Chapter 3 Quantization and the Hydrogen Atom 3.1 What is Quantization?

** Sodium (Na) - Two lines in the yellow region. Each of the sodium D lines represents the energy difference between two discrete energy levels of the sodium atom. ** Quantization – the discrete characteristic of the energy possessed by an atom or molecule. ** Each photon emitted or absorbed is referred as a quantum of radiation.

ΔE = hν = E2 – E1 Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung

1

Chapter 3 Quantization and the Hydrogen Atom Problem 3.1: Use the wavelengths for each of the sodium D lines to obtain the corresponding energy differences, ΔE, in units of eV, cm-1, kcal/mol, kJ/mol, K, J, Hz.

Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung

2

3.2 Quantization of Energy in the Hydrogen Atom Balmer series of lines in the emission spectrum of the hydrogen atom, observed by Balmer in 1885. Balmer series – A series of emission lines of the hydrogen atom converging smoothly to shorter wavelengths. λ = n’2G / (n’2 – 4), n’ = 3, 4, 5, …;.

ν = RH (1/22 - 1/n’2) RH = Rydberg constant =

at 656.3 nm, n’ = 3

mee4/8h3ε02

me : mass of electron e: charge on the electron h : Plank’s constant ε0: vacuum permittivity Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung

3

3.2 Quantization of Energy in the Hydrogen Atom Balmer series of lines in the emission spectrum of the hydrogen atom, observed by Balmer in 1885. Balmer series – A series of emission lines of the hydrogen atom converging smoothly to shorter wavelengths. λ = n’2G / (n’2 – 4), n’ = 3, 4, 5, …;.

ν = RH (1/22 - 1/n’2) RH = Rydberg constant =

at 656.3 nm, n’ = 3

μe4/8h3ε02

μ: reduced mass = memp / (me + mp) me and mp: mass of electron and proton e: charge on the electron h : Plank’s constant ε0: vacuum permittivity Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung

4

3.2 Quantization of Energy in the Hydrogen Atom RH = Rydberg constant =

RH 

9.10939 10



mee4/8h3ε02

31

8  6.62607 10

-34



kg  1.60218 10

  3

19

Js  8.85419 10

C

12



4

Fm



2

 3.2898705 1015 s 1

 2.17989 10 18 J  6.02214 10 23 mol-1  1312.7610 kJ mol-1  13.60569 eV ~

RH  RH / c  1.097373 105 cm 1

2010 CODATA: 1 Hartree = 2RH = 27.211 383 86 (68) eV Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung

5

3.2 Quantization of Energy in the Hydrogen Atom Quantized energy of atomic hydrogen: E = -RHhc /n2 = - 1312.75 kJ mol-1 / n2 = - 13.60569 eV / n2 ~

Selection rule: Δn = 1, ΔE = E2 – E1 2, -1 2 2 = 1312.75·(1/n1 – 1/n2 ) kJ mol 3, 2 2 = 13.60569 ·(1/n1 – 1/n2 ) eV …., ∞ ** The convergence limit at n =  corresponds to ionization of H atom: H + hν  H+ + eThe ejected electron can have any amount of kinetic energy, the energy levels becomes no longer discrete but form a continuum beyond the ionization limit.

Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung

6

3.2 Quantization of Energy in the Hydrogen Atom ** Bohr model in 1913 – The electron in a hydrogen atom moves around the nucleus only in certain allowed circular orbits.

Angular momentum, pθ = nh / 2π ** Photoelectric effect in 1887 by H. Hertz. Quantization of the radiation. hν = Eionization + ½ mev2 -- In 1900, Plank introduced the quantization of energy and the Planck constant, h. -- In 1905, Einstein described light as composed of discrete quanta (photons), E = hν. ** Dual particle – wave nature of waves proposed by de Broglie in 1924: p = h /λ vs p = mev ** Uncertainty principle formulated by Heisenberg in 1927: ΔpxΔx ≧ h / 2π Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung

7

3.3 Results of Applying Quantum Mechanics to the Hydrogen Atom ** Quantum (wave) mechaniscs describes the behavior of an electron as a wave rather than as a particle. ** Schrödinger equation – The basis of the subject of quantum mechanics.

