Collective variables potential energy

If the metadynamics method is applied to the simulation of chemical reactions in conjunction with Car-Parrinello molecular dynamics [36,40,49], the history dependent potential has to force the system to cross barriers of several tenths of kcal/mol in a very short time, usually a few picoseconds. This implies that a lot of energy is injected in the degrees of freedom associated with the collective variables. This might lead to a significant dishomogeneity in the temperature distribution of the system, and possibly to instabilities in the dynamics. 

Keywords Ab initio molecular dynamics simulations Always stable predictor-corrector algorithm Associated liquids Basis set Bom-Oppenheimer molecular dynamics simulations Car-Parrinello molecular dynamics simulations Catalysis Collective variable Discrete variable representation Dispersion Effective core potential Enhanced sampling Fictitious mass First-principles molecular dynamics simulations Free energy surface Hartree-Fock exchange Ionic liquids Linear scaling Metadynamics Nudged elastic band Numerically tabulated atom-centered orbitals Plane waves Pseudopotential Rare event Relativistic electronic structure Retention potential Self consistent field SHAKE algorithm ... 

The potential energy surface E(K) is an energy function where the variables are the internal nuclear coordinates, collected into the symbol K of the nuclear configuration. Note, that different electronic states are associated to different potential energy surfaces. 

From the standpoint of quantum mechanics, the potential energy surface (PES) arises from treating the nuclear variables of a collection of electrons and nuclei, formally described by the Schrodinger Coulomb Hamiltonian, as parameters rather than variables. The basis for this... 

Here, t numbers the time steps, are the collective variables at time t and =fM) is the force at (f. The control parameter 5q specifies the step size. Such a time evolution would, however, head towards the closest free energy minimum and get trapped there. To prevent this from happening, a repulsive Gaussian potential is deposited at every point visited in the space of the collective variables. Each of these potentials acts at all later times such that the total force at time 1 is given by the thermodynamic force plus a sum of the forces stenuning from the Gaussian potentials. 

A variable negative potential on the outer cylinder bends the Auger electrons of a particular kinetic energy E0 back through a second annular aperture on the inner cylinder they are then are focussed at an exit aperture on the analyser axis where they are collected by an electron multiplier. The energy of the transmitted electrons is proportional to the voltage on the outer cylinder (F), and simply scanning the... 

Hohenberg and Kohn have proved generally that the total ground state energy E of a collection of electrons in the presence of an externally applied potential (e.g. the valence electrons in the presence of the periodic potential due to the cores in a lattice), when no net magnetic moment is present, depends only on the average density of electrons n(R). By this proof, n(R) becomes the fundamental variable of the system (as it is in the Thomas-Fermi theory ). Variational minimization of the most general form of E, with respect to n lends to the Hartree-Fock equations formalism. 

Thermodynamic principles arise from a statistical treatment of matter by studying different idealized ensembles of particles that represent different thermodynamic systems. The first ensemble that we study is that of an isolated system a collection of N particles confined to a volume V, with total internal energy E. A system of this sort is referred to as an NVE system or ensemble, as N, V, and E are the three thermodynamic variables that are held constant. N, V, and E are extensive variables. That is, their values are proportional to the size of the system. If we combine NVE subsystems into a larger system, then the total N, V, and E are computed as the sums of N, V, and E of the subsystems. Temperature, pressure, and chemical potential are intensive variables, for which values do not depend on the size of the system. 

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