QP_energies_and_lifetimes.tex

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\begin{document}

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\title{Quasiparticle energies and lifetimes in a metallic chain model
  of a tunnel junction} %Title of paper

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\author{Mark Szepieniec}
\email[]{mark.szepieniec@tyndall.ie}

\author{Irene Yeriskin}

\author{J C Greer}
\affiliation{Tyndall National Institute, University College Cork,
  Dyke Parade, Lee Maltings, Cork, Ireland}

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\date{\today}

\begin{abstract}
As electronics devices scale to sub--10 nanometer lengths, the distinction
between `device' and `electrodes' becomes blurred.
Here, we study a simple model of a molecular tunnel junction, consisting
of an atomic gold chain partitioned into left and right electrodes, and a
central `molecule'.
Using a complex absorbing potential, we are able to reproduce the
single-particle energy levels of the device region including a description
of the effects of the ``semi-infinite'' electrodes.
We then use the method of configuration interaction to explore the effect
of correlations on the system's quasiparticle peaks.
We find that when excitations on the leads are not included, the device's
\ac{HOMO} and \ac{LUMO} quasiparticle peaks when including correlation
are bracketed by their respective values in the Hartree-Fock (Koopmans)
and \dscf approximations.
In contrast, when excitations on the leads are included, the bracketing
property no longer holds and both the positions and the lifetimes of the
quasiparticle levels change considerably, indicating that the combined
effect of coupling and correlation is to alter the quasiparticle spectrum
significantly from that of an isolated molecule.
\end{abstract}

%\pacs{}% insert suggested PACS numbers in braces on next line
\pacs{73.63.Nm, 85.65.+h, 31.15.A-}

\keywords{molecular electronics, configuration interaction,
  complex absorbing potentials}

\maketitle %\maketitle must follow title, authors, abstract and \pacs

% acronyms
\acrodef{MCCI}{Monte Carlo configuration interaction}
\acrodef{CAP}{complex absorbing potential}
\acrodef{CSF}{configuration state function}
\acrodef{CI}{configuration interaction}
\acrodef{DFT}{density functional theory}
\acrodef{NEGF}{nonequilibrium Green's function}
\acrodef{HOMO}{highest occupied molecular orbital}
\acrodef{LUMO}{lowest unoccupied molecular orbital}
\acrodef{IP}{ionization potential}
\acrodef{EA}{electron affinity}
\acrodef{LDA}{local density approximation}
\acrodef{QP}{quasiparticle}
\acrodef{SSP}{source-sink potential}

% Body of paper goes here. Use proper sectioning commands. 

\section{Introduction}

Atomic chains are a prototype for the study of
molecular electronic systems. Atomic nanowires can be fabricated on
inert surfaces \cite{segovia1999nature,nilius2002science} or in
mechanically controlled break junctions \cite{vanruitenbeek1998mcbj}.
Atomic nanowires are of fundamental scientific interest as they form
electronic one-dimensional systems and for technology they may be viewed
as the limit for metal interconnects in use for nanoelectronics.  Atomic
nanowires are also useful for benchmarking one-dimensional quantum
transport formulations as experiments for these systems provide well
defined and understood conductance values.

Some of the earliest theoretical work with explicit treatment of the
electronic structure within a quantum transport study of atomic chains is
due to Lang~\cite{Lang1995prb}, serving as the model for subsequent
studies in molecular tunnel junctions. In this early work, jellium
electrodes are coupled to metal atom chains and the electronic structure
of the system is treated using
\ac{DFT}~\cite{hohenberg_kohn, kohn_sham, kohn_dftreview}.
The Kohn-Sham states in the \ac{LDA} are treated as \acp{QP} and the
electron scattering equations are solved with an external voltage bias
applied to the electrodes. Similar scattering approaches in conjunction
with tight binding Hamiltonians applied to the conductance of molecular
junctions combined with Landauer-B\"uttiker theory for electron transport
\cite{emberlykirczenow1999standingwave,
emberlykirczenow2000molecularwire} have been undertaken. 
Subsequently, similar methods combining a 
\ac{DFT} treatment of the electronic structure with a formal \ac{NEGF}
treatment of transport in open systems have become commonly
used~\cite{damle_ghosh_datta}; more
recently, the {\it GW} approximation~\cite{hedin1965gw} has been used to
improve  the quasiparticle description of atomic scale tunnel junctions
beyond a Kohn-Sham description~\cite{thygesen_rubio,neaton2007amines}.
Use of the Kohn-Sham energies in the NEGF formalism implies a single
determinant or \ac{QP} approximation to the many-electron Green's function,
and similarly the {\it GW} approximation may be viewed as a dynamically
screened version of the Hartree-Fock theory, likewise implying a
\ac{QP} description. Alternatively to these methods, one can use the
ideas of exact diagonalization or \ac{CI} in rate equation formulations
of electron transport~\cite{pedersen_many_body_tunneling}, or through a
many-electron scattering formulation of the transport
problem~\cite{vici2004,greer2011}. In a different approach, the \ac{SSP}
method was developed~\cite{SSP-Ernzerhof,SSP-extension-Ernzerhof}, which
uses carefully tuned complex potential terms in the Hamiltonian to
emulate electrons flowing into and out of the device. This method has
been applied to molecular junctions with a complex variation of
\ac{DFT}~\cite{ComplexDFTErnzerhof}, and to simplified discrete model
systems using a Hubbard Hamiltonian and \ac{CI}~\cite{SSPHubbard-Ernzerhof}.

