Lecture 21 - More on singlet and triplet helium.pdf

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Lecture 21 –More on singlet and triplet helium Lecture 21: More on Singlet and Triplet Helium TTT 2017 Lecture 21/ 2 Singlet and triplet helium square4 We obtain mathematical explanation to the shielding and Hund’s rule ( spin correlation or Pauli exclusion principle ) as they apply to the singlet and triplet states of the helium atom. square4 We discuss spin angular momenta of these states and consider the spin multiplicity of a general atom. TTT 2017 Lecture 21/ 3 Orbital approximation square4 The orbital approximation: an approximate or forced separation of variables square4 We must consider spin and anti-symmetry : Ψ r 1 , r 2 , ° , r n ( ) ϕ 1 r 1 ( ) ϕ 2 r 2 ( ) L ϕ n r n ( ) Ψ r 1 σ 1 , r 2 σ 2 , ° , r n σ n ( ) 1 n ! A ϕ 1 r 1 σ 1 ( ) ϕ 2 r 2 σ 2 ( ) L ϕ n r n σ n ( ) Antisymmetrizer that forms an antisymmetric linear combination of products Spin variable Normalization coefficient Orthonormal TTT 2017 Lecture 21/ 4 Normalized wave functions in the orbital approximation square4 For singlet (1 s ) 2 state of the helium atom: Ψ r 1 σ 1 , r 2 σ 2 ( ) 1 2 A ϕ 1 s r 1 α ( ) ϕ 1 s r 2 β ( ) = 1 2 ϕ 1 s r 1 ( ) α (1) ϕ 1 s r 2 ( ) β (2) - ϕ 1 s r 2 ( ) α (2) ϕ 1 s r 1 ( ) β (1) ( ) = ϕ 1 s r 1 ( ) ϕ 1 s r 2 ( ) 1 2 α (1) β (2) - β (1) α (2) ( ) Orthonormal TTT 2017 Lecture 21/ 5 Normalized wave functions in the orbital approximation square4 For triplet (1 s α ) 1 (2 s α ) 1 state of the helium atom: Ψ r 1 σ 1 , r 2 σ 2 ( ) 1 2 A ϕ 1 s r 1 α ( ) ϕ 2 s r 2 α ( ) = 1 2 ϕ 1 s r 1 ( ) α (1) ϕ 2 s r 2 ( ) α (2) - ϕ 1 s r 2 ( ) α (2) ϕ 2 s r 1 ( ) α (1) ( ) = 1 2 ϕ 1 s r 1 ( ) ϕ 2 s r 2 ( ) - ϕ 2 s r 1 ( ) ϕ 1 s r 2 ( ) ( ) α (1) α (2) TTT 2017 Lecture 21/ 6 Approximate energy E = Ψ * ˆ H Ψ d τ ˆ H = - h 2 2 m 1 2 - 2 e 2 4 πε 0 r 1 ˆ H 1 1 ( ) 1 2 44 3 44 - h 2 2 m 2 2 - 2 e 2 4 πε 0 r 2 ˆ H 2 2 ( ) 1 2 44 3 44 + e 2 4 πε 0 r 12 ˆ H 12 1,2 ( ) 1 2 4 3 4
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Lecture 21 –More on singlet and triplet helium TTT 2017 Lecture 21/ 7 Energy: (1 s ) 2 helium Ψ r 1
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  • Fall '17
  • Do Dinh Khai

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