Response equations field wave functions

To determine the response of the wave function to a given order, we exploit the fact that the variational condition (41) shall hold to any field strength. As a consequence the total derivative of the variational condition with respect to any combination of field strengths will be zero as well. The resulting equations at zero perturbation strength... 

The response equations are usually solved in some iterative manner, in which the explicit construction of Q is avoided, being replaced by the repeated construction of matrix-vector products of the form Q where v is some trial vector . In general, the solution of one set of response equations is considerably cheaper than the optimization of the wave function itself. Moreover, since the properties considered in this chapter involves at most three independent perturbations (corresponding to the three Cartesian components of the external field), the solution of the full set of equations needed for the evaluation of the molecular dipole-polarizability and magnetizability tensors is about as expensive as the calculation of the wave function in the first place. 

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. 

Next, we consider the even weaker second-order perturbation of the three sublevels of T that is due to Recall that the analogous second-order contributions from are neglected in this approach, as they are generally believed to be small, and the first-order contributions have already been included. It is seen from inspection of Equation 3.3 that the operator can mix each of these three triplet wave functions with singlet, other triplet, and quintet wave functions. This interaction has no first-order effect on the energies -D, -Dy, and -D of the three substates, T,, T, and T, respectively, but in second order. Equation 3.14 with V = IP, they will be affected somewhat and become -D , -Dy, and -Z) . If we continue to define the zero-field splitting parameters by D = 3DJ/2 and E = (DJ - Dy )/2, they can be compared with the values observed. In Section 3.3, we noted the difficulties involved in attempts to evaluate these values accurately in this fashion, due to the very large number of states over which the summation in Equation 3.14 is necessary, and we commented on alternative methods of evaluation such as response theory. 

Abstract The modified equation-of-motion coupled cluster approach of Sekino and Bartlett is extended to computations of the mixed electric-dipole/magnetic-dipole polarizability tensor associated with optical rotation in chiral systems. The approach - referred to here as a linearized equation-of-motion coupled cluster (EOM-CCl) method - is a compromise between the standard EOM method and its linear response counterpart, which avoids the evaluation of computationally expensive terms that are quadratic in the field-perturbed wave functions, but still yields properties that are size-extensive/intensive. Benchmark computations on five representative chiral molecules, including (P)-hydrogen peroxide, (5)-methyloxirane, (5 )-2-chloropropioniuile, (/ )-epichlorohydrin, and (75,45)-norbornenone, demonstrate typically small deviations between the EOM-CCl results and those from coupled cluster linear response theory, and no variation in the signs of the predicted rotations. In addition, the EOM-CCl approach is found to reduce the overall computing time for multi-wavelength-specific rotation computations by up to 34%. 

The broadening Fj is proportional to the probability of the excited state k) decaying into any of the other states, and it is related to the lifetime of the excited state as r. = l/Fj . For Fjt = 0, the lifetime is infinite and O Eq. 5.14 is recovered from O Eq. 5.20. Unfortunately, it is not possible to account for the finite lifetime of each individual excited state in approximate theories based on the response equations (O Eq. 5.4). We would be forced to use a sum-over-states expression, which is computationally intractable. Moreover, the lifetimes caimot be adequately determined within a semiclassical radiation theory as employed here and a fully quantized description of the electromagnetic field is required. In addition, aU decay mechanisms would have to be taken into account, for example, radiative decay, thermal excitations, and collision-induced transitions. Damped response theory for approximate electronic wave functions is therefore based on two simplifying assumptions (1) all broadening parameters are assumed to be identical, Fi = F2 = = r, and (2) the value of F is treated as an empirical parameter. With a single empirical broadening parameter, the response equations take the same form as in O Eq. 5.4 with the substitution to to+iTjl, and the damped linear response function can be calculated from first-order wave function parameters, which are now inherently complex. For absorption spectra, this leads to a Lorentzian line-shape function which is identical for all transitions. 

The plane wave equation (101) is parametrized by the electric field strength E. Thus the profile c (p,E) depends on E. Most experimentally accessible is the velocity response function v(e) yielding the dependence of the wave velocity on the electric field as, for example, in Figure 8. 

Proof.- We study the linear response of the system to an electromagnetic wave at frequency Ct). Introduce a space-dependent local dielectric function s(r,(o) to represent the local dielectric response of both the interior of the solid as well as empty space. The theorem is general enough to accommodate a graded dielectric function as well. In the case of a shape of constant dielectric constant e(r,0>) = j(co) if F is inside the solid and s(r,co) = l if F is outside the solid. The potential field (r) obeys the Laplace equation... 

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