Developers Manual:Linear Response
- 1 Static Response
- 2 Dynamic Response
We will consider a Kohn-Sham equation
and a perturbing potential of the form
Then we can expand the Hamiltonian, the wavefunctions and the eigenvalues in terms of
Putting all the expansions until second order in equation (2) we get
we can separate terms according to the dependence on lambda; for the zeroth order we get the ground-state equation.
For the first order we get the Sternheimer equation
The RHS can be written as
where is a projector onto the unoccupied space. We need only the components in the unoccupied space for many applications.
Doing the same for the second-order perturbation we get
Relation to the derivatives
Now we will find the relations between perturbations as defined here and the derivatives of the magnitudes with respect to .
We can see from the second-order equation it's not symmetrical with respect to the index exchange, so . To relate the perturbation terms to the derivatives (which are symmetrical) we must take this in account, taking the average over all index permutations, as well as the factor (which cancels the average factor), so:
An alternative way to do this is to symmetrize the equations, and then solve only for one component.
For the case of a uniform static electric field
The Electron Localization Function
The normalization is
Putting our expansion to first order:
Identifying the form of the ELF in the zeroth-order terms, we get the first-order pertubation ELF
And the normalization:
There is a contribution from the change in :
and with this
We will consider a time-independent ground state system described by a Kohn-Sham equation
and we will apply a perturbative potential which is periodic in time with frequency (and period ).
In particular we will consider
where and are some constant coefficients.
As the perturbed system is time-dependent, we must use the TDDFT Kohn-Sham equations
Due to the form of the potential, we can take the following first-order expansion of the wavefunctions
for the density
and for the hamiltonian (including the perturbative potential)
where the operator is the variation of the hamiltonian with respect to the density, and typically includes the Exchange-Correlation and Hartree terms.
From the expansion for the wavefunctions we can calculate the variation of the density
This gives us
But we can see that
Using this, we can simplify equation (2):
Dynamic current density
Equations for the perturbations
If we take the expansions for the wavefunctions and the hamiltonian and we put them into the TDDFT equation we obtain the equation that the variations obey:
This time there is no projector, so the perturbation has a component onto the occupied space.
As the operator in this equation is the same ground state hamiltonian, we can write the solution in terms of the eigenfunctions (occupied and unoccupied) of the ground state
altough to do this numerically we would have to calculate the unoccupied orbitals.
One of the conditions we have to ask is that the perturbed wavefunctions should be normalized; this is satisfied if
using the exact solution for we know that
Using this, the normalization condition becomes
So which implies that the external potential must be real.
Component of the perturbation on the occupied space
The perturbation can be separated in two perpendicular components
The part in the occupied space is
We will now calculate the contribution to the density perturbation of this part, we have that
using the explicit expresion
and swapping the indexes in the second term
This means that at least for the first order, the part in the occupied space doesn't contribute to the density and we can put a projector in the RHS of the Steinheimer equation. This is not necessarily true if we want to calculate other things that require the perturbations of the orbitals and not the perturbations of the density.
So finally the equations we have to solve are
We will now calculate the polarizability in terms of the perturbation of the density, first we take the dipole moment and we replace the expresion for the density
the second term can be identified as the time dependent polarizability
Now, we take the fourier transform, because this is the frequency-dependent polarizability we are interested in
The time integral can be done explicity giving
and finally this can be written as