Unphysical results

To account for the variation of the dynamics with pressure, the free volume is allowed to compress with P, but differently than the total compressibility of the material [22]. One consequent problem is that fitting data can lead to the unphysical result that the free volume is less compressible than the occupied volume [42]. The CG model has been modified with an additional parameter to describe t(P) [34,35] however, the resulting expression does not accurately fit data obtained at high pressure [41,43,44]. Beyond describing experimental results, the CG fit parameters yield free volumes that are inconsistent with the unoccupied volume deduced from cell models [41]. More generally, a free-volume approach to dynamics is at odds with the experimental result that relaxation in polymers is to a significant degree a thermally activated process [14,15,45]. 

Thus, the initial slope corresponds to that of an A arm performing translational diffusion without knowing about the slower B arm. Similarly the prediction for a small fraction of labelled B arms leads to the unphysical result IXQ)=k TQ /Ni, i again a single arm performing centre of mass diffusion. 

Contrarily to conventional MP2 theory, the original formulation of MP2-R12 theory (3,4) did not provide the same results when canonical or localized molecular orbitals were used. Indeed, for calculations on extended molecular systems, unphysical results were obtained when the canonical Hartree-Fock orbitals were rather delocalized (5). In order to circumvent this problem, an orbital-invariant MP2-R12 formulation was introduced in 1991, which is the preferred method since then (6),... 

The problem with this analysis is that the selection of one or another basis set dramatically affects the calculated charges and occasionally leads to unphysical results [44]. Overlap populations (8.2) are largely responsible for this situation. 

In the spirit of the Anderson model, we separate electrons into two subsystems delocalized electrons for which the LDA is assumed to give reasonable results and localized electrons for which it is well known that the LDA can lead to unphysical results. To treat these states in a better way, and to avoid double counting, we exclude the interaction between localized electrons (/ or d) already taken into account in an average way in the LDA-on-site energy... 

Therefore the fact that 9 is arbitrary in U(l) theory compels that theory to assert that photon mass is zero. This is an unphysical result based on the Lorentz group. When we come to consider the Poincare group, as in section XIII, we find that the Wigner little group for a particle with identically zero mass is E(2), and this is unphysical. Since 9 in the U(l) gauge transform is entirely arbitrary, it is also unphysical. On the U(l) level, the Euler-Lagrange equation (825) seems to contain four unknowns, the four components of , and the field tensor H v seems to contain six unknowns. This situation is simply the result of the term 7/MV in the initial Lagrangian (824) from which Eq. (826) is obtained. However, the fundamental field tensor is defined by the 4-curl ... 

However, the assumption that the fluctuations of neighboring bilayers are totally uncorrelated leads to the unphysical result that they can permeate each other (P(z) > 0 for 2 < 0). One can approximately account for the correlation of the motions of the neighboring bilayers as follows. 

Equation (1-180) can be obtained from a variational calculation of the energy of the dimer with a wave function that is product of determinantal wave functions of the monomers constructed from the optimal (selfconsistent) orbitals. Unfortunately, the optimization of the orbitals of one monomer in the electrostatic field of the other does not prevent the unphysical transfer of electrons from one system to the other, since the antisymmetry of the wave function of the dimer is not preserved, and consequently, the Pauli principle is not satisfied. This may lead to some unphysical results in the short range260. It should also be stressed that the interaction part of Eq. (1-180) cannot be obtained, as proposed in Ref. (261), by taking the expectation value of the interaction operator V with the product of determinantal wave functions of the monomers constructed from the optimal (selfconsistent) orbitals. This would result in an overcounting of the induction terms, already in the second order. 

Comparing eqn (8) with eqn (2) one can observe that the former contains an additional term c,F which, as it will be shown later in the paper, is not in most cases negligible. It is now understandable why eqn (2) provides frequently unphysical results.36 example, there are numerous... 

The problem is, however, that we need to fix the exact conditions of validity of this approximation, this was attempted already in Ref. [Fano 1954], In particular, it has turned out that introduction of the memory effect is a very sensitive issue [Barnett 2001], Highly reasonable but unprecise approximations may lead to non-physical time evolution. An additional problem is that the procedure does not necessarily lead to Master Equations of the Lindblad type, see above. If this is not its form, we may find well known complications, which have to be avoided if we want to escape unphysical results. 

One should not put too much reliance on numbers calculated by population analysis. Mulliken s assignment of half the overlap population to each basis function is arbitrary and sometimes leads to unphysical results (see MulUken and Ermler, Diatomic Molecules, pages 36-38, 88-89). Moreover, a small change in basis set can produce a large change in the calculated net charges. For example, net atomic charges on each H atom in CH4, NH3, H2O, and HF calculated from HF/STO-3G and HF/3-21G wave functions are (Hehre et al.. Section 6.6.2)... 

In the case of the unperturbed hydrogen atom we found that a direct solution at P = 1 gave an unphysical result, but once we introduced appropriate dimensional scalings we found that the energy at P = 1 became equivalent to the solution at P = 3. Let us attempt to... 

The first term of Equation 2.53 is l/A Bianc-While sometimes fine for small deviations from Taianc. these expressions are not generally reliable and may produce erroneous and even unphysical results such as < 0. So unfortunately iterative solution of Equations 2.49 through 2.52 is normally needed. Below we review the applications of these formahsms and relevant measurements. 

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