Physical Chemistry

Quantum Chemistry

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

» 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 t

he electron

• OUTCOME New Rules!!!�

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

– (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 electrochemis

try 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.

» Thomson found that results are independent of (1) cathode material

» (2) residual gas composition » “� electron” is a distinct particle, present in all materi

als! » Classical mechanics force on electron due to defl�

ector voltage:

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

» (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

• Rutherford backscatterng experment

» (1) He2+ nucleus verysmall, << 10-10 m (Rutherford estimated 10-14 m)

• (2) Au atoms are mostly empty!

» 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 emi�tted light = frequency of rotation

» atom should emit light at all frequencies – that is it should produce a continuous spectrum

» 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:

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

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

Planck (~1900) First “quantum” ideas

• (1) The energy of the oscillator frequency

• (2) The energy an integral multiple of

» hbecomes a “quantum” of energy

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

» (b) Photoelectric effect

» 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

» New model of photoelectric effect:

» Comparing to exp’t, value of “h” matches the one found by Planck! This was an extraordinary result !

» 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

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 stat

es 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 .

» 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

» 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

WAVE-PARTICLE DUALITY of LIGHT and MATTER

• MATTER(A)Light (electromagnetic radiation)• Light as a waveFor now neglect polarization vector orie

ntationPropagating in x-direction:

Young’s 2-slit experiment

日本 Hitachi 公司 的 Tonomura 利用

Electron Phase Microscope 所做的雙狹縫實驗 • Double-slit experiment

Fig. 1 Double – slit experiment with single electrons

Fig. 2

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

» Light as a particle

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

• (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

» Consequences (I)

• As Bohr had assumed angular momentum is quantized!!!

WAVE-PARTICLE DUALITY of MATTER

» Consequences (II)

» Heisenberg Uncertainty Principle

» Consider diffraction through a single slit

» 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

» 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!

» The conventional statement of the Heisenberg Uncertainty Principle is

» Implications for atomic structure

» Apply Uncertainty Principle to e-in H atom

» 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??

» 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

» 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

» 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

» We now have the outline of: » a physicalpictureinvolving wave and particle duality

of light and matter ! » a quantitativetheoryallowing calculations of stable st

ates and their properties !