CHAPTER 10 Molecules Why do molecules form? Molecular bonds Rotations Vibrations Spectra Complex planar molecules Johannes Diderik van der Waals (1837 – 1923) http://en.wikipedia.org/wiki/Image:Johannes_Diderik_van_der_Waals.jpg “Life ... is a relationship between molecules.”  Linus Pauling Prof. Rick Trebino, Georgia Tech, www.frog.gatech.edu

Molecules are combinations of atoms. When more than one atom is involved, the potential and the wave function are functions of way more than one position (a position vector for each nucleus and electron): Electrons’ positions Nuclei positions Solving the Schrodinger Equation in this case is even harder than for multi-electron atoms. Serious approximation methods are required. This is called Chemistry! http://www.losgazquez.com/blog/?m=200712

10.1: Molecular Bonding and Spectra Nucleus Electron cloud Nucleus Electron cloud The only force that binds atoms together in molecules is the Coulomb force. But aren’t most atoms electrically neutral? Yes! Indeed, there is no attraction between spherically symmetrical molecules—the positive and negative charges both behave like point sources and so their fields cancel out perfectly! So how do molecules form?

Why Molecules Form + - Most atoms are not spherically symmetrical. For example, these two “atoms” attract each other: + - This is because the distance between opposite charges is less than that between charges of the same sign. http://www.chemsw.com/images/10202c.gif The combination of attractive and repulsive forces creates a stable molecular structure. Force is related to the potential energy surface, F = −dV/dr, where r is the position.

Charge is distributed very unevenly in most atoms. The probability density for the hydrogen atom for three different electron states.

Closed shells of electrons are very stable. Atoms with closed shells (noble gases) don’t form molecules. Ionization energy (eV) Atomic number (Z) Noble gases (difficult to remove an electron)

Atoms like closed electron shells. Atoms with closed shells (noble gases) also have the smallest atomic radii. Atomic number (Z) Atomic radius (nm) But add an extra electron, and it’s weakly bound and far away.

An extra electron or two outside a closed shell are very easy to liberate. Atoms with one or two (or even more) extra electrons will give them up to another atom that requires one or two to close a shell. Atoms with extra electrons are said to be electropositive. Those in need of electrons are electronegative. http://www.thestudentroom.co.uk/showthread.php?t=1173126

Ionic Bonds An electropositive atom gives up an electron to an electronegative one. Example: Sodium (1s22s22p63s1) readily gives up its 3s electron to become Na+, while chlorine (1s22s22p63s23p5) easily gains an electron to become Cl−. http://www.accessexcellence.org/RC/VL/GG/ecb/covalent_ionic_bonds.html http://www.biology.arizona.edu/biochemistry/tutorials/chemistry/page2.html

Covalent Bonds Two electronegative atoms share one or more electrons. Example: Diatomic molecules formed by the combination of two identical electronegative atoms tend to be covalent. Larger molecules are formed with covalent bonds. http://www.accessexcellence.org/RC/VL/GG/ecb/covalent_ionic_bonds.html http://www.everyscience.com/Chemistry/Inorganic/Carbon/a.1189.php Diamond

Metallic Bonds In metals, in which electrons are very weakly bound, valence electrons are essentially free and may be shared by a number of atoms. The Drude model for a metal: a free-electron gas! http://thephysicstutor.com/notes/html/p1-heat.html

Molecular Potential Energy Curve The potential depends on the charge distributions of the atoms involved, but there is always an equilibrium separation between two atoms in a molecule. The energy required to separate the two atoms completely is the binding energy, roughly equal to the depth of the potential well. Vibrations are excited thermally, that is, by collisions with other molecules, or by light, creating superpositions of ground plus an excited state(s).

Molecular Potential An approximation of the force felt by one atom in the vicinity of another atom is: where A and B are positive constants. Because of the complicated shielding effects of the various electron shells, n and m are not equal to 1. One example is the Lennard-Jones potential in which n = 12 and m = 6. The shape of the curve depends on the parameters A, B, n, and m.

Vibrational Motion: A Simple Harmonic Oscillator The Schrödinger Equation can be separated into equations for the positions of the electrons and those of the nuclei. The simple harmonic oscillator accurately describes the nuclear positions of a diatomic molecule, as well as more complex molecules.

