Atomic Structure and Atomic Spectra Chapter 10

Structures of many-electron atoms Because of electron correlation, no simple analytical expression for orbitals is possible Therefore ψ(r1, r2, ….) can be expressed as ψ(r1)ψ(r2)… Called the orbital approximation Individual hydrogenic orbitals modified by presence of other electrons

Structures of many-electron atoms Pauli exclusion principle – no more than two electrons may occupy an atomic orbital, and if so, must be of opposite spin

Structures of many-electron atoms In many-electron atoms, subshells are not degenerate. Why? Shielding and penetration

Fig 10.19 Shielding and effective nuclear charge, Zeff Shielding from core electrons reduces Z to Zeff Zeff = Z – σ where σ ≡ shielding constant

Fig 10.20 Penetration of 3s and 3p electrons Shielding constant different for s and p electrons s-electron has greater penetration and is bound more tightly bound Result: s < p < d < f

Structure of many-electron atoms In many-electron atoms, subshells are not degenerate. Why? Shielding and penetration The building-up principle (Aufbau) Mnemonic:

Order of orbitals (filling) in a many-electron atom 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s

“Fill up” electrons in lowest energy orbitals (Aufbau principle) ? ? Be 4 electrons Li 3 electrons C 6 electrons B 5 electrons B 1s22s22p1 Be 1s22s2 Li 1s22s1 H 1 electron He 2 electrons He 1s2 H 1s1

Structure of many-electron atoms In many-electron atoms, subshells are not degenerate. Why? Shielding and penetration The building-up principle (Aufbau) Mnemonic: Hund’s rule of maximum multiplicity Results from spin correlation

The most stable arrangement of electrons in subshells is the one with the greatest number of parallel spins (Hund’s rule). C 6 electrons O 8 electrons N 7 electrons F 9 electrons Ne 10 electrons Ne 1s22s22p6 O 1s22s22p4 C 1s22s22p2 N 1s22s22p3 F 1s22s22p5

Fig 10.21 Electron-electron repulsions in Sc atom Reduced repulsions with configuration [Ar] 3d1 4s2 If configuration was [Ar] 3d2 4s1

Ionization energy (I) - minimum energy (kJ/mol) required to remove an electron from a gaseous atom in its ground state I1 + X(g) X+(g) + e- I1 first ionization energy I2 + X+(g) X2+(g) + e- I2 second ionization energy I3 + X2+(g) X3+(g) + e- I3 third ionization energy I1 < I2 < I3

Mg → Mg+ + e− I1 = 738 kJ/mol For Mg2+ 1s22s22p6 Mg+ → Mg2+ + e− I2 = 1451 kJ/mol Mg2+ → Mg3+ + e− I3 = 7733 kJ/mol

Fig 10.22 First ionization energies N [He] 2s2 2p3 I1 = 1400 kJ/mol O [He] 2s2 2p4 I1 = 1314 kJ/mol

Spectra of complex atoms Energy levels not solely given by energies of orbitals Electrons interact and make contributions to E

Fig 10.18 Vector model for paired-spin electrons Multiplicity = (2S + 1) = (2·0 + 1) = 1 Singlet state Spins are perfectly antiparallel

Fig 10.24 Vector model for parallel-spin electrons Three ways to obtain nonzero spin Multiplicity = (2S + 1) = (2·1 + 1) = 3 Triplet state Spins are partially parallel

Singlet – triplet transitions Fig 10.25 Grotrian diagram for helium Singlet – triplet transitions are forbidden

Fig 10.26 Orbital and spin angular momenta Spin-orbit coupling Magnetogyric ratio

Fig 10.27(a) Parallel magnetic momenta Total angular momentum (j) = orbital (l) + spin (s) e.g., for l = 0 → j = ½

Fig 10.27(b) Opposed magnetic momenta Total angular momentum (j) = orbital (l) + spin (s) e.g., for l = 0 → j = ½ for l = 1 → j = 3/2, ½

Fig 10.27 Parallel and opposed magnetic momenta Result: For l > 0, spin-orbit coupling splits a configuration into levels

Fig 13.30 Spin-orbit coupling of a d-electron (l = 1) j = l + 1/2 j = l - 1/2

Energy levels due to spin-orbit coupling Strength of spin-orbit coupling depends on relative orientations of spin and orbital angular momenta (= total angular momentum) Total angular momentum described in terms of quantum numbers: j and mj Energy of level with QNs: s, l, and j where A is the spin-orbit coupling constant El,s,j = 1/2hcA{ j(j+1) – l(l+1) – s(s+1) }