Bending Effect of sp-Hybridized Carbon - American Chemical Society

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Bending Effect of sp-Hybridized Carbon (Carbyne) Chains on Their Structures and Properties Yun Hang Hu* Department of Materials Science and Engineering, Michigan Technological University, Houghton, Michigan 49931, United States ABSTRACT: Density functional theory (DFT) calculations were exploited to evaluate the bending of polyynic and polycumulenic atomic carbon chains. It was found that the chains can be easily bent with a small arc-chord ratio. Furthermore, the bending of the chains caused the variation of bond lengths, leading to the change in the chain lengths; namely, the chain was first shortened slightly and then elongated with increasing arcchord ratio. Furthermore, the bending decreased all bond angles as an oscillation manner, in which the larger decrease of bond angles occurred on even-ordinal-number carbon atoms than on corresponding odd-ordinal-number carbons, and the largest decrease was at the middle of the chain. With increasing the bending, the HOMO-LUMO energy gap of the polyynic chain was decreased, whereas the gap of the polycumulenic chain remained almost unchanged. This indicates that bending can increase the electric conductance of the polyynic chain, whereas the conductance of the polycumulenic chain is not affected by bending. PFurthermore, it was found that strain energy due to bending is determined by bent angles (θi) at all carbon atoms as Estrain = 2.54 n-1 i=2 (1 - cos(θi)). Because the strain energy even due to a large bending is still much smaller than the energy required for breaking a carbon bond in a chain, it should be very difficult to break the chain by bending.

1. INTRODUCTION As a third carbon allotrope, the sp-hybridized carbon chain; carbyne;has attracted much attention both experimentally and theoretically due to its significance in science and engineering.1-15 A sp carbon chain can have two possible types of bonding structures: polyyne (-C
C-C 3 3 3 C
C-), which features an alternating triple-single bond pattern, and polycumulene (CdCdC 3 3 3 CdCdC) with nearly equivalent double-bond lengths.7-12 The research for carbyne chains has been focused on the experimental confirmation of their existence. Spectroscopic information of molecules containing such carbon chains was reported for the interstellar space.12 Linear carbon chains capped with some other atom or organic radical have been synthesized and characterized.12 In the recent years, long linear carbon chains inside nanotubes were also reported.3,4 Very recently, a single sp carbon chain was produced and observed by high-resolution transmission electron microscopy (HR-TEM).6 Bending (or curving) plays an important role in the formation of carbon nanostructured materials. Fullerenes and nanotubes can be considered as wrapping and rolling-up graphenes, respectively. Graphene, which consists of perfect sp2-hybridized (trigonal) carbon atoms, is a planar conjugated organic system. In contrast, the curving of planar graphene into nonplanar fullerenes and tubes causes a distortion in π-bonding.16-19 Furthermore, isolated single-walled carbon nanotubes (SWCNTs) are always curved instead of straight, indicating that they are easily bent. For this reason, the bending effects of SWCNTs on their structures r 2011 American Chemical Society

and properties have widely been investigated theoretically and experimentally.20-27 For example, Hansson et al. theoretically studied the effect of bending on the conductance of carbon nanotubes using the Landauer formalism.28 The sp carbon chains are commonly assumed to be “linear” molecules. However, as a thinnest carbon wire, a sp carbon chain, which contains only a single atom in its cross section, should be bent much easier than any carbon nanotube. Indeed, X-ray crystallographic measurements have shown that polyynic molecules often exhibit remarkable static nonlinearity in its solid state and a solution.29-44 Furthermore, the bending of carbon chains could be expected to play a dominant role in fullerene formation.12,39-43 Those results indicate that a bent structure may be an important characteristic of a sp carbon chain. However, to the best of our knowledge, theoretical and experimental work regarding effects of bending on sp carbon chains is very limited. This situation prompted us to investigate, using density functional theory (DFT), how the bending of an sp carbon chain changes its bond structure and properties in this work.

2. CALCULATION METHODS Density functional theories (DFT) are widely exploited to evaluate structures and energies for nanomaterials due to their Received: December 13, 2010 Revised: December 21, 2010 Published: January 10, 2011 1843

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Figure 1. Straigth and bent polycumulenic (C20H4) chains with various arc-chord ratios (τ).

