Computational Physics

166 Solving the Schrödinger equation in periodic solids

(c) Show that

Sextml=







a− ifm=l

− 2

qm−ql

sin

(qm−ql)

2

otherwise

;

and

Sintml=CmCn+DmDn+(CmDn+CnDm)

κ

κ

.

(d) Show that the Hamiltonian matrix is given as

Hml=Hmnint+

1

2

qmqnSextmn+∂Hmn

where

Hmlint=−κsin(κ)(CmDn+CnDm)+κ^2 (CmCn+DmDn),

and where∂His the matrix due to the jumps in derivatives across the barrier

boundaries. Show that∂His given by

∂Hmn=−qnsin[(qn−qm)/ 2 ]

−Cmκsin[(κ−qn)/ 2 ]−Dmκsin[(κ+qn)/ 2 ].

(e) Write a program in which the matricesHmlandSmlare filled and find the zeroes of

the determinant|H−ES|for variousk. Compare the results with the numerically

exact ones, resulting from the previous program.

6.3 In this problem we consider the determination of the local charge density using the
Green’s function. The Green’s function for a Hamiltonian’sHis defined as
(H−E)G(r,r′;E)=δ(r,r′).
(a) Show thatGcan be written as

G(r,r′;z)=

∑∞

n= 1

ψn(r)

1

z−En

ψn(r′).

(b) Show that the electron density (charge density) can be found as

n(r)=^1

2 πi

∫



G(r,r;z)dz,

whereis a closed contour in the complex plane which contains all theoccupied

energy levels (these of course all lie on the real axis).

6.4 [C] As plane waves form an orthogonal basis, it is possible to use the Lowdin
perturbation method discussed inSection 3.4. Write an extension to your
pseudopotential program to incorporate large lattice vectors into the Hamiltonian in a
perturbative manner. Compare the results with those of the direct diagonalisation.
6.5 In this problem we derive the normalisation condition(6.50)from the normalisation
(6.47)of the radial solution inside the muffin tin.