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Modified Schrödinger Equation for Quantum Dynamics of

Radicals. Study of Graphene Nanoflakes

Ernesto Estrada1, Michele Benzi

2

1Department of Mathematics & Statistics, Institute of Complex Systems, University of

Strathclyde, Glasgow G11XQ, UK

2Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia

30322, USA

We develop a theoretical method for studying radicals in conjugated systems. We consider

electron hopping between two-sites and on-site energies equal to the number of nearest

neighbors. Therefore, we modify the Schrödinger equation by replacing the tight-binding

Hamiltonian H by 2H . Using this method we show that the radicals display anomalous

diffusion or ballistic motion in edge-passivated (with hydrogen) graphene nanoflakes (GNFs)

depending on their shapes. Remarkably, we found that the radical density perfectly

reproduces the spin density at carbons in both closed and open-shell GNFs.

PACS numbers: 31.10.+z, 73.63-b, 31.15.bu, 73.22.-f

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Radicals are chemical species with unpaired electrons or open-shell configurations which

play a fundamental role in the physics, chemistry and biology of many molecular systems [1].

In particular, the existence of open-shell configurations in certain graphene nanoflakes

(GNFs) has been identified as the major cause of relevant magnetic properties, such as in the

case of trigonal zigzag GNFs which are predicted to show metallic antiferromagnetism [2-7].

The same electronic property is also responsible for the intriguing properties of functionalized

trigonal GNFs, such as spin transfer, their use as components in molecule-based conductors

and as electrode active materials in secondary batteries [8]. Graphene nanoflakes (GNFs),

also known as nanoislands or nanodisks, are finite-sized graphene fragments of arbitrary size

and shape. When the edges of these fragments are passivated by hydrogen, GNFs are

realizable as all-benzenoid polycyclic aromatic hydrocarbons (PAHs) [9] in which all carbon

atoms have sp2 hybridization. GNFs can possess large spins depending of their shapes, which

include linear, triangular, rectangular, parallelogrammic, hexagonal, bowtie, etc. [2-10]. The

aim of this Letter is to develop a theoretical approach that allows studying the dynamics of

radicals in GNFs and related systems as well as their electronic properties near the zero-

energy states.

Apart from the well-known fact that the zero-energy states in GNFs give rise to

unbounded electrons or radicals, any GNF can be represented as a superposition of various

resonant structures which include a few radical ones. In other words, any GNF has a

radicaloid structure (of polyradicaloid nature in general) even in the case that they do not

have any zero-energy states [8, 11-13]. Such radical character is responsible for the nonlinear

optical (NLO) properties of several GNFs [13]. It is expected to influence the spin density at

the different sites of GNFs and consequently the electronic, optical and magnetic properties

of GNFs. Due to the bipartite nature of graphene lattices, the carbon atoms can be divided

into two sets (starred and unstarred), such that any site from one subset has its neighbors only

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in the other subset. This implies that if j

E is an energy level of a GNF, then j

E is also an

energy level, with the same multiplicity. Consequently, the zero-energy states, if they exist,

lie just in the middle of the spectrum of a GNF. That is, in order to characterize the radical

nature of a GNF we need to ‘see’ its energy levels from the middle out to the extremes.

In order to study the radical character of GNF we modify the tight-binding Hamiltonian

to account for the effect of radical delocalization on the stability and properties of GNF. Let

us consider the triangulene displayed in Figure 1 which has a radical structure (marked here

by an asterisk). In order for this radical to hop to another site, one of the nearest double bonds

must migrate to an adjacent bond in order to complete the valency of the carbon formerly

supporting the radical. Then, the radical must hops not to a nearest neighbour site but to one

separated by two bonds from the original one (see Figure 1). In summary, the radical electron

can hop only from one site to another separated by two bonds from it.

Figure 1. Illustration of some of the resonance structures of a triangulene radical (top), which

illustrates the two-bonds hopping of the radical. Systems with closed-shell configurations like

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antracene (bottom) can display resonance with biradical structures (see [11] and references

therein).