reliable accuracies at affordable calculation cost.45-47 One of the most popular DFT methods is the B3LYP, which is based upon the Becke three-parameter exchange coupled with the LeeYang-Parr (LYP) correlation potential.45 Although the bond lengths of C60 predicted by B3LYP hybrid DFT calculations were in very good agreement with experimental data,48 the B3LYP calculations overestimated the energy gap of C60 between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO).49 In contrast, PW91PW91 DFT calculation with the 6-31g(d) basis set, which is based upon Perdew-Wang’s exchange and correlation,47 predicted a HOMOLUMO energy gap of 1.674 eV for C60, which is in excellent agreement with the experimental value (1.7 eV).50,51 Furthermore, the PW91PW91 DFT calculation can also produce accurate bond lengths of 1.4031 Å (double bond) and 1.4561 Å (single bond) for C60, which are in very good agreement with the experimental values (1.401 and 1.458 Å).52 For this reason, PW91PW91 DFT with the 6-31g(d) basis set was employed for geometric optimizations of atomic carbon chains and their energy calculations in this work. C20 was used as a chain model for all calculations; namely, polyynic and polycumulenic carbon chains were generated by saturating each end of the C20 chain with one and two hydrogen atoms, respectively (Figures 1 and 2). All calculations for geometric optimizations and energies for both straight and bent chains were carried out using the Gaussian 03 program.53 To perform bending, the geometry of the C20 chain was optimized with fixing the distance between its two carbon ends as 20, 18, 16, 14, 12, 10, 8, and 6 Å, respectively, leading to eight bent chains (Figures 1 and 2). The simplest mathematic method to measure the tortuosity of a curved line is arc-chord ratio (τ), which is the ratio of the length (L) of a chain to the distance (D) between its two ends as follows L ð1Þ τ ¼ D

Arc-chord ratio is equal to 1 for a straight line and is greater than 1 for a curved line. The larger the arc-chord ratio, the more a line is curved. For example, arc-chord ratios are 1.11, 1.57, and 3.33 for one-fourth, a half, and three-fourth circles, respectively. Therefore, the arc-chord ratio was employed to express the tortuosities of bent chains in this work (see Figures 1 and 2).

3. RESULTS AND DISCUSSION 3.1. Effect of Bending Chain on Its Structure. The straight polycumulenic C20H4 and polyynic C20H2 chains, which have an arc-chord ratio of 1, were first optimized. As shown in Table 1, one can see that the bond lengths of the polycumulenic carbon chain in its straight form are fairly close, though with very small difference between subsequent bond lengths and slightly greater differences at the ends of the chain due to the saturation with hydrogen atoms. In contrast, different from a polycumulenic chain with nearly equivalent double-bond lengths, the polyynic chain occupies alternating single and triple bonds (Table 2). Those are consistent with reported predictions.7-12 The geometric optimizations of polycumulenic C20H4 and polyynic C20H2 chains were performed with fixed distances between the two carbon ends of each chain, producing bent chains with various arc-chord ratios (Figures 1 and 2). The bending caused variations in bond lengths (Tables 1 and 2). Some of their bond lengths decreased, and others increased, leading to the changes of chain lengths; namely, the chain lengths were slightly shortened first and then elongated with increasing arc-chord ratio (Figures 3). The bending exhibited a great effect on the bond angles. As shown in Figures 4 and 5, one can see that all bond angles of straight polycumulenic and polyynic chains are 180°, but those bond angles were decreased by bending. Furthermore, the decreases are not the same for all bond angles in the chains. The decreases of bond angles are larger at evenordinal-number carbons than those at corresponding odd-ordinalnumber carbons, constituting an oscillation behavior with the 1844

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Figure 2. Straigth and bent polyynic (C20H2) chains with various arc-chord ratios (τ).

