Carbyne is a linear allotrope of carbon that is composed of a chain of sp-hybridized carbon atoms. Through appropriate engineering of the chain termination, carbyne can harbor helical states where the π-electron delocalization twists along the axis of the chain. Herein, we present a comprehensive analysis of these helical states at the tight-binding level. We demonstrate that, in general, the molecular orbital coefficients of the helical states trace out an ellipse, in analogy to elliptically polarized light. Helical states can be realized in a model, inspired by the structure of cumulene, which considers a chain terminated by sp2-hybridized atoms oriented at a nontrivial dihedral angle. We provide a complete analytic solution for this model. Additionally, we present a variation of the model that yields perfect helical states that trace out a circle as opposed to an ellipse. Our results provide a deeper understanding of helical states and lay a foundation for more advanced levels of theory.

Significant efforts have been made to engineer nanomaterials with designer electronic states.1–4 Carbyne is a nanomaterial of interest that comprises a chain of sp-carbon atoms.5–7 While polymeric carbyne has not yet been realized,8,9 short carbyne chains are synthetically accessible as polyynes or cumulenes.10–12 Each sp-carbon atom in the chain possesses two degenerate, mutually perpendicular p orbitals that can be arbitrarily oriented.13 This confers a unique form of electron delocalization to carbyne that may be harnessed to realize novel electronic states. Unlike in conventional sp2-carbon systems where the π-system has a fixed nodal plane, in carbyne, the nodal plane can be oriented at any polar angle with respect to the axis of the chain. It has been shown that through appropriate engineering of the chain termination, carbyne can harbor helical states where the nodal plane of the π-molecular orbitals (MOs) rotates along the axis of the chain.13–16 This electrohelicity17 is associated with ring currents and antiresonances and may be useful for designing nanoscale electronics.18,19 Herein, we provide a detailed analysis of the helical MOs in carbyne systems at the tight-binding level.

The existence of helical MOs in carbyne can be understood by considering an infinite chain where the chain termination is neglected. We will consider a carbyne chain oriented along the z-axis. The π-system of carbyne can be modeled with a basis set comprising of a px and a py orbital for each sp-carbon atom in the chain. Note that the pz orbitals are along the chain direction and, therefore, part of the σ-system and not the π-system. For the tight-binding model, we consider a coupling, t < 0, between adjacent px orbitals as well as adjacent py orbitals. The eigenstates, Ψk, and eigenenergies, Ek, associated with the wavenumber, k, are20,21

Ψkz=ψxzψyz=C1C2eikz,
(1)
Ek=2tcosk,
(2)

where C1 and C2 are complex numbers and z is the integer position of the atomic site along the chain. The components ψx(z) and ψy(z) represent the MO coefficients for the px and py orbitals on atom z, respectively. Since there is no coupling between ψx and ψy, i.e., ψy|H|ψx=0, the complex numbers C1 and C2 can be chosen arbitrarily. Mathematically, Eq. (1) is identical to the Jones vector formulation for elliptically polarized light.22,23 In analogy to light, if C1 and C2 differ by a complex phase, the eigenstate Ψk will be “elliptically polarized.”

The heuristic explanation of helical states can be further developed by incorporating bond alternation into the tight-binding chain. Equations (1) and (2) assume uniform coupling, t, which is only a valid approximation for even [n]cumulenes (an odd number of carbon atoms).12,24,25 For odd [n]cumulenes, polyynes, and polymeric carbyne, due to Peierls instability, the chain is more accurately modeled with alternating coupling, t1, t2 < 0.26–28 With alternating coupling included, the infinite chain has a unit cell that comprises two atoms (i.e., sublattice A and B). We will consider the intracell coupling to be given by t1 and the intercell coupling by t2.29 The eigenstates are described by a Bloch wave,

ΨkABz=C1C2uABeikz,
(3)

where z now indicates the integer position of the unit cell as opposed to the atomic site.

