Electronic Band Structure of Helical Polyisocyanides - The Journal of

Article pubs.acs.org/JPCA

Electronic Band Structure of Helical Polyisocyanides Benoît Champagne,*,† Vincent Liégeois,*,† Joseph G. Fripiat,*,† and Frank E. Harris*,‡,§ †

Laboratory of Theoretical Chemistry, Unit of Theoretical and Structural Physical Chemistry, Namur Institute of Structured Matter, University of Namur, Rue de Bruxelles, 61, 5000 Namur, Belgium ‡ Department of Physics, University of Utah, Salt Lake City, Utah 84112, United States § Quantum Theory Project, University of Florida, Gainesville, Florida 32611, United States S Supporting Information *

ABSTRACT: Restricted Hartree−Fock computations are reported for a methyl isocyanide polymer (repeating unit CNCH3), whose most stable conformation is expected to be a helical chain. The computations used a standard contracted Gaussian orbital set at the computational levels STO-3G, 3-21G, 6-31G, and 6-31G**, and studies were made for two line-group configurations motivated by earlier work and by studies of space-filling molecular models: (1) A structure of line-group symmetry L95, containing a 9-fold screw axis with atoms displaced in the axial direction by 5/9 times the lattice constant, and (2) a structure of symmetry L41 that had been proposed, containing a 4-fold screw axis with translation by 1/4 of the lattice constant. Full use of the line-group symmetry was employed to cause most of the computational complexity to depend only on the size of the asymmetric repeating unit. Data reported include computed bond properties, atomic charge distribution, longitudinal polarizability, band structure, and the convoluted density of states. Most features of the description were found to be insensitive to the level of computational approximation. The work also illustrates the importance of exploiting line-group symmetry to extend the range of polymer structural problems that can be treated computationally.

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INTRODUCTION The helix is among the most fundamental structures of biomacromolecules. These helical structures are at the origin of their functions, as well as of chemical and physical properties. Helical structures were first proposed for α-amylose and then for natural polypeptides whereas Watson and Crick discovered the double-helical structure of DNA (for a review, see ref 1). In many areas, scientists were inspired by these regular, also aesthetic, structures and developed synthetic approaches to build helical structures with targeted properties and functions. The first synthetic polymer found to display a helical structure (in the solid state) was stereoregular isotactic polypropylene, and since then, many have been synthesized, e.g., fluorescent alternating pyridine-pyrimidine oligomers,2 quinoline-derived oligoamides folded into a helix forming a unimolecular capsule to isolate a molecular guest from the solvent,3 similar quinolinederived oligoamides bearing a terminal chiral residue demonstrating handedness-dependent vibrational circular dichroism signatures,4 substituted poly(3,6-phenanthrene)s switching between a coil and a helical structure as a function of the solvent,5 homo- and heterohelicenes exhibiting huge circular dichroism and other chiro-optical properties,6 or more recently 3,4-ethylenedioxythiophene-alkynylpyridine oligomers that exhibit second-order nonlinear optical responses.7 Among these helical polymers, polyisocyanides8,9 present the particularity that every carbon atom of the backbone bears a substituent, resulting in large steric hindrances. To minimize these, the C−C bonds twist so that the −C−C− backbone adopts a © XXXX American Chemical Society

helical structure. Millich proposed that the C backbones of polyisocyanides display a 4-fold screw axis (four repeat units per helical turn). The 4-fold screw axis implies that each rotation through angle 2π/4 around the screw axis is followed by a translation a0/4 in the direction of periodicity, with a0 the lattice constant. In that case, the polymer belongs to the L41 line group, and each symmetric or translational unit contains four asymmetric units. These helical structures have also been the topic of electronic structure calculations in order to understand their structure− property relationships. Then, in the case of stereoregular polymers, translational and helical (screw axis) symmetries can be taken into account to reduce the computational efforts.10 There are published reports on extended Huckel,11 Hartree−Fock, and density-functional-theory (DFT) investigations on a broad range of helical polymers, including polypropylene,12 sulfur and selenium helices,13 polyethylene, poly(oxymethylene) and inorganic polymers,14 polyheteroaromatics including polythiophene and polypyrrole,15 polyethylene,16 polyacetylene,17 and polyethylenimine,18 but to our knowledge, other than an extended Huckel investigation,19 none have addressed polyisocyanides. The present work reports on band structure calculations for two forms of polyisocyanides bearing methyl substituents. Received: July 26, 2017 Revised: September 13, 2017 Published: September 13, 2017 A

DOI: 10.1021/acs.jpca.7b07403 J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

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THEORETICAL BACKGROUND AND COMPUTATIONAL ASPECTS The computations were carried out using the Ewald/Fourierspace approach developed by the authors.22−32 An important feature of our approach, summarized in a recently published review,32 is that, by explicit consideration of an infinite periodic system, it avoids two difficulties that arise when treating large polymers as oligomers with a large but finite number of unit cells: (1) The computational resources that are needed scale with the size of the unit cell, not the size of the oligomer, and (2) end-of-chain effects are avoided. Another feature of our method is that, by using Fourier representations and the Ewald summation procedure, the slow convergence often experienced in direct-space computations is converted (for any periodic system) to an exponential convergence rate. The functional transformations inherent to the procedure lead to mathematical formulas that are less generally familiar than methods based entirely on direct-space molecular integrals, but the effort involved in adapting to the less familiar regime is rewarded with substantially increased computational efficiency, in some cases causing previously impractical studies to become possible. In the present study we work at the restricted Hartree−Fock level of approximation, using hartree atomic units for energy, but expressing lengths in units of the lattice constant (a0 bohr), with periodicity in the z-direction. In these units, the Coulomb repulsion of two unit charges takes the form 1/a0r, the kinetic energy operator is −(1/2a20)∇2, and the lattice constant is the unit vector ẑ. We form atomic orbitals χμp (r) as fixed linear combinations (contractions) of Gaussian-type basis functions (GTOs) centered at point Ap of the cell displaced by μ units from a reference cell (for which μ = 0). For orbitals of nonzero angular momentum we construct contracted Cartesian Gaussians. All contractions are formed using published contraction exponents and coefficients.33 Note that in our units system the contraction exponents are a20 times the values (in 1/bohr2) given in most tabulations. The periodic structure is described by combining atomic orbitals into Bloch functions ηp(k, r), with the Bloch

