# Molecular orbitals of an elastic artificial benzene

A. M. Martínez-Argüello<EMAIL_ADDRESS>Instituto de Ciencias Físicas,
Universidad Nacional Autónoma de México, Apartado Postal 48-3, 62210,
Cuernavaca Mor., Mexico  M. P. Toledano-Marino  Instituto de Ciencias Físicas,
Universidad Nacional Autónoma de México, Apartado Postal 48-3, 62210,
Cuernavaca Mor., Mexico  A. E. Terán-Juárez  Instituto de Ciencias Físicas,
Universidad Nacional Autónoma de México, Apartado Postal 48-3, 62210,
Cuernavaca Mor., Mexico  E. Flores-Olmedo  Departamento de Ciencias Básicas,
Universidad Autónoma Metropolitana-Azcapotzalco, Av. San Pablo 180, Col.
Reynosa Tamaulipas, 02200 Ciudad de México, Mexico.  G. Báez  Departamento de
Ciencias Básicas, Universidad Autónoma Metropolitana-Azcapotzalco, Av. San
Pablo 180, Col. Reynosa Tamaulipas, 02200 Ciudad de México, Mexico.  E.
Sadurní  Benemérita Universidad Autónoma de Puebla, Instituto de Física,
Apartado Postal J-48, 72570 Puebla, México.  R. A. Méndez-Sánchez  Instituto
de Ciencias Físicas, Universidad Nacional Autónoma de México, Apartado Postal
48-3, 62210, Cuernavaca Mor., Mexico

###### Abstract

Benzene, a hexagonal molecule with formula C6H6, is one of the most important
aromatic hydrocarbons. Its structure arises from the $sp^{2}$ hybridization
from which three in-plane $\sigma$-bonds are formed. A fourth $\pi$-orbital
perpendicular to the molecular plane combines with those arising from other
carbon atoms to form $\pi$-bonds, very important to describe the electronic
properties of benzene. Here this $\pi$-system is emulated with elastic waves.
The design and characterization of an artificial mechanical benzene molecule,
composed of six resonators connected through finite phononic crystals, is
reported. The latter structures trap the vibrations in the resonators and
couple them through evanescent waves to neighboring resonators establishing a
tight-binding regime for elastic waves. Our results show the appearance of a
spectrum and wave amplitudes reminiscent of that of benzene. Finite element
simulations and experimental data show excellent agreement with the Hückel
model for benzene.

##  I Introduction

Current experimental techniques allow the controlled fabrication of structures
at a molecular and atomic scale Li2019 ; Carlotti2016 ; Manrique2015 ;
Guedon2012 ; Champagne2005 ; Reichert2002 ; Moresco2001 ; Gimzewski1999 . One
of the main goals in this field of research, inspired by the works of A.
Aviram and M. A. Ratner Aviram1974 , is the use of individual molecules as
basic components of electronic devices Manrique2015 ; Champagne2005 ;
Liang2002 ; Nitzan2003 ; Xu2003 ; Kubatkin2003 ; Joachim2005 ; Tao2006 ;
CunibertiBook ; CuevasBook . In order to do that, high control and
manipulation of the electronic properties of the molecule and of the contacts
it is attached to is required Lastapis2005 ; Zhirnov2006 ; Diez2011 ; Geng2004
; Basch2005 ; Haiss2006 ; Kronemeijer2008 . This still represents a difficult
task to achieve experimentally Carlotti2016 .

Among the several molecules showing electronic functionalities, conjugated
molecules are of great interest Li2019 ; Diez2011 ; Gholami2006 ;
Cardamone2006 ; Yoshizawa2008 ; Andrews2008 ; Chen2008 ; Tada2015  since they
possess delocalized $\pi$-electrons which are responsible for their conducting
properties. Benzene has received special attention due to its size and because
it has transistor functionalities DiVentra2000 . A description and control of
its molecular orbitals are of interest since the coupling of these orbitals to
the outside has implications in the conducting properties of the molecule
Diez2011 ; Tada2015 ; Walczak2004 ; Kornilovitch2001 . Consequently, to get a
further insight into the conducting and electronic properties of
$\pi$-systems, some theoretical models have been used CunibertiBook ;
CuevasBook .