The associated energy levels fall into a lower band (Ek) and an upper band (Ek+) that are separated by a bandgap,30–32 

Ek±=±t12+t22+2t1t2cosk.
(4)

Due to the discrete basis of p orbitals, the periodic component of the Bloch wave is described by discrete values uA and uB for sublattices A and B, respectively. The values of uA and uB are equal in magnitude (|uB| = |uA|), but differ in phase. For the lower band of states (i.e., Ek), uB=eik/2+ηkuA, and for the upper band of states (i.e., Ek+), uB=eik/2+ηkuA, where we have defined ηk=argt1eik/2+t2eik/2.29,33 Incorporating the phase difference between uA and uB into Eq. (3), we see that ΨkA and ΨkB are translationally offset in the z-direction, yielding a double helix structure: one helix for sublattice A and another for sublattice B. Since ΨkA and ΨkB share the same Jones vector, C1C2, these two helices will follow the same ellipse shape. Hence, the core features of the helical states are maintained even when bond alternation is included.

The above analysis demonstrates that carbyne can support helical states in principle. The challenge is to design finite chains with end groups that introduce a phase shift between ψx(z) and ψy(z). We present an analytic model based on cumulene13 that provides an explicit example of how helical states can be achieved. For the sake of simplicity, we only consider uniform coupling since the inclusion of alternating coupling would not affect the main results. In the final section titled “Generalized analysis,” we generalize the results from the three previous sections.

Before discussing the elliptically polarized MOs of the cumulene-based model, it is instructive to first consider finite sp2-carbon chains where the MOs are “linearly polarized.” For a finite chain of N coplanar p orbitals with uniform coupling t, the MOs are standing waves enumerated by quantum number n,33–35 

ψnz=2N+1sinknz for 1zN.
(5)

The MO coefficients are discrete sampling of a sine wave at integer values of z. As an example, the eigenstates for N = 4 (i.e., butadiene) are presented in Fig. 1. Unlike for infinite chains where the wavenumber can freely vary, for finite chains the boundary conditions imposed by the chain termination yields quantized wavenumbers given as

kn=nπN+1 for 1nN.
(6)

This quantization condition is akin to particle-in-a-box, where the electron is confined to a box of length L = N + 1.36 The eigenenergies, En, associated with ψn exhibit the same functional form as for infinite chains [Eq. (2)]: En = 2t cos kn. In fact, Eq. (2) holds for all of the systems discussed herein, although the specific wavenumbers, kn, depend on the boundary conditions of the system.

FIG. 1.

“Linearly polarized” MOs for a tight-binding chain of N = 4 p orbitals. The MO coefficients are indicated by the size of the p orbital and can be viewed as a discrete sampling of a sine wave (purple).

FIG. 1.

“Linearly polarized” MOs for a tight-binding chain of N = 4 p orbitals. The MO coefficients are indicated by the size of the p orbital and can be viewed as a discrete sampling of a sine wave (purple).

Close modal

To realize “elliptically polarized” states, a phase difference between ψx and ψy must be introduced under boundary conditions that asymmetrically couple ψx and ψy. A convenient method to produce this effect is to twist the ends of the chain.15,16,37,38 To explicitly construct helical MOs, we will consider a model that is motivated by the electronic structure of cumulene.13 The Hamiltonian, HN(θ), describes a chain of N − 1 sp-carbon atoms terminated with single p orbitals (i.e., sp2-carbon atoms) oriented at a dihedral angle, θ [Fig. 2(a)]. Including the terminal sp2-carbon atoms, the chain has a total of N + 1 atoms and comprises a total of 2N p orbitals. This definition of N is consistent with that of n in [n]cumulene, which describes the number of cumulated double bonds.12 The Hamiltonian is constructed using the resonance integral,

pαi|pβi+1=tcosαβ,
(7)

where α is the angle of a p orbital on atom z = i and β is the angle of a p orbital on atom z = i + 1. The Hamiltonian for N = 3 is presented in the following equation and Fig. 2(a):

H3θ=  px0px1py1px2py2pθ3px0px1py1px2py2pθ30  t  0000t  0  0t000  0  00t00  t  000t cos θ0  0  t00t sin θ0  0  0t cos θt sin θ0.
(8)

Since the chain comprises N + 1 atoms, for H3(θ), there are 4 atoms labeled as z = 0, 1, 2, 3. The first atom in the chain (z = 0) is sp2-hybridized and provides only one p orbital to the π-system. Without the loss of generality, this p orbital is oriented along the x-axis. The last atom in the chain (z = N) is also sp2-hybridized, with the lone p orbital oriented at an angle θ from the x-axis. This dihedral angle between the terminal p orbitals will be referred to as the twist angle. The atoms in the core of the chain are sp-hybridized and are described by two perpendicular p orbitals that are oriented along the x- and y-axes, respectively [Fig. 2(a)].

