Hyperpolarizability Tensor Research Articles

Spin-unrestricted time-dependent Hartree–Fock theory of frequency-dependent linear and nonlinear optical properties

A self-consistent time-dependent unrestricted Hartree–Fock (TDUHF) theory of linear and nonlinear optical properties is presented. Expressions are derived to calculate the elements of the linear polarizability tensor α, the first-hyperpolarizability tensor β, and the second-hyperpolarizability tensor γ, in terms of spin-adapted perturbed density matrices. For the hyperpolarizability tensors, β and γ, expressions are also derived from the lower-order solutions to the TDUHF equations. A novel feature of the present formulation is that it automatically allows for the separation of the contributions to (hyper)polarizability tensors from individual spin. Results obtained from the calculations of α(ωσ;αa) for H, C, N, O, F, Si, P, S, Cl, O2, NO, and OH and of β(−ωσ;ωa,ωb) corresponding to various second-order nonlinear optical processes for NO and OH radicals are presented. The present results for α show excellent agreement with the literature data. The calculated result for β∥ in the case of the NO radical is too small and has a different sign in comparison to the published experimental data. For both NO and OH, the β tensor for different second-order nonlinear optical processes show the order: β(−2ω;ω,ω)≳β(−ω;0,ω) ≡β(0;ω,−ω)≳β(0;0,0).

Theory of polarization measurements of second-order nonlinear light scattering

We present a complete theory of polarization measurements of second-order nonlinear light scattering in isotropic solutions of nonlinear molecules. The nonlinear interaction between the molecules and input beams at frequencies ω1 and ω2 gives rise to incoherently scattered radiation at frequency ω1+ω2. The fundamental quantity measured by such experiments in the laboratory frame of reference is the quadratic orientational average 〈βiklβjmn*〉 of the hyperpolarizability tensor. The number of independent orientational averages that can be measured is shown to be equal to the number of rotational invariants of sixth rank quantities. The absolute maximum number of independent measurements is 15 and occurs for the most general case in which the hyperpolarizability tensor is complex and the dispersion between all three frequencies is important. This number is reduced to eleven for the case of a real hyperpolarizability tensor and to six and five for the case of hyper-Rayleigh scattering and complex and real tensors, respectively. For the case of planar molecules, these numbers are further reduced to ten, seven, five, and four, respectively. We present explicit expressions that relate the rotational invariants to the components of the hyperpolarizability tensor in the molecular frame of reference. We also present practical measurement schemes that can be used to determine all rotational invariants experimentally and discuss the possibilities and limitations of nonlinear light scattering in determining the values of individual components of the molecular hyperpolarizability tensor.

Novel Chiral Bis-dipolar 6,6‘-Disubstituted Binaphthol Derivatives for Second-Order Nonlinear Optics: Synthesis and Linear and Nonlinear Optical Properties

A number of thermally and optically stable, bis-dipolar chiral molecules based on two geometries of the binaphthol (BN) system with different acceptors/substituents have been synthesized for the first time, and the synthetic routes are reported: optically pure 6,6‘-disubstituted 2,2‘-diethoxy-1,1‘-binaphthyls {R, R‘= −Br, −CHO, −CHC(CN)(COOEt), −CHC(CN)2, −CHCHCN, −CHCH(p-NO2Ph)} and optically pure 9,14-disubstituted dinaphtho[2,1-d:1‘,2‘-f][1,3]dioxepins {R, R‘ = −Br, −CHO, −CH=C(CN)(COOEt), −CHC(CN)2, −CHCHCN, −CHCHSO2CH3, −CHCHCHO, −CHCHCHC(CN)2}. All molecules possess two equal donor−acceptor systems linked together to give a bis-dipolar system. Two mono-dipolar 6-substituted 2-butoxynaphthalene (R = −CHC(CN)2, −CHC(CN)(COOEt)) donor−acceptor systems were prepared as references. The linear optical properties including solvatochromic shifts of absorption and fluorescence revealed strong charge transfer excitations in the new dipolar systems. The molecules show a high first hyperpolarizability β (up to (344−364) × 10-30 esu) as measured by electric-field-induced second-harmonic generation (EFISHG) and hyper-Rayleigh scattering (HRS). A model is developed to express the first hyperpolarizability of the bis-dipolar molecules in terms of the molecular geometry and the β of the monomeric donor−acceptor units. The tensor components were determined experimentally using this model and data from HRS and EFISHG. These techniques probe different combinations of the components of the molecular hyperpolarizability tensor. The results obtained are found to be in excellent mutual agreement.