Table 1. Bond Lengths of Bent Polycumulenic C20H4 Chains bond length (Å) bond

τ = 1a

τ = 1.23

τ = 1.36

τ = 1.53

τ = 1.75

τ = 2.05

τ = 2.46

τ = 3.07

τ = 4.10

C1-C2 C2-C3

1.3269 1.2806

1.3263 1.2801

1.3264 1.2801

1.3266 1.2802

1.3269 1.2803

1.3270 1.2804

1.3273 1.2804

1.3276 1.2805

1.3278 1.2806

C3-C4

1.2947

1.2944

1.2947

1.2949

1.2952

1.2955

1.2957

1.296

1.2963

C4-C5

1.2837

1.2833

1.2833

1.2833

1.2834

1.2835

1.2836

1.2837

1.2838

C5-C6

1.2917

1.2916

1.2920

1.2922

1.2926

1.2929

1.2932

1.2936

1.2940

C6-C7

1.2851

1.2849

1.2850

1.2852

1.2854

1.2856

1.2857

1.2859

1.2861

C7-C8

1.2908

1.2907

1.2911

1.2914

1.2918

1.2922

1.2926

1.2931

1.2937

C8-C9

1.2856

1.2857

1.2859

1.2862

1.2865

1.2867

1.2870

1.2874

1.2878

C9-C10 C10-C11

1.2904 1.2857

1.2903 1.2860

1.2907 1.2863

1.2911 1.2867

1.2915 1.2871

1.2920 1.2874

1.2925 1.2879

1.2930 1.2884

1.2936 1.2890

C11-C12

1.2904

1.2903

1.2906

1.2909

1.2913

1.2916

1.2920

1.2925

1.2930

C12-C13

1.2856

1.2859

1.2862

1.2866

1.2870

1.2875

1.2880

1.2885

1.2891

C13-C14

1.2908

1.2904

1.2907

1.2909

1.2911

1.2914

1.2916

1.2919

1.2921

C14-C15

1.2851

1.2853

1.2856

1.2860

1.2863

1.2867

1.2871

1.2875

1.2880

C15-C16

1.2917

1.2911

1.2912

1.2914

1.2915

1.2917

1.2918

1.2919

1.2921

C16-C17

1.2837

1.2838

1.2841

1.2844

1.2847

1.2850

1.2853

1.2856

1.2860

C17-C18 C18-C19

1.2947 1.2806

1.2940 1.2803

1.2941 1.2805

1.2941 1.2807

1.2942 1.2809

1.2943 1.2811

1.2944 1.2813

1.2944 1.2816

1.2945 1.2818

C19-C20

1.3269

1.3262

1.3263

1.3264

1.3265

1.3267

1.3268

1.3271

1.3272

BLAb

0.0149

0.0146

0.0147

0.0146

0.0146

0.0146

0.0146

0.0146

0.0146

a

τ is arc-chord ratio. It is equal to 1 for a straight chain and is larger than 1 for a bent chain. b Bond Length Alternation (BLA) is defined as average value of bond lengths of CC bonds in “even” position along the chain minus the average value of bond lengths of CC bonds in “odd” position.

ordinal number of carbons. The largest decrease of a bond angle occurs at the middle carbon atom (10th carbon). This indicates that the bonds associated with the middle atom can be bent the easiest.

3.2. Strain Energy of Bent Chains. The energies for bent chains with various arc-chord ratios are listed in Tables 3 and 4. The total system energies of both polyynic and polycumulenic chains increased with increasing arc-chord ratio, indicating that 1845

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Table 2. Bond Lengths of Bent Polyynic C20H2 Chains bond length (Å) bond

τ = 1a

τ = 1.22

τ = 1.36

τ = 1.53

τ = 1.75

τ = 2.04

τ = 2.45

τ = 3.06

τ = 4.08

C1-C2

1.2306

1.2302

1.2304

1.2303

1.2304

1.2305

1.2306

1.2307

1.2308

C2-C3

1.3450

1.3443

1.3444

1.3447

1.3448

1.345

1.3451

1.3452

1.3454

C3-C4

1.2484

1.2481

1.2483

1.2483

1.2485

1.2487

1.2488

1.2490

1.2492

C4-C5

1.3286

1.3281

1.3282

1.3284

1.3285

1.3287

1.3288

1.3290

1.3292

C5-C6

1.2555

1.2554

1.2556

1.2558

1.2560

1.2562

1.2564

1.2567

1.2569

C6-C7

1.3226

1.3223

1.3225

1.3228

1.3231

1.3234

1.3236

1.3239

1.3242

C7-C8

1.2586

1.2585

1.2588

1.2591

1.2594

1.2597

1.2601

1.2604

1.2608

C8-C9 C9-C10

1.3199 1.2599

1.3201 1.2599

1.3204 1.2602

1.3206 1.2606

1.321 1.2609

1.3215 1.2612

1.3219 1.2616

1.3223 1.2620

1.3229 1.2625

C10-C11

1.3192

1.3194

1.3198

1.3202

1.3207

1.3213

1.3219

1.3225

1.3233

C11-C12

1.2599

1.2598

1.2602

1.2604

1.2607

1.2610

1.2613

1.2617

1.2620

C12-C13

1.3199

1.3202

1.3206

1.3210

1.3216

1.3222

1.3228

1.3234

1.3242

C13-C14

1.2586

1.2584

1.2587

1.2588

1.2590

1.2592

1.2594

1.2596

1.2598

C14-C15

1.3226

1.3229

1.3232

1.3236

1.3240

1.3245

1.325

1.3255

1.3261

C15-C16

1.2555

1.2551

1.2553

1.2553

1.2554

1.2555

1.2556

1.2557

1.2559

C16-C17 C17-C18

1.3286 1.2484

1.3286 1.248

1.3289 1.2481

1.3294 1.248

1.3298 1.248

1.3302 1.2481

1.3306 1.2482

1.3309 1.2482

1.3314 1.2483

C18-C19

1.3450

1.3446

1.3447

1.3452

1.3454

1.3457

1.3459

1.3462

1.3465

C19-C20

1.2306

1.2302

1.2303

1.2303

1.2303

1.2305

1.2306

1.2306

1.2308

BLAb

0.0773

0.0775

0.0775

0.0777

0.0779

0.0781

0.0783

0.0784

0.0787

a

τ is arc-chord ratio. It is equal to 1 for a straight chain and is larger than 1 for a bent chain. b Bond Length Alternation (BLA) is defined as average value of bond lengths of CC bonds in “odd” position along the chain minus the average value of bond lengths of CC bonds in “even” position.

