Theoretical Study of Graphene Nanoflakes with Nitrogen and Vacancy

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Theoretical Study of Graphene Nanoflakes with Nitrogen and Vacancy Defects: Effects of Flake Sizes and Defect Types on Electronic Excitation Properties Chih-Kai Lin J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.5b08318 • Publication Date (Web): 09 Nov 2015 Downloaded from http://pubs.acs.org on November 14, 2015

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The Journal of Physical Chemistry

Theoretical Study of Graphene Nanoflakes with Nitrogen and Vacancy Defects: Effects of Flake Sizes and Defect Types on Electronic Excitation Properties Chih-Kai Lin a

a

Center for Condensed Matter Sciences, National Taiwan University, Taipei 10617, Taiwan Email: [email protected]

(Oct. 23, 2015)

ABSTRACT We have established finite-sized model molecules of hexagonal graphene nanoflakes which consisted of up to about 400 carbon atoms and certain defects, and calculated their geometric structures, molecular orbitals and valence electron excitation properties by DFT and TD-DFT approaches. Several types of defects such as single nitrogen-atom dopants, vacancies, combinations of nitrogen atoms and vacancies, etc. have been investigated.

The vertical excitation energies and the

corresponding absorption spectra of all low-lying excited states of all these defects are found strongly depending on nanoflake sizes with few exceptions. The reason is revealed by examining molecular orbital configuration changes upon excitation, where in most cases the electron density does not localize in the defect center but disperses to the edges of the nanoflake instead.

As a result,

graphene nanoflakes with these defects do not present definite spectral features like zero-phonon line in nanodiamond.

The energy gaps and absorption spectra could be tuned by varying nanoflake

sizes and defect types, where the size determines positions of first major absorption peaks and the defect type affects spectral shapes of following bands.

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I. INTRODUCTION It is well known that pristine graphene has unique electronic properties like zero band gap and extremely high carrier mobility owing to its two-dimensional honeycomb lattice extended by the carbon sp2 σ-bond skeleton with the π-conjugated electronic system.1

These properties would be

modified by the existence of native structural defects and manipulations of artificial dopants. Most common structural defects are vacancies and atomic rearrangements such as Stone−Wales defect (a pentagon−heptagon pair) and other topological dislocations.2−6

Mostly applied dopants, on the

other hand, are nitrogen and boron atoms which substitute arbitrary carbon atoms; furthermore, these dopants may aggregate together and/or incorporate with certain structural defects in the lattice to form combinatorial defect centers.7−13 In previous studies of diamond nanocrystals, it was found that electronic excitations and optical properties of a point defect are usually quite localized.

For example, calculated electron densities of

a nitrogen−vacancy defect in nanodiamond concentrates in the center and diminishes away radially by less than three carbon atom shells.14,15

As a result, a relatively small (~100 to 200 carbon atoms)

molecular cluster or supercell is sufficient to describe the defect, making theoretical computations more affordable.16,17

Nevertheless, this simple scheme would be no longer suitable for systems

where strong delocalization of electron is expected such as very large polycyclic aromatic hydrocarbons, carbon nanotubes, and graphene sheets.

The π-electron conjugation in these systems

could propagate the density away from the defect center, and hence the size of model as well as the edge effect should be considered. To investigate electronic properties of defects in graphene, it is intuitive to carry out calculations based on the periodic boundary condition (PBC).2,3,6−13

However, when the sample size is not so

large, e.g. from hundreds to thousands carbon atoms, its properties show significant dependence on crystal size and shape, edge geometry and passivation, and terminal functional groups.4,18−21

The

finite-sized molecular cluster model would therefore be more favorable for dealing with such conditions. In this work we have chosen the latter approach and focused on the size effect on the electronic and optical excitation properties of nitrogen- and vacancy-related defects in graphene nanoflakes. Hexagonal graphene nanoflakes of different sizes, embedding several types of nitrogen and vacancy defects, were constructed and calculated under the density functional theory (DFT) scheme.


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II. COMPUTATIONAL SETTINGS Hexagonal graphene nanoflakes of different sizes were built by extending the graphene unit cell. Each edge of the nanoflake was in the zigzag form and passivated by hydrogen atoms so that all carbon atoms were covalently bonded through sp2 hybridization.

The graphene nanoflakes adopted

in this study were, by ascending order of size, C54H18, C96H24, C150H30, C216H36, C294H42, and C384H48. These structures were “concentric-circle-like” which corresponded to 3 to 8 carbon “circles” as counting outward from the central C6 unit. Several defect types of substitutional nitrogen-atom dopants, vacancies, and combinations of nitrogen atoms and vacancies were embedded in the pristine graphene nanoflakes. were settled in the central C6 circle or its nearest neighbor.

Such defects

The neutral charge state and the lowest

one or two spin states (singlet and triplet for even-electron systems, and doublet for odd-electron ones) were assigned.

For neutral defects with odd electrons, other charge states (+1 and/or −1)

were additionally studied. All these nanoflake structures were fully optimized by spin-unrestricted DFT calculations with the B3LYP functional and the 6-31G(d) basis set. The basis set with diffuse function, 6-31+G(d), was additionally adopted in several negatively-charged systems for a balanced description of the excess charge.

Long-range corrected functionals including CAM-B3LYP, M06-HF, and ωB97X-D have

been further applied to certain models to test the consistency of computational results. Vertical excitation energies, transition dipole moments, oscillator strengths, and leading configurations of low-lying electronic excited states were then computed using time-dependent DFT (TD-DFT) at optimized geometric structures.

For singlet systems, singlet excited states were explored, while for

doublet and triplet ones, spin multiplicities of excited states were not determined due to spin contamination.

All calculations were carried out using the Gaussian 09 package.22

III. RESULTS AND DISCUSSION We have analyzed calculated geometric structures, defect formation energies, molecular orbitals (MOs), and electronic excitation properties of hydrogen-passivated hexagonal pristine graphene nanoflakes and those with nitrogen dopants and/or vacancies. The model species are noted as Cn−x−yNxVy, where n is the total number of carbon atoms in the pristine graphene nanoflake, and x and y refer to numbers of substitutional nitrogen atoms and vacancy sites, respectively. Hydrogen ACS Paragon Plus 3 Environment


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atoms on nanoflake edges are ignored in such notations.