High-order multipolar hyperpolarizabilities with imaginary frequency for H and He

Calculations are reported of X{sub zz,zz,z,z}{sup 3}({minus}i{omega}; i{omega}, 0,0), X{sub zz,zz,x,x}{sup 3} ({minus}i{omega}; i{omega}; 0,0), X{sub zzz,z,z,z}{sup 3}({minus}i{omega}; i{omega},0,0), X{sub zz,z,z,z,z}{sup 4}({minus}i{omega}; i{omega}, 0,0,0), and X{sub zz,x,x,z,z}{sup 4}({minus}i{omega}; i{omega}, 0,0,0) for H and He. These are the independent components of the atomic (quadrupole){sup 2}-(dipole){sup 2}, (octupole)-(dipole){sup 3}, and (quadrupole)-(dipole){sup 4} hyperpolarizability tensors for an imaginary frequency i{omega}. The first three quantities have been used for the determination of the R{sup {minus}8} dispersion contribution to the H-H, H-He, and He-He collision-induced pair polarizability functions. The last two quantities will be needed in future calculations of the leading order (R{sup {minus}7}) dispersion contributions to the collision-induced first hyperpolarizability ({beta}) for the H-He pair. The calculations were based on sum-over-states formulations with near-exact wave functions for H and highly accurate explicitly electron-correlated wave functions for He. For this reason the authors believe them to be of benchmark quality. 8 refs., 3 tabs.

Molecular dynamics simulation of the non-linear optical susceptibility at the phenol/water/air interface

We report results of simulations of phenol at the water/water vapour interface for a number of surface coverages. The density profiles, orientational ordering and surface potential are calculated for the water/adsorbate system and the results are compared with those for the pure water/water vapour interface. The orientational distribution of phenol is quite insensitive to surface coverage and the most probable orientation for the phenol is with the long axis normal to the surface and the hydroxy group pointing to the water. The profiles of the local non-linear optical susceptibility of the phenol–water system are calculated using ab initio estimates of the components of the hyperpolarizability tensor and the orientational ordering predicted from the classical simulation. The ratio of the components of the surface susceptibility can be used in the uniaxial approximation with an exponential form of the distribution to estimate the tilt angle of the phenol molecules. The surface susceptibility of phenol scales linearly with the coverage up to monolayer coverage and it dominates the signal above coverages of 0.005 A–2, where the total susceptibility changes sign. The model, however, does not give an accurate estimate of the surface potential, which reflects the importance of polarization effects.

Variational calculation of the dynamic third-order susceptibility of water

We present the variation-perturbation method for calculating dynamic third-order molecular hyperpolarizabilities. After a summary of the general theory, we report calculations on the water molecule, using the CNDO approximation with an extended basis set, and a truncated expansion of the trial second-order wavefunction that includes only singlyexcited Slater determinants. In the static limit, the results are found to compare well with the best ab initio values. The dispersion of the cubic hyperpolarizability tensor is also calculated.

Cubic response functions in the random phase approximation

Cubic response functions in the random phase approximation have been derived and their use for computations of static and dynamic second hyperpolarizabilities is demonstrated. The performance of implemented computer strategies in terms of direct one-index transformations and of direct atomic orbital constructions of key elements is discussed. A demonstration is given by calculations of various components of the second hyperpolarizability tensor of two linked thiophene rings.

