Optoelectronic Properties of Rectangular Graphene Quantum Dots: PariserParrPople Model Based Computational Study
Abstract
In this paper, we perform largescale electroncorrelated calculations of optoelectronic properties of rectangular graphene quantum dots (RGQDs) containing up to 56 carbon atoms. We consider the influence of increasing the length of RGQDs both along the zigzag, as well as armchair, directions on their optical properties. Theoretical methodology employed in this work is based upon PariserParrPople (PPP) electron model Hamiltonian, which includes longrange electronelectron interactions. Electroncorrelation effects were incorporated using multireference singlesdoubles configurationinteraction (MRSDCI) method, and the ground and excited state wave functions thus obtained were employed to calculate the linear optical absorption spectra of the RGQDs, within the electricdipole approximation. Our results on optical absorption spectra are in very good agreement with the experimental ones wherever available, for the hydrocarbon molecules with the same carbonatom backbone structure as the RGQDs. In addition to the charge gap, spin gap of each quantum dot was also computed using the same methodology. Calculated spin gaps exhibit a decreasing trend with the increasing sizes of the RGQDs, suggesting that the infinite graphene has a vanishing spin gap. This result is consistent with the widespread belief that electroncorrelation effects in graphene are weak.
I Introduction
Despite many attractive properties, graphene has still not found applications in optoelectronic devices, because of the lack of a band gap. Therefore, in recent times, considerable amount of research effort has been directed towards graphene nanostructures such as quantum dots,^{1}; ^{2} and nanoribbons,^{3} which are expected to have band gaps because of quantum confinement. The idealized graphene nanoribbons (GNRs) have either zigzag or armchair edges, with substantially different electronic structure, and related properties. Theoretical studies reveal that zigzag GNRs (ZGNRs) exhibit edge magnetism with possible applications in spintronic devices,^{4}; ^{5} while the armchair GNRs (AGNRs) are direct bandgap semiconductors, with potential optoelectronic applications.^{6}; ^{7} If we consider either an AGNR or a ZGNR of a given width, and hypothetically cut it at two places perpendicular to its width the resultant rectangular structure, referred to as a rectangular graphene quantum dot (RGQD) will have both armchair and zigzag type edges. In this work we perform a computational study of optoelectronic properties of RGQDs, with the aim of understanding as to how they are influenced by the edge structure. Because the electronic properties of ZGNRs and AGNRs are very different from each other, it is of considerable interest as to how the electronic properties of RGQDs, which have both zigzag and armchair edges, evolve with the edge lengths. Such an understanding will help us in tuning the optoelectronic properties of RGQDs by manipulating their edges.
In our theoretical approach we consider graphene quantum dots to be systems whose lowlying excited states are determined exclusively by their electrons, with negligible influence of electrons. As a result we adopt a computational methodology based upon the PariserParrPople (PPP) electron Hamiltonian,^{8}; ^{9} and configuration interaction (CI) approach employed in several of our earlier works on conjugated polymers,^{10}; ^{11}; ^{12}; ^{13}; ^{14}; ^{15}; ^{16} and graphene quantum dots.^{17}; ^{18}; ^{19} We adopt this approach to study RGQDs with the number of carbon atoms ranging from 28 to 56, corresponding to structures with increasing edge lengths in both armchair, and zigzag, directions. We further assume that the dangling carbon bonds at the edges are passivated by hydrogen, thus preventing edge reconstruction, and allowing the RGQDs to retain their symmetric shapes. As a matter of fact, these Hpassivated RGQDs correspond to aromatic hydrocarbon molecules, for most of which experimental optical absorption data exists. Adopting the notation that RGQD denotes a rectangular graphene quantum dot with carbon atoms, the chemical analogs of RGQD28, 30, 36, 40, 42, 50, and 56 are aromatic compounds bisanthenes, terrylene, tetrabenzocoronene, quaterrylene, teranthene, pentarylene and quateranthene, respectively. On comparing our theoretical results to the measured ones on these molecules, we obtain excellent agreement, thus validating our methodology.
Additionally, using the same MRSDCI methodology, we computed the spin gap of each RGQD studied in this work. We find that with the increasing sizes of the RGQDs, their spin gaps are decreasing, suggesting that the spin gap of infinite graphene vanishes.
Remainder of the paper is organized as follows. In the next section we briefly outline the theoretical methodology, while in section III we present and discuss our results. Finally, in section IV we present summary and conclusions.
Ii structure and symmetry
The schematic diagrams of RGQDs considered in this work are shown in Fig. 1. We have assumed that the dangling bonds on the edges of the quantum dots considered have been saturated by hydrogen atoms. Thus, the quantum dots considered here can be treated as planar hydrocarbons, exhibiting conjugation. We have assumed that all the quantum dots lie in the plane, with idealized bond lengths of 1.40 Å, and bond angles of .
Within the PPP model based theoretical methodology adopted here, small variations in bond lengths and angles do not make any significant differences to the calculated optical properties of such structures, as demonstrated by us earlier.^{14}; ^{15} Having assumed the idealized geometry for the RGQDs, their point group symmetry is , as in case of polyacenes studied by us earlier.^{16}; ^{13}; ^{14}; ^{15} Because all the systems considered here have an even number of electrons, their ground state is of symmetry, so that their onephoton dipoleconnected excited states will be of the symmetries: (a) , accessible by absorbing an polarized photon, and (b) , reached through the absorption of a polarized photon.
Iii Theoretical Approach
As discussed in the previous section, with hydrogen passivated edges, the quantum dots considered here are conjugated systems, and, therefore, in this work we employed PariserParrPople (PPP) effective electron model Hamiltonian,^{8}; ^{9}
+  (1) 
where c are creation (annihilation) operators corresponding to the orbital of spin , located on the th carbon atom, while the total number of electrons on the atom is indicated by the corresponding number operator . The first term in Eq. 1 denotes the oneelectron hopping processes connecting th and th atoms, quantified by matrix elements . We assume that the hopping connects only the nearestneighbor carbon atoms, with the matrix element eV, consistent with our earlier calculations on conjugated polymers,^{10}; ^{11}; ^{12}; ^{13}; ^{14}; ^{15}; ^{16} polyaromatic hydrocarbons,^{20}; ^{21} and graphene quantum dots.^{18}; ^{19} The second and the third terms in Eq. 1 represent the electronelectron repulsion terms, with the parameters , and , representing the onsite, and the longrange Coulomb interactions, respectively. The distancedependence of parameters is assumed as per Ohno relationship^{22}