In addition to the L41 conformation, a conformation from a previous investigation20 was considered. It was obtained by modifying the side chains of the most stable structure of the poly(isocyano-L-alanyl-D-alanine methyl ester) (LD-PIAA) polymer, of which the geometry of a segment of 14 repeating units was optimized at the DFT level (B3LYP/6-31G*). To each repeating unit is applied a 9-fold screw axis including a rotation through angle 2π/9 and a translation of 5a0/9; the resulting structure belongs to the L95 line group. Each symmetric unit cell for the L95 structure contains nine asymmetric CNCH3 units: i.e., 54 atoms (C18N9H27) and 198 electrons (99 doubly occupied orbitals). Our calculations used a lattice constant a0 = 765 pm, corresponding to a CC bond length of 149.7 pm. The atomic coordinates of the asymmetric unit are listed in Table 1. The first-mentioned helical Table 1. Atomic Coordinates (in pm) of the Polyisocyanide Asymmetric Unit 1 2 3 4 5 6

atom

x

y

z

C N C H H H

95.727 222.815 308.287 401.354 334.255 264.763

−4.254 −4.613 −93.913 −40.380 −179.449 −132.198

32.809 44.559 −31.931 −53.867 31.568 −124.873

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conformation, which belongs to the L41 line group, has a symmetric unit containing a total of 24 atoms and 88 electrons. The atomic coordinates within the asymmetric unit cell were chosen to be the same as for the L95 geometry. Two alternative lattice constants were studied for the L41 structure; the first was a0 = 254.1 pm, which causes all the bond lengths to be equal to those used for the L95 geometry. The second choice of lattice constant was a0 = 376.5 pm; it is the value of a0 that minimizes the L41 total energy, and results in a CC bond length of 165.0 pm, with all other bond lengths kept equal to those of the L95 geometry. The two polyisocyanide helical structures are represented in Figure 1.

Figure 1. Molecular geometry of polyisocyanide (with three repeating translational units along the z-axis) produced with the program DRAWMOL:21 L41 (on the left) and L95 (on the right) geometries. B


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Figure 2. Unit cell of polyisocyanide in two geometries: L95 (left) and L41 (right). View from the z-axis.

number k in units 1/a0, so the Brillouin zone is of unit length and

enlarging the domain of practical computations. Again, further details may be found in ref 32. Four different atomic basis sets were used in the calculations: STO-3G,33 3-21G,35 as well as 6-31G and 6-31G**.36 For these bases, the number of atomic orbitals per translational unit cell are, for the L41 geometry, 72, 132, 132, and 240, and, for the L95 geometry, 162, 297, 297, and 540, respectively. The Fourier−Ewald transform procedure was implemented in a computer program called FT-1D.37 In this program, the extents of the lattice sums were set to include all overlap terms exceeding 10−12 and all two-electron integrals larger than 10−10. In the STO-3G, 3-21G, and 6-31G calculations, these thresholds cause the lattice summations that depend on the AO overlap to run between −3 and 3 for the L41 case and between −1 and 1 for the L95 case. For other summations these thresholds cause the number of terms explicitly computed in the Ewald formulation not to exceed 7 in direct space and 6 in Fourier space. For the 6-31G** calculation, these limits are the same or at worst one unit larger. For all calculations, the number of k points used in the integration over the first Brillouin zone (−1/2 ≤ k ≤ 1/2) was set to 33.

N

ηp(k , r) = lim (2N + 1)−1/2 N →∞

∑

e2πiμk χpμ (r)

μ =−N

(1)

The restricted Hartree−Fock (RHF) orbitals ϕn(k) are now formed, for each point k of the Brillouin zone, as linear combinations of the ηp, thus ϕn(k , r) =

∑ ηp(k , r)cpn(k) p

(2)

The coefficients cpn are found by solving the RHF equations F(k)c(k) = S(k)c(k)E(k)

(3)

where F is the Fock matrix, E is a diagonal matrix containing its eigenvalues, c is a matrix of elements cpn, and S is the overlap matrix with elements Spq(k) = ⟨ηp(k, r)|ηq(k, r)⟩. The matrix elements of the Fock matrix F are Fpq(k) = Tpq(k) + Vpq(k) + Cpq(k) + X pq(k)

(4)

where p and q refer to ηp and ηq. Here Tpq (the kinetic energy matrix), Vpq (the electron−nuclear interaction), and Cpq and Xpq (the Coulomb and exchange parts of the electron−electron interaction) are, if formulated directly from the definitions, multidimensional sums over the direct-space periodic lattice. While some contributions to these matrix elements converge exponentially due to the exponential decay with distance of the orbital overlaps, others decay with distance only as small inverse powers of the orbital separation, and those that decay as 1/r even generate divergent lattice sums. Use of the Poisson summation theorem, which converts directspace sums into reciprocal (Fourier) space equivalents, facilitates the cancellation of the divergences, but does not entirely eliminate the slow convergence. However, the Ewald process produces a combination of direct-space and Fourier-space summations both of which converge exponentially. For details we refer to ref 32. Another feature of our formulation is its use of symmetry to avoid the repeated computation of equal matrix elements. By extending to line groups the point-group symmetry treatment of molecules as described by Dupuis and King,34 we obtain a procedure most of which scales as the cube (rather than the fourth) power of the orbital basis number, thereby

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RESULTS AND DISCUSSION This section reports the results obtained for helical polyisocyanide chains in two different geometries. First, we describe the line groups, their operations, and irreducible representations (“irreps”) in the notation introduced by Božović and Vujičić.38,39 The L95 and L41 line groups, associated with the isogonal point group C9 or C4,40 have, respectively, the following symmetry operations: ⎧ 1 0 2 5 1⎫ {Ĉ9 |ν} ≡ {E|ν}, ⎨Ĉ9 ν + , Ĉ 9 ν + ⎬ , ⎩ 9 9⎭ ⎧ 1 4 2 2⎫ ⎨Ĉ3 ν + , Ĉ 9 ν + ⎬ , ⎩ 3 9⎭ ⎧ 5 2 7 1⎫ ⎧ 7 8 ⎨Ĉ9 ν + , Ĉ3 ν + ⎬, ⎨Ĉ9 ν + , ⎩ 9 3⎭ ⎩ 9 8 4⎫ Ĉ 9 ν + ⎬ 9⎭

