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CHEM2915

CHEM2915. Introduction to the Electronic Structure of Atoms and Free Ions. A/Prof Adam Bridgeman Room: 222 Email: A.Bridgeman@chem.usyd.edu.au www.chem.usyd.edu.au/~bridge_a/chem2915. Where Are We Going…?. Week 10: Orbitals and Terms

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CHEM2915

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  1. CHEM2915 Introduction to the Electronic Structure of Atoms and Free Ions A/Prof Adam Bridgeman Room: 222 Email: A.Bridgeman@chem.usyd.edu.au www.chem.usyd.edu.au/~bridge_a/chem2915

  2. Where Are We Going…? • Week 10: Orbitals and Terms • Russell-Saunders coupling of orbital and spin angular momenta • Free-ion terms for p2 • Week 11: Terms and ionization energies • Free-ion terms for d2 • Ionization energies for 2p and 3d elements • Week 12: Terms and levels • Spin-orbit coupling • Total angular momentum • Week 13: Levels and ionization energies • j-j coupling • Ionization energies for 6p elements

  3. Ze - r Revision – Atomic Orbitals • For any 1 e- atom or ion, the Schrödinger equation can be solved • The solutions are atomic orbitals and are characterized by n, l and ml quantum numbers Hy = Ey • E is energy of the orbital y • H is the ‘Hamiltonian’ - describing the forces operating: • Kinetic energy due to motion • Potential energy due attraction to nucleus • Total Hydrogen-like Hamiltonian ½ mv2 HH-like

  4. Atomic Orbitals - Quantum Numbers • For any 1-e- atom or ion, the Schrödinger equation can be solved • The solutions are atomic orbitals and are characterized by n, l and ml quantum numbers • Principal quantum number, n = 1, 2, 3, 4, 5, 6, … • Orbital quantum number, l = n-1, n-2, n-3, 0 • Magnetic quantum number, ml = l, l -1, l – 2, …, -l = number of nodal planes l = 2: d l = 0: s l = 1: p

  5. Orbital Quantum Number = number of nodal planes • Orbital quantum number, l = n-1, n-2, n-3, 0 • Magnetic quantum number, ml = l, l -1, l – 2, …, -l • e.g. l = 2 gives ml= 2, 1, 0, -1, -2: so 2l + 1 = five d-orbitals = orientation of orbital

  6. Magnetic Quantum Number • Orbital quantum number, l = n-1, n-2, n-3, 0 • Magnetic quantum number, ml = l, l -1, l – 2, …, -l = number of nodal planes related to magnitude of orbital angular momentum = orientation of orbital related to direction of orbital angular momentum ml = 2 ml = 1 l = 2 ml = 0 ml = -1 ml = -2

  7. Spin Quantum Number • All electrons have spin quantum number, s = +½ • Magnetic spin quantum number, ms = s, s -1 = +½ or –½: 2s+1 = 2 values

  8. Ze - ri Many Electron Atoms • For any 2-e- atom or ion, the Schrödinger equation cannot be solved • The H-like approach is taken for every electron i S S HH-like = ½ mvi2 + i i • Treatment leads to configurations • for example: He 1s2, C 1s2 2s2 2p2 e2 • Neglects interaction between electrons • e- / e- repulsion is of the same order of magnitude as HH-like S - rij i≠j

  9. 1 +½ 1 -½ 0 +½ 0 -½ -1 +½ + -1 -½ ( 1 ) Many Electron Atoms – p1 Configuration • A configuration like p1 represents 6 electron arrangements with the same energy • there are three p-orbitals to choose from as l =1 • electron may have up or down spin ms microstate ml 1 0 -1 ml

  10. Many Electron Atoms – p2 Configuration • A configuration like p2 represents even more electron arrangements • Because of e- / e- repulsion, they do not all have the same energy: • electrons with parallel spins repel one another less than electrons with opposite spins • electrons orbiting in the same direction repel one another less than electrons with orbiting in opposite directions lower in energy than lower in energy than 1 -1 0 1 -1 0 ml ml