A paradigmatic model widely used for describing solid state systems and the
electronic properties of $\pi$-molecular structures is the tight-binding (TB)
model CunibertiBook ; CuevasBook . It takes into account interference effects,
retains atomic and electronic structure details, and also captures important
features like circular currents Andrews2008 ; Mujica1994 ; Kostyrko2002 ;
Walczak2004 ; Rai2010 ; Yuta2014 ; Cattena2010 ; Nozaki2010 ; Nozaki2012 ;
Nozaki2017 ; Rix2019 . In this model the electrons are imagined to be strongly
bonded to atoms and interact weakly with neighboring atoms. Thus the
electronic wave function of a system of interacting atoms is written in terms
of the wave functions of a system of isolated atoms, plus the coupling that is
usually considered to nearest neighbors. This model has been recently emulated
in classical wave systems with the use of optical fibers Kondakci2015  and in
microwave experiments using dielectric resonators Barkhofen2013 , the latter
playing the role of atomic sites, from which artificial analogues of graphene
Bellec2013  and polyacetylene have been constructed Stegmann2017 . The TB
model can also be emulated with mechanical systems. An elastic resonator,
whose wave amplitude is localized in a certain region in space and with an
exponential decay out of it, is required for its implementation. This
localized wave constitutes an _artificial atomic orbital_ Filiberto2020 .
Additionally, to establish the nearest-neighbor tight-binding regime for
elastic waves, at least two such orbitals must be evanescently coupled to each
other. Opposite to the optical and microwave cases the mechanical TB model
allows a high nearest-neighbor coupling control, which is often the most
dominant part; next-nearest neighbor interactions can also be easily
engineered. Here the design and characterization of a mechanical artificial
benzene molecule, in which the analogues of the spectrum and molecular
orbitals of benzene are emulated, is reported. Our experimental results are in
agreement with finite-element numerical simulations of the artificial molecule
and with the simplest tight-binding model for benzene.

The paper is organized as follows. In the next section, we discuss the design
of the relevant artificial elastic atomic orbital. The benzene-like molecule
in the TB regime for elastic waves is shown in section III. The experimental
characterization of the artificial benzene is explained in section IV.
Finally, we present our conclusions in section V.

Figure 1: (Color online) a Elastic artificial atomic site composed of a
hexagonal-shaped resonator of length $L_{h}$ and connected to three bonds with
an angle of $120^{\circ}$ between them. b Finite phononic crystal (FPC),
formed by the repetition of a unit cell, that plays the role of bonds. c The
unit cell is composed of a square-shaped aluminum plate of thick $e$ and side
$L$, and two smaller plates, one on each side of the larger plate, of size
$a/2$ and length $b$. The FPCs give rise to bands and gaps in its level
dynamics (frequency spectrum as a function of $L$), as shown in panel d for a
structure composed of five unit cells. The bandgap is indicated by a red oval.
By choosing a frequency of the resonator in the gap of the FPCs, the vibration
is trapped in the resonator as shown in panels e and f, the maximum (minimum)
amplitude in panel f is shown in red (blue) color. In the lower-right part,
this vibration is shown in 3D. An exponential decay of the amplitude of
vibration along the bonds, in the positions indicated in black-filled circles,
is also evident. This exponentially decaying wave amplitude, panel e in red-
filled circles, emulates a quantum atomic orbital and is used to build the
artificial benzene molecule. The amplitude is measured at the points indicated
in black-filled circles in figure 1 f. An exponential decay of the frequency
level splitting as a function of the number of FPC cells (see panel g) is
obtained, as shown in panel e in blue-filled squares. The dashed lines are
fits to the corresponding data. The geometrical parameters are $e=6.05$ mm,
$L=20.0$ mm, $a/2=2.5$ mm, $b=8.2$ mm, and $L_{h}=30.4$ mm. Figure 2: (Color
online) a Experimental setup. 1 Workstation, 2 Vector network analyzer Anritsu
MS4630B, 3 Cerwin Vega CV-2800 high-fidelity audio amplifier, 4
electromagnetic acoustic transducer (EMAT) exciter, 5 elastic artificial
benzene, 6 EMAT detector. b Numerical frequency spectrum of artificial benzene
as a function of $L_{h}$, the FPCs give rise to forbidden bands that trap the
vibrations in the resonators. For $L_{h}=30.4$ mm, in the interval from 20500
to 21500 Hz, some natural frequencies associated with the resonators (red
horizontal lines) are observed within the forbidden band. c Experimental
frequency spectrum, six resonances are observed: two singlets and two
doublets, the vertical lines correspond to the numerical predictions. An
excellent agreement is observed between the experimental and numerical
results.