}{

}{ }{

}

{

C


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Table 2. Total Energy, Energy of the Top of the Valence Bands, Energy of the Bottom of the Conduction Band, and the Band Gap of Polyisocyanide Obtained for Different Basis Sets for the Two Symmetries atomic basis set

line group

C−C bond length (pm)

total energy per CNCH3 group (hartree)

STO-3G

L95 L41 L41 L95 L41 L41 L95 L41 L41 L95 L41 L41

149.7 149.7 165.0 149.7 149.7 165.0 149.7 149.7 165.0 149.7 149.7 165.0

−130.270629 −130.130867 −130.260010 −131.167152 −131.036572 −131.152659 −131.843859 −131.702769 −131.829431 −131.910522 −131.773523 −131.897721

3-21G

6-31G

6-31G**

top of valence band (eV) −7.1453 −6.0613 −6.7205 −9.0668 −7.9096 −8.5949 −9.2580 −8.0856 −8.8865 −9.4425 −8.1212 −9.0735

(k (k (k (k (k (k (k (k (k (k (k (k

= = = = = = = = = = = =

1/2) 0) 0) 1/2) 3/32) 0) 1/2) 3/32) 0) 1/2) 3/32) 0)

bottom of conduction band (eV) 5.8946 4.3883 6.4882 2.3371 1.1981 2.7252 2.0949 1.0716 2.4927 2.0810 1.0845 2.5163

(k (k (k (k (k (k (k (k (k (k (k (k

= = = = = = = = = = = =

1/2) 7/16) 9/32) 7/16) 7/16) 5/32) 1/2) 7/16) 5/32) 15/32) 7/16) 5/32)

band gap (eV) 13.0399 10.4496 13.2087 11.4039 9.1076 11.3201 11.3529 9.1572 11.3791 11.5235 9.2057 11.5898

and ⎧ 1 0 1 {C4̂ |ν} ≡ {E|ν}, ⎨C4̂ ν + , ⎩ 4 ⎧ 3 3 ⎨C4̂ ν + ⎩ 4

⎫

} {Ĉ ν + 12 ⎬⎭, 2

}

The notation {R̂ |ν + w} denotes an operation where R̂ is a proper or improper rotation around the translational axis, w is a fraction (w ∈ [0,1]), and ν is a signed integer (or zero) generating the translational periodicity. Figure 2 shows the symmetric unit cell for both structures. This figure shows that, in each of the two cases, the atoms can be assigned into six groups related by symmetry operations: • A first group containing the innermost carbon atoms (L95: 1, 7, 13, 19, 25, 31, 37, 43, 49; L41: 1, 7, 13, 19), forming chemical bonds connecting the different asymmetric unit cells. We denote this atom group CI. • A second group (NII) consisting of nitrogen atoms (L95: 2, 8, 14, 20, 26, 32, 38, 44, 50; L41: 2, 8, 14, 20) bonded to the CI carbon atoms. • A third group (CIII) consisting of carbon atoms (L95: 3, 9, 15, 21, 27, 33, 39, 45, 51; L41: 3, 9, 15, 21) that lie on an outer shell. • Three groups that contain hydrogen atoms: (L95: 4, 10, 16, 22, 28, 34, 40, 46, 52; L41: 4, 10, 16, 22), (L95: 5, 11, 17, 23, 29, 35, 41, 47, 53; L41: 5, 11, 17, 23), and (L95: 6, 12, 18, 24, 30, 36, 42, 48, 54; L41: 6, 12, 18, 24), denoted, respectively, HIV, HV, and HVI. Table 2 lists for each of the atomic basis sets the total energy per asymmetric unit cell, the energy of the top of the valence bands (highest occupied crystalline orbital, HOCO), the minimum energy of the lowest unoccupied band (lowest unoccupied crystalline orbital, LUCO), and the band gap. For all basis sets, the L95 chain presents a lower total energy than either of the two L41 chains. The averaged energy difference between the L95 and the lower-energy L41 structure amounts to about 34 kJ mol−1. The L41 structure with the same CICI bond length as in the L95 helix has a much higher energy. Figure 3 displays the 6-31G** band structure of polyisocyanide. The band structures obtained with the STO-3G and 3-21G basis sets are given in the Supporting Information associated with this paper (see Figures S1 and S2). The bands are plotted in the asymmetric Brillouin zone, which encompasses

Figure 3. 6-31G** band structure of polyisocyanide (developed in the asymmetric BZ). All the valence bands and the first four unoccupied bands are shown. Left, L41 geometry with a0 = 376.5 nm; right, L95.

four or nine reciprocal units corresponding to the symmetrical unit cell in the direct space. In Tables 3 and 4, all the valence bands and the lowest unoccupied bands (below +0.7 hartree) are classified according Table 3. Symmetry of the Valence Bands (28−99) and the First 36 Unoccupied Bands (100−135) Obtained with the 6-31G** Basis Set for the L95 Chain irrep

polymer orbitals

{0A0, kA0, 1/2E−4 0 } {0EA1, kA1, 1/2E41} {0EA1, kA−1, {0EA2, kA2,

3 1/2E2}

{0EA2, kA−2, {0EA3, kA3,

1/2A−2}

3 1/2E2}

{0EA3, kA−3, {0EA4, kA4,

−3 1/2E−1}

−3 1/2E−1}

4 1/2E1}

{0EA4, kA−4,

−4 1/2E0 }

28, 37, 46, 59, 68, 77, 82, 91, 104, 107, 120, 127 31, 40, 49, 60, 71, 75, 85, 87, 100, 109, 123, 131 32, 41, 50, 61, 72, 76, 86, 88, 101, 108, 124, 130 35, 45, 53, 58, 69, 80, 94, 98, 111, 112, 126, 135 36, 44, 54, 57, 70, 81, 95, 99, 110, 113, 125, 134 34, 43, 52, 62, 66, 79, 93, 96, 106, 117, 122, 129 33, 42, 51, 63, 67, 78, 92, 97, 105, 116, 121, 128 30, 39, 48, 55, 65, 73, 83, 90, 102, 115, 119, 132 29, 38, 47, 56, 64, 74, 84, 89, 103, 114, 118, 133

to their irreducible representations. The bands listed in these tables are identified by their symmetries at k = 0, 0 < |k| < 1/2, and |k| = 1/2. D


