Second static limit

The coupled cluster expression for frequency-dependent second hyperpolarizability, Eq. (30), can now be expanded in a Taylor series in its frequency arguments around its static limit as ... 

The various response tensors are identified as terms in these series and are calculated using numerical derivatives of the energy. This method is easily implemented at any level of theory. Analytic derivative methods have been implemented using self-consistent-field (SCF) methods for a, ft and y, using multiconfiguration SCF (MCSCF) methods for ft and using second-order perturbation theory (MP2) for y". The response properties can also be determined in terms of sum-over-states formulation, which is derived from a perturbation theory treatment of the field operator — [iE, which in the static limit is equivalent to the results obtained by SCF finite field or analytic derivative methods. 

For a molecule without symmetry reduction the sum-over-states expression for the hyperpolarizability yean often be modeled within the three level approximation (the ground state (index 0) and two excited states (indices 1,2 - not necessarily indicating the hierarchy of the energy eigenvalues)). If we assume po2 Poll f°r substituted molecules with a centrosymmetric backbone as described above, the number of terms reduces leaving only three terms to consider the negative term N > the dipolar term D , and the two-photon term TP, which includes the second excited state 2Ag. In the static limit one obtains for a one-dimensionally conjugated molecule... 

Four frequently used conventions exist for the definition of non-linear optical polarizabilities, leading to confusion in the realm of NLO. This has been largely clarified by Willets et al. (1992) and in their nomenclature we have used the Taylor series expansion (T convention), originally introduced by Buckingham (1967), where the factorials n are explicitly written in the expansion. Here the polarizabilities of one order all extrapolate to the same value for the static limit w— 0. /3 values in the second convention, the perturbation series (B), have to be multiplied by a factor of 2 to be converted into T values. This is the convention used most in computations following the sum-over-states method (see p. 136). The third convention (B ) is used by some authors in EFISHG experiments and is converted into the T convention by multiplication by a factor of 6. The fourth phenomenological convention (X) is converted to the T convention by multiplication by a factor of 4. 

The first integral on the right-hand side of this equation must vanish. (This is the static limit [110] of Eq. (225).) Since Sn is arbitrary, the second integral leads to the identity... 

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... 

For N, there is a sizeable triples contribution to the lowest dipole allowed excitation energy of about 0.07 eV or 0.7%. As a consequence of tliis (unrelaxed) CCSD underestimates the absolute value of the isotropic hyperpolarizability and its dispersion. Tire electronic contribution to the static limit y - has been calculated at the CCSD/t-aug-cc-pVTZ level [32] to be 903.0 a.u. However, as indicated above, the triple- level is often too low for the calculation of second hyperpolarizabilities and the yj obtained at the CCSD/t-aug-cc-pVTZ level turned out to be about 40 a.u. above the CCSD basis-set limit result. Tlie latter has been calculated to be 863.3 3.3 a.u. [35]. Before comparing this result to tlie value extrapolated from experimental results, it has to be corrected for the ZPV contribution, which has been obtained in [32] to 12.0 a.u., thereby yielding 875.3 3.3 a.u. as tlie best estimate for the electronic contribution to y at the CCSD level. Shelton [6] obtained an experimental value of yj , 917 9 a.u., from the extrapolation of tlie results in [2] corrected for the pure rotational and vibrational contributions. Tire discrepancy between this experimental value and the CCSD best estimate is as large as 42 a.u. and makes very clear the importance of tire triples contribution. 

Notice that in the static limit only the second term of equation (14) contrihutes to (there is no memory contrihution) and equation... 

According to eq. (21), transverse field measurements allow, in principle, separation of the static field width (oc o) from the fluctuation rate (1/r) in an intermediate case. In practice, this is rather difficult and the combination of zero and longitudinal field measurements are more powerful in this respect, as will be shown further below. In the static limit (r —> oo) we can extract the second moment of the field distribution (see eq. 20). In the fast fluctuation limit only the product a r appears (eq. 24) and independent information on one of the two quantities is needed. 

A recent review on the nonlinear optical response and ultrafast dynamics in has emphasized on the difficulty of quantum chemistry methods to predict accurate second hyperpolarizabilities of Cgo over the whole frequency range, going from the static limit and small-frequency region to the resonant regions. 

The transition is fully classical and it proceeds over the barrier which is lower than the static one, Vo = ntoColQl- Below but above the second cross-over temperature T 2 = hcoi/2k, the tunneling transition along Q is modulated by the classical low-frequency q vibration. The apparent activation energy is smaller than V. The rate constant levels off to its low-temperature limit k only at 7 < Tc2, when tunneling starts out from the ground state of the initial parabolic term. The effective barrier in this case is neither V nor Vo,... 

The poles con espond to excitation energies, and the residues (numerator at the poles) to transition moments between the reference and excited states. In the limit where cj —> 0 (i.e. where the perturbation is time independent), the propagator is identical to the second-order perturbation formula for a constant electric field (eq. (10.57)), i.e. the ((r r))Q propagator determines the static polarizability. 

A qualitative and somewhat quantitative indication of the case with which the two (or more) fans will parallel is represented by their limit curve of Figure 12-145. This curve is constructed by starting at pressure P corresponding to the system intersection A, and plotting the increments of Volume X, y, etc., to define the limit of available volume that the duct can accept and which the second fan must pick up as it comes on the line. For good parallel operation, the limit curve must intersect the combined static pressure curve, SP, at only one point. 

For an oriented polymer, the magnitude of the observed second moment static magnetic field H0, which can be conveniently defined by the polar and azimuthal angles A, transverse isotropy, to which the following discussion is limited, the observed second moment will depend only on the angle A, there being no preferred orientation in the plane normal to the 3 direction. The treatment follows that originally presented by McBrierty and Ward 9>. 

It should be noted that there is a considerable difference between rotational structure narrowing caused by pressure and that caused by motional averaging of an adiabatically broadened spectrum [158, 159]. In the limiting case of fast motion, both of them are described by perturbation theory, thus, both widths in Eq. (3.16) and Eq (3.17) are expressed as a product of the frequency dispersion and the correlation time. However, the dispersion of the rotational structure (3.7) defined by intramolecular interaction is independent of the medium density, while the dispersion of the vibrational frequency shift (5 12) in (3.21) is linear in gas density. In principle, correlation times of the frequency modulation are also different. In the first case, it is the free rotation time te that is reduced as the medium density increases, and in the second case, it is the time of collision tc p/ v) that remains unchanged. As the density increases, the rotational contribution to the width decreases due to the reduction of t , while the vibrational contribution increases due to the dispersion growth. In nitrogen, they are of comparable magnitude after the initial (static) spectrum has become ten times narrower. At 77 K the rotational relaxation contribution is no less than 20% of the observed Q-branch width. If the rest of the contribution is entirely determined by... 

Debye s theory, considered in Chapter 2, applies only to dense media, whereas spectroscopic investigations of orientational relaxation are possible for both gas and liquid. These data provide a clear presentation of the transformation of spectra during condensation of the medium (see Fig. 0.1 and Fig. 0.2). In order to describe this phenomenon, at least qualitatively, one should employ impact theory. The first reason for this is that it is able to describe correctly the shape of static spectra, corresponding to free rotation, and their impact broadening at low pressures. The second (and main) reason is that impact theory can reproduce spectral collapse and subsequent pressure narrowing while proceeding to the Debye limit. 

---