ABSTRACT
Multiplying to the left of each term by ( )Rr,∗Ψ and integrating over all the electron positions, we obtain the equation for ( )Rϕ as
( ) ( ) ( )2 2 2 pp p
R E R R M
ϕ ϕ− ∇ +∑ = (38.6)
( ) ( ) ( )2 22( , ) ( , ) ( , ) ( , ) 2 p p pp pp p
R E R r R r R dr R r R r R dr
M M ϕϕ ϕ∗ ∗= + Ψ ∇ Ψ + ∇ ⋅ Ψ ∇ Ψ∑ ∑∫ ∫= =
The last two terms on the right-hand side vanish if the electrons can be considered to be perfectly free, because ( )Rr,Ψ becomes independent of the ion coordinates R, so that is zero. Such an approximation can be used for metallic systems. Otherwise, the eigenfunctions
( Rrp ,Ψ∇ ) ( )Rr,Ψ are assumed to be slowly varying
functions of the ionic coordinates R and the second and third terms on the right-hand side of Eq.(38.6) can be neglected to a first approximation. Under this assumption Eq.(38.6) reduces to the equation for ionic motion
( ) ( ) ( ) ( )2 2 2 pp p
R E R R E R M
ϕ ϕ− ∇ + =∑ = ϕ (38.7)
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 electrons, described by Eq.(38.3), or to neglect the electronic interactions when solving the eigenvalue problem (38.7) 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 (38.4) are obtained by first solving the electronic problem for fixed ionic positions and then using the total electronic energy
as a potential function for the ionic motion. The potential energy exhibits a set of minima, corresponding to the equilibrium positions of the ions. For R close to
we may then expand as
( )RE ( )RE 0R
0R
( ) ( ) ( ) ( ) ( ) 0
E 0E R E R R R E R R RR
κ =
⎛ ⎞∂= + − + ≅ + −⎜ ⎟∂⎝ ⎠ …
where the first order derivatives are zero at equilibrium and 0κ stands for the coupling constant, having the dimensions of an elastic constant. Therefore the effect of the electrons appears, to a first approximation, to be an elastic coupling of the ions, with force constants which are dependent on the electronic states.