(2) 
where is the dielectric constant of the system, included to take into account the screening effects, and is the distance (in Å) between the th and th carbon atoms. In the present set of calculations we have used two sets of Coulomb parameters: (a) the “screened parameters”^{23} with eV, , and , and (b) the “standard parameters” with eV and .
We initiate the computations by performing restricted HartreeFock (RHF) calculations for the closedshell singlet ground states of the RGQDs considered here, by employing the PPP Hamiltonian (Eq. 1), using a computer program developed in our group.^{24} The molecular orbitals (MOs) obtained from the RHF calculations are used to transform the PPP Hamiltonian from the site representation, to the MO representation, for the purpose of performing manybody calculations using the CI approach. The correlatedelectron multireference singlesdoubles configuration interaction (MRSDCI) approach was employed in this work, which consists of a CI expansion obtained by exciting up to two electrons, from a chosen list of reference configurations, to the unoccupied MOs..^{25}; ^{26} The reference configurations included in the MRSDCI method depend upon the targeted states, which can be ground state or onephoton excited states. In all the CI calculations, only the configurations consistent with the spin and point group symmetries of the targeted states are included. Nevertheless, in cases of larger dots namely RGQD50, 54, and 56, we had to resort to the frozen orbital approximation to keep the CI expansion tractable. This consists of freezing a few lowest energy occupied MOs, and removing the corresponding chargeconjugation symmetric virtual MOs from the list, as described in our earlier works.^{14}; ^{15}; ^{18}
Once CI calculations are finished, the manybody wave functions obtained are used to compute the transition electric dipole matrix elements connecting onephoton excited states to the ground state. The transition dipole elements, along with the excitation energies of the excited states, are used to compute the optical absorption crosssection , according to the formula
(3) 
In the equation above, denotes frequency of the incident light, represents its polarization direction, is the position operator, denotes the fine structure constant, indices and represent, respectively, the ground and excited states, is the frequency difference between those states, and is the assumed universal line width. The summation over , in principle, is over an infinite number of states which are dipole connected to the ground state. However, in practice, the sum includes only those excited states whose excitation energies are within a certain cutoff, normally taken to be 8 eV.
Iv Results And Discussion
In order to assess the role played by electroncorrelation effects on various properties of RGQDs, it is important first to understand the independent particle results obtained using the tightbinding (TB) model. Therefore, in this section, we first present the result obtained by TB model, followed by those obtained by the PPPmodel.
iv.1 TightBinding Model Results
iv.1.1 Electron Density
Here, we present the electron density associated with the edge atoms, for the highest occupied molecular orbital (HOMO), and the lowest unoccupied molecular orbital (LUMO), for various GQDs, as a function of their size. The reason we choose only HOMO and LUMO is because these orbitals are involved in the first absorption peak of all the RGQDs, thus determining their lowenergy optical properties. The electron density corresponding to the edge atoms, for the th orbitals is defined as
where, denotes an atomic site, is the corresponding MO coefficient, and the “edge” refers either to the armchair edge, or the zigzag edge. Because of the electronhole symmetry in the nearestneighbor TBmodel, the magnitude of the MO coefficients in HOMO and LUMO is the same, so that , where superscripts imply HOMO(LUMO) orbitals. Thus, in the following we refer to a corresponding electron density as , and study it as a function of the size of RGQDs. In order to examine electron densities as a function of dimensions of RGQDs, we use two width parameters and , which are normally utilized for quantifying the widths of graphene nanoribbons.^{27} denotes the number of zigzag lines in the RGQD across the armchair edge, and thus quantifies the length of the armchair edge. Similarly, denotes the number of dimer lines across the zigzag edge, thus representing the length of the zigzag edge.^{27} Therefore, if we denote the dimensions of a given RGQD using the notation , RGQD40 (Fig. 1b) will be denoted as , and RGQD54 (Fig. 1h) can be represented as . Noteworthy point is that the product of and is nothing but the total number of carbon atoms in the quantum dot.
From the definitions of and it is obvious that, when the length of the quantum dot is increased in the zigzag direction holding its width fixed, it corresponds to a fixed value of , and increasing values of . Similarly, increasing length of the quantum dot in armchair direction, for the fixed width implies that is fixed, while is increasing. Behavior of , both for armchair and zigzag edges, for increasing lengths of RGQDs, for various widths is shown in 2. From Fig. 2(a) it is obvious that for a given width, with the increasing zigzag length, resides mostly on the zigzag edges, as compare to the armchair ones. From Fig. 2(b), for various widths and increasing lengths in the armchair directions, the following trends emerge: (a) The value of on armchair edges increases towards saturation for a given width, (b) the electron density on the zigzag edges decreases towards saturation for a given width, (c) for , the saturated value of the electron density of the zigzag edge is less than that on the armchair edge, however, the trend reverses for and . Thus, we conclude that for sufficiently wide RGQDs, the tendency of the electron density of the HOMOLUMO orbitals is to accumulate on the zigzag edges. Therefore, we conclude that impurity substitution on the zigzag edges can be efficiently use to tune the optoelectronic properties of RGQDs .
iv.1.2 Energy Levels
In this section, we present and discuss the behavior of the energy levels and the optical gap, with the increasing sizes of RGQDs. In Fig. 3, we have presented the energy level diagrams for RGQDs of varying sizes. The energy level diagrams are arranged in such a manner that if we move from left to right increases, while is fixed. On examining these, we observe the following trends: (a) The electron and hole levels are mirror symmetric with respect to the Fermi level (), as a consequence of the electronhole symmetry of the nearest neighbor tightbinding model, (b) the lack of any energy level at for any of the RGQDs is consistent with the vanishing density of states at the Fermi level, for infinite graphene.
iv.1.3 Linear Optical Absorption Spectra and Optical Gaps
In Fig. 4, we have presented optical absorption spectra of RGQDs of varying sizes computed using the TB model. The following conclusions can be drawn from this graph:

