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Suppose we have a one dimensional system of lattice constant
, where we impose
PBC over
cells: there are
equally spaced Bloch vectors in the
reciprocal cell
 |
(138) |
The size of the periodically repeated system is
.
The orbitals can be chosen to have the Bloch form
 |
(139) |
where
is a lattice translation, and m is a band index. There are
occupied bands in the Slater determinant wave function, which we write as
 |
(140) |
where A is the antisymmetrizer. A new set of Bloch orbitals can be defined
 |
(141) |
The position expectation value can now be written as
where
is the Slater determinant of the
's.
The overlap among two determinants is equal to the determinant of the overlap matrix of the orbitals:
where
 |
(144) |
Because of the orthogonality of the Bloch functions, the overlap matrix elements vanish
except for
, that is
.
The
determinant can then be factorized into M small determinants
det det |
(145) |
where for the small overlap matrix the notation
 |
(146) |
and
is implicitly understood.
Finally we get for the electric polarization
The total dipole of a system is calculated as the sum of nuclear and electronic contributions
 |
(148) |
Only the total dipole will be independent of the gauge and a proper choice of
reference point will be discussed later.
First we will give the formulas for the electronic contribution in three dimension and
for the case where only the
point of the supercell Brillouin zone is used
The additional factor of 2 comes from the assumed spin degeneracy.
The matrix
is defined as
 |
(150) |
The index
labels the reciprocal-lattice basis vectors
,
is the projection of the electronic dipole moment along the direction
defined by
, and
are the
point Kohn-Sham orbitals.
The IR absorption coefficient
can be calculated from the formula [82]
 |
(151) |
where V is the volume of the supercell, T the temperature,
,
is the refractive index, c is the speed of light in vacuum. The angular brackets indicate
a statistical average. The correlation function
is calculated
classically and quantum effect corrections are taken into account through the factor
tanh
. More sophisticated treatments of quantum effects
are also available. They consist of replacing the classical correlation function with
a more involved procedure [83].
As mentioned above only the total dipole moment is independent of the reference point.
This can cause some problems during a molecular dynamics simulation. The electronic
contribution can only be calculated modulo the supercell size and a unfortunate
choice of reference might lead to frequent changes of the dipole moment by amounts
of the cell size. Therefore it is most convenient to use a dynamic reference
calculated from the nuclear positions and charges. If the reference point is
chosen to be the center of charge of the nuclei,
the nuclear contribution to the dipole will always be zero
 |
(152) |
are the nuclear charges and
the nuclear positions within the supercell.
The electronic contribution is then
where
is the matrix defining the supercell.
Next: Localized Orbitals, Wannier Functions
Up: Calculating the Electronic Structure
Previous: Position Operator in Periodic
Contents
Index
Costas Bekas
2008-09-04