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Correspondence between Gentile Oscillators and N‑Annulenes Yao Shen and Bih-Yaw Jin* Department of Chemistry and Center for Emerging Material and Advanced Devices and Center for Quantum Science and Engineering, National Taiwan University, Taipei 10617, Taiwan ABSTRACT: The cyclic hydrocarbon polyenes with the general formula CNHN are called Nannulenes. In this paper, we discover that Gentile oscillators and N-annulenes in the Hückel approximation have the same energy spectra determined by the contact points of a regular polygon inscribed on a circle. This correspondence is derived from the symmetry of a C(n + 1) rotational group and dihedral group DihN. On the basis of their energy spectra, we further demonstrate that these two kinds of systems have the same partition functions and, thus, the same thermodynamics properties. N-annulenes can, therefore, be viewed as the natural realization of Gentile systems.

1. INTRODUCTION Besides the two kinds of elemental particles, bosons and fermions, G. Gentile introduced a more general kind of particles that allow a finite maximum occupation number in a single-particle state.1,2 Bosons obey Bose−Einstein statistics, in which the maximum occupation number of one state is ∞, while fermions obey Fermi−Dirac statistics, in which one state can be occupied by only one particle at most. In the theory of G. Gentile, the maximum occupation number of a singleparticle state is a finite number n, and the statistics that Gentile particles obey is called Gentile statistics.1−3 Gentile statistics is a kind of intermediate statistics, with two limiting cases corresponding to the Bose−Einstein and Fermi−Dirac statistics. When the value of n is given, the Gentile statistics and the relation of operators are also determined. The operators of bosons and fermions satisfy the usual relations of commutativity and anticommutativity, respectively. However, Gentile particles satisfy the intermediate-statistics quantum bracket, which pick up an additional phase factor exp(iθ) with θ = 2π/(n + 1) when two Gentile operators are commutated.4,5 The intermediate-statistics quantum bracket returns to the standard commutative and anticommutative brackets when the maximum occupation number n becomes ∞ and 1. More specifically, the creation and annihilation operators of the Gentile statistics, a† and b, satisfy [b,a†]n = ba† − exp(iθ) a†b = 1. Here, in general, a† is not the Hermitian conjugate of b. The eigenstates |ν⟩n of a Gentile oscillator where ν is the occupation number can, in principle, be obtained through the intermediate-statistics quantum bracket and the boundary conditions, a†|n⟩n = 0 and b|0⟩n = 0. The Gentile particle model can be used to deal with some composite particle systems. For instance, certain fermion systems can behave like bosons when two of them are strongly bound with each other. However, if they are approaching each other but still not close enough to form boson pairs, they can behave in between bosons and fermions, and thus, the Gentile particle model can be an effective tool for describing composite particles in such a situation.6−12 Besides, this intermediate © 2013 American Chemical Society

statistics model has brought the advantage of being robust against errors in the quantum information process.13 Nonetheless, such systems are only studied by simulation in other systems such as photons14 or nuclear spin systems.15 Direct manipulation and application of intermediate statistical system is significant and highly demanded. Furthermore, whether there exist substances that satisfy the Gentile particle model in nature is still a question. Conjugate polymers as a research of great contention give the probability to realize the Gentile particle model.16−19 To be specific, we wish to show that cyclic hydrocarbon polyenes can be viewed as a realization of Gentile systems. Cyclic hydrocarbon polyenes, or simply N-annulenes, CNHN, denote the family of organic molecules that are monocyclic unsaturated hydrocarbons.20−23 The electron paramagnetic resonance,24,25 aromaticity,26 and fluorescence property27 are well investigated in the annulene systems. The Hückel model, a simple but effective molecular orbital model for describing planar conjugated molecules,28−31 will be used to deal with N-annulenes. There are two possible topologically distinct classes of N-annulenes at the Hückel level, Hückel annulenes, planar structures with parallel pz orbitals, and Möbius annulenes with pz orbitals gradually twisted by 180° like a Möbius band (see Figure1).32−34 Hückel annulenes have two sides to the π surface, while Möbius annulenes have only one. The difference in energy spectra of Hückel and Möbius annulenes is a π/N rotation. In this paper, we show that the energy spectra of Gentile oscillators and N-annulenes in the Hückel approximation are equivalent. We also investigate the connection in thermodynamic properties of N-annulenes and Gentile oscillators through their partition functions, which will be shown to be approximately equivalent to each other in the low- and hightemperature limits. The same energy spectra result in the same partition functions and lead to identical thermodynamic Received: July 2, 2013 Revised: October 2, 2013 Published: October 25, 2013 12540