Mechanisms of Second Harmonic Generation Efficiency Relaxation in Poled Guest/Host Polymers

We have investigated the second harmonic generation (SHG) signals of guest/host system based on 4-(dimethylamino)-4'-nitrostilbene dissolved at 1 wt % in poly(isobutyl methacrylate) and poled in a sandwich configuration of ITO electrodes. By blocking the injection from the ITO surfaces with SiO coatings, a significant impact of charge carriers on the SHG decay pattern can be excluded. Using chloroform as a solvent, an additional contribution to the SHG decay kinetics is observed which is also active in the presence of the external electric poling field. Separating this effect from the pure orientational relaxation of the dopants leads to a highly improved reproducibility. Due to its weak temperature dependence, we tentatively ascribe this solvent-specific effect to the diffusion of dipolar impurities which may affect the local field factors and hyperpolarizability tensors. The resulting corrected SHG time constants related to the pure orientational relaxation still appear to be strongly decoupled from the α-process of the matrix as inferred from dielectric relaxation data of the bulk polymer. This basic discrepancy between SHG decay and α-process is found not to depend on film preparation, sample conductivity, sample geometry, and residual solvent content.

Ab initio studies of the dipole polarizabilities of conjugated molecules: Part 5. The five-membered heterocyclics C4H4E ( E = BH, AlH, CH2, SiH2, NH, PH, O and S)

We report high quality ab initio calculations of the geometries, the static dipole polarizability and the first hyperpolarizability of the electronic ground state for a series of five-membered conjugated heterocyclics C 4 H 4 E, E = BH, AlH, CH 2, SiH 2, NH, PH, O and S (borole, aluminocyclopentadiene, cyclopentadiene, silacyclopentadiene, pyrrole, phosphole, furan and thiophene, respectively). The geometries, fully optimized at HF/6-31G ∗∗ level of theory, show good agreement with reported experimental and/or other theoretical ones. Changes induced by the heteroatom on the structural and electronic properties of these C 4H 4 rings are explained in terms of the heteroatom electronegativity. The sensitivity of the polarizability calculations to the variation of basis set was studied. We show that increasing the number of multiple d and p polarization functions in the 6-31 + G( nd, mp) basis set at Hartree-Fock level leads to values of the polarizability tensor components that are in good agreement with available experimental data. Electron correlation effects on the static dipole polarizability are shown to be non-negligible, when calculated at the MP2/6-31 + G(d,p) level of theory for borole, cyclopentadiene, pyrrole, furan and thiophene. We find that the calculated average dipole polarizability of these molecules is linearly related to the bond distance “ a” C-E of the C 4H 4E ring with a correlation coefficient of r 2 = 0.98, by α ave 10 −40 CV −1m 2 = 1.523 + 4.973a a 0 with phosphole being an exception; this compound shows an enhanced and anomalous polarizability with respect to the rest of the series. In addition, we show that these polarizabilities can be used as an aromaticity scale for explaining the known aromaticity in compounds such thiophene, pyrrole and furan. We also give the first-ever high quality calculation at the HF/6-31 + G(3d,3p) level of theory for the first hyperpolarizability tensors of these molecules.

Characterization of nonlinear optical properties by hyper-scattering techniques

Phenomena of hyper-scattering, i.e., non-linear light scattering at the second-harmonic frequency (hyper-Rayleigh scattering, HRS) or at the sum frequency (parametric light scattering, PLS), have recently become the basis for new methodologies to measure the first hyperpolarizability of molecular species. A major advantage of these new techniques is the experimental simplification and ease of use, as compared to the previous technique (electric-field-induced second-harmonic generation, EFISHG), permitting their application to a broader number of molecules, including octopoles, ionic species, complexes, excited states, biological systems, etc. Moreover, the ease of control and measurement of polarization states of incident and/or hyper-scattered light allows a detailed analysis of the geometrical properties of the hyperpolarizability tensor. New and recently developed approaches, e.g., femtosecond pulse HRS, and typical results obtained with hyper-scattering for a wide range of molecular structures are discussed in this paper.

