Polyatomic chains

Consider a hydrogen polymer consisting of a row of N hydrogen atoms, numbered 1, 2, 3, . . . , N− 2, N − 1, N, and spaced evenly a distance b apart (Figure 10.2). Note that this is an artificial model since such a row would spontaneously dimerize (i.e., form pairs, H2

molecules). That is, it would undergo what is called the Peierls instability. Nevertheless, the model illustrates what happens when one passes from a short to a long chain (Feynman, Leighton, and Sands, 1963).

An equation for the molecular orbital,, of the chain in terms of atomic orbitals, φ (analogous with Equation (10.29)) is:

 = c1φ1+ c2φ2+ c3φ3+ · · · =

N j=1

cjφj (10.34)

where the φj are spherical, s-type amplitude functions. The coefficients cj need to be determined, as well as the energy of the moleculer state,.

To determine the cjand find the energy we need to solve the Schr¨odinger equation:

H = E (10.35)

Substituting Equation (10.34) into Equation (10.35) yields:

N j=1

cjHφj = E

N j=1

cjφj (10.36)

For the dimer, N = 2 (the hydrogen molecule), this becomes:

H (c1φ1+ c2φ2)= E(c1φ1+ c2φ2)

Since the energy at a particular site j depends on the charge density there, and this is proportional to|φ²|, we can evaluate Equation (10.36) by multiplying both sides by the amplitude for a particular site, sayφp, and then integrating both sides (where H is again

Figure 10.2 Linear chain of atoms with spacing b.

the Hamiltonian (energy) operator, E is the energy, and the cjare constants): In doing this we remember that theφ are both normalized and orthogonal. Therefore, for the overlap integrals we have:

φpφpdv = 1 and

φpφjdv = 0. Hence, as j sweeps through the various sites of the right-hand side in Equation (10.37) all of them drop out except the pth one. For example, for p= 1, the right-hand side becomes Ec1. However, if the atoms begin to overlap too much, the overlap integrals,

φpφpdv, no longer equal unity, and the analysis becomes more complex.

On the left-hand side, there are two kinds of matrix elements:

atomic (or on-site) Hpp=

Here it is assumed that hopping only occurs between nearest neighbors. The hopping in-tegrals for longer distances are small, taken here to be zero. So, for various positions,

p= k = 1, 2, 3, . . . , N:

Except for the top and bottom ones, the rest of these equations have the form of the equation for p= j, that is:

cj−1− zcj+ cj+1= 0 (10.38)

where z= (E − )/η.

The special equations at the top and bottom ends of the set can be eliminated by combining the ends so the chain becomes an endless loop of N atoms. The loop is then said to have a periodic boundary condition because a trip around it comes back to the same place (Figure 10.3). The N th position becomes the same as the zeroth position, so:

cN = c0 and cN+1= c1

From one atom of the ring to the next the electron density varies periodically, and the amplitude varies periodically, so the cj must vary periodically. Thus the solutions of Equation (10.37) can be expected to be exponential functions (or, according to Euler’s

Figure 10.3 Loop of N atoms with periodic boundary condition since the Nth atom occupies both the zeroth and the eighth positions so the configuration at each atom is the same as for every other atom.

equation, sines and cosines). Let the position of the j th atom be xj, and its nearest neigh-bors xj− b and xj+ b. The bond length is b. Also, let the wavelength of the probability amplitude beλ, and its wave number k = 2␲/λ. Then solutions of Equation (10.38) have the form: c(xj)= exp(ikxj), c(xj+ b) = exp[ik(xj+ b)], etc. Substituting these into Equation (10.38) gives:

exp[ik(xj− b)] − z exp(ikxj)+ exp[ik(xj+ b)] = 0 Dividing out the exp(ikxj), and using Euler’s equation:

z= −(e^−ikb+ e^ikb)= −2 cos(kb) = (E − )/η so

E = − 2η cos(kb) (10.39)

This indicates that the energy varies from ( − 2η) to ( + 2η) as k goes from zero to

±␲/b. However, the number of atoms in the loop, N, cannot be less than two. For this case, k= 0 or ±␲/b, and there are two values of E : ± 2η. Since η is negative, + 2η is a bound state, while − 2η is anti-bound. As N increases, for each new value of N, one new value of energy appears, alternately higher or lower than . Thus, for large N, the band of energy ± 2η becomes filled with levels with only small increments between the levels (Figure 10.4). Half the band is made up of bonding levels, and half of anti-bonding levels. This is the principal result of extending the simple theory from a diatomic molecule (N = 2) to a polymer molecule of length N. Namely, the set of two energy levels of a diatomic molecule becomes a band with N levels.