##  II Artificial atomic orbital

In order to establish the tight-binding regime for mechanical waves, a
structure whose wave amplitude is spatially localized in a certain region and
with an exponential decay out of it, is required. Furthermore, this structure
has to be able to connect with another structure to form an artificial
molecule. The localized wave amplitude or _artificial elastic atomic orbital_
has to couple in an evanescent way with another artificial atomic orbital to
form _artificial elastic molecular orbitals_. A first 1D example of such
elastic artificial molecule was reported in Filiberto2020 . For its
implementation, a mechanical resonator was connected to finite phononic
crystals (FPCs), playing the role of bonds, which trap the resonator
vibrational wave amplitude in its position and with an exponential decay along
the bonds, emulating in this way an atomic orbital. Furthermore, allowing such
mechanical resonators to weakly interact with each other via the FPCs, a 1D
elastic crystal, which is in complete agreement with a quantum tight-binding
model Filiberto2020 , was constructed. Following this line, we use the
artificial atomic site shown in Fig. 1 a. It is composed of a hexagonal-shaped
two-dimensional elastic resonator of side length $L_{h}$ connected to three
FPCs (bonds), see Figs. 1 b and c. Both, the resonator and FPCs have the same
thickness $e$. The angle between bonds is $120^{\circ}$. This shape allows a
configuration that is typical in molecules with alternating single and double
chemical bonds that yield to an $sp^{2}$ hybridization, and are found in
planar molecules like ethylene and benzene, among others. The FPCs are formed
by the repetition of a unit cell consisting of a square-shaped plate of length
$L$ and two small plates of side $a/2$ and length $b$, one at each side of the
larger plate (see Fig. 1 c). The FPCs give rise to allowed and forbidden bands
in its frequency spectrum, as shown in Fig. 1 d for a structure composed of
five unit cells. By choosing a vibrating frequency $F$ of the resonator in the
bandgap of the FPC, the latter traps the vibration in the resonator and
produces an exponential decay of the amplitude along the bonds as can be
observed in Figs. 1 e and f. The evanescence length $\xi\approx 0.86$, in
units of the maximum amplitude, is in perfect correspondence with the fact
that a localized state decays exponentially by means of an imaginary wave
number $\mathrm{e}^{-\kappa\hat{\kappa}\cdot\vec{x}}$ with $\kappa=1/\xi$.
This trapped vibrational wave amplitude is in agreement with the case of
electrons strongly bonded to their atoms and emulates a single quantum atomic
orbital Filiberto2020 . The structure mentioned above is used to build the
artificial elastic benzene. In figure 1 e the level splitting for two coupled
artificial atomic sites as function of the cells of the coupler is also given.
The coupling between neighboring artificial sites is given by half the level
splitting ($\nu=56$ Hz).

##  III Artificial elastic benzene

The benzene molecule is a hydrocarbon composed of six carbon atoms, bonded to
one hydrogen atom each, and bonded between them by means of an $sp^{2}$
hybridization. In this hybridization the relevant orbitals, concerning the
wave conduction, are the $\pi$-orbitals. These orbitals can be described
within the tight-binding approximation from which an analogous molecule can be
constructed in a mechanical structure. For this purpose, we couple six
identical artificial atomic sites (see figure 1 a) in a hexagonal structure to
form the artificial benzene molecule. The six $\pi$ atomic orbitals, ignoring
the hydrogen atoms, are emulated assuming that the hexagonal structure comes
from the $sp^{2}$ hybridization of the carbon atoms and to the bonding to the
hydrogen atoms. This mechanical analogue reproduces the expected structure of
the frequency spectrum, with the repulsion between the eigenfrequencies of the
resonators, and also the molecular orbitals, or wave amplitudes, that are
reminiscent to that of the benzene molecule. In figure 2 a we show the
artificial elastic benzene molecule. The geometrical parameters are the same
as those of the artificial atomic site of figure 1 a and are determined by
means of finite element simulations using the software COMSOL Multiphysics.
FPCs composed of five unit cells are taken to the outside to avoid edge
effects. FPCs with only two unit cells connect the resonators. This guarantees
that the eigenfrequencies belonging to the resonators are in a bandgap of the
FPCs in a frequency range from $19$ to $22$ kHz, as shown in figure 2 b in red
horizontal lines. As expected, six eigenfrequencies show up: two singlets at
higher and lower frequencies and two doublets in between. The structure of the
eigenfrequency spectrum and the corresponding eigenmodes associated to each
eigenfrequency are in excellent agreement with the ones obtained from the
tight-binding model for benzene, as shown below.

