Size-consistent approach and density analysis of hyperpolarizability: Second hyperpolarizabilities of polymeric systems with and without defects

Abstract

Various size-consistent approaches to the calculation of molecular hyperpolarizabilities are analyzed based on the double perturbation theory. General equations for the nth-order response property with respect to an external time-independent field are derived on the basis of the Rayleigh-Schrödinger perturbation theory ͑RSPT͒ and the coupled-cluster ͑CC͒ theory. The corresponding equations for the time-dependent case are also derived by the CC formalism, which is referred to as the time-dependent CC ͑TDCC͒ method. In order to clarify the spatial characteristics of polarizability and hyperpolarizability, we present an analysis method using a new concept ''the polarizability and hyperpolarizability densities.'' As an application of the size-consistent methods, the static second hyperpolarizabilities ͑␥͒ of -conjugated polymeric systems are calculated by the use of the uncoupled ͑UCHF͒, and coupled-Hartree-Fock ͑CHF͒ methods combined with the semiempirical INDO approximation. Characteristics of ␥ values calculated for regular polyenes, solitonlike polyenes and donor(D) -acceptor(A) disubstituted polyenes are investigated, particularly in relation to the chain-length effect. Further, we employ ␥ density analysis method, in which the third derivatives of the Mulliken charge densities against applied electric fields are plotted for exploring the local contributions of the constituent atoms to ␥ values. Results for the finite polyenes are also extrapolated to an infinity of the chain length to predict the intrinsic ␥ values per unit carbon-carbon ͑CC͒ bond of polymeric chains. Furthermore, the CHFϩMo "ller-Plesset second-order perturbation ͑MP2͒ method in the PPP approximation is applied to polymeric systems with larger chain length in order to elucidate the electron correlation effects on the chain-length dependencies of ␥ values.

NONLINEAR OPTICAL PROPERTIES OF BC5CLUSTERS

“…A estrutura otimizada obtida através dos métodos semiempíricos para a molécula 4, que é octopolar, deve contribuir para os resultados obtidos, visto que o valor de β é influenciado pela geometria molecular de acordo com estudos multidisciplinares. 13,17 Deve ser destacado que a metodologia semiempírica não aborda o efeito de solvente, 26 a formação de pontes de hidrogênio, 27 a transferência de carga intramolecular e intermolecular, 28 os defeitos na cadeia, 29 a correlação eletrônica e efeitos vibracionais 30 que influenciam a magnitude das hiperpolarizabilidades β e γ. Portanto, os resultados obtidos com as técnicas semiempíricas são geralmente inferiores aos experimentais, como observado nas predições dos valores da β 0 (Figura 3) para as moléculas orgânicas investigadas.…”

Visualization of electronic and vibrational polarizabilities and hyperpolarizabilities

“…As formulated, it appears that both methods, i.e., Chopra et al [3] and Nakano et al [5], can only be used when studying local contributions to (hyper)polarizabilities, and it is impossible to obtain an overall picture of how each region affects the global (hyper)polarizability. Indeed, if one considers a planar molecule such as formaldehyde, it is clear that the contour maps obtained on the σ v plane may be considerably different from those on a parallel plane above or below σ v .…”

“…More recently, Geskin and Brédas [4] have opted for the same strategy to investigate the spatial contribution to the second hyperpolarizability γ αβγ δ . Noticing that Chopra et al's [3] definition was origin dependent, Nakano et al [5] have proposed to remove this intrinsic drawback by keeping only the derivatives of the electron density. They used their procedure [6 -12] in a systematic manner for showing the effect of an extended basis set on (hyper)polarizabilities.…”

Experimental and Theoretical Investigation of the Photoelectrical Properties of Tetrabromophenol Blue- and Bromoxylenol Blue-Based Solar Cells

“…Analysis of the First Hyperpolarizability Density. In order to more deeply understand the result of hyperpolarizability, which is studied by hyperpolarizability density analysis and is a powerful technique to characterize contribution to hyperpolarizability from various molecular regions, the electron density of a system ρ r, F may be expressed as Taylor expansion with respect to the externally applied electric field F [55,56]: From the above equation and expansion formula of dipole in the power of F, a component of the first hyperpolarizability can be expressed:…”