Wave functions hydrogen molecule

Fig. 42—2.—Energy curves for the hydrogen molecule-ion (in units e2/2ao), calculated for undistorted hydrogen atom wave functions.

Let us suppose that for a given problem we have selected a certain set of wave functions which are known to represent the system in some extreme configuration. For the hydrogen molecule we might choose a set of hydrogen-atom wave functions which would represent... 

McLean and Yoshimine 58, 59) published an extensive hst of Tables of Linear Molecule Wave Functions which include the diatomic molecules e.g. the hydrogen halides, the oxides (e.g. BeO, MgO), the alkali halides, PN, as well as HCN, HFi, C02, N2O, SCN, C2H2, each for a number of intemuclear configurations. These Tables represent the best ... 

A further simplification is made. The wave functions pi and p2, which are orthogonal and normalized in the hydrogen atom, are assumed to retain their orthonormality in the molecule. Orthonormality requires that... 

Valence bond and molecular orbital theory both incorporate the wave description of an atom s electrons into this picture of H2 but m somewhat different ways Both assume that electron waves behave like more familiar waves such as sound and light waves One important property of waves is called interference m physics Constructive interference occurs when two waves combine so as to reinforce each other (m phase) destructive interference occurs when they oppose each other (out of phase) (Figure 2 2) Recall from Section 1 1 that electron waves m atoms are characterized by their wave function which is the same as an orbital For an electron m the most stable state of a hydrogen atom for example this state is defined by the Is wave function and is often called the Is orbital The valence bond model bases the connection between two atoms on the overlap between half filled orbifals of fhe fwo afoms The molecular orbital model assembles a sef of molecular orbifals by combining fhe afomic orbifals of all of fhe atoms m fhe molecule... 

Consider what happens to the many-electron wave function when two electrons have identical coordinates. Since the electrons have the same coordinates, they are indistinguishable the wave function should be the same if they trade positions. Yet the Exclusion Principle requires that the wave function change sign. Only a zero value for the wave function can satisfy these two conditions, identity of coordinates and an antisymmetric wave function. Eor the hydrogen molecule, the antisymmetric wave function is a(l)b(l)-... 

If a covalent bond is broken, as in the simple case of dissociation of the hydrogen molecule into atoms, then theRHFwave function without the Configuration Interaction option (see Extending the Wave Function Calculation on page 37) is inappropriate. This is because the doubly occupied RHFmolecular orbital includes spurious terms that place both electrons on the same hydrogen atom, even when they are separated by an infinite distance. 

T vo main streams of computational techniques branch out fiom this point. These are referred to as ab initio and semiempirical calculations. In both ab initio and semiempirical treatments, mathematical formulations of the wave functions which describe hydrogen-like orbitals are used. Examples of wave functions that are commonly used are Slater-type orbitals (abbreviated STO) and Gaussian-type orbitals (GTO). There are additional variations which are designated by additions to the abbreviations. Both ab initio and semiempirical calculations treat the linear combination of orbitals by iterative computations that establish a self-consistent electrical field (SCF) and minimize the energy of the system. The minimum-energy combination is taken to describe the molecule. 

Mulliken, R. S., Proc. Natl. Acad. Sci. U.S. 38, 160, "A comparative survey of approximate ground sate wave functions of helium atom and hydrogen molecule/ ... 

Callen, E., J. Chem. Phys. 23, 360, Configuration interaction applied to the hydrogen molecule." A useful table A list of investigations of the wave functions and binding energies of the H2" is also contained. 

Hydrogen molecule, carbon oxide intramolecular energy, 110 clathrates, 12, 20 correlated wave function, 300... 

The original VB wave function was introduced in the treatment of the hydrogen molecule by Heitler and London in 1932. This treatment considered only the one Is orbital on each hydrogen atom and assumed that the best wave function for a system of two electrons on two different atoms is a product of the two atomic Is orbitals i/ — XisXis- This wave function needs to be modified, however, to accommodate the antisymmetry of the wave function and to take into account the spin of the two electrons. 

In order to include the spin of the two electrons in the wave function, it is assumed that the spin and spatial parts of the wave function can be separated so that the total wave function is the product of a spin and a spatial wave function F — iAspace sp n Since our Hamiltonian for the H2 molecule does not contain any spin-dependent terms, this is a good approximation (NB—the complete Hamiltonian does contain spin-dependent terms, but for hydrogen they are rather small and do not appreciably affect the energetics of chemical bonding). For a two-electron system it turns out that there are four possible spin wave functions they are ... 

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