dx.doi.org/10.1021/jp406512n | J. Phys. Chem. A 2013, 117, 12540−12545

The Journal of Physical Chemistry A

Article

εj(N ) = 2β cos

2jπ N

(3)

(2j + 1)π N

(4)

and εj(N ) = 2β cos

respectively, where ⎛ N−1 odd N ⎜ 0, ±1, ±2, ..., ± 2 ⎜ j= ⎜ ⎛N ⎞ N even N ⎜ 0, ±1, ±2, ..., ±⎜ − 1⎟ , ⎝ ⎠ 2 ⎝ 2

For simplicity, the site energy is chosen to be zero because it only causes a trivial shift of the whole molecular orbital spectrum. Hückel annulenes and Möbius annulenes are Nannulenes with different topological configurations. It is interesting to note that the spectrum of a Möbius annulene can be obtained simply by rotating the spectrum of the same Nannulene with the Hückel arrangement with an angle π/N, as shown in the Figure 2. Thus, Hückel annulenes always have a vertex at the bottom, while Möbius annulenes have an edge at the bottom.

Figure 1. Two kinds of annulenes, Hückel and Möbius annulenes.

properties. All of these correspondences are derived from the symmetry of a C(n + 1) rotational group and Dihedral group DihN. Both of them divide the circle into certain parts. From this point of view, N-annulenes are the natural realization of the Gentile particle model. The research of chemical systems using the statistical method has also attracted much attention.35−37 Our work demonstrates that the Gentile oscillator method provides a convenient algebraic way to deal with N-annulene problems. It is worth mentioning the relation between N-annulenes and anyons. An anyon is another kind of particle of intermediate statistics. The relation between properties of Gentile statistics and fractional statistics of the anyon has been discussed,38 including the transformation between the anyon’s winding number representation and the occupation number representation of the Gentile particle, coherent states, and so on. According to ref 38, N-annulenes also can be mapped to the statistics of anyons through the winding number representation. The symmetries of N-annulenes and Gentile oscillators are equal. However, there is no direct symmetric correspondence between N-annulenes and anyons. The direct mapping from Nannulenes to anyons may not be obvious; in this case, Gentile oscillators can act as a bridge between N-annulenes and anyons. This paper is organized as follows. Section 2 gives the energy spectra of those two systems. Section 3 discusses the partition functions. Section 4 concludes the results.

2. THE ENERGY SPECTRUM The cyclic hydrocarbon polyenes that have the general form CNHN are called N-annulenes. They are unsaturated and monocyclic. The configuration of an N-annulene can be viewed as a regular polygon with N sides. The Hamiltonian of Nannulenes in the Hückel approximation can be written as H± = Hlinear polyene ± β(|N ⟩⟨1| + h.c.)

Figure 2. Molecular orbital energies of cyclopropenyl and cyclobutadiene. The left are Hückel types. The right are Möbius types.

