PHY.F20 Molecular and Solid State Physics

Chemical Bonds

This section briefly decribes different kinds of chemical bonds. When two atoms form a bond, there will be a concentration of electron wavefunctions between the nuclei. A more electronegative atom will pull the average position of the electrons in the bond away from an atom with a lower electronegativity. Electronegativity is a measure of the tendency of an atom to attract electrons. The more electronegative an atom is, the more it pulls an electron towards itself.

Ionic bonds
If the electronegativity of one atom is much greater than another atom, then the more electronegative atom takes an electron almost completely away from the less electronegative atom and an almost purely ionic bond is formed. The atom that loses an electron becomes a positive ion and the atom that gains an electron becomes a negative ion. These ions are held together by the electrostatic force between the positive and negative charges. Ordinary table salt, NaCl, is held together by ionic bonds. The binding energy for NaCl is 7.97 eV per bond.

Since an ionic bond is mostly based on an electrostatic interaction, the order of magnitude of the bond strength can be estimated by calculating the energy that is needed to push a positive ion (charge e) away from a negative ion (charge -e). The Coulomb force between these two charges is,

F = e².
4πε0r²

The energy needed to push two charges apart from an interatomic distance of about 0.2 nm is,

E =
0.2 nm
-e²dr = 7 eV.
4πε0r²

When a compound is formed of two elements whose ions are of roughly the same size, they have what is called the interpenetrating simple cubic structure, where two atoms of a different type have individual simple cubic crystals. However, the unit cell consists of the atom of one being in the middle of the 8 vertices, structurally resembling body centered cubic. The most common example is caesium chloride CsCl.

However, if the cation is slightly smaller than the anion (a cation/anion radius ratio of 0.414 to 0.732), the crystal forms a different structure, interpenetrating FCC. When drawn separately, both atoms are arranged in an FCC structure as in NaCl.


The NaCl crystal structure.

Source: http://en.wikipedia.org/wiki/Face-centered_cubic

Covalent bonds
In a covalent bond, atoms share electrons. This reduces the quantum confinement energy of the electrons because the electron wavefunctions spread out over the two atoms. A bond between two identical atoms is always a covalent bond. While electrostatic force also play a role in covalent bond, the most important energy is the quantum confinement energy.

To estimate the strength of a covalent bond, a three-dimensional square well can be used as a simple model for an atom. The lowest energy of an electron in a three-dimensional square well is,

E =  3h².
8mL²

When two such cubic atoms come together to form a molecule with dimensions L×L×2L, the electrons can spread out and their energies become lower. The lowest energy level of the molecule is,

E =  9h².
32mL²

Each of the electrons from the cubic atoms could occupy one of the two spin degenerate ground states of the rectangular molecule. The corresponding decrease in energy would be,

ΔE =  3h².
16mL²

For cubic atoms with a size L = 0.2 nm, this decrease in energy is ΔE = 14 eV.

To properly calculate the energy of a covalent bond, the electrostatic interactions of the electrons and ions would have to be taken into account. However, this calculation describes the essence of covalent bond; the energy is decreased because both electrons can spread out over a larger volume and that decreases the quantum confinement energy. Note that the energy of a covalent bond is approximately equal to the energy of an ionic bond. This is not a coincidence. The size of an atom is determined by a compromise between the electrostatic energy (that is lower when the electron comes closer to the ion) and the quantum confinement energy (that is higher as the electron is confined to a smaller region around the ion). At the length scale of an atomic radius, the electrostatic energy about balances the quantum confinement energy.

Polar bonds
Many bonds have a partly ionic character and partly covalent character. A bond between two atoms where one has a higher electronegativity than the other is called a polar bond because one end of the bond is more positively charge and the other end is more negatively charged. The reduction of confinement energy and electrostatic forces both play a role in a polar bond.

