Variational calculation of static and dynamic vibrational nonlinear optical properties

The vibrational configuration interaction method used to obtain static vibrational (hyper)polarizabilities is extended to dynamic nonlinear optical properties in the infinite optical frequency approximation. Illustrative calculations are carried out on H(2)O and NH(3). The former molecule is weakly anharmonic while the latter contains a strongly anharmonic umbrella mode. The effect on vibrational (hyper)polarizabilities due to various truncations of the potential energy and property surfaces involved in the calculation are examined.     

Comparative study of unscreened and screened molecular static linear polarizability in the Hartree–Fock,hybrid‐density functional,and density functional models

The sum‐over‐states (SOS) polarizabilities are calculated within approximate mean‐field electron theories such as the Hartree–Fock approximation and density functional models using the eigenvalues and orbitals obtained from the self‐consistent solution of the single‐particle equations. The SOS polarizabilities are then compared with those calculated using the finite‐field (FF) method. Three widely used mean‐field models are as follows: (1) the Hartree–Fock (HF) method, (2) the three parameter hybrid generalized gradient approximation (GGA) (B3LYP), and (3) the parameter‐free generalized gradient approximation due to Perdew–Burke–Ernzerhof (PBE). The comparison is carried out for polarizabilities of 142 molecules calculated using the 6‐311++G(d,p) basis set at the geometries optimized at the B3LYP/6‐311G** level. The results show that the SOS method almost always overestimates the FF polarizabilities in the PBE and B3LYP models. This trend is reversed in the HF method. A few exceptions to these trends are found. The mean absolute errors (MAE) in the screened (FF) and unscreened (SOS) polarizability are 0.78, 1.87, and 3.44 ų for the HF, B3LYP, and PBE‐GGA methods, respectively. Finally, a simple scheme is devised to obtain FF quality polarizability from the SOS polarizability. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Theoretical investigation of the (hyper)polarizabilities of pyrrole homologues C4H4XH (X = N, P, As, Sb, Bi). A coupled-cluster and density functional theory study

Geometries, inversion barriers, static and dynamic electronic and vibrational dipole polarizability (alpha), and first (beta) and second (gamma) hyperpolarizability of the pyrrole homologues C(4)H(4)XH (X = N, P, As, Sb, Bi) have been calculated by Hartree-Fock, M?ller-Plesset second-order perturbation theory, coupled-cluster theory accounting for singles, doubles, and noniterative triple excitations methods, as well as density functional theory using B3LYP and PBE1PBE functionals and Sadlej's Pol and 6-311G basis sets. Relativistic effects on the heavier homologues stibole and bismole have been taken into account within effective core potential approximation. The results show that the electronic (hyper)polarizabilities monotonically increase with the atomic number of the heteroatom, consistent with the decrease in the molecular hardness. Ring planarization reduces the carbon-carbon bond length alternation of the cis-butadienic unit, enhancing the electronic polarizability values (alpha(e)) by 4-12% and the (hyper)polarizability values (and gamma(e)) by 30-90%. Pure vibrational and zero-point vibrational average contributions to the (hyper)polarizabilities have been determined within the clamped nucleus approach. In the static limit, the pure vibrational hyperpolarizabilities have a major contribution. Anharmonic corrections dominate the pure vibrational hyperpolarizabilities of pyrrole, while they are less important for the heavier homologues. Static and dynamic electronic response properties of the pyrrole homologues are comparable to or larger than the corresponding properties of the furan and cyclopentadiene homologue series.     

On the performance of long‐range‐corrected density functional theory and reduced‐size polarized LPol‐n basis sets in computations of electric dipole (hyper)polarizabilities of π‐conjugated molecules

Static longitudinal electric dipole (hyper)polarizabilities are calculated for six medium‐sized π‐conjugated organic molecules using recently developed LPol‐n basis set family to assess their performance. Dunning's correlation‐consistent basis sets of triple‐ζ quality combined with MP2 method and supported by CCSD(T)/aug‐cc‐pVDZ results are used to obtain the reference values of analyzed properties. The same reference is used to analyze (hyper)polarizabilities predicted by selected exchange‐correlation functionals, particularly those asymptotically corrected. © 2012 Wiley Periodicals, Inc.     

