Order tensor inversion symmetry

There are different paths to achieving surface specificity. One can exploit optical susceptibilities and resonances that are nonzero only at the surface or only for the molecular species of interest adsorbed on the surface. Examples include the use of second-order nonlinear mixing processes such as second harmonic generation7-9 for which the nonlinear susceptibility tensor is nonzero only where inversion symmetry is broken. Spectroscopic techniques with very high selectivity for molecular resonances such as surface-enhanced infrared or Raman spectroscopy10-12 may also be used. 

Fig. 4. For axially symmetric alignment, a single measured residual dipolar coupling is consistent with orientations of the corresponding internuclear vector, r, which deviate from the principal (z) axis of the order tensor by an angle , thus describing the surface of a cone, and its inverse. If alignment departs from axial symmetry, then the cone of permissible orientations will require further description in terms of an azimuth angle and exhibit some distortion along the x and directions.

Given the interest and importance of chiral molecules, there has been considerable activity in investigating the corresponding chiral surfaces [68, 69 and 70]. From the point of view of performing surface and interface spectroscopy with nonlinear optics, we must first examine the nonlinear response of the bulk liquid. Clearly, a chiral liquid lacks inversion symmetry. As such, it may be expected to have a strong (dipole-allowed) second-order nonlinear response. This is indeed true in the general case of SFG [71]. For SHG, however, the permutation symmetry for the last two indices of the nonlinear susceptibility tensor combined with the... 

The SFG technique probes the second-order nonhnear hyperpolarizability tensor this tensor includes the Raman and IR susceptibihty, which requires that the molecular vibrational modes are both Raman and IR active. Since Raman- and IR-dipole moment transition selection rules for molecules with a center of symmetry indicate that a vibrational mode is either Raman or IR active but not both, only molecules in a non-centrosymmetric environment on the surface interact with the electric fields molecules in the isotropic bulk phase show inversion symmetry where the third rank hyperpolarizability tensor goes to zero [25-27]. 

The second-order susceptibility ax, ox, m) is a third-rank tensor. For materials with inversion symmetry, in the electric dipole approximation for all i,j, k [78]. Ne-... 

Due to the symmetry of T and E, the number of components of the stiffness and compliance tensors is reduced from a total of 81 to 36 independent ones. Thus, it is possible to represent these fourth-order tensors alternatively in the form of (6 x 6) matrices (which, of course, do not have the transformation properties of a tensor), and to express Hooke s law and inverse Hooke s law in direct matrix notation (engineering notation) as... 

Although the order of frequencies in coqOJ — (o ) has no special meaning itself, owing to the intrinsic permutation symmetry, we will keep it fixed (Raman convention), using co and -co as second and third arguments. Besides the intrinsic permutation symmetry, the medium macroscopic symmetry imposes further restrictions on the tensor index of nonvanishing, independent components A very important result is that the nth-order susceptibility vanishes for even n in media showing inversion symmetry and contributes the low-est-order nonlinearity. For isotropic media, it can be shown that vanishes if some Cartesian index appears an odd number of times in the subscript. 

In Cartesian coordinates obviously there are altogether 3" elements in the third-order susceptibility X > ], a fourth-rank tensor, since i, j, k, /) each has three components 1, 2, 3. In an isotropic medium with inversion symmetry, however, it can be shown that there are only four different components, three of which are independent ... 

Symmetry is one of the most important issues in the field of second-order nonlinear optics. As an example, we will briefly demonstrate a method to determine the number of independent tensor components of a centrosymmetric medium. One of the symmetry elements present in such a system is a center of inversion that transforms the coordinates xyz as ... 

Any symmetry operation is required to leave the sign and magnitude of physical properties unchanged and therefore y xxx = 0. The same line of reasoning can be used to show that all tensor components will vanish under inversion. Hence, second-order nonlinear optical properties are not allowed in centrosymmetric media using the electric dipole approximation. The presence of noncentrosymmetry is one of the most stringent requirements in... 

The nonvanishing components of the tensors y a >--eem and ya >-mee can be determined by applying the symmetry elements of the medium to the respective tensors. However, in order to do so, one must take into account that there is a fundamental difference between the electric field vector and the magnetic field vector. The first is a polar vector whereas the latter is an axial vector. A polar vector transforms as the position vector for all spatial transformations. On the other hand, an axial vector transforms as the position vector for rotations, but transforms opposite to the position vector for reflections and inversions.9 Hence, electric quantities and magnetic quantities transform similarly under rotations, but differently under reflections and inversions. As a consequence, the nonvanishing tensor components of x(2),eem and can be different... 

