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 *sp*^{2}-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.

## INTRODUCTION

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 *sp*^{2}-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 *electrohelicity*^{17} 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 *p*_{x} and a *p*_{y} orbital for each *sp*-carbon atom in the chain. Note that the *p*_{z} 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 *p*_{x} orbitals as well as adjacent *p*_{y} orbitals. The eigenstates, Ψ_{k}, and eigenenergies, *E*_{k}, associated with the wavenumber, *k*, are^{20,21}

where *C*_{1} and *C*_{2} are complex numbers and *z* is the integer position of the atomic site along the chain. The components *ψ*_{x}(*z*) and *ψ*_{y}(*z*) represent the MO coefficients for the *p*_{x} and *p*_{y} orbitals on atom *z*, respectively. Since there is no coupling between *ψ*_{x} and *ψ*_{y}, i.e., $\u27e8\psi y|H|\psi x\u27e9=0$, the complex numbers *C*_{1} and *C*_{2} 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 *C*_{1} and *C*_{2} differ by a complex phase, the eigenstate Ψ_{k} 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, *t*_{1}, *t*_{2} < 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 *t*_{1} and the intercell coupling by *t*_{2}.^{29} The eigenstates are described by a Bloch wave,

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 ($Ek\u2212$) and an upper band ($Ek+$) that are separated by a bandgap,^{30–32}

Due to the discrete basis of *p* orbitals, the periodic component of the Bloch wave is described by discrete values *u*_{A} and *u*_{B} for sublattices A and B, respectively. The values of *u*_{A} and *u*_{B} are equal in magnitude (|*u*_{B}| = |*u*_{A}|), but differ in phase. For the lower band of states (i.e., $Ek\u2212$), $uB=\u2212eik/2+\eta kuA$, and for the upper band of states (i.e., $Ek+$), $uB=eik/2+\eta kuA$, where we have defined $\eta k=argt1e\u2212ik/2+t2eik/2$.^{29,33} Incorporating the phase difference between *u*_{A} and *u*_{B} into Eq. (3), we see that $\Psi kA$ and $\Psi kB$ are translationally offset in the *z*-direction, yielding a *double helix* structure: one helix for sublattice A and another for sublattice B. Since $\Psi kA$ and $\Psi kB$ 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 *ψ*_{x}(*z*) and *ψ*_{y}(*z*). We present an analytic model based on cumulene^{13} 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.

## RESULTS and DISCUSSION

### Linearly polarized MOs

Before discussing the elliptically polarized MOs of the cumulene-based model, it is instructive to first consider finite *sp*^{2}-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}

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

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, *E*_{n}, associated with *ψ*_{n} exhibit the same functional form as for infinite chains [Eq. (2)]: *E*_{n} = 2*t* cos *k*_{n}. In fact, Eq. (2) holds for all of the systems discussed herein, although the specific wavenumbers, *k*_{n}, depend on the boundary conditions of the system.

### Elliptically polarized MOs

To realize “elliptically polarized” states, a phase difference between *ψ*_{x} and *ψ*_{y} must be introduced under boundary conditions that asymmetrically couple *ψ*_{x} and *ψ*_{y}. 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, *H*_{N}(*θ*), describes a chain of *N* − 1 *sp*-carbon atoms terminated with single *p* orbitals (i.e., *sp*^{2}-carbon atoms) oriented at a dihedral angle, *θ* [Fig. 2(a)]. Including the terminal *sp*^{2}-carbon atoms, the chain has a total of *N* + 1 atoms and comprises a total of 2*N 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,

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):

Since the chain comprises *N* + 1 atoms, for *H*_{3}(*θ*), there are 4 atoms labeled as *z* = 0, 1, 2, 3. The first atom in the chain (*z* = 0) is *sp*^{2}-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 *sp*^{2}-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)].

