An alternative multipolar expansion for intermolecular potential functions

We have derived a new multipolar expansion for intermolecular potential-energy functions with applications in molecular physics, theoretical chemistry, and mathematical physics. The new formulation employs a separation of radial and angular terms with a simple index structure that leads to computational efficiency and ease of physical interpretation. For the case of the Coulomb interaction, we compare the present formulation with two conventional multipole expansions: the Cartesian tensor and the irreducible spherical tensor expansions. The new formalism leads to efficient numerical algorithms that are useful for general applications beyond intermolecular potentials. In addition to the electrostatic Coulomb interaction, we illustrate the formalism with applications to special function theory and a bipolar expansion involved in potential theory.     

First hyperpolarizability of polymethineimine with long-range corrected functionals

Using the long-range corrected (LC) density functional theory (DFT) scheme introduced by Iikura et al. [J. Chem. Phys. 115, 3540 (2001)] and the Coulomb-attenuating model (CAM-B3LYP) of Yanai et al. [Chem. Phys. Lett. 393, 51 (2004)], we have calculated the longitudinal dipole moments and static electronic first hyperpolarizabilities of increasingly long polymehtineimine oligomers. For comparison purposes Hartree-Fock (HF), Moller-Plesset perturbation theory (MP2), and conventional pure and hybrid functionals have been considered as well. HF, generalized gradient approximation (GGA), and conventional hybrids provide too large dipole moments for long oligomers, while LC-DFT allows to reduce the discrepancy with respect to MP2 by a factor of 3. For the first hyperpolarizability, the incorrect evolution with the chain length predicted by HF is strongly worsened by BLYP, Perdew-Burke-Ernzerhof (PBE), and also by B3LYP and PBE0. On the reverse, LC-BLYP and LC-PBE hyperpolarizabilities are correctly predicted to be positive (but for the two smallest chains). Indeed, for medium and long oligomers LC hyperpolarizabilities are slightly smaller than MP2 hyperpolarizabilities, as it should be. CAM-B3LYP also strongly improves the B3LYP results, though a bit less impressively for small chain lengths. The present study demonstrates the efficiency of long-range DFT, even in very pathological cases.     

Three-dimensional visualization of the first hyperpolarizability tensor

With polarization dependent second harmonic generation (SHG) microscopy becoming a more popular method for investigating the structure of biological materials, there is a need to develop tools with which to understand and interpret the observed SHG properties. Quantum mechanical calculations of the hyperpolarizability tensor have become a popular method for understanding the SHG properties of biomolecules. Visualization of the full hyperpolarizability tensor, termed the unit sphere representation, has been developed to provide insight and intuition on the relationship between SHG properties and molecules. A single vector representation is also presented, which approximates the SHG properties of molecules for certain cases, where the anisotropy is negligible.     

Spherical tensor theory of long-range intermolecular forces

General expressions are given for the electrostatic, induction and dispersion energies of a pair of interacting, molecules in terms of spherical tensor components of the molecular multipole moments and polarizabilities. The orientation dependence is given in terms of the scalar expansion functions S_l1l2l^k1k2.     

Theoretical study of the longitudinal first hyperpolarizability of polysilaacetylene

With the help of ab initio tools taking into account dynamic electron correlation effects, we study the longitudinal electronic first hyperpolarizability of carbon-silicon analogues to polyacetylene. It turns out that the MP2/6-31G(d)//HF/6-31G(d) scheme is suitable to obtain a semiquantitative accuracy for the first hyperpolarizability of long polysilaacetylene oligomers. The conformation of the chain has a crucial impact on its second-order nonlinear optical properties. We also show that, for some chain lengths, the frequency dispersion effects may have a huge impact, even when far away from resonance. These phenomena are rationalized in terms of delocalization and asymmetry.     

Polarizability and second hyperpolarizability evaluation of long molecules by the density functional theory with long-range correction

Polarizabilities and second hyperpolarizabilities of polyacetylene and a hydrogen chain are evaluated by density functional theory (DFT) using a hybrid generalized gradient approximation functional with correct long-range electron-electron interactions. The well known catastrophic overestimate of the hyperpolarizabilities for molecular systems of enhanced length is corrected by the two-electron repulsion operator decomposition technique, integrating the distance-dependent nonlocal exchange effects for long-range interaction, while neither the asymptotically corrected exchange functional for long-range interaction nor ordinary hybrid methods seem to be capable of overcoming the serious drawback of the DFT in polarizability/hyperpolarizability evaluation.     