According to their energy spectra, N-annulenes can be further grouped into four families, that is, N = 4t, 4t + 1, 4t + 2, and 4t + 3, where t = 0, 1, 2, ... The smallest annulene is the cyclopropenyl radical (N = 4 × 0 + 3 = 3, t = 0), followed by cyclobutadiene (N = 4 × 1 = 4, t = 1), the cyclopentadienyl radical (N = 4 × 1 + 1 = 5, t = 1), benzene (N = 4 × 1 + 2 = 6, t = 1), and so on. Their energy spectra come from the dihedral group DihN because in the Hü ckel model, the cyclic hydrocarbon polyenes are simply regular polygons. To be more precise, the spectrum has a C(N) rotational symmetry, which is a subgroup of the dihedral group. Therefore, a simple mnemonic to get the molecular orbital energy spectrum of a Hückel annulene is to construct a regular polygon (with a vertex down) inscribed on a circle with radius 2|β|. The contact points of the circumscribed circle correspond to the eigenvalues of the Hückel Hamiltonian. On the other hand, the Möbius

(1)

where Hlinear polyene is the Hamiltonian for the linear polyenes N−1

Hlinear polyene = β

∑ (|j⟩⟨j + 1| + h.c.) (2)

j=1

(5)

and the constant β < 0 is the resonance integral. Depending on how the two ends of a linear polyene are connected, we can get two classes of topologically distinct annulenes corresponding to Hückel annulenes (H+) and Möbius annulenes (H−). The molecular orbital energies for a single electron in the case of Hückel and Möbius annulenes are 20,39

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+ 1 by identifying N = n + 1, where t,t′ = 0, 1, 2,... In this case, the spectra of aromatic N-annulenes are

annulene has a molecular orbital energy spectrum corresponding to contact points of the circumscribed circle of a regular polygon with a side down. For the Gentile statistics, the lemma is also dividing 2π into n + 1 equal parts, where n is the maximum occupation number of one state. Therefore, its group has also C(n + 1) rotational symmetry. The oscillator that obeys the Gentile statistics is called the Gentile oscillator. The Hamiltonian of a Gentile oscillator can be constructed as5 H=

1 [α(n)a†b + β(n)ba† + h.c.] 4

εj(N ) = 2β cos

π 1 π 2kπ − csc cos N 2 N N

Ek(n) = cos2

k = 0, 1, ...,

N 2 (14)

1 2

(7)

HF = a†a −

1 2

(8)

The differences in the multiple and constant shifts of the spectrum in eqs 13 and 14 are ignored because they can be easily handled by a redefinition of the scale and origin of the system. Therefore, the aromatic N-annulenes and Gentile oscillators with maximum occupation number n = 4t′ + 1 have the same energy spectra of the form cos (2jπ/N) and the same number of the available energy levels, that is N/2 + 1. We take a benzene molecule as an example. For a benzene molecule, we have N = 6 (t = 1), which corresponds to a Gentile oscillator with n = 5 (t′ = 1), that is, the maximum occupation number of one quantum state is 5. The spectrum of a benzene molecule is jπ εj(6) = 2β cos j = 0, ±1, ±2, 3 (15) 3

By solving their energy spectra, we find that Gentile oscillators exhibit a peculiar period of four in n, that is, the spectra of Gentile oscillators are also divided into four cases, n = 4t′ + 1, n = 4t′ + 2, n = 4t′ + 3, and n = 4t′ + 4, where t′ ⩾ 0 and n ⩾ 1, and n is the maximum occupation number of one quantum state n = 4t′ + 1

where j = ±1 and ±2 are doubly degenerate. While the energy levels for the corresponding Gentile oscillator with n = 5 (t′ = 1) are Ek(5) = cos2

(4k − n − 1)π π π 1 = cos + csc sin n+1 n+1 2(n + 1) 2 n+1 where k = 0, 1, ..., (9) 2 2

π 1 π kπ − csc cos 6 2 6 3

k = 0, 1, 2, 3 (16)

There is a one-to-one correspondence in the energy spectra, (6) (5) (0) (5) (6) E(5) 0 ≈ ε0 , E1 ≈ ε±1 (degeneracy 2), E2 ≈ ε±2 (degeneracy (6) 2), and E(5) ≈ ε . 3 3 These two spectra can be presented graphically in Figure 3. We can see that in the figure, every vertex of the regular

n = 4t′ + 2 (2k − n)π π 1 π + csc sin n+1 2 n+1 2(n + 1)

where k = 0, 1, ..., n

⎛N ⎞ N ⎜ − 1⎟ , ⎝2 ⎠ 2

while the spectrum for the n = 4t′ + 1 Gentile oscillator is

(6)