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Table 6. Character Table of the Line Group L41, in the Notation of Božović and Vujičić38,a

Table 4. Symmetry of the Valence Bands (13−44) and the First 16 Unoccupied Bands (45−60) Obtained with the 6-31G** Basis Set for the L41 Chain (C−C Length = 164 nm) polymer orbitals

0A0

13, 17, 21, 26, 31, 32, 37, 42, 47, 48, 53, 58

0EA1

14, 18, 22, 27, 29, 34, 38, 40, 45, 50, 54, 60

0A2

irrep {0A0, kA0, 1/2E10} {0EA1, kA1, 1/2E10} {0EA1,

2 kA−1, 1/2E−1}

kA0

15, 19, 23, 28, 30, 35, 39, 41, 46, 49, 55, 59

{0A2, kA2, 1/2E2−1}

{Ê |ν}

irrep

kA1

16, 20, 24, 25, 33, 36, 42, 43, 44, 51, 56, 57

kA−1 kA2

For reference, we provide in Tables 5 and 6 the character tables of the L95 and L41 line groups. The band structures for the STO-3G and 3-21G basis sets are listed in the Supporting Information associated with this paper (see Tables S1−S4). We found that there are no significant differences in the band structures obtained with the 3-21G and the 6-31G** basis sets except for an overall shift to lower energies in the 6-31G** basis. On the other hand, a comparison between the STO-3G bands and those from the 3-21G and 6-31G** bases shows more significant differences in the band ordering in addition to an overall shift to lower energies for the 3-21G and 6-31G** bands. The STO-3G, 3-21G, and 6-31G** calculations are in agreement that the highest occupied band (L95: band 99, L41: band 44) belongs, respectively, to the {0EA2, kA−2, 1/2A−2}(L95) or {0EA2, kA2, 1/2E2−1}(L41) irreducible representations. However, the 6-31G** and the 3-21G calculations indicate that the lowest unoccupied band for the L95 structure has symmetry {0EA1, kA1, 1/2E41} while the STO-3G calculation assigns this band the symmetry {0EA4, kA4, 1/2E41}. For the L41 structure, there are no such differences among the three calculations. On the basis of the character tables, it is possible to form symmetry-adapted combinations of the Bloch orbitals for the points k = 0 and k = 1/2 of the Brillouin zone. The formulas are

1 1/2E0 2 1/2E−1

1

1

{Ĉ2 ν + 2 }

3

3

{C4̂ ν + 4 }

1 2

1 0

1 −2

1 0

1 c0(k) c0(k) c0(k) c0(k) 2(−1)ν 2(−1)ν

−1 c1(k) a1c1(k) a−1c1(k) a2c1(k) √2 −√2

1 c2(k) a2c2(k) a−2c2(k) a4c2(k) 0 0

−1 c3(k) a3c3(k) a−3c3(k) a6c3(k) √2 √2

cm(k) = exp −2π ik⎡⎣ν +

(

a

1

{C4̂ ν + 4 }

m⎤ 4⎦

), a

m

= exp(2π im/4).

rather lengthy and are presented in full in the Supporting Information (Tables S7−S10). Tables 7−9 list the contribution of different Bloch functions to the polymer orbitals for the 6-31G** basis set. These contributions are analyzed from the magnitudes of the LCAO coefficients with the help of the expressions for the symmetry orbitals (found in the Supporting Information). An arbitrary threshold was chosen and set to 0.08 for the L95 structure and 0.1 for the L41 structure. All LCAO coefficients (in absolute value) below these thresholds were considered not to make important bonding contributions. These tables list also the bands which belong to each “asymmetric” band and indicate the bonding character of the CIN, N−CIII, and CIIIH bands at the point k = 0 of the asymmetric Brillouin zone. In these tables, the notation CI(2s, 2p∥, 2p⊥, 2pz) indicates that the Bloch sums of the 2s, 2p∥, 2p⊥, 2pz atomic orbitals centered on an atom belonging to the first group of carbons contribute to this polymer orbital. The notations ∥ and ⊥ denote 2p symmetry orbitals, respectively, parallel and perpendicular to the

Table 5. Character Table of the Line Group L95, in the Notation of Božović and Vujičić38,a irrep 0 A0

{Ĉ

9

ν+

5 9

} {Ĉ

2 9

ν+

1 9

} {Ĉ

3

ν+

2 3

}

{Ĉ

4 9

ν+

2 9

} {Ĉ

5 9

ν+

7 9

} {Ĉ

2 3

ν+

1 3

} {Ĉ

7 9

ν+

8 9

} {Ĉ

8 9

ν+

4 9

0EA1

1 2

1 C2

1 C4

1 −1

1 −C1

1 −C1

1 −1

1 C4

1 C2

0EA2

2

C4

−C1

−1

C2

C2

−1

−C1

C4

0EA3

2

−1

−1

2

−1

−1

2

−1

−1

0EA4

2

−C1

C2

−1

C4

C4

−1

C2

−C1

k A0

c0(k) c0(k) c0(k) c0(k) c0(k) c0(k) c0(k) c0(k) c0(k) 2(−1)ν 2(−1)ν 2(−1)ν 2(−1)ν (−1)ν

c5(k) a1c5(k) a8c5(k) a2c5(k) a7c5(k) a3c5(k) a6c5(k) a4c5(k) a5c5(k) −D2 −D4 (−1)ν D1 −(−1)ν

c1(k) a2c1(k) a7c1(k) a4c1(k) a5c1(k) a6c1(k) a3c1(k) a8c1(k) a1c1(k) −D4 D1 (−1)ν −D2 −(−1)ν

c6(k) a3c6(k) a6c6(k) a6c6(k) a3c6(k) c6(k) c6(k) a3c6(k) a6c6(k) −(−1)ν −(−1)ν 2(−1)ν −(−1)ν (−1)ν

