Kinetic energy total

Most of the better approximate functionals give reasonable, albeit unspectacular, results for the kinetic energy when accurate atomic and molecular densities are used. Flowever, when the functionals are used in the variational principle for the electron density, the results deteriorate catastrophically, giving kinetic energies, total energies, and electron densities that are qualitatively incorrect. These failings may be attributed to the A-representability problem for density functionals the kinetic energy functional does not correspond to an acceptable A-electron system. 

The total kinetic energy then decomposes into the sum... 

Because NH has an electron affinity of 0.4 eV, the total energies of the above two states can be equal only if the kinetic energy KE carried away by the ejected electron obeys... 

This gives the total energy, which is also the kinetic energy in this case because the potential energy is zero within the box , m tenns of the electron density p x,y,z) = (NIL ). It therefore may be plausible to express kinetic energies in tenns of electron densities p(r), but it is by no means clear how to do so for real atoms and molecules with electron-nuclear and electron-electron interactions operative. 

Flere we distinguish between nuclear coordinates R and electronic coordinates r is the single-particle kinetic energy operator, and Vp is the total pseudopotential operator for the interaction between the valence electrons and the combined nucleus + frozen core electrons. The electron-electron and micleus-micleus Coulomb interactions are easily recognized, and the remaining tenu electronic exchange and correlation... 

In Eqs. (5) and (6), M is the total mass of the nuclei and is the mass of one electron. By using Eq. (2), the system s internal kinetic energy operator is given in terms of the mass-scaled Jacobi vectors by... 

A typical molecular dynamics simulation comprises an equflibration and a production phase. The former is necessary, as the name imphes, to ensure that the system is in equilibrium before data acquisition starts. It is useful to check the time evolution of several simulation parameters such as temperature (which is directly connected to the kinetic energy), potential energy, total energy, density (when periodic boundary conditions with constant pressure are apphed), and their root-mean-square deviations. Having these and other variables constant at the end of the equilibration phase is the prerequisite for the statistically meaningful sampling of data in the following production phase. 

Figure 7-15. Healing and equilibration phase of a typical MD simulation, In the ideal case, the temperature should fluctuate around the desired value (here 298 K), and the potential energy should remain constant. Remember that the total energy is the sum of potential and kinetic energy, the latter being directly coupled to the temperature of the system,...

HyperChern immediately scales (adjusts) the velocities of each atom so that the total kinetic energy is equal lo V.,kT, where k is the Boltzmann constant and f is the specified starting ternpera-... 

Kinetic energy (EKfX i Potential energy (KPO I ) Total energy (ETOT) Tern perature (THMP)... 

In th is eon text, K is the total classical energy including kinetic energy. You can then investigate the potential energy surface in a purely classical way using the positions (Rj) and velocities (V = dKi/dt) of the con stitiieri t atom s. 

The canonical ensemble is the name given to an ensemble for constant temperature, number of particles and volume. For our purposes Jf can be considered the same as the total energy, (p r ), which equals the sum of the kinetic energy (jT(p )) of the system, which depends upon the momenta of the particles, and the potential energy (T (r )), which depends upon tlie positions. The factor N arises from the indistinguishability of the particles and the factor is required to ensure that the partition function is equal to the quantum mechanical result for a particle in a box. A short discussion of some of the key results of statistical mechanics is provided in Appendix 6.1 and further details can be found in standard textbooks. 

In a canorrical ensemble the total temperature is constant. In the microcanonical ensemble, however, the temperature will fluctuate. The temperature is directly related to the kinetic energy of the system as follows ... 

---