The energy levels of *H*_{N} are given by the roots of the characteristic polynomial, det(*H*_{N} − *EI*), where *I* denotes the identity matrix. The characteristic polynomial takes a simple form if the energy, *E*, is expressed as *E*_{k} = 2*t* cos *k* [Eq. (2)] as follows:^{39,40}

Setting Eq. (9) to zero yields a trancendental equation describing the allowed *k*_{n} for a given twist angle *θ*,

There are two limiting cases of Eq. (10) that are worth noting: *θ* = 0° and *θ* = 90°. In both of these situations, *H*_{N} 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)*k*_{n} sin *Nk*_{n} = 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 sin^{2}(*N* + 1)*k*_{n} = 0, which describes two chains of length *N* [Fig. 2(a)]. The limiting cases manifest in the energy spectrum for *H*_{3}(*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 *sp*^{2}-carbon molecules. At *θ* = 0°, the energy levels correspond to butadiene (*N*_{x} = 4, red) and ethylene (*N*_{y} = 2, blue), while at *θ* = 90°, the energy levels are doubly degenerate and correspond to the allyl radical (*N*_{x} = *N*_{y} = 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 *p*_{x} and *p*_{y} π-systems are inextricably linked. In Fig. 3, the helical states of *H*_{3}(*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 *H*_{N} [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 ≤ *z* ≤ *N* − 1, are plotted in the vector form, $\Psi nz=\psi xz\psi yz$, 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:

where *a*_{n} and *b*_{n} 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, *δ*_{n}, is needed so that *ψ*_{y}(0) = 0 since the *p*_{x} orbital at *z* = 0 cannot have a *ψ*_{y} component. This condition is satisfied by the formula tan *δ*_{n} = −*a*_{n}/*b*_{n} tan(*θ*/2).

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 *φ*_{n} of the eigenstates Ψ_{n}:

where *φ*_{n} = *φ*^{+}(*k*_{n}) when *n* is odd, and *φ*_{n} = *φ*^{−}(*k*_{n}) 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 *k*_{n} are given by Eq. (10) and can be determined graphically [Fig. 4(a)]. The ellipticity angles *φ*_{n} can then be calculated from *k*_{n} as indicated in Fig. 4(b). In this way, the length of the chain only determines the specific *φ*_{n} but not the overall trends of *φ*^{±}(*k*).

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*^{−}(*π* − *k*) = −*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).

### Circularly polarized MOs

A slight modification to *H*_{N} [Eq. (8)] simplifies the mathematics considerably and yields “circularly polarized” states. In Eq. (13), we consider a modified Hamiltonian, $HN\u2032$, where the coupling to the terminal lone *p* orbitals is increased to $2t$ [Fig. 5(a)],

The characteristic polynomial for $HN\u2032$ is presented in the following Eq. (14), where the substitution *E* = *E*_{k} [Eq. (2)] has been made,

The roots of the characteristic polynomial give the allowed wavenumbers, *k*_{n}, listed as follows for even and odd *n*:

Remarkably, the eigenstates for $HN\u2032$ are “circularly polarized” and are given as

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.

### Generalized analysis

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 *p*_{x} orbital on atom *i* is described by $10\u2297|i$and the *p*_{y} orbital on atom *i* is described by $01\u2297|i$. 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 2*N* × 2*N*,

The left (*L*) and right (*R*) end groups are incorporated into *H* by means of 2 × 2 matrices $\Sigma LE=VLE\u2212HL\u22121VL\u2020$ and $\Sigma RE=VRE\u2212HR\u22121VR\u2020$, where *H*_{L(R)} is the Hamiltonian of the end group and *V*_{L(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 Σ_{L} and Σ_{R} do not commute. Otherwise Σ_{L} and Σ_{R} 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,

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: *a*^{2} + *b*^{2} = 1, which is equivalent to the condition tr[*M*^{T}*M*] = 1. These constraints leave *M* with 2 free parameters that can be the orientation angle, *ϕ*, and the ellipticity angle, *φ*,

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

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

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 *E*_{k} = 2*t* cos *k* [Eq. (2)]. Equation (21) demonstrates that, in general, energy levels for which |*E*_{n}| ≤ 2 *t* will be associated with helical eigenstates. It should be noted that for certain Σ_{L(R)}, there can exist energy levels |*E*_{n}| > 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:

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

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

Equations (26) and (27) allow us to determine the relationship between *M* and Σ_{L(R)} without concerning ourselves with the specifics of Ψ_{n}. 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:

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 (Σ_{L(R)}). One can use Eqs. (26) and (27) to engineer designer helical states through the appropriate selection of Σ_{L(R)}.

## CONCLUSIONS

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 [Σ_{L}, Σ_{R}] ≠ 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 *k*_{n}. 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 AVAILABILITY

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

## ACKNOWLEDGMENTS

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.

## REFERENCES

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