Operator electrostatic potential energy

The Hamiltonian of the Coulomb term involves electrostatic potential energy operator for the interaction of all electrons and nuclei of donors with those of acceptors. The electrostatic potential can be expanded into multipole terms of the type... 

Where T and Tei are the kinetic energy operators for the nuclei and electrons, and Vmi, Vne and Vee are the electrostatic potential energies arising from intemuclear, nucleus-electron, and interelectronic interactions. At this point, the Born-Oppenheimer approximation is invoked, i.e., the assumption that the movement of electrons is much faster than that of the nuclei and therefore the two can be decoupled from one another. This is mathematically represented by a separation of variables in the wavefunction, Y(R,r)= (r(R)) %(R), where <1> and % are the electronic and nuclear wavefunctions, respectively, and O is a function of r parameterized by R. Thus, with some rearrangement, Equation (21) becomes... 

With Stevens operator equivalents technique (where one replaces the terms in the expansion of the electrostatic potential by suitable angular-momentum operators) the potential energy of the 4f electrons with the polar axis along... 

Basically, the operator for the interaction of the magnetic ion with the CEF potential is the classical electrostatic potential energy... 

The concept of adsorption potential comes from work with high-purity, synthetic microporous carbon, which relies solely on van der Waals dispersive and electrostatic forces to provide the energy for adsorption. The polymeric microporous adsorbents that operate solely through van der Waals dispersive and electrostatic forces often cannot provide the surface potential energy needed to trap compounds that are gases under ambient conditions, and for very volatile compounds the trapping efficiency can be low for similar reasons. 

Our initial, small models of an isolated cellulose chain ranged from the dimer (cellobiose) to the octamer. The dynamics of these fragments have thus far been simulated only in vacuum, using different potential energy functions such as those of MM2(85) (9) and AMBER (10), with and without contributions from electrostatic terms and hydrogen bonds, etc. (The program DISCOVER, customized for carbohydrates and for operation on the Alliant FX/80 computer, has been used (12).) Generally, the time span for the simulations has been of the order of several hundred picoseconds to 1 nanosecond. 

Because of the complexity of the forces operating in the covalent bond, it is not possible to write a simple potential energy function as for the electrostatic forces such as ion—ion and dipole-dipole. Nevertheless, it is possible to describe the covalent energy qualitatively as a fairly short-range force (as the atoms are forced apart, the overlap decreases). 

Let us emphasize that in single-configurational approach the terms of the Hamiltonian describing kinetic and potential energies of the electrons as well as one-electron relativistic corrections, contribute only to average energy and, therefore, are not contained in, which in the non-relativistic approximation consists only of the operators of electrostatic interaction e and the one-electron part of the spin-orbit interaction so, i.e. 

Between the wall of the cell and any ions (H+, H30+, H502+) forces of supermolecular hydrogen P-bonds and electrostatic y-bonds operate (see Fig. 2). Surfaces of intermolecular potential energy have been calculated by density functional method stated in our paper [6], Necessary data about spatial distributions of electron charge density inside framework of aqua multiparticle had been taken from calculations of aquatic ions and the ring of water (H20)n by using of standard molecular orbital method in the minimal basis set (STO-3G). Results of calculations are shown in Table 1. 

For MD and/or QM/MM geometry optimizations gradients of the energies are needed. They follow naturally from the energy expressions by replacing electrostatic potential and field operators by, respectively, the corresponding field and field gradient operators. 

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