Z L 0 φ∗n,s(x, [u]) ∂2 ∂u2φn,r(x, [u]) (4) A

(1)

2

in the same way we can define B¯_s,rⁿ [u] =

Z L 0

φ^∗_n,s(x, [u]) ∂²

∂u²φn,r(x, [u]) (4)

A. Born-Oppenheimer scheme

H = H^el(r, u) + P²

2M, (5)

H^el(r, u) = p²

2m + V^el−ion(r, u) + V^ion(u), (6)

where r is electronic coordinate, m the electron mass and u and M are respectively the ion displacement from the lattice equilibrium and the reduced mass associated to a particular vibrational mode.

We formally indicate the solution of the total Hamiltonian as

H|Ψ^µi = E^µ|Ψ^µi (7)

In the spirit of the “Born-Oppenheimer” scheme we can solve the purely electronic quan- tum problem

H^el|ϕⁿ[u]i = ǫ^eln[u]|ϕⁿ[u]i (8) where we indicate with [u] a dependence from an operator. {|ϕⁿ[u]i} represents an orthonor- mal basis with fixed u.

Now we solve the adiabatic problem

"

P²

2M + ǫ^el_n[u]

#

|φ^α,ni = E^α,n|φ^α,ni (9)

obtaining the eigenstates |φ^α,ni

X

α |φ^α,nihφ^α,n| = 1. (10)

We can define the B-O basis as {|ϕⁿ[u]i|φ^α,ni}; it is complete and hortonormal

hφ^β,m|hϕ^m[u]|ϕⁿ[u]i|φ^α,ni = δ^m,nδα,β. (11) In this basis

|Ψ^µi =^X

α,n

γ_α,n^µ |ϕⁿ[u]i|φ^α,ni (12)

(2)

3

H|Ψ^µi =^X

α,n

γ_α,n^µ

"

P²

2M + ǫ^el_n[u]

#

|ϕⁿ[u]i|φ^α,ni (13)

hφ^β,m|hϕ^m[u]|H|Ψ^µi =^X

α,n

γ_α,n^µ hφ^β,m|hϕ^m[u]|

"

P²

2M + ǫ^el_n[u]

#

|ϕⁿ[u]i|φ^α,ni = E^µγ_β,m^µ (14) remember that

[f [u], P ] = i ∂

∂uf[u] (15)

and

[f [u], P²] = i

"

P ∂

∂uf[u] + ∂

∂uf[u]P

#

(16) so that

hϕ^m[u]|P²

2M|ϕⁿ[u]i = hϕ^m[u]|ϕⁿ[u]iP² 2M

− i

2M

"

hϕ^m[u]|P ∂

∂u|ϕⁿ[u]i + hϕ^m[u]| ∂

∂u|ϕⁿ[u]iP

#

= δn,m

P² 2M − i

2M

"

2hϕ^m[u]| ∂

∂u|ϕⁿ[u]iP − ihϕ^m[u]| ∂²

∂u²|ϕⁿ[u]i

#

(17) Coming back to Eq.(14)

(Eβ,m− E^µ) γ_β,m^µ = i 2M

X

α,n

γ^µ_α,n

"

2hφ^β,m|hϕ^m[u]| ∂

∂u|ϕⁿ[u]iP |φ^α,ni+

− ihφ^β,m|hϕ^m[u]| ∂²

∂u²|ϕⁿ[u]i|φ^α,ni

#

=

= ^X

α,n

γ_α,n^µ [aβ,m,α,n+ bβ,m,α,n] (18)

with

A_m,n(u) = 1

√M

Z

dxϕ^∗_m(x; u) ∂

∂uϕ_n(x; u) (19)

Bm,n(u) = 1 2M

Z

dxϕ^∗_m(x; u) ∂²

∂u²ϕn(x; u) (20)

aβ,m,α,n = 1

√M

Z

duAm,n(u)φ^∗_β,m(u) ∂

∂uφα,n(u) (21)

bβ,m,α,n =

Z

duBm,n(u)φ^∗_β,m(u)φα,n(u) (22) The Born-Oppenheimer approximation consists in neglecting Am,n(u) and Bm,n(u). In this limit the eigenstates are

|Ψ^α,ni ≃ |ϕⁿ[u]i|φ^α,ni (23)

having splitted the µ index into α, n