Second-order property instances

We have established that, for a fully variational wave function, we may calculate the first-order properties from the zero-order response of the wave function (i.e., from the unperturbed wave function) and the second-order properties from the first-order response of the wave function. In general, the 2n -f 1 rule is obeyed For fully variational wave functions, the derivatives (i.e., responses) of the wave function to order n determine the derivatives of the energy to order 2n+ 1. This means, for instance, that we may calculate the energy to third order with a knowledge of the wave function to first order, but that the calculation of the energy to fourth order requires a knowledge of the wave-function response to second order. 

Now, it is one issue whether functional properties are identical with their first-order realizing properties it is another whether every instance of a functional property is identical with some instance of one of its first-order realizing properties. The claim that every instance of a functional property is identical with some instance of one of its first-order realizing properties is compatible with the conception of events as property exemplifications. On the conception of properties as universals, instances of properties are just things that have them. Thus, a red truck is an instance of the universal (the property) redness. Since events are things that can have second-order properties, they can be instances of such properties and so can be typed as such. A property exemplification theory of events can, however, distinguish properties the exemplification of which are events from properties that are possessed by events. And it is compatible with such a view that the events that have functional properties (such as the property of being an occupant of a role) are exemplifications of physical properties. Thus, it is compatible with the property exemplification conception of events that even if no functional property is a physical property, every instance of a functional property is a physical event. 

Although incompatible with Kim s denial that there are second-order properties, this combined view is compatible with his position that every mental event is either a physical event or an epiphenomenon. For it entails his position. This view, moreover, is compatible with Kim s position that events have causal effects only in virtue of being exemplifications of physical properties. For the property in virtue of which an event has causal effects is, arguably, its constitutive property, and on this view, the constitutive properties of events are physical properties. If the properties in virtue of which events have casual effects are constitutive properties, then, on this view, although instances of mental properties have causal effects, they do not have them in virtue of being instances of mental event types rather, they have causal effects in virtue of being instances of physical event types. 

This is in fact the view Jaegwon Kim has advanced in several places about instances of second-order properties and instances of their first-order realizers.As Kim has noted, such an identification requires a revision of his property-exemplification account of events assuming that mental properties are second-order properties, it requires the exclusion of mental properties as constitutive properties of events. This instance-identity thesis is supposed to support reductionism about the mental. But there is a tension between this thesis and Kim s formulation in several places of his causal inheritance principle, which says that the causal powers of an instance of a higher-order property are identical with or are a subset of [emphasis mine] the causal powers of the instance of its realizer. Clearly, if the causal powers of the realized property instance were... 

What is meant by the claim that functional properties are second-order properties is importantly different depending on whether we take it as definitive of second-order properties that their instances are identical with those of the first-order properties that are their realizers. If we do not take it as definitive, second-order properties simply are properties things have in virtue of having properties not identical with them it is then uncontroversial that functional properties are second order and also uncontroversial that determinables are second order. If we do take it as definitive, it is not uncontroversial that functional properties are second order - and I think it is false. 

On matrix form the non-unitary transformations (27) and (30) of the previous section are easily extended to the complete Hamiltonian and have therefore allowed relativistic and non-relativistic spin-free calculations of spectroscopic constants and first-order properties at the four-component level (see, for instance. Refs. [45 7]). In this section, we consider the elimination of spin-orbit interaction in four-component calculations of second-order electric and magnetic properties. Formulas are restricted to the Hartree-Fock [48] or Kohn-Sham [49] level of theory, but are straightforwardly generalized. 

The properties of these equations for the g tensor can be seen more clearly if they are simplified by assuming l = 1, X < A E, and neglecting second-order terms. This is quite reasonable since for many instances 2 < 0.1 OA. The equations then take the form... 

In the present paper, we have discussed the ab-initio evaluation of the static polarizabilities and magnetizabilities of molecular systems, with emphasis on the principles underlying such calculations. With the recent widespread availability of powerful computers, these second-order molecular properties may nowadays be calculated a priori for large molecular systems, allowing us to explore, for instance, the relationship between the properties and molecular structure. Such calculations complement the experimental work in the area and may help in reassessing and improving on the empirical schemes... 

Electron correlation plays a role in electrical response properties and where nondynamical correlation is important for the potential surface, it is likely to be important for electrical properties. It is also the case that correlation tends to be more important for higher-order derivatives. However, a deficient basis can exaggerate the correlation effect. For small, fight molecules that are covalently bonded and near their equilibrium structure, correlation tends to have an effect of 1 5% on the first derivative properties (electrical moments) [92] and around 5 15% on the second derivative properties (polarizabilities) [93 99]. A still greater correlation effect is possible, if not typical, for third derivative properties (hyperpolarizabilities). Ionic bonding can exhibit a sizable correlation effect on hyperpolarizabilities. For instance, the dipole hyperpolarizability p of LiH at equilibrium is about half its size with the neglect of correlation effects [100]. For the many cases in which dynamical correlation is not significant, the nondynamical correlation effect on properties is fairly well determined with MP2. For example, in five small covalent molecules chosen as a test set, the mean deviation of a elements obtained with MP2 from those obtained with a coupled cluster level of treatment was 2% [101]. 

