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<section id="simulations-of-photo-induced-properties-using-litesoph">
<h1>Simulations of Photo-Induced Properties using LITESOPH<a class="headerlink" href="#simulations-of-photo-induced-properties-using-litesoph" title="Permalink to this heading"></a></h1>
<p>Photophysical processes are initiated by the absorption of light by the molecules.
After light irradiation molecules get promoted to the electronic excited state.
In the excited state, molecules can undergo a reaction on the same PES or they can
deactivate to a ground electronic state via radiative channels like <a class="reference external" href="https://en.wikipedia.org/wiki/Fluorescence">fluorescence</a> and
<a class="reference external" href="https://en.wikipedia.org/wiki/Phosphorescence">phosphorescence</a> or nonradiative channels like internal conversion ( <a class="reference external" href="https://en.wikipedia.org/wiki/Internal_conversion#:~:text=Internal%20conversion%20is%20a%20non,(ejected)%20from%20the%20atom">IC</a> ), and intersystem
crossing (<a class="reference external" href="https://en.wikipedia.org/wiki/Intersystem_crossing">ISC</a>). To understand the photophysics and photochemistry of a system,
we have to first perform an <a class="reference internal" href="#absorption-spectrum"><span class="std std-ref">Absorption Spectrum</span></a> to get an idea about the optically accessible bright
and inaccessible dark states respectively. Then the state of interest can be targeted to understand the
photo dynamics in the excited state.</p>
<section id="absorption-spectrum">
<span id="id1"></span><h2>Absorption Spectrum<a class="headerlink" href="#absorption-spectrum" title="Permalink to this heading"></a></h2>
<p>The absorption spectrum provides information about the energies of the electronic excited state and
their transition probabilities. The absorption spectrum can be calculated using different theories like
<a class="reference external" href="https://en.wikipedia.org/wiki/Time-dependent_density_functional_theory#:~:text=Time%2Ddependent%20density%2Dfunctional%20theory,as%20electric%20or%20magnetic%20fields.">TDDFT</a>, <a class="reference external" href="https://pubs.acs.org/doi/10.1021/acs.jctc.2c00368">CASPT2</a>, <a class="reference external" href="https://en.wikipedia.org/wiki/Coupled_cluster#:~:text=For%20example%2C%20the%20CCSD(T,many%2Dbody%20perturbation%20theory%20arguments.">CCSD</a>, and <a class="reference external" href="https://adc-connect.org/v0.15.13/theory.html">ADC</a> (2) <a class="footnote-reference brackets" href="#id16" id="id2" role="doc-noteref"><span class="fn-bracket">[</span>1<span class="fn-bracket">]</span></a>. Among all methods, <a class="reference external" href="https://en.wikipedia.org/wiki/Time-dependent_density_functional_theory#:~:text=Time%2Ddependent%20density%2Dfunctional%20theory,as%20electric%20or%20magnetic%20fields.">TDDFT</a> offers a good compromise
between accuracy and computational cost for a large system <a class="footnote-reference brackets" href="#id17" id="id3" role="doc-noteref"><span class="fn-bracket">[</span>2<span class="fn-bracket">]</span></a> <a class="footnote-reference brackets" href="#id18" id="id4" role="doc-noteref"><span class="fn-bracket">[</span>3<span class="fn-bracket">]</span></a> <a class="footnote-reference brackets" href="#id19" id="id5" role="doc-noteref"><span class="fn-bracket">[</span>4<span class="fn-bracket">]</span></a> <a class="footnote-reference brackets" href="#id20" id="id6" role="doc-noteref"><span class="fn-bracket">[</span>5<span class="fn-bracket">]</span></a> <a class="footnote-reference brackets" href="#id21" id="id7" role="doc-noteref"><span class="fn-bracket">[</span>6<span class="fn-bracket">]</span></a> . Thus <a class="reference external" href="https://en.wikipedia.org/wiki/Time-dependent_density_functional_theory#:~:text=Time%2Ddependent%20density%2Dfunctional%20theory,as%20electric%20or%20magnetic%20fields.">TDDFT</a> is widely used to compute
the absorption spectrum. Both linear response (LR)-TDDFT and real time (RT)-TDDFT can be used
to calculate spectrum in frequency and time domain respectively. In our tools, we have only
