ABSTRACT
A solid consists of a sufficiently large collection of atoms so that they exhibit the characteristic behaviour of the bulk material. In a solid the atom may lose some of its tightly bound electrons, leaving the core electrons. Such an entity is known as an ion. Hence, we will consider a system of ions with position vectors R , masses Af , and
charges Z pe and valence electrons ,me9 — e, described by the Hamiltonian operator:
h = - — Y v 2 - y -*i-v2+Iy y f t 2me f J f 2 M V p t j rjk
Z nel 1 „ Z nZ,e l-II I i p ^ p q*p ^pq
I * ) - 1 — V 2p + U « ( f ) + Uti(r ,R) + Utt(R) (1.1) 2m, j p 2 M p
The first two terms describe the kinetic energy, where p is a running index over all the ions, the index j extends over all the electrons, whereas Uee(r ), Uei( r ,R ) and UU(R) are the potential energies of the electron-electron, electron-ion and ion-ion interaction respectively, expressed in terms of e0 = e/^47t€0, while e is given in SI units. In the
Schrodinger equation for the system of atoms, the motion of the valence electrons may be separated from that of the ions by introducing in E q.(l.l) the Hamiltonian He which describes the electron motion for a set of fixed ionic positions, denoted collectively by R , in the form:
H<b(r,R) = \ H e - Y ^ - V 2 W ( r ,R ) = E ^ ( r , R ) (1.2) I p 2 M r J
The eigenvalue problem for the electronic motion reads:
where ¥ (? ,/? ) and E{R) are the eigenfunctions and eigenvalues of He with R as a fixed parameter. The electron coordinates, denoted collectively by r , are the dynamical variables in Eq.(1.3). Assuming that the eigenfunctions T ^r,/?) are known, the wave
functions 0 (r ,/? ) which solve the Schrodinger equation (1.2) are usually approximated by the following expression:
O (r ,/0 = ¥(r,£)<p(£) (1-4)
which is known as the adiabatic approximation. Substituting Eq.(1.4), Eq.(1.2) becomes:
- 1 + 2V, ^ F’ V X p(p(R) + 'F (r, R )V 2p<p(R)}
+ E(R) R) cp(R) = £'xF(r, R) cp(R) (1.5)
Then multiplying to the left of each term by ¥*(? ,/?) and integrating over all electron
positions, we obtain an equation for cp(R):
where the electron eigenfunctions are assumed to be normalized. If the valence electrons can be considered perfectly free, the last two terms on the right-hand side vanish because T^r,/?) becomes independent of the ion coordinates R , so that V H^r,/?) is zero. Such an approximation is appropriate for metallic systems. For non-metals, it can be shown that the eigenfunctions T^r,/?) are slowly varying functions of the ionic coordinates. It is therefore always possible, to a first approximation, to reduce Eq.(1.6), for the ionic motion, to the form:
- ' Z ^ - ^ l < p ( R ) + E(R)(p(R) = E(p(R) (1.7) p 2 M p
As a result of the adiabatic approximation to the many-body eigenvalue problem of a system of atoms, it is possible either to suppress the ionic interactions while considering the motion of the valence electrons, described by Eq.(1.3) or to neglect the electronic interactions when solving the eigenvalue problem (1.7) for the motion of the ions. In other words, ionic motion can be ignored when describing the binding of the electrons to the nucleus, whereas the chemical bond may be regarded as a parameter when considering the ionic motion. The state functions introduced by Eq.(1.4) are obtained by first solving the electronic problem for fixed ionic positions and then using the total electronic energy E{R) as a potential function for the ionic motion. The potential
energy E(R ) exhibits a minimum, corresponding to the equilibrium positions R0 of the ions, which move in potential wells as illustrated in Figure 1.1, where the horizontal line at E corresponds to the constant energy of a hypothetical bound state of the solid. Therefore the effect of the electrons appears as a coupling interaction of the ions, with forces which are dependent on the electronic states. Any change in the electronic states
modifies the potential energy E(R) felt by the ions, and so modifies their coupling.