Dynamic hyperpolarizability calculations of large systems: the linear-scaling divide-and-conquer approach

We report a linear-scaling computation method for evaluating the dynamic first hyperpolarizability β based on the divide-and-conquer (DC) method. In the present scheme, we utilized the quasi-density-matrix expression derived from Wigner's (2n + 1) rule for β, where the quasi-density matrices are constructed from the solution obtained via the DC time-dependent self-consistent field (TD-SCF) method [T. Touma, M. Kobayashi, and H. Nakai, Chem. Phys. Lett. 485, 247 (2010)]. Numerical evaluation of π-conjugated and saturated organic chain systems verified that the present scheme considerably reduces the computational time for the β evaluation with a slight loss of accuracy, even around the singular frequency appearing at the electronic excitation energy. This evaluation indicates that the present linear-scaling TD-SCF scheme can also be used to estimate the molecular excitation energy. Furthermore, we succeeded in accurately evaluating the macroscopic second-harmonic generation coefficient of the polyvinylidene fluoride from the molecular (hyper)polarizabilities.     

Linear optical transmission measurements and computational study of linear polarizabilities, first hyperpolarizabilities of a dinuclear iron(III) complex

The optical transmission spectral study of a dinuclear metal complex [FeL(MeOH)Cl]2 (L=N-(5-methylphenyl)-3-methoxysalicylaldimine) has been carried out. The linear optical characterization shows that this molecule has transparency in the visible range. To understand linear optical and microscopic second-order nonlinear optical (NLO) behavior of the compound, we have computed the electric dipole moment (mu) and dispersion-free first hyperpolarizabilities (beta) using Finite Field second-order M?ller-Plesset perturbation (FF MP2) procedure; and also dispersion-free linear polarizabilities (alpha), frequency-dependent linear polarizabilities within the linear optical spectrum and first hyperpolarizabilities at lambda=730-1050 and 1000-1400 nm wavelength areas using time-dependent Hartree-Fock (TDHF) method. The ab-initio calculation results reveal that the examined complex might have linear optical and microscopic NLO behavior with non-zero values.     

TTF+1-π共轭桥-6-氧四联氮阳离子双自由基NLO性质的理论研究

采用量子化学UPBE1PBE结合有限场(FF)方法,对系列TTF+?-π共轭桥-6-氧四联氮阳离子双自由基体系的稳定性,极化率αs和第一超极化率βtot进行研究.结果表明,TTF+?-6-氧四联氮阳离子双自由基引入共轭桥后,随体系共轭性增强,αs和βtot值均增大(体系2S的βtot值除外).自旋多重度和构象对双自由基体系的极化率和第一超极化率都有影响,双自由基体系由单重态转变为三重态时,极化率减小,而第一超极化率明显增大.以体系2为例,在单重态时αs和βtot值随构象变化较小,而三重态时αs和βtot值随二面角θ1和θ2的增加而减小.     

Doping-enhanced hyperpolarizabilities of silicon clusters: a global ab initio and density functional theory study of Si10 (Li, Na, K)n (n=1, 2) clusters

A global theoretical study of the (hyper)polarizabilities of alkali doped Si(10) is presented and discussed. First, a detailed picture about the low lying isomers of Si(10)Li, Si(10)Na, Si(10)K, Si(10)Li(2), Si(10)Na(2), and Si(10)K(2) has been obtained in a global manner. Then, the microscopic first (hyper)polarizabilities of the most stable configurations have been determined by means of ab initio methods of high predictive capability such as those based on the M?ller-Plesset perturbation and coupled cluster theory, paying extra attention to the (hyper)polarizabilities of the open shell mono-doped systems Si(10)Li, Si(10)Na, Si(10)K, and the influence of spin contamination. These results were used to assess the performance of methods of low computational cost based on density functional theory (DFT) in the reliable computation of these properties in order to proceed with an in-depth study of their evolution as a function of the alkali metal, the cluster composition, and the cluster structure. The most interesting outcomes of the performed (hyper)polarizability study indicate that while alkali doping leaves the per atom polarizability practically unaffected, influences dramatically the hyperpolarizabilities of Si(10). The lowest energy structures of the mono-doped clusters are characterized by significantly enhanced hyperpolarizabilities as compared to the analogue neutral or charged bare silicon clusters Si(10) and Si(11), while, certain patterns governed by the type and the number of the doping agents are followed. The observed hyperpolarizability increase is found to be in close connection with specific cluster to alkali metal charge transfer excited states and to the cluster structures. Moreover, an interesting correlation between the anisotropy of the electron density, and the hyperpolarizabilities of these systems has been observed. Finally, it is important to note that the presented method assessment points out that among the various DFT functionals used in this work, (B3LYP, B3PW91, BhandHLYP, PBE0, CAM-B3LYP, LC-BLYP, LC-BPW91) only B3PW91 and PBE0 out of the seven provided a consistent quantitative performance for both polarizabilities and hyperpolarizabilities with respect to the ab initio methods utilized here. On the other hand, the long range corrected functionals LC-(U)BLYP and LC-(U)BPW91 (μ = 0.47) failed to supply quantitatively accurate hyperpolarizability results in all the studied clusters while the CAM-(U)B3LYP functional performs satisfactory only in the case of the Na and K doped systems.     

