Derivatives response function approach

The MCSCF gradient expression was first given by Pulay (1977). The MCSCF Hessian and first anharmonicity expressions were derived by Pulay (1983) using a Fock-operator approach, and by Jprgensen and Simons (1983) and Simons and Jorgensen (1983) using a response function approach. 

A method closely related to the CCSD linear response function approach but derived differently is the equation-of-motion coupled cluster approach (EOM-CCSD) (Sekino and Bartlett, 1984 Geertsen et al, 1989 Stanton and Bartlett, 1993). The EOM-CCSD excitation energies are identical to the excitation energies obtained from the CCSD linear response function, but the transition moments and second-order properties, like frequency-dependent polarizabihties of spin-spin coupling constants, differ somewhat. 

In the next section we derive the Taylor expansion of the coupled cluster cubic response function in its frequency arguments and the equations for the required expansions of the cluster amplitude and Lagrangian multiplier responses. For the experimentally important isotropic averages 7, 7i and yx we give explicit expressions for the A and higher-order coefficients in terms of the coefficients of the Taylor series. In Sec. 4 we present an application of the developed approach to the second hyperpolarizability of the methane molecule. We test the convergence of the hyperpolarizabilities with respect to the order of the expansion and investigate the sensitivity of the coefficients to basis sets and correlation treatment. The results are compared with dispersion coefficients derived by least square fits to experimental hyperpolarizability data or to pointwise calculated hyperpolarizabilities of other ab inito studies. 

In the last three decades, density functional theory (DFT) has been extensively used to generate what may be considered as a general approach to the description of chemical reactivity [1-5]. The concepts that emerge from this theory are response functions expressed basically in terms of derivatives of the total energy and of the electronic density with respect to the number of electrons and to the external potential. As such, they correspond to conceptually simple, but at the same time, chemically meaningful quantities. 

Prof. Fleming, the expressions you are using for the nonlinear response function may be derived using the second-order cumulant expansion and do not require the use of the instantaneous normal-mode model. The relevant information (the spectral density) is related to the two-time correlation function of the electronic gap (for resonant spectroscopy) and of the electronic polarizability (for off-resonant spectroscopy). You may choose to interpret the Fourier components of the spectral density as instantaneous oscillators, but this is not necessary. The instantaneous normal mode provides a physical picture whose validity needs to be verified. Does it give new predictions beyond the second-order cumulant approach The main difficulty with this model is that the modes only exist for a time scale comparable to their frequencies. In glasses, they live much longer and the picture may be more justified than in liquids. 

A mixed quantum classical description of EET does not represent a unique approach. On the one hand side, as already indicated, one may solve the time-dependent Schrodinger equation responsible for the electronic states of the system and couple it to the classical nuclear dynamics. Alternatively, one may also start from the full quantum theory and derive rate equations where, in a second step, the transfer rates are transformed in a mixed description (this is the standard procedure when considering linear or nonlinear optical response functions). Such alternative ways have been already studied in discussing the linear absorbance of a CC in [9] and the computation of the Forster-rate in [10]. 

Potentiometric titration curves normally are represented by a plot of the indicator-electrode potential as a function of volume of titrant, as indicated in Fig. 4.2. However, there are some advantages if the data are plotted as the first derivative of the indicator potential with respect to volume of titrant (or even as the second derivative). Such titration curves also are indicated in Figure 4.2, and illustrate that a more definite endpoint indication is provided by both differential curves than by the integrated form of the titration curve. Furthermore, titration by repetitive constant-volume increments allows the endpoint to be determined without a plot of the titration curve the endpoint coincides with the condition when the differential potentiometric response per volume increment is a maximum. Likewise, the endpoint can be determined by using the second derivative the latter has distinct advantages in that there is some indication of the approach of the endpoint as the second derivative approaches a positive maximum just prior to the equivalence point before passing through zero. Such a second-derivative response is particularly attractive for automated titration systems that stop at the equivalence point. 

When the external electric field is time-dependent, there is no well-defined energy of the molecular system in accordance with Eq. (100), and the wave function response can thus not be retrieved from a variational condition on the energy as in the analytic derivative approach described above. Instead the response parameters have to be determined from the time-dependent Schrodinger equation, a procedure which was illustrated in Section 3 for the exact state case. In approximate state theories, however, our wave function space only partially spans the 7V-electron Hilbert space, and the response functions that correspond to an approximate state wave function will clearly be separate from those of the exact state wave function. This fact is disregarded in the sum-over-states approach, and, apart from the computational aspect of slowly converging SOS expressions, it is of little concern when highly accurate wave function models are used. But for less flexible wave function models, the correct response functions should be used in the calculation of nonlinear optical properties. 