c2(k) a4c2(k) a5c2(k) a8c2(k) a1c2(k) a3c2(k) a6c2(k) a7c2(k) a2c2(k) −D1 D2 −(−1)ν D4 (−1)ν

c7(k) a5c7(k) a4c7(k) a1c7(k) a8c7(k) a6c7(k) a3c7(k) a2c7(k) a7c7(k) D1 −D2 (−1)ν −D4 −(−1)ν

c3(k) a6c3(k) a3c3(k) a3c3(k) a6c3(k) c3(k) c3(k) a6c3(k) a3c3(k) (−1)ν (−1)ν −2(−1)ν (−1)ν −(−1)ν

c8(k) a7c8(k) a2c8(k) a5c8(k) a4c8(k) a3c8(k) a6c8(k) a1c8(k) a8c8(k) D4 −D1 −(−1)ν D2 (−1)ν

c4(k) a8c4(k) a1c4(k) a7c4(k) a2c4(k) a6c4(k) a3c4(k) a5c4(k) a4c4(k) D2 D4 −(−1)ν −D1 (−1)ν

k A1 kA−1 k A2 kA−2 k A3 kA−3 k A4 kA−4 −3 1/2E−1 −4 1/2E0 4 1/2E1 3 E 1/2 2 1/2A−2

a

{Ê |ν}

}

Cm = 2cos(mπ/9), cm(k) = exp(−2πik [ν + ((m)/(9))]), am = exp(2πi m/9), Dm = 2(−1)ν cos(mπ/9). E


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The Journal of Physical Chemistry A Table 7. Analysis of the Valence Bands (28−99) As Obtained with the 6-31G** Basis Set for the L95 Structurea “asymmetric” band number

k-range

band(s) Bloch orbitals

main contributors in the asymmetric cell

1

0−1/18 1/18−1/6 1/6−5/18 5/18−7/18 7/18−1/2 0−1/18 1/18−1/6 1/6−5/18 5/18−7/18 7/18−1/2 0−1/18 1/18−1/6 1/6−5/18 5/18−7/18 7/18−1/2 0−1/18 1/18−1/6 1/6−5/18 5/18−7/18 7/18−1/2 0−1/18 1/18−1/6 1/6−5/18 5/18−7/18 7/18−1/2 0−1/18 1/18−1/6 1/6−5/18 5/18−7/18 7/18−1/2 0−1/18 1/18−1/6 1/6−5/18 5/18−7/18 7/18−1/2 0−1/18 1/18−1/6 1/6−5/18 5/18−7/18 7/18−1/2

28 29, 30 31, 32 33, 34 35, 36 37 38, 39 40, 41 42, 43 45, 44 46 47, 48 49, 50 51, 52 53, 54 59 56, 55 60, 61 63, 62 58, 57 68 64, 65 71, 72 67, 66 69, 70 77 74, 73 75, 76 78, 79 80,81 82 84, 83 85, 86 92, 93 94, 95 91 89, 90 87, 88 97, 96 98, 99

CI(2s), NII(2s) CI(2s), NII(2s) CI(2s), NII(2s) NII(2s) NII(2s) CI(2s) CI(2s), CIII(2s) CI(2s), CIII(2s) NII(2p∥), CIII(2s) NII(2p∥), CIII(2s) CI(2p∥), CIII(2s) CI(2p∥), NII(2s), CIII(2s) CI(2s, 2p∥), NII(2s) CI(2s, 2p⊥, 2pz), NII(2s) CI(2p⊥, 2pz) CI(2p∥), NII(2s, 2p∥), CIII(2p⊥) CI(2p∥), NII(2s, 2p∥), CIII(2p⊥, 2pz) CI(2p∥, 2pz), NII(2s, 2p∥), CIII(2p⊥, 2pz), HVI(1s) CI(2p∥), NII(2p∥) NII(2p∥)),CIII(2p⊥) NII(2pz), CIII(2pz) NII(2pz), CIII(2p⊥, 2pz), HV(1s) NII(2p⊥), CIII(2p∥, 2p⊥), HIV(1s) CIII(2p⊥, 2pz), HV(1s) CIII(2p⊥, 2pz), HV(1s) CIII(2p∥), HIV(1s) NII(2p⊥), CIII(2p∥),HV(1s), HVI(1s) CIII(2p∥, 2pz),HIV(1s) CIII(2p∥) CI(2p⊥), CIII(2p∥), HIV(1s) NII(2pz) CI(2p⊥, 2pz), NII(2p⊥, 2pz), CIII(2pz) NII(2p∥, 2p⊥, 2pz), CIII(2p⊥) CI(2s, 2pz), NII(2p⊥), CIII(2p⊥) CI(2p⊥), NII(2pz) CI(2p⊥), NII(2p⊥) NII(2s, 2p⊥, 2pz) CI(2p⊥, 2pz), NII(2pz) CI(2p⊥), NII(2s, 2pz) CI(2pz), NII(2s, 2p⊥)

2

3

4

5

6

7

8

bonding character at k = 0 (CINII)

(CINII)*, (NIICIII)

(CINII), (NIICIII)*

(CINII)

(NIICIII)

(NIICIII), (CIIIHIV)

(CINII), (NIICIII)*

(CINII)

a “Main contributing atomic orbitals” are those with an LCAO coefficient larger in absolute value than 0.08 at k = 0. In the last column, the asterisk indicates a band with an antibonding character. Without asterisk, the band has a bonding character.

