Molecular first hyperpolarizability tensor

In formula (34), /u, (f) = (0 /Ai(f) 0 ) is the ground state, permanent, electric dipole moment of species f, while ) is the molecular first hyperpolarizability tensor defined by... 

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. 

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. 

As will be explained in Section 8.3.3, the reduction spectrum of rank-three tensors contains, in the general case, one (pseudo)scalar, three vectors, two (pseudo)devia-tors, and one septor [13]. Here, we only have to deal with the vectorial parts. They transform as a vector. Hence, the scalar product of the dipole moment vector with a vector part of the first hyperpolarizability tensor can be written as the simple product of two scalars, when we designate 9 as the angle between the molecular axis (z-axis) and the dipole moment axis (Eq. (14)). 

Heterogeneous dielectric media models have included the developments of Jprgensen et al. [7-9] (reviewed here) and Corni and Tomasi [52,53], Generally, the number of methods for determining frequency-dependent molecular electronic properties, such as the polarizability or first- and second hyperpolarizability tensors of heterogeneously solvated molecules, is very limited. 

In the present contribution we will discuss the direction of the changes of the NLO response and the solvatochromic behavior as a function of solvent polarity of the D-tt-A chromophores. The best starting point for these considerations seems to be the simple two-state model for the first-order hyperpolarizability (/ ) [8]. To avoid the extreme complexity of the sum-over-states (SOS) expression [101], Oudar and Chemla proposed the relation between the dominant component of 13 along the molecular axis (let it be the x-axis) and the spectroscopic parameters of the low-lying CT transition [8]. The use of the two-level approximation in the static case (ru = 0.0) has lead to the following expression for the static component of the first-order hyperpolarizability tensor ... 

In the above equations is the component of the molecular first-order hyperpolarizability tensor p along the principal molecular axis, and are angular averages which describe the degree of polar order,/"/ " are the local field correction factors at [Pg.125]

The first hyperpolarizability of configurationally locked trienes (CLT) has been calculated using the finite field method with the aim of addressing the relationship between the molecular and crystal second-order NLO responses. In particular, the high performance of the 2-(5-methyl-3-(4-(pyrrolidin-l-yl)styryl)cyclohex-2-enylidene)malononitrile (MH2) (Fig. 1,6) species has been attributed to optimal orientation of the dominant tensor components with respect to the crystal polar axis for phase matching. 

An interesting application of the first case (i.e., an external oscillating field) is the study of the nonlinear properties of molecules in condensed matter. Once the approximate solutions of the corresponding time-dependent SchrOdinger equation are found, the frequency-dependent electric response functions (polarizability and hyperpolarizabilities tensors) of the molecular solute are easily calculated. 

The summation runs over repeated indices, /r, is the i-th component of the induced electric dipole moment and , are components of the applied electro-magnetic field. The coefficients aij, Pijic and Yijki are components of the linear polarizability, the first hyperpolarizability, and the second hyperpolarizability tensor, respectively. The first term on the right hand side of eq. (12) describes the linear response of the incident electric field, whereas the other terms describe the nonhnear response. The ft tensor is responsible for second order nonlinear optical effects such as second harmonic generation (SHG, frequency AotAAin, frequency mixing, optical rectification and the electro-optic effect. The ft tensor vanishes in a centrosymmetric envirorunent, so that most second-order nonlinear optical materials that have been studied so far consists of non-centrosyrmnetric, one-dimensional charge-transfer molecules. At the macroscopic level, observation of the nonlinear optical susceptibility requires that the molecular non-symmetry is preserved over the physical dimensions of the bulk stmcture. 

A computer program for the theoretical determination of electric polarizabilities and hyperpolarizabilitieshas been implemented at the ab initio level using a computational scheme based on CHF perturbation theory [7-11]. Zero-order SCF, and first-and second-order CHF equations are solved to obtain the corresponding perturbed wavefunctions and density matrices, exploiting the entire molecular symmetry to reduce the number of matrix element which are to be stored in, and processed by, computer. Then a /j, and iap-iS tensors are evaluated. This method has been applied to evaluate the second hyperpolarizability of benzene using extended basis sets of Gaussian functions, see Sec. VI. 

We have performed a series of semiempirical quantum-mechanical calculations of the molecular hyperpolarzabilities using two different schemes the finite-field (FF), and the sum-over-state (SOS) methods. Under the FF method, the molecular ground state dipole moment fJ.g is calculated in the presence of a static electric field E. The tensor components of the molecular polarizability a and hyperpolarizability / are subsequently calculated by taking the appropriate first and second (finite-difference) derivatives of the ground state dipole moment with respect to the static field and using... 

The oriented gas model was first employed by Chemla et al. [4] to extract molecular second-order nonlinear optical (NLO) properties from crystal data and was based on earlier work by Bloembergen [5]. In this model, molecular hyperpolarizabilities are assumed to be additive and the macroscopic crystal susceptibilities are obtained by performing a tensor sum of the microscopic hyperpolarizabilities of the molecules that constitute the unit cell. The effects of the surroundings are approximated by using simple local field factors. The second-order nonlinear response, for example, is given by... 

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