In the following, we apply the method of \ac{CI} to a simple model of
electrodes coupled to an atomic chain to investigate the influence of
electron correlations on the \ac{QP} states of the device
region. We explicitly consider the influence of electrode excitations as
they couple to the device region, since these were previously shown to have
a significant effect on conductance
characteristics~\cite{galperin_nitzan2006leadexcitations}.
Opening of the system by the extended
electrodes is modeled by a \ac{CAP} which has been constructed specifically
to yield the same complex energy eigenvalues as a conventional electrode
self-energy produces~\cite{henderson}. This approach enables us to study
the junction in an uncorrelated limit (single determinant approximation), and
to systematically include electron correlations on the device region
to study how the \ac{QP} states evolve. 
We find that including lead excitations causes significant changes in the
shifts and broadenings of \ac{QP} levels, indicating that it is important
to capture these effects for an accurate treatment of electron transport
in atomic chain systems.

\section{Methods}
\label{sec:method}

\subsection{Model system}
\label{subsec:modelsystem}

The extended device we study is built from a simple atomic chain
consisting of 68 gold atoms with an interatomic separation of 0.28 nm
approximating the geometry of gold chains studied experimentally in
ref.~(\onlinecite{nilius2002science}) (see~\fref{fig:chaincapdevice}).
Within the explicit chain region, two interatomic spacings are chosen
larger than all other atomic spacings to define a `molecule' between two
leads. The explicit left and right leads are each composed of 24 atoms,
and the central molecular region contains 20 atoms. The width of the gap
between the leads and molecule determines the strength of the
molecule-electrode coupling; varying the spacing allows different
coupling strengths to be investigated. 

The gold atoms are described by a single 6$s$ electron and a 68-electron
effective core potential~\cite{CRENBS} yielding a single half filled
one-dimensional band. The Fock and overlap matrices in the Hartree-Fock
approximation are calculated  using the TURBOMOLE program
package~\cite{turbomole}, and these matrices are used to calculate
electron transmission in the absence of electron correlations with the
TiMeS  program~\cite{times}.
The one- and two-electron integrals of the Hartree-Fock molecular orbitals
are generated using a modified version of TURBOMOLE and then used to
construct the Hamiltonian matrix elements for the CI calculations.

\begin{figure}
	\begin{center}
		\includegraphics[width=0.9\linewidth]{figure1}
	\end{center}
	\caption{Extended device region studied in this work: The molecule
	consists of 20 Gold atoms, while the explicit electrode regions
	each consist of 24 atoms. The spacing between molecules and electrodes
	determines the coupling of the molecular states to the electrodes.}
	\label{fig:chaincapdevice}
\end{figure}

\subsection{Complex absorbing potential}
\label{subsec:CAP}

In defining models for atomic-scale transport junctions, a partitioning
of the system into semi-infinite electrodes and device region is typically
performed. The device region is given by an explicit model that includes
the device (i.e. atomic chain, molecule, nanowire) as well as a region
that defines the bonding of the device to electrodes, resulting in an
`extended device' or `extended molecule'. The portion of the
electrode not explicitly included in the extended device region in
most theoretical studies is then treated by electrode self-energies. In this
approximation, electronic excitations in the electrode region are not
considered. Exceptions include ref.~\cite{galperin_nitzan2006leadexcitations}
where the effect of coupling of lead excitations to a two state model of
a molecular tunnel junction is explored, and a recent formulation
of \ac{NEGF} that allows for electrode interactions to be explicitly
treated~\cite{ness_dash_prb}.
With our approach, excitations from the portions of the electrodes that
are included in the explicit device region can be included.

In this study, the regions outside of the explicit atomic region are
described by a \ac{CAP} in analogy to the use of electrode self-energies.
The details of the \ai \ac{CAP} used are described below and its use has
advantages when including electron correlations into the calculations.
As our 0$^{th}$ order approximation, we use the Hartree-Fock
orbitals to calculate the transmission spectrum for the model system using
a Green's function approach  as implemented in the TiMeS program~\cite{times};
the electrodes in these (uncorrelated) scattering calculations are
described using standard electrode  self-energies. This scattering
calculation delivers the Hartree-Fock \ac{QP} spectrum including
the energy shifts and state broadenings that arise from coupling the molecular
region to electrodes, and serves as the reference point for discussing
subsequent calculations where we introduce correlations.

