Abstract
Various periodic piecewise linear potentials for extracting the electronic response of an infinite periodic system to a uniform electrostatic field are examined. It is shown that discontinuous potentials, such as the sawtooth, cannot be used for this purpose. Continuous triangular potentials can be successfully employed to determine both even- and odd-order (hyper)polarizabilities, as demonstrated here for the first time, although the permanent dipole moment of the corresponding long finite chain remains out of reach. Moreover, for typical highly polarizable organic systems, the size of the repeated unit has to be much larger than that of the finite system in order to obtain convergence with respect to system size. All results are illustrated both through extensive model calculations and through ab initio calculations on poly- and oligoacetylenes.
This work was supported by the German Research Council (DFG) through Project No. Sp439/20. Moreover, two of the authors (M.F. and M.S.) are very grateful to the International Center for Materials Research, University of California, Santa Barbara, for generous hospitality.
I. INTRODUCTION
II. THEORETICAL ANALYSIS
A. Foundations
B. Effect of the external potential
C. In practice
D. Charge distribution
III. MODEL CALCULATIONS
IV. AB INITIO CALCULATIONS
V. CONCLUSIONS
Key Topics
- Ab initio calculations
- 11.0
- Electric dipole moments
- 9.0
- Electrostatics
- 9.0
- Polarizability
- 4.0
- Polarization
- 4.0
Figures
Schematic representation of various types of sawtooth and triangular potentials as experienced by a linear chain. The chain is lying along the axis, and each repeated unit consists of two different atoms (marked with different colors). The lattice constant is and the two atoms per unit are displaced and away from the equidistant positions, so that the alternating nearest-neighbor distances become , as shown. The BvK zone is marked in the figure. [(a) and (b)] show sawtooth potentials with (a) the BvK periodicity and (b) the lattice periodicity. [(c) and (d)] show different continuous, symmetric triangular potentials, whereas (e) shows an asymmetric, continuous, triangular potential. Discontinuous triangular potentials, being either (f) asymmetric or (g) symmetric, are shown in the last two panels. Finite chains are assumed to have the length of the BvK zone. In the present work we consider explicitly only case (a) and case (e). Case (e) shows triangular potentials if the two segments contain and unit cells, with and fixed, and triangular potentials if the two segments contain and unit cells, respectively.
Schematic representation of various types of sawtooth and triangular potentials as experienced by a linear chain. The chain is lying along the axis, and each repeated unit consists of two different atoms (marked with different colors). The lattice constant is and the two atoms per unit are displaced and away from the equidistant positions, so that the alternating nearest-neighbor distances become , as shown. The BvK zone is marked in the figure. [(a) and (b)] show sawtooth potentials with (a) the BvK periodicity and (b) the lattice periodicity. [(c) and (d)] show different continuous, symmetric triangular potentials, whereas (e) shows an asymmetric, continuous, triangular potential. Discontinuous triangular potentials, being either (f) asymmetric or (g) symmetric, are shown in the last two panels. Finite chains are assumed to have the length of the BvK zone. In the present work we consider explicitly only case (a) and case (e). Case (e) shows triangular potentials if the two segments contain and unit cells, with and fixed, and triangular potentials if the two segments contain and unit cells, respectively.
Extracted values of (left panels) and (right panels) from the model calculations and as a function of the length, , of the finite chain or of the BvK zone. is given in units of atom pairs. In each case, two curves are given corresponding to fitting the total energies in a fourth or a sixth order series in the field strength. Moreover, the dashed lines mark the converged finite-chain values. For the results labeled , both (symmetric) (1,1) and (3,3) triangular potentials are considered, whereas for those labeled asymmetric (2,1), (3,1), and (3,2) triangular potentials are considered. In both cases, the curves that start for the lowest are the results for the lowest values of . Finally, we also show results for the asymmetric triangular potentials. For a precise definition of the potential shapes, see the text and Fig. 1.
Extracted values of (left panels) and (right panels) from the model calculations and as a function of the length, , of the finite chain or of the BvK zone. is given in units of atom pairs. In each case, two curves are given corresponding to fitting the total energies in a fourth or a sixth order series in the field strength. Moreover, the dashed lines mark the converged finite-chain values. For the results labeled , both (symmetric) (1,1) and (3,3) triangular potentials are considered, whereas for those labeled asymmetric (2,1), (3,1), and (3,2) triangular potentials are considered. In both cases, the curves that start for the lowest are the results for the lowest values of . Finally, we also show results for the asymmetric triangular potentials. For a precise definition of the potential shapes, see the text and Fig. 1.
As Fig. 2, but for . Only those cases are shown that result in meaningful results.
As Fig. 2, but for . Only those cases are shown that result in meaningful results.
Extracted values of (upper panel) and (lower panel) for PA oligomers and polymers from the ab initio calculations as a function of the length of the finite chain or of the BvK zone. is given in units of atom pairs.
Extracted values of (upper panel) and (lower panel) for PA oligomers and polymers from the ab initio calculations as a function of the length of the finite chain or of the BvK zone. is given in units of atom pairs.
Tables
The total charge (given in number of electrons relative to that of the neutral case) of the left part and the right part of a segment containing atoms for different systems and field strengths. The values with the extra index 0 are those for the field-free case. In those cases where results are given for only , the ones for are identical to those for .
The total charge (given in number of electrons relative to that of the neutral case) of the left part and the right part of a segment containing atoms for different systems and field strengths. The values with the extra index 0 are those for the field-free case. In those cases where results are given for only , the ones for are identical to those for .
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