(Hyper)polarizability density analysis for open-shell molecular systems based on natural orbitals and occupation numbers

We have developed a method for analyzing the (hyper)polarizabilities of open-shell molecular systems. This method employs the (hyper)polarizability densities based on the natural orbitals and occupation numbers, which enables us to analyze the contributions of odd electrons having various open-shell (diradical) characters. Within broken-symmetry, i.e., spin-unrestricted, single-determinant molecular orbital and density functional theory approaches, we can also remove the spin contamination effects on these quantities through spin projection. To do that, an approximate spin projected method has been elaborated and applied to the analysis of the (hyper)polarizability of multi-radical systems. As examples, typical open-shell singlet systems, 1,3-dipoles and rectangular graphene nanoflakes, are examined.     

Dipole polarizabilities of germanium clusters

Dipole polarizabilities of Ge_n clusters with 2–25 atoms are calculated using finite field (FF) method within density functional theory. The dipole moments and polarizabilities of clusters are sensitively dependent on the cluster geometries and electronic structures. The clusters with low symmetry and large HOMO–LUMO gap prefer to large dipole moments. The polarizabilities of the Ge_n clusters increase rapidly in the size range of 2–5 atoms and then fluctuate around the bulk value. The larger HOMO–LUMO gap may lead to smaller polarizability. As compared with the compact structure and diamond structure, the prolate cluster structure corresponds to a larger polarizability.     

Calculation of electric dipole (hyper)polarizabilities by long-range-correction scheme in density functional theory: a systematic assessment for polydiacetylene and polybutatriene oligomers

The long-range correction (LC) for treating electron exchange in density functional theory, combined with the Becke-Lee-Yang-Parr (BLYP) exchange-correlation functional, was used to determine (hyper)polarizabilities of polydiacetylene/polybutatriene oligomers. In comparison with coupled-cluster calculations including single and double excitations as well as a perturbative treatment of triple excitations, our values indicate that the tendency of conventional functionals to result in a catastrophic overshoot for these properties is alleviated but not eliminated. No clear-cut preference for LC-BLYP over Hartree-Fock values is obtained. This analysis is consistent with the calculations of Sekino et al. [J. Chem. Phys. 126, 014107 (2007)] on polyacetylene and molecular hydrogen oligomers. Thus, the performance of LC-BLYP with regard to (hyper)polarizabilities of quasilinear conjugated systems is now well characterized.     

含不同共轭桥的TCNQ开闭壳层体系NLO性质的DFT研究

采用密度泛函理论(DFT)结合有限场(FF)方法,对不同共轭桥连接的四氰代二甲基苯醌(TCNQ)开壳层和闭壳层电子态的非线性光学(NLO)系数进行计算,并以乙烯桥为例讨论共轭链长度与NLO性质的关系.结果表明:开壳层体系的极化率和二阶超极化率值都大于闭壳层体系,且共轭桥的共轭性越强,体系的极化率和二阶超极化率越大;在自由基体系中,单重态的二阶超极化率随双自由基成分y和自旋多重度的增加而增大.体系的共轭链增长,BLA(BondLength Alternation,共轭分子中相邻单、双键键长差的平均值)逐渐减小,双自由基成分y逐渐增大,体系的二阶超极化率也逐渐增大.     