Table 8. Analysis of the First 18 Unoccupied Bands (100−117) As Obtained with the 6-31G** Basis Set for the L95 Structurea “asymmetric” band number

k-range


main contributors in the asymmetric cell

bonding character at k = 0

9

0−1/18 1/18−1/6 1/6−5/18 5/18−7/18 7/18−1/2

104 103, 102 100, 101 105, 106 111,110

CI(2pz), NII(2p⊥, 2pz), CIII(2s) CI(2s, 2p⊥, 2pz), NII(2s, 2p⊥, 2pz) CI(2s, 2p∥, 2p⊥, 2pz), NII(2s, 2p∥, 2p⊥, 2pz), HV(1s) CI(2s, 2p∥, 2p⊥, 2pz), NII(2s, 2p⊥, 2pz), CIII(2s, 2p⊥, 2pz), HIV,V,VI(1s) CI(2s, 2p∥, 2p⊥, 2pz), NII(2pz), CIII(2s, 2p∥, 2p⊥, 2pz)), HIV,V,VI(1s)

(CINII)*, (NIICIII)*

10

0−1/18 1/18−1/6 1/6−5/18 5/18−7/18 7/18−1/2

107 114,115 109, 108, 116, 117 112,113

CIII(2s, 2p∥, 2p⊥), HIV,V,VI(1s) CI(2s, 2p⊥), NII(2s, 2p∥, 2p⊥), CIII(2s, 2p∥, 2p⊥, 2pz)), HIV,V,VI(1s) CI(2s, 2p∥, 2p⊥), NII(2s, 2p∥), CIII(2s, 2p∥, 2p⊥)), HIV, V, VI(1s) NII(2s, 2p∥, 2pz), CIII(2s, 2p∥, 2p⊥, 2pz), HIV,V,VI(1s) CI(2s, 2p∥, 2p⊥, 2pz), NII(2s, 2p∥, 2p⊥, 2pz), CIII(2s, 2p∥, 2p⊥, 2pz)), HIV,V,VI(1s)

(CINII)*, (NIICIII)*

“Main contributing atomic orbitals” are those with an LCAO coefficient larger in absolute value than 0.08 at k = 0. In the last column, the asterisk indicates a band with an antibonding character. a

F


Article


Table 9. Analysis of the Valence Bands (13−44) and the First 8 Unoccupied Bands (45−52) As Obtained with the 6-31G** Basis Set for the L41 Structure (C−C Length = 164 nm)a “asymmetric” band number 1

2

3

4

5

6

7

8

9

10

k-range


0−1/8 1/8−3/8 3/8−1/2 0−1/8 1/8−3/8 3/8−1/2 0−1/8 1/8−3/8 3/8−1/2 0−1/8 1/8−3/8 3/8−1/2 0−1/8 1/8−3/8 3/8−1/2 0−1/8 1/8−3/8 3/8−1/2 0−1/8 1/8−3/8 3/8−1/2 0−1/8 1/8−3/8 3/8−1/2 0−1/8 1/8−3/8 3/8−1/2 0−1/8

13 14, 16 17 18, 20 21 22, 24 25 28, 26 31 29, 33 32 34, 36 37 38, 43 42 40, 44 47 45, 51 48

1/8−3/8

50, 49

3/8−1/2

52

15

19

23

27

30

35

39

41

46

main contributors in the asymmetric cell CI(2s), NII(2s) CI(2s, 2p∥), NII(2s) CI(2s, 2p∥), NII(2s) CI(2s), CIII(2s) CI(2s), NII(2p∥), CIII(2s) NII(2p∥), CIII(2s) CI(2p∥), NII(2s), CIII(2s) CI(2s, 2p∥), NII(2s) CIII(2p∥), NII(2s, 2p∥) CI(2p⊥, 2pz), NII(2p⊥), CIII(2p⊥) CI(2p∥),NII(2p∥), CIII(2p⊥), NII(2s, 2p∥) CI(2p∥), NII(2p∥), CIII(2p⊥) NII(2p⊥), CIII(2p∥, 2p⊥, 2pz), HIV(1s) NII(2pz), CIII(2p⊥, 2pz), HIV(1s) CIII(2p∥, 2pz), HV(1s) NII(2pz), CIII(2p⊥, 2pz), H(1s) CIII(2p⊥, 2pz), H(1s) CI(2p⊥), CIII(2p∥, 2pz), HIII(1s) NII(2s, 2pz), CIII(2pz), HVI(1s) CI(2p⊥, 2pz), NII(2s, 2p⊥, 2pz), CIII(2p⊥) CI(2p⊥), NII(2pz) CI(2p⊥, 2pz), NII(2p⊥) CI(2s, 2p∥, 2p⊥), NII(2p∥, 2pz) CI(2p⊥, 2pz), NII(2s, 2p⊥) CI(2p⊥, 2pz), NII(2p⊥, 2pz) CIII(2s), HIV,VI(1s) CI(2s, 2p⊥, 2pz), NII(2s, 2p∥, 2p⊥), HV(1s) NII(2p∥, 2p⊥), CIII(2s, 2p∥, 2p⊥), HIV,V,VI(1s) CI(2s, 2p⊥), NII(2s, 2p∥), CIII(2s,p∥, 2p⊥), HIV,V,VI(1s) CI(2s, 2p∥, 2p⊥, 2pz), NII(2s, 2p∥, 2p⊥), HV,VI(1s) CI(2p⊥, 2pz), NII (2s, 2p⊥; 2pz), CIII(2s, 2pz), HV,VI(1s)

bonding character at k = 0 and k = 1/2 (CINII)

(NIICIII) (CINII), (NIICIII)* (NIICIII)* (CINII) (CINII)*, (NIICIII) (NIICIII), (CIIIHIV) (CIIIHV) (NIICIII) (CIIIHIII) (NIICIII)*, (CIIIHVI)* (CINII) (CINII) (CINII)* (CINII)*, (CIIIHIV,VI)* (NIICIII)*, (CIIIHIV,V,VI)* (CINII)*, (NIICI), (CIH)*

(CINII)*, (NIICIII)*, (NIICIII), (CIIIHVI)*, (CIIIHVI)

“Main contributing atomic orbitals” are those with an LCAO coefficient larger in absolute value than 10−1 at k = 0. In the last column, the asterisk indicates a band with an antibonding character. Without asterisk, the band has a bonding character. a

Figure 4. 6-31G** convoluted density of states of polyisocyanide. Left, L41 geometry; right, L95.