\subsection{Electronic structure}
To treat electron correlations, we apply the method of \ac{CI}, which is
the application of the  Rayleigh-Ritz linear variational
principle to the calculation of quantum many-electron  energies. Within
this approach, the many-electron wavefunction is expanded in terms  of
Slater determinants (or spin-coupled sums of determinants referred to as
\acp{CSF}).  Variation of the \ac{CI} expansion coefficients leads to a
matrix eigenvalue problem
\begin{equation}
\um{H} \vec{c} = E \um{S} \vec{c},
\end{equation} 
where $\um{H}$ is the matrix representation of the many-electron
Hamiltonian in the CSF basis, $\um{S}$ is the metric for the CSF basis,
$E$ is a many-electron energy and $\vec{c}$ is the vector of the
expansion coefficients or CI vector. The matrix elements of $\um{H}$
and $\um{S}$ are constructed from one- and two-electron integrals
defined in the single particle basis used to construct the CSFs, and for
example can be Hartree-Fock molecular orbitals. However, in the CI treatment
of the many-electron problem, no reference is made to single particle or
\ac{QP} energies. Hence the inclusion of the electrodes into the
CI problem using electrode self-energies is complicated by the fact that
the self-energies are functions of the \ac{QP} energies on the
electrodes. We employ instead energy-independent
\acp{CAP} that serve as replacement to the self-energy terms, eliminating
reference to single particle energies into the resulting \ac{CI}
problem~\cite{riss_meyer,santracederbaum,muga2004,henderson,varga2008}.
Note however, that by opening the system using either electrode self-energies
or \acp{CAP},  the Hermitian \ac{CI} generalized eigenvalue
problem becomes a complex symmetric generalized eigenvalue problem. The
procedure we follow for construction of an energy-independent \ai \ac{CAP} 
from the energy-dependant self-energy has been previously
presented~\cite{henderson}; we briefly cover the essential points
in the next section.

\subsection{Construction of the \ac{CAP}}
We first consider a system with bare \ac{QP} Hamiltonian $\um{h}_0$ in the
Hartree-Fock approximation, coupled to two electrodes described by known
left $\um{\Sigma}_L(\omega)$ and right $\um{\Sigma}_R(\omega)$ self-energies.
These self-energies are added to the single particle
Hamiltonian to yield
\begin{subequations}
\begin{equation}
\label{eq:capseq}
        [\um{h}_0 + \lambda \um{\Sigma_L}(\omega_i^\lambda)
            +\lambda \um{\Sigma_R}(\omega_i^\lambda)] \ket{U_i^\lambda}
        = \omega_i^\lambda \ket{U_i^\lambda}
\end{equation}
\begin{equation}
\label{eq:adiabaticap}
        \bra{V_i^\lambda} [\um{h}_0 + \lambda \um{\Sigma_L}(\omega_i^\lambda)
            +\lambda \um{\Sigma_R}(\omega_i^\lambda)]
        = \omega_i^\lambda \bra{V_i^\lambda},
\end{equation}
\end{subequations}
where the parameter $\lambda$ is introduced to allow for an adiabatic
transition between eigenstates $\ket{X_i}$ of $\um{h}_0$ at $\lambda = 0$,
and the true, finite-lifetime resonant states of the system at
$\lambda = 1$. The Hartree-Fock states for the open system are given by
the complex energies $\omega_i$, right eigenvectors $\ket{U_i}$ and left
eigenvectors $\bra{V_i}$; distinct left and right eigenvectors are a
consequence of introducing the complex symmetric self-energy matrices into
the eigenvalue problem.
The adiabatic introduction of the self-energies allows
us to `label' the molecular states and to follow their evolution as the
system is opened. This enables us to select those eigenstates (and
eigenvalues) of the coupled system that correspond to states of the
uncoupled device region. At $\lambda=1$, the real part of $\omega_i$
gives the energy of the $i$th resonance including the shift due to
electrode coupling and the imaginary part is proportional to the level
broadening.

The goal is to build an energy-independent complex potential
$\um{W} = \um{W_L} + \um{W_R}$
such that the Hamiltonian $\um{h}_0 + \um{W}$ has the same eigenvalues
as when calculated using explicit self-energies.
Such a Hamiltonian is not readily constructed since when we select the
different $\omega_i$, we are as a consequence selecting different Hamiltonians
$\um{h}_0 + \um{\Sigma_L}(\omega_i) + \um{\Sigma_R}(\omega_i)$
and the biorthogonality for the left and right eigenvectors does not
necessarily hold ($\bracket{V_i}{U_j} \ne \delta_{ij}$). We consider instead
constructions which yield eigenvalues that approximate the $\omega_i$  and
eigenvectors that approximate the left and right eigenvectors while
satisfying biorthogonality. Given such a set of approximate eigenvectors
we write
\begin{equation}
    \um{W} = \sum_i  \ket{U^\prime_i} \omega_i \bra{V^\prime_i} - \um{h}_0
    \label{eq:capsdefUV}
\end{equation}
which for biorthogonal $\ket{V^\prime_i}$ and $\bra{U^\prime_i}$ will by
construction give an operator $\um{h}_0 + \um{W}$ with the correct eigenvalues
and approximate eigenvectors. In practice we work in the (real) eigenbasis
of $\um{h}_0$ and calculate
\begin{equation}
    \um{W} = \um{S} \um{X} \boldsymbol{\omega}\um{X}^\dagger - \um{h}_0
    \label{eq:capsdefX}
\end{equation}
in matrix form, where $\um{S}$ is the overlap matrix for the atomic
orbital basis used to describe the system, $\um{X}$ is the matrix of the
eigenvectors of the Hamiltonian $\um{h}_0$, and $\boldsymbol{\omega}$ is a
diagonal matrix with the eigenvalues $\omega_i$ calculated at
$\lambda = 1$ along the diagonal.
In practice, since the self-energy is a property of the electrode only
(for properly defined electrodes and device regions), the \acp{CAP} are
generated on a single principal layer of the lead, and then transformed to the
molecular orbital basis of the extended device region, which contains a
principal layer on either side.

