Second Order Frequencies

In transition metal complexes, proton hfs are normally 20 MHz so that the corresponding second order contributions, which amount to 10 kHz, may usually be neglected. For nitrogen ligands, however, the second order corrections produce frequency shifts up to 200 kHz. Since hf interactions of central ions can amount to several hundred megacycles, the terms in AE become very important for a correct description of the ENDOR spectra. 

In some compounds the assumption 9 Ez %fsor is not fulfilled. In these 

If two magnetically nonequivalent nuclei I and K are present in the spin system, the transition frequency of nucleus I is shifted by an additional second order term48 55) 

The cross-term described by (3.18) produces shifts or splittings in the ENDOR spectrum. Shifts are observed if the hfs of nucleus K is resolved in the EPR spectrum, i.e, if EPR transitions with different mg may be used as observers. This is often the case in 

Since the shifts produced by the cross-terms between ligand and central ion nuclei are often significant, the strikingly broad ENDOR lines ( 1 MHz) observed for transition metal ions61,62) may be traced back, at least in part, to unresolved splittings due to second order interactions of numerous ligands with the central ion. ENDOR frequencies up to second order for an I = 1 nucleus in the presence of a second ligand nucleus with a spin K = 1 are tabulated in Appendix B, Eqs. (B4). 

In second order, however, eight ENDOR frequencies are obtained for each ms-state. The transition frequencies tabulated in Appendix B, Eqs. (B 5) are again described by ai, a2 and as defined in (3.12). If the hfs is resolved in the EPR spectrum, the number of induced transitions depends on the mp-value of the saturated line in the EPR quintet. For mp = 0 six transitions, for Imp] = 1 four transitions, and for mp = 2 one transition are observed in the ENDOR spectrum of each ms-state .  

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In the case of noninteger spin nuclei, such as 27A1 or 23Na, we observe only the central —transition, as the other transitions are spread over too wide a frequency range. The first- and second-order frequency shifts are... 

Nonlinear optics is the interaction of laser radiation with a substance to produce new radiation which is altered in phase, frequency, and amplitude from the incident radiation. There are several types of nonlinear effects but the most important are second-order frequency doubling and reverse saturable absorption. 

Equation (16) is very important because it imcovers two central ideas. First, the second-order frequency splitting depends inversely on the Larmor frequency, thus the importance of this term diminishes with increasing external magnetic field strength. Second, the k=0 term has no orientation dependence (Do 0 = 1) or, in other words, it is an isotropic term. This means that the isotropic shift observed in the NMR spectriun of a quadrupolar nucleus has, in addition to the usual isotropic chemical shift, a contribution from the quadrupole cou-phng, which is given by... 

To get some experience in using the PP to express second-order frequency-dependent and -independent properties and to indicate some problems that may appear when using the PP in finite basis set calculations, we now derive alternative but formally equivalent expressions for the frequency-dependent... 

TABLE V Performance for Selected Second-Order Frequency Conversion Interactions... 

The second-order shift increases with Vq and is inversely proportional to the magnetic field strength. Since the dispersion of the chemical shift, which is what we normally wish to measure, is proportional to it is advantageous to work at high fields, where the chemical shift effects make the maximum contribution to the spectrum. As the second-order frequency shift is always present for all transitions, the feasibility of obtaining useful spectra depends on the magnitude of Vq. 

Second-order effects include experiments designed to clock chemical reactions, pioneered by Zewail and coworkers [25]. The experiments are shown schematically in figure Al.6.10. An initial 100-150 fs pulse moves population from the bound ground state to the dissociative first excited state in ICN. A second pulse, time delayed from the first then moves population from the first excited state to the second excited state, which is also dissociative. By noting the frequency of light absorbed from tlie second pulse, Zewail can estimate the distance between the two excited-state surfaces and thus infer the motion of the initially prepared wavepacket on the first excited state (figure Al.6.10 ). 

Figure Bl.5.3 Magnitude of the second-order nonlinear susceptibility x versus frequency co, obtained from the anliannonic oscillator model, in the vicinity of the single- and two-photon resonances at frequencies cOq and coq 2> respectively.

Similarly, we can define the corresponding frequency-dependent second-order, and third-order,... 

The second-order nonlinear susceptibility tensor ( 3> 2, fOj) introduced earlier will, in general, consist of 27 distinct elements, each displaying its own dependence on the frequencies oip cci2 and = oi 012). There are, however, constraints associated with spatial and time-reversal symmetry that may reduce the complexity of for a given material [32, 33 and Ml- Flere we examine the role of spatial synnnetry. 

The second-order nonlinear optical processes of SHG and SFG are described correspondingly by second-order perturbation theory. In this case, two photons at the drivmg frequency or frequencies are destroyed and a photon at the SH or SF is created. This is accomplished tlnough a succession of tlnee real or virtual transitions, as shown in figure Bl.5.4. These transitions start from an occupied initial energy eigenstate g), pass tlnough intennediate states n ) and n) and return to the initial state g). A fiill calculation of the second-order response for the case of SFG yields [37]... 

Figure Bl.5.5 Schematic representation of the phenomenological model for second-order nonlinear optical effects at the interface between two centrosynnnetric media. Input waves at frequencies or and m2, witii corresponding wavevectors /Cj(co and k (o 2), are approaching the interface from medium 1. Nonlinear radiation at frequency co is emitted in directions described by the wavevectors /c Cco ) (reflected in medium 1) and /c2(k>3) (transmitted in medium 2). The linear dielectric constants of media 1, 2 and the interface are denoted by E2, and s, respectively. The figure shows the vz-plane (the plane of incidence) withz increasing from top to bottom and z = 0 defining the interface.

The interpretation of MAS experiments on nuclei with spin / > Fin non-cubic enviromnents is more complex than for / = Fiuiclei since the effect of the quadnipolar interaction is to spread the i <-> (i - 1) transition over a frequency range (2m. - 1)Vq. This usually means that for non-integer nuclei only the - transition is observed since, to first order in tire quadnipolar interaction, it is unaffected. Flowever, usually second-order effects are important and the angular dependence of the - ytransition has both P2(cos 0) andP Ccos 9) terms, only the first of which is cancelled by MAS. As a result, the line is narrowed by only a factor of 3.6, and it is necessary to spin faster than the residual linewidth Avq where... 

In this section we consider the classical equations of motion of particles in cases where the highest-frequency oscillations are nearly harmonic The positions y t) = j/i (t) evolve according to the second-order system of differential equations... 

A different long-time-step method was previously proposed by Garci a-Archilla, Sanz-Serna, and Skeel [8]. Their mollified impulse method, which is based on the concept of operator splitting and also reduces to the Verlet scheme for A = 0 and admits second-order error estimates independently of the frequencies of A, reads as follows when applied to (1) ... 

Orbital-based methods can be used to compute transition structures. When a negative frequency is computed, it indicates that the geometry of the molecule corresponds to a maximum of potential energy with respect to the positions of the nuclei. The transition state of a reaction is characterized by having one negative frequency. Structures with two negative frequencies are called second-order saddle points. These structures have little relevance to chemistry since it is extremely unlikely that the molecule will be found with that structure. 

Materials for Frequency Doubling. Second-order NLO materials can be used to generate new frequencies through second harmonic generation (SHG), sum and difference frequency mixing, and optical parametric oscillation (OPO). The first, SHG, is given in equation 3. 

Commercial frequency doublers have rehed on inorganic materials. The commercial future of doublers depends on not only the improvement in second-order materials but also the development of diode lasers capable of operating in the visible frequency domain. 

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