• The second set (bands 37−45 or 17−20) is characterized by significant contributions from 2s Bloch orbitals on carbons CI, the 2s or 2p∥ orbitals on NII, and the 2s orbitals on CIII. • In the interval k = (0, 5/18) of the L95 Brillouin zone, the third set (bands 46−54) consists mainly of 2p∥ Bloch orbitals on carbons CI, 2s orbitals on nitrogens NII, and 2s orbitals on carbons CIII. In the interval k = (5/18, 1/2),

C−N bonds. Similar notations are used for the other atoms. Figure 3 and Tables 7−9 show that the occupied bands can be classified into eight sets spanning different energy ranges corresponding to the ”asymmetric” bands: • The Bloch sums of largest contribution to the first set (bands 28−36 for the L95 geometry or 13−16 for the L41 geometry) are mainly those involving the 2s atomic orbitals centered on carbons CI and nitrogens NII. G


Article

The Journal of Physical Chemistry A 2p⊥ starts to contribute to the polymer orbitals. For the L41 structure (bands 21−24), the contribution comes mainly from the 2p∥ orbitals on CI and the 2s orbitals on NII. In chemical language, the three first sets of bands are of σ bond type within individual asymmetric units. • Contributions from the 2p⊥ and 2pz orbitals on CIII appear in the fourth set of bands (55−63 or 25−28), and the 2p∥ Bloch sums centered on CI and NII also contribute to form delocalized molecular orbitals. • The fifth and the sixth sets of bands are characterized by a weak participation of Bloch orbitals centered on carbons CI to the delocalized molecular orbitals. In other words, these bands contribute weakly to the σ bonds connecting asymmetric units. • In the seventh and eighth sets (bands 82−99 or bands 37−44), the 2p⊥ and 2pz Bloch sums centered on CI and NII are major contributors to the delocalized molecular orbitals. For this reason, we identify these bands as providing σ bonds between asymmetric units. • The ninth and tenth sets of bands consist of unoccupied molecular orbitals. Figure 4 displays the convoluted density of states as determined using the 6-31G** basis set, characterized by a first peak at −0.4 hartree (−10.88 eV) and a maximum intensity at −0.56 hartree (−15.24 eV). Table 10 lists the formal charges on the atoms obtained with the four different basis sets. These data show that the first shell

Table 11. Bond Properties

Table 10. Atomic Charges

and CIIIHVI obtained with the four different basis sets. Except for the CI−NII bonds, the overlap populations and bond lengths are consistent with the identification of all bonds as being normal single-bond character. The CINII bonds, with bond length 127.6 nm and overlap population between 0.517 and 0.580 depending on the basis set, have values that are in the normal range for double bonds; the standard length for the CN double bond is 129 nm. Table 12 lists the longitudinal polarizability per unit cell and per unit length obtained with the four atomic basis sets. The observed trend between the STO-3G basis and the splitvalence basis sets is the same as that observed in polyethylene and polysilane,42,43 where the polarizabilities obtained with the split-valence basis are almost twice the STO-3G values. This is consistent with the fact that the split-valence basis sets give more flexibility to the wave function and to the electronic density than is provided by the STO-3G basis set. Considering the most stable L95 structure, the polarizability per unit length evaluated with the 6-31G** basis set is 81% larger than for polyethylene but 16% smaller than for polysilane, demonstrating the weak electron delocalization along the helix.

overlap population

I II III IV V VI I II III IV V VI I II III IV V VI

atom symbol

STO-3G

3-21G

6-31G

L95 Geometry (C−C length = 149.7 pm) C 0.094 0.319 0.216 N −0.233 −0.586 −0.478 C −0.103 −0.419 −0.306 H 0.073 0.212 0.169 H 0.083 0.231 0.187 H 0.086 0.243 0.212 L41 Geometry (C−C length = 149.7 pm) C 0.084 0.364 0.240 N −0.246 −0.649 −0.527 C −0.102 −0.406 −0.286 H 0.070 0.233 0.198 H 0.068 0.213 0.190 H 0.127 0.245 0.186 L41 Geometry (C−C length = 164 pm) C 0.101 0.325 0.237 N −0.244 −0.599 −0.499 C −0.097 −0.402 −0.283 H 0.079 0.224 0.182 H 0.076 0.218 0.175 H 0.084 0.234 0.188

CICI

149.7

154

0.382

0.236

0.205

0.342

CINII

127.6

129

0.517

0.519

0.563

0.580

NIICIII

145.4

147

0.318

0.235

0.189

0.248

CIIIHIV

109.6

109

0.382

0.370

0.389

0.407

CIIIHV

109.7

109

0.377

0.365

0.385

0.400

CIIIHVI

109.5 109 0.379 0.342 0.379 L41 Geometry (CC Length = 149.7 pm)

0.396

CICI

149.7

154

0.371

0.101

0.080

0.304

CINII

127.6

129

0.513

0.524

0.518

0.537

NIICIII

145.4

147

0.291

0.149

0.039

0.129

CIIIHIV

109.6

109

0.387

0.384

0.408

0.427

CIIIHV

109.7

109

0.356

0.348

0.373

0.384

CIIIHVI

109.5 109 0.371 0.338 0.364 L41 Geometry (CC Length = 164 pm)

0.402

CICI

165.0

154

0.331

0.264

0.265

0.335

CINII

127.6

129

0.521

0.511

0.525

0.567

NIICIII

145.4

147

0.316

0.230

0.188

0.243

CIIIHIV

109.6

109

0.383

0.377

0.394

0.412

CIIIHV

109.7

109

0.377

0.360

0.383

0.398

CIIIHVI

109.5

109

0.379

0.352

0.381

0.398

standard bond length (pm)41

STO3G

3-21G

6-31G 6-31G**

L95 Geometry (CC Length = 149.7 pm)

atomic charge group

bond type

bond length (pm)

6-31G** 0.234 −0.480 −0.164 0.120 0.136 0.154 0.268 −0.549 −0.160 0.143 0.132 0.165 0.235 −0.484 −0.139 0.131 0.124 0.133

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CONCLUSIONS The analysis applied in this paper demonstrates that the use of line-group symmetry is effective in enlarging the range of polymer electronic-structure problems that can be treated computationally. The present study confirms earlier indications that an L95 helical conformation is energetically favorable for the polyisocyanide chain here under study, with that structure computed at several levels of approximation as more stable than the L41 structure by amounts of a few eV per CNCH3 unit.

containing the carbon atoms CI and the outer shells with the hydrogen atoms are slightly positive while the nitrogens and the carbons CIII are negative. Table 11 gives the overlap population and the bond lengths for the bonds CICI, CINII, NIICIII, CIIIHIV, CIIIHV, H


Article


Table 12. Uncoupled (SOS) Longitudinal Polarizability of Polyisocyanide per Unit Cell, αzz/(2N + 1), and per Unit Length, αzz/(2N + 1)a0, Obtained with Different Basis Sets (in au) L95 geometry C−C length = 149.7 pm

