Order properties

If ai 4 1 this completes the proof. If equality holds then it follows that i = 3 and 2 = 4 or that/ (S, ) =/ (Sf) and/ (S ) =/i,(S ).The functions / and / cross at the origin and at most one other point. Since 0 S2 S, the two functions are exactly the same. This contradiction completes the proof of the lemma.  

In the preceding sections, the possible rest points for the gradostat equations were determined and their stability analyzed. The problem that remains is to determine the global behavior of trajectories. In this regard, the theory of dynamical systems plays an important role. First of all, some information can be obtained from the general theorem on inequalities discussed in Appendix B. We illustrate this with an application to the gradostat equations. 

for ail t 0, Theorem B.l (the basic comparison theorem) states that 

However, if (5.2a) is satisfied then iim, co a ,(0 = 0 i — 1,2), so the omega limit point p must be of the form (0,0, , ) that is, the u population becomes extinct. Similarly, if (5.2b) holds then p is of the form (, , 0,0) and the v population is eliminated. Note that this is independent of initial conditions and hence is a global result. The component marked will be zero or positive depending on the other inequality. 

The system of interest, (2.4), is neither cooperative nor competitive with respect to the obvious order. The U1-U2 populations cooperate, as do the V1-V2 populations, but u and v compete. In ecological terms, the problem involves families of competing mutualists [S2]. Let x = ( i, 2, Ui, V2) and y = (Ui,U2,Vi,V2). Define the partial order relation x xy by Uj Uj, Vi Vi. We shall also usex /f to mean Ui u, 0,. (This corresponds to the notation x K y used by some authors.) The system (2.4) is monotone (see Appendix C) with respect to this order (see also the discussion 

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Figures 4.14a and b are the analogs of Figs. 4.3a and b they schematicaUy describe second- and first-order transitions, respectively. It is the discontinuity in these second-order properties that characterizes a second-order transition.

In ternary mixtures of oil, water, and surfactant the ordering properties of the system follow from the vectorial character of the interactions of the surfactant molecules with both the oil and the water molecules. The typical size of the ordered domains, much larger than the molecular size, justifies the application of the mesoscopic Landau-Ginzburg approach to the ordering. In the simplest approach of Gompper and Schick [3,12], which we call here the basic Landau-Ginzburg model, the orientational degrees of free-... 

The nth-order property is the nth-order derivative of the energy, d EjdX" (the factor 1 /n may or may not be included in the property). Note that the perturbation is usually a vector, and the first derivative is therefore also a vector, the second derivative a matrix, the third derivative a (third-order) tensor etc. 

Derivative techniques consider the energy in the presence of the perturbation, perform an analytical differentiation of the energy n times to derive a formula for the nth-order property, and let the perturbation strength go to zero. 

In such cases the expression from fii st-order perturbation theory (10.18) yields a result identical to the first derivative of the energy with respect to A. For wave functions which are not completely optimized with respect to all parameters (Cl, MP or CC), the Hellmann-Feynman theorem does not hold, and a first-order property calculated as an expectation value will not be identical to that obtained as an energy derivative. Since the Hellmann-Feynman theorem holds for an exact wave function, the difference between the two values becomes smaller as the quality of an approximate wave function increases however, for practical applications the difference is not negligible. It has been argued that the derivative technique resembles the physical experiment more, and consequently formula (10.21) should be preferred over (10.18). 

The Lagrange technique may be generalized to other types of non-variational wave functions (MP and CC), and to higher-order derivatives. It is found that the 2n - - 1 rule is recovered, i.e. if the wave function response is known to order n, the (2n + l)th-order property may be calculated for any type of wave function. 

The second-order property, the dipole polarizability, as given by the derivative formula eq. (10.31), is... 

Here (r - Rc) (r - Rq) is the dot product times a unit matrix (i.e. (r — Rg) (r — Rg)I) and (r - RG)(r — Rg) is a 3x3 matrix containing the products of the x,y,z components, analogous to the quadrupole moment, eq. (10.4). Note that both the L and P operators are gauge dependent. When field-independent basis functions are used the first-order property, the HF magnetic dipole moment, is given as the expectation value over the unperturbed wave funetion (for a singlet state) eqs. (10.18)/(10.23). 

Frequency-dependent higher-order properties can now by obtained as derivatives of the time-average of the quasienergy WIt- with respect... 

This raises a dilemma in treating second- and higher-order properties in coupled-cluster theory. In the EOM-CC approach, which is basically a Cl calculation for a non-Hermitian Hamiltonian H= that incorporates... 

We study three different approximations for removing unlinked diagrams in EOM-CC and show that these models provide second-order properties and transition probabilities that are close to those provided by CCLR in isolated molecular systems, but in a more convenient computational structure. 

Table 3 Second Order Properties Calculated by EOM-CCSD Methods...

An alternative approach is to apply stronger fields and only use energies calculated for positive field strengths in generating the polynomial fit. In this case the energy is a function of both odd and even powers in the polynomial fit. We will show that the dipole moments derived from our non-BO calculations with the procedure that uses only positive fields and polynomial fits with both even and odd powers match very well the experimental results. Thus in the present work we will show results obtained using interpolations with even- and odd-power polynomials. Methods other than the finite field method exist where the noise level in the numerical derivatives is smaller (such as the Romberg method), but such methods still do not allow calculation of odd-ordered properties in the non-BO model. 

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 theory of statistical mechanics provides the formalism to obtain observables as ensemble averages from the microscopic configurations generated by such a simulation. From both the MC and MD trajectories, ensemble averages can be formed as simple averages of the properties over the set of configurations. From the time-ordered properties of the MD trajectory, additional dynamic information can be calculated via the time correlation function formalism. An autocorrelation function Caa( = (a(r) a(t + r)) is the ensemble average of the product of some function a at time r and at a later time t + r. 

Analogous considerations apply, for example, to the calculation of second-order properties, for which a very similar computational problem must be addressed. For typical applications this step constitutes 95-100% of the total computational effort, and a successful parallelization will therefore reflect directly on overall performance. 

As stated above, for a molecnle to demonstrate bnlk second order properties it is necessary for the dipole of all the molecnles to lie in the same direction. This situation is very difficnlt to attain in small molecnles where the tendency is for the molecules to crystallise with their dipoles opposing each other. This fact has led to a switch of interest to polymers that incorporate the small NLO molecnle. 

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