Hyperpolarizability of tetraorganotin compounds determined by the hyper-Rayleigh scattering technique

The hyperpolarizability of non-polar tetraorganotin molecules with T d symmetry has been investigated by the hyper-Rayleigh scattering technique. The results show that, in spite of the absence of a dipolar contribution to the hyperpolarizability tensor, the value of β is significantly high.

Ab initio studies of the dipole polarizabilities of conjugated molecules Part 4. The dipole polarizability and first hyperpolarizability of o-benzyne and related molecules

We report ab initio SCF and Møller-Plesset MP2 calculations of the static dipole polarizability and SCF calculations of the first hyperpolarizability tensors of o-benzyne. The standard STO/6–31G basis set was augmented with s and p diffuse functions and multiple d polarization functions on the carbon atoms. Further p functions were also added on the hydrogen atomic centres. We compare the results for this molecule with those for ethyne, allene and s-cis-butadiene, for which we also report new polarizability tensor calculations. Comparison is made with experiment where possible. Using the largest basis set STO/6–31 + G(3d,3p) at the SCF level of theory, the α tensor components for o-benzyne were α xx = 41.80 u, α yy = 70.37 u and α zz = 82.8 u. The corresponding components of β were 146.2, 36.3 and 9.7 u respectively. The MP2 procedure gives negligible improvements on the SCF average dipole polarizability predictions by using the STO/6–31 + G(d,p) basis set. We show that o-benzyne should be considered as a potential reagent for the design and synthesis of new non-linear optical materials. Our results could also be useful for the identification of this transient molecule in non-linear optical experiments.

Parametric light scattering

We observe a nonlinear light-scattering process in which incident photons at two different frequencies give rise to scattered radiation at the sum frequency in an isotropic medium. We utilize this process to determine the ratio β322/β333 of the dominant components of the molecular hyperpolarizability tensor β for para-nitroaniline.

Missing-orbital analysis of molecular hyperpolarizability β calculated by a simplified sum-over-states method: “multi-state” model for β zxx

Missing-orbital analysis is applied to the diagonal and off-diagonal components of the first-order hyperpolarizability tensor β evaluated by a simplified sum-over-states method. Various molecules assumed to have C 2v symmetry are examined: benzene, pyridine-oxide and polyynes substituted with amino and either cyano or nitro groups. The missing-orbital analysis shows that the diagonal component β zzz ( z-axis in the donor-to-acceptor direction) for most of the molecules is essentially dominated by only one strongly charge-transfer excited configuration, and is describable by the conventional two-state model. For the off-diagonal component β zxx , it is revealed that the excitations from a 2 and b 2 orbitals cooperatively make an important contribution. This indicates that, even for the molecules whose β zzz is described well by the two-state model, β zxx is necessarily modeled by a “multi-state” model consisting of the ground state and at least two excited configurations belonging to two different symmetries.

Missing-orbital analysis of molecular hyperpolarizability β calculated by a simplified sum-over-states method: “multi-state” model for β zxx

Missing-orbital analysis is applied to the diagonal and off-diagonal components of the first-order hyperpolarizability tensor β evaluated by a simplified sum-over-states method. Various molecules assumed to have C 2v symmetry are examined: benzene, pyridine-oxide and polyynes substituted with amino and either cyano or nitro groups. The missing-orbital analysis shows that the diagonal component β zzz (z-axis in the donor-to-acceptor direction) for most of the molecules is essentially dominated by only one strongly charge-transfer excited configuration, and is describable by the conventional two-state model. For the off-diagonal component β zxx , it is revealed that the excitations from a 2 and b 2 orbitals cooperatively make an important contribution. This indicates that, even for the molecules whose β zzz is described well by the two-state model, β zzz is necessarily modeled by a “multi-state” model consisting of the ground state and at least two excited configurations belonging to two different symmetries.

