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Calculation of longitudinal polarizability and second hyperpolarizability of polyacetylene with the coupled perturbed Hartree-Fock/Kohn-Sham scheme: Where it is shown how finite oligomer chains tend to the infinite periodic polymer
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10.1063/1.3690457
/content/aip/journal/jcp/136/11/10.1063/1.3690457
http://aip.metastore.ingenta.com/content/aip/journal/jcp/136/11/10.1063/1.3690457

Figures

Image of FIG. 1.
FIG. 1.

LDA polarizability α xx (in 105 a.u.) and second hyperpolarizability γ xxxx (in 1016 a.u.) of PA as functions of the shrinking factor S. The asymptotic values are and . Highlighted areas include α and γ values converged to better than 1% with respect to the asymptotes (solid lines).

Image of FIG. 2.
FIG. 2.

HF and PBE0 polarizability α xx and second hyperpolarizability γ xxxx of PA as functions of the thresholds T x controlling the truncation of the exact exchange integrals series (see text for details). α xx in 102 a.u. (PBE0 and HF); γ xxxx in 109 (PBE0) and 106 (HF) a.u. The asymptotic values are and (PBE0); and (HF). Highlighted areas include α and γ values converged to better than 1% with respect to the asymptotes (solid lines).

Image of FIG. 3.
FIG. 3.

0D − 1D trends of the polarizability α xx (left) and the second hyperpolarizability γ xxxx (right) at the HF (bottom), PBE0 (center), and LDA (top) level of theory. α in 102 (HF and PBE0) and 103 (LDA) a.u.; γ in 106 (HF), 109 (PBE0), and 1010 (LDA) a.u. Fitting functions defined as the polynomials and . The asymptotic values, α and γ (where existent), are compared with α pol and γ pol calculated for the polymer. The minimum m value providing convergence to better than 1% with respect to the asymptotes (solid lines) is highlighted.

Image of FIG. 4.
FIG. 4.

Trends of the longitudinal polarizability and second hyperpolarizability (in a.u.) of PA oligomers, (C2H2) m , with m up to 22. Comparison between results from the present work and from Ref. 23 (obtained at the HF level) is made.

Image of FIG. 5.
FIG. 5.

Longitudinal polarizability α xx (top) and second hyperpolarizability γ xxxx (bottom) of PA (in a.u.) as functions of the energy gap E g (eV) values obtained using different percentages of exact exchange within the PBE functional. Series R min (circles) refers to CPKS calculations performed after geometry relaxation; series R PBE (triangles) refers to CPKS calculations performed on the PBE relaxed geometry. LDA, B3LYP, and HF values of α xx and γ xxxx are indicated with colored points.

Tables

Generic image for table
Table I.

Total energy and equilibrium geometry of PA as functions of the shrinking factor S. E g is the energy gap (in eV) and ΔE (in microhartree) the energy difference with respect to the most accurate results, i.e., −76.86124747 hartree (HF), −77.29556515 hartree (PBE0), and −76.67436338 hartree (LDA). Interatomic distances (L) in Å and Mulliken bond populations (BP) in |e|. A 6-31G type basis set has been used. Other computational parameters (see text for details): T E = 11, T C = 10, and T x = 30.

Generic image for table
Table II.

Polarizability α xx and second hyperpolarizability γ xxxx (in a.u.) of PA as functions of the shrinking factor S. γ xxxx in 106 (HF), 109 (PBE0), and 1016 (LDA) a.u. T CP = 4. Basis set and other computational parameters as in Table I. Bold lines define α xx and γ xxxx values converged to at least 1%.

Generic image for table
Table III.

Polarizability α xx (× 102 a.u.) and second hyperpolarizability γ xxxx (in a.u.) of PA as functions of the parameter T C controlling the truncation of the Coulomb series (see text for details). E g is the energy gap (in eV) and ΔE (in microhartree) the energy difference with respect to the most accurate results, i.e., −76.86126625 hartree (HF), −77.29557482 hartree (PBE0), and −76.67435270 hartree (LDA). Shrinking factor S is set to 300 (LDA), 50 (PBE0), and 30 (HF) for geometry optimizations and to 1200 (LDA), 100 (PBE0), and 30 (HF) for CPHF/KS calculations. Other computational parameters as in previous tables.

