Quantum Chemistry 4th Year Lectures PDF

Quantum Chemistry Chemistry Department 4th year lectures References 1- “Quantum Chemistry”, McQuarrie, Donald A. 2nd ed. Sausalito, CA: University Science Books, 2007. 2- “Molecular Quantum Mechanics”, Atkins, P., and Friedman, R. 4...

Quantum Chemistry Chemistry Department 4th year lectures References 1- “Quantum Chemistry”, McQuarrie, Donald A. 2nd ed. Sausalito, CA: University Science Books, 2007. 2- “Molecular Quantum Mechanics”, Atkins, P., and Friedman, R. 4th ed. Oxford University, 2005. 3- “Physical Chemistry“, Atkins, P., and J. de Paula.. 7th ed. New York, NY: W.H. Freeman and Company, 2001. 4- “Atoms and Molecules, An Introduction for Students of Physical Chemistry”, Karplus, M., and R. Porter Reading, MA: Addison Wesley, 1970. 1  Quantum Chemistry 1st lecture Historical development - Prior to quantum mechanics (QM) * Newton’s Laws apply universally * Atoms are basic constituents of matter * The world is deterministic - Classical mechanics (CM): In (CM), given initial positions , velocities , and all forces , all the future can be predicted:- - Physics was complete excepting * Newtonian mechanics explained macroscopic behaviour of matter - universal motion, fluid flow, etc. * Thermodynamics had its first two laws and most of their consequences * Basic statistical mechanics had been applied to chemical systems * Light was explained as an electromagnetic wave. 2 ⎯ Experiments not explained by classical physics! * Black-body radiation * Heat capacities * Photoelectric effect * Discrete atomic spectra * The electron as a subatomic particle ⎯ Conclusions * Atoms are not the most microscopic objects * Newton’s laws do not apply to the microscopic world of the electron 3 - Quantum Mechanics * Describes rules that apply to electrons in atoms and molecules * Non-deterministic, probabilistic! A new philosophy of nature * Explains unsolved problems of late 19th century physics * Explains bonding, structure, and reactivity in chemistry - The DEMISE of CLASSICAL PHYSICS * Discovery of the Electron In 1897 by Thomson and measures (e/me) Then ⇒ “electron” is a distinct particle, present in all materials! In CM ⇒ force on electron due to deflector voltage ** In 1909 Milliken oil drop experiment determines e, me separately. - What’s the structure of the atom? Assumptions underlying the Rutherford’s model atom * Rutherford backscattering experiment ⇒ (1) Nucleus very small (2) Atoms are mostly empty! Rutherford planetary model: CM model of atomic structure ⇒ Coulomb attraction plays the role of gravity 4 This is stable compared to separated electron & nucleus BUT model not consistent with classical electrodynamics: Accelerating charge emits radiation! (centripetal acceleration = v2/r) And since light has energy, E must be getting more negative with time Then ⇒ r must be getting smaller with time ⇒ Electron spirals into nucleus in ~ 10-10 s ! Also, as r decreases, v should increase Frequency ν of emitted light = frequency of rotation ⇒ atom should emit light at all frequencies – that should produce a continuous spectrum BUT emission from atoms was known to be discrete, not continuous. 5 For the H atom, Rydberg showed that the spectrum was consistent with the simple formula: Where n1 = 1, 2,…. & n2 = n1+1, n1+2,….. R (Rydberg constant) = 109,737 cm-1 Rutherford’s model of the atom (1) is not stable relative to collapse of electron into nucleus (2) the atom should emit light at all frequencies(continuous spectrum) not discrete (3) Does not explain the Rydberg formula 6 (1) Black-body radiation * the radiation in thermal equilibrium with a body that absorbs and emits without favouring particular frequencies Fig. A black-body emitter can be simulated by a heated container with a pinhole in the wall * When heated all objects emit light! Classically: (1) Radiation from a black-body is the result of electrons oscillating with frequency ν Oscillating charged particle ⇔ antennae (2) The electrons can oscillate (& radiate) equally well at any frequency ⇒ Rayleigh-Jeans Law for spectral density ρ(ν), where intensity of emitted light in frequency range from ν to ν + dν is I(ν)~ ρ(ν)dν where dρ(ν,T) = density of radiative energy in frequency range from ν to ν + dν at temperature T. k = Boltzmann’s constant c = speed of light 7  Fig. The Rayleigh-Jeans ρν∝ν2 ⇒ divergence at high freq. ⇒ lots of emission in UV, x-ray regions! “UV catastrophe” – not observed in exp’t Planck (~1900) ⇒ First “quantum” ideas (1) The energy of the oscillator ∝ frequency (E ∝ν) (2) The energy ∝ an intger multiple of ν E ∝ nν ⇒ E = nhν where h constant hν becomes a “quantum” of energy Planck used statistical mechanics to derive the expression for black body radiation 8  Fig.The Planck distribution. Fitting experiment to model ⇒ h = 6.626 x 10-34 J-sec Planck’s constant 2) Heat capacities * “the atoms of all simple bodies have exactly the same heat capacity” of about 25 J K-1 mol-1 - Dulong and Petit (classical physics) * were done at room temperature. * In classical physics each atom acts as a classical oscillator in three dimensions. The heat capacity, CV,m of a monatomic solid should be equal to 3R=24.94 J/ K mol. * unfortunate for classical physics when measurements were extended to lower temperatures. It was found that all elements had heat capacities lower than predicted by Dulong and Petit’s law and that the values tended towards zero as T → 0. 