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An extra charge state is shown as + or − if

applicable.

1. Geometric Structures All these hydrogen-passivated hexagonal graphene nanoflakes, either with or without defects, have planar structures at their electronic ground state after full geometric relaxation.

We have

intentionally dragged certain atoms out-of-plane and re-optimized the structures, finding all nanoflakes went back to the planar geometry.

These results are consistent with previous reports

when the temperature effect was excluded.18,20,23 All pristine hexagonal graphene nanoflakes have the point group of D6h, and all those with defects are of C2v.

(The local structure of defect may belong to a higher symmetry, e.g. D3h for a vacancy, a

single N dopant, or an N3V defect, but the whole molecular model has a lower symmetry because the defect is slightly off-center.) The optimized geometric structures and important bond lengths around defect centers are drawn in Figure 1. The standard C−C bond length in pristine graphene is 1.42 Å.

When a single vacancy exists, the three surrounding carbon atoms slightly move outward

from their original lattice positions, and their bond lengths with neighbors shorten to 1.39 Å.

When

a divacancy exists, the four carbon atoms around the vacant space move inward, forming two pseudo-bonds of 1.68 Å and an overall symmetric pentagon-octagon-pentagon (5−8−5) structure. A single substitutional nitrogen atom makes minor distortion to the lattice structure with the C−N bond length of 1.41 Å. geometry.

An extra charge, either positive or negative, brings negligible effect to the

The combination of one nitrogen atom and one vacancy, i.e. an N−V defect center,

makes the structure much distorted, where the C−N bond lengths shorten to 1.32 Å and the opposite site forms a pentagon with the C−C pseudo-bond of 1.76 Å. The N2V, N3V, and N4V2 defect centers, on the other hand, almost keep the honeycomb geometry although the C−N bond lengths are all remarkably shorter than a normal C−C bond.

This shortening could be attributed to the lone pair

electrons located on nitrogen atoms, which stretch into the vacancy and hence expand the vacant space as well as compress the C−N bonds.

2. Formation Energies The formation energy, Ef, of the defective species Cn−x−yNxVy is defined by Ef = ED − EPG + nC EC − nN EN ACS Paragon Plus 4 Environment

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where ED and EPG are total energies of defective and pristine graphene nanoflakes, respectively; nC = x + y and nN = x are numbers of carbon atoms removed from and nitrogen atoms added into the model, respectively; EC is the average energy of a single carbon atom in graphene, and EN is that of nitrogen atom in N2 molecule.

The results are illustrated in Figure 2, and detailed values are

listed in Supporting Information.

Generally, the formation energies show roughly same values for a

same defect in different-sized models.

It could be seen that creating a single vacancy or a

divacancy defect (Cn−1V or C n−2V2) in the pristine graphene nanoflake requires quite high energy (~7−8 eV). A single nitrogen dopant (Cn−1N) in the nanoflake center raises the total energy by ~1.3 eV. Combinations of nitrogen dopants with vacancies could stabilize the models to different extents. A set of values from a previous theoretical study using periodic models7 is also addressed in the figure, showing similar relative stability of several defective species yet different values which should be attributed to computational methods. Extra charge states of certain models are further considered. Adding one more electron to Cn−1N, Cn−2NV, and Cn−4N3V causes total energy lowering of ~2.2−3.1 eV, which corresponds to the electron affinity.

Removing one electron, in contrast, makes destabilization by ~4.2−5.2 eV which marks the

ionization potential of the defect.

Cn−1N− is the only model in this study that could be more stable

than the pristine graphene nanoflake, and is obviously an exception whose formation energy shows a noticeable size-dependent trend. The trend could be understood that the extra pair of electrons (with respect to pristine graphene) is merging into the π-conjugated system, and a larger nanoflake would benefit its stabilization.

On the other hand, the total energy difference between Cn−2NV and

Cn−2NV−, ranging from 2.8 to 3.1 eV, is found comparable to the recombination energy of NV → NV− in nanodiamond reported as 2.96 eV.24

3. Molecular Orbitals and Excited States Calculated molecular orbital (MO) energy levels as well as energy gaps between HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital) of selected models are illustrated in Figure 3, and a complete list is given in Supporting Information.

For different defects

in the same-sized nanoflake, there seems no obvious regulation between them.

However, a

common feature is that defective species always have smaller HOMO−LUMO energy gaps than pristine graphene nanoflakes (1.27 eV for the C216 model) as shown in Figure 3a.

It could be

regarded that doping atoms and/or vacancies create extra levels in the original gap. For ACS Paragon Plus 5 Environment


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different-sized nanoflakes with the same defect, on the other hand, the HOMO−LUMO energy gaps decrease smoothly as the sizes increase, which trend is found not only in pristine graphene18 but also in all types of defects.

The gaps of Cn−3N2V models, as a demonstration in Figure 3b, decrease

from 1.47 eV to 0.71 eV from the smallest to the largest nanoflakes.

The influences of MO gaps

and electron density distributions in each type of defect are analyzed in the following.

Pristine graphene nanoflakes (Cn) The hexagonal pristine graphene nanoflakes have their zigzag edges passivated by single hydrogen atoms through sp2 bonding, so they could be topologically regarded as extremely large polycyclic aromatic hydrocarbons.4 The spin-unrestricted (U) B3LYP solutions reduce to spin-restricted (R) B3LYP ones as reported previously,25 and the singlet ground states are much more stable than triplet ones.

The point group of each model keeps D6h after optimization, giving the 1Ag symmetry for the

electronic ground state. HOMO and LUMO as well as HOMO−2 and LUMO+2 are doubly degenerate, while HOMO−1 and LUMO+1 are non-degenerate. It should be noticed that the electron densities of all these orbitals distribute mainly on the edge carbon atoms of the nanoflake as shown in Figure 4a.

