upper H 2"/> with two electrons. In covalently bonded solids, each atom forms bonds to several neighbors and the total energy gain per atom can be higher than for . In silicon, for example, the energy gain per atom (or cohesive energy) is 4.6 eV.
Apart from a high cohesive energy, a characteristic feature of covalent bonding is the directional nature of the bonds. The preference for certain bond directions governs the crystal structures of covalently bonded solids, and these are more complex than the close‐packed structures typically encountered in metals. The treatment of the molecular ion has demonstrated the idea of constructing bonding and antibonding molecular orbitals from a linear combination of atomic orbitals, but since we combined two isotropic 1s orbitals in this case, the directional character of covalent bonding did not emerge.
Figure 2.3 demonstrates the consequences of using directional orbitals, such as p orbitals, for the formation of wave functions in molecules or solids. Figure 2.3a represents the case that we have just treated. Two s orbitals are combined to produce a bonding and an antibonding orbital by being added either in phase or out of phase (as indicated by the shading and the sign on the wave function). Due to the symmetry of the s orbital, the bond direction is not important. Combining an s orbital with a p orbital along the interatomic axis works in the same way (see Figure 2.3b). The p orbital is strongly anisotropic, but still the direction between the atoms does not matter in this case, because we define the intermolecular axis as the axis and then use the orbital aligned in this direction to form bonding and antibonding states. However, in this arrangement, no bonding interactions are possible between the s orbital of the left atom and the and orbitals of the right atom. This is illustrated in Figure 2.3c. No matter what linear combination coefficients we use, the overlap of the wave functions contains equal and mutually canceling bonding and antibonding contributions. This dilemma is the source of the directional preference in covalent bonding. In order to achieve the highest energy gain, it is often favorable to use linear combinations of the orbitals in one atom before combining them with other atoms. Examples for this are the and hybrid orbitals found in carbon‐based solids such as diamond or graphite, which directly explain the preferred bonding directions shown in Figure 1.7. The orbitals in graphene are also shown in Figure 6.15a. All these orbitals are highly directional.
Figure 2.3 Linear combination of orbitals on neighboring atoms. (a) Two s orbitals as in the molecular ion. (b) An s orbital and a orbital (the choice of the direction is arbitrary). (c) An s orbital with a or orbital. Due to symmetry, the total overlap is zero and no bonding or antibonding orbitals can be formed.
In more complex solids, several bonding types can coexist. For instance, the diamond structure of Figure 1.7 is also found for materials that mix group III and group V elements, such as GaAs, or group II and group VI elements, such as InP. In these, the different atoms occupy alternating sites but still adopt the bonding arrangement. This is accompanied by an electron transfer from the higher‐group atom to the lower‐group atom, resulting in a bonding situation with both covalent and ionic contributions.
We now return to the Schrödinger equation for the full Hamiltonian from Eq. (2.3). This has to be solved by a wave function containing the coordinates and of two electrons (we still keep the positions of the nuclei fixed). This makes the problem much harder and we can only begin to imagine the difficulty of finding a wave function describing all the electrons in a solid! Fortunately, even in a solid, most cases can be described quite well by considering one electron moving in the potential of the ions and some averaged potential arising from the other electrons. In this book, there are only two cases where this simple description will fail: magnetism and superconductivity. The following discussion will mainly be useful for the treatment of magnetism in Chapter 8. Understanding these details is not crucial at this point, and you could decide to jump to Section 2.4 and return here later.
Solving the Schrödinger equation for the Hamiltonian from Eq. (2.3) would be greatly simplified if we could somehow “switch off” the electrostatic interaction between the two electrons, because then the Hamiltonian could be written as the sum of two parts, one for each electron. Indeed, the Hamiltonian in Eq. (2.3) could essentially be the sum of two Hamiltonians similar to the one in Eq. (2.4), one for each electron (but with the term appearing only once). If a Hamiltonian containing two electronic coordinates could be separated into a sum of two Hamiltonians that contain only one electronic coordinate each, the corresponding Schrödinger equation could be solved by a product of the two wave functions that are solutions to the two individual Hamiltonians. We could therefore start an attempt to solve Eq. (2.3) based on what we have already learned for the ion. However, we will start from an even simpler point of view: Without any interaction between the electrons, and for a large distance , the electron near nucleus A will not feel the potential of nucleus B and vice versa. In this case, Eq. (2.3) would simply turn into the sum of the Hamiltonians for two hydrogen atoms and we could approximate the two‐electron wave function by
