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An Introduction to Molecular Orbital Theory

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An Introduction to Molecular Orbital Theory Objectives of the course • Wave mechanics / Atomic orbitals (AOs) – The basis for rejjgecting classical mechanics ( the Bohr Model) in treating An introduction to electrons – Wave mechanics and the Schrödinger equation Molecular Orbital Theory – Representatifion of atom ibilic orbitals as wave fifunctions – Electron densities and radial distribution functions – Understanding the effects of shielding and penetration on AO energies 6 Lecture Course • Bonding – Review VSEPR and Hybridisation Prof G. W. Watson – Linear combination of molecular orbitals (LCAO), bonding / antibonding Lloyd d Ins titut e 2 .36 – La be lling o f mo lecu lar or bital s (MO s) ( σ, π and)d g, u) [email protected] – Homonuclear diatomic MO diagrams – mixing of different AO’s – More complex molecules (CO , H 2O)O ….) – MO diagrams for Inorganic complexes 2 Lecture schedule Literature Lecture 1 Revision of Bohr model of atoms • Book Sources: all titles listed here are available in the Hamilton Library Lecture 2 Schrödinger equation, atomic wavefunctions and radial – 1. Chemical Bonding, M. J. Winter (Oxford Chemistry primer 15) distribution functions of s orbitals Oxford Science Publications ISBN 0 198556942 – condensed text, excellent diagrams Lecture 3 More complex wavefunctions and radial distribution functions and electron shielding – 2. Basic Inorggy(y),,anic Chemistry (Wiley) F.A.Cotton, G. Wilkinson, P. L. Gaus – comprehensive text, very detailed on aufbau principle Lecture 4 Lewis bonding, Hybridisation, and molecular orbitals – 3. Inorgan ic Chemi stry (P renti ce H all) C . H ousecrof t, A. G. Sharpe – Lecture 5 Labelling MO’s. 1st row homonuclear diatomics comprehensive text with very accessible language. CD contains interactive energy diagrams Lecture 6 MO approach to more complex molecules and CO bonding in transition metals complexes – Additional sources: h//ihttp://winter.group.sh hfk/bi/ef.ac.uk/orbitron/ - gallery o f AOs an d MOs 3 4 Tutorials • Expectation – Tutorials are to gggpo through problems that students are having with the course An introduction to – Tutorilials are NOT for th e l ecturer to gi ive you th e answers to t he Molecular Orbital Theory questions – or to give you another lecture. – All student must BEFORE the tutorial • Look at the notes for the course and try to understand them Lecture 1 The Bohr Model • Attempt the questions set – and hence find out what you can not do! • Bring a list of questions relating to aspects of the course which you could not understand (either from looking at the notes or attempting the Prof G. W. Watson questions) Lloyd Institute 2.05 [email protected] • It is a waste of both the lecturers and students time if the tutorial to ends up being a lecture covering questions. 5 Bohr model of the atom (1913) Adsorption / Emission spectra for Hydrogen http://www.youtube.com/watch?v=R7OKPaKr5QM Johann Balmer (1885) measured line spectra for hydrogen Assumptions 364.6 nm (uv), 410.2 nm (uv), 434.1 nm (violet), 486.1 nm (blue), and 1) Rutherford (1912) model of the atom (Planetary model with central 656.3 nm (red). nucleus + electrons in orbit) 2) Planck (1901) , Einstein (1905) – the energy electromagnetic waves is quantised into packets called photons (particle like property). E = hν Balmer discovered these lines occur in a series - both absorption and emission - where ℜ is the Rydberg constant (3.29 ×1015 Hz) Velocity, c Wavelength, λ ic rr ⎛ 1 1 ⎞ v = ℜ⎜ − ⎟ ⎜ 2 2 ⎟ etic field n n ing elect 1 2 tt ⎝ ⎠ nn Balmer series n1=2 and n2=n1+1, n1+2, n1+3 ….. /mag Fluctua OtOtherher seseriesries fforor n1=11 ((LymanLyman – UV), n1=3 ((PaschenPaschen – IRIR)) etc. An stationary observer counts } i.e. frequency = v Hz , cycles/sec, ν waves passing per second Electrons must have specific energies – no model of the atom could explain this sec-1 7 8 Bohr model of the atom Bohr model of the atom Speed of electromagnetic waves (c) is constant (ν and λ vary) 3) Electron assumed to travel in circular orbits. c = ν λ, ν = c / λ, E = h ν, E = h c / λ 4) Apply quantisation to orbits - only orbits allowed have quantised angular momentum (comes from observation of spectra) As frequency increases, wavelength decreases. Given λ Æ ν ⎛ h ⎞ mvr = n⎜ ⎟ ⎝ 2π ⎠ ege.g. radiowaves: λ = 01m0.1 m X-rays: λ = 1x101 x 10-12 m ν = 3 x 109 Hz ν = 3 x 1020 Hz 5) Classical electrodynamical theory rejected (charged particles E = 2 x 10-24 J E = 2 x 10-13 J undergoing acceleration must emit radiation) 6) Radiation adsorbed or emitted only when electrons jump from one orbit to another E – energy (J), h – Plancks constant (J s), ν – frequency (Hz), ΔE = E − E c – speed of light (m-1), λ – wavelength (m) a b where a and b represent the energy of the start and finish orbits 9 10 Bohr model – calculating the energy and radius Bohr model NOT EXAMANABLE Electron travelling around nucleus in circular 2 • Energy − Ze 1 2 orbits – must be a balance between attraction = − mv 8πε0r 2 to nucleus and flying off (like a planets orbit) ⎛ h ⎞ mvr = n⎜ ⎟ Electron feels two forces – must be balanced • Quantised angular momentum ⎝ 2π ⎠ − Ze2 − Ze2 1) Centrapedal (electrostatic) F = PE = − Ze2 1 − ()mvr 2 − n2h2 2 = − mv2 = = 4πε0r 4πε0r 2 2 2 • Combining the two 8πε0r 2 2mr 8π mr 2) Centrafugal mv2 1 F = KE = mv2 r 2 2 2 2 2 2 r − n h 8πε0 n h ε0 • Rearranging to give r = 2 2 r = 2 Equalize magnitude of forces Resulting energy r 8π m (− Ze ) πmZe 2 2 2 2 2 2 2 4 mv Ze 2 Ze 1 2 − Ze − Ze 1 2 − Ze − mZ e = 2 ⇒ mv = E = mv + = = − mv • Substitute r into energy gives = r 4πε r 4πε r 2 4πε0r 8πε0r 2 2 2 2 0 0 8πε0r 8n h ε0 2 2 -Ze – nuclear charge, e – electron charge, ε0- permittivity of free space, • Energy is dependent on n and Z (2s and 2p the same – only true for 1 r - radius of the orbit, m – mass of electron, v – velocity of the electron electron systems 11 12 Energy levels of Hydrogen Energy levels of Hydrogen n=∞ Substitute quantised momentum into energy expression and rearrange n=5 in terms of r (radius) (see previous slide) n=4 nn3=3 n2h2ε n2a FhdFor hydrogen (Z1)(Z=1) r = 0 = 0 energy n=2 π mZe2 Z 2 −13.6056 n=1 a0 (Bohr) radius of the 1s electron on Hydrogen 52.9 pm (n =1, Z =1) r = n a0 En = n2 1 2 Radius (()r) dep ends on n and Z n energy (eV) nucleus Substitute r back into energy expression gives 1 -13.6056 − mZ 2e4 2 2 -3.4014 E = 13.6056× Z n 2 2 2 = 2 (in eV) 3 -1.5117 8n h ε0 n 4 -0. 8504 Energy of 1s electron in H is 13.6056 eV = 0.5 Hartree (1eV = 1.602 × 10-19 J) 5 -0.3779 1 ∞ 0.0000 Energy (E) depends on and Z2 Note. The spacing reflects the energy n2 Ionization energy = -13.6056 eV not the radius of the orbit. 13 14 n=4 Radius of orbits Emission spectra (httppy://www.youtube.com/watch?v=5z2ZfYVzefs) hv Energy of emission is Einitial -Efinal= For hydrogen (Z=1) n=3 ⎛ 1 1 ⎞ distance ΔE =13.6056 ⎜ − ⎟ ⎜ 2 2 ⎟ ⎝ ninital n final ⎠ 2 −13.6056 r = n a0 En = n=2 n2 Same form as fitted to emission specta n=1 Balmer series (Æ nfinal=2) 5 4 3 2 1 n energy (eV) r (pm) nucleus 1 -13.6056 52.9 2 -3. 4014 211 3 -1.5117 476 n=3 Æ n=2 Æ λ = 656 nm 4 -0.8504 847 n=4 Æ n=2 Æ λ = 486 nm 5 -0.3779 1322 n=5 Æ n=2 Æ λ = 434 nm ∞ 0.0000 ∞ Note. The spacing reflects the radius of the Note. The spacing reflects the energy Orbit – not the energy. ℜ = 13.6056 eV / c = 3.29 ×1015 Hz not the radius of the orbit. 15 16 Problems with the Bohr Model Wave / particle duality httppy://www.youtube.com/watch?v=IsA __ oIXdF 8 • Only works for 1 electron systems de Broglie (1923) – E.gg,. H, He+, Li2+ Byyp this time it was accepted that EM radiation can have wave and p article properties (photons) • Can not explain splitting of lines in a magnetic field – Modified Bohr-Sommerfield (elliptical orbits - not satisfactory) dlidhilldhde Broglie proposed that particles could have wave properti i(/es (wave / particle duality). Particles could have an associated wavelength (l) • Can not apply the model to interpret the emission spectra of complex atoms hc h E = mc2 , E = ⇒ λ = λ mc • Electrons were found to exhibit wave-like ppproperties – e.g. can be diffracted as they pass through a crystal (like x-rays) No experimental at time. – considered as classical particles in Bohr model 1925 Davisson and Germer showed electrons could be diffracted according to Braggs Law (used for X-ray diffraction) Numerically confirm de Broglie’s equation 17 18 Wave Mechanics Wave mechanics and atoms • For waves: it is impossible to determine the position and momentum of the • What does this mean for atoms electron simultaneously – Heisenberg ‘Uncertainty principle’ • Electrons in “orbits” must have an integer number of • Use probability of finding an electron from ψ2 (actually ψ*ψ – but functions wavelengths we will deal with are real) • E.g. n=4 and n=5 are allowed Where ψ is is a wave function and a solution of the Schrödinggqer equation – These create continuous or standing waves (like on a (1927).
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