HB = a†a +

Ek(n) = cos2

j = 0, ±1, ..., ±

(13)

where α(n) and β(n) are coefficients satisfying Re α(∞) = 1, Re β(∞) = 1, Re α(1) = 1, and Re β(1) = −1. When n = ∞ and 1, the Hamiltonian returns to boson and fermion cases, respectively, that is

Ek(n)

2jπ N

(10)

n = 4t′ + 3 (4k − n + 1)π π π 1 + csc sin 2(n + 1) n+1 n+1 2 n−1 where k = 0, 1, ..., (11) 2

Ek(n) = cos2

Figure 3. Molecular orbital energy spectrum of a benzene molecule (left) and energy levels of the n = 5 Gentile oscillator (right).

n = 4t′ + 4 Ek(n) = cos2

(2k − n)π π π 1 + csc sin 2(n + 1) n+1 2 n+1

where k = 0, 1, ..., n

hexagon represents a single-particle quantum state, that is, a single molecular orbital for a benzene molecule. Two vertices with the same height are degenerate because they have identical energies. Because two graphic representations are identical, we can conclude that the benzene molecule can be viewed potentially as a realization of the Gentile oscillator with n = 5. Antiaromatic N-Annulenes. Annulenes with N = 4t + 4 belong to the family of antiaromatic compounds that are usually quite unstable. Here, we identify that there is a connection between the Gentile oscillator with n = 4t′ + 3 and the Möbius annulenes with N = 4t + 4, where t,t′ = 0, 1, 2,... The energy spectra of antiaromatic Hückel N-annulenes are given by

(12)

Exactly similar to the situations that occur in the energy spectra of N-annulenes, the phase in commutativity of the Gentile statistics also comes from the circumscribed circle by a regular polygon.4,5 Considering this analogy, these two kinds of systems must have certain correspondence. Aromatic N-Annulenes. We first discuss the simplest annulenes with N = 4t + 2, which belong to the most stable aromatic family. We discover that the Hückel annulenes with N = 4t + 2 correspond exactly to the Gentile oscillator with n = 4t′ 12542


The Journal of Physical Chemistry A εj(N ) = 2β cos

2jπ N

Article

⎛N ⎞ N j = 0, ±1, ..., ±⎜ − 1⎟ , ⎝2 ⎠ 2

εj(N ) = 2β cos (17)

j = 0, ±1, ..., ±

N−1 2

(21)

and ⎛ kπ 1 π π π ⎞ ⎟ cos⎜ − csc + ⎝N 2 2N ⎠ N N 2(N − 1) k = 0, 1, ..., (22) 2 respectively. In the case of annulenes with N = 4t + 3 and Gentile oscillators with n = N − 1 = 4t′ + 2, we also have the same energy spectra as those given by eqs 21 and 22. Equations 21 and 22 seem to be different, but they have certain correspondence actually. A regular polygon with N edges has a one-to-one connection to another regular polygon with 2N edges. If we rotate the regular polygon with 2N edges by an angle π/2N, such that one edge is located at the bottom, the resulting energy levels will become doubly degenerate and can be viewed as the spectrum of a Möbius 2N-annulene. Here, only half (either left or right half) of the spectra in Möbius 2Nannulenes correspond to the spectra of suitable Gentile oscillators. Here, we use cyclopropenyl and cyclopentadienyl radicals as examples (see Figures 5 and 6). In the case of the

while the energy levels of the corresponding Gentile oscillators with n = N − 1 = 4t′ + 3 are ⎛ 2kπ π π π⎞ 1 − csc + ⎟ cos⎜ ⎝ N N N N⎠ 2 N −1 k = 0, 1, ..., 2

2jπ N

Ek(n) = cos2

Ek(n) = cos2

(18)

Ignoring the multiple and constant in eq 18, we can get the same spectra by performing a clockwise π/N rotation on the spectra of antiaromatic Hückel N-annulenes, which are just the spectra of the same annulenes with a Möbius arrangement. As an example, let us discuss a cyclobutadiene molecule with N = 4 (t = 0) and the corresponding Gentile oscillator with n =

Figure 4. Molecular orbital energy spectrum of cyclobutadiene (left) and energy levels of the n = 3 Gentile oscillator (right).

