Potential energy electron-nuclear

The total energy in an Molecular Orbital calculation is the net result of electronic kinetic energies and the interactions between all electrons and atomic cores in the system. This is the potential energy for nuclear motion in the Born-Oppenheimer approximation (see page 32). 

Ue in (6.7), the value at the equilibrium geometry of the potential energy for nuclear motion. The convention is to denote quantities of the upper electronic state of a transition by single primes, and quantities of the lower electronic state by double primes. The symbols Te, G, and F are used for the electronic, vibrational, and rotational energies in cm-1 ... 

According to the Born-Oppenheimer approximation, the electronic wave function and the potential energy governing nuclear motion are independent of nuclear mass. What does depend on nuclear mass is the zero-point energy,... 

In our subsequent development we shall take the origin of coordinates to be at the centre of mass of the two nuclei, although we could equally well have chosen the molecular centre of mass as origin. Setting aside the translational motion of the molecule, we use equation (2.28) to represent the kinetic energy of the electrons and nuclei. To this we add terms representing the potential energy, electron spin interactions, and nuclear spin interactions. We subdivide the total Hamiltonian Xx into electronic and nuclear Hamiltonians,... 

The potential energy function C/(ri---rjv) expresses the energy of an assembly of N atoms or ions as a function of the nuclear coordinates ri - rjv. The Bom-Oppenheimer approximation is, of comse, implicit in the use of such fimctions but there is no explicit inclusion of the effects of the electronic stractme of the system such effects are subsumed into the potential function. The energy zero for such functions is normally taken to be that of the component atoms (or ions) at rest at infinity, that is, the self energies (electron-nuclear) of the atoms (or ions) are not included in U. 

The potential energy of nuclear-electronic and intemuclear interactions is... 

Some progress can be made by decomposing the energy into contributions from kinetic energy, electron-nuclear attraction potential energy, and electron-electron repulsion potential energy... 

Solving Eq. (7-76) for i eiec for every value of R allows us to plot the electronic and also the total energy of the system as a function of R. But Et tc R) -h 1/7 is just the potential energy for nuclear motion. Therefore, this quantity can be inserted as the potential in the hamiltoiuan operator for nuclear motion ... 

V = one-electron potential energy operator (nuclear-electron attraction)... 

The result is that, to a very good approxunation, as treated elsewhere in this Encyclopedia, the nuclei move in a mechanical potential created by the much more rapid motion of the electrons. The electron cloud itself is described by the quantum mechanical theory of electronic structure. Since the electronic and nuclear motion are approximately separable, the electron cloud can be described mathematically by the quantum mechanical theory of electronic structure, in a framework where the nuclei are fixed. The resulting Bom-Oppenlieimer potential energy surface (PES) created by the electrons is the mechanical potential in which the nuclei move. Wlien we speak of the internal motion of molecules, we therefore mean essentially the motion of the nuclei, which contain most of the mass, on the molecular potential energy surface, with the electron cloud rapidly adjusting to the relatively slow nuclear motion. 

Note the stnicPiral similarity between equation (A1.6.72) and equation (Al.6.41). witii and E being replaced by and the BO Hamiltonians governing the quanPim mechanical evolution in electronic states a and b, respectively. These Hamiltonians consist of a nuclear kinetic energy part and a potential energy part which derives from nuclear-electron attraction and nuclear-nuclear repulsion, which differs in the two electronic states. 

The fitting parameters in the transfomi method are properties related to the two potential energy surfaces that define die electronic resonance. These curves are obtained when the two hypersurfaces are cut along theyth nomial mode coordinate. In order of increasing theoretical sophistication these properties are (i) the relative position of their minima (often called the displacement parameters), (ii) the force constant of the vibration (its frequency), (iii) nuclear coordinate dependence of the electronic transition moment and (iv) the issue of mode mixing upon excitation—known as the Duschinsky effect—requiring a multidimensional approach. 

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. 

Oppenlieimer potential which is defined as the electronic ground-state energy for nuclear configuration T, including the niiclear-niiclear repulsion. 

---