Gaussian theories

An initial equilibrium structure is obtained at the Hartree-Fock (HF) level with the 6-31G(d) basis [47]. Spin-restricted (RHF) theory is used for singlet states and spin-unrestricted Hartree-Fock theory (UHF) for others. The HF/6-31G(d) equilibrium structure is used to calculate harmonic frequencies, which are then scaled by a factor of 0.8929 to take account of known deficiencies at this level [48], These frequencies are used to evaluate the zero-point energy Ezpe and thermal effects. 

The equilibrium geometry is refined at the MP2(fu)/6-31 G(d) level, using all electrons for the calculation of correlation energies. This is the final equilibrium geometry in the theory and is used for all single-point calculations at higher levels of theory in step 3. Except where otherwise noted by the symbol (fu), these subsequent calculations include only valence electrons in the treatment of electron correlation. 

A series of single-point energy calculations is carried out at higher levels of theory. The first higher-level calculation is the complete fourth-order Mpller-Plesset perturbation theory [13] with the 6-31G(d) basis set, i.e. MP4/6-31G(d). For convenience of notation, we represent this as MP4/d. This energy is then modified by a series of corrections from additional calculations  

EG3Large = EMP2(fu)/G3Large EMP2/2df,p EmP2/p1us + EMP2/d  

The largest basis set, denoted as G3Large [21] includes some core polarization functions as well as multiple sets of valence polarization functions. It should be noted that MP2 calculation with the largest basis set in Eq. (3.4) is carried out at the MP2(fu) level. 

The Gl method is seldom used since G2 yields an improved accuracy of results. G2 has proven to be a very accurate way to model small organic molecules, but gives poor accuracy when applied to chlorofiuorocarbons. At 

Jensen, Introduction to Computational Chemistry John Wiley Sons, New York (1999). 

Atkins, R. S. Friedman, Molecular Quantum Mechanics Oxford, Oxford (1997). 

Molecular Modelling Principles and Applications Longman, Essex (1996). 

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Gaussian theory (Gl, G2, G3) a method for extrapolating from ah initio results to an estimation of the exact energy Gaussian-type orbital (GTO) mathematical function for describing the wave function of an electron in an atom... 

Figure 18.17 shows that the characteristics of the stress-strain curve depend mainly on the value of n the smaller the n value, the more rapid the upturn. Anyway, this non-Gaussian treatment indicates that if the rubber has the idealized molecular network strucmre in the system, the stress-strain relation will show the inverse S shape. However, the real mbber vulcanizate (SBR) that does not crystallize under extension at room temperature and other mbbers (NR, IR, and BR at high temperature) do not show the stress upturn at all, and as a result, their tensile strength and strain at break are all 2-3 MPa and 400%-500%. It means that the stress-strain relation of the real (noncrystallizing) rubber vulcanizate obeys the Gaussian rather than the non-Gaussian theory. 

The result, Eq. (41), is remarkable also because it takes the form anticipated intuitively by the multi-Gaussian theory, Eq. (19). The observation that ps = Y P w/ (I + X n> I °V>w) makes that clear. Here... 

The molecular theories of networks to be presented in the following paragraphs are based on the Gaussian picture of the individual network chains. With reference to the form of the distribution function, these theories are referred to as "Gaussian theories". 

The networks studied were prepared from reactions carried out at different initial dilutions. Aliquots of reaction mixtures were transferred to moulds, which were maintained at the reaction temperature under anhydrous conditions, and were allowed to proceed to complete reaction(32). Sol fractions were removed and shear moduli were determined in the dry and equilibrium-swollen states at known temperatures using uniaxial compression or a torsion pendulum at 1Hz. The procedures used have been described in detail elsewhere(26,32). The shear moduli(G) obtained were interpreted according to Gaussian theory(33 34 35) to give values of Mc, the effective molar mass between junction points, consistent with the affine behaviour expected at the small strains used(34,35). 

The role of chain entangling in cross-linked elastomers is an old issue which has not yet been settled. The success of Flory s new rubber elasticity theory 0-5) in describing some of the departures from the simple Gaussian theory has acted as a strong catalyst for new work in this area. 

The non-Gaussian theories of rubber elasticity have the disadvantage of containing parameters which generally can be determined only by experiment. Recently,... 

Curro and Mark 38) have proposed a new non-Gaussian theory of rubber elasticity based on rotational isomeric state simulations of network chain configurations. Specifically, Monte Carlo calculations were used to determine the distribution functions for end-to-end dimensions of the network chains. The utilization of these distribution functions instead of the Gaussian function yields a large decreases in the entropy of the network chains. 

The Mukherji-Prins approach represents a possible refinement in the Gaussian theory, but does not lead to a C2 term. 

Such structuring is necessarily an intermolecular effect. The simplest type of an intermolecular effect, which should be treated first, is due to the crosslinks between the chains themselves. Dobson and Gordon (50) have remarked that most crosslinks are actually short chains of one or several links, which upon straining the network, become oriented but cannot be stretched. As a result an additional entropy force should arise, which has not yet been accounted for in the Gaussian theory. This force can be calculated on the basis of the Kuhn and Grun (114) chain vector orientation argument, which yields in extension... 

In many swollen networks the Gaussian theory seems to apply, i.e. there is no C2-term. This is corroborated by a number of additional, more complex experimental techniques described in Chapter III. From these experiments it follows that at least there is no contradiction with the consequences of the Gaussian theory. 

In the derivation of the Gaussian theory several alternative paths have been used, of which the one of Flory and Wall is favored by the present authors on theoretical grounds. Recent experimental evidence obtained on carefully prepared networks also points in this direction. 

As the scope of the present chapter is to stress limits and background for the Gaussian theory, a discussion of the said viscosity-molecular weight relationships is omitted. Some general remarks will be made in the discussion of the next section. 

C. Menduina, C. McBride, and C. Vega (2001) Correctly averaged Non-Gaussian theory of rubber-like elasticity - application to the description of the behavior of poly(dimethylsiloxane) bimodal networks. Phys. Chem,. Chem. Phys. 3, p. 1289... 

Following the suggested Gaussian theory, all systems were reoptimized at the MP2 level without any frozen core orbitals. 

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