Figure 3. Lengths of bent polyynic (C20H2) and polycumulenic (C20H4) chains vs arc-chord ratios.

Figure 5. Bond angles of bent polyynic (C20H2) chains with various arc-chord ratios (τ) vs ordinal numbers of carbon (τ is arc-chord ratio).

Table 3. Energies of Bent Polycumulenic C20H4 Chains

Figure 4. Bond angles of bent polycumulenic (C20H4) chains with various arc-chord ratios (τ) vs ordinal numbers of carbon. 1846

arc-chord

total energy

strain energy

HOMO-LUMO

ratio

(Hartree)

(eV)

gap (eV)

1

-762.500398

0.000000

0.8768

1.23

-762.490589

0.271465

0.8803

1.36

-762.485913

0.398770

0.8814

1.53

-762.481009

0.531978

0.8822

1.75

-762.475849

0.672239

0.8833

2.05 2.46

-762.470387 -762.464579

0.820821 0.978642

0.8836 0.8838

3.07

-762.458356

1.147314

0.8838

4.10

-762.451555

1.330578

0.8817

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Table 4. Energies of Bent Polyynic C20H2 Chains arc-chord

total energy

strain energy

HOMO-LUMO

ratio

(Hartree)

(eV)

gap (eV)

1

-762.500398

0.000000

1.3813

1.22

-762.490589

0.266923

1.3775

1.36

-762.485913

0.394164

1.3736

1.53

-762.481009

0.527622

1.3712

1.75

-762.475849

0.668026

1.3677

2.04

-762.470387

0.816666

1.3647

2.45 3.06

-762.464579 -762.458356

0.974714 1.144036

1.3603 1.3557

4.08

-762.451555

1.329119

1.3505

the bending decreased their stabilities. The energy change of a sp carbon chain from its straight form to the bent one can be defined as its strain energy (Estrain); namely Estrain ¼ Ebent - Estraight

ð2Þ

where Estrain, Estraight, and Ebent are strain energy due to bending, energy of the straight chain, and energy of the bent chain, respectively. The strain energies were calculated and listed in Tables 3 and 4. One can see that the strain energy is only 0.27 eV for a bent chain with small arc-chord ratio of 1.22. This indicates that a sp carbon chain can be easily bent to a small arc-chord ratio, which requires a small force to overcome small strain energy. However, strain energy can increase from 0.27 to 1.5 eV with increasing arc-chord ratio from 1.22 (small bending, Figure 1b and 2b) to 4.1 (large bending, Figures 1i and 2i). Furthermore, we calculated the energy difference between a straight chain and its broken one (between 10th and 11th carbon atoms), which is 5.91 eV (for polycumulenic C20H4) and 6.74 eV (for polyynic C20H2). This indicates that the strain energy (1.5 eV) due to a large bending (arc-chord ratio of 4.1) is three times smaller than the energy (5.91-6.74 eV) required to break the chain. Therefore, it is very difficult to break an sp carbon chain by bending. In other words, sp carbon chains should be considered as a special soft material. The primary reason causing the strain in an sp carbon chain is the deviation of bond angles from the optimized one (180°). In general, there are two types of chemical bonds for a compound: electrostatic bond and covalent bond.54 Because electrostatic interaction is independent of orientation, the strength of the electrostatic bond is not affected by bond angles.55 In contrast, covalent bonding constitutes highly directional bonds via sharing a pair of electrons between two nearest atoms, which requires the match in both energies and orientations of orbitals.56 As a result, the strengths of covalent bonds are strongly dependent on bond angles.55 However, as pointed out by Pauling,54 even for a covalent, there is also some electrostatic interaction. Furthermore, while most atomic orbitals have orientations, some orbitals (such as the s orbital) do not have orientations. Therefore, a covalent bond should contain two parts: orientation-dependent component (Fd) and orientation-independent component (Fnd). In other words, the total bonding ability (F) of an atom to its nearest atom in a covalent molecule can be simply divided into two parts: orientation-dependent component (Fd) and orientationindependent component (Fnd); namely F ¼ Fd þ Fnd

ð3Þ

Figure 6. Strain energies of polycumulenic (C20H4) and polyynic P (C20H2) chains vs n-1 i=2 (1 - cos(θi)).