(Please refer to Supporting Information for more MO density maps of this

model and models with defects hereinafter.)

For the first several excited states the major

configurations concern the above-mentioned orbitals as listed in Table 1, and all these states are symmetry-forbidden except the 1 1E1u state which has very large oscillator strengths (~15 Debye for each transition dipole moment component). energies are depicted in Figure 5.

These low-lying states and their vertical excitation

As could be anticipated from the edge-related orbital densities, a

significant size-dependence that excitation energies decrease smoothly by enlarging the nanoflake is observed. The trend of HOMO−LUMO energy gap, consistent with a previous computational report,18 is expected to converge to the infinitesimal gap toward the infinite graphene sheet.

Single vacancy defects (Cn−1V) Vacancy defects are common in carbon materials which can be formed natively in the crystal growth process and/or artificially by high-energy particle bombardments.26,27

A single neutral

vacancy in diamond yields the optical transition band of the zero-phonon line (ZPL) at 1.673 eV, called GR1 center.28−30

In this study we started the exploration of defect effects with such a single

vacancy located in the center of graphene nanoflake, seeing if there is also a definite optical ACS Paragon Plus 6 Environment

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transition in the two-dimensional carbon material. Calculated MOs and excitations of neutral-charge, singlet-spin single vacancy defects in graphene nanoflakes are plotted in Figure 4b and listed in Table 1. group and the 1A1 electronic ground state. orbitals which localize around the vacancy.

All model molecules have the C2v point

The densities of LUMO and LUMO+1 are “pure defect” Different from all neighboring π-conjugated MOs,

these two orbitals represent in-plane σ bonds between carbon atoms around the vacancy.

HOMO

and HOMO−2 are partially related to edges, while HOMO−1 and other orbitals are purely edge-distributed like pristine graphene nanoflake.

As a result, only first two excited states, 1 1B1

and 1 1A2, concerning major configurations of HOMO → LUMO and LUMO+1 show size-independent excitation energies of ~0.4 eV, while all other low-lying excited states display significant size-dependence as shown in Figure 6a.

By selection rule all 1A1, 1B1, and 1B2 excited

states are symmetry-allowed, but only 1 1B2 and 2 1A1 states which concern HOMO → LUMO+2 and LUMO+3 configurations are optically observable among the first ten excited states. However, it is intuitive that unpaired electrons would appear when removing one carbon atom, and hence the most stable spin multiplicity of the single-vacancy defect might not be singlet. Calculation of the triplet state shows that it is more stable than the singlet by ~0.8 eV (Figure 2). Consequently, the true ground state of the single-vacancy graphene nanoflake appears to be 3B1. There are two singly occupied MOs assigned to the α-spin, but most transitions to low-lying excited states start off from the highest occupied MOs of β-spin.

Unlike the singlet model, the σ-bond

orbitals now move far away from LUMO, leaving all relevant orbitals π-conjugated and more or less edge-related.

As a result, the size-dependent trend of excitation energies (Figure 6b) is not

surprising. In summary, all symmetry-allowed low-lying electronic excitation energies of the single vacancy defect in graphene vary remarkably by nanoflake sizes, and there is no absorption band with fixed energy in the region studied.

Divacancy defects (Cn−2V2) A divacancy defect forms when two adjacent carbon atoms are simultaneously absent.

It is found

the singlet state is much more stable than other spin multiplicity, and hence the ground state is 1A1 of the C2v point group.

Referring to Figure 4c, LUMO is π-bonds concerning pentagons in the 5−8−5

structure and mainly concentrating in the divacancy, while other MOs are much less related to the defect but strongly correlated with edges. HOMO and HOMO−1 are nearly degenerate, causing the ACS Paragon Plus 7 Environment


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first two excited states close in energy in each model.

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Considering excitation configurations,

excitation energy decreasing as size increasing of all low-lying excited states is expected as shown in Table 1 and Figure 6c, where a small zigzag feature is attributed to interchanged orbital symmetries of HOMO and HOMO−1 in different-sized models.

Single nitrogen atom dopants (Cn−1N, Cn−1N+, Cn−1N−) As the most common doping element in graphene, a single neutral nitrogen atom dopant brings one extra electron into the π-conjugated system and yields the doublet spin multiplicity.

The point

group is C2v and the ground state is 2B1 for all optimized Cn−1N models carried in this study. The MO densities, demonstrated in Figure 7a, reveal that HOMO and LUMO+1 of α-spin partially allocate on the nitrogen atom and its nearest carbon neighbors where the unpaired electron resides, while adjacent MOs are irrelevant to the nitrogen atom but distribute on edges.

The low-lying

excited states have their major configurations of HOMO → LUMO, LUMO+1, LUMO+2, etc. and a size-dependent trend of excitation energies is presented in Table 1 and Figure 8a. When removing the unpaired electron, the system becomes singlet Cn−1N+ which is isoelectronic with pristine graphene. The ground state becomes 1A1.

The MO densities are essentially the same

as Cn−1N except that occupations and energies are shifted (see Supporting Information).

It is

noticed that HOMO and HOMO−1, from which orbitals the major excitation configurations occur, are nearly irrelevant to the nitrogen dopant and become almost degenerate (with the energy difference of ~0.02 eV) just like the case in pristine graphene nanoflake. The excitation energies therefore show the same size-dependent trend (Figure 8b, cf. pristine graphene in Figure 5).

It is

also noticed that many low-lying excited states of Cn−1N+ nanoflakes are intensely symmetry-allowed with transition dipole moments of ~10 Debye, making them very strong chromophores in the near-infrared to visible region. Things go in a different way if an extra electron is added to form singlet Cn−1N−.

Compared to

pristine graphene, there is one more pair of electrons which resides in the new HOMO and becomes very versatile.

The MO densities resemble those of Cn−1N+ but the HOMO−LUMO gap now turns

very small (Figure 3), and hence all low-lying excited states start off from HOMO with lower excitation energies.

An overall size-dependent trend is also met, where some zigzag seen in Figure

8c is due to interchanged arrangements of unoccupied MOs in different-sized models.