incorporated the <a class="reference external" href="https://nwchemgit.github.io/RT-TDDFT.html">RT-TDDFT</a> approach to compute the absorption spectrum as well as other properties
because this method is more rigorous and applicable to all kinds of systems.</p>
<hr class="docutils" />
<hr class="docutils" />
<section id="calculations-of-absorption-spectrum">
<h3>Calculations of Absorption Spectrum<a class="headerlink" href="#calculations-of-absorption-spectrum" title="Permalink to this heading"></a></h3>
<p>The <a class="reference internal" href="#absorption-spectrum"><span class="std std-ref">Absorption Spectrum</span></a> is calculated based on these three <a class="reference internal" href="about_LITESOPH.html#engines"><span class="std std-ref">Engines Interfaced with LITESOPH</span></a>. Here are the results of some molecular systems using LITESOPH-GUI.</p>
<ul class="simple">
<li><p><a class="reference internal" href="GPAW/GPAW_Calculation.html"><span class="doc">TDDFT calculations using GPAW</span></a></p></li>
</ul>
</section>
</section>
<section id="kohn-sham-decomposition-ksd">
<h2>Kohn-Sham Decomposition (KSD)<a class="headerlink" href="#kohn-sham-decomposition-ksd" title="Permalink to this heading"></a></h2>
<p>Time-dependent density functional theory (<a class="reference external" href="https://en.wikipedia.org/wiki/Time-dependent_density_functional_theory#:~:text=Time%2Ddependent%20density%2Dfunctional%20theory,as%20electric%20or%20magnetic%20fields.">TDDFT</a>) built on top of Kohn-Sham (KS) density functional theory (DFT) is a
powerful tool in computational physics and chemistry for accessing the light-matter interaction. Starting from the seminal work
on jellium nanoparticles, <a class="reference external" href="https://en.wikipedia.org/wiki/Time-dependent_density_functional_theory#:~:text=Time%2Ddependent%20density%2Dfunctional%20theory,as%20electric%20or%20magnetic%20fields.">TDDFT</a> has become an important tool for modelling plasmonic response from a quantum mechanical
perspective and proven to be useful for calculating the response of individual nanoparticles and their compounds as well as other
plasmonic materials. Additionally, a number of models and concepts have been developed for quantifying and understanding
plasmonic character within the TDDFT approach. TDDFT in the linear-response regime is often formulated in frequency space
in terms of the Casida matrix expressed in the Kohn-Sham electron-hole space.The Casida approach directly enables a decomposition of the electronic excitations into the underlying KS electron-hole transitions, which easily provides quantum-mechanical
understanding of the plasmonic response. By contrast, real-time <a class="reference external" href="https://en.wikipedia.org/wiki/Time-dependent_density_functional_theory#:~:text=Time%2Ddependent%20density%2Dfunctional%20theory,as%20electric%20or%20magnetic%20fields.">TDDFT</a> (RT-TDDFT) (an alternative but computationally efficient approach with favorable scaling with respect to system size and is also applicable to the nonlinear regime) results are often
limited to absorption spectra or the analysis of the induced densities or fields. However, recently, Rossi et al. have developed KS
decomposition ( <a class="reference external" href="https://materialsmodeling.org/publications/2017-Kohn-Sham-decomposition-in-real-time-time-dependent-density-functional-theory-An-efficient-tool-for-analyzing-plasmonic-excitations/">KSD</a> ) tool based on the RT-TDDFT code that is available in the free GPAW code. The underlying RT-TDDFT
code uses the linear combination of atomic orbitals ( <a class="reference external" href="https://en.wikipedia.org/wiki/Linear_combination_of_atomic_orbitals#:~:text=A%20linear%20combination%20of%20atomic,atoms%20are%20described%20as%20wavefunctions.">LCAO</a> ) method and enable calculations involving hundreds of noble metal
atoms.</p>
<p>In order to analyze the response in terms of the KSD, the decomposition is presented as a transition contribution map ( <a class="reference external" href="http://allie.dbcls.jp/pair/TCM;transition+contribution+map.html">TCM</a> ),
which is an especially useful representation for plasmonic systems in which resonances are typically superpositions of many
