ENERGY QUANTISATION

Dr Neil Allan

• Any atom or molecule can ONLY HAVE CERTAIN FIXED ENERGIES, energy is QUANTISED and an atom or molecule can exist only in certain discrete levels.

In many ways electrons, atoms and molecules behave like nothing like we might expect!

e.g., a vibrating molecule oscillates like a pendulum,

• There are 10¹² oscillations per second, i.e., the frequency of vibration is 10¹² s^-1
• The molecule has vibrational energy and quantum chemistry tells us the molecule can ONLY have vibrational energy,

where n ("nu") is the frequency of vibration and h is Planck's constant (approx. 6.6 x 10^-34 J s).

So in this case the allowed energies are:

• NOTE: Because the minimum energy is ½hn the molecule is always vibrating. It is never at rest! It has ZERO-POINT energy, which must be allowed for in many applications.

MOLECULAR ENERGY LEVELS

Molecules can exist only in certain discrete energy states.

• Why is energy quantization important? Becauseof the link between the energy levels and the spectra observed experimentally.

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SPECTROSCOPY AND ENERGY LEVELS

The difference in energy between any two different energy levels determines the frequency of light absorbed when there is a transition between the two levels.

• The Bohr-Einstein condition is:

where n ("nu") is the frequency of the light absorbed/emitted and h is Planck's constant. hn is a quantum ("packet") of energy - called a photon.

If have many energy levels get many transitions - hence get SPECTRUM characteristic of the atom or molecule studied (molecular "fingerprint").

Since light is so central to chemistry we now consider light in more detail.

LIGHT

• Exhibits both wave and particle properties! This wave-particle "duality" is a feature of the quantum world.

Light as a Wave:

• Light consists of an oscillating electric field and an oscillating magnetic field that travels through space with speed c (2.998 x 10⁸m s^-1 in vacuum).
• Often also encounter the wavenumber

         eq.(1.2)

Units of are m^-1, though usually quoted in cm^-1.

Example: Street lights are yellow due to emission from excited Na atoms: in fact have 2 spectral lines centred on

Find n and .

= 1 / l = 1,697,000 m^-1 = 16,970 cm^-1.

Light as particle:

• Sometimes, though, light behaves as if it was a stream of particles. (e.g., eq. (1.1))
• E.g., the photoelectric effect (basis of photoelectron spectroscopy)

See handout

• Light as wave and particle

Also: (!)

• Electrons and atoms also behave like particles and waves

Electrons as waves:

• For light, energy of a photon, E = hn

Einstein also showed E = mc²

So mc² = hn = hc / l, and therefore l = h / mc

• By analogy, de Broglie suggested for any particle moving with velocity v

For given v: if m is large, have small l.

For given m: if v is large, have small l.

• Electrons can be diffracted just like light! A Ni crystal will diffract a beam of electrons. The spacing between the Ni atoms is comparable to the wavelength of the electrons. The de Broglie equation is confirmed.
• Electrons exhibit wave properties. Electron waves Y("psi") in atoms and molecules called orbitals. We must use a model for atoms and molecules that incorporates the wave properties of electrons.
• Atoms can be diffracted too!

To sum up:

• Light: behaves like particles and waves
• Electrons: behave like particles and waves.

More about electron "waves"

• Because of its wavelike character cannot say precisely where an electron is when it is travelling along a path. This is expressed in Heisenberg's Uncertainty Principle:

• The more precisely we know the position of a particle the less we can say about the speed (and vice versa). If we know the position of a particle of mass m to within a range Dx then its momentum must be uncertain by at least an amount Dp where

          eq. (3.2)

(h is Planck's constant)

Example: What is the minimum uncertainty in the speed of an electron Dv in an H atom given the uncertainty in its position, Dx, is the diameter of the atom (10^-10 m)?

Method: Estimate Dv from Dp since p=mDv

From eq. (3.2)

and

• If Dx is small, Dp must be very large!
  If Dp is small, Dx must be very large!

• Pictures of electrons in precise orbits cannot be valid because the electron would then have a definite position and momentum at any instant. Need a completely different picture incorporating wave properties!

• Electron waves (more correctly wavefunctions) Y ("psi") in atoms and molecules are called orbitals.
• The interpretation of electron wavefunctions (orbitals) is that we talk about the probabilities of an electron being at a particular place, e.g., being at a given distance from a nucleus.
• If the probability of finding an electron at some point is high then the electron density is large at that point.
• How is the probability linked to the wavefunction Y? Basic idea is due to Born. He suggested ("postulated") the probability of finding a particle in a small region of volume dV is

If Y² is large at a particular point there is a high probability of finding the electron at that point.

• For each energy level there is an associated orbital, Y e.g., for the H atom have 1s, 2s, 2p etc.
• How do we find out the allowed energy levels E and corresponding Y for atoms and molecules? We use a famous equation suggested by Schrödinger.

Input: All the interactions involving the electrons and the nuclei (classical electrostatics)

Output: The allowed E and corresponding Y.

