Sum and Projection Methods

as described in more detail in Appendix C, this wave function, which configurationally corresponds to an a electron in the orbital and a p electron in the ( 2 orbital, is neither a singlet nor a triplet, but a 50 50 mixture of the two, and this point is emphasized by die left superscript on 4/ in Eq. (14.14). While the wave function does not represent a pure spin state, we may take advantage of the prevailing situation by noting that we may write 

To provide a specific example of die method, near UV experiments have led to assignments of the vertical and adiabatic excitation energies for die I B PAg transition in A-diazene (HN=NH), where the Bg state is open-shell. Table 14.4 compares sum-method predictions at the UHF and BLYP levels of theory to diese experimental values, and also to published results at the MRCI level of theory. For diis system, die HF results are systematically too high, and the DFT too low (cf. the sum method prediction for A2 phenylnitrene in Table 14.1), but are competitive with the much more expensive MRCI results. Note that all three levels do quite well at predicting the difference in verdcal and adiabatic excitation energies. 

To reduce or eliminate spin contamination problems in unrestricted wave functions, spin projection methods have been developed diat annihilate the contributions of certain spin states higher than the desired one. As derived in Appendix C, the PUHF energy for a wave function that has had contamination from the next higher spin state annihilated is computed as 

In application to typical UHF wave functions, the second term on the r.h.s. of Eq. (14.18) provides a small correction that improves the estimate of the state energy for slightly contaminated cases. In principle, however, there is no reason the formalism cannot be applied to a 50 50 wave function. The results of such an application to the already discussed diazene excitation are listed in Table 14.4. 

At the UHF level, the excitation energies from spin annihilation represent a fairly severe underestimation of the excited-state energies, and disagree significantly with the sum method results from that same level of theory. UHF spin annihilation is generally not a worthwhile method to apply unless spin contamination effects are fairly small, which obviously is not the case here. 

---

The largest drawback of the spin annihilation procedure is similar to that of the sum method. That is, while the spin-annihilated wave function which results from the application of Aj+i to the 50 50 antecedent is in principle spin pure, it is expressed in the MOs that were optimized for the 50 50 case. These MOs minimize the energy of the contaminated state, but not that of the spin pure state, and errors can be significant. Nevertheless, the speed of the sum and projection methods, and their utihty in many if not all instances, makes them useful for rough applications prior to resort to more expensive and sophisticated models. 

Standard methods are used to propagate each Om in time. For the z and Z coordinates we make use of the fast fourier transform [99], and for the p coordinate we use the discrete Bessel transform [100]. The molecular component of asymptotic region at each time step, and projected onto the ro-vibrational eigenstates of the product molecule, for a wide range of incident energies included in the incident wave packet [82]. The results for all ra-components are summed to produce the total ER reaction cross section, a, and the internal state distributions. 

Even so, the EUAR method is best suited to those situations where (as in this case) the undiscounted net cash flows are constant. However, when they are not constant, each NCF must be discounted back to year zero, summed, and annualized by multiplying it by the CRF. Finally, this annualized NCF must be added to the capital recovery cost. By the time the analyst has done all of this, he or she could just as well have calculated the net present worths or internal rates of return of the competing projects. 

The BS plus spin projection method discussed here is closely connected to the simple open-shell singlet method for optical excitations based on the Slater sum rule and ASCF (self-consistent-field total energy difference method). The mixed spin excited state is like the BS state, also of mixed spin. The Slater sum rule method" " is also quite effective for multiplet problems for excited states of transition metal complexes as shown in the work of Dahl and Baerends. ... 

This is seen to be similar to the photoabsorption formula except that (i) it involves a half Fourier transform, (ii) (t) is projected onto 1- n = representing the final state of interest, and (iii) the integral is squared. Since we have just discussed tractable ways of finding (j)(t), using equation (14) to calculate Raman scattering amplitudes is just as easy as the case of photoabsorption. The time-dependent technique will be far easier than the sum-over-states method of equation (7), and again it focusses our attention on the physically relevant dynamics. For low-lying excited states is still localized to the Franck-Condon vicinity. 