Moreover, it

is considerable whether the diffuse function is necessary to describe the extra negative charge. ACS Paragon Plus 8 Environment

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When applying the 6-31+G(d) basis set to either atoms around the defect or all atoms in the system, the results are found quite similar to those calculated by 6-31G(d).

The excitation energies and

oscillator strengths differ by no more than 0.05 eV and 0.005, respectively, indicating that diffuse functions make negligible effect on this system.

Nitrogen−vacancy defects (Cn−2NV, Cn−2NV−) An NV defect center consists of one substitutional nitrogen atom and one adjacent vacancy in the carbon lattice.

In artificial nanodiamonds it is a well-known red fluorescence chromophore, where

neutral NV and NV− have apparent ZPL at 2.156 eV and 1.945 eV, respectively.14,26,27,31

In

graphene nanoflakes, however, neither ZPL nor absorption band with definite energy could be found. The defect shows common size-dependent excitation energies just like others presented above. Both Cn−2NV and Cn−2NV− have the C2v symmetry with the ground state of the former being 2B1 while the latter 1A1.

For the neutral doublet NV defect, the HOMO−LUMO gap of α-spin is much

larger than β-spin in each model (Figure 2), causing most low-lying excitation states occur with

β-spin electron transitions (Table 1). The density of β-HOMO distributes mostly in the defect with a π-character, while all other relative MOs are much more edge-related as seen in Figure 7b.

The

low-lying excited states consequently show systematic size-dependent excitation energies, summarized in Figure 9a.

It is found that an occupied “pure defect” MO lies rather deeply, e.g.

β-HOMO−17 whose orbital energy is 2.0 eV lower than HOMO in C214NV. This orbital, unlike all valence orbitals mentioned above, consists of pyridinyl lone-pair electrons and neighboring σ bonds, and hence has the in-plane character. Excitation from this defect orbital to β-LUMO is essentially an nπ* transition, and the energy is expected size-independent because of edge-irrelevance. However, its value (~2.0 eV) is relatively high and does not list among the first dozens of ππ* excited states in large nanoflakes. For the singlet NV− defect, HOMO is rather defect-related while other MOs disperse to edges (see Supporting Information). LUMO and LUMO+1 are nearly degenerate with a difference < 0.05 eV, making the first two excited states quite close in energy. excitation energies, Figure 9b, is fairly smooth.

The overall trend in size-dependent

Applying the basis set with diffuse function,

6-31+G(d), to the system makes negligible differences comparing with the 6-31G(d) results.

All

these transitions involve π-conjugated orbitals which have symmetry terms of either b1 or a2, and hence all excited states are either 1A1 or 1B2.

It is also noticed that these states have significant



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transition dipole moments up to ~12 Debye.

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Similar to the neutral C214NV model, the pure-defect

pyridinyl nonbonding orbital of C214NV− takes place as HOMO−13 (see Supporting Information), which has the symmetry term of a1 and the orbital energy is lower than HOMO by 1.9 eV. Excitations from this very orbital to LUMO and LUMO+1 yield rare nπ* 1A2 and 1B1 states, but the excitation energies (~2.8 eV) are much higher than first dozens of low-lying excited states. This situation is much different from that in nanodiamond, whose pure-defect orbitals composed of nitrogen lone-pair electrons exist as HOMO and contribute to the first excited state.14,17

Multiple-nitrogen−single-vacancy defects (Cn−3N2V, Cn−4N3V, Cn−4N3V+, Cn−4N3V−) Defect centers of multiple nitrogen atom substitutions combined with single vacancies like N2V and N3V have been found in nanodiamonds and carbon nanotubes experimentally.15,26,32,33

More

recently, a theoretical study of the periodic graphene system indicated localized electronic densities of these defects.7 In this work, however, it demonstrations that in finite-sized models most valence MOs associate with edges of nanoflakes while just few localize in the defect center. In the case of Cn−3N2V, the optimized structure has the C2v symmetry and the ground state is 1A1. Plotted in Figure 10a, the density of LUMO concentrates on the two nitrogen and one carbon atoms surrounding the vacancy with the in-plane σ-character, while all other neighboring MOs disperse to edges with the π-character.

The trend that excitation energy decreases as size increases is then not

surprising as shown in Figure 11a.

Like the NV defects, the occupied pure-defect pyridinyl MOs

of N2V defects lie relatively deeply, e.g. HOMO−19 and HOMO−57 (2.3 eV and 3.7 eV lower than HOMO) in C213N2V, and hence have nothing to do with first dozens of excited states. In the case of neutral Cn−4N3V, three nitrogen atoms surrounding one vacancy yield also the C2v symmetry, but the spin is doublet and the ground state is 2B1. The HOMO−LUMO gap of α-spin is much larger than β-spin (Figure 2) and therefore first several excited states involve just transitions of

β-spin electrons similar to neutral Cn−2NV. β-LUMO has a large distribution on the defect center, while neighboring MOs are almost irrelevant to the defect as shown in Figure 10b.

The occupied

pure-defect pyridinyl MOs, composed of lone-pair electrons of nitrogen atoms, appear as

β-HOMO−6 and β-HOMO−7 (and a more deep-lying β-HOMO−34) in C212N3V. Unlike NV and N2V defects, these defect orbitals are not too low to participate in the first several excited states. Excitations from these specific MOs to β-LUMO result in B1 and A2 states which exhibit size-independent energies of ~0.9 eV as shown in Figure 11b, since their major configurations ACS Paragon Plus 10 Environment

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involve orbitals mostly distributed to the defect but not to edges.

All other low-lying excited states,

on the other hand, show a common size-dependent trend. When removing one electron to form Cn−4N3V+, the previous α-HOMO turns to empty LUMO. Excitations from pure-defect MOs (HOMO−7 and HOMO−9 in the C212N3V+ model, see Supporting Information) to this defect-relevant LUMO cause the same size-independent excitation energies of 1 1

B1 and 1 1A2 states around 1.0 eV illustrated in Figure 11c. Instead, when adding one extra

electron to form Cn−4N3V−, the defect-relevant MO is fulfilled.

