Molecules and solids

Bonds. Energy States and Spectra of Molecules. Bonding in Solids.

Free-Electron Theory of Metals. Band Theory of Solids. Electrical Conduction in Metals, Insulators, and Semiconductors. Semiconductor Devices. Superconductivity.

model a molecule should account for two known features of

molecular bonding:
1. The force between atoms is repulsive at very small separation distances. When two atoms are brought close to each other, some of their electron shells overlap, resulting in repulsion between the shells. This repulsion is partly electrostatic in origin and partly the result of the exclusion principle. Because all electrons must obey the exclusion principle, some electrons in the overlapping shells are forced into higher energy states and the system energy increases as if a repulsive force existed between the atoms.
2. At somewhat larger separations, the force between atoms is attractive. If that were not true, the atoms in a molecule would not be bound together.
Taking into account these two features, the potential energy for a system of two atoms can be represented by an expression of the form

system is graphed in Figure. At large separation distances between

the two atoms, the slope of the curve is positive, corresponding to a net attractive force. At the equilibrium separation distance, the attractive and repulsive forces just balance. At this point, the potential energy has its minimum value and the slope of the curve is zero.

ions.

in which electrons supplied by either one or both atoms

are shared by the two atoms.

molecules would not interact by means of the electric force

because they each have zero net charge. They are attracted to each other, however, by weak electrostatic forces called van der Waals forces.

There are three types of van der Waals forces. The first type, called the dipole–dipole force, is an interaction between two molecules each having a permanent electric dipole moment. For example, polar molecules such as HCl have permanent electric dipole moments and attract other polar molecules.

force, results when a polar molecule having a permanent electric

dipole moment induces a dipole moment in a nonpolar molecule. In this case, the electric field of the polar molecule creates the dipole moment in the nonpolar molecule, which then results in an attractive force between the molecules.
The third type is called the dispersion force, an attractive force that occurs between two nonpolar molecules. In this case, although the average dipole moment of a nonpolar molecule is zero, the average of the square of the dipole moment is nonzero because of charge fluctuations. Two nonpolar molecules near each other tend to have dipole moments that are correlated in time so as to produce an attractive van der Waals force.

expected to form a covalent bond with only one other

atom within a molecule. A hydrogen atom in a given molecule can also form a second type of bond between molecules called a hydrogen bond.

DNA molecules are held together by hydrogen bonds

mass of the molecule

the rotational quantum number

the vibrational quantum number

any molecule can be calculated from this expression, and each

level is indexed by the two quantum numbers v and J.

In general, a molecule vibrates and rotates simultaneously. To a first approximation, these motions are independent of each other, so the total energy of the molecule for these motions is

a photon with the appropriate energy, the vibrational quantum number

v increases by one unit while the rotational quantum number J either increases or decreases

Therefore, the molecular absorption spectrum consists of two groups of lines:

the Madelung constant

for an ionic solid, where U0 is the ionic cohesive

energy and r0 is the equilibrium separation distance between ions

This minimum energy U0 is called the ionic cohesive energy of the solid, and its absolute value represents the energy required to separate the solid into a collection of isolated positive and negative ions.

The atomic cohesive energy of NaCl is

or covalent bonds. The outer electrons in the atoms of

a metal are relatively free to move throughout the material, and the number of such mobile electrons in a metal is large. The metallic structure can be viewed as a “sea” or a “gas” of nearly free electrons surrounding a lattice of positive ions

energy E is occupied by one of the electrons in

a solid is

f(E) is called the Fermi–Dirac distribution function and EF is called the Fermi energy.

Statistical physics can be applied to a collection of particles in an effort to relate microscopic properties to macroscopic properties. In the case of electrons, it is necessary to use quantum statistics, with the requirement that each state of the system can be occupied by only two electrons (one with spin up and the other with spin down) as a consequence of the exclusion principle.

function.

in sodium as a function of the separation distance r

between atoms.

represents energy bands occupied by the sodium electrons when the

atom is in its ground state. Gold represents energy bands that are empty.

Band theory allows us to build simple models to understand the behavior of conductors, insulators, and semiconductors as well as that of semiconductor devices, as we shall discuss in the following sections.

an electrical conductor. At T=0 K, the Fermi energy lies

in the middle of the band.

called the valence band, and the upper, empty band is

the conduction band. (The conduction band is the one that is partially filled in a metal.) It is common to refer to the energy separation between the valence and conduction bands as the energy gap Eg of the material.

of band structure as an insulator, but the energy gap

is much smaller, on the order of 1 eV.

superconductors whose electrical resistance decreases to virtually zero below a

certain temperature Tc called the critical temperature.