Generic image for table
Table IV.

Total energy and equilibrium geometry of PA as functions of the parameter T x , controlling the truncation of the exchange series (HF and PBE0). E g is the energy gap (in eV) and ΔE (in microhartree) the energy difference with respect to the most accurate result, i.e., −76.8612902 hartree (HF) and −77.2956558 hartree (PBE0). Symbols, units, and other computational parameters as in previous tables.

Generic image for table
Table V.

Polarizability α xx and second hyperpolarizability γ xxxx (in a.u.) of PA as functions of the thresholds, , controlling the truncation of the exchange series (see text for details). M is the number of direct lattice vectors involved in the exchange series summations, R is the radius (in Å) of this exchange zone. Symbols, units, and other computational parameters as in previous tables.

Generic image for table
Table VI.

0D → 1D convergence of C1 = C2 and C2 − C3 bond lengths (Å) and total energy E tot (hartree). Oligomer structures have been cut from the polymer, saturated, and geometry optimized. Bond lengths refer to the chain center and E tot is evaluated as the difference (m is the number of −C2H2− monomers), in order to eliminate border effects.

Generic image for table
Table VII.

0D → 1D convergence of the polarizability α xx and the second hyperpolarizability γ xxxx (in a.u.) of PA. E g (eV) is the energy gap. α in 103 a.u. (LDA), 102 a.u. (PBE0 and HF); γ in 1010 a.u. (LDA), 109 a.u. (PBE0) and 106 a.u. (HF). Oligomer structures have been cut from the polymer, saturated, and geometry optimized. Differences and , without border effects, are reported (m is the number of −C2H2− monomers). Basis set and computational parameters as in previous tables.

Generic image for table
Table VIII.

Bond lengths C1 = C2 and C2 − C3 (Å), cell parameter a (Å), energy gap E g (eV) and the coupled perturbed polarizability α xx and second hyperpolarizability γ xxxx (in a.u.) of PA as functions of the level of theory adopted for calculations. Sum over state (SOS) values for α xx and γ xxxx are also shown. Basis set and computational parameters as in previous tables. PBE calculations performed using computational parameters as set for LDA; B3LYP as for PBE0.

Generic image for table
Table IX.

The effect of geometry on the calculation of the polarizability α xx and the second hyperpolarizability γ xxxx of PA. Values of the optical properties at different band gaps E g (eV) have been obtained using i. the Hamiltonians indicated in parentheses (column E g ) at the relaxed geometry - column Calc(R min ); ii. the fitting functions α xx (E g ) and γ xxxx (E g ) defined at the minimum PBE energy structures for variable exact exchange percentages 0 < X HF < 100% values - Fit(R min ); iii. and the fitting functions α xx (E g ) and γ xxxx (E g ) defined for X HF = 0% - Fit(R PBE). Columns 10 x report the orders of magnitude relative to α xx and γ xxxx .

Generic image for table
Table X.

Effect of the basis set on the calculation of the polarizability α xx and the second hyperpolarizability γ xxxx of PA. Columns 2–7 provide the exponents (bohr−2) of the polarization functions added to the 6-31G and DZP sets (see Refs. 54 and 55 for a complete definition). The exponents of the most diffuse functions are s H = 0.16 and sp C = 0.17 for the 6-31G set and s H = 0.12, s C = 0.16 and p C = 0.12 for the DZP set. γ xxxx in 109 (PBE0) and 106 (HF) a.u. Energy gaps E g in eV. Calculations have been performed at the optimized geometries. Other computational parameters as in previous tables.

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/content/aip/journal/jcp/136/11/10.1063/1.3690457
2012-03-15
2014-04-19
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Scitation: Calculation of longitudinal polarizability and second hyperpolarizability of polyacetylene with the coupled perturbed Hartree-Fock/Kohn-Sham scheme: Where it is shown how finite oligomer chains tend to the infinite periodic polymer
http://aip.metastore.ingenta.com/content/aip/journal/jcp/136/11/10.1063/1.3690457
10.1063/1.3690457
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