The remained unoccupied MOs are

all edge-related, and therefore the excitation energies follow the common size-dependent trend as shown in Figure 11d.

It is mentioned by the way that Cn−4N3V− has the largest total oscillator

strength among three analogues, where transition dipole moments of up to ~15 Debye have been obtained.

Multiple-nitrogen−divacancy defects (Cn−6N4V2) The N4V2 defect center is formed by replacing four carbon atoms by nitrogen atoms around a divacancy defect.

The ground state is found singlet, i.e. 1A1 of the C2v symmetry, while the triplet

and quintet states are unstable by more than 1 eV, indicating good incorporation of single electrons of nitrogen atoms into the π-conjugated system like pyridine. Depicted in Figure 10c, The HOMO and HOMO−1 are nearly degenerate, making the first two excited states quite close in energy. Similar to the N3V defect, there are pure-defect pyridinyl MOs composed of nitrogen lone-pair electrons residing as HOMO−8 and HOMO−9 (and the more deep-lying HOMO−31 and HOMO−36) in the C210N4V2 model.

Excitations from these two orbitals result in approximately

size-independent nπ* 1A2 state at ~1.8 eV and 1B1 state at ~2.0 eV, but they are quickly overwhelmed by other size-dependent low-lying ππ* excited states when the nanoflake size grows as demonstrated in Figure 12.

1 1A2 state is symmetry-forbidden and 1 1B1 state has negligible oscillator strength,

but many other states have noticeable strong transitions with dipole moments up to ~14 Debye.

4. Absorption Spectra As seen previously in Figure 3, molecular orbital levels and HOMO−LUMO energy gaps of graphene nanoflakes depend on both defect types and nanoflake sizes.

This dependency concretely

visualizes as differences in corresponding absorption spectra. The absorption spectra of selected defective models, as shown in Figure 13, are simulated according to calculated vertical excitation ACS Paragon Plus 11 Environment


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energies and transition dipole moments.

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In each spectrum the first 50 excited states are included

with a full width at half maximum (FWHM) of 0.1 eV for each absorption peak.

Note that the

simulation is based on pure (symmetry-allowed) electronic transitions without any vibronic contribution. For a series of the same defect in different-sized nanoflakes, the red-shift of absorption peaks with increasing model sizes is intuitively expected and here demonstrated by Cn−3N2V models in Figure 13a.

The first several major peaks of small nanoflakes such as C51N2V and C93N2V fall in the

visible range (1.7−3.2 eV), while those of larger ones further red-shift to the infrared region. The intensities of major peaks also grow up with increasing sizes, which would be attributed to longer distances between defect centers and nanoflake edges that result in larger transition moments. For different defects in same-sized nanoflakes, the absorption spectra exhibit a pronounced diversity as shown in Figure 13b. Pristine graphene nanoflake C216 possesses three distinct peaks at 1.32, 1.90 and 2.28 eV (all belonging to 1E1u excited states), while all defects display various continuous absorption bands in this region.

All of these C216-based models have intense peaks

around 1.2−1.4 eV despite of defect types, but they show quite different fingerprint-like absorption features in other parts that would be distinguishable.

Consequently, the position of the first major

absorption peak could reveal the size of nanoflake, and the shapes of following bands imply the defect types.

Among these models Cn−1N+, Cn−2NV−, and Cn−6N4V2 have the largest total

absorbance; nevertheless, others do not fall behind too much in intensity. It shows that all studied defective graphene nanoflakes could be good chromophores through near-infrared (< 1.7 eV) to visible region, and their colors could be tunable by varying the sizes.

5. Effect of Different Functionals It is known that B3LYP functional lacks long-range exchange corrections.34,35

Consequently, its

description on excitations with distant transitions, e.g. density transfer from the central defect to edges of large nanoflakes, might be inaccurate. For this sake we have applied several long-range corrected functionals including CAM-B3LYP, M06-HF and ωB97X-D with the same 6-31G(d) basis set to check the validity of obtained properties. Geometric structures optimized with different functionals are nearly identical where only tiny variances in bond lengths (within 1%) are seen.

MO energy levels and excitation energies, however,

present noticeable divergence. Figure 14 illustrates comparison of low-lying excited state energies ACS Paragon Plus 12 Environment

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of selected defective models including N+, NV−, and N4V2 calculated by the four functionals.

It

shows that three corrected functionals cause apparently larger HOMO−LUMO gap and hence higher excitation energies ranging from 0.5 to 1.0 eV than B3LYP for each state of each defect.

The major

configurations of excited states are reasonably alike, but the oscillator strengths are noticeably different between functionals (see Supporting Information). The larger amount of HF exchange in the XC functional (e.g. 100% in M06-HF vs. 20% in B3LYP) causes larger excitation energies, transition dipole moments, and oscillator strengths, which is consistent with previous reports on valence excitations of aromatic compounds.36

Despite of these variations associated with

functionals, the general size-dependent phenomena associated with edge-related MOs, as well as the few size-independent excitation of localized defect density found in N4V2, are essentially the same in all these calculations. The B3LYP values might be somewhat underestimated, but they are undoubtedly consistent with the common trend.

Further experimental investigations or higher-level

computations such as coupled-cluster calculations would be required to determine the accurate values. Another issue concerning different functionals is the existence of open-shell singlet (OSS) ground states.