The numerical frequency spectrum for the artificial molecule is

$\displaystyle f_{1}$ $\displaystyle=$ $\displaystyle 20989\,\mathrm{Hz},$
$\displaystyle f_{2}$ $\displaystyle=$ $\displaystyle 21042\,\mathrm{Hz},\quad
f_{2^{\prime}}=21043\,\mathrm{Hz},$ $\displaystyle f_{3}$ $\displaystyle=$
$\displaystyle 21153\,\mathrm{Hz},\quad
f_{3^{\prime}}=21155\,\mathrm{Hz},\quad\mathrm{and}$ $\displaystyle f_{4}$
$\displaystyle=$ $\displaystyle 21209\,\mathrm{Hz}$ (1)

where we observe that $f_{2}\approx f_{2^{\prime}}$ and $f_{3}\approx
f_{3^{\prime}}$, i.e., the modes of vibration, or artificial elastic molecular
orbitals, associated to those eigenfrequencies are degenerate, and represent
doublets in the molecule. Thus, the artificial system shows six frequency
levels, two of them degenerate with a frequency difference $\Delta f\approx
53$ Hz between singlets and doublets and $2\Delta f$ ($\approx 111$ Hz)
between doublet and doublet. This structure shown by the frequency levels is
observed in the energy spectrum of the benzene molecule in the quantum
mechanical case. Furthermore, the modes of vibration associated to those
eigenfrequencies also reproduce the symmetry of the molecular orbitals of
benzene. In the nearest-neighbor tight-binding approximation for the
artificial benzene the following Hamiltonian is assumed

$H=\left(\begin{array}[]{cccccc}F&-\nu&0&0&0&-\nu\\\ -\nu&F&-\nu&0&0&0\\\
0&-\nu&F&-\nu&0&0\\\ 0&0&-\nu&F&-\nu&0\\\ 0&0&0&-\nu&F&-\nu\\\
-\nu&0&0&0&-\nu&F\end{array}\right),$ (2)

where all site frequencies, analog to the site energies, are equal to
$F=21099$ Hz. The hopping frequencies, analog to the hopping energies, are all
$\nu=56$ Hz. The wave amplitudes $\psi_{n}(\vec{r})$ of interacting artificial
atoms are expanded in terms of those of isolated artificial atoms,
$\phi_{m}(\vec{r})$ as

$\psi_{n}(\vec{r})=\sum_{m}c_{nm}\phi_{m}(\vec{r}),$ (3)

where $c_{nm}$ are the coefficients in the expansion. More details of the
tight-binding approximation, and of the solutions, can be found in Appendix A
and in Ref. Hernandez2020 .

Figure 3: (Color online) Wave amplitudes from tight-binding model, artificial
mechanical benzene, and experiments, in left, middle and right columns of
panels a and b. The color scale is the same of Fig. 1. The blue-dashed circles
in the center and right columns of a and b indicate equivalent points and help
to identify the symmetry of the artificial orbital. The “E” in the
experimental orbitals indicate the position of the EMAT exciter.

Figure 3 shows the molecular orbitals obtained from the tight-binding model
for benzene of (2), in left columns of panels a and b. An excellent
approximation to both the structure of the frequency spectrum and the
symmetries of the molecular orbitals of the benzene molecule is observed
CuevasBook . Furthermore, we note that the modes of vibration obtained from
the numerical artificial molecule also reproduce the symmetries of the
molecular orbitals of benzene, as shown in the same panels of figure 3 in the
middle columns. In the molecular orbitals, $\varphi_{j}(x,y)$, each atomic
site of the molecule in red (blue) color corresponds to equivalent points in
the artificial benzene. These points are indicated in blue-dashed circles in
figure 3, panels a and b. As can be observed, the mechanical artificial
orbitals $M_{j}(\textbf{r})$ reproduce the symmetries of the molecular
orbitals $\varphi_{j}(x,y)$. These results show the successful emulation of
the underlying properties of the benzene molecule like its spectrum structure
and its molecular orbitals.

##  IV Experiment

The predictions obtained from the numerical simulations for the artificial
benzene can be tested experimentally with the setup of Figure 2 a. The
artificial benzene molecule, with geometrical parameters given in Fig. 1,
machined from a single aluminum plate (Young modulus $E=71.1$ GPa, mass
density $\rho=2.708\mathrm{kg}/\mathrm{m}^{3}$, and Poisson’s ratio
$\nu=0.36$), was held up by nylon threads attached to support stands. The
threads are tensioned by the weight of the structure and maintain minimal
contact with the system avoiding an alteration of the vibrations of the
artificial molecule.