Furthermore, because an sp-hybridized carbon atom can form two optimized bonds at its perfect bonding orientations, which constitute a bond angle of 180°, the orientation-dependent bonding strength of an sp carbon atom to its nearest atom is equal to Fd at the perfect orientation. If the orientation-dependent bonding strength of an sp carbon at the perfect orientation is broken down into two parts, Fdx (at x-axis orientation) and Fdy (at y-axis orientation), one can obtain the following expression Fdx ¼ Fd cosðθÞ

ð4Þ

where θ is the angle between the perfect bonding orientation and x-axis. If the bonding orientation of an sp carbon is deviated from its perfect bonding orientation by θ angle, the total bonding strength of the sp carbon atom to its nearest atom can be expressed as F ¼ Fnd þ F dx ð5Þ The combination between eqs 4 and 5 gives F ¼ Fnd þ Fd cosðθÞ

ð6Þ

Therefore, the change (ΔF) of bonding strength due to the deviation of bond orientation from its perfect one by the θ angle can be expressed as ΔF ¼ ðFnd þ Fd Þ - ðFnd þ Fd cosðθÞÞ

ð7Þ

ΔF ¼ Fd ð1 - cosðθÞÞ

ð8Þ

Thus If an sp-hybridized chain is bent at a carbon atom by θ degree, strain energy (Estrain) for this atom should be equal to the change (ΔF) of bonding strength due to the bending. Because bending a chain can cause a change of bond angles at all carbon atoms except two ending carbon atoms, the total strain energy of a bent chain can be expressed as n-1 X Estrain ¼ Fd ð1 - cosðθi ÞÞ ð9Þ i¼2

where Estrain is strain energy, i the ordinal number of carbons, θi the bent angle at the ith carbon, and n the total number of carbon atoms in a chain. This equation indicates that Estrain should have a 1847

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Figure 7. HOMO-LUMO energy gap of bent polycumulenic (C20H4) chains vs arc-chord ratios.

Figure 8. HOMO-LUMO energy gap of bent polyynic (C20H2) chains vs arc-chord ratios.

P linear relationship with n-1 i=2 (1 - cos(θi)) for bent sp-hybridized carbon chains. Indeed, as shown in Figure 6, one can Psee a perfect linear relationship between the strain energy and n-1 i=2 (1 - cos(θi)), leading to the following linear equation Estrain ¼ 2:54

n-1 X

ð1 - cosðθi ÞÞ

ð10Þ

i¼2

This indicates that the orientation-dependent bonding strength (Fd) of an sp carbon to its nearest carbon atom is 2.54 eV. By using this equation, one can easily calculate the strain energy from bent angles for any sp carbon chain. 3.3. Effect of Bending a Chain on Its HOMO-LUMO Energy Gap. One of the potential applications of carbyne chains is a molecular wire. The energy gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) is a key parameter determining the conductance of a molecular wire.57,58 Although conductance is also dependent on the position of the HOMO and LUMO relative to the Fermi energy of electrodes, the smaller HOMOLUMO gap of a molecular wire favors a higher conductance. The energy gap, with which the electrons have a difficulty to move, is typically greater than 5 eV. In contrast, for the realization of “charge transfer” between HOMO and LUMO bands, the energy difference between them must be small compared with the bandwidth. Since the bandwidth for an ordinary organic metal is about 0.5-1 eV, the HOMO-LUMO energy gap must be less than 0.5 eV.59 Therefore, a material with the energy gap larger than 5 eV is defined as an insulator, whereas one with the energy gap smaller than 0.5 eV is called a conductor. Furthermore, a material with the energy gap between 0.5 and 3.5 eV is a semiconductor. It would be important to examine how the bending of an sp hybrid (carbyne) chain affects its HOMOLUMO energy gap, which can allow one to evaluate the conductance of the bent chains. The energy gaps were calculated for polyynic C20H2 and polycumulenic C20H4 chains with various arc-chord ratios (Figures 7 and 8). The straight polycumulenic and polyynic chains have the energy gaps of 0.87 and 1.38 eV, respectively, indicating their characteristics as a semiconductor. Furthermore, one can see that the energy gap of the polycumulenic C20H4 chain remained almost unchanged with increasing arc-chord ratio from 1 to 4.1 (Figure 7). In contrast, the energy gap of the polyynic C20H2 chain decreased from 1.38 to 1.35 eV with increasing arc-chord ratio from 1 to 4.1 (Figure 8). Bond length alternation (BLA), which is defined as average value

Figure 9. HOMO-LUMO energy gap of bent polyynic (C20H2) chains vs bond length alternation (BLA).

of bond lengths of CC bonds in “even” position along the chain minus the average value of bond lengths of CC bonds in “odd” position, is directly correlated to the electronic gap.14d For the polycumulenic C20H4 chain, its unchanged HOMO-LUMO gap with increasing arc-chord ratio implies that its BLA is constant. Indeed, the BLA of the bent polycumulenic C20H4 chain remained almost unchanged with increasing arc-chord ratio from 1 to 4.1 (Table 1). Furthermore, the direct correlation between the HOMO-LUMO gap and BLA was also observed for the polyynic C20 H 2 chain. As shown in Figure 9, the HOMO-LUMO gap of the polyynic C20H2 chain linearly decreased with increasing BLA. Because the electrical conductance of a molecular wire is reversely proportional to its HOMO-LUMO energy gap, the bending of a polyynic chain, which decreased its HOMOLUMO gap (Figure 8), should lead to an increase in its conductivity, whereas the bending has no effect on the conductance of a polycumulenic chain. For this reason, the polyynic chains may be a potential material as pressure sensors, in which pressure can cause the bending of the chains, leading to a change of their conductance.