We start by considering the tight-binding Hamiltonian

† †ˆ ˆ ˆ ˆi i i ij i j

i ij

H c c t c c (1)

where the operators †

ic (

ic ) create (annihilate) an electron at carbon atom i ;

i and

ijt are the

site and transfer energies, respectively and the second sum is carried out over all pairs of

nearest neighbor sites i and j . We now modify (1) by replacing ij

t by ik

t , which is the

transfer energy between sites i and k separated by two bonds. In GNFs there are sites which

are adjacent to two or to three other sites. It is known that the stability of a radical depends on

the coordination of the carbon atom supporting it. If we consider for instance the spin density

in the triangulene radical (1) it is known that most of it is found on the sites having only two

nearest neighbors, while the spin density appearing on the sites with three nearest neighbors

are marginal and only due to spin polarization effects [8]. Then, in order to differentiate

between these two carbon coordinations we use the same approach employed for the tight-

binding method when there are heteroatoms in the molecules. That is, we substitute i by

i ik in the Hamiltonian (1). Here we select ik to be the number of nearest neighbors of the

corresponding carbon atom. Because the values of ik are positive, the effect of placing a

radical on a carbon atom is considered a destabilizing one, with most of the destabilization

being produced by a radical at a site with three nearest neighbors. The Hamiltonian is then

expressed as

† †ˆ ˆ ˆ ˆi i i i ik i k

i ik

H k c c t c c

, (2)

where the second sum is carried out over all pairs of sites separated by two bonds. As it is

typical in the tight-binding approach we select 0i for all carbon sites and 1

ikt for all

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pairs of sites at two bonds. Then, it is straightforward to realize the following important

relationship between both Hamiltonians: 2H H .

Using the time-independent Schrödinger equation we can obtain the energy levels for

the radical migration in the GNF, i.e.,

2H E . (3)

Obviously, the energy levels E are the square of those obtained from H E . This

implies that the energy levels corresponding to 2H appear as pairs of positive values plus the

zero-energy states. In addition, according to the spin alternation rule [14], which states that

the singlet spin pairing is preferred solely between sites in different subsets, the free valences

on the starred and unstarred sites might be identified with “up” and “down” spin.

Consequently, if we assume H as the adjacency matrix of a graph, such graph is formed by

two disconnected components, one representing carbon sites with spin up and the other

representing sites with spin down.

If we consider that a radical is located at the site 0

x of a GNF at the time 0t , the

dynamics of the radical hopping through the sites of the GNF can be understood by studying

the evolution of the electronic wave packet given by the time-dependent Schrödinger

equation with the radical Hamiltonian ( 1 )

2

ti H t

t

, (4)

with the initial condition 0,

0x x x

t . The solution of (4) is 2exp 0t itH .

The amplitude that the radical that resided at site q at time 0t ends up at the site p

due to the quantum dynamics is given by 2expp itH q . By continuation from the real

time t to the imaginary time, we obtain the thermal Green’s function as

2exppq

G p H q where 1

kT

is the inverse temperature. Here we always

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assume that 0 . Obviously, the amplitude that the radical returns to its original site is

given by

2

2 2

1exp exp

n

p pp j jjR G p H p p E

. (5)

Notice that the larger the value of p

R the larger the chances of finding a radical at the

corresponding site in the molecule. For instance, when ( 0T ) the index

2

i u

u

R i where ui are the eigenfunctions corresponding to zero-energy states. In

this context, the probability that a GNF is in a state with energy j

E is given by

exp /j j r

p E Z , where 1exp

n

r jjZ E

. It is straightforward to realize that the

highest probability occurs for the zero-energy states or for those close to them if they do not

exist. We can write the ‘radical hopping’ partition function as

22ˆ

1

j

nE

r

j

Z tr e e

H

. (6)

Then, it is evident that 2expj

E is a Gaussian function having its peak at the zero-energy

states (or those near to them in case they do not exist). The probability that a given molecule

is in a zero-energy state is then simply 1/r

Z , which means that the smaller the value of the

‘radical hopping’ partition function the larger the probability of finding the molecule in a

zero-energy, or a near to it, state. That is, the higher the value of r

Z the larger the

delocalization of the radical(s) in the molecular structure and consequently more stable the

radical(s) should be. Consequently, we consider r

Z as an index of radical delocalization in a

molecular structure. It should be noticed that as , r

Z , where is the number of

zero-energy states or ‘nullity’ of the lattice [15], which plays a role in understanding the

topological frustration in GNFs [6].