FIG. 2.

(a) Visual representation of Hamiltonian HN(θ) for N = 3. The coupling between adjacent p orbitals is given by Eq. (7). Note that the pz orbitals involved in the σ-system are not shown. For θ = 0° and θ = 90°, HN can be partitioned into two subsystems indicated in red and blue. (b) Energy spectrum for HN(θ) for N = 3. At θ = 0° and θ = 90°, the energy levels for the two subsystems are indicated in red and blue, corresponding to (a).

FIG. 2.

(a) Visual representation of Hamiltonian HN(θ) for N = 3. The coupling between adjacent p orbitals is given by Eq. (7). Note that the pz orbitals involved in the σ-system are not shown. For θ = 0° and θ = 90°, HN can be partitioned into two subsystems indicated in red and blue. (b) Energy spectrum for HN(θ) for N = 3. At θ = 0° and θ = 90°, the energy levels for the two subsystems are indicated in red and blue, corresponding to (a).

Close modal

The energy levels of HN are given by the roots of the characteristic polynomial, det(HNEI), where I denotes the identity matrix. The characteristic polynomial takes a simple form if the energy, E, is expressed as Ek = 2t cos k [Eq. (2)] as follows:39,40

detHNEkI=t2Nsin2N+1ksin2kcos2θ.
(9)

Setting Eq. (9) to zero yields a trancendental equation describing the allowed kn for a given twist angle θ,

sin2N+1kn=cos2θsin2kn for 1n2N.
(10)

There are two limiting cases of Eq. (10) that are worth noting: θ = 0° and θ = 90°. In both of these situations, HN can be partitioned into two uncoupled π-systems indicated by red and blue in Fig. 2(a). The eigenstates for these uncoupled π-systems are “linearly polarized” (e.g., Fig. 1). Accordingly, Eq. (10) reduces to particle-in-a-box quantization [Eq. (6)]. For θ = 0°, Eq. (10) can be re-expressed as sin(N + 2)kn sin Nkn = 0, which yields the condition of Eq. (6) for chains of length N + 1 and N − 1. Similarly, for θ = 90°, Eq. (10) reduces to sin2(N + 1)kn = 0, which describes two chains of length N [Fig. 2(a)]. The limiting cases manifest in the energy spectrum for H3(θ) presented in Fig. 2(b). At θ = 0° and θ = 90°, the energy levels correspond to the energy levels of Hückel models for standard sp2-carbon molecules. At θ = 0°, the energy levels correspond to butadiene (Nx = 4, red) and ethylene (Ny = 2, blue), while at θ = 90°, the energy levels are doubly degenerate and correspond to the allyl radical (Nx = Ny = 3).

Between the two limiting cases mentioned above, i.e., for nontrivial twist angles: 0° < θ < 90°, the energy levels correspond to helical states where the px and py π-systems are inextricably linked. In Fig. 3, the helical states of H3(θ = 60°) are presented. Unlike Fig. 1, where the MOs were depicted from the side (z-axis pointing to the right), in Fig. 3, the MOs are depicted from the front, with the positive z-axis pointing out of the page. Following from the definition of HN [Fig. 2(a)], the MO coefficients at z = 0 and z = N are depicted on the x- and θ-axes, respectively. The MO coefficients for the core of the chain, 1 ≤ zN − 1, are plotted in the vector form, Ψnz=ψxzψyz, and can be arbitrarily oriented. Figure 3 demonstrates that, for 0° < θ < 90°, the MOs are no longer confined to a single plane as in Fig. 1. Instead, the MOs are “elliptically polarized” and can be parameterized in the following form:

Ψnz=cosθ/2sinθ/2sinθ/2cosθ/2ancosknz+δnbnsinknz+δn,
(11)

where an and bn are the semi-major/minor axes of the ellipse. The left matrix in Eq. (11) is a 2 × 2 rotation matrix that rotates the ellipse to an orientation angle of ϕ = θ/2. This orientation angle follows from the symmetry of the chain. An additional phase factor, δn, is needed so that ψy(0) = 0 since the px orbital at z = 0 cannot have a ψy component. This condition is satisfied by the formula tan δn = −an/bn tan(θ/2).

