? ?𝑥 2 2 ) + ?𝑥 4 ? ?𝑥 2 2 ]?𝑥 = 2 2? [? ( 𝜋 2 ) 1 2 − ? 2 ( 1 2? ) ( 𝜋 ? ) 1 2 ] + ? ( 3 4? 2 ) ( 𝜋 ? ) 1 2 = 2 ? 2? ( 𝜋 ? ) 1 2 + 3? 4? 2 ( 𝜋 ? ) 1 2
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∫ 𝜙 𝜙 ?𝜏 = ∫ ? −?𝑥 2 ?𝑥 = ( 𝜋 ? ) 1 2 ? 𝜙 = ( ? 𝜋 ) 1 2 [ 2 ? 2? ( 𝜋 ? ) 1 2 + 3? 4? 2 ( 𝜋 ? ) 1 2 ] = 2 ? 4? + 3? 4? 2 Now minimize with respect to the parameter in our trial wave function, ? : ?? 𝜙 ?? = 2 4? 3?
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