9 * Einstein proposed: 1- each atomic oscillator required a certain minimum energy before it would actively oscillate and hence contribute to the heat capacity, then at low temperatures some would be inactive and the heat capacity would be smaller than expected. 2- applied Planck’s suggestion for electromagnetic oscillators to the material, atomic oscillators of the solid Where ɵE Einstein temperature Fig. The Einstein and Debye molar heat capacities. 3- the matter is quantized. 10 3) Photoelectric effect * The emission of electrons from metals when they are exposed to ultraviolet radiation. * Expt. * Classically expect - Electromagnetic radiation is an electric field oscillating perpendicular to its direction of propagation and the intensity is proportional to the square of the amplitude of the electric field. -The electron at the surface of the metal should oscillate along with field ⇒ increase intensity. - The kinetic energy of oscillating electron depends on the intensity. - Photoelectric effect occurs for any frequency in high intensity. * Expt Opposite 11 Einstein proposed: (1) Light is made up of energy “photons” (2) All the observations fell into place if the electromagnetic field was quantized, and that it consisted of bundles of energy of magnitude hv. (3) The photon energy proportional to the light frequency E = hν where h is called Planck’s constant New model of photoelectric effect: ∴ K.E. = hν – φ =hν –hν0 = h(ν – ν0) Comparing to exp’t, value of “h” matches the one found by Planck Summary: (1) Structure of atom can’t be explained classically (2) Discrete atomic spectra and Rydberg’s formula can’t be explained (3) Black-body radiation can be “explained” by quantifying energy of oscillators E = hν (4) In heat capacities the matter is quantized 12 (5) Photoelectric effect can be “explained” by quantifying energy of light E=hν Outward -The ATOM of NIELS BOHR * Assumptions underlying the Bohr model of atom (1) Atoms can exist in stable “states” without radiating. The states have discrete energies En, n = 1, 2,... The number “n” is an integer, a quantum number that labels the state. (2) Transitions between states can be made with the absorption or emission of a photon of frequency ν where ν = ΔE /h These two assumptions “explain” the discrete spectrum of atomic vapor emission. Each line in the spectrum corresponds to a transition between two particular levels. This is the birth of modern spectroscopy. (3) Angular momentum is quantized: Angular momentum For circular motion: 13 I= mrv is a constant of the motion Other useful properties Recall the moment of inertia Note: 14 Kinetic energy Rotational kinetic energy * -Introduce Bohr’s quantization into the Rutherford’s planetary model. * For a 1-electron atom with a nucleus of charge +Ze L= mevr = nћ 15 the Bohr radius - the radius is quantized (for H atom with n = 1, r = a0 = 0.529 Å) Take Rutherford’s energy and put in r, Substitute r-giving by above equation in the previous eq. Energies are quantized For H atom, emission spectrum - The duality of matter - Light (electromagnetic radiation) - Light as a wave For now we neglect the polarization vector orientation propagating in x-direction: 16 Define k =2π/λ (“wavevector” magnitude) At some fixed time, say t =0,can write simply ξ(x,t =0)=ξ0(sin kx) - Superposition principle Constructive interference: Destructive interference 17 This leads to many interference phenomena (1) on Bohr atom Criterion for stability: As Bohr had assumed angular momentum is quantized! 18 - Young’s two-slit experiment exemplify wave-particle duality This experiment displays the wave properties of light via interference of two beams of light. The beam of electrons or any others atomic will produce an interference pattern. - Light as a particle * Compton experiment If just a wave, expect light to scatter off electron 19 The backscattered wave is red-shifted (λ′>λ), i.e. less energy/photon. Energy (and momentum) transferred to the electron. Need relativistic mechanics to solve Light is a particle with energy In summary:- Light can behave both as a wave and as a particle - Matter Matter as particles ⇒ obvious from everyday experience Matter as waves ⇒ (deBroglie) Same relationship between momentum and wavelength for light and for matter Amazing notion! But wavelength only observable for microscopic momentum 20 21