L41 geometry C−C length = 149.7 pm

per unit cell

per unit length

per unit cell

per unit length

per unit cell

per unit length

STO-3G 3-21G 6-31G 6-31G**

74.248 149.146 159.436 175.255

5.136 10.317 11.029 12.123

44.054 81.797 84.920

9.174 17.033 17.684

36.310 69.050 73.433 80.438

5.103 9.704 10.320 11.305

(4) Ducasse, L.; Castet, F.; Fritsch, A.; Huc, I.; Buffeteau, T. Density Functional Theory Calculations Chevaliernd Vibrational Circular Dichroism of Aromatic Foldamers. J. Phys. Chem. A 2007, 111, 5092−5098. (5) Vanormelingen, W.; Smeets, A.; Franz, E.; Asselberghs, I.; Clays, K.; Verbiest, T.; Koeckelberghs, G. Conformational Steering in Substituted Poly(3,6-phenanthrene)s: A Linear and Nonlinear Optical Study. Macromolecules 2009, 42, 4282−4287. (6) Gingras, M. One Hundred Years of Helicene Chemistry. Part 3: Applications and Properties of Carbohelicenes. Chem. Soc. Rev. 2013, 42, 1051−1095. (7) Chevallier, F.; Charlot, M.; Mongin, F.; Champagne, B.; Franz, E.; Clays, K.; Blanchard-Desce, M. Synthetic, Optical and Theoretical Study of Alternating Ethylenedioxythiophene-Pyridine Oligomers: Evolution from Planar Conjugated to Helicoidal Structure towards a Chiral Configuration. ChemPhysChem 2016, 17, 4090−4101. (8) Millich, F. Polymerization of Isocyanides. Chem. Rev. 1972, 72, 101−113. (9) Nolte, R. J. M. Helical Poly(isocyanides). Chem. Soc. Rev. 1994, 23, 11−19. (10) Hirata, S. Quantum Chemistry of Macromolecules and Solids. Phys. Chem. Chem. Phys. 2009, 11, 8397−8412. (11) Glassey, W. V.; Hoffmann, R. Band Structure Representations of the Electronic Structure of One-Dimensional Materials with Helical Symmetry. Theor. Chem. Acc. 2002, 107, 272−281. (12) André, J. M.; Vercauteren, D. P.; Bodart, V. P.; Fripiat, J. G. Ab Initio Calculations of the Electronic Structure of Helical Polymers. J. Comput. Chem. 1984, 5, 535−547. (13) Springborg, M.; Jones, R. O. Sulfur and Selenium Helices: Structure and Electronic Properties. J. Chem. Phys. 1988, 88, 2652− 2658. (14) Cui, C. X.; Kertész, M. Conformation Study of Helical MainGroup Polymers: Organic and Inorganic, Trans and Gauche. J. Am. Chem. Soc. 1989, 111, 4216−4224. (15) Cui, C. X.; Kertész, M. Two Helical Conformations of Polythiophene, Polypyrrole, and their Derivatives. Phys. Rev. B: Condens. Matter Mater. Phys. 1989, 40, 9661−9670. (16) Mintmire, J. W. Local-Density Functional Electronic Structure of Helical Chain Polymers. Density Functional Methods in Chemistry; Springer: New York, 1991; pp 125−137. (17) Springborg, M.; Calais, J. L.; Goscinski, O.; Eriksson, L. A. Linear-Muffin-Iin-orbital Method for Helical Polymers: A Detailed Study of Trans-Polyacetylene. Phys. Rev. B: Condens. Matter Mater. Phys. 1991, 44, 12713−12736. (18) Herlem, G.; Lakard, B. Ab Initio Study of the Electronic and Structural Properties of the Crystalline Polyethyleneimine Polymer. J. Chem. Phys. 2004, 120, 9376−9381. (19) Kollmar, C.; Hoffmann, R. Polyisocyanides: Electronic or Steric Reasons for their Presumed Helical Structure? J. Am. Chem. Soc. 1990, 112, 8230−8238. (20) Schwartz, E.; Liégeois, V.; Koepf, M.; Bodis, P.; Cornelissen, J. J. L.; Brocorens, P.; Beljonne, D.; Nolte, R. J. M.; Rowan, A. E.; Woutersen, S.; et al. Beta Sheets with a Twist: The Conformation of Helical Polyisocyanopeptides Determined by Using Vibrational Circular Dichroism. Chem. - Eur. J. 2013, 19, 13168−13174. (21) Liégeois, V. DRAWMOL; Université de Namur: Belgium. 2014. (22) Ewald, P. P. Die Berechnung optischer und elektrostatischer Gitterpotentiale. Ann. Phys. 1921, 369, 253−287.

An important feature of methods with full utilization of the symmetry is that the computation time depends primarily on the size of the periodically repeating unit, and not on the overall system size.

■

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.7b07403. STO-3G and 3-21G data for the L95 and L41 structures: symmetry of the bands and band structure developed in the asymmetric BZ; symmetry-adapted combinations of C, N, and H Bloch orbitals corresponding to irreducible representation of the L95 and L41 line groups at k = 0 and k = 1/2 (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: *E-mail: *E-mail: *E-mail:

[email protected]. [email protected]. [email protected]. [email protected]fl.edu.

ORCID

Benoît Champagne: 0000-0003-3678-8875 Joseph G. Fripiat: 0000-0002-3600-8843 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS V.L. thanks the Fund for Scientific Research (FRS.-FNRS) for his Research Associate position. F.E.H. was supported by U.S. National Science Foundation Grant PHY-0601758. Part of this research has been funded by BELSPO (IAP P7/05 network “Functional Supramolecular Systems”) and by the Francqui Foundation. The calculations were performed on the computing facilities of the Consortium des Équipements de Calcul Intensif (CÉCI), in particular those of the Plateforme Technologique de Calcul Intensif (PTCI) installed in the University of Namur, for which we gratefully acknowledge financial support from the FNRS-FRFC (Conventions 2.4.617.07.F and 2.5020.11) and from the University of Namur.

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L41 geometry C−C length = 164 pm

basis set

REFERENCES

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