electron-hole excitations. The TCM represents the KSD weight at a fixed energy of excitation in the two-dimensional plane
spanned by the energy axes for occupied and unoccupied states. For more details, refer to <a class="footnote-reference brackets" href="#id22" id="id8" role="doc-noteref"><span class="fn-bracket">[</span>7<span class="fn-bracket">]</span></a>.</p>
<p>More specifically, the 2D plot is defined by</p>
<div class="math notranslate nohighlight">
\[\begin{equation}
M^{TCM}_{\omega}(\varepsilon_o, \varepsilon_u) = \sum_{ia}w_{ia}(\omega)g_{ia}(\varepsilon_o, \varepsilon_u)
\tag{1}
\label{eq:1}
\end{equation}\]</div>
<p>where indices <span class="math notranslate nohighlight">\(i\)</span> and <span class="math notranslate nohighlight">\(a\)</span> correspond to the occupied and unoccupied states, respectively. The function <span class="math notranslate nohighlight">\(g_{ia}\)</span> is a
2D broadening function for the discrete KS <span class="math notranslate nohighlight">\(i \rightarrow a\)</span> transition contributions. By employing the 2D Gaussian
function, it is written as</p>
<div class="math notranslate nohighlight">
\[\begin{equation}
g_{ia}(\varepsilon_o, \varepsilon_u) = \frac{1}{2\pi\sigma^2}\exp{\bigg [-\frac{(\varepsilon_o - \varepsilon_i)^2 + (\varepsilon_u - \varepsilon_a)^2}{2\sigma^2}\bigg ]}
\tag{2}
\label{eq:2}
\end{equation}\]</div>
<p>where <span class="math notranslate nohighlight">\(\sigma\)</span> is the broadening parameter, which is generally taken to be the same as used for the spectral broadening.</p>
<p>The weight <span class="math notranslate nohighlight">\(w_{ia}(\omega)\)</span> in Eq. <span class="math notranslate nohighlight">\(\ref{eq:1}\)</span> is obtained from the absorption decomposition normalized by the total absorption, i.e.</p>
<div class="math notranslate nohighlight">
\[\begin{equation}
w_{ia}(\omega) = S^x_{ia}(\omega)/S_x(\omega)
\label{eq:3}
\tag{3}
\end{equation}\]</div>
<p>We note here is that the above equation is given particularly for the x-polarized light. However, the same thing can also be evaluated for y or z-polarized light.</p>
<p>The KS decomposition of the absorption spectrum in Eq. <span class="math notranslate nohighlight">\(\ref{eq:3}\)</span> is defined as</p>
<div class="math notranslate nohighlight">
\[\begin{equation}
S^x_{ia}(\omega) = -\frac{4\omega}{\pi}Im[\mu^{x^\star}_{ia}\delta\rho^x_{ia}(\omega)]
\label{eq:Sx_ia}
\tag{4}
\end{equation}\]</div>
<p>where the dipole matrix element is obtained as</p>
<div class="math notranslate nohighlight">
\[\begin{equation}
\mu^{x}_{ia} = \int \psi_i^{(0)*}( r)x\psi_a^{(0)}( r)d\bf r
\label{eq:mux_ia}
\tag{5}
\end{equation}\]</div>
<p>where <span class="math notranslate nohighlight">\(\psi_j^{(0)}\)</span> corresponds to the KS wave function for <span class="math notranslate nohighlight">\(j = i,a\)</span>.</p>
<p>The linear response of the real part of the KS density matrix in the electron-hole space is written as</p>
<div class="math notranslate nohighlight">
\[\begin{equation}
\delta\rho^x_{ia}(\omega) = \frac{1}{K_x}\int_0^\infty Re[\rho^x_{ia}(t) - \rho_{ia}(0^{0-})]e^{i\omega t}dt + O(K_x)
\label{eq:delta}
\tag{6}
\end{equation}\]</div>
<p>where <span class="math notranslate nohighlight">\(K_x\)</span> is the laser strength and <span class="math notranslate nohighlight">\(\rho_{ia}(0^{0-})\)</span> is the initial density matrix before the <span class="math notranslate nohighlight">\(\delta\)</span>-pulse perturbation,
and the superscript <span class="math notranslate nohighlight">\(x\)</span> indicates the direction of the perturbation.</p>
<p>Transition contribution map for the photoabsorption decomposition of Au55 metallic nanoparticle at 3.02 eV resonance
energy calculated using LCAO <a class="reference external" href="https://en.wikipedia.org/wiki/Time-dependent_density_functional_theory#:~:text=Time%2Ddependent%20density%2Dfunctional%20theory,as%20electric%20or%20magnetic%20fields.">TDDFT</a> approach in GPAW code is shown in <a class="reference internal" href="#au55-tcm"><span class="std std-ref">Fig.1</span></a>.</p>
<figure class="align-center" id="id23">
