Tempering parameter

In the scheme which we label (a), the same basis set is employed for both the ground and excited state. Therefore, the same integrals over basis functions are used for both states. The values of the even-tempered parameters a and /3 for M = 6 are those which were optimized for the ground state of the atom as reported by Schmidt and Ruedenberg [9]. These values are given in Table 1 of reference [9j. Larger basis sets were generated by means of the recursion formulae [10] ... 

In scheme (6) the basis set is optimized by invoking the variation principle for each state considered. For the ground state the optimized values of the even-tempered parameters ao and / o given by Schmidt and Ruedenberg [9] are used. We add the subscript 0 to distinguish ground state values. For the excited state optimal ai and / i values for a sequence of Mi values are determined. 

TABLE 2. Self-Consistent Field energies (in hartree) of excited states of the Li atom as a function of the size, M, of the even-tempered basis set used to parametrize the orbitals. In the colmnn headed (a) the same even-tempered basis set optimized for the ground state, is used for all states (b) the even-tempered basis set is optimized for each state (c) the even-tempered parameters a and f3 are optimized for each basis set for the smallest basis set (M = 6) and larger basis sets are generated using the recursions. ls 3s... 

TABLE 4. Optimized even-tempered parameters a and / for excited states of He as a function of size of basis set. 

In Tables 8 and 9, the even-tempered parameters for the two excited states of the beryllium atom considered in this work are presented, with Table 8 giving the parameters obtained according to scheme (b) in which the parameters were fully optimized and Table 9 giving the parameters obtained by means of scheme (c) m which the parameters a and (3 were only optimized for the smallest basis set, i.e. M = 6. 

Figure 3.8 Tempering parameter vs tensile strength for quenched and tempered 21/4Cr-1 Mo steel.5...

Parameter estimation is a procedure for taking the unit measurements and reducing them to a set of parameters for a physical (or, in some cases, relational) mathematical model of the unit. Statistical interpretation tempered with engineering judgment is required to arrive at realistic parameter estimates. Parameter estimation can be an integral part of fault detection and model discrimination. 

It is the rate of temperature inerease (i.e., power output) between the set pressure and the maximum allowable pressure, whieh determines the vent size and not the peak rate. Boiling is attained before potential gaseous deeomposition (i.e., the heat of reaetion is removed by the latent heat of vaporization). The reaetion is tempered, and the total pressure in the reaetor is equal to the vapor pressure. The prineipal parameter determining the vent size is the rate of the temperature rise at the relief set pressure. 

Even-tempered basis sets have the same ratio between exponents over the whole range. From chemical considerations it is usually preferable to cover the valence region better than the core region. This may be achieved by well-tempered basis sets. The idea is similar to the even-tempered basis sets, tire exponents are generated by a suitable formula containing only a few parameters to be optimized. The exponents in a well-tempered basis of size M are generated as ... 

For steels that are most frequently used in the annealed or normalised condition the most important structural parameter that can be influenced by heat treatment is the grain size, although the extensive use of welding as a means of fabricating mild steels means that martensitic and tempered martensitic... 

Intrinsic viscosity measurements revealed a conformational transition upon heating from 26 to 40 °C, while the UV absorbance of the solution was insensitive to the change. The entropy parameters for PA were also discussed in light of the Flory-Krigbaum correlation between the second virial coefficient and theta temper-... 

These data show that both models identify important variables that affect 5 Obody w.ier and 8 Ophospha in mammals. Both serve to identify the dikdik as an outlier which may be explained by their sedentary daytime pattern. On the other hand, the body-size model (Bryant and Froelich 1995), which may reliably predict animal 5 0 in temperate, well-watered regions, does not predict 8 Opho,phaw in these desert-adapted species. The second model (Kohn 1996), by emphasizing animal physiology independent of body size, serves to identify species with different sensitivities to climatic parameters. This, in conjunction with considerations of behavior, indicate that certain species are probably not useful for monitoring paleotemperature because their 5 Obodyw er is not tied, in a consistent way, to The oryx, for example, can... 

The rate of convergence of expansions in the basis (1.2) has received little attention except for purely numerical studies [3,7,8,9,16] which indicated that the convergence is at least (unlike for bais set of type) not frustratingly slow. Rather detailed studies were performed for the even-tempered basis set, i.e. for exponents constructed from two parameters and /di (for each /)... 

We were able to show analytically - in an unexpectedly tricky way (the mathematical ingredients of which are in the appendix) - that the error of an expansion of the function 1/r in terms of an even-tempered Gaussian basis of dimension n goes as exp(-ci/n) provided that the two parameters of the even-tempered basis are optimized. 

The worst operating condition in a common design practice consists of overly conservative assumptions on the hot-channel input. These assumptions must be realistically evaluated in a subchannel analysis by the help of in-core instrumentation measurements. In the early subchannel analysis codes, the core inlet flow conditions and the axial power distribution were preselected off-line, and the most conservative values were used as inputs to the code calculations. In more recent, improved codes, the operating margin is calculated on-line, and the hot-channel power distributions are calculated by using ex-core neutron detector signals for core control. Thus the state parameters (e.g., core power, core inlet temper-... 

One of the most popular such techniques is parallel tempering in the canonical ensemble, for which the index parameter is temperature [13-15], While parallel tempering (or replica exchange) strategies had been independently proposed on multiple occasions in various scientific areas, perhaps the earliest seed of the idea can be found in early work by Swendsen and Wang [37]. 

Another practical limitation in complex applications lies in the fact that, if temperature is used as a control parameter, one needs to worry about the integrity of a system that is heated too much (e.g., water-membrane systems or a protein heated above its denaturation temperature). When issues such as those mentioned above are addressed, parallel tempering can be turned into a powerful and effective means of enhanced conformational sampling for free energies over a range of temperatures for various systems. 

F S. Root mean square order parameter plotted vs. temper-... 

The common multicanonical techniques such as replica-exchange or simulated tempering have been described and reviewed extensively in different contexts [124], They interface naturally with MC simulations as they are cast as (biased or unbiased) random walks in terms of a control parameter — usually temperature. They work by exchanging information between the different conditions, thereby allowing increased barrier crossing and quicker convergence of sampling at all conditions of interest. 

Conversely, if it is assumed that the exponents do form a geometric progression and the parameters a and [3 are optimized for atoms then there is found to be little lost in accuracy. Basis sets developed in this way are termed even-tempered basis sets (for a discussion see [2]) and open up the possibility of constmcting the large and flexible basis sets that are inevitably required for calculations of high precision. 

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