The static part

It was to surmount this problem of the inability to calculate the effects of the dipolar Hamiltonian that Van Vleck [7] proposed the method of moments. We can define the nth moment (M ) of a resonance absorptionias 

Effectively, we divide the lineshape into vertical strips and take the integral, weighting each strip by the nth power of its deviation from the centre of the line. For a symmetric line, the odd moments are therefore all 0. The signal, following a pulse can then be defined in terms of the moments [8] 

This series converges slowly, which implies that the moments of the system must be known to high order if the FID and lineshape is to be completely determined. Van Vleck gives the method for calculating the moments from the structure of a rigid solid, using traces of commutators that reduce to a series of sums over internuclear vectors. The second moment involves summing the square of an interaction term for every pair of spins. The fourth moment involves the square of that sum, the sum of the fourth power of the interaction 

In liquids, where molecular motion is rapid and isotropic, the effects of the static part of the dipolar Hamiltonian are averaged to zero, giving rise to an exponential FID and hence a Lorentzian lineshape. In this case, as introduced in chapter 1, we can define a transverse relaxation time, T2, by 

However, in a rigid solid, the definition of a transverse relaxation time is more difficult since the shape of the decay is more complex. T2 is therefore often taken as a characteristic time of the FID, usually the time taken to decay to 1/e of its initial value, and as such can be used as an effective lifetime for the transverse magnetisation. If we consider the effects of increasing the temperature of a solid sample, we find that as various motions come into play, the second moment decreases and the FID lengthens. Eventually there comes a point where there is no residual static interaction on the timescale of the 

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RHF to UHF, or to a TCSCF, is almost pure static correlation. Increasing the number of configurations in an MCSCF will recover more and more of the dynamical correlation, until at the full Cl limit, the correlation treatment is exact. As mentioned above, MCSCF methods are mainly used for generating a qualitatively correct wave function, i.e. recovering the static part of the correlation. 

The static part deals with the information we have about the state of an object at ary given moment. At a given level of time granularity, we describe static attributes, relationships, and constraints between objects. Chapter 2, Static Models Object Attributes and Invariants, is about modeling static aspects using abstract attributes. 

Models can be divided into static, dynamic, and interactive parts dealing with, respectively, what is known about an object at any one moment, how this information changes dynamically with events, and how objects interact with one another. This chapter discusses the static part of a model, in which you characterize the state of an object by describing the information known about it at any point in time. It uses the type model diagram to capture the static model and snapshot diagrams to show instantaneous configurations of object state. 

The strict answer is that the static model, without actions, does not tell us enough. The only real test is whether the system we re modeling behaves (responds to actions) as a client would expect from reading the whole model, actions and all the static part merely sets a vocabulary for the rest. This strict view allows some implementations to conform that might not otherwise. For example, suppose we never specified any actions that used the balance. By the retrieval function rule, we would still have to implement that attribute even though it would make no perceptible difference to clients whether or not it was implemented. 

In other cases, there is no strong need to construct a business model before you get on with a component spec, particularly if the software focuses on only one aspect of a very general business. Sometimes it is useful to create the static part of the business model, without any collaborations. For example, before designing a word processor, you first define what a document is. This is the business world in this case—the subject matter with which the software will deal but modeled separately from any concerns of what the software will do with it. This approach is particularly useful when there are standards of interchange between components Web pages, RTF files, floppy disks, and TV pictures all have standard models of the objects without designing the equipment that handles them. 

The ZFS is assumed to be cylindrically symmetric (only the /q component is different from zero) and of constant magnitude. The static part of the Hzfs is obtained by averaging the Wigner rotation matrix Dq q[ pm(0] over the anisotropic distribution function, Pip pj. The principal axis of the static ZFS is, in addition, assumed to coincide with the dipole-dipole IS) axis. Eq. (48) becomes equivalent to Eq. (42), with the /q component scaled by Z)q q[ 2pm(0] The transient part of the Hzfs can be expressed in several ways, the simplest being 92) ... 

The static part is simply the energy of polarization of a by the static field of P and... 

Recall that the /4 are the static part of the AOM parameters el, and that to obtain the experimentally accessible parameters ei, we must subtract off the parameter ei. 

Debye, on the assumption of quadrupole attraction ( 43.VII C), calculated the static part, %, of the surface tension, i.e. the part independent of molecular motion, and found the molecular diameter d=3-2%o/4,. 

The spectroscopy described above is effectively the static part of the interaction, but there are dynamical processes that determine how rapidly the spins lose coherence and... 

So far no approximation for a form of the optical potential U has been made. At this point we assume static-exchange approximation defined by the eq. (4). First two terms of eq. (4) describe electrostatic interaction consisting of nuclear attraction and static repulsion with the bound electrons. Corresponding matrix elements for the static part Us come out as a Fourier transform of the charge density ... 

The database is subdivided into a static and a dynamic part. The static part comprises the elements that change very little with time (e.g., the definition of analytical methods), whereas the dynamic part relates to clients, samples, planning, and results. 

A modern LIMS includes a module designed for the administration of recipes for the various products. This includes specifying exact quantities or relationships of individual components of a product. With a recipe administration module, each recipe is subdivided into two different manufacturing phases a static part (i.e., gross-mixing ratio) and a dynamic part. The static part describes which quantities of the individual components are used for the manufacturing of the product. The quantities are specified as absolute units or as ratios. The dynamic part describes dependencies of the quantities of the components on different parameters, such as time, temperature, pH-value, or pressure. With the recipe module, various calculations and evaluations can be carried out. 

We will now take a closer look at the static part of the Green s function optical Hamiltonian... 

The matrix is what we call the static part of the Feshbach Hamiltonian. It is defined by the matrix elements... 

Thus the kinetic energy and the influence of external forces is already described exactly by the static part of the optical Hamiltonian and independent of approximations for the target wavefunction. Note that the possibility for two particles labelled by r and s to exchange can be seen explicitely in the kinetic energy and external field parts of Eqs. (119) and (120) by the terms that antisymmetrise the matrix with respect to interchange of these labels. An expression for the interaction part contained in the static self energy is... 

Like in the case of single-particle scattering, there are other possibilities of defining optical potentials. Without going into detail we want to mention that an analogous expression to the Feshbach effective Hamiltonian (109) can be derived for two-particle scattering. The static part has now the matrix elements [H, o ,a, ] o ). For positrons as projectile... 

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