\subsection{Verification of the \ac{CAP}}
A comparison can be performed for this approximation of a \ac{CAP} 
against the energy level shifts and broadening
obtained from explicit calculations using the self-energy. To accomplish
this comparison, we first calculate the electron transmission through
the device region using the Green's function
approach. The calculation is then repeated but in the definition of the
Green's function ${\bf G}$ and spectral densities ${\bf \Lambda}_{L/R}$ the
self-energy terms are replaced with the \acp{CAP}, $\um{W}_{L/R}$, leading
to the following expressions ($\um{h}_D$ is the Hamiltonian of the extended
device region):
\begin{subequations}
\begin{eqnarray}
	  \um{G}^{-1} = [\boldsymbol{\omega}\um{S} - (\um{h}_D + \um{W})], \\
	  \um{\Lambda}_{L/R} = \text{i} \left( \um{W}_{L/R} - \um{W}^\dagger_{L/R}
    \right).
  \label{eq:glambdadef}
\end{eqnarray}
\end{subequations}

The transmission is then calculated as
\begin{equation}
	T = \text{Tr} \left[ \um{\Lambda}_L \um{G} \um{\Lambda}_R \um{G}^\dagger \right].
  \label{eq:transmfromG}
\end{equation}
The comparison obtained in this manner is shown in~\fref{fig:transdat}
and reveals that the \ac{CAP} provides an accurate energy-independent
representation of the self-energies. In particular, excellent agreement
is achieved on the energy range around the \ac{HOMO}-\ac{LUMO} gap
which is the focus of our subsequent calculations.

\begin{figure} 
	\begin{center}
		\includegraphics[width=0.9\linewidth]{figure2a_2b}
	\end{center}
	\caption{Transmission through the chain junctions obtained in
	         scattering calculations with a Green's functions formalism
		 and with lead-molecule separations of 0.45 nm (a) and 0.40
		 nm (b). The black curve is a transmission spectrum obtained
		 using a conventional self-energy, the red curve is
		 obtained using a \ac{CAP}.}
	\label{fig:transdat}
\end{figure}

Having verified that the \acp{CAP} are able to reproduce the electron
transmission to a good approximation compared to explicit
electrode self-energies, we next examine whether the transmission resonances
can be well approximated as Lorentzian resonances. This question is
motivated by the fact that in our subsequent many-body treatment,
\ac{QP} energies will be calculated as differences of complex
many-electron state energies. If we may approximate the transmission
resonances as Lorentzian peaks, then the complex \ac{QP} energies 
can be used to generate Lorentzian transmission functions for direct
comparison to the scattering function transmissions (there is some
indication of Fano-type resonances indicating that there is a degree of
coupling between electrodes, but these small effects do not influence
our analysis). To investigate this point, Lorentzian  resonances are
constructed using the complex eigenvalues $\omega_i$ obtained from
\eref{eq:capseq} with $\lambda=1$, 
\begin{equation}
        f(\varepsilon;\omega_i)
        = \frac{\left( \Gamma/2 \right)^2}
               {(\varepsilon - \mathrm{Re}(\omega_i))^2
               + \left( \Gamma/2 \right)^2},
        \label{eq:lobro}
\end{equation}
where $\Gamma = 2 \mathrm{Im}(\omega_i)$ (i.e., the width of the
Lorentzian peak) is inversely proportional to the state lifetime.
These Lorentzian resonances are compared to the transmission
spectrum of the device obtained using explicit self-energies and a
scattering calculation; the results of the comparison are shown in
fig.~\ref{fig:13evals} for the chain system with molecule-lead separation
of 0.45 nm (weak coupling case). The Green's function transmission (which
we will denote by `Koopmans' in the rest of the paper, since it is derived
from the complex Hartree-Fock problem) (black) and the peaks derived
from the complex eigenvalues (red) agree extremely well, both in terms
of the position of the transmission peak and resonance width. The inset
in~\fref{fig:13evals} displays the region around the \ac{HOMO} in
detail, and reveals that the transmission peaks are well described by a
Lorentzian profile and that the CAPs are able to reproduce peak positions
and widths to the accuracy required for our study.

\begin{figure} 
	\begin{center}
		\includegraphics[width=0.9\linewidth]{figure3}
	\end{center}
	\caption{Comparison of transmission data obtained using a Green
	function method	(black) with Lorentzian broadened complex eigenvalues
	from \eref{eq:capseq} (red) for a model system as described
	in~\sref{subsec:modelsystem}, with device-lead gap of 0.45 nm.
	The inset shows a close-up view of the \ac{HOMO} (the black and red
	curves overlap perfectly), with the dashed line indicating the HF
	\ac{HOMO} for reference.
	}
	\label{fig:13evals}
\end{figure}

\subsection{Complex symmetric \ac{CI} problem}

All \ac{CI} calculations  are performed using a version of the \ac{MCCI}
program~\cite{mcci1995, mcci1998} modified to treat the complex
symmetric generalized eigenvalue problem that arises when adding the
\ac{CAP} to the many-electron Coulomb Hamiltonian. The \ac{CAP} is
introduced as a one-body operator describing the extended device region 
and its interaction with the electrodes into the \ac{CI} Hamiltonian. 
\ac{MCCI} uses a modified Lanczos method for matrix diagonalization and
in the complex symmetric version of the program, a projection
method~\cite{tarantelli_csd} is introduced to solve the complex
symmetric generalized eigenvalue problem. 

