^{1,a)}, Michel Rèrat

^{2}, Roberto Orlando

^{1}, Mauro Ferrero

^{2}and Roberto Dovesi

^{1}

### Abstract

The longitudinal polarizability, α_{ xx }, and second hyperpolarizability, γ_{ xxxx }, of polyacetylene are evaluated by using the coupled perturbed Hartree-Fock/Kohn-Sham (HF/KS) scheme as implemented in the periodic CRYSTAL code and a split valence type basis set. Four different density functionals, namely local density approximation (LDA) (pure local), Perdew-Becke-Ernzerhof (PBE) (gradient corrected), PBE0, and B3LYP (hybrid), and the Hartree-Fock Hamiltonian are compared. It is shown that very tight computational conditions must be used to obtain well converged results, especially for γ_{ xxxx }, that is, very sensitive to the number of points in reciprocal space when the band gap is small (as for LDA and PBE), and to the extension of summations of the exact exchange series (HF and hybrids). The band gap in LDA is only 0.01 eV: at least 300 points are required to obtain well converged total energy and equilibrium geometry, and 1200 for well converged optical properties. Also, the exchange series convergence is related to the band gap. The PBE0 band gap is as small as 1.4 eV and the exchange summation must extend to about 130 Å from the origin cell. Total energy,band gap, equilibrium geometry, polarizability, and second hyperpolarizability of oligomers −(C_{2}H_{2})_{ m }−, with *m* up to 50 (202 atoms), and of the polymer have been compared. It turns out that oligomers of that length provide an extremely poor representation of the infinite chain polarizability and hyperpolarizability when the gap is smaller than 0.2 eV (that is, for LDA and PBE). Huge differences are observed on α_{ xx } and γ_{ xxxx } of the polymer when different functionals are used, that is in connection to the well-known *density functional theory(DFT) overshoot*, reported in the literature about short oligomers: for the infinite model the ratio between LDA (or PBE) and HF becomes even more dramatic (about 500 for α_{ xx } and 10^{10} for γ_{ xxxx }). On the basis of previous systematic comparisons of results obtained with various approaches including DFT,HF, Moller-Plesset (MP2) and coupled cluster for finite chains, we can argue that, for the infinite chain, the present HF results are the most reliable.

The authors wish to acknowledge Professor Bernard Kirtman, from the Department of Chemistry and Biochemistry of UCSB - Santa Barbara, for his important help in orienting through the wide literature on the subject and in commenting on early drafts of the paper.

I. INTRODUCTION

II. THE CPHF/CPKS METHOD TO FOURTH ORDER

III. COMPUTATIONAL DETAILS

A. Convergence with respect to the number of points

B. Convergence with respect to the two-electron series range

IV. RESULTS

A. From the molecule to the polymer:Structures, energetics, and optical properties

B. Effect of the Hamiltonian on periodic properties

C. Effect of the basis set

V. CONCLUSIONS

### Key Topics

- Polymers
- 34.0
- Laser Doppler velocimetry
- 26.0
- Band gap
- 25.0
- Density functional theory
- 14.0
- Polarizability
- 13.0

## Figures

LDA polarizability α_{ xx } (in 10^{5} a.u.) and second hyperpolarizability γ_{ xxxx } (in 10^{16} 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).

LDA polarizability α_{ xx } (in 10^{5} a.u.) and second hyperpolarizability γ_{ xxxx } (in 10^{16} 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).

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 10^{2} a.u. (PBE0 and HF); γ_{ xxxx } in 10^{9} (PBE0) and 10^{6} (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).

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 10^{2} a.u. (PBE0 and HF); γ_{ xxxx } in 10^{9} (PBE0) and 10^{6} (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).

0*D* − 1*D* 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 10^{2} (HF and PBE0) and 10^{3} (LDA) a.u.; γ in 10^{6} (HF), 10^{9} (PBE0), and 10^{10} (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.

0*D* − 1*D* 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 10^{2} (HF and PBE0) and 10^{3} (LDA) a.u.; γ in 10^{6} (HF), 10^{9} (PBE0), and 10^{10} (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.

Trends of the longitudinal polarizability and second hyperpolarizability (in a.u.) of PA oligomers, (C_{2}H_{2})_{ m }, with *m* up to 22. Comparison between results from the present work and from Ref. 23 (obtained at the HF level) is made.

Trends of the longitudinal polarizability and second hyperpolarizability (in a.u.) of PA oligomers, (C_{2}H_{2})_{ m }, with *m* up to 22. Comparison between results from the present work and from Ref. 23 (obtained at the HF level) is made.

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.

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

Total energy and equilibrium geometry of PA as functions of the shrinking factor *S*. *E* _{ g } is the energy gap (in eV) and Δ*E* (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.

Total energy and equilibrium geometry of PA as functions of the shrinking factor *S*. *E* _{ g } is the energy gap (in eV) and Δ*E* (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.

Polarizability α_{ xx } and second hyperpolarizability γ_{ xxxx } (in a.u.) of PA as functions of the shrinking factor *S*. γ_{ xxxx } in 10^{6} (HF), 10^{9} (PBE0), and 10^{16} (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%.

Polarizability α_{ xx } and second hyperpolarizability γ_{ xxxx } (in a.u.) of PA as functions of the shrinking factor *S*. γ_{ xxxx } in 10^{6} (HF), 10^{9} (PBE0), and 10^{16} (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%.

Polarizability α_{ xx } (× 10^{2} 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 Δ*E* (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.

Polarizability α_{ xx } (× 10^{2} 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 Δ*E* (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.

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 Δ*E* (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.

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 Δ*E* (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.

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.

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.

0*D* → 1*D* convergence of C_{1} = C_{2} and C_{2} − C_{3} 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 −C_{2}H_{2}− monomers), in order to eliminate border effects.

0*D* → 1*D* convergence of C_{1} = C_{2} and C_{2} − C_{3} 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 −C_{2}H_{2}− monomers), in order to eliminate border effects.

0*D* → 1*D* 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 10^{3} a.u. (LDA), 10^{2} a.u. (PBE0 and HF); γ in 10^{10} a.u. (LDA), 10^{9} a.u. (PBE0) and 10^{6} 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 −C_{2}H_{2}− monomers). Basis set and computational parameters as in previous tables.

0*D* → 1*D* 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 10^{3} a.u. (LDA), 10^{2} a.u. (PBE0 and HF); γ in 10^{10} a.u. (LDA), 10^{9} a.u. (PBE0) and 10^{6} 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 −C_{2}H_{2}− monomers). Basis set and computational parameters as in previous tables.

Bond lengths C_{1} = C_{2} and C_{2} − C_{3} (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.

Bond lengths C_{1} = C_{2} and C_{2} − C_{3} (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.

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 }.

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 }.

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 10^{9} (PBE0) and 10^{6} (HF) a.u. Energy gaps *E* _{ g } in eV. Calculations have been performed at the optimized geometries. Other computational parameters as in previous tables.

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 10^{9} (PBE0) and 10^{6} (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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