HΨ = EΨ E (Eigenvalue): discrete energy levels of the system Ψ(Eigenfunction): the wave function corresponding to each eigenvalue - Schrödinger equation to obtain wave function corresponding to the known energy levels from spectroscopic measurements. - Assumed forms of wave functions to predict the energy levels of an atom or molecules. H (Hamiltonian) = potential energy + kinetic energy = -(e2/4πε0 r) - ( h2 / 2μ) 2

Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung

8

3.3 Results of Applying Quantum Mechanics to the Hydrogen Atom ** 90% probability of finding the electron within the boundary surface.

** Principle quantum number n = 1, 2, 3, …. Azimuthal quantum number (l) l = 0, 1, 2, 3, 4, 5, …, (n-1) s, p, d, f, g, h, … = orbital angular momentum Magnetic quantum number (ml) ml = 0, ±1, ±2, ±3, …, ± l (2l+1 values)

Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung

9

3.3 Results of Applying Quantum Mechanics to the Hydrogen Atom **Orbital angular momentum of the electron is a vector quantity. **In the presence of a magnetic field, the vector of the orbital angular momentum can lie only in certain directions to the field, ml = -l, -(l-1),…, -1, 0, 1, …, (l-1), l. In the absence of a magnetic field, all ml are degenerate. ** The Schrödinger equation can be solved exactly for H (He+ and Li2+). **Two further angular momenta: -- electron spin: ms = s, s-1, ……, -(s-1), -s +1/2 and -1/2 for one electron  Important in photoexcitation spectroscopy. -- nuclear spin: 2H, I = 1; 12C, I = 0, 13C, I = ½ . mI = I, I-1, ……, -(I-1), -I  Important in nuclear magnetic resonance (NMR) spectroscopy. Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung

L  l (l  1)h Lz  ml h 10

Problems 1. Calculate the angular momentum of the electron in the hydrogen atom when the electron is in the (a) n = 5 and (b) n = 100 orbital. 2. Calculate the wavelength of a proton travelling in a straight line with a velocity of 6034 m s-1. (mp = 1.672621777(74)×10−27 kg) 3. In the absence of a magnetic field or electric field, what is the degree of degeneracy of (a) f orbitals and (b) h orbitals in the hydrogen atom? 4. Calculate the wavelength of the first line (n’ = 2. n” = 1) of the Lyman series for (a) 1H and (b) 6Li2+.(-13.60 eV vs -122.4 eV) Dependency : Z2 Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung

11

Chapter 4

Quantization and Polyelectronic Atoms

4.1 Effects of More than One Electron in an Atom ** For more than one electron in an atom, the Schrödinger equation is no longer exactly soluble: approximations must be made. (Coulombic attraction + electron-electron repulsion) ** The (2l + 1) degeneracy of the s, p, d, … orbitals is removed. ** Aufbau principle – Electrons are fed into the available orbitals, in order of ** increasing Pauli exclusion energy.principle – No two electrons have the same set of quantum numbers n, l, ml, and ms. Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung

12

4.1 Effects of More than One Electron in an Atom ** Aufbau + Pauli principles Periodic table to give the ground configuration for each element. Noble gases Alkali metals Alkaline metals Transition metals 1st transition series (3d) 2nd transition series (4d) Lanthanide series (4f) Actinide series (5f)

** Orbitals with the same “n” principle quantum number comprise a shell. n = 1, 2, 3, 4 are abbreviated to K, L, M, N … shells. ** Orbitals with the same values of n and l are referred to as sub-shells. Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung

13

4.2 The Helium Atom ** Helium: A prototype for the spectroscopic behavior of other closed-shell polyelectronic atoms and som molecules. ** He: 1s2, no net orbital angular momentum, i. e. L = 0. (H: l = 0) L = 0 1 2 3 4 ….. S P D F G …... no net electron spin angular momentum, i. e. S = 0. (H: s = ½, doublet ) Electron spin multiplicity = 2S + 1 Ms = S, (S-1), …., -S  The term symbol of the ground state Helium is 1S, a singlet term. A statistical weight (i.e., number of possible microstates) of (2S+1)(2L+1). ** Promotion of one of the electron from the 1s to a higher energy orbital:

Ground Excited singlet singlet Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung

Ground triplet

14

4.2 The Helium Atom

The selection rules for transitions in He atom: Δn is unrestricted, Δl = ±1, ΔS = 0. ** For transitions that one electron remains in the 1s orbital, no transitions are allowed between singlet and triplet terms. ** For terms differ only in their multiplicity, the term of highest multiplicity lies lowest in energy. ** 2s1S and 2s3S are metastables. ** Each series converges at high energy to the ionization limit. ** The ΔS = 0 selection rule is very rigidly obeyed in the helium atom. Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung

15

4.3 Other Polyelectronic Atoms 4.3.1 Coupling of Orbital Angular Momenta ** For two (or more electrons) with l ≠ 0, their orbital angular momenta are coupled to give the total orbital angular momentum . Example: Excited carbon atom: 1s22s22p13d1 2p, l1 =1; 3d, l2 = 2 Total orbital angular momentum, L = l1 + l2, (l1 + l2-1), …., |l1 - l2| = 3, 2, 1 F, D, P ** The magnitude of orbital angular momentum, [l (l+1)]1/2 h = l* h ** The magnitude of the resulted vector, [L(L + 1)]1/2 h = L* h

L* = 121/2 h

Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung

61/2 h

21/2 h

16

4.3 Other Polyelectronic Atoms 4.3.2 Coupling of Electron Spin Angular Momenta ** Total spin angular momentum, S = s1 + s2, s1 + s2 - 1, …, |s1 – s2| For two electron system, S = 1, 0. ** Electron spin angular momentum, [s(s+1)]1/2 h = s* h ** The magnitude for the coupling of two electron spin angular momenta to give (a) a triplet and (b) a singlet term. [S(S+1)]1/2 h = S* h

** Hund’s rule – For ground configurations which give rise to states of different multiplicity, the higher multiplicity lies low in energy. Example: P [Ne]3s23p3

Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung

17

4.3 Other Polyelectronic Atoms 4.3.2 Coupling of Electron Spin Angular Momenta Example: What are the possible multiplicities of ground state Mo? Mo

[Kr]4d55s1 Septet Quintet Quintet Triplet Triplet Singlet

Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung

18

4.3 Other Polyelectronic Atoms 4.3.3 Terms Arising From Two or More Electrons in the Same Sub-Shell

Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung

19

4.3 Other Polyelectronic Atoms 4.3.3 Terms Arising From Two or More Electrons in the Same Sub-Shell Homework: Terms of d2 and d3. d2: 1S, 3P, 1D, 3F, and 1G d3: 2P, 4P, 2D (2 terms), 2F, 4F, 2G, and 2H ** Of the terms arising from equivalent electrons, those with the highest multiplicity lie lowest in energy. ** Of these, the lowest is that with the highest value of L. ** A vacancy in an orbital behaves, in respect of the terms which arises, like an electron. Ground term of p4 = p2  3P d2 = d8  3F

Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung

20

4.3 Other Polyelectronic Atoms 4.3.4 Multiplet Structure in Atomic Spectra Russell – Saunders coupling - Any coupling between the orbital angular momentum of an electron to its own spin angular momentum (ls coupling) is completely neglected. ** The total orbital angular momentum and the total spin angular momentum couple together to give the total angular momentum. J = L + S, L + S – 1, ……., |L – S| 3D: 3D 3P: 3S: 4P: 5D: 2F:

3,

3D , 3D 2 1 3P , 3P , 3P 2 1 0 3S 1 4P , 4P3 , 4P 5/2 /2 1/2 5D , 5D , 5D , 5 D , 5 D 4 3 2 1 0 2F , 2F 7/2 5/2

[J(J+1)]1/2 h = J* h Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung

21

4.3 Other Polyelectronic Atoms 4.3.4 Multiplet Structure in Atomic Spectra Russell – Saunders coupling - Any coupling between the orbital angular momentum of an electron to its own spin angular momentum (ls coupling) is completely neglected. Spin – orbit coupling splits apart the states with different values of J arising from a particular term. Lande interval rule: The rule holds for atoms in which spin-orbit (ls) coupling is small. (The interval between successive levels is proportional to the larger of their total angular momentum values. ) ΔE = EJ - EJ-1 = AJ A > 0, normal multiplet A < 0, inverted multiplet ** The ls coupling is proportional to Z4, where Z is the charge on the nucleus, the rule of the Russell – Saunders coupling holds best for atoms with low Z, i. e. light atoms.

Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung

22

4.3 Other Polyelectronic Atoms 4.3.4 Multiplet Structure in Atomic Spectra Inverted case Ground state Carbon

For multiplets arising from ground terms: • When an orbital is less than half full, the multiplet that arises is normal. • When an orbital is more than half full, the multiplet that arises is inverted. • When half filled, the ground state is always an S term with only one J value. Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung

23

4.4

Selection Rules in Spectra of Polyelectronic Atoms

For polyelectronic atoms with multiplet splitting, the selection rules are: • • •

Δn is not restricted. ΔL = 0, ±1 except that the L= 0 to L = 0 transition is not allowed. ΔJ = 0, ±1 except that the J = 0 to J = 0 transition is not allowed.