\ac{MCCI} computes energy estimates by an iterative process, starting from
a set of \acp{CSF} (which initially may be a single \ac{CSF}). In each step,
additional \acp{CSF} are randomly generated by single and double excitations
relative to the \acp{CSF} in the trial vector. The \ac{CI} matrix
diagonalization problem is solved on the subspace defined by the resulting
expanded vector. This vector is then screened or `pruned' by removing
\acp{CSF} whose associated coefficient in the \ac{CI} eigenvector has a
magnitude lower than a given threshold. This pruned vector serves as
trial vector for the generation of a new set of randomly generated
\acp{CSF} in a subsequent cycle, and  this sequence is repeated until
convergence in the energy and CI vector length is reached. This scheme
makes it possible to gradually increase the amount of correlation that
the program will find, by gradually decreasing the \ac{CSF} inclusion
threshold, while keeping the computations tractable.
Details of the \ac{MCCI} method have been previously
reported.~\cite{mcci1995,mcci1998,mcci2000,multiref}.

The use of \ac{MCCI} in the present study allows us to follow the
evolution of \ac{QP} energies as electron correlations are
increased; this is achieved by starting from a single determinant picture
and then systematically including electron correlations by
performing \ac{CI} calculations at different values of the coefficient
selection threshold. As the coefficient selection threshold is reduced,
more \acp{CSF} are included in the calculations leading to improving
descriptions of electron correlations.

\subsection{CI approximations}

In our analysis, we consider several different approximations for the CI
wavefunction expansions. The first set of approximations are derived from
CI expansions consisting of single excitations 
\begin{equation} 
\label{eq:singles}
  \ket{\Psi_0} + \ket{\Psi_{S}}
  =
  \left(1 + \sum_{i,a} c_i^a \hat{a}_a^\dagger \hat{a}_i \right) \ket{\Psi_0},
\end{equation} 
with $\ket{\Psi_0}$ the Hartree-Fock reference state, $\ket{\Psi_{S}}$
consists of the singly excited CSFs, $i$ is an index over occupied
orbitals, $a$ is an index over unoccupied states, $c_i^a$ is the CI
coefficient for a singly excited CSF, and $\hat{a}^\dagger$ and $\hat{a}$
are creation and annihilation operators, respectively. The use of this
form of the CI wavefunction allows for the investigation of the effect
of optimizing the single particle orbitals. This may be understood by
considering Thouless' theorem~\cite{thouless} which relates two arbitrary
determinants $\ket{\Psi_A}$ and $\ket{\Psi_B}$ through  an exponential
operator as
\begin{equation}
\label{eq:thouless}
\ket{\Psi_A} = \exp \left( \sum_{i,a} c_i^a a_a^\dagger a_i \right) \ket{\Psi_B}.
\end{equation}     
The exponential operator acts as a rotation to the single particle basis.
For appropriately selected coefficients $c_i^a$, $\ket{\Psi_A}$ can be
chosen as the single particle determinant which optimizes, for example,
the energy on the extended device region. If the reference weight of
$\ket{\Psi_0}$ in the singles wavefunction is large, then the expansion
coefficients $c_i^a$ will be small and~\eref{eq:singles} becomes an
approximation to the Thouless form~\eref{eq:thouless}, accurate to first
order in the $c_i^a$. 

From a many-electron standpoint, energy differences between many-electron
states determine \ac{QP} excitations, as well as electron affinities
and ionization energies, or respectively addition and subtraction energies. 
In this study, the focus is on the electron affinities and ionization
potentials of the {\it explicit} device region as these can be directly
related to the quality of an electronic structure description for transport
calculations~\cite{golden}. For example, the \ac{EA} is given
by the difference in complex energies for the $N$-electron state $\Omega^N$
and the $(N+1)$-electron state $\Omega^{N+1}$
\begin{subequations}
\label{eq:EAIP}
\begin{equation}
\label{eq:EA}
\omega_{\rm EA} = \Omega^{N+1} -\Omega^N,
\end{equation}
and similarly the \ac{IP}
\begin{equation}
\label{eq:IP}
\omega_{\rm IP} =  \Omega^N - \Omega^{N-1}
\end{equation}
\end{subequations}
is the energy difference between the $(N-1)$- and $N$-electron states.
The inclusion of the set of singly excited configurations allows a
comparison on the explicit device region between a Koopmans approximation 
(no orbital relaxation) and a \dscf approximation (inclusion of
orbital relaxations).  For example, in the transmission spectra displayed
in~\fref{fig:transdat}, the resonances correspond to the eigenvalues of
the $N$ particle Fock matrix that are shifted in energy and broadened due
to the presence of the electrode self-energies. Hence the \ac{HOMO} state
corresponds to the \ac{IP} for the extended device region but excluding the
effects of relaxation of the other single electron states that occurs
when an electron is removed. Likewise the LUMO measures the energy gained
by attaching an electron onto the device region with all other electrons
frozen. In this sense, the resonances are identified as Koopmans \ac{IP}
and \ac{EA}. If the \ac{IP} and \ac{EA} are calculated from a many-electron
viewpoint and including all single excitations into the CI wavefunction,
all other electrons can relax on the extended device region as the charge
state is changed. As Thouless' theorem tells us that the CI singles
energies $\Omega^{N-1}$ and $\Omega^{N+1}$ are approximations to the optimized
single determinant energies with orbital relaxations for the device region,
the \acp{EA} and \acp{IP} from \eref{eq:EAIP} correspond to a \dscf
approximation. 