In this chapter, we will not be concerned with the detailed expressions of the response functions that we find for the standard electronic structure methods in theoretical chemistry. However, we will briefly outline the basic elements in two alternative formulations of response tlieory, namely the polarization propagator and the quasi-energy derivative approaches. 

The approach outlined above combines the calculation of response functions (i.e. of frequency-dependent properties) with the theory of analytic derivatives developed for static higher-order properties. In the limit of a static perturbation all equations above reduce to the usual equations for (unrelaxed) coupled cluster energy derivatives. This is an invaluable advantage for the implementation of frequency-dependent properties in quantum chemistry programs. 

In principle, this formalism can be extended to quadratic and cubic response which allows the derivation of higher order gradient terms. In practice, however, the limited present knowledge on the required response functions restricts the usefulness of this approach. 

In the derivation of this term the solvent distribution is assumed to be uniform (but it is possible to extend the method to non uniform distributions, to describe e.g. dielectric saturation effects for cations [19], and electrostriction effects for supercritical liquids [20, 21]), and the response function is modeled in terms of a matrix partitioning approach to the calculation of intermolecular potentials [22]. With the aid of a formal re-elaboration of the whole model, and by introducing a few reasonable approximations, the related matrix is reduced to the following form ... 

If the system is cooled isobarically along a path above the critical pressure Pc (Fig. 5b, path a), the state functions continuously change from the values characteristic of a high-temperature phase (gas) to those characteristic of a low-temperature phase (liquid). The thermodynamic response functions, which are the derivatives of the state functions with respect to temperature (e.g., C ), have maxima at temperatures denoted Pmax (P) Remarkably these maxima are still prominent far above the critical pressure [31], and the values of the response functions at Pmax(P) (e-g-, C max) diverge as the critical point is approached. The lines of the maxima for different response functions asymptotically approach one another as the critical point is approached, since all response functions become expressible in terms of the correlation length. This asymptotic line is sometimes called the Widom line, and is often regarded as an extension of the coexistence line into the one-phase regime. ... 

Different response functions such as Cp and Up show maxima, and these maxima increase and seem to diverge as the critical pressure is approached, consistent with the Widom line picture discussed for other water models in the sections above. Moreover, the temperature derivative of the number of hydrogen bonds dAnB/dT displays a maximum in the same region where the other thermodynamic response functions have maxima, suggesting that the fluctuations in the number of hydrogen bonds is at a maximum at the Widom line temperature Tw. To further test whether this model system also displays a dynamic crossover as found in the other models of water, the total spin relaxation time of the system as a function of T for different... 

Needless to say, equation (21.8) is a bit cumbersome and its original derivation is rather lengthy. However, many subsequent treatments of the macroscopic theory are now available which provide both a more readily understandable approach and many useful approximate expressions. In fact, by using the method of Parsegian and Ninham to determine the dielectric response function from absorption data and reflectance measurements, it is now quite straightforward to calculate dispersion forces from Lifshitz s theory. 

As derivation of response functions is easier than describing, in an analytical manner, the intricacies of the processes responsible for the loss of depth resolution (cascade mixing and so on) this is the general approach of choice. Even so, response functions exhibit a complex dependence on the incoming ion, the substrate, the signal recorded and the instrumental conditions used. 

We are going to rewrite the three linear response functions now as ground-state expectation values similar to the derivations in Section 5.9. However, here we wiU not proceed via the sum-over-states expressions for the response function, but want to illustrate an alternative approach via the equation of motion of the polarization propagator for zero frequencies, Eq. (3.141). Recalhng that O p is the canonical conjugate... 

The formulation of approximate response theory based on an exponential parame-trization of the time-dependent wave function, Eq. (11.36), and the Ehrenfest theorem, Eq. (11.40), can also be used to derive SOPPA and higher-order Mpller-Plesset perturbation theory polarization propagator approximations (Olsen et al., 2005). Contrary to the approach employed in Chapter 10, which is based on the superoperator formalism from Section 3.12 and that could not yet be extended to higher than linear response functions, the Ehrenfest-theorem-based approach can be used to derive expressions also for quadratic and higher-order response functions. In the following, it will briefly be shown how the SOPPA linear response equations, Eq. (10.29), can be derived with this approach. 

The same problem with the pole structure appears also for coupled cluster response functions, if one defines them as derivatives of a time-average quasi-energy Lagrangian including orbital relaxation, ft is therefore preferable also in the analytical derivative approach like in Section 11.4 to derive coupled cluster response functions as derivatives of a time-dependent quasi-energy Lagrangian without orbital relaxation... 

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