There have been reports on pristine and edge-functionalized graphene nanoflakes indicated

the OSS ground states, instead of usual closed-shell states, of these systems.4,25,37,38

In this work

pristine graphene nanoflakes and selected defective model species with stable singlet ground states are examined by the expectation value of the square magnitude of the total spin angular momentum,

S 2 , which are listed in Table 2. The spin-unrestricted B3LYP optimized wavefunctions always show

S 2 = 0 , designating the closed-shell nature of these ground states which are identical to

spin-restricted calculations; so do M06HF results. However, the other two functionals, CAM-B3LYP and ωB97X-D, give certain non-zero values. While

S 2 = 0 are obtained by all

means for C54-based models, one non-zero case emerges for specific C96-based species, and almost all become non-zero for C150 and increase up to ~3 for C216 calculated by the two functionals. These OSS ground states, in terms of energy, are just slightly more stable than closed-shell states by less than 0.1 eV for C150 and ~0.4 eV for C216 models (please refer to Supporting Information for detailed values). As to their MO shapes and energies, pronounced features such as edge-related distributions and defective pyridinyl orbitals are generally consistent with closed-shell ones although there exist certain different arrangements, e.g. some orbitals have the α-spin components reside on three edges of the hexagon while the corresponding β-spin ones on the other three. Comparisons of ACS Paragon Plus 13 Environment


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absorption spectra of closed-shell vs. open-shell C150-based singlet models are demonstrated in Figure 15. The closed-shell B3LYP and CAM-B3LYP spectra possess roughly same peaks, but the latter are ~0.5 eV higher in excitation energies (cf. Figure 14). The OSS spectra obtained by spin-unrestricted CAM-B3LYP additionally blue-shift by ~0.1−0.2 eV while keep similar excitation configurations and absorption bands. It should be addressed that the OSS character is found highly dependent on the choice of functional, and thus further theoretical and experimental investigations are demanded.

IV. CONCLUSION We have carried out DFT and TD-DFT calculations in this work to study geometric structures, molecular orbitals, and valence electronic excitation properties of finite-sized hexagonal graphene nanoflakes embedding different types of defect centers which consist of single vacancies and divacancies, substitutional nitrogen atom dopants, and combinations of nitrogen dopants and vacancy sites. Due to the π-conjugated electronic system of graphene, HOMO and LUMO as well as neighboring MOs of most defective species have their densities disperse onto edges of the nanoflake instead of localizing in the defect, causing a general size-dependent trend that excitation energies of low-lying excited states decrease as the model size increases. The pure-defect MOs with the in-plane σ-character are still found in N−V defects, and excitations concerning these orbitals have size-independent energies. However, such orbitals are few and usually residing much lower than HOMO, and hence their excitations are easily overwhelmed by other size-dependent excited states especially when the model size grows large. Computations applying different functionals have obtained somewhat different values while common tendencies of excitation properties. Certain open-shell singlet ground states have been searched in specific models; this character was found functional-dependent and requires further exploration. The size-dependent excitation features are anticipated to converge smoothly to the limit of infinite lattice; in the finite-size region, however, it is clearly shown that nanoflake sizes do affect these properties. Calculations of defects in graphene by using molecular cluster models therefore should be a suitable choice for samples of hundreds to thousands atoms. All defect types carried in this study are found good chromophores in the near-infrared to visible wavelengths, among which N+, NV−, and N4V2 possess the strongest absorption intensities. Unlike the case in nanodiamond, positions of absorption bands are not definite for defects in graphene ACS Paragon Plus 14 Environment

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nanoflakes. The positions of first major peaks are relevant to nanoflake sizes, and the fingerprint-like spectral shapes of following bands imply defect types. The energy gaps and absorption spectra therefore could be tuned by varying sizes of nanoflakes, and these features would find applications in optical and optoelectronic nanodevices. Based on this preliminary work with typical finite-sized models, further investigations on different defect positions (in place of on-center), interplays between multiple defects, different shapes and edges, and association with additional functional groups would bring interesting phenomena.

Although it would be difficult to control

sizes of nanoflakes as well as types and amounts of defects in a real-world experiment, we are confident this theoretical study has provided more insights to the characteristics of graphene nanoflakes with nitrogen and vacancy defects that could benefit further practical utilization.

ASSOCIATED CONTENT Supporting Information Detailed lists of excitation properties of each nanoflake size and defect. Selected molecular orbitals involved in low-lying excited states. This material is available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION Corresponding Author E-mail: [email protected].

Notes The author declares no competing financial interest.

ACKNOWLEDGEMENTS We are grateful to the National Center for High-performance Computing for computer time and facilities. The author thanks Dr. M. Hayashi for his helpful suggestions. This research is supported by a grant from Ministry of Science and Technology (Grant No. MOST 103-2113-M-002-020-MY2) of Taiwan, ROC.



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Figure 1. DFT-optimized geometric structures of each defect. The point group of local structure and important bond lengths (in Å) are given. PG represents pristine graphene, V indicates vacancy, and N denotes substitutional nitrogen atom (blue-colored).

Figure 2. Comparison of formation energies, Ef, of each defect type in hexagonal graphene nanoflakes calculated by B3LYP/6-31G(d). Values of certain defects obtained from periodic models (Ref. 7) are also noted as the right-most data points.


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Figure 3. Molecular orbital energy levels of (a) C216-based models with different defects and (b) series of Cn−3N2V models. The HOMO energy levels of α-spin are adjusted to 0 eV, and the HOMO−LUMO energy gaps are indicated (in eV).



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Figure 4. Selected molecular orbitals of hydrogen-passivated hexagonal graphene nanoflakes: (a) singlet pristine graphene nanoflake C216H36, (b) singlet C215V model, and (c) singlet C214V2 model. Dark gray and light gray balls denote carbon and hydrogen atoms, respectively. Red and green clouds represent positive and negative components, respectively, of molecular orbitals with the isosurface value of 0.02.


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Figure 5. Vertical excitation energies low-lying excited states of hydrogen-passivated hexagonal pristine graphene nanoflakes.

Figure 6. Vertical excitation energies of graphene nanoflakes with single vacancy and divacancy.



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Figure 7. Selected molecular orbitals of (a) doublet C215N model and (b) doublet C214 NV model, where the blue ball denotes nitrogen atom.

Figure 8. Vertical excitation energies of graphene nanoflakes with single nitrogen dopant.


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Figure 9. Vertical excitation energies of graphene nanoflakes with single-nitrogen−single-vacancy defect.



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Figure 10. Selected molecular orbitals of (a) singlet C213N2V model, (b) doublet C212N3V model, and (c) singlet C210N4V2 model.


