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\label{sec:sternheimer}
%David, Xavier
In textbooks, perturbation theory is formulated in terms of
sum sums over
states and response functions. These are useful theoretical
constructions that permit a good description and understanding of the
underlying physics. However, this is not always a good description for
numerical applications, since it involves the calculation of a large
number of eigenvectors, infinite sums over these eigenvectors, and functions that depend on two or more spatial variables.
An interesting
approach, approach that avoids the problems
mentioned
above, above is the formulation of perturbation theory
in terms of differential equations for the variation of the
wave-functions, This wave-functions. In the literature, this is usually
named in called
the
literature as Sternheimer equation~\cite{Sternheimer_1951} or density functional perturbation theory (DFPT)~\cite{Baroni_2001}. Although a
perturbative technique, it avoids the use of empty states, and has a
favorable scaling with the number of atoms.
...
resonance~\cite{Andrade_2007}. The method is suited for linear and
non-linear response; higher-order Sternheimer equations can be
obtained for higher-order variations. For second-order response,
however, we apply the
\(2N\,+\,1\) \(2n\,+\,1\) theorem (also known as Wigner's
\(2N\,+\,1\) \(2n\,+\,1\) rule)~\cite{Gonze_1989,Corso_1996} to get the
coefficients directly from first-order response variations.
In the Sternheimer
formalism formalism, we consider the response to
monochromatic perturbative field \(\lambda
\delta{\hat v}(\vec{r})\cos\left(\omega{t}\right)\). This perturbation induces a variation in the time-dependent Kohn-Sham (KS) orbitals,
that which we denote $\delta\varphi_{n}(\vec r, \omega)$. These variations allow us to calculate response observables, for example, the frequency-dependent
polarization. polarizability.
In order to calculate the variations of the orbitals we need to solve a linear equation that only depends on the occupied orbitals (atomic units are used throughout)
%\begin{multline}
...
\,,
\end{equation}
%
were where the variation of the time-dependent density, given by
%
%\begin{multline}
\begin{equation}
\label{eq:varrho}
\delta{n}(\vec r, \omega) =
\sum_k^{\rm occ.} \sum_k f_k \Big\{
\left[\varphi_n(\vec{r})\right]^*\delta\varphi_{n}(\vec r, \omega)\\
+ \left[\delta\varphi_{n}(\vec r, -\omega)\right]^*\varphi_n(\vec{r})
\Big\}\ ,
\end{equation}
%\end{multline}
needs to be calculated self-consistently. The
first order first-order variation of the KS Hamiltonian is
\begin{multline}
\label{eq:h1}
\delta{\hat H}(\omega)=
...
%
\(\mathrm{\hat P}_c\) is a projector operator, and
\(\eta\) a positive in\-fi\-ni\-te\-si\-mal, essential to obtain the
correct position of the poles of the causal response function,
and, and
consequently
to obtain the imaginary part of the
po\-la\-ri\-za\-bi\-li\-ty and remove the divergences of
the equation for resonant frequencies. In the usual implementation of
DFTP, DFPT, $\hat P_c = 1 - \sum^{\rm occ}_n \left| \varphi_n \right> \left< \varphi_n \right|$ which effectively removes the components of \(\delta\varphi_{n}(\vec r, \pm\omega)\) in the
subspace of the occupied ground-state wave-functions. In linear
response, these components do not contribute to the variation of the
density.
...
for large systems, it saves time overall since it increases the condition number of the matrix for the linear solver,
and thus reduces the number of solver iterations required to attain a given precision.
% chloroform nearly degenerate perturbation theory? smearing minimal projector = delta function?
We also have implemented the Sternheimer formalism when non-integer occupations are used, as appropriate for metallic systems. In this case weighted projectors are added to both sides of the eq.~(\ref{eq:sternheimer}).
%cite De Geroncoli.
Apart from semiconducting smearing (\textit{i.e.} the original equation above, which corresponds to the zero temperature limit),