Theoretical study on nonlinear optical properties of the Li+[calix[4]pyrrole]Li−dimer,trimer and its polymer with diffuse excess electrons

The static (hyper)polarizabilities of the dimer and trimer with diffuse excess electrons, [Li⁺[calix[4]pyrrole]Li^?]_n, are firstly investigated by the DFT(B3LYP) method in detail. For the dimer and trimer, a Li atom inside each calix[4]pyrrole unit is ionized to form a diffuse excess electron. The results show that the dimer and trimer containing two and three excess electrons, respectively, have very large first hyperpolarizablities as 2.3 × 10⁴ and 4.0 × 10⁴ au, which are 30 and 40 times larger than that of the corresponding [calix[4]pyrrole]_n (n = 2, 3) without Li atom. Also, β values of dimer and trimer are twice and four times as large as that of monomer containing one excess electron. Obviously, not only excess electron but also the number of excess electron plays an important role in increasing the first hyperpolarizability. Moreover, the (hyper)polarizabilities of the [Li⁺[calix[4]pyrrole]Li^?]_n polymer are investigated at ab initio level by using the elongation finite‐field (elongation FF) method. All the oligomers of the [Li⁺[calix[4]pyrrole]Li^?]_n with many excess electrons exhibit very large first hyperpolarizability and large second hyperpolarizability. The present investigation shows that by introducing several and more excess electrons into the nonlinear optical (NLO) materials will be an important strategy for improving their NLO properties, which will be helpful for design of NLO materials. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2010     

Electronic structures and nonlinear optical properties of supramolecular associations of benzo-2,1,3-chalcogendiazoles by the elongation method

The investigations of one-dimensional periodic associations of benzo-2,1,3-chalcogendiazoles have been carried out by the elongation method. S, Se and Te chalcogens have been in use. For the polymers, band structures and local densities of states were extracted from ab initio calculations of electronic structure of the 15 units’ oligomers. Static electric (hyper)polarizabilities were studied by the elongation finite-field method.     

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.     

Electrostatic calculation of linear and non-linear optical properties of ice Ih,II, IX and VIII

Macroscopic first- and third-order susceptibilities of ice Ih, ice II, ice IX and ice VIII are calculated using static and frequency-dependent electronic and static vibrational molecular (hyper)polarizabilities at the MP2 level. The molecular properties are in good agreement with experiment and with high-level ab initio calculations. Intermolecular electrostatic and polarization effects due to induced dipoles are taken into account using a rigorous local-field theory. The electric field due to permanent dipoles is used to calculate effective in-crystal (hyper)polarizabilities. The polarizability depends only weakly on the permanent field, but the dipole moment and the hyperpolarizabilities are strongly affected. The calculated linear susceptibility is in good agreement with available experimental data for ice Ih, and the third-order susceptibility for a third harmonic generation experiment is in reasonable agreement with experimental values for liquid water. The molecular vibrational contributions have a small effect on the susceptibilities. The electric properties of a water tetramer are calculated and used to estimate the effect of non-dipolar interactions on the susceptibilities of ice Ih, which are found to be small.     

Basis set and electron correlation effects on the polarizability and second hyperpolarizability of model open-shell pi-conjugated systems

The basis set and electron correlation effects on the static polarizability (alpha) and second hyperpolarizability (gamma) are investigated ab initio for two model open-shell pi-conjugated systems, the C(5)H(7) radical and the C(6)H(8) radical cation in their doublet state. Basis set investigations evidence that the linear and nonlinear responses of the radical cation necessitate the use of a less extended basis set than its neutral analog. Indeed, double-zeta-type basis sets supplemented by a set of d polarization functions but no diffuse functions already provide accurate (hyper)polarizabilities for C(6)H(8) whereas diffuse functions are compulsory for C(5)H(7), in particular, p diffuse functions. In addition to the 6-31G(*)+pd basis set, basis sets resulting from removing not necessary diffuse functions from the augmented correlation consistent polarized valence double zeta basis set have been shown to provide (hyper)polarizability values of similar quality as more extended basis sets such as augmented correlation consistent polarized valence triple zeta and doubly augmented correlation consistent polarized valence double zeta. Using the selected atomic basis sets, the (hyper)polarizabilities of these two model compounds are calculated at different levels of approximation in order to assess the impact of including electron correlation. As a function of the method of calculation antiparallel and parallel variations have been demonstrated for alpha and gamma of the two model compounds, respectively. For the polarizability, the unrestricted Hartree-Fock and unrestricted second-order M?ller-Plesset methods bracket the reference value obtained at the unrestricted coupled cluster singles and doubles with a perturbative inclusion of the triples level whereas the projected unrestricted second-order M?ller-Plesset results are in much closer agreement with the unrestricted coupled cluster singles and doubles with a perturbative inclusion of the triples values than the projected unrestricted Hartree-Fock results. Moreover, the differences between the restricted open-shell Hartree-Fock and restricted open-shell second-order M?ller-Plesset methods are small. In what concerns the second hyperpolarizability, the unrestricted Hartree-Fock and unrestricted second-order M?ller-Plesset values remain of similar quality while using spin-projected schemes fails for the charged system but performs nicely for the neutral one. The restricted open-shell schemes, and especially the restricted open-shell second-order M?ller-Plesset method, provide for both compounds gamma values close to the results obtained at the unrestricted coupled cluster level including singles and doubles with a perturbative inclusion of the triples. Thus, to obtain well-converged alpha and gamma values at low-order electron correlation levels, the removal of spin contamination is a necessary but not a sufficient condition. Density-functional theory calculations of alpha and gamma have also been carried out using several exchange-correlation functionals. Those employing hybrid exchange-correlation functionals have been shown to reproduce fairly well the reference coupled cluster polarizability and second hyperpolarizability values. In addition, inclusion of Hartree-Fock exchange is of major importance for determining accurate polarizability whereas for the second hyperpolarizability the gradient corrections are large.     