Vibrational States Dn = ±1 n is called the vibrational quantum number. Don’t confuse it for n, the principal quantum number of the electronic state. The energy levels are those of a quantum-mechanical oscillator. Vibrational-transition selection rule: Dn = ±1 The only spectral line is w ! However, deviations from a perfect parabolic potential allow other transitions (~2w, ~3w, …), called overtones, but they’re much weaker.

Vibrational Frequencies for Various Bonds Different bonds have different vibrational frequencies (which are also affected by other nearby atoms). Wavenumber (cm-1) ← Higher energy (frequency) Notice that bonds containing Hydrogen vibrate faster because H is lighter.

Water’s Vibrations Movies from http://www.lsbu.ac.uk/water/vibrat.html

Rotational States Consider diatomic molecules. A diatomic molecule may be thought of as two atoms held together with a massless, rigid rod (rigid rotator model). In a purely rotational system, the kinetic energy is expressed in terms of the angular momentum L and rotational inertia I.

Rotational States L is quantized. where ℓ can be any integer. The energy levels are Erot varies only as a function of the quantum number ℓ. = ħ2/I

Rotational transition energies And there is a selection rule that Dℓ = ±1. Transitions from ℓ +1 to ℓ : Emitted photons have energies at regular intervals:

Vibration and Rotation Combined Note the difference in lengths (DE) for larger values of ℓ. Note the similarity in lengths (DE) for small values of ℓ. DE increases linearly with ℓ. Most transitions are forbidden by the selection rules that require Dℓ = ±1 and Dn = ±1.

Vibration and Rotation Combined The emission (and absorption) spectrum spacing varies with ℓ. The higher the starting energy level, the greater the photon energy. Vibrational energies are greater than rotational energies. For a diatomic molecule, this energy difference results in band structure. The line strengths depend on the populations of the states and the vibrational selection rules, however. Weaker overtones Dn = 3 Dn = 0 Dn = 1 Dn = 2 Energy or Frequency → Dℓ = -1 Dℓ = 1

Vibrational/Rotational Spectrum In the absorption spectrum of HCl, the spacing between the peaks can be used to compute the rotational inertia I. The missing peak in the center corresponds to the forbidden Dℓ = 0 transition. ℓi- ℓf = -1 ℓi- ℓf = 1 ni- nf = 1

Frequencies in Atoms and Molecules Electrons vibrate in their motion around nuclei High frequency: ~1014 - 1017 cycles per second. Nuclei in molecules vibrate with respect to each other Intermediate frequency: ~1011 - 1013 cycles per second. Nuclei in molecules rotate Low frequency: ~109 - 1010 cycles per second.

Including Electronic Energy Levels A typical large molecule’s energy levels: E = Eelectonic + Evibrational + Erotational 2nd excited electronic state Lowest vibrational and rotational level of this electronic “manifold.” Energy 1st excited electronic state Excited vibrational and rotational level Transition There are many other complications, such as spin-orbit coupling, nuclear spin, etc., which split levels. Ground electronic state As a result, molecules generally have very complex spectra.

Studying Vibrations and Rotations Infrared spectroscopy allows the study of vibrational and rotational transitions and states. But it’s often difficult to generate and detect the required IR light. It’s easier to work in the visible or near-IR. DE Input light Output light Raman scattering: If a photon of energy greater than DE is absorbed by a molecule, another photon with ±DE additional energy may be emitted. The selection rules become: Δn = 0, ±2 and Δℓ = 0, ±2

Modeling Very Complex Molecules Sometimes more complex is actually easier! Many large organic (carbon-based) molecules are planar, and the most weakly bound electron is essentially free to move along the perimeter. We call this model the Perimeter Free-Electron Orbital model. plus inner elec-trons This is just a particle in a one-dimensional box! The states are just sine waves. The only difference is that x = L is the same as x = 0. So y doesn’t have to be zero at the boundary, and there is another state, the lowest-energy state, which is a constant:

Auroras http://www.itp.uni-hannover.de/~zawischa/ITP/atoms.html

Typical Aurora Emission Spectrum Intensity http://www.itp.uni-hannover.de/~zawischa/ITP/atoms.html

Species Present in the Atmosphere http://www.itp.uni-hannover.de/~zawischa/ITP/atoms.html

Constituents Contributing to Auroras H N http://www.itp.uni-hannover.de/~zawischa/ITP/atoms.html