Two types of CI singles approximations are considered. The first set are
referred to as molecular singles, in these calculations only one-electron
or singles excitations to and from orbitals derived from the molecular
region are allowed. In the second set of singles, all orbitals on the
extended device region whether arising from the molecular or the explicit
electrode regions are allowed. This latter approximation is referred to
as the device region singles. The CI singles as discussed approximate a
single particle description of the tunnel junction due to Thouless' theorem.
To include correlations, MCCI calculations with coefficient thresholds
of 0.003 and 0.001 are performed. Again, two such sets of calculations
are performed. In the first set, only excitations to and from the molecular
derived states are allowed. These calculations are labelled as `CI(0.003)
molecule' and `CI(0.001) molecule'. For the extended device region, a
single MCCI calculation is performed with a coefficient cutoff of 0.003
with excitations to and from all orbitals permitted. This set of calculations
is labelled `CI(0.003) device region'. 

\section{Quasiparticle states and lifetimes}

Through the introduction of the \acp{CAP} describing the electrodes, the
device region becomes an open system resulting in complex many-electron
energies. Hence the \ac{QP} energies $\omega_i$, calculated  as the
difference in energies between many-electron states, are also complex
valued. It is the energy shift (related to the real part of $\omega_i$)
and finite lifetime (related to the imaginary part of $\omega_i$) due to
the opening of the system which we study for \acp{QP} on the
tunnel junction as electron correlations are introduced.

\subsection{Molecular excitations}
We start our analysis by considering the model with weaker coupling
defined by a 0.45 nm gap between the electrodes and molecular region,
and initially only including electronic excitations for orbitals localized
on the molecular region. As can be seen in~\fref{fig:13evals}, the
main effect of the electrode coupling at the single particle level is an
energy shift in the HOMO level of 32 meV and a state broadening indicating
a lifetime $\tau$ of approximately 223 fs, as calculated by
$\tau = \hbar / \Gamma$.

Performing a molecular singles calculation with only molecule orbital
excitations, we see the HOMO peak shifts upward in energy by 288 meV and
the LUMO peak shifts downward in energy by 260 meV as seen in the red
curves within figs.~\ref{fig:all45Ahomo} and \ref{fig:all45Alumo}, hence
the overestimation of the bandgap typical of the Hartree-Fock approximation
is compensated in the \dscf calculation.
The electron lifetimes decrease substantially relative to the Koopmans
states with the HOMO state lifetime becoming 59 fs and the LUMO state
becoming 17 fs when self-consistency is allowed, indicating an increased
molecule-electrode coupling. 
Introducing electron correlations using the molecule-like orbital
set and a coefficient threshold of 0.003 in the MCCI calculations shifts
the molecular HOMO downward and the LUMO upward, increasing the gap
relative to the \dscf approximation. Further increasing
the correlation allowed with use of a MCCI coefficient threshold of 0.001,
but still only using molecule-like excitations, shifts both peaks back
slightly in the opposite direction, decreasing the gap.
These findings for the weakly coupled molecular state are consistent with
typical findings for molecular systems: the Koopmans \acp{EA} and \acp{IP}
are  underestimated and overestimated, respectively, whereas the \dscf
procedure tends to overestimate and underestimate \acp{EA} and \acp{IP},
respectively.
Introducing electron correlations then renormalizes the fundamental gap
with the correlated energies for the HOMO and LUMO states bracketed by
the Koopmans and \dscf estimates.
However, the added correlation is seen not to cause any additional broadening
beyond that seen in the \dscf approximation.
Hence in the more weakly coupled case and including only excitations to and
from molecule-like orbitals, we find that the molecular region behaves
very much like an isolated molecule, with the additional features of an
energy shift and a finite state lifetime due to the molecule-electrode
coupling.