As has been demonstrated by numerous studies, the accuracy of properties calculated using the GGA DFT methods is, in most cases, comparable to or better than those from ab initio MP2 (Mpller-Plesset second-order perturbation) or CISD (configuration interaction with single and double excitations) methods. In fact, the accuracy of the DFT results in some instances matches those obtained from the much more costly (but, in principle, more exact) CCSD(T) (coupled cluster singles and doubles with a perturbative inclusion of connected triple excitations) method (29) and the ab initio G1 procedure (30). 

In general, the physical properties of an electron system are defined by referring to a specific perturbation problem and can be classified according to the order of the perturbation effect. For instance, the electric dipole moment is associated with the first-order response to an applied electric field (i.e. the perturbation), the electric polarizability with the second-order response, hyperpolarizabilities with higher-order terms. In addition to dipole moments, there is a number of properties which can be calculated as a first-order perturbation energy and identified with the expectation value... 

Polyparaphenylenevinylene based macro-initiator 2 was used for NMRP of various monomers (styrene, methyl aciylate, butyl acrylate). From this compound various well defined rod-coil blocks copolymers with polystyrene and polyaciylate based coil blocks have been obtained. Furthermore, in each case, it is possible to random copolymeiize a second monomer for instance chloromethylstiyrene. The first monomer determines mechanical properties and phase transitions of the coil block, for example, bytulacrylate based coils have low Tg and can provide easy processabihty towards thin films. The second monomer (between 5% and 10% in molar ratio) provides the introduction of functional moieties which are necessary for a further modification in order to tune the electronic properties of the copolymer. NMRP from DEH-PPV macroinitiator 2 is schematically presented in Figure 2. 

Direct reaction of castor oil with 4,4 -methylenebisphenyl diisocyanate gave standard polyurethane material [52]. In efforts to modify the physical properties of the polymer, castor oil was mixed with trimethylol propane (a common industrial building block) and the stoichiometric ratios of the diisocyanates involved were varied [53], The reaction in this instance displayed second order kinetics. Polycarbonate-urethanes were also produced from... 

The fact that the EUE theory [1, 5, 15] can be chiefly founded on the one-electron RDM is remarkable per se. However, electron correlation effects are at least two-electron in nature, and it is no wonder that the second-order RDM was applied for quantifying EUE and related electron-correlation properties. Seemingly, the first investigation in this direction was presented in book [19] where in Sect 6.5 a special operator named correlation operator was introduced. Actually, in [19] the two-electron counterpart of D was examined. In this section we will denote RDMs of order k by Z °. The superscript so shows that the fid RDM (in spin-orbital basis) is considered. For instance, and Dj are the conventional one-electron and two-electron RDMs. 

The above simple analysis now elucidates how small contributions from 4,- and la are essentially suppressed in the [2] and Afjj p[2] indices. As a rule, these small contributions appear mainly from dynamical correlations. Eor instance, MP2 (the Moller-Plesset second-order perturbation theory) normally produce the contributions of this kind. Evidently, they have no direct relation to diradicahty and polyradicality, and the [2] and Ajj p[2] indices should be rather small without a significant contribution from non-dynamical correlation. This is a good property of the generalized indices such as (6.94) and (C8), and apparently, this is the basic reason why [2] is systematically employed in papers [9, 11, 122, 124] for analyzing the unpaired electrons in large PAHs. At the same time, the dynamical correlation cannot fully ignored, and the problem of an optimal quantification... 

A precise theoretical and experimental determination of polarizability would provide an important probe of the electronic structure of clusters, as a is very sensitive to the presence of low-energy optical excitations. Accurate experimental data for a wide range of size-selected clusters are available only for sodium, potassium [104] and aluminum [105, 106]. Theoretical predictions based on DFT and realistic models do not cover even this limited sample of experimental data. The reason for this scarcity is that the evaluation of polarizability by the sum rule (46) requires the preliminary computation of S(co), which, with the exception of Ref. [101], is available only for idealized models. Two additional routes exist to the evaluation of a, in close analogy with the computation of vibrational properties static second-order perturbation theory and finite differences [107]. Again, the first approach has been used exclusively for the spherical jellium model. In this case, the equations to be solved are very similar to those introduced in Ref. [108] for the computation of atomic polarizabilities. Applications of this formalism to simple metal clusters are reported, for instance, in Ref. [109]. 

It should be pointed out that there are alternative methods to calculate properties other than direct differentiation. For instance, the second-order perturbation theory correction to the energy for a perturbing electric field yields an expression for the dipole polarizability tensor, a. 

A simple relationship between microscopic hyperpolarizability (fi, y, etc.) of the active molecules and NLO macroscopic properties of the materials (x " ) can be established under the assumption of weak intermolecular interaction. This approximation is known as the oriented gas model [18]. For instance, the second order bulk susceptibility can be regarded as the statistical average of individual molecular hyperpolarizability... 

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