The first peak in the absorption spectra for all the RGQDs is polarized, and corresponds to excitation of an electron from HOMO to LUMO, leading to the excited state . Therefore, this peak corresponds to the optical gap of the concerned RGQDs, and it is the most intense peak in the spectra.

The intensity of the first peak in the spectra increases significantly with the increasing size of RGQDs.

We also note that all polarized peaks are nondegenerate, and correspond to excited states of symmetry , as per the selection rules of point group. All polarized peaks are doubly degenerate, as a consequence of electronhole symmetry of the nearest neighbor TB model, and correspond to excited states of symmetry. For example, in RGQD30, first peak is polarized, and it is due to nondegenerate excitation , while the second peak is polarized, and is due to doubly degenerate excitations and . As mentioned earlier, notations imply HOMO/LUMO. Similarly, notations ( imply th orbital below HOMO (th orbital above LUMO).

With the increasing length of the RGQDs along a given orientation (zigzag or armchair), the optical gap decreases.
Next, we investigate the nature of optical gap of RGQDs as a function of their size. The optical gaps for various widths, and increasing lengths of RGQDs are presented in Fig. 5, from which it is obvious that for a given width, band gaps of all the RGQDs decrease towards zero, with the increasing lengths. The rate of decrease of the gap with increasing length is faster for ribbons of the larger widths. It is also obvious that for a given length, the gaps decrease with the increasing widths.
In Table 1, we compare the HOMOLUMO gap obtained at the tightbinding level, width the experimental results, wherever available. For the sake of comparison, we also present the values of optical gaps obtained using the PPPCI approach to be discussed in the next section. From the table it is obvious that the gaps obtained using the TB model are much smaller than the experimental values. On the other hand, the PPPCI values of the optical gaps, are generally in much better agreement with the experiments. Therefore, it is obvious that the TB model cannot provide good quantitative agreement with the experiments, because it ignores the electroncorrelation effects.
System  optical gap (eV)  
TB Model  PPPCI  Experimental  
SCR  STD  
RGQD28  0.85  2.00  2.21  1.80^{28}, 2.02^{29}, 2.15^{30} 
RGQD30  1.16  2.11  2.43  2.14^{31}, 2.21^{32}; ^{33}, 2.22^{34}, 
2.35^{35}, 2.36^{36}; ^{37}  
RGQD36  0.45  2.11  2.30   
RGQD40  0.89  2.02  2.30  1.84^{38}, 1.87^{34}; ^{32}, 1.91^{39}, 
2.03^{36}, 2.04^{35}; ^{37}  
RGQD42  0.44  1.86  2.04   
RGQD50  0.72  1.72  1.98  1.66^{32}; ^{34} 
RGQD54  0.17  1.63  2.09   
RGQD56  0.24  1.50  1.91  1.35^{40} 
iv.2 PPP Model Based CI Results
In this section, we present our results obtained from the PPPmodel based CI calculations. First, we present the result on the spin gaps of different RGQDs, followed by their linear optical absorption spectra.
iv.2.1 Spin Gaps
Spin gap of an electronic system is the energy difference between the lowest triplet and singlet states. For RGQDs, the lowest singlet state is ground state, while the lowest triplet state is the state, whose spatial part of the wave function consists predominantly of the single excitation , just as in the case of state. Thus, at the TB level, and will be degenerate, and, therefore their spin and optical gaps will be identical. However, if the two gaps are found to be different for RGQDs, it can only be due to electronelectron interactions. Therefore, difference in the spin and optical gaps is a measure of the electron correlation effects in RGQDs. With this in mind, we computed the spin gaps of all RGQDs studied in this work, using our PPP model based MRSDCI approach. Because, to compute the spin gaps we needed energies of and states which are in different symmetry manifolds, we managed to perform very largescale MRSDCI calculations, as is evident from Table 2. Therefore, we believe that our results on spin gaps of RGQDs presented in Table 3 are fairly accurate. From the inspection of Table 3 it is obvious that spin gap of RGQDs is decreasing with their increasing sizes. Thus, this result suggests that the spin gap of infinite graphene is zero, consistent with the widespread assumption that graphene is a weaklycorrelated material. It will also be interesting to compare the experimental values of the spingaps of individual hydrocarbon molecules corresponding to these RGQDs, with our calculated values. Therefore, we urge the experimentalists to measure the spin gaps of hydrogen passivated counterparts of various RGQDs studied in this work.
MRSDCI method with screened parameters.  

MRSDCI method with standard parameters. 
System  (eV)  

Scr  Std  
RGQD30  1.11  1.30 
RGQD40  0.97  1.16 
RGQD50  0.79  0.94 
RGQD28  0.76  0.75 
RGQD42  0.40  0.36 
RGQD56  0.15  0.11 
RGQD36  0.37  0.34 
RGQD54  0.13  0.07 
RGQD72  0.05  0.02 
iv.2.2 Linear optical absorption spectrum
In this section, we present optical absorption spectra of RGQD, with ranging from 30 to 56, computed using the PPP model, and the MRSDCI approach. Before discussing the results of our calculations, in Table 4 we give the sizes of the CI matrices, for different symmetry spaces of various RGQDs. The fact that the sizes of the CI matrices were in the range , implies that these calculations were indeed large scale, and, therefore, should be fairly accurate. For RGQD, to , all the MO’s of the systems were included in the MRSDCI calculations. However, from to , it was no longer possible to perform accurate calculations with all the orbitals. Therefore, for these cases, some lowlying occupied MO’s, were frozen, while their unoccupied counterparts were deleted.
n  
MRSDCI method with screened parameters.  
MRSDCI method with standard parameters. 
The calculated spectra of these RGQDs are presented in Fig. 6, while the important information regarding the excited states contributing to various peaks in the spectra are presented in Tables IXVI of Supporting Information.
Before discussing the spectra of individual RGQDs, we discuss the general trends observed in our calculation:

For each RGQD, the absorption spectrum obtained using the PPPCI approach is blueshifted in comparison to the TB model.