The setup consists of a workstation that controls and extracts the data from a
vector network analyzer (VNA) Anritsu MS4630B. The VNA generates a harmonic
signal of frequency $f_{0}$ which is sent to a Cerwin-Vega CV-2800 high-
fidelity audio amplifier. The intensified signal is sent to an electromagnetic
acoustic transducer (EMAT), consisting of a coil and a permanent magnet
Morales2001 . The EMAT was located close to a vertex of one resonator of the
artificial molecule, without physical contact. The former excites out-of-plane
waves when the dipole moment axis of both the coil and permanent magnet
coincide with the normal to the plate. A second EMAT, of smaller dimensions,
with the same configuration of coil and permanent magnet as the exciter,
located close to the farthest resonator, detects the response that is sent
back to the VNA. The data are analyzed in the workstation.

To obtain the artificial molecular orbitals we do the following procedure.
Firstly, a set of measurements on a grid of a sixth of one resonator is
performed. Secondly, due to the symmetry of the mode in the resonator (see
figure 1 f), this sixth is reflected to build the full artificial atomic
orbital. Finally, to obtain the artificial molecular orbital in any resonator
a measure of its amplitude in equivalent points is multiplied by the atomic
orbital, previously obtained. This gives a total of four modes each one
associated to its corresponding eigenfrequency. The other two modes,
associated to the degeneracies, are obtained as before, but the vibrations in
two opposite resonators are suppressed by clamps. These modes of vibration
correspond to the artificial benzene molecular orbitals and reproduce the
symmetry of those of benzene obtained from the TB model and other more
sophisticated models like density-functional theory CuevasBook .

Figure 2 c shows the measured frequency spectrum. As expected six resonances
show up, two of them degenerate: two singlets and two doublets. This is the
analogous spectrum to the energy spectrum of the benzene molecule. An
asymmetry of these resonances, due to the Fano effect Angel2015 , is observed.
In the right columns of panels a and b of figure 3 the measured modes are
shown. These are the analogous orbitals reminiscent to the orbitals of the
benzene molecule.

##  V Conclusions

We have presented a design and experimental realization of a benzene-like
elastic artificial molecule. The analogous spectrum and $\pi$-orbitals which
are reminiscent to that of benzene were emulated. The experimental frequency
spectrum and wave amplitudes agree with the theoretical predictions from
finite element simulations and the simplest tight-binding model for benzene.

In the literature, artificial analogues have been used to get a further
understanding of the wave behavior of molecular systems with the advantage of
an easy manipulation of the systems. They also provide experimental
realizations to verify theoretical predictions that are difficult to attain at
quantum level. Furthermore, elastic systems in particular are of interest
since it has been found that the quality factor in these systems is sometimes
orders of magnitude higher than those found in microwave systems. The
successful emulation of $\pi$-orbitals and the tight-binding regime for
elastic waves opens the door to study more complicated molecules and of
paradigmatic models of solid state physics. In addition, our results provide
high control over the nearest-neighbor and of the orbitals, which are of
interest since the coupling of these orbital to the outside contacts has
strong implications in the conducting properties of molecules. Here, by
manipulating the geometrical parameters of the resonators or bondings allows
to control the coupling strength of the amplitudes of vibrations as required.

###### Acknowledgements.

This work was supported by CONACYT and DGAPA-UNAM under projects
CB-2016/184096 and PAPIIT IN111021, respectively. AMM-A acknowledges financial
support from DGAPA-UNAM through the POSDOC program. GB was supported by
CONACYT under project CB2017-2018/AI-S-33920.

##  Appendix A Nearest-neighbor tight-binding approximation for benzene

In the TB model, the frontier electrons of atoms are assumed to be well
localized around the ions. The electronic wave functions of interacting atoms
are expanded in terms of the wave functions of isolated atoms, or individual
atomic orbitals. That is,

$\psi_{n}(\vec{r})=\sum_{m}c_{nm}\phi_{m}(\vec{r}),$ (4)

where $c_{nm}$ are the coefficients of the expansion in the base of atomic
orbitals $\\{\phi_{m}\\}$. This is in essence the method known as linear
combination of atomic orbitals, or LCAO.

