Schrodinger equation for diatoms

The concepts which we need for understanding the structural trends within covalently bonded solids are most easily introduced by first considering the much simpler system of diatomic molecules. They are well described within the molecular orbital (MO) framework that is based on the overlapping of atomic wave functions. This picture, therefore, makes direct contact with the properties of the individual free atoms which we discussed in the previous chapter, in particular the atomic energy levels and angular character of the valence orbitals. We will see that ubiquitous quantum mechanical concepts such as the covalent bond, overlap repulsion, hybrid orbitals, and the relative degree of covalency versus ionicity all arise naturally from solutions of the one-electron Schrodinger equation for diatomic molecules such as H2, N2, and LiH. 

Our treatment of the nuclear Schrodinger equation for diatomic molecules has shown that the wave function for nuclear motion can be separated into rotational, vibrational, and translational wave functions ... 

In the last section, we obtained an approximate solution to the nuclear Schrodinger equation for diatomic molecules. We now improve the accuracy of that solution. In going from (4.11) to (4.23), we neglected the smaller terms in the expansions (4.12) and (4.21). We shall deal with the... 

Most of this chapter deals with the electronic Schrodinger equation for diatomic molecules, but this section examines nuclear motion in a bound electronic state of a diatomic molecule. From (13.10) and (13.11), the Schrodinger equation for nuclear motion in a diatomic-molecule bound electronic state is... 

To describe the orientations of a diatomic or linear polyatomic molecule requires only two angles (usually termed 0 and ([)). For any non-linear molecule, three angles (usually a, P, and y) are needed. Hence the rotational Schrodinger equation for a nonlinear molecule is a differential equation in three-dimensions. 

As given in Chapter 3, the Schrodinger equation for the angular motion of a rigid (i.e., having fixed bond length R) diatomic molecule is... 

Including higher-order terms leads to anharmonie correetions to the vibration, sueh effects are typically of the order of a few %. The energy levels obtained from the Schrodinger equation for a one-dimensional harmonie oscillator (diatomic system) are given by... 

Schrodinger equation for each molecular electronic state, we seek an approximation that will represent U reasonably well for most diatomics. For a bound electronic state, we know that U has the general appearance of the solid curve in Fig. 4.2. We expect the nuclei to vibrate about the position of minimum potential energy therefore, we expand U in a Taylor series (Section 1.2) about Re, the equilibrium internuclear separation ... 

In quantum mechanics (18,19) the vibration of a diatomic molecule can be treated as a motion of a single particle having mass n whose potential energy is expressed by (1-21). The Schrodinger equation for such a system is written as... 

In chapter 2 we discussed at length the separation of nuclear and electronic coordinates in the solution of the Schrodinger equation. We described the Born-Oppenheimer approximation which allows us to solve the Schrodinger equation for the motion of the electrons in the electrostatic field produced by fixed nuclear charges. There are certain situations, particularly with polyatomic molecules, when the separation of nuclear and electronic motions cannot be made satisfactorily, but with most diatomic molecules the Born-Oppenheimer separation is acceptable. The discussion of molecular electronic wave functions presented in this chapter is therefore based upon the Born-Oppenheimer approximation. 

Schrodinger equation for a one-dimensional harmonic oscillator (diatomic system) are... 

If the energies of the atomic orbitals of the two atoms of a diatomic molecule or ion are quite different, the MO diagram may be unlike that known for any homonuclear species. Its unique MO diagram is constructed by combining the Schrodinger equations for the two atoms. Construction of the MO diagram for CO is a complex case, beyond the coverage in this textbook. 

The above demonstrated possibility of obtaining numerical virtual orbitals indicate that the FD HF method can also be used as a solver of the Schrodinger equation for a one-electron diatomic system with an arbitrary potential. Thus, the scheme could be of interest to those who try to construct exchange-correlation potential functions or deal with local-scaling transformations within the functional density theory (32,33). 

The Schrodinger equation for the one-electron diatomic (homo- and hetero-nuclear) is also soluble but the solutions are not of wide applicability essentially because the solutions are numerical in one of the three dimensions. 

We shall see in Section 13.1 that to an excellent approximation one can treat separately the motions of the electrons and the motions of the nuclei of a molecule. (This is due to the much heavier mass of the nuclei.) One first imagines the nuclei to be held stationary and solves a Schrodinger equation for the electronic energy U. U also includes the energy of nuclear repulsion.) For a diatomic (two-atom) molecule, the electronic energy U depends on the distance R between the nuclei, U = U R), and the U versus R curve has the typical appearance of Fig. 13.1. 

We start with diatomic molecules, the simplest of which is Hj, the hydrogen molecule ion, consisting of two protons and one electron. Just as the one-electron H atom serves as a st u ting point in the discussion of many-electron atoms, the one-electron H2 ion furnishes many ideas useful for discussing many-electron diatomic molecules. The electronic SchrOdinger equation for is separable, and we can get exact solutions for the eigenfunctions and eigenvalues. 

Figure 3.15 The potential of a diatomic molecule AB as a function of the distance, shotild solve the one-electron Schrodinger equation for the AB molecule, namely...

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