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Lecture 21 More on singlet and triplet helium

Lecture 21 More on singlet and triplet helium.

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Lecture 21 More on singlet and triplet helium

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  1. Lecture 21More on singlet and triplet helium (c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has been developed and made available online by work supported jointly by University of Illinois, the National Science Foundation under Grant CHE-1118616 (CAREER), and the Camille & Henry Dreyfus Foundation, Inc. through the Camille Dreyfus Teacher-Scholar program. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the sponsoring agencies.

  2. Singlet and triplet helium • We obtain mathematical explanation to the shielding and Hund’srule (spin correlation or Pauli exclusion principle) as they apply to the singlet and triplet states of the helium atom. • We discuss spin angular momenta of these states and consider the spin multiplicity of a general atom.

  3. Orbital approximation • The orbital approximation: an approximate or forced separation of variablesWe must consider spin and anti-symmetry: Normalization coefficient Orthonormal Spin variable Antisymmetrizerthat forms an antisymmetriclinear combination of products

  4. Normalized wave functions in the orbital approximation • For singlet (1s)2 state of the helium atom: Orthonormal

  5. Normalized wave functions in the orbital approximation • For triplet (1sα)1(2sα)1 state of the helium atom:

  6. Approximate energy

  7. Energy: (1s)2 helium

  8. Energy: (1s)2 helium 1 by normalization 0 by orthogonality

  9. Energy: (1s)2 helium (1s) energy of electron 1 (1s) energy of electron 2 Coulomb repulsion of electrons 1 and 2 – Shielding effect Probability density of electrons 1 and 2

  10. Energy: (1sα)1(2sα)1 helium

  11. Energy: (1sα)1(2sα)1 helium 1 by normalization 0 by orthogonality

  12. Energy: (1sα)1(2sα)1 helium (1s) energy of electron 1 (2s) energy of electron 2 Coulomb or Shielding effect Exchange term– lowers the energy only when two spins are the same (Hund’srule)

  13. Total spins of singlet and triplet • Singlet • Triplet Sym. Antisym. Antisym. Sym.

  14. Spin operators • Spin angular momentum operators • Totalz-component spin angular momentum operator:

  15. Total spin of singlet

  16. Total spin of singlet 2s 1s Singlet

  17. Total spins of triplet

  18. Total spin of triplet 2s 1s Triplet

  19. Spin multiplicity • S = 0 : singlet (even number of electrons) • S = ½ : doublet (odd) • S = 1 : triplet (even) • S = 1½ : quartet (odd) • All radiative transitions between states with different spin multiplicities are forbidden. • Atoms with S > 0 are magnetic and highly degenerate.

  20. Summary • The expectation value of the Hamiltonian in the normalized, antisymmetricwave function of the helium atom is a good approximation to its energy. • It mathematically explains the shieldingand spin correlation effects. • Total spin angular momenta of the helium atom in the singlet and triplet states are obtained. The concept of the spin multiplicity is introduced.

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