Lecture 21 more on singlet and triplet helium
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Lecture 21 more on singlet and triplet helium
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.


Singlet and triplet helium
Singlet and triplet helium

  • We obtain mathematical explanation to the shielding and Hunt’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
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


Normalized wave functions in the orbital approximation
Normalized wave functions in the orbital approximation

  • For singlet (1s)2 state of the helium atom:

Orthonormal


Normalized wave functions in the orbital approximation1
Normalized wave functions in the orbital approximation

  • For triplet (1sα)1(2sα)1 state of the helium atom:



Energy 1 s 2 helium
Energy: (1s)2 helium


Energy 1 s 2 helium1
Energy: (1s)2 helium

1 by normalization

0 by orthogonality


Energy 1 s 2 helium2
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 1 s 1 2 s 1 helium
Energy: (1sα)1(2sα)1 helium


Energy 1 s 1 2 s 1 helium1
Energy: (1sα)1(2sα)1 helium

1 by normalization

0 by orthogonality


Energy 1 s 1 2 s 1 helium2
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 (Hunt’s rule)


Total spins of singlet and triplet
Total spins of singlet and triplet

  • Singlet

  • Triplet

Sym.

Antisym.

Antisym.

Sym.


Spin operators
Spin operators

  • Spin angular momentum operators

  • Totalz-component spin angular momentum operator:



Total spin of singlet1
Total spin of singlet

2s

1s

Singlet



Total spin of triplet
Total spin of triplet

2s

1s

Triplet


Spin multiplicity
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
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.