Symmetry Properties of Second Order Hyperpolarizabilities

Abstract Time-reversal symmetry property of a second order nonlinear optical hyperpolarizability is examined in terms of the Manley-Rowe power relation to find relationship to the overall permutation symmetry. An extended symmetry relation is discovered reducing the independent number of the hyperpolarizability tensor components describing the second order nonlinear optical processes. Experimental implications are discussed.

Transient optically induced non-centrosymmetry in a solution of octupolar molecules

The transient polar orientation of a solution of the octupolar molecule ethyl violet is achieved by optical means in an experimental geometry of phase conjugation. The corresponding resonant six-wave mixing process is described in terms of anisotropic nonlinear absorption. Orientation results from the consistent octupolar symmetry of both the molecular species and the orienting field tensor. The 26 ps decay time of the induced non-centrosymmetry is limited by electronic excitation relaxation. A 510×10 −30 esu second-harmonic hyperpolarizability at 1064 nm fundamental wavelength with 13 D transition dipole moment between excited states is inferred. Octupolar nature of the hyperpolarizability tensor symmetry is verified by polarization analysis of the induced second-harmonic susceptibility components.

Theoretical and experimental investigations of the nonlinear optical properties of vanillin, polyenovanillin, and bisvanillin derivatives

The potential interest of vanillin, isovanillin, thiovanillin, and isothiovanillin as starting units for the design of hyperpolarizable molecules such as polyenovanillin (Ar-(CH=CH) n -CH=O) and bisvanillin derivatives was investigated by a CNDO/S method using a specific program that allowed the calculation of all the components of the first-ordcr molecular hyperpolarizability tensor β in nonpolar solvents. The reliability of the calculations was assesscd by comparison with EFISH measurements on model systems. Among the vanillin derivatives, thiovanillin, where the sulfur group is in para to the aldehyde group, showed the largest hyperpolarizability, β

Determination of higher electric polarizability tensors from unrelaxed coupled cluster density matrix calculations of electric multipole moments

The CCSDT-1 method is used to calculate electric dipole, quadrupole, and octopole moments for the HF and H2O molecule and the F− anion, together with the associated higher-order polarizabilities. All multipole moments are computed as expectation values. Externally perturbed moments are used to determine the components of the electric tensors of second and higher order. It is shown that the correlated electric moments are in good agreement with experiment and that they may be used to obtain the electric polarizability and hyperpolarizability tensors in an effective way. © 1994 John Wiley & Sons, Inc.

Ab initio studies of the dipole polarizabilities of conjugated molecules: Part 2. Monocyclic azines

We report high quality ab initio calculations of the dipole polarizability and first hyperpolarizability of the following single monoeyclic azines: pyridine, pyridazine, pyrimidine, pyrazine, s-triazine and s-tetrazine. A study of the sensitivity of such calculated properties to the variation of basis set shows that increasing the number of multiple d and p diffuse functions in the 6-31 + G( nd, mp) basis set at Hartree-Fock level gives values of the polarizability tensor components that are comparable with the experimental ones, and consistent values of hyperpolarizability tensor components. We have also found that correlation effects are negligible, provided that such calculations are performed at the SCF level with very extended basis sets. We show that standard sp basis sets need to be extended in the sp space, to have extra diffuse functions added and also to include suitable d and p gaussian primitives in order to be useful for the calculation of response functions. We find that the average dipole polarizability (α ave) is related to the number of nitrogen atoms N in the ring by α ave/10 −40(CV −1 m 2) = 10.90–0.90 N We also find that the calculated average and anisotropy polarizability components are both linearly related to the sum of the HOMO and LUMO orbital energies. We also give the first ever high-quality calculation of the hyperpolarizability tensors for these molecules. Electric dipole and quadrupole moments agree well with experiment, as do the asymmetry parameter and the nitrogen quadrupole coupling constants.