Characterization of Molecular Hyperpolarizabilities Using Third Harmonic Generation. Third harmonic generation (THG) is the generation of light at frequency 3co by the nonlinear interaction of a material and a fundamental laser field at frequency co. The process involves the third-order susceptibility x 3K-3 , , ) where —3 represents an output photon at 3 and the three s stand for the three input photons at . Since x(3) is a fourth (even) rank tensor property it can be nonzero for all material symmetry classes including isotropic media. This is easy to see since the components of x(3) transform like products of four spatial coordinates, e.g. x4 or x2y2. There are 21 components that are even under an inversion operation and thus can be nonzero in an isotropic medium. Since some of the terms are interrelated there are only four independent terms for the isotropic case. 

In this equation, po is the permanent dipole moment of the molecule, a is the linear polarizability, 3 is the first hyperpolarizability, and 7 is the second hyperpolarizability. a, and 7 are tensors of rank 2, 3, and 4 respectively. Symmetry requires that all terms of even order in the electric field of the Equation 10.1 vanish when the molecule possesses an inversion center. This means that only noncentrosymmetric molecules will have second-order NLO properties. In a dielectric medium consisting of polarizable molecules, the local electric field at a given molecule differs from the externally applied field due to the sum of the dipole fields of the other molecules. Different models have been developed to express the local field as a function of the externally applied field but they will not be presented here. In disordered media,... 

Just as in the case of the conventional anisotropic approximation, the maximum number of displacement parameters is only realized for atoms located in the general site position (site symmetry 1). In special positions some or all of the displacement parameters will be constrained by symmetry. For example, 7333, Yi 13, Y223 and Y123 for an atom located in the mirror plane perpendicular to Z-axis are constrained to 0. Furthermore, if an atom is located in the center of inversion, all parameters of the odd order anharmonic tensors (3, 5, etc.) are reduced to 0. 

Since the electro-optic tensor has the same symmetry as the tensor of the inverse piezoelectric effect, the linear electro-optic (Pockels) effect is confined to the symmetry groups in which piezoelectricity occurs (see Table 8.3). The electro-optic coefficients of most dielectric materials are small (of the order of 10 m V ), with the notable exception of ferroelectrics such as potassium dihydrogen phosphate (KDP KH2PO4), lithium niobate (liNbOs), lithium tantalate (LiTaOs), barium sodium niobate (Ba2NaNb50i5), or strontium barium niobate (Sro.75Bao.25Nb206) (Zheludev, 1990). For example, the tensorial matrix of KDP with symmetry group 42m has the form... 

So far we have considered the order parameter for a cylindrically symmetric liquid crystal phase formed by cylindrically symmetric molecules. If either the phase or the molecules are not cylindrically symmetric it is necessary to specify the orientational ordering using tensors. A second-rank tensor describes, to lowest order, the orientation of phases with an inversion plane of symmetry (N, SmA,...). Second-rank tensors were defined in Section... 

It is often said that group 432 is too symmetric to allow piezoelectricity, in spite of the fact that it lacks a center of inversion. It is instructive to see how this comes about. In 1934 Neumann s principle was complemented by a powerful theorem proven by Hermann (1898-1961), an outstanding theoretical physicist with a passionate interest for symmetry, whose name is today mostly connected with the Hermann-Mau-guin crystallographic notation, internationally adopted since 1930. In the special issue on liquid crystals by ZeitschriftfUr Kristal-lographie in 1931 he also derived the 18 symmetrically different possible states for liquid crystals, which could exist between three-dimensional crystals and isotropic liquids [100]. His theorem from 1934 states [101] that if there is a rotation axis C (of order n), then every tensor of rank rcubic crystals, this means that second rank tensors like the thermal expansion coefficient a, the electrical conductivity Gjj, or the dielectric constant e,y, will be isotropic perpendicular to all four space diagonals that have threefold symme-... 

The molecular point group G char acterizes the symmetry exhibited by the location of nuclei. TURBOMOLE has been designed from the very beginning to take symmetry properly into account. This has two advantages it guarantees an accurate representation of the symmetry behavior of computed molecular properties (electronic states, vibrational modes, polarizability tensors, etc.), and it reduces computational work which is typically proportional to G , the inverse of the order of the point group G. The proper symmetry behavior is achieved in TURBOMOLE by a transformation of the original GTO basis into a symmetry adapted basis which is then used to represent all MOs. 

The application of perturbation theory to a nondegenerate state (for ko = 0) will involve no linear term in k (by symmetry) and the first significant term, which gives the curvature of the band, is given by second order and conventionally expressed as an effective (inverse) mass tensor. If the unperturbed state is denoted by 0> and all other k = 0 eigenstates by n>, the result for this is... 

The tensor scattering amplitude satisfies a useful symmetry property which is referred to as reciprocity. As a consequence, reciprocity relations for the amplitude, phase and extinction matrices can be derived. Reciprocity is a manifestation of the symmetry of the scattering process with respect to an inversion of time and holds for particles in arbitrary orientations [169]. In order to derive this property we use the following result if H and E2 H2 are the total fields generated by the incident fields E i, ffei and E 2, He2, respectively, we have... 

---