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

Yüklə 474 b.
 tarix 02.03.2018 ölçüsü 474 b. #28592 • ## Schrödinger equation for

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

• Singlet and triplet states.
• Exchange energy.

• ## The eigenfunctions of H1 and H2 can be written as the product: • ## Can be used as a first approximation to two interacting particles. Can then use perturbation theory to include interaction. • ## For first excited state, n(1) = 1, n(2) = 2 => E =-68 eV. • ## where • ## Zeff is the effective nuclear charge and nl is the shielding constant. This gives rise to the shell model for multi-electron atoms. • ## As electrons are fermions (spin 1/2), the wavefunction of a multi-electron atom must be anti-symmetric with respect to particle exchange. • ## As electrons are indistinguishable =>  must be anti-symmetric. See table for allowed symmetries of spatial and spin wave functions. • ## where is a normalisation factor. • ## There are three possible symmetric spin eigenfunctions: • ## The Sz value is indicated by the quantum number for ms, which is obtained by adding the ms values of the two electrons together. • ## This has a strong effect on the energies of the allowed states. • ## 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)
• ## But are they both allowed quantum mechanically? • ## 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
• ## 11S1 is therefore disallowed by Pauli principle as ms quantum numbers are the same. • ## 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

• ## *Violate Pauli Exclusion Principle (See Eisberg & Resnick, Appendix P) • ## Singlet states result when S = 0.

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

• Orthohelium. • ## where ER = 13.6 eV is called the Rydberg energy. • ## The resulting energy is E12 ~ 2.5 ER. Note that in the exchange integral, we integrate the expectation value of 1/r12 with each electron in a different shell. See McMurry, Chapter 13. • ## 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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