Energy eigenfunctions electronic

If now the nuclear coordinates are regarded as dynamical variables, rather than parameters, then in the vicinity of the intersection point, the energy eigenfunction, which is a combined electronic-nuclear wave function, will contain a superposition of the two adiabatic, superposition states, with nuclear... 

If, on the other hand, the electron had been prepared in a state /(x,y) that is not a pure eigenstate (i.e., cannot be expressed as a single energy eigenfunction), then the time evolution is more complicated. For example, if at t=0 / were of the form... 

Fig. 4.2 Electronic energy of H2 as a function of the internuclear distance, for the lower energy eigenfunctions. (Adapted with permission from ref. 26.)...

I is in general no direct relation between such functions and ionization energies or electron excitation this is because they are not eigenfunctions of a hamiltonian, hence they cannot be associated with an energy. For that reason, we kept the usual designation localized molecular orbitals but with [ the last word in inverted commas orbitals . However, for the interpretation of some other molecular properties, the minimized residual interactions i between quasi-localized molecular orbitals are not very importaint and, so, the direct use of a localized bond description is quite justified. That is the [ Case for properties such as bond energies and electric dipole moments, as well as the features of the total electron density distribution with which those properties are directly associated. 

Show that (a rfc (-iV ) + r -iVk) >) = ii ab a rkr b) for two energy eigenfunctions o) and b), where Uab = Ea Eb in the units with Planck s constant = 27t, and electron mass=l. Note the relation to the electric quadrupole moment operator for an electronic system f dfrrq(f). Note the dyadic notation ff for the tensor quantity ... 

The phenomenon is closely related to the resonance phenomenon treated by Heisenberg. While in resonance however, electrons with a different motion in one and the same eigenfunction series exchange their energy here electrons of the same excitation stages (the same energy) but in different eigenfunction systems (V and... 

The square of the orbital angular momentum of the electron and one component of this angular momentum can have predictable values if a hydrogen atom is in a state corresponding to an energy eigenfunction. The angular momentum values are different from the Bohr theory. 

In the perturbation method the Hamiltonian is written as + H, where corresponds to a Schrodinger equation that can be solved. The perturbation term H is arbitrarily multiplied by a fictitious parameter k, so that A. = 1 corresponds to the actual case. The method is based on representations of energy eigenvalues and energy eigenfunctions as power series in A. and approximation of the series by partial sums. The method can be applied to excited states. In the helium atom treatment the electron-electron repulsive potential energy was treated as the perturbation term in the Hamiltonian operator. 

The electronic energy eigenfunctions of a molecule can be eigenfunctions of the symmetry operators that commute with the electronic Hamiltonian. The symmetry operators that commute with the electronic Hamiltonian are said to belong to the molecule. In order for a symmetry operator to commute with the electronic Hamiltonian, it must leave the potential energy unchanged when applied to the electron s coordinates. Otherwise, a different result would occur if the symmetry operator were applied to an electronic wave function after application of the Hamiltonian than if the operators were applied in the other order. 

As with atoms, term symbols are used to designate values of angular momentum quantum numbers for homonuclear diatomic molecules. The scheme is slightly different from the Russell-Saunders scheme that is used for atoms, since L is not a good quantum number. The operator does commute with the electronic Hamiltonian and an energy eigenfunction can be an eigenfunction of Lj ... 

If an atom is confined in a rectangular box its center of mass cannot move completely up to the walls of the box because of the electrons in the atom. However, if the box is much larger than the size of an atom it will be an excellent approximation to apply the formulas that apply to a particle of zero size to the translation of an atom in a box. The translational energy eigenfunctions would be represented by the normalized version of Eq. (15.3-21) ... 

In addition the density of states changes with dimensionality. This is already so in the case of the quasi-free electron approximation. A three-dimensional box with dimensions Lx x Ly x L, in which the energy eigenfunction has, on account of the factorizability of the wave function, the general form... 

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