Hyperpolarizabilities systems

The polarizability expresses the capacity of a system to be deformed under the action of electric field it is the first-order response. The hyperpolarizabilities govern the non linear processes which appear with the strong fields. These properties of materials perturb the propagation of the light crossing them thus some new phenomenons (like second harmonic and sum frequency generation) appear, which present a growing interest in instrumentation with the lasers development. The necessity of prediction of these observables requires our attention. 

This calculation has shown the importance of the basis set and in particular the polarization functions necessary in such computations. We have studied this problem through the calculation of the static polarizability and even hyperpolarizability. The very good results of the hyperpolarizabilities obtained for various systems give proof of the ability of our approach based on suitable polarization functions derived from an hydrogenic model. Field—induced polarization functions have been constructed from the first- and second-order perturbed hydrogenic wavefunctions in which the exponent is determined by optimization with the maximum polarizability criterion. We have demonstrated the necessity of describing the wavefunction the best we can, so that the polarization functions participate solely in the calculation of polarizabilities or hyperpolarizabilities. 

Besides the elementary properties of index permutational symmetry considered in eq. (7), and intrinsic point group symmetry of a given tensor accounted for in eqs. (8)-(14), much more powerful group-theoretical tools [6] can be developed to speed up coupled Hartree-Fock (CHF) calculations [7-11] of hyperpolarizabilities, which are nowadays almost routinely periformed in a number of studies dealing with non linear response of molecular systems [12-35], in particular at the self-consistent-field (SCF) level of accuracy. 

In a previous work [1,2], we were interested in the calculation of second order hyperpolarizabilities of eonjugated systems including substituted benzenes, pyridine N-oxydes and vinyl oligomers, in relation with non linear optical activity [3]. We showed that MNDO ealeulations were in good agreement with SCF ab initio results obtained using a double zeta basis set plus polarization and diffuse orbitals. 

Before closing this section, it is worth mentioning that the hyperpolarizability tensors are complex quantities usually given in the old cgs system of units of esu (electrostatic units). The transformation into the International System is readily obtained with the relationship ... 

Based on the fundamental dipole moment concepts of mesomeric moment and interaction moment, models to explain the enhanced optical nonlinearities of polarized conjugated molecules have been devised. The equivalent internal field (EIF) model of Oudar and Chemla relates the j8 of a molecule to an equivalent electric field ER due to substituent R which biases the hyperpolarizabilities (28). In the case of donor-acceptor systems anomalously large nonlinearities result as a consequence of contributions from intramolecular charge-transfer interaction (related to /xjnt) and expressions to quantify this contribution have been obtained (29). Related treatments dealing with this problem have appeared one due to Levine and Bethea bearing directly on the EIF model (30), another due to Levine using spectroscopically derived substituent perturbations rather than dipole moment based data (31.) and yet another more empirical treatment by Dulcic and Sauteret involving reinforcement of substituent effects (32). 

The values for the dipoles, polarizabilities, and hyperpolarizabilities of the H2 series were obtained using (a) a 16-term basis with a fourfold symmetry projection for the homonuclear species and (b) a 32-term basis with a twofold symmetry projection for the heteronuclear species. These different expansion lengths were used so that when combined with the symmetry projections the resulting wave functions were of about the same quality, and the properties calculated would be comparable. A crude analysis shows that basis set size for an n particle system must scale as k", where k is a constant. In our previous work [64, 65] we used a 244-term wave function for the five-internal-particle system LiH to obtain experimental quality results. This gives a value of... 

We have considered scalar, vector, and matrix molecular properties. A scalar is a zero-dimensional array a vector is a one-dimensional array a matrix is a two-dimensional array. In general, an 5-dimensional array is called a tensor of rank (or order) s a tensor of order s has ns components, where n is the number of dimensions of the coordinate system (usually 3). Thus the dipole moment is a first-order tensor with 31 = 3 components the polarizability is a second-order tensor with 32 = 9 components. The molecular first hyperpolarizability (which we will not define) is a third-order tensor. 

It should be noted that polarizabilities of various orders can be defined in an alternative way in the SI system of units to that discussed previously. A quantity having the dimension of volume a = a/47re0 can be considered to be an SI analogue of the cgs polarizability. Analogously, y = -y/47re0 (or y = y/eo) can be used as the third-order hyperpolarizability in the SI system, with y having the units of m5 V 3. The presence or absence of the factor of An in the definition of the hyperpolarizability is, unfortunately, not always obvious in literature data. 

Most numerical methods for calculating molecular hyperpolarizability use sum over states expressions in either a time-dependent (explicitly including field dependent dispersion terms) or time-independent perturbation theory framework [13,14]. Sum over states methods require an ability to determine the excited states of the system reliably. This can become computationally demanding, especially for high order hyperpolarizabilities [15]. An alternative strategy adds a finite electric field term to the hamiltonian and computes the hyperpolarizability from the derivatives of the field dependent molecular dipole moment. Finite-field calculations use the ground state wave function only and include the influence of the field in a self-consistent manner [16]. 

Methods that are known to calculate transition matrix elements reliably for the systems of interest (e.g., 7r-electron systems) have been used extensively [13,17]. Especially for /3 calculations, where relatively few electronic states often dominate the hyperpolarizability, numerical methods are reliable. However, 7 calculations are more complicated because of the larger number of contributing terms and the possibility of subtle cancellations that can occur only when the full series is summed. General aspects of / and 7 calculations are discussed in the next section. 

Hydrogenic atoms (one electron bound by a nuclear charge Z) have 7 proportional to the seventh power of the orbital radius [29]. Square well 1-D potentials with infinitely high walls and an appropriate number of filled states give 7 proportional to the 5th power of the well width [29]. There is clearly a rapid increase expected in the second hyperpolarizability with system size for delocalized systems. 

This paper summarizes the theoretical analysis of some new molecules with methylsulfonyl group as the electron acceptor group, describes the syntheses of new stilbene and azobenzene systems, and presents the measurements of their optical spectra, ground-state dipole-moments, and molecular hyperpolarizability coefficients, p. We compare theoretical and experimental results and comment on the potential usefulness of these chromophores as components for NLO materials. The incorporation of sulfonyl-containing chromophores into polymers, and the NLO properties of the resulting materials, will be discussed in our forthcoming paper (9). 

Since the hyperpolarizability of a given molecule is a function of the donor and the acceptor properties, and nature of the conjugation path between them, we turned to the biphenyl system and analyzed the 4-amino-4 -methylsulfonylbiphenyl (V). The calculated ground state dipole moment of this molecule is smaller than expected for such an increase in the distance between the donor and the acceptor. 

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