• Δl = ±1.  Transitions are forbidden between states arising from the same configuration. Example: Excited carbon with 1s22s22p13d1 configuration give rises to 1F, 1D, 1P, 3F(3F , 3F , 3F ), 3D(3D , 3D , 3D ), and 3P. 4 3 2 3 2 1 The 3F3 – 3D2 transition is allowed by ΔL, ΔJ, ΔS selection rules, but violates the Δl selection rule, therefore is a forbidden transition. • ΔS = 0. Light atoms obey this rule. The rule is broken down by spin-orbit coupling, which is proportional to Z4.

Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung

24

4.4

Selection Rules in Spectra of Polyelectronic Atoms Na (3s1  3p1)

• Unrestricted Δn; ΔL = 0, 1; ΔJ = 0, 1; Δl = 1; ΔS = 0 Multiplet structures and transitions Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung

25

Chapter 5 Electronic States of Diatomic and Polyatomic Molecules 5.1 Homonuclear Diatomic Molecules 5.1.1 Molecular Orbitals and Ground Electron Configurations ** Solution of the Schrödinger equation with LCAO wave functions:

E± = (EA ± β) / (1 ± S)

EA: The relative energies associated with the AO on atom A. β: Resonance energy, a negative quantity. S: Overlap integral, a measure of the extent to which the two AOs overlap. For total overlap, S = 1, typically S = 0.2. ** S is far from negligible, it can be neglected in a very approximate calculation of MO energies.  E± = EA ± β ** Antibonding orbital is more antibonding than the bonding orbital is bonding.

E   E H 1s  E   E H 1s Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung

26

5.1 Homonuclear Diatomic Molecules 5.1.1 Molecular Orbitals and Ground Electron Configurations MOs formed from 1s, 2s and 2p AOs for a homonuclear diatomic molecule: ** σ-bond: Cylindrical symmetry wrt the internuclear axis. A nondegenerate state. ** π-bond: Having one nodal plane through the z-axis. A doubly degenerate state. ** Symmetric MO through invervsion  “gerade” (even), g. Antisymmetric MO through inversion  “ungerade” (odd), u. Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung

27

5.1 Homonuclear Diatomic Molecules 5.1.1 Molecular Orbitals and Ground Electron Configurations ** All electrons are fed into the orbitals in order of increaing energy, according to the aufbau principle. ** B2, C2, N2: σg2s and σg2p MO have the same symmetry, they push each other apart, thereby losing the symmetetrical disposition of the bonding and antibonding energy levels. Can explain the paramagnetic property of B2. ** O2 and F2: The pushing apart of σg2s and σg2p MOs is less profound, the order of σg2s and πu2p MOs is reversed. Can explained the triplet multiplicity and paramagnetic property of O2. ** Bond order = (net number of bonding electrons) / 2

Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung

28

5.1 Homonuclear Diatomic Molecules 5.1.1 Molecular Orbitals and Ground Electron Configurations

Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung

29

5.1 Homonuclear Diatomic Molecules 5.1.2 Ground and Excited Electronic States Russell – Saunders coupling approximation of atoms Hund’s case (a) coupling approximation of molecules ** The coupling of the orbital angular momentum to the internuclear axis is very strong, L is not a good quantum number. The spin is not so strongly coupled to the axis, S remains a good quantum number. ** Introdcution of a second nucleus, introduces two new quantum number, Λ and Σ. Orbital angular momentum along the internuclear axis: Λ = 0 1 2 3 4 …… Σ Π Δ Φ Γ…… Spin angular momentum, Σ = S, S-1, …., -S ** Term symbol for electronic states: 2Σ+1Λ

g/u

** For Σ terms, reflection through any plane containing the internuclear axis: Symmetric: “+” Antisymmetric: “-” National University of Kaohsiung Grace H. Ho, Department of Applied Chemistry,