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In order to understand the dynamics of the radical hopping in GNFs we consider the

time evolution of the wave packet in four types of GNF systems: linear (acenes), hexagonal

(coronenes), trigonal (triangulenes) and bowtie GNFs (see Figure 2). The first two types of

systems are examples of closed-shell GNFs and the last two represent systems with open-

shells.

Figure 2. Representative structures of GNFs with linear (A), hexagonal (B), triangular (C)

and bowtie (D) shapes. The number of cycles B in the base is 4, 2, 3 and 3, respectively. The

total number of carbon atoms can be calculated as: 4 2B , 26B , 2 4 1B B and

22 4 2B B , respectively.

In order to compare the dynamic properties of the four systems under study we analyze

the scaling behavior of the temporal autocorrelation function averaged over different

nonequivalent initial positions of the wave packet. The temporal autocorrelation function is

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defined by 2

0

1 t

ppC t t dt

t , where 2exp

pq

pq

t itH

and indicates the

average over the nonequivalent sites. The integrand of this function 2

p ppP t t is known

as the return probability of the wave packet initially localized at the site p . It is known that

the temporal autocorrelation function scales as ~C t t , where 0 1 [16]. As it is well

known 0 indicates localization of the wave packet, 0 1 indicates anomalous

diffusion and 1 indicates ballistic motion. Thiem and Schreiber [17] have recently

suggested the use of the scaling behavior of the return probability instead of that of the

autocorrelation function. In this case, ~P t t and it is expected that . However, it is

usually the case that and can be even larger than one, which according to Zhong

and Mosseri [18], makes the integral in the autocorrelation function convergent, yielding

~1/C t t . In Figure 3 we illustrate the scaling behavior of the return probability and the

temporal autocorrelation function for the four systems studied. The sizes of the systems are

498, 486, 526, and 500, respectively. It is observed in the four GNF systems that C t

decays with a power-law behavior before approaching a constant value due to finite size

effects. The quantity C t decays strongly for the GNFs with closed-shell systems, i.e.,

linear and hexagonal, indicating a more extended nature of the wave functions in these

systems. That is, the values of for the four systems are, respectively 0.92H

, 0.92L

,

0.89B

and 0.88T , where H, L, B and T stand for hexagonal, linear, bowtie and

triangular, respectively. This indicates that in the four systems there is anomalous quantum

diffusion with larger localization of the radical wave packet in those systems with open-

shells. In general, it is difficult to extract the scaling behavior for the return probability in

these systems due to the short period of decay before the wave packet starts to oscillate

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around a constant value due to the small sizes of the systems considered here. The only

exception is the linear GNF where it was possible to check that 1.02~P t t , indicating a

ballistic motion of the radical along the linear graphene nanostructure. Therefore, the same

kind of ballistic motion should be expected for the hexagonal system.

Linear

Hexagonal

Triangular

Bowtie

Figure 3. Scaling behavior of the return probability and the temporal autocorrelation function

for the dynamics of a radical in the four types of GNFs studied in this Letter.

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Now, we can use the radical density index i

R to find the sites with the highest radical

densities in the studied GNFs. First, we investigate how reliable is this index as a predictor of

the radical character of a given site. For that we analyze its relation with the bond dissociation

enthalpies (kcal/mol at 298K) for the cleavage of the C-H bond in PAHs. In Figure 4 we

show the radical densities at the nonequivalent sites of naphthalene, anthracene and pyrene

together with their bond dissociation enthalpies calculated at the B3LYP/6-31G(d) level by

Barckholtz et al. [19]. As can be seen the i

R index perfectly predicts the sites with the largest

enthalpies for cleavaging the C-H bond, which correspond to those with the highest radical

densities. This result can be explained as follows. The site with the highest i

R already has a

radical nature. Then removing a hydrogen attached to this site is quite energetic as it will

leave two radicals at the same carbon atom.

