Classical potentials

Alternatively, if the momenta are integrated out, an effective distribution of the coordinates is obtained. In this distribution the Boltzmann factor is sampled with the classical potential energy replaced by a quantum effective potential... 

The classical potential energy of an harmonic oscillator, V = kx2 = (47r2mv2x2), and... 

Thermal fluctuations, however, may contribute to the classical potential to restore the broken symmetry when the thermal energy is sufficient enough to overcome the potential energy barrier between the true and false vacua. The renormalized effective potential for the background field [Pg.277]

Molecular dynamics simulations, with quantum-mechanically derived energy and forces, can provide valuable insights into the dynamics and structure of systems in which electronic excitations or bond breaking processes are important. In these cases, conventional techniques with classical analytical potentials, are not appropriate. Since the quantum mechanical calculation has to be performed many times, one at each time step, the choice of a computationally fast method is crucial. Moreover, the method should be able to simulate electronic excitations and breaking or forming of bonds, in order to provide a proper treatment of those properties for which classical potentials fail. 

In Equation 4.56, the real quantities p, v, and j are the charge density, velocity field, and current density, respectively. The above equations provide the basis for the fluid dynamical approach to quantum mechanics. In this approach, the time evolution of a quantum system in any state can be completely interpreted in terms of a continuous, flowing fluid of charge density p(r,t) and the current density j(r,t), subjected to forces arising from not only the classical potential V(r, t) but also from an additional potential VqU(r, t), called the quantum or Bohm potential the latter arises from the kinetic energy and depends on the density as well as its gradients. The current... 

The classical potential energy term is just a sum of the Coulomb interaction terms (Equation 2.1) that depend on the various inter-particle distances. The potential energy term in the quantum mechanical operator is exactly the same as in classical mechanics. The operator Hop has now been obtained in terms of second derivatives with respect to Cartesian coordinates and inter-particle distances. If one desires to use other coordinates (e.g., spherical polar coordinates, elliptical coordinates, etc.), a transformation presents no difficulties in principle. The solution of a differential equation, known as the Schrodinger equation, gives the energy levels Emoi of the molecular system... 

Molecules consist of electrons and nuclei the principal difference between a molecule and an atom is that the latter has only one particle of the nuclear sort. Classical potential theory, which in this case works for quantirm mechanics, says that Coulomb s law operates between charged particles. This asserts that the potential energy of a pair of spherical, charged objects is... 

The approach with the partitioning of the system into a QM and a classical molecular mechanical (MM) part, thus usually termed hybrid QM/MM procedure, provides a reasonable reduction of the computational effort by restricting the time-consuming QM calculation of forces to the most relevant part of the liquid system. The main error sources in this approach are a too small choice of the QM region, an inadequate level of theory for the QM calculation, the choice of suitable potentials for the MM part of the system, and smooth transitions of particles between QM and MM region. In conventional QM/MM procedures, the whole system is first evaluated at MM level and then corrected by the QM data. This means that classical potential functions (with all their problems and difficulty of construction) are needed for all components of the system. A recently developed methodology can reduce the need for such potentials to the solvent only, as will be outlined below. 

The physical description of the simulated systems in Odyssey is via classical potential functions that have been developed for research applications. In many areas relevant to teaching, the description is at least qualitatively correct. This is all that is required from a pedagogical standpoint. Nevertheless, the models do fail on occasion, even qualitatively. Rather than being a drawback, this can well be considered a compelling illustration of the fact that eventually all scientific models have intrinsic limitations. As teachers of science (rather than of scientific facts), we should be conveying this to our students in any case Going back to the laboratory is eventually the only way to find out ... 

The Aharonov-Bohm effect is self-inconsistent in U(l) electrodynamics because [44] the effect depends on the interaction of a vector potential A with an electron, but the magnetic field defined by = V x A is zero at the point of interaction [44]. This argument can always be used in U(l) electrodynamics to counter the view that the classical potential A is physical, and adherents of the received view can always assert in U(l) electrodynamics that the potential must be unphysical by gauge freedom. If, however, the Aharonov-Bohm effect is seen as an effect of 0(3) electrodynamics, or of SU(2) electrodynamics [44], it is easily demonstrated that the effect is due to the physical inhomogeneous term appearing in Eq. (25). This argument is developed further in Section VI. 

These results show that including quantum mechanical electronic rearrangement in dynamics calculations of the configurations of water on a metal surface can reveal effects that are not present in classical models of the water metal interface which treat the interaction of water with the surface as a static, classical potential energy function. For example, in classical calculations of the behavior of models of water at a paladium surface the interaction with one water molecule with the surface had a similar on-top binding site, a clas-... 

Classical molecular simulation methods such as MC and MD represent atomistic/molecular-level modeling, which discards the electronic degrees of freedom while utilizing parameters transferred from quantum level simulation as force field information. A molecule in the simulation is composed of beads representing atoms, where the interactions are described by classical potential functions. Each bead has a dispersive pair-wise interaction as described by the Lennard-Jones (LJ) potential, ULj(Ly) ... 

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