Three-propeller-blade-shaped electride: remarkable alkali-metal-doped effect on the first hyperpolarizability

Significant alkali-metal-doped effects on the structure and the first hyperpolarizability (β ₀) of effective multi-nitrogen complexant tris[(2-imidazolyl)methyl]amine (TIMA) are investigated. Three imidazoles of TIMA like three blades of propeller connect with methyls by the C–C single bonds. Because of the three C–C single-bond cooperative rotations, the TIMA behaves with great flexibility, and it is a high-performance multi-nitrogen complexant for the alkali metal doping. Thus, the new complexes A_m-TIMA (A_m = Li, Na, and K) with electride characteristic have diffuse excess electron than the reported electride-type system due to the strong interaction between the complexant TIMA and alkali metal. For the first hyperpolarizability, three engaging electrides A_m-TIMA with the diffuse excess electrons exhibit considerably large β ₀ values using the MP2 (full) method and the β ₀ values of new electrides are greatly larger (3,464–29,705 times) than that (338 au) of TIMA. Surprisingly, the K-TIMA sets a new record β ₀ value to be 1.00 × 10⁷ au which far exceeds than that (3,694–76,978 au) of the reported electride-type system Li@calix[4]pyrrole (J Am Chem Soc 127:10977–10981, 2005) and Li_n−H−(CF₂−CH₂)₃−H (n = 1, 2) (J Am Chem Soc 129:2967–2970, 2007) and 31,123 au of the organometallic system (J Am Chem Soc 121:4047–4053, 1999) Ru(trans-4,4′-diethylaminostyryl-2,2′-bipyridine)₃²⁺, as well as 1.23 × 10⁶ au of the large donor-CNT systems (Nano Lett 8:2814–2818, 2008). Clearly, the alkali-metal-doped effect on the first hyperpolarizability is very dramatic for the high-performance multi-nitrogen complexant TIMA. Considering simple possibility from molecule to material, the β ₀ values of optimized Li-TIMA-dimer and Li-TIMA-tetramer are investigated by BHandHLYP method. Interestingly, results show that the order of β ₀ value is Li-TIMA-monomer < Li-TIMA-dimer < Li-TIMA-tetramer. So the new three-propeller-blade-shaped electrides can be considered as candidates for high-performance nonlinear optical materials.     

Electronic and vibrational contributions to first hyperpolarizability of donor-acceptor-substituted azobenzene

In this study we report on the electronic and vibrational (hyper)polarizabilities of donor-acceptor-substituted azobenzene. It is observed that both electronic and vibrational contributions to the electric dipole first hyperpolarizability of investigated photoactive molecule substantially depend on the conformation. The contributions to the nuclear relaxation first hyperpolarizability are found to be quite important in the case of two considered isomers (cis and trans). Although the double-harmonic term is found to be the largest in terms of magnitude, it is shown that the total value of the nuclear relaxation contribution to vibrational first hyperpolarizability is a result of subtle interplay of higher-order contributions. As a part of the study, we also assess the performance of long-range-corrected density functional theory in determining vibrational contributions to electric dipole (hyper)polarizabilities. In most cases, the applied long-range-corrected exchange-correlation potentials amend the drawbacks of their conventional counterparts.     

Role of long-range intermolecular forces in the formation of inorganic nanoparticle clusters

An understanding of the role played by intermolecular forces in terms of the electron density distribution is fundamental to the understanding of the self-assembly of molecules in the formation of a molecular crystal. Using ab initio methods capable of describing both short-range intramolecular interactions and long-range London dispersion interactions arising from electron correlation, analyses of inorganic dimers of As(4)S(4) and As(4)O(6) molecules cut from the structures of realgar and arsenolite, respectively, reveal that the molecules adopt a configuration that closely matches that observed for the crystal. Decomposition of the interaction energies using symmetry-adapted perturbation theory reveals that both model dimers feature significant stabilization from electrostatic forces as anticipated by a Lewis acid/Lewis base picture of the interaction. London dispersion forces also contribute significantly to the interaction, although they play a greater role in the realgar structure near equilibrium than in arsenolite.     