For all the RGQDs, absorption spectra obtained using the screened parameters are redshifted compared to those obtained using the standard parameters.

In all cases, the first peak of the spectrum is due to optical excitation from the ground state to excited state, and corresponds to the optical gap. As per electric dipole selection rules, this peak is polarized.

For all the RGQDs, the first peak is not the most intense peak. For a number of RGQDs, several high energy peaks are more intense than the first one. This result is in sharp contrast with the TB model results.

Dominant configurations in the wave functions of the excited states corresponding to the lower energy peaks are single excitations, while those in the higher energy peaks are dominated by double and higher excitations.
A comparison of calculated peak positions with the experimentally measured values, and the theoretical results of other authors, for RGQD28, RGQD30, RGQD36, RGQD40, RGQD42, RGQD50 and RGQD56 is presented in Tables 5, 6, 7, 8, 9, 10, and 12, respectively. For RGQD54, we could not locate any previous experimental or theoretical data.
Rgqd28
Clar and Schmidt^{29}, Arabei ^{30}, and Konishi ^{28} have reported the measurements of the absorption spectrum of bisanthene, and its derivatives, the structural analogs of RGQD. In Fig. 6 (a) and (b), we present our calculated spectra using the screened and standard parameters, respectively, within the PPPCI approach. If we compare the relative intensity of the first peak of the experimental spectra, we find that results of Arabei .^{30} are in perfect agreement with our results in that the first peak is not the most intense. However, Konishi ^{28} report that the first peak is the most intense one, in complete disagreement with our results. Our calculated location of the first peak corresponding to the optical gap, was found to be eV with the screened parameters, and eV for the standard parameters. As is obvious from Table 5, the experimental values of the optical gap range from eV to eV. Thus, we find that both our screened and standard parameter of optical gap are quite close to the range of experimental values. We also note that our screened parameter of eV is in almost perfect agreement with the value of optical gap eV, measured by Clar and Schmidt^{29}. As far as higher energy peaks are concerned, Arabei report a peak at eV for which there are no counterparts in our calculated spectra. Konishi .^{28} measured a peak at eV, while Arabei ^{30} measured one at eV. Comparing our calculations to these, we find there is no matching peak at eV, however, our screened parameters spectrum has a peak at eV, in perfect agreement with Arabei ^{30}. Next experimental peak located at 4.05 eV, reported by Konishi ^{28}, is in good agreement with our standard parameter peak computed at 4.14 eV, while the corresponding screened parameter candidate at 4.19 eV is somewhat higher. The highest measured peak located at 4.80 eV, reported by Arabei ^{30} is in reasonable agreement with the standard parameter value of 4.68 eV, while screened parameter value of 4.49 eV is significantly smaller. Furthermore, we have computed several higher energy peaks as well, for which no experimental results exist. We hope that in future measurements of the absorption spectrum of bisanthenes, energy range of 5 eV, and beyond, will be explored.
On comparing our results to calculations by other authors, we find that the TDDFT value of the optical gap computed by Malloci ^{41} is significantly smaller than our results, as well as experiments. However, Parac and Grimme ^{42} report a TDDFT value of the optical gap at 1.78 eV, which is in very good agreement with the lowest measured experiment value of the optical gap, 1.80 eV.^{28} They also report a TDPPP value of the optical gap which is in very good agreement with our screened parameter result, as well as other experimental results. As far as higher energy peaks computed by Malloci are concerned, our PPP model values are generally in good agreement with them.
The manyparticle wave function of the state corresponding to the optical gap is dominated by the single excitation , where and , respectively, denote the HOMO and the LUMO of the system. Peak VI is the most intense peak in the absorption spectra computed using both the screened as well as the standard parameters. The most intense peak computed using the screened parameters located at 5.63 eV, corresponds to a state with symmetry, whose wave function is dominated by excitations, where denotes the charge conjugated configration. However, the standard parameter calculations predict the most intense intense peak to be due to a state, located at 5.95 eV, along with a small mixture of a state located at 6.17 eV, with their wave functions dominated by single excitations , and , respectively. The detailed wave function analysis of all the excited states contributing to various peaks in the calculated spectra of RGQD28, is presented in Tables I and II of the Supporting Information.
Experiments  Theory (others)  This work  
Screened  Standard  
1.80^{28},2.02^{29},2.15^{30},  1.47^{41}  2.00  2.21 
1.98^{28},2.43^{30},  1.78^{42}, 1.98^{42}  
  2.83^{42}, 2.96^{42}     
3.64^{28},3.87^{30}  3.76^{41}  3.87   
4.05^{28}    4.19  4.14 
4.80^{30}  4.47^{41}  4.49  4.68 
  5.39^{41}  5.22  5.09 
    5.63  5.41 
    6.05  6.06 
  6.35^{41}  6.41   
  7.09^{41}  7.03  6.80 
TDDFT method, TDPPP method 
Rgqd30
Koch ^{34}, Ruiterkamp ^{35} and Halasinski ^{37} have reported the measurements of the absorption spectrum of terrylene, the structural analog of RGQD, and its derivatives. However, Clar ^{36}, Kummer ^{31}, Biktchantaev ^{33} and Baumgarten ^{32} reported only the optical gap of terrylene. In Fig. 6 (a) and (b), we present our calculated spectra using the screened and standard parameters, respectively, within the PPPCI approach. If we compare the relative intensity of the first peak of the experimental spectra, we find that the results of Koch ^{34}, Ruiterkamp ^{35} and Halasinski ^{37} are in perfect agreement with our results in that the first peak is not the most intense. The calculated location of the first peak of the absorption spectrum, which defines the optical gap, was found to be eV, and eV, from our standard, and screened parameter based calculations, respectively. As it is obvious from Table 6, that the experimental values of the optical gap range from eV to eV. Thus, we find that both our screened and standard parameter of optical gap are quite close to the range of experimental values. We also note that our screened parameter value of eV is in almost perfect agreement with the value of optical gap eV, measured by Kummer ^{31}. As far as higher energy peaks are concerned, we have a peak at 4.07 eV from our screened parameter calculations, for which there are no available counterparts in the experiments, or the theoretical works of other authors. Halasinski ^{37} measured a peak at 4.33 eV, close to which our calculations predict no peak. However, Koch ^{34} measured a peak at 4.47 eV, which is in good agreement with a peak at 4.58 eV, predicted by our screened parameter calculations. Next experimental peak located at 4.71 eV, reported by Halasinski ^{37} and Ruiterkamp ^{35} , is in good agreement with our standard parameter peak computed at 4.75 eV. Koch ^{34}, Ruiterkamp ^{35} and Halasinski ^{37} have also reported several higher peaks which are in reasonably good agreement with our computed screened and standard results. Furthermore, we have computed several higher energy peaks as well, for which no experimental results exist. We hope that in future measurements of the absorption spectrum of terrylene, energy range of 7 eV and beyond will be explored.
On comparing our results to the theoretical works by other authors, we find that the value of the optical gap reported by Malloci ^{43}, Minami ^{44} and Karabunarliev ^{45} are slightly larger than our screened parameter results. On the other hand, the gaps reported by Halasinski ^{37} and Viruela ^{46} are slightly smaller than our screened parameter results. As far as higher energy peaks computed by Malloci ^{43} are concerned, our PPP model values are generally in good agreement with them.
The manyparticle wave function of the state corresponding to the optical gap is dominated, as expected, by the single excitation . In the spectra computed using the screened parameters, IV peak is most intense, and it is due to a state, located at 5.12 eV, whose wave function is dominated by the excitation. For the standard parameter calculations, peak V is the most intense one, due to a state located at 6.01 eV, along with a small mixture of state located at 5.87 eV, with wave functions dominated by configurations , and , respectively. The detailed wave analysis of all the excited states contributing peaks in the computed spectra is presented in Tables III and IV of the Supporting Information.
Experiments  Theory (others)  This work  