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Figure 11. Vertical excitation energies of graphene nanoflakes with multiple-nitrogen−single-vacancy defect.



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Figure 12. Vertical excitation energies of graphene nanoflakes with multiple-nitrogen−divacancy defect.

Figure 13. Simulated absorption spectra of (a) Cn−3N2V models of different sizes and (b) C216-based models with different defects. Note that only the first 50 excited states are counted in each spectrum, and the high-energy tail is cut since higher states are not included.


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Figure 14. Comparison of vertical excitation energies of selected defective species calculated using different functionals.



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Figure 15. Comparison of absorption spectra of closed-shell and open-shell singlet ground states of C150-based models calculated by B3LYP and CAM-B3LYP functionals.


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Table 1. Low-Lying Excited States with Vertical Excitation Energies (ΔE in eV), Oscillator Strengths, and Major MO Configurations of Hydrogen-Passivated Hexagonal Graphene Nanoflakes with Different Defects Based on the C216 Model a C216

C215V singlet

C215V triplet

C214V2

state

ΔE (osc. str.)

config.b

state

ΔE (osc. str.)

config.

state

ΔE (osc. str.)

config.c

state

ΔE (osc. str.)

config.

1 1B2u

0.989 (0.000)

H→L, H'→L'

1 1B1

0.348 (0.000)

H→L

1 A1

0.514 (0.001)

β H-1→L

2 1A1

0.677 (0.041)

H→L

1

1

1

1 B1u

1.137 (0.000)

H→L', H'→L

1 A2

0.380 (0.000)

H→L+1

1 B2

0.539 (0.001)

β H→L

1 B2

0.723 (0.082)

H-1→L

1 1E1u

1.322 (1.311)

H→L', H'→L

2 1A2

0.727 (0.000)

H-1→L

2 A1

0.716 (0.001)

β H-1→L

3 1A1

1.060 (0.023)

H-1→L+1, H-2→L

1

1

1

1 E2g

1.376 (0.000)

H-1→L', H→L+1

2 B1

0.739 (0.000)

H-1→L+1

2 B2

0.781 (0.011)

β H-2→L

4 A1

1.091 (0.040)

H-2→L

1 1A2g

1.473 (0.000)

H-2→L', H-2'→L

3 1A2

0.850 (0.000)

H-2→L+1

3 A1

0.876 (0.000)

α H-1→L+2

2 1B2

1.093 (0.001)

H-3→L

1

1

1

2 E2g

1.496 (0.000)

H-2→L, H-2'→L'

3 B1

0.890 (0.000)

H-2→L

3 B2

0.904 (0.004)

β H-3→L

3 B2

1.175 (0.149)

H→L+1

2 1A2g

1.517 (0.000)

H→L+2', H'→L+2

1 1B2

0.929 (0.055)

H→L+2

4 B2

0.942 (0.004)

β H-3→L

4 1B2

1.325 (0.766)

H-1→L+2

H-2→L, H-2'→L'

1

2 A1

α H→L

1

5 B2

1.389 (0.107)

H→L+3

H-1→L, H-1→L'

1

4 A2

α H-1→L

1

5 A1

1.306 (0.055)

H-4→L

H-1→L, H-1→L'

1

β H-4→L

1

1.347 (0.922)

H→L+2, H-1→L+1

1

2 A1g 3 1E2g 1

4 E2g

1.543 (0.000) 1.548 (0.000) 1.622 (0.000)

4 B1

1.049 (0.065) 1.127 (0.000) 1.130 (0.000)

H→L+3 H-2→L+1 H-2→L+2

4 A1 5 B2 5 A1

0.969 (0.015) 1.052 (0.005) 1.086 (0.000)

C215N−

C215N+

C215N

6 A1

state

ΔE (osc. str.)

config.c

state

ΔE (osc. str.)

config.

state

ΔE (osc. str.)

config.

1 B2

0.241 (0.000)

α H→L

1 1B2

0.914 (0.031)

H→L

1 1B2

0.066 (0.000)

H→L

0.304 (0.000)

α H→L+1

1

2 A1

H-1→L

1

2 A1

0.379 (0.032)

H→L+1

0.473 (0.064)

α H→L+2

1

2 B2

H-1→L+1, H→L+2

1

2 B2

0.499 (0.025)

H→L+2, H→L+3

0.581 (0.026)

α H→L+1, β H→L+1

1

3 A1

H→L+1

1

3 B2

0.635 (0.239)

H→L+2, H→L+3

0.673 (0.001)

α H→L+3

1

3 B2

H-2→L

1

3 A1

0.918 (0.016)

H→L+4

0.757 (0.004)

β H-2→L, α H-1→L+2

1

4 A1

H-1→L+2

1

4 B2

0.982 (0.002)

H→L+5

0.842 (0.076)

α H→L+2, β H→L

1

4 B2

H→L+2

1

4 A1

1.041 (0.000)

H→L+6

0.898 (0.000)

α H→L+4

1

5 A1

H-2→L+1

1

5 B2

1.163 (0.841)

H-1→L

β H→L

1

5 B2

H-1→L+4

1

5 A1

1.189 (0.000)

H→L+7

β H→L+1, α H-1→L

1

H-3→L, H-1→L+4

1

1.322 (0.109)

H-2→L+1

1 A1 2 B2 2 A1 3 B2 3 A1 4 B2 4 A1 5 B2 5 A1

0.914 (0.144) 0.942 (0.070)

6 B2

1.019 (0.056) 1.137 (0.212) 1.203 (0.483) 1.240 (0.011) 1.349 (0.493) 1.331 (0.702) 1.367 (0.008) 1.430 (0.107) 1.446 (0.090)


6 B2


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

C214NV−

C214NV state

ΔE (osc. str.)

config.

state

ΔE (osc. str.)

config.