For molecules like benzene formed by atoms of the same species, the nearest-
neighbor tight-binding model consists in taking equal energies, $\varepsilon$,
for each atomic site, and that the electrons can only “hop” between nearest
neighbors with a hopping energy $\nu$. Then, the TB model for benzene can be
written as CuevasBook

$H=\left(\begin{array}[]{cccccc}\varepsilon&-\nu&0&0&0&-\nu\\\
-\nu&\varepsilon&-\nu&0&0&0\\\ 0&-\nu&\varepsilon&-\nu&0&0\\\
0&0&-\nu&\varepsilon&-\nu&0\\\ 0&0&0&-\nu&\varepsilon&-\nu\\\
-\nu&0&0&0&-\nu&\varepsilon\end{array}\right),$ (5)

where we note that the matrix element $H_{16}=H_{61}$, which is due to the
cyclicity of the molecule. This tight-binding model for benzene is completely
equivalent to the so-called _Hückel approximation_ , which considers only the
atomic $\pi$ orbitals since the electrons in the $\sigma$ orbitals are
strongly bonded to the atom and they have very weak interactions to one
another.

The diagonalization of matrix (5) yields the eigenenergies

$\displaystyle E_{1}$ $\displaystyle=$ $\displaystyle\varepsilon-2\nu,$
$\displaystyle E_{2}=E_{2^{\prime}}$ $\displaystyle=$
$\displaystyle\varepsilon-\nu,$ $\displaystyle E_{3}=E_{3^{\prime}}$
$\displaystyle=$ $\displaystyle\varepsilon+\nu,$ $\displaystyle E_{4}$
$\displaystyle=$ $\displaystyle\varepsilon+2\nu,$ (6)

which consist of six levels: two singlets, $E_{1}$ and $E_{4}$, and two doubly
degenerate levels or doublets, $E_{2}=E_{2^{\prime}}$ and
$E_{3}=E_{3^{\prime}}$. These degeneracies are due to the ring structure of
the molecule as well as time-reversal symmetry Hernandez2020 . The energy
difference between singlets and doublets, and between doublet and doublet, is
$\Delta E=-\nu$ and $2\Delta E=-2\nu$, respectively. This structure of the
energy spectrum is a characteristic feature of benzene. A common way of
writing the eigenstates is CuevasBook

$\displaystyle|\varphi_{1}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{6}}(|\phi_{1}\rangle+|\phi_{2}\rangle+|\phi_{3}\rangle+|\phi_{4}\rangle+|\phi_{5}\rangle+|\phi_{6}\rangle),$
$\displaystyle|\varphi_{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{12}}(2|\phi_{1}\rangle+|\phi_{2}\rangle-|\phi_{3}\rangle-2|\phi_{4}\rangle-|\phi_{5}\rangle+|\phi_{6}\rangle),$
$\displaystyle|\varphi_{2^{\prime}}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}(|\phi_{2}\rangle+|\phi_{3}\rangle-|\phi_{5}\rangle-|\phi_{6}\rangle),$
$\displaystyle|\varphi_{3}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{12}}(2|\phi_{1}\rangle-|\phi_{2}\rangle-|\phi_{3}\rangle+2|\phi_{4}\rangle-|\phi_{5}\rangle-|\phi_{6}\rangle),$
$\displaystyle|\varphi_{3^{\prime}}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}(|\phi_{2}\rangle-|\phi_{3}\rangle+|\phi_{5}\rangle-|\phi_{6}\rangle),$
$\displaystyle|\varphi_{4}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{6}}(|\phi_{1}\rangle-|\phi_{2}\rangle+|\phi_{3}\rangle-|\phi_{4}\rangle+|\phi_{5}\rangle-|\phi_{6}\rangle).$
(7)

Each molecular orbital, $|\varphi_{j}\rangle$
($j=1,2,2^{\prime},3,3^{\prime},$ and 4), is a combination of the atomic
orbitals ($\pi$-orbitals), $|\phi_{n}\rangle$ ($n=1,2,\ldots,6$), and can be
expressed in configuration space, where a reasonable assumption is to consider
that the $\pi$ atomic orbitals decay exponentially out of its position, with
an evanescence length $\xi$. Under this assumption the atomic orbitals take
the form

$\phi_{j}(r)=A\,\mathrm{exp}\Big{(}-\frac{|x_{j}-x|}{\xi}-\frac{|y_{j}-y|}{\xi}\Big{)},$
(8)

where $(x_{j},y_{j})$ indicate the position coordinates of the atomic site $j$
and $A$ is a normalization constant. Considering the molecular structure of
benzene, with a distance between sites of $1.4$ Å, and only the $\pi$
orbitals, a reasonable estimate to the decay length leads to $\xi\approx
0.176$ Å Hernandez2020 .

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