\begin{figure}
	\begin{center}
		\includegraphics[width=0.9\linewidth]{figure4a_4b}
	\end{center}
	\caption{(a) HOMO region \ac{QP} peaks of the 0.45 nm separated
	         system, with excitations restricted to the `molecule' region,
		 at different approximations: Koopmans (black),
		 \dscf (red), and MCCI with threshold parameters of
		 0.003 (blue) and 0.001 (green). (b) Density of states on the
		 lead.}
	\label{fig:all45Ahomo}
\end{figure}

\begin{figure}
	\begin{center}
		\includegraphics[width=0.9\linewidth]{figure5a_5b}
	\end{center}
	\caption{(a) LUMO region \ac{QP} peaks of the 0.45 nm separated
	         system, with excitations restricted to the `molecule' region,
		 at different approximations: Koopmans (black),
		 \dscf (red), and MCCI with threshold parameters of
		 0.003 (blue) and 0.001 (green). (b) Density of states on the
		 lead.}
	\label{fig:all45Alumo}
\end{figure}

Next, the effect of stronger coupling is considered by reducing the gap
between the electrodes and molecular regions to 0.40 nm (figures
\ref{fig:all40Ahomo} and \ref{fig:all40Alumo}).
A similar behavior is seen as compared to the weaker coupling case. The
\dscf approximation again shifts the energies of the HOMO and LUMO
resonances reducing the energy gap, and the state broadening relative to
the Koopmans resonances is significantly altered with state lifetimes
decreasing to approximately 17 fs from 71 fs, and to 7 fs from 26 fs for
the HOMO and LUMO, respectively.
Again it is found that increasing correlation shifts the energies of the
resonances but does not significantly alter the lifetime broadening beyond
that introduced at the \dscf level. Introducing electron correlations
again renormalizes the energy gap, with the correlated resonant states
bracketed by the Koopmans and \dscf values.

\begin{figure}
	\begin{center}
		\includegraphics[width=0.9\linewidth]{figure6a_6b}
	\end{center}
	\caption{(a) HOMO region \ac{QP} peaks of the 0.40 nm separated
	         system, with excitations restricted to the `molecule' region,
		 at different approximations: Koopmans (black),
		 \dscf (red), and MCCI with threshold parameters of
		 0.003 (blue) and 0.001 (green). (b) Density of states on the
		 lead.}
	\label{fig:all40Ahomo}
\end{figure}

\begin{figure}
	\begin{center}
		\includegraphics[width=0.9\linewidth]{figure7a_7b}
	\end{center}
	\caption{(a) LUMO region \ac{QP} peaks of the 0.40 nm separated
	         system, with excitations restricted to the `molecule' region,
		 at different approximations: Koopmans (black),
		 \dscf (red), and MCCI with threshold parameters of
		 0.003 (blue) and 0.001 (green). (b) Density of states on the
		 lead.}
	\label{fig:all40Alumo}
\end{figure}

\subsection{Molecular and electrode excitations}
We now investigate the effect of including excitations from the
explicit electrode region included within the definition of the extended
device region. Until this point, we have constrained the CI space to only
\acp{CSF} including excitations from and to orbitals localized on the
molecular region. However, in a previous work it has been shown that the
coupling of electrode excitations to the molecular excitations can have
significant effects on current flow through a correlated two-level model
system~\cite{galperin_nitzan2006leadexcitations}. 

We start again with the more weakly coupled model defined by a 0.45 nm gap
between the electrodes and the molecular region;
the results are displayed in \fref{fig:cidevregion}(a). 
Since self-consistency is introduced by approximating Thouless' theorem
with all single excitations on the extended device region, the energy gap
between HOMO and LUMO narrows with respect to the Koopmans values, as
previously was the case. However, in this case the \dscf results
in much larger energy shifts with respect to a \dscf approximation
relying on excitations from only molecule-like orbitals. There is also a
significant difference in the resonant broadening, the HOMO state lifetime
decreases from the Koopmans estimate of 223 fs to 18 fs as obtained from
the \dscf approximation. 
However, much less resonant broadening is seen in this example for the
LUMO state, with a lifetime of 85 fs from the Koopmans approximation
and 121 fs from the \dscf calculation. The results of introducing
correlation with a MCCI vector generated with a coefficient threshold of
0.003 is seen in the red curve of \fref{fig:cidevregion} and again a
renormalization of the energy gap is seen.

However, as electrode excitations are allowed to couple to the
molecule-like excitations, the correlated HOMO and LUMO resonance energies
are no longer bracketed by the Koopmans and \dscf peaks.
There is also a difference in the behavior for the HOMO and LUMO resonances.
For the HOMO peak, the lifetime stays approximately the same as in the
\dscf approximation whereas electron correlations broaden the LUMO
peak from a lifetime of 121 fs in the \dscf approximation to 21
fs. 

As the electrode-molecule distance is reduced to 0.40 nm, the increased
coupling in combination with the inclusion of electrode excitations
introduces quantitatively and qualitatively different behavior compared
to all previous cases. Including the electrode interactions
with the \dscf approximation as for the 0.45 nm coupling case,
significantly reduces the \ac{QP} gap and the magnitude of the
correction to the gap energy for the 0.40 and 0.45 nm cases are similar.
For the state lifetimes, the HOMO state is approximately the same as for
the 0.45 nm case, whereas the LUMO state is significantly broader than
all calculations previously presented with a lifetime of $\approx 2$ fs.
In this case, the CI singles approximation to the $N+1$-electron state
becomes multi-reference with two CSFs predominantly contributing in the
expansion. One of these CSFs has the additional electron in the molecule's
LUMO, whereas the other consists of an additional electron in an electrode
orbital. Hence in the \dscf approximation to the LUMO, the electrode
and molecule regions are strongly coupled leading to a broad resonance.
Introducing electron correlations with the MCCI vector obtained from a
coefficient threshold of 0.003 leads to further anomalous behavior. The
QP gap is again renormalized by the electron correlations, but the
HOMO state remains approximately unchanged although there is
slightly more broadening and a small energy shift. In contrast, the character
of the LUMO state relative to the \dscf approximation changes: the
electron correlations help to decouple the state from the electrodes
resulting in a single dominant CSF in the CI expansion that describes the
added electron as a molecule-like state. The energy broadening decreases,
leading to a lifetime of $\approx 7$ fs, and the state has shifted upward
by approximately 1 eV.
 