Screened  Standard  
2.14^{31}, 2.21^{32}; ^{33},  2.02^{37}, 2.03^{46},  2.11  2.43 
2.22^{34}, 2.35^{35},  2.21/2.22^{43}, 2.29^{44},  
2.36^{36}; ^{37}, 2.39^{34},  2.52^{45}, 2.98^{37},  
2.57^{34}, 2.76^{34},  3.31^{37}, 3.40^{37},  
  3.84^{37},  
    4.07   
4.33^{37}, 4.47^{34}    4.58  4.64 
4.71^{37}; ^{35}  4.7^{43}    4.75 
    5.12  5.03 
5.20^{37}, 5.27^{35},    5.35  5.53 
5.41^{37}, 5.48^{34}    
    5.95  5.87 
6.10^{34}, 6.19^{37},  6^{43}  6.08  6.01 
6.44^{37},    6.47  6.24 
6.69^{37},  6.8^{43}  6.86  6.76 
TDDFT method, DFT(KohanSham) method 
Rgqd36
The hydrogen passivated structural analog of RGQD36 is tetrabenzocoronene, for which we were unable to locate any experimentally measured optical absorption spectrum. Therefore, we can only compare our calculations to the theoretical works of other authors, for which also we could find just one TDDFT based computation of the optical absorption spectra by Malloci ^{47}. In Fig. 6 (a) and (b), we present our calculated spectra using the screened and standard parameters, respectively, within the PPPCI approach. If we compare the relative intensity of the first peak in the spectra, we find that the results of Malloci ^{47} are in perfect agreement with our results in that the first peak is not the most intense. In Table 7, we have compared the locations of various peaks reported by Malloci with our computed results. We find that the value of optical gap reported by Malloci ^{43} is 0.95 eV, which significantly smaller than our computed results of 2.11 eV (screened) and 2.30 eV (standard). Given such severe disagreement between two theoretical calculations, it will be really useful if an experiment is performed on this molecule, or another theoretical calculation is done. Given the fact that our results on optical gaps on smaller RGQDs were in excellent agreement with the experiments, we speculate that the TDDFT calculation of Malloci ^{47} has significantly underestimated the optical gap of RGQD36. As far as higher peaks are concerned Malloci ^{47} have report a peak at 3.16 eV, for which there is no counterparts is available in our computed spectra. Our next peak using screened parameter is located at 3.63 eV, which is in perfect agreement with the value 3.64 reported by Malloci ^{47}. As far as higher energy peaks computed by Malloci ^{47} are concerned, our PPP model values are generally in good agreement with them.
The manyparticle wave function of the state corresponding to the optical gap is dominated by the single excitation , similar to the case of smaller RGQDs. For both the screened as well as the standard parameters computed absorption spectra, peak VIII is the most intense one. In the spectra computed using the screened parameters, the most intense peak is located at 5.99 eV, and is due to states of symmetries and contributing almost equally to the oscillator strength, with their manyparticle wave functions dominated by configurations , and , respectively. In the spectrum computed using the standard parameters, the most intense peak is located at 6.57 eV, due to a state, with wave function dominated by excitations. The detailed wave analysis of excited states contributing peaks in the spectra computed by the screened, and the standard parameters is presented in Tables V and VI of the Supporting Information.
Theory (others)^{47}  This work  