0.835 (0.076)

H→L

1 1B1

0.362 (0.000)

H→L

H→L+1

1

1 A2

0.709 (0.000)

H-1→L

H-1→L+1, H→L+2

1

2 B1

0.925 (0.000)

H-3→L

H→L+2

1

1 B2

0.933 (0.051)

H→L+1

H-1→L

1

2 A1

1.055 (0.063)

H→L+2

H-2→L, H→L+3

1

2 A2

1.103 (0.000)

H-2→L

H→L+3

1

3 A2

1.196 (0.000)

H-4→L

H-2→L+1

1

2 B2

1.215 (0.390)

H-1→L+2

H-1→L+2, H-3→L

1

3 A1

1.264 (0.802)

H-1→L+1

H→L+4, H-4→L

1

1.307 (0.014)

H→L+3

ΔE (osc. str.)

config.

1 B2

0.378 (0.000)

β H→L

1 1B2

0.471 (0.007)

β H-1→L

1

2 A1

0.676 (0.001)

β H-1→L, α H→L+1

1

2 B2

0.689 (0.051)

β H-2→L

1

3 B2

0.792 (0.001)

β H-3→L

1

3 A1

0.860 (0.000)

α H-3→L+1, α H→L+2

1

4 B2

0.867 (0.013)

α H→L, β H-2→L+2

1

5 B2

0.963 (0.001)

β H-4→L

1

4 A1

α H-1→L, β H-4→L

1

5 A1

α H-1→L+1, β H→L+2

1

2 A1 2 B2 3 B2 3 A1 4 B2 4 A1 5 A1 5 B2

1.029 (0.024) 1.059 (0.066)

6 A1

a b c

0.925 (0.076) 1.101 (0.038) 1.140 (0.188) 1.201 (0.608) 1.305 (0.371) 1.350 (0.256) 1.353 (0.210) 1.367 (0.062) 1.402 (0.031)

ΔE (osc. str.)

config.c

state

1 B2

0.545 (0.004)

β H→L

1

1 B2

1 A1

0.547 (0.001)

β H-1→L

2 1A1

2 A1

0.731 (0.000)

β H-1→L

1

2 B2

2 B2

0.800 (0.015)

β H-2→L

3 1B2

3 A1

0.904 (0.000)

α H-1→L+2

1

3 A1

3 B2

0.911 (0.009)

β H-2→L

4 1B2

2 B1

0.956 (0.000)

β H-7→L

1

4 A1

1 A2

0.975 (0.000)

β H-6→L

5 1A1

4 A1

0.995 (0.005)

α H→L

1

1 A2

4 B2

1.000 (0.002)

β H-3→L

1 1B1

ΔE (osc. str.)

3 B2 C212N3V−

C212N3V+

C212N3V state

C213N2V

c

state

1 A1

Page 28 of 34

config.

state

0.172 (0.002)

H→L

1

1 B2

0.367 (0.003)

H-1→L

2 1A1

0.670 (0.196)

H-2→L

1

2 B2

0.696 (0.008)

H-3→L

3 1B2

0.986 (0.002)

H-4→L

1

3 A1

1.098 (0.057)

H-1→L+1

4 1B2

1.126 (0.042)

H→L+1

1

4 A1

1.175 (0.216)

H-5→L

5 1B2

1.010 (0.000)

H-7→L

1

1 A2

1.080 (0.000)

H-9→L

1 1B1

ΔE (osc. str.)

C210N4V2 config.

state

ΔE (osc. str.)

config.

0.632 (0.047)

H→L

1

2 A1

0.900 (0.007)

H-1→L

0.739 (0.070)

H→L+1

1 1B2

0.970 (0.141)

H→L

0.955 (0.043)

H→L+2

1

3 A1

1.217 (0.792)

H→L+1

1.074 (0.049)

H-1→L+1

2 1B2

1.250 (0.513)

H-1→L+1

1.166 (0.294)

H-1→L

1

4 A1

1.268 (0.103)

H-2→L

1.194 (0.001)

H→L+3

3 1B2

1.316 (0.278)

H-3→L

1.324 (0.140)

H→L+4

1

4 B2

1.356 (0.264)

H→L+3

1.347 (1.112)

H-2→L

5 1B2

1.397 (0.013)

H-1→L+2

1.420 (0.000)

H-5→L

1

1 A2

1.716 (0.000)

H-8→L

1.462 (0.000)

H-6→L

1 1B1

1.885 (0.000)

H-9→L

Please refer to Supporting Information for a complete list of different-sized models. H and L denote HOMO and LUMO, respectively, and an apostrophe represents a degenerate orbital.

α and β indicate different spin components in an open-shell system. ACS Paragon Plus 28 Environment

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Table 2.


S2

Valuesa of Most Stable Singlet States Calculated by Different Functionals 54

96

150

216

B3LYP

0.00

0.00

0.00

0.00

M06HF

0.00

0.00

0.00

0.00

ωB97X-D

0.00

0.00

0.98

3.20

CAM-B3LYP

0.00

0.00

1.25

3.20

B3LYP

0.00

0.00

0.00

0.00

M06HF

0.00

0.00

0.00

0.00

ωB97X-D

0.00

0.00

0.84

2.98

CAM-B3LYP

0.00

0.00

1.09

3.11

B3LYP

0.00

0.00

0.00

0.00

M06HF

0.00

0.00

0.00

0.00

ωB97X-D

0.00

0.00

0.54

2.71

CAM-B3LYP

0.00

0.00

0.84

2.82

B3LYP

0.00

0.00

0.00

0.00

M06HF

0.00

0.00

0.00

0.00

ωB97X-D

0.00

0.00

0.32

2.83

CAM-B3LYP

0.00

0.00

0.76

2.98

B3LYP

0.00

0.00

0.00

0.54

M06HF

0.00

0.00

0.00

0.00

ωB97X-D

0.00

0.01

0.00

3.26

CAM-B3LYP

0.00

0.54

1.03

3.45

n

Cn

Cn−1N+

Cn−2NV−

Cn−3N2V

Cn−6N4V2

a

Zero values represent closed-shell, while non-zero ones indicate open-shell singlet ground states.



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38


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Theoretical Study of Graphene Nanoflakes with Nitrogen and Vacancy

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