4. CONCLUSION In conclusion, density functional theory (DFT) calculations demonstrated that the bending of the chains caused unique changes in their structures and properties. Bending an sp hybrid carbon chain can lead to the variation of bond lengths and thus change the chain lengths; namely, the chain was first shortened slightly and then elongated with increasing arc-chord ratio. Furthermore, the bending decreased 1848

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The Journal of Physical Chemistry C all bond angles as an oscillation manner, in which the larger decrease of bond angles occurred on even-ordinal-number carbon atoms than on corresponding odd-ordinal-number carbons, and the largest decrease was at the middle of the chain. The HOMO-LUMO energy gap of a polyynic chain was decreased by bending, whereas the gap of a polycumulenic chain remained almost unchanged during bending. This indicates that bending can increase the electric conductance for a polyynic chain, whereas the effect of the bending on the conductance of a polycumulenic chain is negligible. Both polycumulenic and polyynic chains are easily bent to a small arc-chord ratio. The strain energy due to bending is determined Pn-1 by bonding angles (θi) at all carbon atoms as Estrain = 2.54 i=2 (1 - cos(θi)). Furthermore, because the strain energy even due to a large bending is much smaller than the energy required to break a carbon chain, it should be very difficult to break the chain by bending. Therefore, sp hybrid carbon (carbyne) chains can be considered as a special soft material.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ REFERENCES (1) (a) Henning, T.; Salama, F. Science 1998, 282, 2204–2210. (b) Giesen, T. F.; Vanorden, A.; Hwang, H. J.; Fellers, R. S.; Provencal, R. A.; Saykally, R. J. Science 1994, 265, 756–759. (2) Lagow, R. J.; Kampa, J. J.; Wei, H. C.; Battle, S. L.; Genge, J. W.; Laude, D. A.; Harper, C. J.; Bau, R.; Stevens, R. C.; Haw, J. F.; Munson, E. Science 1995, 267, 362–367. (3) Zhao, X.; Ando, Y.; Liu, Y.; Jinno, M.; Suzuki, T. Phys. Rev. Lett. 2003, 90, 187401. (4) Wang, Z.; Ke, X.; Zhu, Z.; Zhang, F.; Ruan, M.; Yang, J. Phys. Rev. B 2000, 61, R2472. (5) (a) Cazzanelli, E.; Castriota, M.; Caputi, L. S.; Cupolillo, A.; Gialombardo, C.; Papagno, L. Phys. Rev. B 2007, 75, 121405. (b) D’Urso, L.; Grasso, G.; Messina, E.; Bongjorno, C.; Scuderi, V.; Scalese, S.; Puglisi, O.; Spoto, G.; Compagnini, G. J. Phys. Chem. C 2010, 114, 907–915. (6) Jin, C.; Lan, H.; Peng, L.; Suenaga, K.; Ijima, S. Phys. Rev. Lett. 2009, 102, 205501. (7) (a) Yang, S. J.; Kertesz, M. J. Phys. Chem. A 2008, 112, 146–151. (b) Yang, S. J.; Kertesz, M. J. Phys. Chem. A 2006, 110, 9771. (c) Kertesz, M.; Koller, J.; Azman, A. J. Chem. Phys. 1978, 68, 2779. (8) (a) Kim, H. J.; Oh, S.; Kim, K. S.; Zhang, Z. Y.; Cho, J. H. Phys. Rev. B 2010, 82, 041401. (b) Rodrigues, K. R.; Williams, S. M.; Young, M. A.; Teeters-Kennedy, S.; Heer, J. M.; Coe, J. V. J. Chem. Phys. 2006, 125, 194716. (9) (a) Jacquemin, D.; Champagne, B. Int. J. Quantum Chem. 2000, 80, 863–870. (b) Poulsen., T. D.; Mikkelsen, K. V.; Fripiat, J. G.; Jacquemin, D.; Champagne, B. J. Chem. Phys. 2001, 114, 5917–5922. (10) Hu, Y. H. J. Phys. Chem. C 2009, 113, 17751–17754. (11) Hu, Y. H. Phys. Lett. A 2009, 373, 3554–3557. (12) Van Orden, A.; Saykally, R. J. Chem. Rev. 1998, 98, 2313–2357. (13) Tommasini, M.; Fazzi, D.; Milani, A.; Zoppo, M. D.; Castiglioni, C.; Zerbi, G. J. Phys. Chem. A 2007, 111, 11645–11651. (14) (a) Milani, A.; Tommasini, M.; Fazzi, D.; Castiglioni, C.; Zoppo, M. D.; Zerbi, G. J. Raman Spectrosc. 2008, 39, 164–168. (b) Milani, A.; Tommasini, M.; Zerbi, G. J. Chem. Phys. 2008, 128, 064501. (c) Milani, A.; Tommasini, M.; Zoppo, M. D.; Castiglioni, C.; Zerbi, G. Phys. Rev. B 2006, 74, 153418. (d) Milani, A.; Tommasini, M.; Zerbi, G. J. Raman Spectrosc. 2009, 40, 1931–1934. (15) (b) Scemama, A; Chaquin, P; Gazeau, M. C.; Benilan, Y. Chem. Phys. Lett. 2002, 361, 520–524.