N − 1 = 3 (t′ = 0). In Figure 4, we show the spectrum of a cyclobutadiene molecule jπ εj(4) = 2β cos j = 0, ±1, 2 (19) 2 and the energy levels of an n = 3 Gentile oscillator Ek(3) = cos2

⎛ kπ π π π⎞ 1 − csc cos⎜ + ⎟ ⎝2 4 2 4 4⎠

Figure 5. Molecular orbital energy spectrum of cyclopropenyl (left) and energy levels of the n = 2 Gentile oscillator (right).

k = 0, 1 (20)

Again, each vertex of the square in Figure 4 stands for a singleparticle quantum state. Similarity in these two situations is apparent. Namely, upon a rotation of the energy square of a cyclobutadiene molecule by π/4 clockwise, which is in fact the spectrum of a Möbius cyclobutadiene molecule, we obtain the energy levels of an n = 3 Gentile oscillator. For a cyclobutadiene molecule, there is a vertex of the square located (4) (4) at the bottom. We have three energy levels ε(4) 0 , ε±1 , and ε2 . (4) The degeneracy of ε±1 is 2, and the other two are nondegenerate. When we rotate it by π/4 clockwise, an edge lies at the bottom; therefore, we have two energy levels this (3) time, E(3) 0 and E1 ; their degeneracies are 2. The spectrum of a Möbius cyclobutadiene molecule is thus mapped to that of an n = 3 Gentile oscillator. Nonaromatic N-Annulenes. N-annulenes with N = 4t + 1 and 4t + 3 belong to the nonaromatic family. Similar to the antiaromatic N-annulenes, the connection to the Gentile oscillators is also of Möbius type. The annulenes with N = 4t + 1 can be mapped to the Gentile oscillators with n = N − 1 = 4t′ (or 4t′ + 4). Their energy spectra can be written as

Figure 6. Molecular orbital energy spectrum of cyclopentadienyl (left) and energy levels of the n = 4 Gentile oscillator (right).

cyclopropenyl radical with N = 3, the energy spectrum is a regular triangle. There are two energy levels and three singleparticle quantum states εj(3) = 2β cos

2jπ j = 0, ±1 3

(23)

Their degeneracies are 1 and 2, respectively, where one of the triangular vertices is located at the bottom. The corresponding Gentile oscillator is n = 2 with the energy spectrum Ek(2) = cos2

⎛ kπ π π π⎞ 1 − csc cos⎜ + ⎟ ⎝ 3 2 3 3 6⎠

k = 0, 1, 2 (24)

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which can be specified by a regular hexagon (2N = 6) with one edge located at the bottom. Note that eq 24 contains only half of regular hexagon spectrum, which can either be the left or right half, as shown in Figure 5. Here, if we rotate eq 23 by π/6 clockwise, we have eq 24. Similarly, when N = 5, the cyclopentadienyl radical is a regular pentagon; it has three energy levels and five single-particle quantum states εj(5) = 2β cos

2jπ 5

j = 0, ±1, ±2

(25)

and the degeneracies are 1, 2, and 2, respectively. Also, one of the pentagonal vertices lies at the bottom. While the corresponding Gentile oscillator is a regular decagon (n = 4, 2N = 10), with the energy spectrum Ek(4) = cos2

Figure 7. oscillators ln( f(10,x) ln( f(20,x)

⎛ kπ π π π ⎞ 1 ⎟ − csc cos⎜ + ⎝ 5 2 5 5 10 ⎠

k = 0, 1, 2, 3, 4

In the classical limit, both N-annulenes and Gentile oscillators obey the Boltzmann distribution. The partition function of the Boltzmann distribution is

(26)

which is just half (for example, the left half) of the regular decagonal spectrum with one of the edges located at the bottom.