FIG. 3.

The helical MOs of HN(θ) for N = 3 and θ = 60°. The MOs are plotted head-on, with the positive z-axis pointing out of the page. The MO coefficients follow an elliptical path (purple) around the axis of the chain. The handedness of the helix alternates between right-handed (n = 1, 3, 5) and left-handed (n = 2, 4, 6). The x- and θ-axes are indicated by the dashed lines.

FIG. 3.

The helical MOs of HN(θ) for N = 3 and θ = 60°. The MOs are plotted head-on, with the positive z-axis pointing out of the page. The MO coefficients follow an elliptical path (purple) around the axis of the chain. The handedness of the helix alternates between right-handed (n = 1, 3, 5) and left-handed (n = 2, 4, 6). The x- and θ-axes are indicated by the dashed lines.

Close modal

To characterize the helical states, it is convenient to introduce an ellipticity angle, −π/2 < φ < π/2, where the semi-major/minor axes are given by a = r cos φ and b = r sin φ (r is determined by the normalization of the MO). When |φ| < π/4, a is the major axis, whereas when |φ| > π/4, b is the major axis (Fig. 4). The sign of φ dictates the handedness of the helical state, with positive and negative φ yielding right-handed and left-handed helices, respectively. We derive the following formula for the ellipticity angle φn of the eigenstates Ψn:

φ±k=±12cos1±cosk1cos2θsin2k,
(12)

where φn = φ+(kn) when n is odd, and φn = φ(kn) when n is even. The functions φ+(k) and φ(k) are plotted in Fig. 4(b). Remarkably, these two functions do not depend on N and only depend on the properties of the chain termination, in this case, the twist angle (θ). For a specific N, the allowed wavenumbers kn are given by Eq. (10) and can be determined graphically [Fig. 4(a)]. The ellipticity angles φn can then be calculated from kn as indicated in Fig. 4(b). In this way, the length of the chain only determines the specific φn but not the overall trends of φ±(k).

FIG. 4.

Mathematical description of helical states of HN(θ) for N = 3 and θ = 60°. (a) Graphical representation of Eq. (10). The allowed wavenumbers kn are given by the points of intersection. (b) The ellipticity angle as a function of wavenumber [Eq. (12)], given by φ+(k) (teal) and φ(k) (orange). The specific φn for N = 3 are indicated by the black markers and correspond to the points of intersection in (a). The vertical sticks are included to guide the eye. (c) The ellipses described by φ+(k) (teal) and φ(k) (orange) for k = 0.35π. The ellipses are oriented at an angle ϕ = θ/2.

FIG. 4.

Mathematical description of helical states of HN(θ) for N = 3 and θ = 60°. (a) Graphical representation of Eq. (10). The allowed wavenumbers kn are given by the points of intersection. (b) The ellipticity angle as a function of wavenumber [Eq. (12)], given by φ+(k) (teal) and φ(k) (orange). The specific φn for N = 3 are indicated by the black markers and correspond to the points of intersection in (a). The vertical sticks are included to guide the eye. (c) The ellipses described by φ+(k) (teal) and φ(k) (orange) for k = 0.35π. The ellipses are oriented at an angle ϕ = θ/2.

Close modal

Since φ+(k) is positive and φ(k) is negative, these ellipticity angles describe right-handed and left-handed helices, respectively. Therefore, the handedness of the helical states alternates with increasing n (e.g., Fig. 3). For a given k, φ+(k) and φ(k) differ by π/2 and produce two perpendicularly oriented ellipses [Fig. 4(c)]. Last, we note that φ(πk) = −φ+(k), which is a manifestation of the Coulson–Rushbrooke pairing theorem.41 This pairing phenomenon can be seen in Fig. 3 for the MO pairs: n = 1,6; n = 2,5; and n = 3,4. For each MO pair (e.g., n = 1,6), both MOs follow the same ellipse shape but with opposite handedness (e.g., n = 1 is right-handed and n = 6 is left-handed).