A molecular design view on the first hyperpolarizability of salicylideneaniline derivatives

Electron correlation effect was incorporated into the two-level framework through configuration interaction (CI) calculation including both occupied and unoccupied frontier orbitals, to evaluate the first hyperpolarizabilities of title compounds. Theoretical results are in excellent agreement with experimental data obtained by the solvatochromic method. Some detailed chemical and electronic information of the electron excitation process related to the nonlinear optical (NLO) properties were produced as well. Based on these data, effects of the character, number and position of donor and acceptor groups on the NLO properties are discussed from the viewpoint of molecular design. Not only the experimental data but also the theoretical analysis have suggested that a large number of intensive and properly located donor and acceptor groups may yield the optimal hyperpolarizabilities. Received: 22 May 1996 / Accepted: 9 December 1996     

An effective method for accurate prediction of the first hyperpolarizability of alkalides

The proper theoretical calculation method for nonlinear optical (NLO) properties is a key factor to design the excellent NLO materials. Yet it is a difficult task to obatin the accurate NLO property of large scale molecule. In present work, an effective intelligent computing method, as called extreme learning machine-neural network (ELM-NN), is proposed to predict accurately the first hyperpolarizability (β(0)) of alkalides from low-accuracy first hyperpolarizability. Compared with neural network (NN) and genetic algorithm neural network (GANN), the root-mean-square deviations of the predicted values obtained by ELM-NN, GANN, and NN with their MP2 counterpart are 0.02, 0.08, and 0.17 a.u., respectively. It suggests that the predicted values obtained by ELM-NN are more accurate than those calculated by NN and GANN methods. Another excellent point of ELM-NN is the ability to obtain the high accuracy level calculated values with less computing cost. Experimental results show that the computing time of MP2 is 2.4-4 times of the computing time of ELM-NN. Thus, the proposed method is a potentially powerful tool in computational chemistry, and it may predict β(0) of the large scale molecules, which is difficult to obtain by high-accuracy theoretical method due to dramatic increasing computational cost.     

Environmental and excitonic effects on the first hyperpolarizability of polar molecules and related dimers

We investigate the second-order nonlinear optical properties of a push-pull chromophore in different external and supramolecular environments, through a combined experimental and theoretical approach. In particular, we compare the first hyperpolarizability (beta) of a model dipolar and polarizable chromophore with that of a charged analogue and of a molecular dimer based on the chromophore itself. We find that the beta value of the model chromophore in solutions of low-polarity solvents is strongly affected by association effects, already at concentrations of 10 (-3) M. The presence of a positive charge in close proximity to the chromophore is found to lead to a 100% increase of the beta response of the model push-pull chromophore. This effect is of major importance for biological applications, in particular when chromophores are used as markers in charged anisotropic environments. Finally, excitonic effects, beyond the Frenkel exciton approximation, are discussed for the dimer and found to be more important the higher the order of nonlinearity is.     

After the electronic field: Structure,bonding, and the first hyperpolarizability of HArF

In this work, we add different strength of external electric field (E_ext) along molecule axis (Z‐axis) to investigate the electric field induced effect on HArF structure. The H‐Ar bond is the shortest at E_ext = ?189 × 10^?4 and the Ar‐F bond show shortest value at E_ext = 185 × 10^?4 au. Furthermore, the wiberg bond index analyses show that with the variation of HArF structure, the covalent bond H‐Ar shows downtrend (ranging from0.79 to 0.69) and ionic bond Ar‐F shows uptrend (ranging from 0.04 to 0.17). Interestingly, the natural bond orbital analyses show that the charges of F atom range from ?0.961 to ?0.771 and the charges of H atoms range from 0.402 to 0.246. Due to weakened charge transfer, the first hyperpolarizability (β_tot) can be modulated from 4078 to 1087 au. On the other hand, make our results more useful to experimentalists, the frequency‐dependent first hyperpolarizabilities were investigated by the coupled perturbed Hartree‐Fork method. We hope that this work may offer a new idea for application of noble‐gas hydrides. © 2013 Wiley Periodicals, Inc.     

Evaluation of alternative sum-over-states expressions for the first hyperpolarizability of push-pull pi-conjugated systems

A dipolar-free sum-over-states expression for the diagonal components of the first hyperpolarizability (beta) tensor has been proposed by Kuzyk [Phys. Rev. A 72, 053819 (2005)] as an alternative to the traditional expression. We examine both alternatives for the longitudinal beta of four typical push-pull pi-conjugated systems using the ab initio CIS and CIS(D) schemes to approximate the excited state properties. Since they are each evaluated approximately the two SOS expressions yield different values for beta and it is found that (i) they evolve symmetrically as the number of excited states is increased so that their average is nearly constant; (ii) in the static limit, the two values agree better with one another when their average is close to the "exact" correlated result; and (iii) frequency dispersion can affect the agreement between the alternative expressions. On the basis of (i) and (ii) it appears best for typical push-pull pi-conjugated systems to estimate the static beta, and the error in the value so obtained, by averaging the Kuzyk and traditional results.     

