Second hyperpolarizability response

Dispersion coefficients for second hyperpolarizabilities using coupled cluster cubic response theory... 

In the next section we derive the Taylor expansion of the coupled cluster cubic response function in its frequency arguments and the equations for the required expansions of the cluster amplitude and Lagrangian multiplier responses. For the experimentally important isotropic averages 7, 7i and yx we give explicit expressions for the A and higher-order coefficients in terms of the coefficients of the Taylor series. In Sec. 4 we present an application of the developed approach to the second hyperpolarizability of the methane molecule. We test the convergence of the hyperpolarizabilities with respect to the order of the expansion and investigate the sensitivity of the coefficients to basis sets and correlation treatment. The results are compared with dispersion coefficients derived by least square fits to experimental hyperpolarizability data or to pointwise calculated hyperpolarizabilities of other ab inito studies. 

Our present focus is on correlated electronic structure methods for describing molecular systems interacting with a structured environment where the electronic wavefunction for the molecule is given by a multiconfigurational self-consistent field wavefunction. Using the MCSCF structured environment response method it is possible to determine molecular properties such as (i) frequency-dependent polarizabilities, (ii) excitation and deexcitation energies, (iii) transition moments, (iv) two-photon matrix elements, (v) frequency-dependent first hyperpolarizability tensors, (vi) frequency-dependent polarizabilities of excited states, (vii) frequency-dependent second hyperpolarizabilities (y), (viii) three-photon absorptions, and (ix) two-photon absorption between excited states. 

Nonlinear optical activity reflects the nonlinear response of /r, (f- ) to electromagnetic radiation, which Eq. (7) shows to be governed by the first and second hyperpolarizabilities, p and y. A high level of such activity can have important applications in a variety of electro-optical devices,82,86,87 such as frequency converters, modulators, switches, etc. 

The coeflScients a, P, and y are the second, third, and fourth rank tensors and are referred to as the polarizability, first hyperpolarizability, and second hyperpolarizability, respectively. The hyperpolarizability terms are responsible for the nonlinear response of the molecule to impinging radiation. These coefiBcients are not very large, and the associated nonlinear optical effects are usually studied by taking advantage of the high optical field obtainable with laser beams. 

Norman and Jensen27 have implemented a method for obtaining second order response functions within the four component (relativistic) time-dependent Hartree-Fock scheme. Results are presented for the first order hyperpolarizabilities for second harmonic generation, />(—2o o),o ) for CsAg and CsAu. A comparison of the results with those of non-relativistic calculations implies that the nonrelativistic results are over-estimated by 18% and 66% respectively. In this method transitions that are weakly-allowed relativistically can lead to divergences in the frequency-dependent response, which would be removed if the finite lifetimes of the excited states could be taken into account. 

Liquid Phase Hyperpolarizability Calculations. In contrast to the case of the linear polarizability, the differences in hyperpolarizabilities between gas and liquid phases are very marked. The most characteristic feature is the reversal of the sign of the second order response functions on going from the gas to the liquid. Some results from recent calculations are shown in Table 11. 

Lamanna et al.S9,9° attempt to find suitable STO basis sets for the computation of the response functions, taking the first and second hyperpolarizabilities of FI20, CFI4 and NH3 as calculated by TDFIF theory as examples. 

The interaction of long-chain molecules such as polymers is a problem area where the nature of polarization response can be a significant concern on its own. An example is from a study of parallel hexatriene molecules carried out to represent a truncated form of solid-state polyacetylene [192]. This smdy included both ab initio calculations and an electrostatic model using polarizability, a, and second hyperpolarizability, y, tensors distributed to the carbon centers. The ab initio calculations on a single hexatriene molecule were used to find the distributed tensors for the electrical analysis. The objective in this smdy was not the interaction energy, but the effect on each molecule s polarizability and hyperpolarizability due to intermolecular interaction. The ab initio evaluations benchmarked the electrostatic model calculations both for... 

Zhang [200] incorporated the cations into their treatment and determined the SHG response of the isomorphous. aClO3 and NaBrOa crystals. As in Ref. 193, they reproduced the experimental fact that, although of the same chirality, these two crystals possess values of opposite sign. In another study, Xue et al. [201] investigated the effects of magnesium-doping on the SHG responses of lithium niobate. Recently, Xue and Bishop [202] have extended the treatment of Refs. 182, 187, and 193 to evaluate the crystal macroscopic THG second hyperpolarizability and applied it to diamond- and quartz-type crystals. 

In the next section we summarize the theoretical background for coupled cluster response theory and discuss certain issues related to their actual implementation. In Sections 3 and 4 we describe the application of quadratic and cubic response in calculations of first and second hyperpolarizabilities. The use of response theory to calculate magneto-optical properties as e.g. the Faraday effect, magnetic circular dichroism, Buckingham effect, Cotton-Mouton effect or Jones birefringence is discussed in Section 5. Finally we give some conclusions and an outlook in Section 6. 

Naively, one would expect that second hyperpolarizabilities y are theoretically and experimentally more difficult to obtain than first hyperpolarizabilities (3. From a computational point of view the calculation of fourth-order properties requires, according to the 2n + 1-rule, second-order responses of the wavefunction and thus the solution of considerably more equations than needed for j3 (cf. Section 2.3). However, unlike (3 the second dipole hyperpolarizability y has two isotropic tensor... 

Since second hyperpolarizabilities depend in addition to the first-order also on the second-order response of the wavefunction, the minimal requirements with respect to the choice of basis sets are for y somewhat higher than for the linear polarizabilities a and the first hyperpolarizabilities j8, in particular for atoms and small molecules. For the latter at least doubly-polarized basis sets augmented with a sufficient number of diffuse functions (e.g. d-aug-cc-pVTZ or t-aug-cc-pVTZ) are needed to obtain qualitatively correct results. Highly accurate results at a correlated level will in general only be obtained in quadruple- or better basis sets. 

Coupled cluster response calculaAons are usually based on the HF-SCF wave-function of the unperturbed system as reference state, i.e. they correspond to so-called orbital-unrelaxed derivatives. In the static limit this becomes equivalent to finite field calculations where Aie perturbation is added to the Hamiltonian after the HF-SCF step, while in the orbital-relaxed approach the perturbation is included already in the HF-SCF calculation. For frequency-dependent properties the orbital-relaxed approach leads to artificial poles in the correlated results whenever one of the involved frequencies becomes equal to an HF-SCF excitation energy. However, in Aie static limit both unrelaxed and relaxed coupled cluster calculations can be used and for boAi approaches the hierarchy CCS (HF-SCF), CC2, CCSD, CC3,... converges in the limit of a complete cluster expansion to the Full CI result. Thus, the question arises, whether for second hyperpolarizabilities one... 

Section 4.1.1 reviews second harmonic generation (SHG) for para-nitroaniline (PNA), Section 4.1.2 the polarizability and second hyperpolarizability of nitrogen and benzene, Section 4.1.3 the second hyperpolarizability of Cgo, Section 4.2 the excited state polarizability of pyrimidine and r-tetrazine. Section 4.3 three-photon absorption, and finally, in Section 4.5 the electronic g-tensor and the hyperfine coupling tensor are reviewed as examples of open shell DFT response properties. 

---