30

5.2 Heteronuclear Diatomic Molecules 5.2.1 Molecular Orbitals and Ground Electron Configurations

Formation of MOs from • 2px and 3dxz AOs

•

Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung

2pz and 3dz2 AOs

31

5.2 Heteronuclear Diatomic Molecules 5.2.2 Ground and Excited Electronic States ** The highest occupied molecular orbital (HOMO) The lowest unoccupied molecular orbital (LUMO) CO, (σ2p)2  CO*, (σ2p)1(π*2p)1;

1Π, 3Π

NO, (σ2p)2(π*2p)1  NO*, (σ2p)2(σ*2p)1 ; (σ2p)2(π*2p)1  NO*, (σ2p)1(π*2p)2 ;

2Σ+

HCl, (π)4

 HCl*, (π)3(σ*)1; 1Π, 3Π

Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung

32

5.3 Polyatomic Molecules 5.3.1 Molecules Containing a Carbon – Carbon Multiple Bond ** Ethyne (acetylene), H-C≡C-H Ground: ….. (πu2p)4

Term symbol 1Σ + g

Excited: …..(πu2p)3(πg2p)1

1,3Σ +, 1,3Σ -, 1,3Δ u u u

Hybrid orbital ** Propyne, CH3-C≡C-H: No center nor cylindrical symmetry, excitation from π-type component to π*-type component. ** Ethene (ethylene), H2C=CH2 Local MOs: 4 σ-type C-H bonds 1 σ-type component of the C=C bond 1 π-type component of the C=C bond Ground (πu)2  Excited (πu)1 (πg)1 ** For molecules containing ethynyl or ethenyl group, as in CH3C≡CH, CH3CH2C≡CH, CH3CH=CH2, the first electronic excitation involves the localized MOs ππ* Grace H. Ho, Departmenttransition. of Applied Chemistry, National University of Kaohsiung

33

5.3 Polyatomic Molecules

Singlet excited states: S1, S2, … Triplet excited states: T1, T2, …

5.3.2 Molecules Containing a Carbonyl Group Examples of molecules contain a C=O group but do not contain any other πelectron-containing group: MOs of methanal (formaldehyde)

Local MOs in the C=O group: 1 σ-type component from a sp2 hybrid orbital on carbon and a 2p orbital on oxygen. 1 π-type component from a p orbital on carbon and a 2p orbital on oxygen. 1 non-bonding pair from the remaining two 2p electrons on oxygen. Excitation: S0  The nπ* state (T1, the lowest energy excited state, and S1). The ππ* states are at higher energies. The n  π* transition Other examples Ethanal (acetaldehyde), CH3CHO Propanone (acetone), CH3COCH3 Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung

34

5.3 Polyatomic Molecules 5.3.2 Molecules Containing a Carbonyl Group

H2CO, C2v symmetry

Strong π*-π transition

Weak π*- n transition vibronic coupling

Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung

5.3 Polyatomic Molecules 5.3.2 Molecules Containing a Carbonyl Group X

H2CO, C2v symmetry

** Forbidden electronic transition: A2 – A1 ν2(a1)

ν3(a1)

ν1(a1) Y ν4(b1)

ν (b )

ν (b )

-- Vibronic coupling: A2b1,ν4μb2,YA1 A2b2,ν5,6μb1,XA1 Which one is favorable?  A long progression in ν4 is observed.

Grace H. Ho, Department of Applied National University 6 2 of Kaohsiung 5 Chemistry, 2

5.3 Polyatomic Molecules 5.3.3 Molecules Containing Conjugated π-Electron Systems ** ----

buta-1,3-diene Four 2p AOs to form four MOs Ground state S0, (π1)2(π2)2 The lowest excited state S1, (π1)2(π2)1(π3)1

** ----

Benzene Six 2p AOs to form six MOs Ground state S0, (π1)2(π2,3)4 The lowest excited state S1, (π1)2(π2,3)3(π4,5)1



π  π* transitions

Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung

5.3 Polyatomic Molecules 5.3.3 Molecules Containing Conjugated π-Electron Systems Question: How about C5H5N? •

N atom: 3 sp2 hybrid MO, 2 bonded to adjacent carbon atoms, and the remaining sp2 in the plane takes up 2 electrons to form a non-bonding MO. One 2p orbital perpendicular to the plane takes part in πMO.

•

The lower symmetry of pyridin results in the loss of the double degeneracies of the π MOs.

•

HOMO: n LUMO: π* Ground state: S0 Excited state, promoted to LUMO: T1, S1



n  π* transition

Grace H. Ho, Department of Applied Chemistry, National University of Kaohsiung