1. Physical Chemistry Quantum Chemistry

2. Quantum Mechanics – Historical Background »Physics in the Late 19th Century (prior to quantum mechanics (QM)) Atoms are basic constituents of matter Newton’s Laws apply universally The world is deterministic »Physics was complete except for a few decimal places ! Newtonian mechanics explained macroscopic behavior of matter --planetary motion, fluid flow, elasticity, 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

3. »However there were several experiments that could not be explained by classical physics and the accepted dogma ! Blackbody radiation Photoelectric effect Discrete atomic spectra The electron as a subatomic particle » � Inescapable conclusions would result from these problems Atoms are not the most microscopic objects Newton’s laws do not apply to the microscopic world of the electron OUTCOME � New Rules!!!

4. 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

5. The DEMISE of CLASSICAL PHYSICS –(a) Discovery of the Electron »In 1897 J.J. Thomson discovers the electron and measures (eme) (and inadvertently invents the cathode ray (TV) tube) »Faraday (1860’s – 1870’s) had already shown using electrochemistry that amounts of electric current proportional to amounts of some substances could be liberated in an electrolytic cell. The term “electron” was suggested as a natural “unit” of electricity. »But Thomson experimentally observes electrons as particles with charge & mass.

6. »Thomson found that results are independent of (1) cathode material »(2) residual gas composition » � “electron” is a distinct particle, present in all materials! »Classical mechanics � force on electron due to deflector voltage:

8. »(b) 1909 Mulliken oil drop experiment determines e, me separately

9. »(c) Where are the electrons? What’s the structure of the atom? »Angstrom (10-10 m) atomic size scale already inferred from gas kinetics First “jellium” model didn’t last long

10. Rutherford backscatterng experment »(1) He2+ nucleus verysmall, << 10-10 m (Rutherford estimated 10-14 m) (2) Au atoms are mostly empty!

11. »BUT model not consistent with classical electrodynamics: Accelerating charge emits radiation! (centripetal acceleration = v2/r) And since light has energy, Emust be getting more negative with time »R must be getting smaller with time! »Electronspiralsintonucleusin~1010s! »Also, as rdecreases, vshould increase Frequency � of emitted light = frequency of rotation »atom should emit light at all frequencies – that is it should produce a continuous spectrum

12. »BUT emission from atoms was known to be discrete, not continuous! »For the H atom, Rydberg showed that the spectrum was consistent with the simple formula:

13. Summary: Rutherford’s model of the atom (1) Is not stable relative to collapse of electron into nucleus (2) Does not yield discrete emission lines, (3) Does not explain the Rydberg formula

14. The DEMISE of CLASSICAL PHYSICS (cont’d) (a) Blackbody radiation --All things emit light when heated up! »Classically: (1) Radiation from a blackbody is the result of electrons oscillating with frequency 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 k = Boltzmann’s constant [= R/NA (gas constant per molecule)] c = speed of light

16. Planck (~1900) First “quantum” ideas (1) The energy of the oscillator frequency (2) The energy an integral multiple of »hbecomes a “quantum” of energy

17. »Planck used statistical mechanics (5.62) to derive the expression for black body radiation

18. »(b) Photoelectric effect

19. »Einstein (1905) proposed: (1) Light is made up of energy “packets: “photons” (2) The energy of a photon is proportional to the light frequency E = hv h Planck’s constant

20. »New model of photoelectric effect: »Comparing to exp’t, value of “h” matches the one found by Planck! This was an extraordinary result !

21. »Summary: (1) Structure of atom can’t be explained classically (2) Discrete atomic spectra and Rydberg’s formula can’t be explained (3) Blackbody radiation can be “explained” by quantifying energy of oscillators E = h (4) Photoelectric effect can be “explained” by quantifying energy of light E = h

22. The ATOM of NIELS BOHR »Niels Bohr, a Danish physicist who established the Copenhagen school. –(a) Assumptions underlying the Bohr atom (1) Atoms can exist in stable “states” without radiating. The states have discrete energies En, n= 1, 2, 3,..., where n= 1 is the lowest energy state (the most negative, relative to the dissociated atom at zero energy), n= 2 is the next lowest energy state, etc. 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 where ν = ΔE/ h.

23. »These two assumptions “explain” the discrete spectrum of atomic vapor emission. Each line in the spectrum corresponds to a transition between two particular levels. Thisisthebirthofmodernspectroscopy. »(3) Angular momentum is quantized: l = nh where h = h /2π Angular momentum

25. »For H atom withn= 1, r=a0= 5.29x10-11m = 0.529 Å (1 Å = 10- 10m) »Take Rutherford’s energy and put in r, »Energies are quantized!!! »For H atom, emission spectrum

26. WAVE-PARTICLE DUALITY of LIGHT and MATTER MATTER(A)Light (electromagnetic radiation) Light as a waveFor now neglect polarization vector orientationPropagating in x-direction:

28. Young’s 2-slit experiment

29. 日本 Hitachi 公司 的 Tonomura 利用 Electron Phase Microscope 所做的雙狹縫實驗 Double-slit experiment Fig. 1Double – slit experiment with single electrons

30. Fig. 2 Single electron events build up to from an interference pattern in the double-slit experiments.

31. »Light as a particle Light can behave both as a wave and as a particle!! Which aspect is observed depends on what is measured.

32. (B) Matter Matter as particles obvious from everyday experience Matter as waves (deBroglie, 1929, Nobel Prize for his Ph.D. thesis!) »Same relationship between momentum and wavelength forlightandformatter

33. »Consequences (I) As Bohr had assumed angular momentum is quantized!!!

34. WAVE-PARTICLE DUALITY of MATTER »Consequences (II) »Heisenberg Uncertainty Principle »Consider diffraction through a single slit

35. »Now consider a beam of electrons with de Broglie wavelength.The slit restricts the possible positions of the electrons in the x direction: at the slit, the uncertainty in the electron x-position is »This means the electrons must go through the slit with some range of velocity components Vx

36. »So the position and momentum of a particle cannot both be determined with arbitrary position! Knowing one quantity with high precision means that the other must necessarily be imprecise!

37. »The conventional statement of the Heisenberg Uncertainty Principle is »Implications for atomic structure »Apply Uncertainty Principle to e-in H atom

38. »Basically, if we know the e-is in the atom, then we can’t know its velocity at all! »Bohr had assumed the electron was a particle with a known position and velocity. To complete the picture of atomic structure, the wavelike properties of the electron had to be included. »So how do we properly represent where the particle is??

39. »Schrodinger (1933 Nobel Prize) »A particle in a “stable” or time-independent state can be represented mathematically as a wave, by a “wavefunction” (x) (in 1-D) which is a solution to the differential equation

40. »We cannot prove the Schrödinger equation. But we can motivate why it might be reasonable. »Similarly, a left-traveling wave can be represented as » Both are solutions to the wave equation

43. »As in a vibrating violin string, the node positions are independent of time. Only the amplitude of the fixed waveform oscillates with time. »More generally, we can write wave equation solutions in the form

44. »We now have the outline of: » a physicalpictureinvolving wave and particle duality of light and matter ! » a quantitativetheoryallowing calculations of stable states and their properties !