ARTICLE

(16) Schmalz, T. G.; Seitz, W. A.; Klein, D. J.; Hite, G. E. J. Am. Chem. Soc. 1988, 110, 1113–1127. (17) Bakowies, D.; Thiel, W. J. Am. Chem. Soc. 1991, 113, 3704– 3714. (18) Haddon, R. C. Science 1993, 261, 15451550. (19) Haddon, R. C. J. Am. Chem. Soc. 1986, 108, 2837–2842. (20) Hansson, A.; Paulsson, M.; Stafstr€om, S. Phys. Rev. B 2000, 62, 7639. (21) Chibotaru, L. F.; Bovin, S. A.; Ceulemans, A. Phys. Rev. B 2002, 66, 161401(R). (22) Zhang, Z.; Guo, W. Appl. Phys. Lett. 2007, 90, 053114. (23) Wang, B.; Gupta, A. K.; Huang, J.; Vedala, H.; Hao, Q.; Crespi, V. H.; Choi, W.; Eklund, P. C. Phys. Rev. B 2010, 81, 115422. (24) Rochefort, A.; Avouris, P.; Lesage, F.; Salahub, D. R. Phys. Rev. B 1999, 60, 13824. (25) Nardelli, M. B.; Bernholc, J. Phys. Rev. B 1999, 60, R16338. (26) Mai, A.; Svizhen, A.; Anantram, M. P. Phys. Rev. Lett. 2002, 88, 126805. (27) Farajian, A. A.; Mizuseki, H.; Kawazoe, Y. Phys. E 2004, 22, 675. (28) Hansson, A.; Paulsson, M.; Stafstr€om, S. Phys. Rev. B 2000, 62, 7639. (29) Szafert, S.; Gladysz, J. A. Chem. Rev. 2006, 106, PR1–PR33. (30) Eisler, S.; Slepkov, A. D.; Elliott, E.; Luu, T.; McDonald, R.; Hegmann, F. A.; Tykwinski, R. R. J. Am. Chem. Soc. 2005, 127, 2666– 2676. (31) Luu, T.; Elliott, E.; Slepkov, A. D.; Eisler, S.; McDonald, R.; Hegmann, F. A.; Tykwinski, R. R. Org. Lett. 2005, 7, 51–54. (32) Dembinski, R.; Lis, T.; Szafert, S.; Mayne, C. L.; Bartik, T.; Gladysz, J. A. J. Organomet. Chem. 1999, 578, 229–246. (33) Mohr, W.; Stahl, J.; Hampel, F.; Gladysz, J. A. Chem.;Eur. J. 2003, 9, 3324–3340. (34) Antonova, A. B.; Bruce, M. I.; Elis, B. G.; Gaudio, M.; Humphrey, P. A.; Jevric, M.; Melino, G.; Nicholson, B. K.; Perkins, G. J.; Skelton, B. W.; Stapleton, B.; White, A. H.; Zaitseva, N. N. Chem. Commun. 2004, 960–961. (35) Maier, W. F.; Lau, G. C.; McEwen, A. B. J. Am. Chem. Soc. 1985, 107, 4724–4731. (36) Santiago, C.; Houk, K. N.; DeCicco, G. J.; Scott, L. T. J. Am. Chem. Soc. 1978, 100, 692–696. (37) Liang, C.; Allen, L. C. J. Am. Chem. Soc. 1991, 113, 1873– 1878. (38) Santiago, C.; Houk, K. N.; DeCicco, G. J.; Scott, L. T. J. Am. Chem. Soc. 1978, 100, 692. (39) Faust, R. Angew. Chem., Int. Ed. 1998, 37, 2825–2828. (40) Goroff, N. S. Acc. Chem. Res. 1996, 29, 77–83. (41) Kiang, C. H.; Goddard, W. A. Phys. Rev. Lett. 1996, 76, 2515– 2518. (42) Hunter, J. M.; Fye, J. L.; Roskamp, E. J.; Jarrold, M. F. J. Phys. Chem. 1994, 98, 1810–1818. (43) Weltner, W.; Vanzee, R. J. Chem. Rev. 1989, 89, 1713–1747. (44) Lucotti, A.; Tommasini, M.; Fazzi, D.; Zoppo, M. D.; Chalifoux, W. A.; Ferguson, M. J.; Zerbi, G.; Tykwinski, R. R. J. Am. Chem. Soc. 2009, 131, 4239–4244. (45) Lee, C.; Yang, W.; Parr, R. G. Phys. Rev. B 1988, 37, 785. (46) (a) Zhao, Y; Truhlar, D. G Phys. Chem. Chem. Phys. 2005, 7, 2701–2705. (b) Zhao, Y.; Lynch, B. J.; Truhlar, D. G. J. Phys. Chem. A 2004, 108, 4786–4791. (c) Zhao, Y.; Lynch, B. J.; Truhlar, D. G. J. Phys. Chem. A 2004, 108, 2715–2719. (d) Lynch, B. J.; Truhlar, D. G. J. Phys. Chem. A 2001, 105, 2936–2941. (47) Perdew, P.; Wang, Y. Phys. Rev. B 1992, 45, 13244. (48) (a) Hu, Y. H.; Ruckenstein, E. J. Am. Chem. Soc. 2005, 127, 11277–11282. (b) Hu, Y. H.; Ruckenstein, E. J. Chem. Phys. 2004, 120, 7971–7975. (c) Hu, Y. H.; Ruckenstein, E. J. Chem. Phys. 2003, 119, 10073–10080. (49) El-Barbary, A. A.; Lebda, H. I.; Kamel, M. A. Comput. Mater. Sci. 2009, 46, 128–132. (50) Lu, Z. H.; Lo, C. C.; Huang, C. I.; Dharm-Wardana, M. W. C.; Zqierski, M. Z. Phys. Rev. B 2005, 72, 155440. 1849