A slight modification to HN [Eq. (8)] simplifies the mathematics considerably and yields “circularly polarized” states. In Eq. (13), we consider a modified Hamiltonian, HN, where the coupling to the terminal lone p orbitals is increased to 2t [Fig. 5(a)],

H3θ=  px0px1py1px2py2pθ3px0px1py1px2py2pθ302t00002t00t000000t00t0002t cos θ00t002t sin θ0002t cos θ2t sin θ0.
(13)

The characteristic polynomial for HN is presented in the following Eq. (14), where the substitution E = Ek [Eq. (2)] has been made,

detHNEkI=2t2Ncos2Nkcos2θ.
(14)

The roots of the characteristic polynomial give the allowed wavenumbers, kn, listed as follows for even and odd n:

Nkn=θ,θ+π,θ+2π,for n=1,3,5,,πθ,2πθ,3πθ,for n=2,4,6,.
(15)

Remarkably, the eigenstates for HN are “circularly polarized” and are given as

Ψnz=1Ncosknz±sinknz for 1zN1,
(16)

where (+) and (−) describe odd and even n, respectively. As before, the handedness of the helix alternates with increasing n. In Figs. 5(b) and 5(c), the frontier MOs for H′(θ = 60°) when N = 6 are presented. The MO coefficients of the core atoms (1 ≤ z ≤ 5) uniformly twist around the axis of the chain, following a circular path. However, due to the modified coupling, the terminal p orbitals do not obey Eq. (16), and the MO coefficients for z = 0 and z = N are a factor of 1/2 lower in magnitude.

FIG. 5.

(a) Visual representation of Hamiltonian HNθ for N = 3. [(b) and (c)] Circularly polarized states of HN(θ=60) for a seven-carbon chain (N = 6) for (b) n = 6 (HOMO) and (c) n = 7 (LUMO). Note that we present a longer chain length in (b) and (c) than in (a) to better demonstrate the circular polarization.

FIG. 5.

(a) Visual representation of Hamiltonian HNθ for N = 3. [(b) and (c)] Circularly polarized states of HN(θ=60) for a seven-carbon chain (N = 6) for (b) n = 6 (HOMO) and (c) n = 7 (LUMO). Note that we present a longer chain length in (b) and (c) than in (a) to better demonstrate the circular polarization.

Close modal

In the above cumulene-based model, helical states arise from a twist angle between terminal p orbitals. The simple chain termination, i.e., lone p orbitals at z = 0 and z = N, allowed us to derive a complete analytic description of the helical states. However, for more complicated chain terminations, which may better model realistic molecules,18 analytic expressions cannot be obtained, and a more general framework is needed to analyze the helical states. In the following discussion, we generalize the results from the previous sections. We analyze the eigenstates of a finite carbyne chain that is terminated by arbitrary end groups. As we will show, the eigenstates for any tight-binding sp-carbon chain will be elliptically polarized with the familiar linearly polarized states only being a limiting case.

For the generalized analysis, it is convenient to invoke a direct product notation, where the px orbital on atom i is described by 10|iand the py orbital on atom i is described by 01|i. We will consider an sp-carbon chain of N atoms that is capped by some arbitrary end groups. Using the Löwdin partitioning technique,42 the full Hamiltonian can be reduced to an effective Hamiltonian H of dimension 2N × 2N,

H=ΣLE11+ΣRENN+z=1N1tIzz+1+H.c.
(17)

The left (L) and right (R) end groups are incorporated into H by means of 2 × 2 matrices ΣLE=VLEHL1VL and ΣRE=VREHR1VR, where HL(R) is the Hamiltonian of the end group and VL(R) is the coupling between the end group and the terminal atoms of the chain.43,44 In order for H to yield helical states, it is necessary that ΣL and ΣR do not commute. Otherwise ΣL and ΣR could be simultaneously diagonalized, and H could be separated into two uncoupled subsystems.

