Operators potential-energy

H=Kinetic energy operator + Potential energy operator... 

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. 

The potential energy part is diagonal in the coordinate representation, and we drop the hat indicating an operator henceforth. The kinetic energy part may be evaluated by transfonning to the momentum representation and carrying out a Fourier transform. The result is... 

The f operators are the usual kinetic energy operators, and the potential energy V(r,R) includes all of the Coulomb interactions ... 

We follow Thompson and Mead [13] to discuss the behavior of the electronic Hamiltonian, potential energy, and derivative coupling between adiabatic states in the vicinity of the D31, conical intersection. Let A be an operator that transforms only the nuclear coordinates, and A be one that acts on the electronic degrees of freedom alone. Clearly, the electronic Hamiltonian satisfies... 

We envision a potential energy surface with minima near the equilibrium positions of the atoms comprising the molecule. The MM model is intended to mimic the many-dimensional potential energy surface of real polyatomic molecules. (MM is little used for very small molecules like diatomies.) Once the potential energy surface iias been established for an MM model by specifying the force constants for all forces operative within the molecule, the calculation can proceed. 

We assume that the nuclei are so slow moving relative to electrons that we may regard them as fixed masses. This amounts to separation of the Schroedinger equation into two parts, one for nuclei and one for electrons. We then drop the nuclear kinetic energy operator, but we retain the intemuclear repulsion terms, which we know from the nuclear charges and the intemuclear distances. We retain all terms that involve electrons, including the potential energy terms due to attractive forces between nuclei and electrons and those due to repulsive forces... 

The sum of two operators is an operator. Thus the Hamiltonian operator for the hydrogen atom has — j as the kinetic energy part owing to its single election plus — 1/r as the electiostatic potential energy part, because the charge on the nucleus is Z = 1, the force is atrtactive, and there is one election at a distance r from the nucleus... 

By extension of Exercise 6-1, the Hamiltonian for a many-electron molecule has a sum of kinetic energy operators — V, one for each electron. Also, each electron moves in the potential field of the nuclei and all other electrons, each contiibuting a potential energy V,... 

One more quantum number, that relating to the inversion (i) symmetry operator ean be used in atomie eases beeause the total potential energy V is unehanged when all of the eleetrons have their position veetors subjeeted to inversion (i r = -r). This quantum number is straightforward to determine. Beeause eaeh L, S, Ml, Ms, H state diseussed above eonsist of a few (or, in the ease of eonfiguration interaetion several) symmetry adapted eombinations of Slater determinant funetions, the effeet of the inversion operator on sueh a wavefunetion P ean be determined by ... 

Hamiltonian quantum mechanical operator for energy, hard sphere assumption that atoms are like hard billiard balls, which is implemented by having an infinite potential inside the sphere radius and zero potential outside the radius Hartree atomic unit of energy... 

Fig. 6. Van der Waals potential energy-distance curve showing regions of operation of contact, noncontact, and intermittent contact or tapping-mode afm...

There are examples of each of these mechanisms, and a three-dimensional potential energy diagram can provide a useful general framework within which to consider specific addition reactions. The breakdown of a tetrahedral intermediate involves the same processes but operates in the opposite direction, so the principles that are developed will apply equally well to the reactions of the tetrahedral intermediates. Let us examine the three general mechanistic cases in relation to the energy diagram in Fig. 8.3. 

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