Partitioned energy terms

The partitioned energy terms can thus be derived separately for each contributing monomer (M). The sum of these terms per each subunit corresponds to a (partitioned) total energy quantity for the given subsystem (EM). 

The results presented in this work show that in the linear structured water dimer the partitioned energy terms calculated for the proton donor and acceptor molecules are significantly different (except the kinetic energy). The electron structure of the proton donor molecule was found more compact than that of the acceptor subsystem, when compared their (partitioned) total energy EM values. This result is in an excellent agreement with our pre-vious results obtained on the separated molecular orbital energies [17]. 

For small p the contribution of paths with large x (n 0) to the partition function Z is suppressed because they are associated with large kinetic-energy terms proportional to v . That is why the partition function actually becomes the integral over the zeroth Fourier component Xq. It is therefore plausible to conjecture that the quantum corrections to the classical TST formula (3.49a) may be incorporated by replacing Z by... 

While thermodynamics does not describe the nature of this internal energy, it is helpful to consider the insights gained from kinetic molecular theory. According to this theory, the internal energy can be partitioned into kinetic and potential energy terms associated with various motions and positions of the nuclei of the atoms or molecules that make up the gas, and with energies associated with their electrons. 

MMl represents the mass and moment-of-inertia term that arises from the translational and rotational partition functions EXG, which may be approximated to unity at low temperatures, arises from excitation of vibrations, and finally ZPE is the vibrational zero-point-energy term. The relation between these terms and the isotopic enthalpy and entropy differences may be written... 

Fig. 4.1 The zero point energy or low temperature approximation As temperature drops and u increases above u 4 the harmonic oscillator partition function Q (Harm. Osc.) is better and better approximated by the zero point energy term, exp(—u/2). For a typical CH stretching frequency, v = 3000 cm-1, u 4 at 1050 K and it is reasonable to use the ZPE approximation for that frequency at temperatures below 1000 k...

The total stabilization energy of a cluster rarely exceeds 25 kcal mol , i.e., a small fraction of a strong covalent bond energy (ca. 100 kcal mol ). Its partitioning into electrostatic, induction, and dispersion terms differs from cluster to cluster. In some cases, one particular energy term is dominant. More typically, many attractive terms contribute to the overall stabilization of non-covalent clusters, as it often happens to hydrogen-bonded complexes. Nevertheless, the electrostatic interaction plays a dominant role, and in the case of polar subsystems. 

In equation 11.148, AE is regarded as an Arrhenius-slope energy term that has a value of 1 kJ/mol, and (1000/r) ln(gVg°) is the reduced partition function ratio for the reference phase (quartz). 

In Fenske and Hall s parameter-free SCF calculations (80-84), the He1t 1-electron operator is substituted by a model 1-electron operator that has a kinetic energy and potential energy term for each atomic center in the complex. This approach assumes that the electron density may be assigned to appropriate centers. The partitioning of electron density is done through Mulliken population analyses (163) until self-consistency is obtained. The Hamiltonian elements are evaluated numerically, and the energies of the MO s depend only on the choice of basis functions and the intemuclear distance. 

Before we go on and apply Eq. 3-35 to describe the partitioning of a compound i between two phases, a few comments are necessary on the various partial molar free energy terms included in Eq. 3-35. First, we rewrite Eq. 3-35 by splitting the term that expresses the difference in partial molar free energy of a compound i between its actual situation in a given solution and its situation in the reference state ... 

Assuming that the adsorbed molecules have lost their three translational degrees of freedom, we calculate S from Eq. (32) of Chapter 5, taking the energy term as zero. The partition function for the molecules on the surface is given by Eqs. (71) and (72) ... 

An a-helix bundle may become a second-order cooperative folding unit if the interaction energy terms are such that the intermediate terms in the partition function become negligibly small [Eq. (14)] and the entire partition function reduces to a two-state partition function (i.e., a partition function of the form 1 + e G/RT). If such is the case, the a-helix bundle will be either completely folded or unfolded. Higher order cooperative folding units can be constructed from lower order ones following the same rules. The most immediate application of this approach is to proteins exhibiting pure a-helical structural motifs. 

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