Expansions for

Bell R J 1970 Multipolar expansion for the non-additive third-order interaction energy of three atoms J. 

The same result can also be obtained directly from the virial equation of state given above and the low-density fonn of g(r). B2(T) is called the second virial coefficient and the expansion of P in powers of is known as the virial expansion, of which the leading non-ideal temi is deduced above. The higher-order temis in the virial expansion for P and in the density expansion of g(r) can be obtained using the methods of cluster expansion and cumulant expansion. 

In the thennodynamic limit (N x, F -> oo withA7F= p), this is just the virial expansion for the pressure, with 7,(7) identified as the second virial coefficient... 

Iiifomiation about the behaviour of the 3D Ising ferromagnet near the critical point was first obtained from high- and low-temperatnre expansions. The expansion parameter in the high-temperatnre series is tanli K, and the corresponding parameter in the low-temperatnre expansion is exp(-2A ). A 2D square lattice is self-dual in the sense that the bisectors of the line joining the lattice points also fomi a square lattice and the coefficients of the two expansions, for the 2D square lattice system, are identical to within a factor of two. The singularity occurs when... 

Stell G 1964 Cluster expansions for classical systems In equilibrium The Equilibrium Theory of Classical Fluids ed H L Frisch and J L Lebowitz (New York Benjamin)... 

Chandler D and Andersen H C 1972 Optimized cluster expansions for classical fluids II. Theory of molecular liquids J. Chem. Phys. 57 1930... 

In 1972 Wegner [25] derived a power-series expansion for the free energy of a spin system represented by a Flamiltonian roughly equivalent to the scaled equation (A2.5.28). and from this he obtained power-series expansions of various themiodynamic quantities around the critical point. For example the compressibility... 

B2.2.6.7 PARTIAL WAVE EXPANSION FOR TRANSPORT CROSS SECTIONS... 

The close-coupling equations are also applicable to electron-molecule collision but severe computational difficulties arise due to the large number of rotational and vibrational channels that must be retained in the expansion for the system wavefiinction. In the fixed nuclei approximation, the Bom-Oppenlieimer separation of electronic and nuclear motion pennits electronic motion and scattering amplitudes f, (R) to be detemiined at fixed intemuclear separations R. Then in the adiabatic nuclear approximation the scattering amplitude for ... 

Undoubtedly the most successful model of the nematic-smectic A phase transition is the Landau-de Gennes model [201. It is applied in the case of a second-order phase transition by combining a Landau expansion for the free energy in tenns of an order parameter for smectic layering with the elastic energy of the nematic phase [20]. It is first convenient to introduce an order parameter for the smectic stmcture, which allows both for the layer periodicity (at the first hannonic level, cf equation (C2.2A)) and the fluctuations of layer position ur [20] ... 

In the derivation we used the exact expansion for X t), but an approximate expression for the last two integrals, in which we approximate the potential derivative by a constant at Xq- The optimization of the action S with respect to all the Fourier coefficients, shows that the action is optimal when all the d are zero. These coefficients correspond to frequencies larger than if/At. Therefore, the optimal solution does not contain contributions from these modes. Elimination of the fast modes from a trajectory, which are thought to be less relevant to the long time scale behavior of a dynamical system, has been the goal of numerous previous studies. 

We now introduce the atomic orbital expansion for the orbitals i/), and substitute for the corresponding spin orbital Xi into the Hartree-Fock equation,/,(l)x,(l) = X (1) ... 

Table 2.3 Coefficients and e-xponents for best-fit Gaussian expansions for the Is Slater type orbital [Hehre et al. 1969].

The convergence sphere of the multipole expansion for a molecule such as butane may be penetrated by r molecule. 

Substituting this expansion for /k into the preeeeding equation gives... 

The molecular quantities can be best understood as a Taylor series expansion. For example, the energy of the molecule E would be the sum of the energy without an electric field present, Eq, and corrections for the dipole, polarizability, hyperpolarizability, and the like ... 

Stand-alone computer systems, usually based on a personal computer (PC) or programmable logic controller (PLC), provide a separate computer system for each pilot plant. This allows for economical expansion for new units, separates pilot plants completely for maintenance and troubleshooting, and often has the lowest initial cost. Standardization can be a problem and software control, data gathering, and storage packages can be limited in size, scope, and capabiUty these are usually acceptable trade-offs. 

MiscelDneous. Other important properties are resistance to thermal shock, attack by slag, and, in the case of refractories (qv), thermal expansion. For whiteware, translucency, acceptance of glazes, etc, may be extremely important. These properties depend on the clay mineral composition, the method of manufacture and impurity content. 

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