Hill-type potentials

From electronic structure theory it is known that the repulsion is due to overlap of the electronic wave functions, and furthermore that the electron density falls off approximately exponentially with the distance from the nucleus (the exact wave function for the hydrogen atom is an exponential function). There is therefore some justification for choosing the repulsive part as an exponential function. The general form of the Exponential - R Ey w function, also known as a ""Buckingham " or ""Hill" type potential is... 

The energy minimum for a given atomic pair is described by the potential depth, Eij, and position, R j. Other force fields model van der Waals interactions using a modified Hill equation, which replaces the twelfth power term in Eq. (28) with an exponential term [42,43]. Different approaches are also used to describe nonbonded interactions between those atoms that may form hydrogen bonds. Some force fields model these interactions using only Coulombic terms, whereas other force fields employ special functions, such as a modified 10-12 Lennard-Jones-type potential term [46], as shown in Eq. (29). 

There are two potential solutions to this problem. First, we can pre-specify a single test (group) on the primary endpoint at a single point in time, much in line with Hill s views. Or, we can attempt a statistical solution based on adjusting the type-1 error for the individual tests. 

Baker DG, West SA, Orth DN, Hill KK, Nicholson WE, Ekhator NN, Bruce AB, Wortman MD, Keck PE, Geracioti JD (1997) Cerebrospinal fluid and plasma beta endorphin in combat veterans with post traumatic stress disorder. Psychoneuroendocrinology 22 517-529 Bauer EP, Schafe GE, LeDoux JE (2002) NMDA receptors and L-type voltage-gated calcium channels contribute to long-term potentiation and different components of fear memory formation in the lateral amygdala. J Neurosci 22 5239-5249... 

In view of the Hessian character (10.20) of the thermodynamic metric matrix M(c+2), the eigenvalue problem for M(c+2) [(10.23)] can be usefully analogized with normal-mode analysis of molecular vibrations [E. B. Wilson, Jr, J. C. Decius, and P. C. Cross. Molecular Vibrations (McGraw-Hill, New York, 1955)]. The latter theory starts from a similar Hessian-type matrix, based on second derivatives of the mechanical potential energy Vpot (cf. Sidebar 2.8) rather than the thermodynamic internal energy U. 

A useful extension of this method consists in assuming that the membrane acquires a Boltzmann-like distribution of type (B.5) even for an arbitrary, nonharmonic potential /( ), which has a minimum at u = 0 [24,25]. When an external pressure Peat is applied to the undulating membrane, the relevant statistical potential is the enthalpy hill) = /(ii) + pextu and the membrane distribution is provided by ... 

It is helpful to contrast the view we adopt in this book with the perspective of Hill (1986). In that case, the normative example is some separable system such as the polyatomic ideal gas. Evaluation of a partition function for a small system is then the essential task of application of the model theory. Series expansions, such as a virial expansion, are exploited to evaluate corrections when necessary. Examples of that type fill out the concepts. In the present book, we establish and then exploit the potential distribution theorem. Evaluation of the same partition functions will still be required. But we won t stop with an assumption of separability. On the basis of the potential distribution theorem, we then formulate additional simplified low-dimensional partition function models to describe many-body effects. Quasi-chemical treatments are prototypes for those subsequent approximate models. Though the design of the subsequent calculation is often heuristic, the more basic development here focuses on theories for discovery of those model partition functions. These deeper theoretical tools are known in more esoteric settings, but haven t been used to fill out the picture we present here. 

A theoretical treatment of aqueous two-phase extraction at the isoelectric point is presented. We extend the constant pressure solution theory of Hill to the prediction of the chemical potential of a species in a system containing soivent, two polymers and protein. The theory leads to an osmotic virial-type expansion and gives a fundamentai interpretation of the osmotic viriai coefficients in terms of forces between species. The expansion is identical to the Edmunds-Ogston-type expression oniy when certain assumptions are made — one of which is that the solvent is non-interacting. The coefficients are calculated using simple excluded volume models for polymer-protein interactions and are then inserted into the expansion to predict isoelectric partition coefficients. The results are compared with trends observed experimentally for protein partition coefficients as functions of protein and polymer molecular weights. 

Wheeler and collaborators [3], in the context of nuclear physics, showed at that time that the limit in the variational procedure potential itself was not reached. Indeed, the Rayleigh-Ritz (RR) variational scheme teaches us how to obtain the best value for a parameter in a trial function, i.e., exponents of Slater (STO) or Gaussian (GTO) type orbital, Roothaan or linear combination of atomic orbitals (LCAO) expansion coefficients and Cl coefficients. Instead, the generator coordinate method (GCM) introduces the Hill-Wheeler (HW) equation, an integral transform algorithm capable, in principle, to find the best functional form for a given trial function. We present the GCM and the HW equation in Section 2. 

Another possibility, of course, is that the cooperative response, whether it is manifested as a structural phase transition or some higher-order collective interaction, e.g., a Frohlich type transition, or some combination of both, does not necessarily have to directly involve the membrane potential or membrane field, E, as defined by Hill. 

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