^{1,a)}, Roberto Car

^{1}and Raffaele Resta

^{2}

### Abstract

A theorem for end-charge quantization in quasi-one-dimensional stereoregular chains is formulated and proved. It is a direct analog of the well-known theorem for surfacecharges in physics. The theorem states the following: (1) Regardless of the end groups, in stereoregular oligomers with a centrosymmetric bulk, the end charges can only be a multiple of and the longitudinal dipole moment per monomer can only be a multiple of times the unit length in the limit of long chains. (2) In oligomers with a noncentrosymmetric bulk, the end charges can assume any value set by the nature of the bulk. Nonetheless, by modifying the end groups, one can only change the end charge by an integer and the dipole moment by an integer multiple of the unit length . (3) When the entire bulk part of the system is modified, the end charges may change in an arbitrary way; however, if upon such a modification the system remains centrosymmetric, the end charges can only change by multiples of as a direct consequence of (1). The above statements imply that—in all cases—the end charges are uniquely determined, *modulo an integer*, by a property of the bulk alone. The theorem’s origin is a robust topological phenomenon related to the Berry phase. The effects of the quantization are first demonstrated in toy LiF chains and then in a series of *trans*-polyacetylene oligomers with neutral and charge-transfer end groups.

This material is based on the work supported in part by the U.S. Army Research Office Multidisciplinary University Research Initiative (ARO/MURI) under Grant No. W911NF-04-1-0170 and by ONR Grant No. N00014-07-1-1095.

I. INTRODUCTION

II. STATEMENT OF THE QUANTIZATION THEOREM

III. A TOY MODEL: 1D BINARY CHAIN

IV. PROOF OF THE QUANTIZATION THEOREM

A. Mapping the finite quantum system onto a system of classical point charges

B. The infinite system

C. Relating the finite system to the infinite one

D. The most general case

E. The correlated case

V. CALCULATIONS FOR A CASE OF CHEMICAL INTEREST

VI. CONCLUSIONS

### Key Topics

- Polymers
- 32.0
- Geometric phases
- 16.0
- Electric dipole moments
- 12.0
- Wave functions
- 10.0
- Charge transfer
- 6.0

## Figures

(Color online) Two states of a prototypical push-pull system. The long insulating chain of alternant polyacetylene has a “donor” and “acceptor” (COOH) groups attached at the opposite ends. The charge transfer occurring in such systems upon some physical or chemical process is simulated here by moving a proton from the COOH to groups: in (a) we show the “neutral” structure and in (b) the “charge-transfer” one. The two structures share the same “bulk,” where the cell (or repeating monomer) is , and the figure is drawn for .

(Color online) Two states of a prototypical push-pull system. The long insulating chain of alternant polyacetylene has a “donor” and “acceptor” (COOH) groups attached at the opposite ends. The charge transfer occurring in such systems upon some physical or chemical process is simulated here by moving a proton from the COOH to groups: in (a) we show the “neutral” structure and in (b) the “charge-transfer” one. The two structures share the same “bulk,” where the cell (or repeating monomer) is , and the figure is drawn for .

Simple binary chains, where four monomer units are shown. (a) centrosymmetric bulk and [(b) and (c)] noncentrosymmetric bulk. Notice that (b) and (c) share the same bulk but have different terminations. We have actually run RHF and UHF calculations for such chains, choosing F as the gray atom and Li as the black atom, at variable sizes .

Simple binary chains, where four monomer units are shown. (a) centrosymmetric bulk and [(b) and (c)] noncentrosymmetric bulk. Notice that (b) and (c) share the same bulk but have different terminations. We have actually run RHF and UHF calculations for such chains, choosing F as the gray atom and Li as the black atom, at variable sizes .

Longitudinal dipole moment per cell , defined in Eq. (5), of the LiF chain as a function of . The labels (a), (b), and (c) refer to the geometries schematized in Fig. 2. The short double arrow indicates the half-quantum difference of between the RHF and the UHF for the centrosymmetric system (a); the long double arrow indicates 1 quantum difference of between RHF calculations for systems (b) and (c).

Longitudinal dipole moment per cell , defined in Eq. (5), of the LiF chain as a function of . The labels (a), (b), and (c) refer to the geometries schematized in Fig. 2. The short double arrow indicates the half-quantum difference of between the RHF and the UHF for the centrosymmetric system (a); the long double arrow indicates 1 quantum difference of between RHF calculations for systems (b) and (c).

Longitudinal dipole moment per monomer of the *trans*-polyacetylene oligomers, exemplified in Fig. 1, as a function of : diamonds for the neutral structure [NN] [Fig. 1(a)] and squares for the charge-tranfer structure [+⋯−] [Fig. 1(b)]. The double arrow indicates their difference, which is exactly equal to one quantum.

Longitudinal dipole moment per monomer of the *trans*-polyacetylene oligomers, exemplified in Fig. 1, as a function of : diamonds for the neutral structure [NN] [Fig. 1(a)] and squares for the charge-tranfer structure [+⋯−] [Fig. 1(b)]. The double arrow indicates their difference, which is exactly equal to one quantum.

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