Quantum-mechanical flux, model

Figure 4. a. Quantum-mechanical flux in a tunneling transition in a model system shown in Fig. 3. 

This is the naive picture on which many tentative models of chemical reactions used in the past were based. The material model is reduced to the minimal reacting system (A+B in the example presented above) and supplemented by a limited number of solvent molecules (S). Such material model may be studied in detail with quantum mechanical methods if A and B are of modest size, and the number of S molecules is kept within narrow limits. Some computational problems arise when the size of reactants increases, and these problems have been, and still are, the object of active research. This model is clearly unsatisfactory. It may be supplemented by a thermal bath which enables the description of energy fluxes from the microscopic to the outer medium, and vice versa, but this coupling is not sufficient to bring the model in line with chemical intuition and experimental evidence. 

As with the equilibrium solvation models introduced earlier, it is also possible to incorporate quantum mechanical effects into the non-equilibrium transport model. Our motivation is to account for non-equilibrium ion fluxes and induced response in the electronic structure of the solute or membrane protein. To this end, we combine our DG-based DFT model with our DG-based PNP model as illustrated in Fig. 12.4 to develop a free energy functional and derive the associated governing equations. 

The existence of fairly large delocalized electron flow was demonstrated in H2O, BH3, NH3, CH4, CH3-CH3, HsO-, CH3+, and NH/, by plots of quantum mechanical current density and by current susceptibilities, Eq. (7.55), calculated by accurate ab initio methods [102, 145]. The latter, also referred to as current strengths measured in nano ampere per tesla, are defined via flux integrals evaluated over suitably chosen molecular domains [76]. Simple procedures, allowing for ideal current models based on the BS law [46], have been applied to predict the ability of a certain molecule to support magnetic-field induced electron currents flowing through an interatomic circuit [77, 141-143]. 

Thus physically the definition of an expectation value and of its change for any system, closed or open, requires that the system be a bounded region of real space. Mathematically the boundary comes about via the imposition of a boundary condition in a variational procedure. Modeling an atom or its properties in a molecule in terms of functions that extend over all space, as is done in orbital models, precludes the use of quantum mechanics in determining its properties. The presence of the surface flux contribution imparts new and important features to an open system not present in the mechanics of the total molecule, features that play an essential role in relating the physics of an atom to its chemical behavior. 

Recently, a QUAPI procedure was developed suitable for evaluating the full flux correlation function in the case of a one-dimensional quantum system coupled to a dissipative harmonic bath and applied to obtain accurate quantum mechanical reaction rates for a symmetric double well potential coupled to a generic environment. These calculations confirmed the ability of analytical approximations to provide a nearly quantitative picture of such processes in the activated regime, where the reaction rate displays a Kramers turnover as a function of solvent friction and quantum corrections are small or moderate, They also emphasized the significance of dynamical effects not captured in quantum transition state models, in particular under small dissipation conditions where imaginary time calculations can overestimate or even underestimate the reaction rate. These behaviors are summarized in Figure 7. 

Since a localized bond model requires information neither about the distribution of electron density nor about the energy of the molecule, quantum mechanics is not needed. Bond valence theory determines the number of bonding electrons (atomic valence) by counting how many electrons have a small ionization energy, and it develops the bond picture using the electrostatic field rather than the electrostatic potential. In the ionic model representation of bond valence theory, a bond exists between any two atoms linked by lines of electric field and the number of these lines linking two atoms (the electrostatic flux) is a measure of the number of electrons used to form the bond. This is the foundation on which the model is built. 

The BCS theory, however, developed in 1957 by three physicists, John Bardeen, Leon Cooper, and Robert Schrieffer, does estabhsh a model for the mechanism behind superconductivity. Bardeen, Cooper, and Schrieffer received the Nobel Prize in physics in 1972 for their theory. It was known that the flux quantum was inversely proportional to twice the charge of an electron, and it had also been observed that different isotopes of the same superconducting element had different critical temperatures. Actually, the heavier the isotope, the lower the critical temperature is. The critical temperature, in K, of an isotope with an atomic mass, M, expressed in kg.moT can be predicted by the following equation ... 

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