Proton configurations

Fig. 3. A schematic view of intruder states near Z=82. The normal proton configurations for Au and Pt are nOh) and 7c(4h) respectively. The proton intruder configurations are Tt(lp-4h) and it(2p-6h) respectively, where p=particle and h=hole.

The region around 100Sn also offers exciting possibilities. Studies of nearby nuclei have been unable to determine the applicable coupling scheme or the interplay of the nearly symmetric neutron-proton configurations. Furthermore, heavy-ion reaction cross sections pose a severe limit in extending the previous studies in this region. 

Kamb B, Hamilton WC, LaPlaca SJ, Prakash A (1971) Ordered proton configuration in ice II from single-crystal neutron diffraction. J Chem Phys 55 1934-1945... 

Calculate a statistical/spectroscopic value of Sg(p, T ), compare this with your thermodynamic value, and report the discrepancy. For most substances, these two Sg values agree and the third law is valid for the solid as Tapproaches 0 K. However, you should find that 5g(spectroscopic) > (thermodynamic) for H2O, which means that ice does not have the perfect order at 0 K required by the third law. The explanation for this in terms of the disorder in the proton configurations in ice was first given by Pauling and is well described by Davidson. ... 

The curvatures across the energy minima of a number of the classic pair-wise potentials have been calculated (see Table 1). The force constants, of equation (22), have been extracted for the four proton configurations, namely A, B, C and D for ice Ih shown in Fig. 8. These values can be compared with the force constants used in the classic LD calculations of section 5.1. As one can see from Table 1, the ratios amongst the four force constants for the configurations for the potentials listed are all less than 1.3 which is considerable small than the value of 1.9 required for reproducing the observed INS spectrum for ice Ih as (more details discussed in section 6.1). 

From the simulation results, we can conclude that the classic pair-wise potentials are unlikely to reproduce the double peak structure of the observed translational spectrum. This implies that the differences between the forces of the different proton configurations are too small and the potentials are, therefore, much too isotropic, as we illustrated in Table 1. 

From the simulations, we conclude that two hydrogen bonding force constants are a basic requirement for reproducing the measured spectrum. If a water-water potential generates sufficiently large force constant differences for the different proton configurations (or the different relative dipole-dipole orientations in water or ice), it should produce the same effect as seen in the LR model. The anisotropic properties of the classic potentials are a result of charge interaction and this anisotropy should increase in the polarisable potentials and hence they produce a broad optic peak. This broad peak indicates that the orientational variation of the potential function has been increased considerably but it may still be less than the critical value of 1.5 as we indicated in the section 6.1. One would, therefore, expect that a better polarisable potential would, eventually, be able to reproduce the split optic peaks in the measured INS spectrum. 

Figure 7. Proton configurations near ionic groups and the proton energies. (From Ref. 136.)...

Here (b a) = (w — ) — ( — W) is the difference between energies of proton configurations at the first and last ionic groups (A and B) and an internal potential well (C). The value a = w + 6 + 6 determines the energy of domain walls at the ends of the chain (due to the fact that the boundary of the chain is characterized by other parameters than the internal part). The other parameters a) are the following ... 

The appearance of high-energy proton configurations, D and L defects, has occurred at 8 > 82, which is caused by the increasing the dipole moment of the proton subsystem with the gain of the number of hydrogen bonds. 

Terms To and T describe unordered proton configurations in the bonds, so that the probability of single-domain chain oscillations becomes a+lV- 2 cos [(Ei — Eo)tjh. When To and Ti are small in comparison with the a terms, one can treat the chain repolarization as that of a single domain. Contribution of To and Ti in (442) becomes more appreciable at the increasing of the integral J that leads to the destroying the correlated proton dynamics. For example, in the case of a chain consisting only of two bonds, which is characterized by periodical boundary conditions, we have... 

Here w, w, and e are the energies of proton configurations in the minima of the potential near an ionic group nRj(n/j) is the proton number operator that characterizes the occupation of the right (left) well of Ith hydrogen bond. The Hamiltonian //(ur, in expression (472) describes the tunnel transition between two proton states in the same hydrogen bond ... 

Fig. 2. Ne = Np = 16, Vs = 1.31. Dependence of total energy, variance and energy difference for a pair of proton configurations S, S ) on the RQMC projection time. The study is performed for Te = 0.02Dotted lines represent the variational estimates with their error bars. In panel b) and c) the lines are exponential fits to data and in panel d) the continnons line is a linear fit in the region < 0.005. Black circles (3BF-A) are resnlts obtained with the analitical three-body and backflow trial wave functions discnssed earlier, the red triangle is a variational resnlt with a Slater-Jastrow trial function with simple plane wave orbitals and the blue squares are results from a trial function with LDA orbitals and an optmized two-body Jastrow...

The total number of various proton configurations in ice-like systems is rather large. By analogy with well-known Pauling s formula we obtained the following expression for residual entropy of PWCs ... 

The comparison with results of high level quantum-chemical calculations proves the utility of the simple discrete models of molecular interaction for predicting the most stable topologies of water cycles and PWCs. Based on these discrete models an effective enumerating techniques was developed for hierarchical classification of proton configurations. In spite of the fact that PWCs are very complex systems with complicated interactions, the discrete models of inter-molecular interaction help us to see the wood for the trees (Fig. 3). 

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