Lecture 21 More on singlet and triplet helium

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Lecture 21 More 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.

Singlet and triplet helium 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. We discuss spin angular momenta of these states and consider the spin multiplicity of a general atom.

Orbital approximation The orbital approximation: an approximate or forced separation of variables We must consider spin and anti-symmetry: Normalization coefficient Orthonormal Spin variable Antisymmetrizer that forms an antisymmetric linear combination of products

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

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

Approximate energy

Energy: (1s)2 helium

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

Probability density of electrons 1 and 2 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

Energy: (1sα)1(2sα)1 helium

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

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’s rule)

Total spins of singlet and triplet Sym. Antisym. Antisym. Sym.

Spin operators Spin angular momentum operators Total z-component spin angular momentum operator:

Total spin of singlet

Total spin of singlet 2s 1s Singlet

Total spins of triplet

Total spin of triplet 2s 1s Triplet

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.

Summary The expectation value of the Hamiltonian in the normalized, antisymmetric wave function of the helium atom is a good approximation to its energy. It mathematically explains the shielding and 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.