# Lectures 10-11: Multi-electron atoms Schrödinger equation for

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• ## Schrödinger equation for

• Two-electron atoms.
• Multi-electron atoms.
• ## Helium-like atoms.

• Singlet and triplet states.
• Exchange energy.

• ## If s1 = +1/2 and s2 = -1/2 => s = 0.

• Therefore ms = 0 (singlet state)
• ## If s1 = +1/2 and s2 = +1/2 => s = 1.

• Therefore ms = -1, 0, +1 (triplet states)

• ## Orbital angular momentum: l1=l2 = 0 => L = l1 + l2 = 0

• Gives rise to an S term.
• ## Spin angular momentum: s1 = s2 = 1/2 => S = 0 or 1

• Multiplicity (2S+1) is therefore 2(0) + 1 = 1 (singlet) or 2(1) + 1 = 3 (triplet)
• ## J = L + S, …, |L-S| => J = 1, 0.

• Therefore there are two states: 11S0 and 13S1 (also using n = 1)

• ## Consider the 11S0 state: L = 0, S = 0, J = 0

• n1 = 1, l1 = 0, ml1 = 0, s1 = 1/2, ms1 = +1/2
• n2 = 1, l2 = 0, ml2 = 0, s2 = 1/2, ms2 = -1/2

• ## Now consider the 13S1 state: L = 0, S = 1, J = 1

• n1 = 1, l1 = 0, ml1 = 0, s1 = 1/2, ms1 = +1/2
• n2 = 1, l2 = 0, ml2 = 0, s2 = 1/2, ms2 = +1/2

• ## Orbital angular momentum: l1= 0, l2 = 1 => L = 1

• Gives rise to an P term.
• ## Spin angular momentum: s1 = s2 = 1/2 => S = 0 or 1

• Multiplicity (2S+1) is therefore 2(0) + 1 = 1 or 2(1) + 1 = 3
• ## For L = 1, S = 1 => J = L + S, …, |L-S| => J = 2, 1, 0

• Produces 3P3,2,1

• ## For L = 1, S = 0 => J = 1

• Term is therefore 1P1
• Allowed from consideration of Pauli principle

• ## Orbital angular momentum: l1 = l2=1 => L = 2, 1, 0

• Produces S, P and D terms

• ## Singlet states result when S = 0.

• Parahelium.
• ## Triplet states result when S = 1

• Orthohelium.

• ## Note:

• Exchange splitting is part of gross structure of He - not a small effect. The value of 2J is ~0.8 eV.
• Exchange energy is sometimes written in the form
• which shows explicitly that the change of energy is related to the relative alignment of the electron spins. If aligned = > energy goes up.

• ## Orthohelium states are lower in energy than the parahelium states. Explanation for this is:

• Parallel spins make the spin part of the wavefunction symmetric.
• Total wavefunction for electrons must be antisymmetric since electrons are fermions.
• This forces space part of wavefunction to be antisymmetric.
• Antisymmetric space wavefunction implies a larger average distance between electrons than a symmetric function. Results as square of antisymmetric function must go to zero at the origin => probability for small separations of the two electrons is smaller than for a symmetric space wavefunction.
• If electrons are on the average further apart, then there will be less shielding of the nucleus by the ground state electron, and the excited state electron will therefore be more exposed to the nucleus. This implies that it will be more tightly bound and of lower energy.

Yüklə 474 b.

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