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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 Li B5 C 3 64electrons electrons 22s 222s 22p 12 1 BBe Li1s1s 1s 2s H He12electron electrons He H 1s 1s12 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). Ne97 C N O F 6 810 electrons electrons electrons 22s 222p 22p 5 246 3 Ne C N O F 1s 1s222s 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 Mg+ → Mg2+ + e− I2 = 1451 kJ/mol Mg2+ → Mg3+ + e− I3 = 7733 kJ/mol For Mg2+ 1s22s22p6 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 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 for l=0→j=½ 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 El,s,j = 1/2hcA{ j(j+1) – l(l+1) – s(s+1) } where A is the spin-orbit coupling constant