Wavelength dependence of first molecular hyperpolarizability of a dendrimer in solution

The frequency dependence of the first molecular hyperpolarizability of a dendrimer incorporated with thiophene-stilbene based charge-transfer chromophores is investigated by using a nanosecond 1907 nm laser and a number of wavelengths ranging from 1160 to 1760 nm emitted from an optical parametric amplifier pumped by a 1 kHz 130 fs Ti:sapphire laser. The measured hyperpolarizabilities are compared with those calculated from the charge-transfer absorption spectrum involving a Kramers-Kronig transformation scheme. The Kramers-Kronig transformation analysis provides a satisfactory account of the dispersion of the first molecular hyperpolarizability over the entire excitation wavelength range measured. The Kramers-Kronig technique extends the Oudar-Chemla two-level model previously proposed for the first molecular hyperpolarizability and it can be used in the nonresonance as well as the resonance region where the Oudar-Chemla model fails. The Kramers-Kronig transformation scheme allows a consistent intrinsic hyperpolarizability beta(0) to be obtained from the measured beta(HRS) using different excitation wavelengths for the dendrimer. The comparison of beta(0) for the dendrimer, which contains three chromophores, with that of corresponding monomer chromophore suggests that the chromophores inside the dendrimer are independent. This gives the evidence of the site isolation effect of the dendrimer and substantiates the larger macroscopic optical nonlinearity recently obtained for the dendrimer.     

Pure surface plasmon resonance enhancement of the first hyperpolarizability of gold core-silver shell nanoparticles

We report the optical second harmonic (SH) response from gold core-silver shell nanoparticles supported at a liquid-liquid interface in the spectral region where the second harmonic generation (SHG) frequency is resonant with the surface plasmon (SP) resonance excitation of the nanoparticles. We compare these results with that obtained by classical linear optical absorption spectroscopy and show that the nonlinear optical response is dominated by the SP resonance enhancement with negligible contributions from the interband transitions. As a result, the SH spectrum exhibits two clear SP resonance bands attributed to the two SP resonances of the composite nanostructure formed by the gold core-silver shell nanoparticles. Absolute values of the hyperpolarizabilities are measured by hyper Rayleigh scattering (HRS) and compared that of pure gold nanoparticles. The hyperpolarizability measured at a harmonic energy of 3.0 eV, enhanced through excitation of the high energy SP resonance of the nanoparticle, increases with the silver content whereas the hyperpolarizability measured at a harmonic energy of 2.4 eV, enhanced through the excitation of the low energy SP resonance of the nanoparticle, decreases because of the shift of this resonance away from the harmonic frequency. The hyperpolarizability determined by HRS and the square root of the SHG intensities, scaling with the nanoparticle hyperpolarizability, have similar trends with respect to the silver content indicative of closely related adsorption properties yielding similar surface concentrations at the liquid-liquid interface.     

Ab initio assessment of the first hyperpolarizability of saturated and unsaturated polyaminoborane/polyphosphinoborane copolymers

The linear and nonlinear optical properties of trans-ciso?d saturated and unsaturated polyaminoborane/polyphosphinoborane alternating copolymers are studied at the MP2/6-311G(2d)//PBE0/6-31G(2d) level of approximation, by using the increasingly large oligomer series methodology. We report the evolution with chain length of geometry parameters, partial atomic charges, electric dipole moments, polarizabilities and first hyperpolarizabilities of both phosphorus- and nitrogen-terminated chains. It turns out that for the saturated copolymer, linear and nonlinear optics properties of the saturated copolymer correspond to an average of the response of polyaminoborane and polyphosphinoborane. For the conjugated copolymer, the response is guided by the least delocalizable component.     

The static polarizability and first hyperpolarizability of the water trimer anion: ab initio study

This work predicts the extraordinary hyperpolarizability of inorganic clusters: two water trimer anions. The first hyperpolarizabilities of (H2O-)(3) are considerable, beta(0)=1.715 x 10(7) a.u. for configuration A and beta(0)=1.129 x 10(7) a.u. for configuration B at MP2/d-aug-cc-pVDZ+x level. The first hyperpolarizabilities of (H2O-)(3) (configuration A) and related systems [(H2O)(3) and (H2O)(3)F-] are compared at the MP2/d-aug-cc-pVDZ+x level. These results are beta(0)=1.715 x 10(7) a.u. for (H2O-)(3), beta(0)=35 a.u. for (H2O)(3) [the neutral core of (H2O-)(3)], and beta(0)=46 a.u. for (H2O)(3)F-). Comparing the beta(0) values of related systems, we find that the dipole-bound excess electron is the key factor in the extraordinary first hyperpolarizability of (H2O-)(3) species. It will provide a future in the development of some materials with the excess electron (e.g., electrides) that exhibit large nonlinear optical response.