<span id="au55-tcm"></span><a class="reference internal image-reference" href="_images/Au55_tcm_3.02.png"><img alt="alternate text" src="_images/Au55_tcm_3.02.png" style="width: 500px; height: 400px;" /></a>
<figcaption>
<p><span class="caption-text"><strong>Fig.1: Transition contribution map for the photoabsorption decomposition of Au55 metallic nanoparticle
at 3.02 eV resonance energy calculated using LCAO TDDFT approach in GPAW code. The KS eigenvalues are
given with respect to the Fermi level. The constant transition energy line</strong> <span class="math notranslate nohighlight">\(\varepsilon_u - \varepsilon_o = \omega\)</span>
<strong>is the analysis energy (solid line). Red and blue colors indicate positive and negative values of the
photoabsorption, respectively. The density of states (DOS) are also shown.</strong></span><a class="headerlink" href="#id23" title="Permalink to this image"></a></p>
</figcaption>
</figure>
</section>
<section id="molecular-orbital-mo-population">
<h2>Molecular Orbital (MO) Population<a class="headerlink" href="#molecular-orbital-mo-population" title="Permalink to this heading"></a></h2>
<p>The Kohn-Sham (<a class="reference external" href="https://pubs.acs.org/doi/10.1021/ja9826892">KS</a>) orbital can be expanded using gaussian basis function <span class="math notranslate nohighlight">\(\phi_\mu\)</span> as:</p>
<div class="math notranslate nohighlight">
\[\psi_i (\mathbf r, \mathbf t) = \sum_{\mu = 1}^{N_{AO}} \mathbf {C_{\mu i}} (\mathbf t) \phi_\mu (\mathbf r)
\label{Eq:psi_i}
\tag{7}\]</div>
<p><a class="reference external" href="https://en.wikipedia.org/wiki/Density_matrix">Density matrix</a> can be obtained from the products of time-dependent coefficient:</p>
<div class="math notranslate nohighlight">
\[\mathbf {P_{\mu \nu}} = \sum_i^{N_{MO}} \mathbf {C_{\mu i}^*}(\mathbf t) \mathbf C_{i \nu}(\mathbf t)
\label{Eq:P_nu}
\tag{8}\]</div>
<p>Time dependent orbital population computed by projecting the density matrix on to the ground state
orbitals:</p>
<div class="math notranslate nohighlight">
\[n_{\bf k} (\bf t) = \mathbf {C_k^*} \mathbf P (\mathbf t)\mathbf C_k(\mathbf t)
\label{Eq:n_k}
\tag{9}\]</div>
</section>
<section id="laser-simulation">
<h2>LASER Simulation<a class="headerlink" href="#laser-simulation" title="Permalink to this heading"></a></h2>
<p>Once we have the photoabsorption spectrum of any system, we get an information about the excitation frequencies
present in the system. Using any frequency of excitation, we can hit the system with a laser and study the time
dynamics of any physical observable of our interest.</p>
<p>The Gaussian light pulse is defined as</p>
<div class="math notranslate nohighlight">
\[\begin{equation}
\varepsilon(t) = \varepsilon_0\sin(\omega_0t)e^{-(t-t_0)^2/2\sigma^2}
\label{eq:laser_pulse}
\tag{10}
\end{equation}\]</div>
<p>that induces a real time dynamics in the system. The pulse frequency <span class="math notranslate nohighlight">\(\omega_0\)</span>
is tuned to the frequency of excitation of interest, the pulse duration is determined
by <span class="math notranslate nohighlight">\(\sigma\)</span>, and the pulse is centered at <span class="math notranslate nohighlight">\(t_0\)</span>. The pulse strength <span class="math notranslate nohighlight">\(\varepsilon_0\)</span> is
generally kept weak, to ensure that the system is in the linear response regime. In the
frequency space, the pulse should be wide enough to cover the whole frequency of excitation.</p>
<p>The determination of the pulse width and its center is crucial for the light-matter interaction.
However, these quantities can be evaluated from the <a class="reference external" href="https://en.wikipedia.org/wiki/Envelope_(waves)">envelope</a> function in Eq. <span class="math notranslate nohighlight">\(\ref{eq:laser_pulse}\)</span>,
which is defined as</p>
<div class="math notranslate nohighlight">
\[\begin{equation}
G(t) = \exp(-(t-t_0)^2)/2\sigma^2)
\label{eq:gaussian_envelope}
\tag{11}
\end{equation}\]</div>
<p>The Gaussian envelope in time will also yield the Gaussian envelope in frequency (Gaussians are eigenfunctions of a
Fourier transform), with width equal to the inverse of the width in time.