Z=

∞

∏

1 − e−βT(Ei − μ)

(27)

∞

∏ (1 + e−β (E − μ)) T

i=1

i

(28)

Notice that Ei is the energy of the ith molecular state, in other words, Ei is the many-electron energy. We need to fill in the molecular orbital with π electrons of N-annulenes and Gentile oscillators. Because the system is stable when the temperature is low, we only need to consider the ground state of N-annulenes. When T → 0, we have exp(−βTE) → 0. Even at the ambient temperature, the energy differences of N-annulenes are on the order of the resonance integral |β| ≈ 3.6 eV ≫ kBT, and thus, we have exp(−βTE) → 0. Furthermore, in quantum statistics where exp(−βTμ) ≫ 1, we should have exp(−βT(Ei − μ)) → 0. This requires βT(Ei − μ) → ∞. We define βT(Ei − μ) = x and f (N , x ) =

1 − e−Nx 1 − e −x

(30)

4. CONCLUSION In conclusion, we found an interesting correspondence between Gentile oscillators and N-annulenes in the Hückel limit. The peculiar period of four in the Gentile oscillators is shown to be related to similar periodicity of the cyclic hydrocarbon polyenes, that is, N-annulenes. We also recognize that this correspondence results from the fact that the C(n + 1) group is the subgroup of dihedral group DihN. We have also shown that in the low-temperature limit, partition functions of N-annulenes and the Gentile oscillators are approximately equal, and in the high-temperature, classical limit, both systems approach the same Boltzmann statistics due to their same energy spectra. Thus, both kinds of systems have the same thermodynamic properties. From this point of view, N-annulenes can be seen as the natural realization of Gentile systems. This suggests that we can really find a kind of system that behaves like Gentile oscillators. Finally, we also want to point out that due to the inevitable strong electron−phonon coupling, N-annulenes will exhibit the Peierls instability as N increases, which will lead to the bond length alternation in their ground states. However, the correspondence between Gentile oscillators and N-annulenes will not be affected by this kind of instability because the influence of Peierls instability for N-annulenes can, in principle, be treated within the Gentile oscillator representation of this family of compounds by introducing a suitable perturbation and is still under study in our group. We believe that the Gentile oscillator representation of N-annulenes in its second quantized form can provide an effective algebraic method to solve the problems of N-annulenes even with electron−electron and electron−phonon interactions included.

where βT = 1/(kBT), Ei is the energy of the ith state, ni is the maximum occupation number of the ith state, and μ is the chemical potential.3 In the Hü ckel approximation, the independent π electrons of N-annulenes, which determine the chemical properties of the molecules, obey the Fermi−Dirac statistics. Therefore, the grand partition function of Nannulenes is ΞA (T , V , μ) =

exp( −βTEi)

In this case, eqs 27 and 28 become eq 30. We discover that Nannulenes and Gentile oscillators have the same energy spectrum. Because of the same energy spectrum, they also have the same partition functions. From this point of view, they also have the same thermodynamical properties.

1 − e−βT(ni + 1)(Ei − μ)

i=1

∑ i

3. PARTITION FUNCTION The grand partition function of Gentile particles is ΞG(T , V , μ) =

The differences between partition functions of Gentile with different N. The blue thick line represents Δf1 = − f(2,x)), and the red dashed line represents Δf 2 = − f(10,x)).

(29)

Figure 7 shows the differences between partition functions of Gentile oscillators with different N, that is, Δf1 = ln( f(10,x) − f(2,x)) and Δf 2 = ln( f(20,x) − f(10,x)). Specifically, N = 2 corresponds to the Fermi−Dirac case, and the corresponding partition function is eq 28. In this figure, Δf1 and Δf 2 monotonically decrease as x is increased, and both of them are small enough to be neglected. Therefore, we can conclude that the partition functions with different N are approximately equal, especially when x is sufficiently large.

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Corresponding Author

*E-mail: [email protected]. 12544



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Notes

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The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Yi-Hsien Liu for useful discussions. The research was supported by the National Science Council, Taiwan and the Center of Theoretical Sciences of the National Taiwan University.

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