\begin{figure}
	\begin{center}
		\includegraphics[width=0.9\linewidth]{figure8a_8c}
	\end{center}
	\caption{CI results with inclusion of lead excitations for the
	         0.45 (a) and 0.40 nm (b) systems, with the density
		 of states of the leads plotted in (c).}
	\label{fig:cidevregion}
\end{figure}

Finally, as an illustration of how the changes in transmission curves would
affect the devices' conductance properties, we calculate the $I$-$V$ curves
for the 0.40 nm separated junction in the different approximations using
the Landauer formula.
In \fref{fig:iv}(a), we have plotted the results for the \dscf (both
molecule and device region) and Koopmans transmissions. The wide LUMO
peak which was noted earlier for the `device region' case causes an earlier
onset and higher final value of the current than in the `molecule' case.
The current from the Koopmans peaks (using only the HOMO and LUMO)
has a later onset, due to the larger bandgap.
\Fref{fig:iv}(b) shows the current from the two CI
transmission spectra. Due to the peaks in both cases being roughly the same
width, the current reaches similar values for high voltages, but the
higher position of both `device region' peaks relative to their `molecule'
counterparts causes a different onset behavior.

\begin{figure}
	\begin{center}
		\includegraphics[width=0.9\linewidth]{figure9a_9b}
	\end{center}
	\caption{Current versus voltage for the various transmission
	         spectra of the 0.45 nm separated system, using the
		 Landauer formula. (a) Comparison of the two \dscf results,
		 molecule (black) and device region (red) to the current
		 obtained using the HOMO and LUMO of the Koopmans
		 transmission (dashed line). (b) Same as (a), but for the
		 two CI(0.003) spectra.
		 The equilibrium Fermi
		 energy was chosen to lie in the middle of the band gap
		 of the leads, at $-4.42$ eV, and the electrochemical
		 potentials in the electrodes are taken to move apart
		 symmetrically as voltage is applied.}
	\label{fig:iv}
\end{figure}
 
\section{Conclusions}
\label{sec:conclusions}

This study has focused on the examination of the electronic structure
treatment of molecular tunnel junctions, and in particular has studied
the impact of the electronic structure treatment on \ac{QP}
energies and lifetimes. A simple model of a tunnel junction was introduced,
consisting of an atomic gold chain partitioned into left and right
electrodes, and a central `molecule'. The electrode self-energies were
approximated using \acp{CAP}, enabling the use of a \ac{CI} method.
The use of the \acp{CAP} was
validated against calculations using explicit electrode self-energies.
The junction model was investigated using a non-self consistent single
particle treatment or Koopmans treatment of \ac{QP} states, an
approximate self-consistent treatment based on a CI singles expansion,
and including electron correlations within truncated many-body expansions.
Two values of electrode coupling were studied by varying the distance
between the electrode and molecular regions. The ability to predict
electron affinities and ionization potentials was assessed as it has been
shown that the quality of a quantum transport calculation may be directly
related to the ability to predict electronegativity within the electronic
structure description of the junction~\cite{golden}. 

The calculations were further divided into two forms: those that did not
allow electronic excitations from the electrodes into the CI calculations,
and those that did. In the case that electrode excitations were excluded
from the \dscf and CI calculations, and for the range of electrode
couplings we considered, it was found that the molecule in the junction
behaved analogously to a free molecule, similar to the findings
of Goyer and Ernzerhof~\cite{SSPHubbard-Ernzerhof} for linear chains.
The Koopmans approximation overestimated the QP energy gap, the \dscf
approximation underestimated the gap, and including explicit electron
correlations renormalized the energy gap with the HOMO and LUMO states
being bracketed by the Koopmans and \dscf estimates. However, these
findings are fundamentally altered when electrode excitations are included,
consistent with a previous study of transport through a simplified two-level
system~\cite{galperin_nitzan2006leadexcitations}.
The junction's electronic structure becomes less like that of a free
molecule and the QP states and lifetimes are heavily influenced by both
the amount of correlation and the degree of coupling. Their combined
effect is difficult to predict, implying that detailed treatments of the
electronic structure are required to accurately model electronic behavior
such as charge transport and photoexcitations in a molecular junction.


\begin{acknowledgments}
This work was supported by Science Foundation Ireland via grant 06/IN.1/I857.
IY acknowledges support from an IRCSET postgraduate scholarship. 
\end{acknowledgments}


\begin{thebibliography}{37}%
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