Metals have different physical properties from either ionic compounds or covalent compounds: high melting and boiling points, hard and crystalline but malleable and ductile, conducts electricity in solid or melt but not soluble nor an electrolyte.
Need a new bonding model in order to account for these properties: use the crystalline structure as a starting point but since we do not have ions we cannot use simple electrostatics; this means that we must use orbitals and overlap as the cohesive force. Consider building a covalent lattice one particle at a time (like we built the ionic lattice):
The orbital diagram is called a quasicontinuum of states or a band. The energy between orbitals is ~ bandwidth/N0 ~ 0 hence called continuous.
A typical metal bandwidth is ~10 eV so the energy between orbitals is ~10 eV/6×1023 ~ 1.7×10-23 eV/state or, in terms of temperature, ~ 1.9×10-19 K/state; this means that electrons can move from orbital to orbital very readily at any temperature.
The bandwidth is related to the amount of overlap: greater overlap leads to a wider band.
What do the orbitals look like?
The lowest energy orbitals are all bonding; the highest are all antibonding. If the band is only partially filled, there is a large bonding energy explaining the high mp and bp.
Metals always have partially filled bands.
Degeneracy of orbitals in the band can exist which leads to the idea of density of states: the number of states found in a given energy interval (mathematically, dN/dE)
Density of States is usually represented graphically as follows:
These represent the solid state equivalent of molecular orbitals
Metallic properties arise from partially filled bands called conduction bands.
Energetically, there are a huge number of bonding orbitals filled and fewer antibonding orbitals so the "bond order" is large which means that metals are hard and have high mp and bp. Mechanical properties: deforming the sample is not traumatic because the electrons can easily move into orbitals that accommodate the changes. Finally, charge transport occurs because an electric field can excite electrons into orbitals delocalized over the whole sample (or at least long distances) which allows motion across the whole sample. All of this depends on the availablity of many (i.e., a high density of states) orbitals at low energy above the highest filled orbital.
The energy of the highest filled orbital (at 0 K) is called the Fermi level; this is equivalent of the chemical potential.
Why is there any electrical resistance?
Defects in the lattice disrupt the delocalization of wavefunctions. If a charge is being moved from one end of a wire to the other end but hits a defect site, the direction of charge motion is changed (i.e., the charge carrier is scattered) and the charge either takes longer to get to the other end or never does - this is resistance. "Defects" can be created by vibrations. If a vibration occurs at the same time that a charge arrives at a particular site, there will be a scattering event. Since the number of vibrations that are active depends on temperature, thus the resistivity must also depend on temperature.
In general, conductivity = = ne, where n is the number of charge carriers, e is the electronic charge, and is the mobility (which is dependent on the number of scattering events)