Born-Oppenheimer approximation momentum

We restrict ourselves to the clamped-nucleus or Born-Oppenheimer approximation [30,31] because essentially all the work done to date on electron momentum densities has relied on it. Therefore we focus on purely electronic wavefunctions and the electron densities that they lead to. 

Eq. (5.34). However, it is possible to construct approximate wavefunctions that lead to electron momentum densities that do not have inversion symmetry. Within the Born-Oppenheimer approximation, the total electronic system must be at rest the at-rest condition... 

Within the Born-Oppenheimer approximation, the nuclei are at rest and have zero momentum. So the electron momentum density is an intrinsically one-center function that can be expressed usefully in spherical polar coordinates and expanded as follows [162,163]... 

From the general considerations presented in the previous section, one can expect that the many-body non-adiabatic wave function should fulfill the following conditions (1) All particles involved in the system should be treated equivalently (2) Correlation of the motions of all the particles in the system resulting from Coulombic interactions, as well as from the required conservation of the total linear and angular momenta, should be explicitly incorporated in the wave function (3) Particles can only be distinguishable via the permutational symmetry (4) The total wave function should possess the internal and translational symmetry properties of the system (5) For fixed positions of nuclei, the wave functions should become equivalent to what one obtains within the Born-Oppenheimer approximation and (6) the wave function should be an eigenfunction of the appropriate total spin and angular momentum operators. 

Here im is the effective mass of the i th vibration and Pi is the momentum conjugate to the corresponding normal vibrational coordinate Qi. The first two terms transform the electronic levels into potential energy manifolds in the coordinates of the octahedral normal modes Qi with vibrational frequencies m,- = yZ T/I/", and the complete wave functions in the Born-Oppenheimer approximation can be written as a product of the electronic and vibrational parts. The third term describes the distortions produced by the vibrations and can be interpreted in terms of a force Fi, which acts along the vibrational mode Qi associated with the electronic state E ... 

The above approximation corresponds to neglecting the coupling between the nuclear and electronic velocities, i.e. the nuclei are stationary from the electronic point of view. The electronic wave function thus depends parametrically on the nuclear coordinates, since it only depends on the position of the nuclei, not on their momentum. To a good approximation, the electronic wave function thus provides a potential energy surface upon which the nuclei move, and this separation is known as the Born-Oppenheimer approximation. 

Only the kinetic energy of the electrons is considered within the Born-Oppenheimer approximation, and the generalized momentum becomes (q = -1) eq. (10.64). 

One notes also that the quantities (3 J /9S.) are invariant under isotopic substitution (Born -Oppenheimer approximation, provided no vibrational angular momentum results ( Ay = 0). 

Here p, is the reduced mass, and are the Hamiltonians defined in Equation 11.1 for each atom, and Vint is the effective interaction potential depending on the relative position of the atoms, r. For many applications, such as the description of broad scattering resonances and their associated Feshbach molecules, it is sufficient to include in Vint only the rotationally symmetric singlet and triplet Born-Oppenheimer potentials, Vs=o and V5=i, respectively. Their labels 5 = 0 and 5=1 refer to the possible values of the angular-momentum quantum number associated with the total spin of the two atomic valence electrons, S = si -E S2. In this approximation, the interaction part of Equation 11.4 can be represented by [8,29]... 

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