  11. Many Electron Atoms – L • For a p2 configuration, both electrons have l = 1 but may have ml = 1, 0, -1 • L is the total orbital angular momentum • Lmax = l1 + l2 = 2 • Lmin = l1 - l2 = 0 • L = l1 + l2,l1 + l2– 1, …l1–l2 = 2, 1 and 0 • For each L, ML = L, L-1, … -L ml1 = 1 ml1 = 1 ml2 = 1 ml = -1 L: 0, 1, 2, 3, 4, 5, 6 … code: S, P, D, F, G, H, I …

  12. Many Electron Atoms – S • Electrons have s = ½ but may have ms = + ½ or - ½ • S is the total spin angular momentum • Smax = s1 + s2 = 1 • Smin = s1 - s2 = 0 • S = s1 + s2,s1 + s2– 1, …s1–s2 = 1 and 0 • For each S, MS = S, S-1, … -S

  13. Many Electron Atoms – p2 • L = 2, 1, 0 • for each L: ML = L, L -1, L – 2, …, -L • for each L, there are 2L+1 functions • S = 1, 0 • for each S: MS= S, S-1, S-2, …. –S • for each S, there are 2S+1 functions L: 0, 1, 2, 3, 4, 5, 6 … code: S, P, D, F, G, H, I … Wavefunctions for many electron atoms are characterized by L and S and are called terms with symbol: L “singlets”: 1D, 1P, 1S 2S+1 “triplets”: 3D, 3P, 3S

  14. +/- +/- ( ml1, ml2 ) Microstates – p2 • For example, 3D has L = 2 and S = 1 so: • ML = 2, 1, 0, -1, -2 and MS = 1, 0, -1 • five ML values and three MS values: 5 × 3 = 15 wavefunctions with the same energy MS = ms1 + ms2 ML = ml1 + ml2

  15. Pauli Principle and Indistinguishabity • The Pauli principle forbids two electrons having the same set of quantum numbers. Thus for p2 • Microstates such as and are not allowed • Electrons are indistinguishable • Microstates such as and are the same BUT • Microstates such as and are different • For example, for p2 • 6 ways of placing 1st electron, 5 ways of placing 2nd electron (Pauli) • Divide by two because of indistinguishabillty:

  16. Pauli forbidden Indistinguishable

  17. Working Out Allowed Terms • Pick highest available ML: there is a term with L equal to this ML • Highest ML = 2 L = 2  D term • For this ML: pick highest MS: this term has S equal to this M2 • Highest MS = 0  S = 0  2S+1 = 1: 1D term • Term must be complete: • For L = 2, ML = 2, 1, 0, -1, -2 • For S = 0, Ms = 0 • Repeat 1-3 until all microstates are used up • Highest ML = 1  L = 1  P term • Highest MS = 1  S = 1  2S+1 = 3: 3P term • Strike out 9 microstates (MS = 1, 0, -1 for each ML = 1, 0, -1) • Left with ML = 0  L = 0  S term • This has MS = 0  S = 0  2S+1 = 1: 1S term } for each value of Ms, strike out microstates with these ML values

  18. 1D 3P 1S

  19. Check • The configuration p2 gives rise to 15 microstates • These give belong to three terms: • 1D is composed of 5 states (MS = 0 for each of ML = 2, 1, 0, -1, -2 • 3P is composed of 9 states (MS = 1, 0, -1 for each of ML = 1, 0, -1 • 1S is composed of 1 state (MS = 0, ML = 0) • 5 + 9 + 1 = 15 • The three terms differ in energy: • Lowest energy term is 3P as it has highest S (unpaired electrons)

  20. Summary Configurations • For many electron atoms, the HH-like gives rise to configurations • Each configuration represents more than one arrangements • Terms • The arrangements or microstates are grouped into terms according to L and S values • The terms differ in energy due to interelectron repulsion • Next week • Hund’s rules and ionization energies • Task! • Work out allowed terms for d2

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