Screened  Standard  
0.95     
  2.11  2.30 
3.16     
3.64  3.63   
  3.77  3.87 
  4.01   
  4.35  4.41 
4.83  4.94  4.88 
    5.09 
  5.65  5.60 
  5.99  5.86 
6.22  6.50  6.57 
Rgqd40
Ruiterkamp ^{35}, Koch ^{34}, and Halasinski ^{37} have reported the measurements of the absorption spectrum of quaterrylene, the hydrogen passivated structural analog of RGQD, and its derivatives. However, Clar ^{36}, Former ^{38}, Gudipati ^{39}, and Baumgarten ^{32} reported only the optical gap of quaterrylene. In Fig. 6 (a) and (b), we present our calculated spectra using the screened and the standard parameters, respectively, within the PPPCI approach. If we compare the relative intensity of the first peak of the experimental spectra, we find that results of Ruiterkamp ^{35} and Halasinski ^{37} are in perfect agreement with our results in that the first peak is not the most intense. However, Koch ^{34} report that the first peak is the most intense one, in disagreement with our results, as well those of other experimentalists. In Table 8, we present the locations of important peaks obtained from our calculations, and compare them to the experimental results, as well as theoretical calculations of other authors. As is obvious from the table, the experimental values of the optical gap range from eV to eV, implying that our screened parameter result of optical gap (2.02 eV) is quite close to the range of experimental values, while the optical gap obtained using the standard parameters (2.30 eV) is somewhat larger. We also note that our screened parameter value of the optical gap, eV, is in almost perfect agreement with 2.03 eV, the value of the optical measured by Clar and Schmidt^{29}. As far as the higher energy peaks are concerned, Koch .^{34} report peaks at eV and 3.85 eV, Halasinski ^{37} report a peak at 3.78 eV, while Ruiterkamp ^{35} report one at eV. Although, our calculated spectra have no peaks precisely at these locations, however, the screened parameters spectrum has a peak at 4.06 eV, in reasonable agreement with Ruiterkamp ^{35}, and the higher energy peak of Koch .^{34}. After that the experimental peaks located at 4.71 eV and 4.83 eV, reported by Halasinski ^{37}, are in good agreement with our standard parameter peak computed at 4.85 eV. The next higher measured peaks reported by Koch ^{34} at 5.27 eV is in good agreement with our screenedparameter based peak at 5.16 eV. Halasinski ^{37} report a peak at 5.39 eV, which is in good agreement with the peak at 5.48 eV, predicted by standard parameter calculations. Furthermore, Halasinski ^{37} have measured four more peaks in the range 5.82 — 6.63 eV, each of which is in good agreement with our calculated peaks (see Table 8).
On comparing our results to the calculations by other authors, we find that values of optical gap reported by Viruela ^{46} and Halasinski ^{37} are significantly smaller than our results as well as experiments. However, TDDFT calculations of Malloci ^{43}, Gudipati ^{39} and Minami ^{44}, and PM3/CI results of Karabunarliev et al. ^{45} are in reasonably good agreement with the experiment, and our calculated values, of the optical gaps. As far as higher energy peaks computed by Malloci ^{43} are concerned, our PPP model values are generally in good agreement with them.
The manyparticle wave function of the state corresponding to the optical gap is, quite expectedly, dominated by the single excitation . In the spectra computed using the screened parameters, peak IV is most intense, and it is due to a state of symmetry, located at 5.16 eV, along with a small contribution from a state, located at 5.17 eV. The wave functions of these states are dominated by single excitations , and , respectively. For the standard parameter calculations, peak VI is the most intense one, corresponding again to a mixture of a state (at 6.23 eV), and a state (at 6.17 eV), with wave functions dominated by excitations and , respectively. The detailed wave function analysis of the excited states contributing to various peaks in the spectra calculated by the screened and standard parameters, respectively, is presented in Tables VII and VIII the Supporting Information.
Experiment  Theory (others)  This work  

Screened  Standard  
1.84^{38}, 1.87^{32}; ^{34},  1.65^{46},1.67^{37},  2.02  2.30 
1.91^{39}, 1.99^{38},  1.79/1.83^{43},  
2.03^{36}, 2.04^{35}; ^{37},  1.87^{39}, 1.88^{44},  
2.18^{45}, 2.97^{37},  
3.13^{37}, 3.25^{37},  
3.40^{37},  
3.71^{34}, 3.78 ^{37},  3.60^{43}  4.06   
3.85^{34}, 3.86^{35},  
  4.40^{43}  4.53  4.36 
4.71^{37}, 4.83 ^{37}      4.85 
5.27^{34}, 5.39 ^{37}  5.30^{43}  5.16  5.48 
5.82 ^{37}    5.88  5.85 
6.32 ^{37}  6.00^{43}  6.33  6.19 
6.50 ^{37}      6.45 
6.63 ^{37}  6.60 ^{43}  6.85  6.73 
DFT(KohanSham) method, TDDFT method 
Rgqd42
Teranthene is the hydrogensaturated structural analogue of RGQD42, for which no experimental, or theoretical data is available, as far as optical absorption spectrum is concerned. However, Konishi ^{40} have reported the measurement of the absorption spectrum of teranthene with tertiarybutyl group attached on its edge atoms, and the results of their experiments, along with those obtained from our calculations, are summarized in Table 9. In Fig. 6 (a) and (b), we present our calculated spectra of RGQD42 using the screened, and the standard parameters, respectively, within the PPPCI approach. If we compare the relative intensity of the first peak of the experimental spectra, we find that results of Konishi ^{40} are in perfect agreement with our results in that the first peak is not the most intense. However, as is obvious from Table 9, quantitatively speaking, our theoretical results and experimental results of Konishi ^{40} disagree completely in the lowenergy region. Konishi ^{40} have reported two lowlying excited energy peaks located at 1.17 eV, and 1.21 eV, for which there are no counterparts in our computed spectra. The calculated location of the first peak, which also corresponds to the optical gap, was found to be eV from our screened parameter calculation, and eV for the standard parameter calculation, as against significantly smaller values 1.17–1.21 eV measured by Konishi ^{40}. As far as higher energy peaks are concerned, Konishi ^{40} report peaks at 2.96 eV and 3.19 eV, for which again our calculations have no counterparts. The highest measured peak by Konishi ^{40} is located at 3.87 eV, which is in good agreement both with a screened parameter peak at 3.96 eV, and a standard parameter peak at 3.80 eV. Our calculations predict several peaks in energy region of 4 eV and beyond, which Konishi et al.^{40} have not probed. The only possible reason we can think of behind the disagreement between the theory and the experiments in the lower energy region is that the experiments were performed on teranthene saturated with tbutyl group, while our results are valid for hydrogensaturated material. Nevertheless, we hope that more groups will perform measurements of optical absorption spectra of teranthenes so as to be sure about the value of their optical gap.
Quite expectedly, the manyparticle wave function of the state corresponding to the optical gap, obtained from both the standard and the screened parameter calculations, is dominated by the single excitation . In the spectrum computed using the screened parameters, peak III is most intense, corresponding to a state located at 3.96, whose wave function is dominated by single excitations. For the standard parameter calculation, peak VII is most intense, corresponding again to a state, but located at 5.47 eV, with a small mixture of state located at 5.33 eV. The wave functions of the two states are dominated by single excitations, , and , respectively. The detailed wave analysis of all the excited states contributing to peaks in the calculated spectra using the screened and the standard paremeters, is presented in Tables IX and X, respectively, the Supporting Information..
Experiment^{40}  This work  