The full width at half maximum ( <a class="reference external" href="https://en.wikipedia.org/wiki/Full_width_at_half_maximum">FWHM</a> ) of Gaussian pulse is given as</p>
<div class="math notranslate nohighlight">
\[\begin{equation}
\text{FWHM} = 2\sqrt{2\ln 2}\sigma_\omega
\label{eq:fwhm}
\tag{12}
\end{equation}\]</div>
<p>The center of the pulse is obtained as</p>
<div class="math notranslate nohighlight">
\[\begin{equation}
t_0 = t-\sqrt{-2\sigma^2\ln G(t)}
\label{eq:t_0}
\tag{13}
\end{equation}\]</div>
<section id="laser-masking-to-study-energy-transfer-using-rt-tddft-approach">
<h3>LASER Masking <strong>to study energy transfer using RT-TDDFT approach</strong><a class="headerlink" href="#laser-masking-to-study-energy-transfer-using-rt-tddft-approach" title="Permalink to this heading"></a></h3>
<p>Energy transfer is one of the most fundamental processes on the molecular scale, governing light-harvesting in biological
systems and energy conversion in electronic devices such as organic solar cells or light-emitting diodes. The design principles
of natural light-harvesting complexes have found considerable interest, as scientists believe that the principles realized in nature
can be mimicked in the design of artificial organic devices.One standard method to interpret experimental data of excitation energy transfer between a donor (D) and an acceptor (A)
molecule separated by the distance R is the so-called Forster resonance energy transfer ( <a class="reference external" href="https://en.wikipedia.org/wiki/F%C3%B6rster_resonance_energy_transfer">FRET</a> ) theory. <span class="math notranslate nohighlight">\(Forster\)</span> theory describes ¨
the nonradiative energy transfer mediated by a (quantum-mechanical) coupling between the transition dipoles of the donor and
acceptor molecules. One of the central assumptions in FRET is that the coupling between D and A
can be described by a (point)-dipole-dipole interaction, falling as 1/R3.
Furthermore, <a class="reference external" href="https://en.wikipedia.org/wiki/F%C3%B6rster_resonance_energy_transfer">FRET</a> theory is formulated for the weak
coupling regime (i.e., the isolated D and A excited states do not change significantly on coupling).</p>
<p>The energy transfer rate is written as</p>
<div class="math notranslate nohighlight">
\[\mathbf K^\text{ET} = 2\pi {\lvert V\rvert}^2\int_{0}^\infty\!\mathrm{d}\varepsilon {\mathbf J(\varepsilon)}
\label{Eq:K^ET}
\tag{14}\]</div>
<p>where <span class="math notranslate nohighlight">\(\mathbf J(\varepsilon)\)</span> is the spectral overlap between the normalized donor emission and acceptor absorption spectra.</p>
<p>The factor V in Eq. <span class="math notranslate nohighlight">\(\ref{Eq:K^ET}\)</span> is the electronic coupling matrix element which is obtained using
Davydov splitting. In the case two same molecules, the
Davydov splitting <span class="math notranslate nohighlight">\(\Delta\Omega\)</span> equals the energy splitting <span class="math notranslate nohighlight">\(\Delta E\)</span> of the (nearly) degenerate excitation energies of the monomers
in the supermolecule.</p>
<div class="math notranslate nohighlight">
\[\begin{equation}
V = \frac{\Delta\Omega}{2}
\label{eq:Davydov_splitting}
\tag{15}
\end{equation}\]</div>
<p>There is however, an another alternative way of evaluating the required information by observing the <span class="math notranslate nohighlight">\(D\)</span> and <span class="math notranslate nohighlight">\(A\)</span> dipole moments separately.
It significantly reduces the computational cost. We note that the Davydov splitting and consequently also the coupling matrix element <span class="math notranslate nohighlight">\(V\)</span>
manifests itself as a frequency <span class="math notranslate nohighlight">\(\omega_{beat}\)</span> in the time-dependent <span class="math notranslate nohighlight">\(D\)</span> (and <span class="math notranslate nohighlight">\(A\)</span>) dipole moment. This frequency can be extracted as
a beat frequency <span class="math notranslate nohighlight">\(\omega_{beat}\)</span> of an oscillation between <span class="math notranslate nohighlight">\(D\)</span> and <span class="math notranslate nohighlight">\(A\)</span>.</p>
<div class="math notranslate nohighlight">
\[\begin{equation}
V = \omega_{beat}
\label{eq:V_beat}
\tag{16}
\end{equation}\]</div>
</section>
</section>
<section id="references">
<span id="ref"></span><h2>References<a class="headerlink" href="#references" title="Permalink to this heading"></a></h2>
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</aside>
</aside>
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