EXAMPLES:

(i) H atom

(ii) large conjugated organic molecules

HYDROGEN ATOM

• Experiment: Spectrum of atomic H consists of series of narrow lines - major evidence for the quantization of energy.
• Each series looks like:

• Quantum theory shows allowed energy levels of the H atom are:

      eq. (4.1)

h is Planck's constant, c speed of light. R is the Rydberg constant (109,678 cm^-1).

n is the PRINCIPAL QUANTUM NUMBER. n can take integer values 1,2,3..........

• Energy levels are widely separated at low values of n but they converge as n increases.
• Zero of energy (n=infinity) corresponds to the infinitely separated and stationary electron and nucleus, i.e., the ionised H atom. Relative to this all energies described by eq. (4.1) are negative: the states it describes are the bound states of the atom.
• The lowest energy state is n=1, E₁ = -hcR         GROUND STATE
  First excited state is n=2, E₂=-hcR/4.
  The ionisation energy is the minimum energy required to remove the electron from the H atom in its ground state (n=1). E₁ = -hcR = -2.18x10^-18 J so the ionisation energy is +2.18x10^-18J (1312 kJ mol^-1)

• Explains the spectrum of the H atom quantitatively! Each line corresponds to a transition between two energy levels.

For transition from level with quantum number n₁ to level with quantum number n₂   (n₂> n₁)

• All the lines in the spectrum fit this formula eq. (4.2).

Lyman series: n₁=1      n₂=2,3,4,5,6.....

Balmer series: n₁=2     n₂=3,4,5,6.....

• Absorption spectrum of H atom only consists of one series of lines (n₁=1 to n₂=2,3,4...) since all atoms at room temperature are in the ground state (n₁=1). If pass electric discharge through H atoms, excite atoms to many excited states and in emission spectrum we see several series of lines as electrons jump down to lower levels from higher levels.

• Associated with each energy level is a wavefunction Y. For one-electron systems the Y are called orbitals. When an electron is described by one of these wavefunctions we say that it occupies that orbital.

e.g., the n=1 level corresponds to the 1s orbital.
the n=2 to the 2s, 2p orbitals.
the n=3 to the 3s, 3p, 3d orbitals.

In the H atom 2s, 2p orbitals are DEGENERATE (=have same energy), similarly 3s, 3p, 3d orbitals have same energy, i.e., energy depends ONLY on the principal quantum number n (eq. (4.1)). ONLY true for one-electron systems!

• In many-electron atoms 2 electrons (with opposite spin) can occupy any orbital. Similarly in molecules 2 electrons can occupy any molecular orbital which can extend over many atomic centres.

CONJUGATED ORGANIC COMPOUNDS

• "Particle-in-a-box" model for p-electrons in molecules such as H₂C=CH-CH=CH₂.
• The p-electrons are confined to the molecule - assume the potential energy of the electrons is zero throughout the molecule.
• The key to explaining the energy levels of the electrons is their wave properties. The electrons are confined, and so the electron waves are "standing" waves.
• Compare waves on a guitar string which is fixed at both ends. Only certain wavelengths are possible for vibrations - the wavelengths of the allowed vibrations must be such that a whole number of half-wavelengths are equal to the length of the box.
• Let the molecule (length L) lie along the x-axis between x=0 and x=L.

e.g.,

• So any electron wave/wavefunction Y must be zero for x<=0 and x>=L. Zero probability of finding electron outside box. Wavelengths l can ONLY be

In general

      n=1,2,3,4      eq. (5.1)

From De Broglie,

      (eq. 3.1), and

So substituting for l from eq. (5.1),

• Finally

         n=1,2,3,4....eq.(5.2)

where n is a quantum number.

• The corresponding wavefunctions Y are:

   for 0 <=x <=L,     n=1,2,3...eq.(5.3)

See handout.

• Energy quantized! So we have a set of allowed energy levels and corresponding wavefunctions Y (orbitals).
• As n increases, the energy increases and the number of nodes in Y increases. A node is where passes through zero. Y for n=0 has zero nodes, Y for n=1 has 1 node (in the middle), Y for n=2 has 2 nodes. Probability of finding electron at a node is 0.

N.B. The edges of the box where Y=0 are not nodes - Y does not pass through zero at edges.

• The lowest possible energy is not zero, but is the energy for n=1 (E=h²/8mL²). This is the zero-point energy.

• EXAMPLE: Apply model to butadiene,

We have 4 p-electrons, which fill up the lowest 2 p-orbitals. Length of molecule is L.

n=1 and n=2 are the occupied p-orbitals. n=2 is the HOMO (highest occupied molecular orbital), n=3 is the LUMO (lowest unoccupied molecular orbital).

• Lowest energy electronic transition is when an electron is excited from the HOMO to the LUMO (here n=2 to n=3). This transition occurs in the ultra-violet region of the spectrum.

• In general consider transition between adjacent levels - from level with quantum number n to the level with n+1. Energy gap, DE, given by:

• So DE decreases as L increases. As the chain length increases, i.e., as go down series,

then HOMO-LUMO separation DE decreases.

• Since DE=hn then the frequency of light absorbed when this transition takes place decreases as L increases, and for large conjugated carbon chains this transition takes place in the visible region. Hence COLOUR. Basis of dye industry.
• The multiple choice questions "Basic ideas in physical science" are on the Web.