Bondlength (nm) and bond energy (eV)

Bond

Length

Energy

Bond

Length

Energy

H--H

0.074

4.52

H--C

0.109

4.28

C--C

0.154

3.61

H--N

0.101

4.05

C=C

0.134

6.36

H--F

0.92

5.89

C≡C

0.120

8.70

H--O

0.96

3.79

C--O

0.143

3.73

H--Cl

0.127

4.48

C--S

0.182

2.82

H--Br

0.141

3.79

C--F

0.135

5.06

H--I

0.161

3.09

C--Cl

0.177

3.42

N--N

0.145

1.76

C--Br

0.194

2.98

I--I

0.267

1.57

C--I

0.214

2.24

O--O

0.148

1.50

C--N

0.147

3.19

O=O

0.121

5.16

N--N

0.145

1.76

N≡N

0.110

9.79

O--O

0.148

1.50

Cl-Cl

0.199

2.52

F--F

0.142

1.64

Br-Br

0.228

2.00

Quantum mechanics can be used to precisely calculate these bond lengths and energies. However, these calculations are mathematically complicated. A good approximation to potential of a molecular bond is the Morse potential.

U(r) = U0(e-2(r - r0)/a - 2e-(r - r0)/a)

Here U0 is the bond energy and r0 is the bond length that can be read from the above table. The parameter a describes the width of the Morse potential and is typically between r0/2 and r0/15.

The Morse potential with U0 = 3 eV, r0 = 0.15 nm, and a0 = 0.075 nm.

The minimum of the Morse potential occurs are r = r0. The potential rises sharply for small bondlengths due to the Coulomb repulsion of the positive nuclei when they get too close together.

Sigma bonds
In a
sigma bond, the electrons wavefunctions that form the bond are concentrated along the line between the two nuclei. Examples of overlaping are shown below. Hybrid sp, sp2, and sp3 orbitals can also form sigma bonds. Two parts of a molecule that are connected by a single sigma bond can rotate with respect to each other.

 →  A sigma bond formed by two s-orbitals.
 →  A sigma bond formed by an s-orbital and a p-orbital.
 →  A sigma bond formed by two p-orbitals.

Pi bonds
A pi bond forms when two side by side p-orbitals overlap. The electron wavefunction overlap in two places above and below the line connecting the two nuclei. Two parts of a molecule connected by a pi bond cannot rotate with respect to each other.

 →  A pi bond formed by two p-orbitals.

Single bond
In a single bond, one pair of electrons is shared between two atoms. Single bonds are always sigma bonds.

Double bonds
A double bond is a sigma bond and a pi bond. The pi part of a double bond does not allow for rotation.

Triple bonds
The triple bond is made up of one sigma bond and two pi bonds. It does not allow rotation.

Metallic bonds
In a metal, the electron wavefunctions of the valence electrons spread out over the whole crystal. This results in a large decrease in confinement energy and makes a strong bond. The electrons in a metal are delocalized. This makes metals good electrical and thermal conductors. The transition metals have high melting points because the d-electrons delocalize as well as the s-electrons and they all form metallic bonds.

Hydrogen bonds
Hydrogen tends to form polar bonds which leave the hydrogen atom with a net positive charge. The positively charged hydrogen can then be attracted to a negatively charged region of another molecule forming a hydrogen bond. Hydrogen bonds tend to be weaker than covalent or ionic bonds. The hydrogen bond formed between water molecules has a dissociation energy of 0.24 eV. A hydrogen bond has is approximately 90% ionic and 10% covalent.

Van der Waals bonds
Van der Waals bonds are weak bonds with a dissociation energy of about 0.01 eV. The binding force comes from charge fluctuations. All atoms exhibit charge fluctuations where the electrons are not symmetrically distributed around the nucleus. This results in a fluctuating electric dipole moment. Two nearby atoms can reduce their total energy if they fluctuate in unison so that the positive end of one of the atoms is located by the negative end of the other atom. This results in a dipole-dipole force that holds the atoms together.