Screened  Standard  
1.17,1.21,     
1.41, 1.57  1.86  2.04 
2.96     
3.19  3.56   
3.87  3.96  3.80 
    4.02 
  4.15  4.21 
  4.53  4.52 
Rgqd50
For RGQD50 and larger structures, it would have been computationally very tedious to perform MRSDCI calculations retaining all the MOs, therefore, we decided to freeze a few lowestlying occupied orbitals, and delete their electronhole symmetric highest virtual orbitals. For the case of RGQG50, we froze/deleted four occupied/virtual orbitals, so that the total number of MOs involved in the calculations reduced to fortytwo, same as in case of RGQD42. In order to demonstrate that this act of freezing and deleting the MOs does not affect the calculated optical absorption spectra, we have performed the calculations for RGQD50, with four and seven frozen/deleted orbitals, leading to 42/36 active MOs. From the calculated absorption spectra presented in Fig. 7, it is obvious that except for the intensity of the highest energy peak IX in the standard parameter calculations, the spectra remain the same for both the cases, implying that the convergence has been achieved within an acceptable tolerance.
Hydrogen saturated structural analog of RGQD50 is pentarylene, for which, to the best of our knowledge, no experimental data of optical absorption exists. However, Koch ^{34} have reported the measurements of the absorption spectrum of pentarylene saturated with the tbutyl group, while Baumgarten ^{32} measured only its optical gap. We present our calculated absorption spectra for RGQD50 in Fig. 6 (a) and (b), while the comparison of important peak locations resulting from our calculations, with the experiments, and other theoretical works is presented in Table 10. The value of the optical gap measured by both the groups^{34}; ^{32} is 1.66 eV, which is in very good agreement with the value 1.72 eV computed using the screened parameters, while the corresponding standard parameter value 1.98 eV is on the higher side. If we compare the relative intensity of the first peak of the experimental spectra corresponding to the optical gap, we find that Koch ^{34} report that the first peak to be the most intense one, in disagreement with our results. However, the noteworthy point is that the first peak computed using the screened parameter, is quite intense, and is only somewhat lesser in intensity than the most intense peak (peak V) of the computed spectrum. As far as higher energy peaks are concerned, Koch .^{34} report peaks at 3.28 eV and 3.45 eV, which are close to a screened parameter peak calculated at 3.39 eV. Next peak reported by Koch .^{34} is at 4.62 eV, which is in good agreement with our standard parameter peak located at 4.71 eV. After that the experimental peak located at 4.80 eV^{34}, is somewhat close to our screened parameter peak located at 4.97 eV. The next higher measured peak reported by Koch ^{34} is located at 5.29 eV, is in very good agreement with the standard parameter value of 5.34 eV, while screened parameter value of 5.41 eV is slightly larger. Furthermore, we have computed several higher energy peaks as well, for which no experimental results exist. We hope that in future measurements of the absorption spectrum of pentarylene, energy range beyond 5.3 eV will be explored.
On comparing our results to the calculations by other authors, we find that the value of the optical gaps reported by ViruelaMartín ^{46} (1.40 eV) using the valenceeffective Hamiltonian approach, Malloci ^{43} (1.54 eV) using the TDDFT method are significantly smaller than our results, as well as experiments. However, Minami ^{44} report a TDDFT value which is in good agreement with the experiment value of the optical gap, but about 0.1 eV lower than our result. Karabunarliev ^{45} computed the optical gap to be 1.97 eV using PM3 semiempirical method, is in perfect agreement with our standard parameter result located at 1.98 eV, but significantly higher than the experimental value, as well as our screened parameter value. As far as higher energy peaks computed by Malloci are concerned, our PPP model values are in reasonable agreement with them.
The manyparticle wave function of the state corresponding to the optical gap is dominated by the single excitation . In the spectra computed using the screened parameter, peak V is most intense, and is due to a state located at 5.04 eV, along with a small intensity due to a state located at 4.90 eV. The wave functions of the two states are dominated by configurations , and excitations, respectively. In the standard parameter spectrum, peak VIII is most intense, and is mainly due to a state located at 6.38 eV, along with a small contribution of a state located at 6.45 eV. The dominant contributions to the manyparticle wave functions of these two states are from single excitation , and the double excitation , respectively. The detailed wave function analysis of all the excited states contributing peaks in the spectra computed using the screened and the standard parameters, are presented in Tables XI, and XII, respectively, the Supporting Information.
Experiments  Theory (others)  This work  
Screened  Standard  
1.66^{32}; ^{34}  1.40^{46}, 1.51/1.54^{43} ,  1.72  1.98 
1.60^{44},  
1.97^{45}  
3.28^{34}, 3.45^{34}    3.39   
  4.0^{43}  3.91  3.84 
    4.21  
4.62^{34}  4.5^{43}    4.71 
4.80^{34}  5.2^{43}  4.97  5.12 
5.29^{34}    5.41  5.34 
    5.73  5.62 
  6.1^{43}  5.95  5.99 
    6.23  6.41 
  7.4^{43}    6.96 
DFT(KohanSham) method, TDDFT method 
Rgqd54
For the case of RGQD54, we performed MRSDCI calculations after freezing/deleting six occupied/virtual MOs, i.e., with forty two active MOs. For this molecule, we were not able to locate any experimental results, or other theoretical calculations, thus, making our results the first ones. We hope that our calculations will give rise to future theoretical and experimental works on this system.
Calculated optical absorption spectra for RGQD54 are presented in Figs. 6 (a) and (b), obtained using the screened and standard parameters, respectively. The locations of important peaks, and the symmetries of excited states giving rise to them, are summarized in Table 11. The first peak corresponding to the optical gap, is a very weak peek from both sets of calculations, and was found to be at 1.63 eV with the screened parameters, and 2.09 eV with the standard parameters. Given the pattern observed for smaller RGQDs discussed in the previous sections, we expect the screened parameter value of the optical gap to be closer to the experimental value. For both sets of calculations, the optical gap corresponded to a transition to the excited state , whose wave function is dominated by the singlyexcited configuration .
In the screened parameter calculations, next we find a group of three wellseparated peaks, with strong, and almost equal, intensities, located at located at 2.56 eV, 2.83 eV, and 3.09 eV. The first of these peaks corresponds to an polarized transition, while the next two are polarized. In the standard parameter spectrum as well, the next three peaks are quite strong, and well separated, but they have their intensities in the ascending order, while the middle peak (peak III) appears as a shoulder of peak IV. The locations of these peaks are blueshifted compared to their screened parameter counterparts, and are 3.20 eV, 3.69 eV, and 3.98 eV. The polarization characteristics are also different, with two of the peaks exhibiting mixed polarization.
At higher energies, in the screened parameter spectrum there are well defined highintensity peaks at energies 3.71 eV ( polarized), 3.95 eV ( polarized), 5.14 eV ( polarized), and 5.40 eV ( polarized). Out of these the peak located at 5.14 eV (peak IX) is the most intense peak of the computed spectrum. This peak is due to a state, whose wave function is dominated by the singlyexcited configuration.
In the standard parameter spectrum, beyond 4 eV, there are a number of lowintensity peaks or shoulders, except for a peak located at 6.56 eV (peak XII), which is the most intense one, and exhibits mixed polarization. This peak is due to a state located at 6.51 eV, along with a smaller contribution to the intensity from a state located at 6.61 eV. The wave functions of the two states are dominated by singly excited configurations, , and , respectively.
The detailed wave function analysis of all the excited states contributing peaks in the spectra computed using the standard parameters, and the screened parameters, are presented in Tables XIII, and XIV, respectively, the Supporting Information.
This work  

