# Embedding Combined with the Algebraic Diagrammatic Construction: Tackling Excited States in Biomolecular Systems

Yesterday the journal of chemical theory and computation
accepted our manuscript titled **Polarizable Embedding Combined with the
Algebraic Diagrammatic Construction: Tackling Excited States in Biomolecular Systems**.
In this work we describe the combination of the polarisable embedding (PE) scheme for
including environmental effects with the
algebraic-diagrammatic construction (ADC) approach.

The main idea of **polarisable embedding**
is to model the effect an environment
has on a core system by means of modelling the classical electrostatic
interactions between both.
For this the potential generated by the electron density of the environment
is expanded in a multipole series, which is typically truncated
at the level of dipoles. Furthermore effect of the core
on the environment itself is taken into account by allowing for
induced dipoles in the environment next the aforementioned static
dipoles of the multipole expansion.
Instead of modelling the full details, the environment
is thus reduced to a set of static dipole moments and polarisabilities,
which are located at some predefined sites,
typically the position of the nuclei.
During the SCF procedure of the core region
the dipoles induced by the core are optimised self-consistently
along the SCF, such that after the SCF procedure
a consistent electrostatic potential of the environment
is included inside the core density.

The reduction of the environment to a set of dipoles and polarisabilities
has the advantage, that these parameters can be determined
*a priori* based on a comparatively cheap computational method
and a standardised
fitting procedure (e.g. LoProp).
This implies that one can reduce the size of the system,
which needs to be modelled by a more expensive method.
One example would be the
**algebraic-diagrammatic construction scheme for the polarisation propagator**
for computing so-called excited states,
which we connect with polarisable embedding in this work.
In order to achieve this properly one has to do a little more
than self-consistently finding the electrostatic embedding potential.
Namely, one has to keep in mind that an electronic excitation
changes the electronic structure of the core, thus changes
the polarisation on the embedding and thus again the electronic
structure of the core. Taking this effect fully into account
would require another self-consistent iteration back and forth
between ADC and the PE scheme.
One can avoid this by treating this effect only approximately
via linear response as well as perturbative corrections
on top of an ADC calculation.

Apart from describing these corrections and their connections to physical quantities, we also hint at another important aspect of the simulation procedure, namely finding sensible arrangements for the environment atoms. Since the environment often consists of a solvent or a protein, it tends to be much more flexible compared to the core system. For this reason one cannot merely pick a single random molecular arrangement for the environment, but one needs to sample and perform the full calculation (that is ADC on the core) multiple times in order to gain some statistical results, which are a meaningful model for the exact physical system.

Last but not least we demonstrate the applicability of PE-ADC using three examples, namely a water cluster environment around para-nitroaniline, lumiflavin in bulk water and lumiflavin inside a dodecin protein. For each of the cases a detailed analysis of the results and comparison with existing methods is presented including statistical analysis of the sampling employing box plots.

The full abstract of the paper reads as follows:

We present a variant of the algebraic diagrammatic construction (ADC) scheme by combining ADC with the polarizable embedding (PE) model. The presented PE-ADC method is implemented through second and third order and is designed with the aim of performing accurate calculations of excited states in large molecular systems. Accuracy and large-scale applicability are demonstrated with three case studies, and we further analyze the importance of both state-specific and linear-response-type corrections to the excitation energies in the presence of the polarizable environment. We demonstrate how our combined method can be readily applied to study photo-induced biochemical processes as we model the charge-transfer (CT) excitation which is key to the photoprotection mechanism in the dodecin protein with PE-ADC(2). Through direct access to state-of-the-art excited state analysis, we find that the polarizable environment plays a decisive role by significantly increasing the CT character of the electronic excitation in dodecin. PE-ADC is thus suited to decipher photoinduced processes in complex, biomolecular systems at high precision and at reasonable computational cost.

Tags: electronic structure theory theoretical chemistry algebraic-diagrammatic construction and embedding