Goddard expansion

Bird, R. B., Hassager, O., and Abdel-Khalik, S. I., Co-rotational rheological models and the Goddard expansion, AIChEJ., 20, 1041-1066 (1974). 

Acrivos and Goddard (1965) developed an asymptotic expansion for the large-Pe problem and obtained... 

Robert Jastrow, the head of NASA s Goddard Institute for Space Studies, called this the most powerful evidence for the existence of God ever to come out of science. Other amazing parameters abound. If all of the stars in the universe were heavier than three solar masses, they would live for only about 500 million years, and life would not have time to evolve beyond primitive bacteria. Stephen Hawking has estimated that if the rate of the universe s expansion one second after the Big Bang had been smaller by even one part in a hundred thousand million million, the universe would have recollapsed.The universe must live for billions of years to permit time for intelligent life to evolve. On the other hand, the universe might have expanded so rapidly that protons and electrons never united to make hydrogen atoms. 

Another Class I charge model that is also sensitive to geometry is the QEq charge equilibration model of Rappe and Goddard (1991). From representing the energy m of an isolated atom k as a Taylor expansion in its charge truncated at second order, one can derive... 

Goddard (27) expressed the notion of the simple fluid constitutive equation in a co-rotational integral series. The integral series expansion had been used in the co-deformational frame by Green and Rivlin (28) and Coleman and Noll (29). The co-rotational expansion takes the form ... 

Acrivos, A. and Goddard, J.D. (1965), Asymptotic expansions for laminar-forced convection heat and mass transfer, J. Fluid Mech., 23, 273-291. 

Intermediate between the spin-coupled expansion in N orthonormal sets and the MO expansion in a single set is an expansion in just two distinct sets. The first term of (17) now becomes the electron-pair wavefunction of Hurley, Lennard-Jones and Pople " , in which pairs of orbitals (i/ i,i/ 2X (i/ j, 1/ 4),..., overlap with each other, but different pairs are (strongly) orthogonal. Much work has been carried out on this model by Goddard and coworkers under the nomenclature generalized valence bond (GVB) wavefunction . ... 

Higher-order terms in this expansion are given explicitly by Acrivos and Goddard (Al). 

Fig 6.25 The hierarchy of cells used in the cell multipole method For an atom in the black cell, the interactions with atoms in the 26 nearby cells (N) are calculated explicitly Interactions with the atoms in cells labelled A and B are calculated using a Taylor series multipole expansion. (Figure adapted from Ding H-Q, N Karasaxva and W A Goddard III, 1992b The Reduced Cell Multipole Method for Coulomb Interactions in Periodic Systems with Million-Atom Unit Cells Chemical Physics Letters 196-6-10)... 

Limiting solutions based on pertubatlon methods have also been discussed in the literature. Goddard et al. ( ), Kreuzer and Hoofd (53). and Smith et al. (5j() all used matched asymptotic expansions to develop criteria for reactive boundary layer zones within facilitated transport membranes. These results can also be used to calculate solute fluxes. For systems of Interest, the reaction boundary layer will be negligible and an analysis of this detail is unnecessary. 

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