Screened  Standard 
1.63  2.09 
2.56   
2.83   
3.09  3.20 
3.71  3.69 
3.95  3.98 
4.15  4.22 
4.31  4.60 
  4.97 
5.14  5.14 
5.40  5.41 
5.60   
5.82  5.97 
  6.22 
  6.56 
Rgqd56
Again, due to a large number of electrons in the system, for RGQG56 we froze/deleted seven occupied/virtual orbitals, so that the total number of active MOs involved in the calculations reduced to fortytwo, same as in case of RGQD42, RGQD50, and RGQD54.
The hydrogen saturated analog of RGQD56 is quateranthene, for which no experimental measurements of optical absorption spectrum exist. However, Konishi ^{40} measured the absorption spectrum of quateranthene, with tbutyl groups attached to its edge carbon atoms, with which we will compare our calculated spectra. In Figs. 6 (a) and (b), we present our calculated spectra using the screened and standard parameters, respectively, within the PPPMRSDCI approach. In Table 12, we present the locations of various peaks in the calculated spectra, and compare them to the measured values of Konishi ^{40}. If we compare the relative intensity of the first peak of the experimental spectra, we find that the results of Konishi ^{40} are in perfect agreement with ours in that the first peak is not the most intense. The calculated location of the first peak, which also corresponds to the optical gap, from our calculations was found to be 1.50 eV with the screened parameters, and 1.91 eV with the standard parameters. The experimental value of optical gap reported by Konishi ^{40} is 1.35 eV, which is about 0.15 eV lower than our screened parameter value, but significantly smaller than the value obtained from the standard parameter calculations. As far as higher energy peaks are concerned, Konishi ^{40} report several peaks in the range 2.012.32 eV, while our calculations do not predict any peaks in that energy region. Next experimental peak located at 3.21 eV, reported by Konishi ^{40}, is in good agreement with our standard parameter peak computed at 3.35 eV. The highest measured peak reported by Konishi ^{40} is located at 3.46 eV, and it is in reasonably good agreement with a screened parameter peak at 3.61 eV. Furthermore, we have computed several higher energy peaks as well, for which no experimental results exist. We hope that in future measurements of the absorption spectrum of quateranthene, energy range beyond 3.50 eV will be explored.
The manyparticle wave function of the state corresponding to the optical gap is, as in all previous cases, dominated by the single excitation . In the spectra computed using screened parameter, peak III is the most intense one, and it is due to a state located at 2.76 eV, with a smaller contribution to the intensity from a state located at 2.82 eV. The wave functions of the two states are dominated by doublyexcited configurations , and , respectively. In the standard parameter spectrum, peak V is most intense, and is entirely due to a state located at 4.34 eV, whose wave function derives most important contribution from the singlyexcited configuration . The detailed wave function analysis of the excited states contributing to varioius peaks in the spectra computed using the screened and standard parameters is presented in Tables XV, and XVI, respectively, the Supporting Information.
Experiment^{40}  This work  

Screened  Standard  
1.35, 2.01, 2.10,  1.50  1.91 
2.20, 2.27, 2.32     
  2.79   
3.21    3.35 
3.46  3.61   
  3.92  3.87 
V Conclusions
In this paper, we have presented the results of our very largescale correlatedelectron calculations of spin gaps and optical absorption spectra of rectangular graphene quantum dots, with the number of carbon atoms in the range 28–56, using PPP model Hamiltonian, and the MRSDCI approach. Results of our calculations on the spin gaps of these RGQDs, when extrapolated to infinite graphene, suggest that it has a vanishing spin gap, implying weak electron correlation effects. This result is consistent with the widespread assumption that graphene is a weaklycorrelated material.
For the case of optical absorption spectra, we generally found very good agreement with the experiments performed on hydrogensaturated structural analog of each RGQD, wherever experimental data was available. In certain cases, where no experimental data was available for the Hpassivated molecule, the comparison was instead made with the measurements performed on tbutyl group saturated systems, and some quantitative disagreements were encountered, most severe of which were for RGQD42. It will be very interesting if future experimental measurements could be performed on the Hpassivated molecules in those cases. For the case of RGQD36, and RGQD54, no experimental measurements exist, while for the case of RGQD54, even prior theoretical calculations do not exist. Thus, results of our calculations on these molecules could be tested in future measurements of their absorption spectra.
Acknowledgements.
This research was supported in part by Department of Science and Technology, Government of India, under project no. SB/S2/CMP066/2013.References
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