+ 2η

∋

1 2 3 4 5 6 7 8 9 10 11 12 20

NUMBER OF ATOMS IN CHAIN

ENERGY ∋

∋ -- 2η

Figure 10.4 Schematic distribution of overlap energies for sets of N atoms each having one s-state.

The number of levels equals the number of atoms for each state. The zero of the energy scale is at the level for one atom. This splits into two levels when the s-amplitude function of this atom overlaps with the amplitude function of another identical atom. One of these is lower (bonding), and one higher (anti-bonding) than the energy of the non-bonded single atom. When another atom is added to the chain (N = 3), the two levels split further to create three energy levels (bonded, non-bonded, and anti-bonded). The splitting continues as N increases, with less magnitude at each step so the total range of energies asymptotically approaches a constant value with increasing N , thereby generating a band of energies of width 4η. For even values of N, half the levels are bonding and half anti-bonding.

In the presence of electrons, each state may be occupied by two electrons (one spin up, and one spin down) starting at the lowest unoccupied state, and ranging up to the Fermi energy. The strength of the polymer chain is essentially the same as that of the dimer. It is determined by the magnitude of the hopping integralη which is, in turn, determined by the bond length b.

The case we have considered which allowed only nearest-neighbor hopping leads to one energy band. If hopping between next-nearest neighbors is allowed in addition to the nearest-neighbor hopping, then an additional band of energies appears. The second may, or may not, overlap the first one. If not, there will be an energy gap in which there are no allowed energy levels.

If the chain of N atoms that we have considered is not a closed ring, but is a straight line of finite length, then the atoms near the two ends will have different environments from those in the middle of the chain. This will create energy levels that lie outside of the main band of energy levels. These are known as Tamm, or surface, states. They will be considered when fracture is discussed in Chapter 21.

Figure 10.5 Comparison of experimental and theoretical momentum spectra for SiC. Left, scattered intensity showing absence of intensity near a binding energy of 10 eV and gap of about 3 eV. Right, theoretical line with experimental points superimposed. After Vos and McCarthy (1995).

The existence of the bands of energy levels separated by gaps for periodic arrays of atoms has been confirmed experimentally by means of electron scattering (Figure 10.5). In the case of a chain that alternates between two kinds of atoms (such as Si and C in silicon carbide) two bands can be detected with an energy gap between them (Vos and McCarthy, 1995).

References

Atkins, P.W. (1983). Molecular Quantum Mechanics, 2nd edn. Oxford: Oxford University Press.

Burdett, J.K. (1995). Chemical Bonding in Solids. Oxford: Oxford University Press.

Chen, C.J. (1991). Attractive interatomic force as a tunneling phenomenon, J. Phys.: Condens.

Matter, 3, 1227.

Coppens, P. (1997). X-Ray Charge Densities and Chemical Bonding. New York: Oxford University Press.

Ferrante, J., Smith, J.R., and Rose, J.H. (1983). Diatomic molecules and metallic adhesion, cohesion, and chemisorption: a single binding-energy relation, Phys. Rev. Lett., 50 (18), 1385.

Feynman, R.P., Leighton, R.B., and Sands, M. (1963). Feynman Lectures on Physics, Volume III.

Reading, MA: Addison-Wesley.

Keyes, R.W. (1958). An antibonding Morse potential, Nature, 182, 1071.

Kimball, G.E. (1950). Lecture Notes. New York: Columbia University.

Morse, P.M. (1929). Phys. Rev., 34, 57.

Parr, R.G. and Yang, W. (1989). Density-Functional Theory of Atoms and Molecules. Oxford:

Clarendon Press.

Pauling, L. (1931). J. Am. Chem. Soc., 53, 1367.

Pauling, L. and Wilson, E.B. (1983). Introduction to Quantum Mechanics. New York: McGraw-Hill, 1935. Republished by Dover Publications, New York.

Pettifor, D.G. (1995). Bonding and Structure of Molecules and Solids. Oxford: Clarendon Press.

Slater, J.C. (1963). Quantum Theory of Molecules and Atoms, Volume 1, Electronic Structure of Molecules. New York: McGraw-Hill.

Slater, J.C. (1965). Quantum Theory of Molecules and Atoms, Volume 2, Symmetry and Energy Bands in Crystals. New York: McGraw-Hill.

Sutton, A.P. (1993). Electronic Structure of Materials. Oxford: Clarendon Press.

Vos, M. and McCarthy, I.E. (1995). Observing electron motion in solids, Rev. Mod. Phys., 67, 713.

11