Performance of density functional theory in computing nonresonant vibrational (hyper)polarizabilities

A set of exchange‐correlation functionals, including BLYP, PBE0, B3LYP, BHandHLYP, CAM‐B3LYP, LC‐BLYP, and HSE, has been used to determine static and dynamic nonresonant (nuclear relaxation) vibrational (hyper)polarizabilities for a series of all‐trans polymethineimine (PMI) oligomers containing up to eight monomer units. These functionals are assessed against reference values obtained using the Møller–Plesset second‐order perturbation theory (MP2) and CCSD methods. For the smallest oligomer, CCSD(T) calculations confirm the choice of MP2 and CCSD as appropriate for assessing the density functionals. By and large, CAM‐B3LYP is the most successful, because it is best for the nuclear relaxation contribution to the static linear polarizability, intensity‐dependent refractive index second hyperpolarizability, static second hyperpolarizability, and is close to the best for the electro‐optical Pockels effect first hyperpolarizability. However, none of the functionals perform satisfactorily for all the vibrational (hyper)polarizabilities studied. In fact, in the case of electric field‐induced second harmonic generation all of them, as well as the Hartree–Fock approximation, yield the wrong sign. We have also found that the Pople 6–31+G(d) basis set is unreliable for computing nuclear relaxation (hyper)polarizabilities of PMI oligomers due to the spurious prediction of a nonplanar equilibrium geometry. © 2013 Wiley Periodicals, Inc.     

Theoretical study of static (Hyper)polarizabilities of twisted intramolecular charge transfer chromophores

Density functional theory calculations are performed to study the (hyper)polarizabilities of a series of planar and twisted intramolecular charge transfer molecules (tictoids) with different electron‐donating and electron‐withdrawing groups. Both similarity and difference between the planar and twisted molecules are noted in their (hyper)polarizability variation with respect to substituent and solvent dielectric constant. When compared with dramatic enhancements resulting from geometry twist and solvent effect, substitution to D/A pairs leads to relatively moderate variation in (hyper) polarizability. In addition, tictoids with different substituents may exhibit different solvent effects in their hyperpolarizability magnitudes. Our calculations suggest that the nonlinear optical response of the tictoids could be tuned by appropriately selecting the donor/acceptor pairs and solvents. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

Static and dynamic coupled perturbed Hartree-Fock vibrational (hyper)polarizabilities of polyacetylene calculated by the finite field nuclear relaxation method

The vibrational contribution to static and dynamic (hyper)polarizability tensors of polyacetylene are theoretically investigated. Calculations were carried out by the finite field nuclear relaxation (FF-NR) method for periodic systems, newly implemented in the CRYSTAL code, using the coupled perturbed Hartree-Fock scheme for the required electronic properties. The effect of the basis set is also explored, being particularly important for the non-periodic direction perpendicular to the polymer plane. Components requiring a finite (static) field in the longitudinal direction for evaluation by the FF-NR method were not evaluated. The extension to that case is currently being pursued. Whereas the effect on polarizabilities is relatively small, in most cases the vibrational hyperpolarizability tensor component is comparable to, or larger than the corresponding static electronic contribution.