A Monoatomic Linear Chain

All atoms are equivalent by translation a. Although this is only a simplified ideal situation, it is used as a first example in many introductory textbooks on physical chemistry and solid state physics, because it is a problem simple enough to be treated analytically, especially if an easy approximation such as Hiickel s model Flamiltonian is applied. Several important simplifications in Hiickel s model make the calculation very easy, while preserving the main topological characteristics of the system. In this simple model, only one pz AO is considered for each atom. The different orbitals will be identified by the g lattice vector of the cell in which they are centered and denoted as p. Hiickel s approximation prescribes simple rules for the determination of the overlap and the Hamiltonian matrices with two parameters, a and p  

Before evaluating H(k) and S(k) following Hiickel s prescriptions, a basis set of Bloch functions must be defined in the local basis of the p AOs in the polymer 

Because the basis set is minimal (only one Bloch function per k point), this Bloch function is itself an eigenfunction of the Hamiltonian. With this basis set, S takes the form  

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Figure 5 Schematic representation of a Bloch function for a monoatomic linear chain. A black circle represents a AO.

Figure 6 Hiickel n-band of a monoatomic linear chain in the first Brillouin zone.

Fig.4.17. Dispersion curves for a monoatomic linear chain with adiabatically moving bond charges, calculated according (4.125)...

Consider a monoatomic linear chain of atoms with mass m and nearest-neighbour interactions of the Lennard-Jones type with (p(r) given by (4.2). Prove that the zero-point energy... 

A perturbation treatment of anharmonic systems was first performed by DEBYE in 1914 [5.3]. In 1925, PAULI calculated the damping constant r for infrared absorption by a monoatomic linear chain of the form (5.15) but composed of atoms with alternating positive and negative charges. He obtained the result [5.4]... 

In Fig. 21.1 this condition is visualized as the intersection of the straight line cq with the dispersion curve (Om q) = 2cm m ql2). Going back to the original units, cOm can be identified as the dispersion of a monoatomic chain of masses M = + M2 with a linear in-... 

Fig.5.2. a) Linear monoatomic chain at T = 0 and K = 0 with N lattice spacing ao- b) Chain at temperature T under the influence of an external force K with lattice spacing a = a(T,K)... 

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