Figure 4. Illustration of the radical densities, i

R and the bond dissociation enthalpies

(kcal/mol at 298K) for the cleavage of the C-H bond at different radical sites of three PAHs.

The radical densities are plotted as circles whose radii are proportional to i

R and the color

correspond to spin up or down.

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In a similar way we have analyzed the total radical character of a GNF as measured by

rZ . In this case we considered 11 GNFs previously studied by Murray et al. [11] by using the

average local ionization energies measured on the molecular surfaces of the GNFs, obtained

from first-principle methods. In their analysis Murray et al. [11] have considered that the

plausible causes for the lowest values of this property for molecules like 3,4-

benzphenanthrene, 1,2-benzanthracene, tetracene and 3,4-benzpyrene is due to the existence

of relatively stable biradical structures contributing to the total stability of these molecules.

The values of the average local ionization energies decay linearly with r

Z (data not shown)

indicating that the average energy needed to remove an electron from a PAH decays linearly

with the feasibility of radical delocalization of that molecule as measured by r

Z . The above

mentioned molecules are the ones displaying the largest values of r

Z , indicating the largest

contribution from the radical in their structures. In addition, the biradical structures proposed

by Murray et al. [11] coincide exactly with the sites in which there is the largest contribution

of the radical density index.

At this point we study the local and global radical character of the four GNF systems

studied in this Letter. As can be seen in Figure 5 for small representative of the GNFs studied

the index i

R predicts the highest radical densities at the edges of the GNFs in a way that

coincides with the spin densities calculated for these systems using first principle methods

[13]. This is a remarkable finding as our Hamiltonian does not consider electron spin

explicitly and indicates that most of the spin densities in GNFs are produced by the radicaloid

character of these structures, even in closed-shell configurations. In the case of the

triangulene illustrated in Figure 5 the sites with the highest radical densities are those that

support radicals in the molecules obtained experimentally so far [20]. All these findings

indicate that the index i

R is an excellent indicator of the radical density at sites in GNFs,

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which is responsible for most of the spin densities in these systems. Finally, we consider the

radical delocalization entropy in order to study the total radical delocalization in the GNFs

studied. It is defined as lnr i ii

S p p , where 2exp /i i r

p E Z . Here we consider only

the case 1 . We then consider the average entropy per site r

S . The results indicate that

the maximum radical delocalization per site is present in the hexagonal system and the

highest localization is obtained for the triangular GNF, with the linear and bowtie GNFs

occupying an intermediate position: 11.4r H

S , 11.2r L

S , 11.1r B

S and 10.7r T

S

. The highest localization in the triangular system is expected from the fact that it has an open

shell configuration in which a few radicals (20 in this case) are found in some sites with high

probability. The same is also expected for the bowtie nanoflake, which indeed has the second

largest localization according to r

S . The surprise then comes from the similarity in the

localization index between the linear system and those with open-shell configurations.

Despite the fact that linear GNFs are expected to have closed shell configurations, there are

important evidences that when the number of fused rings is larger than 7 the ground states of

these nanoflakes are antiferromagnetic ones [21], which agrees with our prediction about the

similarity between this system and the open-shell ones.

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Figure 4. Illustration of the radical densities, i

R at the sites of some small representatives of

the four types of GNFs studied. The radical densities are plotted as circles whose radii are

proportional to i

R and the color correspond to spin up or down.

Finally, we wish to remark that the extension of the current approach to account for

different characteristics of graphene nanostructures is straightforward. For instance, it is easy

to consider GNFs or nanoribbons with edges passivated in different ways by changing the on-

site energies of all atoms at the edges of these nanostructures [22]. In a similar way it can be

considered the doping of graphene nanostructures with other types of chemical elements [23].

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On the other hand, the current model can be applied to the study of radical hopping in other

types of -conjugated systems, ranging from polyenes to fullerenes and nanotubes.

EE acknowledge the University of Strathclyde New Professor’s Fund and MB thanks the

National Science Foundation, grant DMS1115692, for partial financial support.

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