^{1,a)}, Robert M. Parrish

^{1}, John S. Sears

^{1}, C. David Sherrill

^{1}and Jean-Luc Brédas

^{1,b)}

### Abstract

Predicting accurate bond-length alternations (BLAs) in long conjugated molecular chains has been a major challenge for electronic-structure theory for many decades. While Hartree-Fock (HF) overestimates BLA significantly, second-order perturbation theory and commonly used density functional theory(DFT) approaches typically underestimate it. Here, we discuss how this failure is related to the many-electron self-interaction error (MSIE), which is inherent to both HF and DFT approaches. We use tuned long-range corrected hybrids to minimize the MSIE for a series of polyenes. The key result is that the minimization of the MSIE alone does not yield accurate BLAs. On the other hand, if the range-separation parameter is tuned to yield accurate BLAs, we obtain a significant MSIE that grows with chain length. Our findings demonstrate that reducing the MSIE is one but not the only important aspect necessary to obtain accurate BLAs from density functional theory.

The Center for Computational Molecular Science and Technology is funded through a National Science Foundation (NSF) CRIF award (Grant No. CHE-0946869) and by the Georgia Institute of Technology. This work was partly supported by the AFOSR through the COMAS MURI program (Agreement No. FA9550-10-1-0558). T.K. thanks the AvH Foundation for financial support through the Feodor-Lynen program. R.P. thanks the U.S. Department of Energy for funding through the Computational Science Graduate Fellowship Program (DE-FG02-97ER25308). C.D.S. acknowledges support from the US National Science Foundation (Grant No. CHE-1011360).

I. INTRODUCTION

II. THEORETICAL METHODS

III. RESULTS

A. Standard LRC-hybrids

B. IP-optimized LRC-hybrids

IV. DISCUSSION

V. CONCLUSIONS

### Key Topics

- Density functional theory
- 25.0
- Eigenvalues
- 8.0
- Optical properties
- 4.0
- Ionization potentials
- 3.0
- Molecular electronic properties
- 3.0

## Figures

BLA at the central carbon atoms of polyenes H–(HC=CH)_{ n }–H as calculated with CCSD(T), HF, PBE, and PBEh. HF and DFT calculations employed a cc-pVTZ basis. CCSD(T) geometries employed a 6-31G(d) basis set.

BLA at the central carbon atoms of polyenes H–(HC=CH)_{ n }–H as calculated with CCSD(T), HF, PBE, and PBEh. HF and DFT calculations employed a cc-pVTZ basis. CCSD(T) geometries employed a 6-31G(d) basis set.

Deviation of total energy from that of the neutral molecule (top), deviation of total energy from straight line (ΔE) (top inset), and HOMO eigenvalue (bottom) as a function of the fractional deviation of the number of electrons from that of the neutral molecule δ, calculated for C_{16}H_{18} using HF, PBEh, and PBE and a jun-cc-pVTZ^{43} basis set.

Deviation of total energy from that of the neutral molecule (top), deviation of total energy from straight line (ΔE) (top inset), and HOMO eigenvalue (bottom) as a function of the fractional deviation of the number of electrons from that of the neutral molecule δ, calculated for C_{16}H_{18} using HF, PBEh, and PBE and a jun-cc-pVTZ^{43} basis set.

Central BLA of polyenes H–(HC=CH)_{ n }–H from the standard (open symbols) and IP-tuned (filled symbols) long-range corrected hybrids ωPBE (squares), ωPBEh (triangle up), ωB97 (circles), and ωB97X (triangle right).

Central BLA of polyenes H–(HC=CH)_{ n }–H from the standard (open symbols) and IP-tuned (filled symbols) long-range corrected hybrids ωPBE (squares), ωPBEh (triangle up), ωB97 (circles), and ωB97X (triangle right).

Deviation of total energy from straight line (ΔE) as a function of the fractional deviation of the number of electrons from that of the neutral molecule δ, calculated for H–(HC=CH)_{ n }–H with *n* = 8 (top) and *n* = 16 (bottom) using the standard (ω = 0.4 bohr^{−1}) and IP-optimized (ω = 0.186 bohr^{−1} for *n* = 8, ω = 0.136 bohr^{−1} for *n* = 16) ωPBE.

Deviation of total energy from straight line (ΔE) as a function of the fractional deviation of the number of electrons from that of the neutral molecule δ, calculated for H–(HC=CH)_{ n }–H with *n* = 8 (top) and *n* = 16 (bottom) using the standard (ω = 0.4 bohr^{−1}) and IP-optimized (ω = 0.186 bohr^{−1} for *n* = 8, ω = 0.136 bohr^{−1} for *n* = 16) ωPBE.

Central BLA of C_{16}H_{18} from ωPBE (red dashed line) and the global hybrid PBEα (blue solid line) as a function of ω and α, respectively. The dotted lines indicate the BLAs obtained from PBE, HF, and the CCSD(T) benchmark. The red and blue dots mark the standard values for ω and α in ωPBE and PBEh, respectively.

Central BLA of C_{16}H_{18} from ωPBE (red dashed line) and the global hybrid PBEα (blue solid line) as a function of ω and α, respectively. The dotted lines indicate the BLAs obtained from PBE, HF, and the CCSD(T) benchmark. The red and blue dots mark the standard values for ω and α in ωPBE and PBEh, respectively.

IP-tuned range-separation parameter ω for H–(HC=CH)_{ n }–H for ωPBE (squares) and ωPBEh (triangles) as a function of *n*.

IP-tuned range-separation parameter ω for H–(HC=CH)_{ n }–H for ωPBE (squares) and ωPBEh (triangles) as a function of *n*.

Deviation of total energy from straight line (ΔE) (top) and HOMO eigenvalue (bottom) as a function of the fractional number of electrons (neutral molecule for *N* = 58) calculated for C_{8}H_{10} using PBEh (dots) as well as the standard (squares) and IP-optimized (triangles) ωPBE.

Deviation of total energy from straight line (ΔE) (top) and HOMO eigenvalue (bottom) as a function of the fractional number of electrons (neutral molecule for *N* = 58) calculated for C_{8}H_{10} using PBEh (dots) as well as the standard (squares) and IP-optimized (triangles) ωPBE.

(Top) Central BLA of polyenes H–(HC=CH)_{ n }–H from CCSD(T) (diamonds) and the “BLA-tuned” ωPBE (squares) and PBEα (triangles) as a function of *n*. (Bottom) Corresponding HOMO MSIE (see text) as a function of *n*.

(Top) Central BLA of polyenes H–(HC=CH)_{ n }–H from CCSD(T) (diamonds) and the “BLA-tuned” ωPBE (squares) and PBEα (triangles) as a function of *n*. (Bottom) Corresponding HOMO MSIE (see text) as a function of *n*.

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