Expenditure on corrosion prevention is an investment and appropriate accountancy techniques should be used to assess the true cost of any scheme. The main methods used to appraise investment projects are payback, annual rate of return and discounted cash flow (DCF). The last mentioned is the most appropriate technique since it is based on the principle that money has a time value. This means that a given sum of money available now is worth more than an equivalent sum at some future data, the difference in value depending on the rate of interest earned (discount rate) and the time interval. A full description of DCF is beyond the scope of this section, but this method of accounting can make a periodic maintenance scheme more attractive than if the time value of money were not considered. The concept is illustrated in general terms by considering a sum of money P invested at an... 

Lately it has become fashionable to compute an arbitrary projection of the scattering data, a so-called sum of the WAXS or sum of the SAXS . Variation of the resulting number is then discussed in terms of structure variation. Such numbers are computed by simply summing the intensity readings from every pixel of the detector. Obviously this number cannot be related to structure and application of this method reveals lack of basic analytical skills. 

A complete description of the method requires a procedure for selecting the initial conditions. At t 0, initial values for the complex basis set coefficients and the parameters that define the nuclear basis set (position, momentum, and nuclear phase) must be provided. Typically at the beginning of the simulation only one electronic state is populated, and the wavefunction on this state is modeled as a sum over discrete trajectories. The size of initial basis set (N/it = 0)) is clearly important, and this point will be discussed later. Once the initial basis set size is chosen, the parameters of each nuclear basis function must be chosen. In most of our calculations, these parameters were drawn randomly from the appropriate Wigner distribution [65], but the earliest work used a quasi-classical procedure [39,66,67], At this point, the complex amplitudes are determined by projection of the AIMS wavefunction on the target initial state (T 1)... 

The DFT values for the At state derive from the sum method or projection techniques presented in Section 14.4, and discussion of those values is deferred to that point. As for the 2 A state, although no experimental measurement is available, comparison to other... 

As was already mentioned, in theoretical atomic spectroscopy, while considering complex electronic configurations, one has to cope with many sums over quantum numbers of the angular momentum type and their projections (3nj- and ym-coefficients). There are collections of algebraic formulas for particular cases of such sums [9, 11, 88]. However, the most general way to solve problems of this kind is the exploitation of one or another versions of graphical methods [9,11]. They are widely utilized not only in atomic spectroscopy, but also in many other domains of physics (nuclei, elementary particles, etc.) [13],... 

The essence of the DCF process is that it does take into account the passage of time during the life of the project. It works by modifying all sums of money, costs or revenues, to give their nominal values at a particular point in time, so that a balance can be struck between the sum of all costs and the sum of all revenues at that point. The method works equally well at any point in time over the project s life, and the first decision to be taken is to select the most meaningful point to use. 

The work breakdown structure is therefore a method of describing the work to be done in the project in terms of deliverables and the tasks to be accomplished. It involves defining major deliverables and accomplishments during the project and fisting a hierarchy of sub-deliverables and accomplishments. The deliverables and accomplishments are a sum of their sub components and the WBS is a sum of all the elements. A more detailed description of this can be found in any of the general books on Project Management [D-13]. 

Crowell s Expression for Lattice Sum. The Original Application. Crowell (2) has shown how to evaluate the lattice sums, s n, for a layer-lattice structure by an analytical method. The lattice is approximated by a set of layer planes, each with a uniform distribution of matter, and separated by the interplanar distance, d. The lattice sum is approximated by integrating over the planes and forming the sum of the resulting terms. Let the adsorbate, at a distance z from the surface, be separated by a distance Sy from any point in the mtn plane below the surface. The vertical distance of the adsorbate from this plane is (z + md) let y be the projection of s on the mth plane. Then the summation of sj1 for the mth plane is... 

One example of a cost that can vary during the life of a project is depreciation cost. If straight-line depreciation is used, this cost will remain constant however, it may be advantageous to employ a declining-balance or sum-of-the-years-digits method to determine depreciation costs, which will immediately result in variations in costs and profits from one year to another. Other predictable factors, such as increasing maintenance costs or changing sales volume, may also make it necessary to estimate year-by-year profits with... 

---