The eigenstates of H can be computed using an ansatz that describes an elliptically polarized wavefunction,

Ψz=MRkzv^0,
(18)

where v^0 is some unit vector, R is the 2 × 2 rotation matrix, and M is a real, symmetric matrix characterizing the elliptical polarization. The matrix M possesses a pair of orthonormal eigenvectors v^1 and v^2 and associated eigenvalues a and b. The unit vector, Rkzv^0, rotates along the unit circle for increasing z and is mapped by M onto an ellipse whose major/minor axes are given by av^1 and bv^2.45,46 The handedness of the helical state is given by the sign of det[M] = ab. For right-handed helices, det[M] is positive, whereas for left-handed helices, det[M] is negative. We will consider the normalization condition: a2 + b2 = 1, which is equivalent to the condition tr[MTM] = 1. These constraints leave M with 2 free parameters that can be the orientation angle, ϕ, and the ellipticity angle, φ,

M=Rϕcosφ00sinφRϕ.
(19)

For the ansatz [Eq. (18)] to describe the eigenstates of H, the ansatz must satisfy

EΨz=z|H|Ψ
(20)

for all z. Within the bulk of the chain (2 ≤ zN − 1), Eq. (20) yields

EΨz=tΨz1+tΨz+1=tMRk+RkRkzv0=2tcoskMRkzv^0,
(21)

where we have made use of the relation: R(−k) + R(k) = 2 cos k I. Therefore, the ansatz correctly describes eigenstates of H with eigenenergies Ek = 2t cos k [Eq. (2)]. Equation (21) demonstrates that, in general, energy levels for which |En| ≤ 2 t will be associated with helical eigenstates. It should be noted that for certain ΣL(R), there can exist energy levels |En| > 2 t, which cannot be satisfied by Eq. (21). In such instances, the associated eigenstates will not be helical.

Equation (20) must also be satisfied at the ends of the chain. The application of Eq. (20) to z = 1 and z = N yields, respectively, the following equations:

EkΨ1=ΣLEkΨ1+tΨ2=ΣLEkM+tMRkRkv^0,
(22)
EkΨN=tΨN1+ΣREkΨN=tMRk+ΣREkMRNkv^0.
(23)

Substituting the relation EkM = tM(R(−k) + R(k)) into the left-hand side of Eqs. (22) and (23) leads to the simplified following equations:

tMRkΣLEkMRkv^0=0,
(24)
tMRkΣREkMRNkv^0=0.
(25)

It follows that the operator in the square brackets for both Eqs. (24) and (25) has a nonzero kernel and consequently a zero determinant,

dettMRkΣLEkM=0,
(26)
dettMRkΣREkM=0.
(27)

Equations (26) and (27) allow us to determine the relationship between M and ΣL(R) without concerning ourselves with the specifics of Ψn. For a given value of k, these equations represent a system of quadratic equations that can be solved for M(k). We find, in general, four solutions to these equations, two of which are unique. The two solutions can be expressed in the following form:

M+k=Rϕa00bRϕ,Mk=Rϕb00aRϕ.
(28)

The matrices M±(k) represent helical states of opposite handedness, for which the associated ellipses are perpendicularly oriented [i.e., Fig. 4(c)]. This is a nice generalization of φ±(k) [equation (12)]. In accordance with φ±(k), M±(k) do not depend on N and solely depend on the properties of the chain termination (ΣL(R)). One can use Eqs. (26) and (27) to engineer designer helical states through the appropriate selection of ΣL(R).

In conclusion, we have demonstrated that the eigenstates of sp-carbon chains are “elliptically polarized,” drawing analogy to elliptically polarized light. Helical states emerge when the ends of the chain are asymmetrically coupled, which can be made mathematically precise under the condition that [ΣL, ΣR] ≠ 0. The relationship between the helical states and the wavenumber [i.e., M±(k)] depends solely on the chain termination and not on the chain length. The chain length only determines the specific allowed wavenumbers kn. These ideas are made more concrete within a cumulene-based model, for which we provide the complete analytical solution. We hope that the underlying patterns identified herein, at the tight-binding level, will guide future theoretical and experimental efforts into the electronic structure of carbyne.

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

S.G. acknowledges support from the NSF Graduate Research Fellowship under Grant No. DGE-1644869. L.V. acknowledges support from the NSF under Grant No. DMR-1807580.

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