Internal potential energy

The internal (potential) energy is a direct sum of energies, which is normally given as a sum over pairwise interactions (i.e. van der Waals and electrostatic contributions in a force field description). 

Here, Pp = mpR is a Cartesian bead momentum, U is the internal potential energy of the system of interest, Xi,.. .,Xk are a set of AT Lagrange multiplier constraint fields, which must be chosen so as to satisfy the K constraints, and is the rapidly fluctuating force exerted on bead p by interactions with surrounding solvent molecules. The corresponding Hamiltonian equation of motion is... 

The Hamiltonian is the total energy operator for a system, and is written as the sum of the kinetic energy of all the components of the system and the internal potential energy. In an atom or molecule, comprised of positive nuclei and negative electrons, the potential energy is simply that due to the coulombic interactions present. Thus for the kinetic energy in a system of M nuclei and N electrons ... 

One example of non-IRC trajectory was reported for the photoisomerization of cA-stilbene.36,37 In this study trajectory calculations were started at stilbene in its first excited state. The initial stilbene structure was obtained at CIS/6-31G, and 2744 argon atoms were used as a model solvent with periodic boundary conditions. In order to save computational time, finite element interpolation method was used, in which all degrees of freedom were frozen except the central ethylenic torsional angle and the two adjacent phenyl torsional angles. The solvent was equilibrated around a fully rigid m-stilbene for 20 ps, and initial configurations were taken every 1 ps intervals from subsequent equilibration. The results of 800 trajectories revealed that, because of the excessive internal potential energy, the trajectories did not cross the barrier at the saddle point. Thus, the prerequisites for common concepts of reaction dynamics such TST or RRKM theory were not satisfied. 

Here, the internal potential energy of the thermodynamic system as a whole takes the form of the following expansion ... 

The hydrogen molecnlar ion is sketched in Fignre 6.1. The two nuclei, for convenience labeled A and B, are separated by the distance Rab along the internnclear axis, chosen by convention to be the x-axis. The electron is located at distance Ta from nucleus A and at distance tb from nucleus B. The angle describes rotation about the internuclear axis. For a fixed value of Rab> the position of the electron is more conveniently specified by the values of (ta, tb, 4>) than by (x, y, z) because the former set reflects the natural symmetry of the system. The internal potential energy is given by... 

As usual, the internal potential energy U(rN) of the system is calculated from a specified intersite potential-energy function, generally by assuming pairwise additive interactions. The kinetic energy of the system is given by... 

Potential energies for the nuclear motions in a polyatomic system can be obtained from the Born-Oppenheimer separation of electronic and nuclear motions, for each adiabatic electronic state. Their values E can be separated into asymptotic contributions giving internal potential energies and Vg, and a remainder term V describing the interaction potential. 

The term describes here a transition where the last interaction occurs only within the cluster C. and is coupled to similar operators except itself. It can be obtained by first calculating the simpler auxiliary operators 7 which only contain the interaction potential, and the free-motion propagator G including the internal potential energies of A and B,... 

Let y be the potential of an external (classical) conservative body force and U be the internal potential energy of the fluid due to interparticle interactions. We assnme that the Lagrangian has the same form as in the classical theory of ideal flnids, except for the fnnctional dependence of U this depends on p(g) and its first derivatives and,... 

Special attention should be paid to the fact that the supersystem should include especially those solvent molecules which are expected to play an active role in the chemical process studied. The inclusion of statistical methods in simulating medium influences opens the possibility of changing from an internal potential energy profile to a free energy (enthalpy) one. 

Fig. 2 - Internal potential energy of isotactic P(S)3MP. In the parts A and B are reported the regions of the multidimensional energy surface corresponding both to a left--handed helix for the two different conformations of the lateral group observed in the crystalline state. In the part C the minimum energy region corresponding to a right-handed helix is reported. The energies, referred to the absolute minimum (part B e, - 180 and 0 6O ), are in Real (mol of CRU) " .

Of course, this electronic rearrangement will also affect the internal potential energy. However, these changes are dominated by the kinetic energy (Figures 6c and 6f) and will not be discussed further. 

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