dx.doi.org/10.1021/jp111851u |J. Phys. Chem. C 2011, 115, 1843–1850

The Journal of Physical Chemistry C

ARTICLE

(51) Heath, J. R.; Curl, R. F.; Smalley, R. E. J. Chem. Phys. 1987, 87, 4236. (52) (a) Hawkins, J. M.; Meyer, A.; Lewis, T. A.; Loren, S. D.; Hollander, F. J. Science 1991, 252, 312. (b) Fagan, P. J.; Calabrese, J. C.; Malone, B. Science 1991, 252, 1160. (c) Liu, S.; Lu, Y. J.; Kappes, M. M.; Ibers, J. A. Science 1991, 254, 408. (d) Yannoni, C. S.; Bernier, P. P.; Bethune, D. S.; Meijer, G.; Salem, J. R. J. Am. Chem. Soc. 1991, 113, 3190. (53) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Montgomery, J. A., Jr.; Vreven, T.; Kudin, K. N.; Burant, J. C.; Millam, J. M.; Iyengar, S. S.; Tomasi, J.; Barone, V.; Mennucci, B.; Cossi, M.; Scalmani, G.; Rega, N.; Petersson, G. A.; Nakatsuji, H.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Klene, M.; Li, X.; Knox, J. E.; Hratchian, H. P.; Cross, J. B.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Ayala, P. Y.; Morokuma, K.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Zakrzewski, V. G.; Dapprich, S.; Daniels, A. D.; Strain, M. C.; Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Clifford, S.; Cioslowski, J.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Gonzalez, C. Pople, J. A. Gaussian 03, revision D.01; Gaussian, Inc.: Wallingford CT, 2004. (54) Pauling, L. The Nature of the Chemical Bond, 3rd ed.; Cornell University Press: New York, 1960; pp 3-27. (55) West, A. R. Solid State Chemistry and Its Applications; John Wiley & Sons: Chichester, NY, 1984; pp 263-317. (56) Langmuir, I. J. Am. Chem. Soc. 1919, 41, 868. (57) Hoffmann, R. Angew. Chem., Int. Ed. 1987, 26, 846–878. (58) Parr, R. G.; Pearson, R. G. J. Am. Chem. Soc. 1983, 105, 7512– 7516. (59) Ouahab, L. Yagubskii, E. Organic conductors, superconductors and magnets: from synthesis to molecular electronics; Kluwer Academic Publishers: Netherlands, 2004; pp 81-98.

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dx.doi.org/10.1021/jp111851u |J. Phys. Chem. C 2011, 115, 1843–1850