Electronic wave function for the H2 molecule

In the H2 molecule the lowest energy electron configuration is obtained by placing both electrons in the aR s molecular orbital with their spins paired so as to satisfy the Pauli exclusion principle. The Slater determinant for this arrangement is 

If we now represent the erg l,v molecular orbital as a linear combination of the Is atomic orbitals on the two atoms, we can expand (6.86) as follows  

Note that, in this two-electron system, the spatial and spin parts of the wave function can be separated. 

The electronic Hamiltonian for H2, including the nuclear repulsion term is 

This Hamiltonian can be combined with the wave function given in (6.87) and, using the hydrogen atom l,v atomic orbitals once more, Coulson [19] obtained a calculated dissociation energy of 2.681 eV and an equilibrium intemuclear separation of R = 0.850 A, compared with the experimental values of 4.75 eV and 0.740 A. 

If we now represent the a ls molecular orbital as a linear combination of the I5 atomic 

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Kotos, W., Roothaan, C. C. J. (1960). Accurate electronic wave functions for the H2 molecule. 

The wave functions just described are one-electron wave functions, but the H2 molecule has two electrons to be dealt with. In the methods of molecular orbital theory, a wave function for the... 

All of the successful wave functions for the H2 molecule lead to the conclusion that there is a piling-up of electron density between the two nuclei. This is illustrated in Figure 1.12c. As a result of the high probability of finding an electron between the nuclei, the electron cloud forming the bond screens the nuclei from each other, thus reducing the repulsion between the nuclei. This high electron density between the nuclei is characteristic of the covalent bond. 

The other wavefunction from equation 12.43 is for the antibonding orbital. Because both electrons can be described with this spatial wavefunction, the spatial wave-function for the H2 molecule is the product of two such... 

Expansions such as eq. (4.76) are known as Hylleraas type wave functions. For the hydrogen molecule, it is possible to converge the total energy to 10 au, which is more accurate than what can be determined experimentally. In fact, the prediction that the experimental dissociation energy for H2 was wrong, based on calculations, was one of the first hallmarks of quantum chemistry. " Such wave functions unfortunately become impractical for more than 3-4 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 ... 

The simplest diatomic molecule consists of two nuclei and a single electron. That species, H2+, has properties some of which are well known. For example, in H2+ the internuclear distance is 104 pm and the bond energy is 268kJ/mol. Proceeding as illustrated in the previous section, the wave function for the bonding molecular orbital can be written as... 

Like Hund, Mulliken developed the basic Schrodinger equation in the direction of establishing the electron charge density resulting from a combination of the attractions of two or more nuclei and the averaged repulsions of other electrons in the system. This is a method that favors some particular region of space and disfavors others. In contrast to the Heitler-London method, it over-emphasizes, rather than underemphasizes, the ionic character of a molecule. For example, for the H2 molecule, Hund s wave function equation assumes that it is just as probable to have two electrons around the same nucleus as to have one electron around each nucleus. For a molecule made up of identical nuclei, this treatment is a considerable exaggeration of the ionic character of the molecule. 

Before discussing the problems of many-electron wave functions for molecules, it is instructive to consider the special cases of the Hj and H2 molecules, containing one and two electrons respectively. The electronic wave function for H2 can actually be calculated exactly within the Bom-Oppenheimer approximation, not analytically but by using series expansion methods an excellent description of the calculation has been given by Teller and Sahlin [17], and we give a summary of the method in appendix 6.1. We will, however, use the II2 and H2 molecules to illustrate the l.c.a.o. method in the next two subsections. 

This theory starts from Heitler and London s 1927-description of the electron-pair bond in the hydrogen molecule H2. The wave function for the electron-pair bond is the covalent function a( )h(2) + h( )a(2), where a and h are Is atomic orbitals on the two atomic centres. Spin is included by multiplying with the spin singlet... 

The one-electron molecular orbitals thus formed consist of a bonding molecular orbital (i/it) and an antibonding molecular orbital (i/ij. If we allow a single electron to occupy the bonding molecular orbital (as in H2, for example), the approximate wave function for the molecule is... 

An approximate wave function for a diatomic molecule is a product of LC AOMOs similar to those of hJ. The ground state of the H2 molecule corresponds to the electron configuration (ag Is). A wave function with a single configuration can be improved by adding terms corresponding to different electron configurations. This procedure... 

The first applications of natural orbitals were the factorization of the two electron wave function of He and H2. Iterative natural orbitals were used to develop accurate wave functions for many small molecules in ground and excited states. Natural orbitals provide a powerful tool for reducing wave functions expanded using an arbitrary set of molecular orbitals to a more compact canonical form that shows rapid convergence. Used in this way, they are a useful interpretive tool. 

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