The Hydrogen Molecule Molecular Orbitals

When two hydrogen atoms come together, they form a molecule, H2. In the molecule, two electrons move in the field of two protons (Fig. 1.10) here again we can get an approximate representation of the system in terms of individual one-electron functions or orbitals by averaging the interelectronic repulsions, i.e., by neglecting electron correlation. Each orbital represents the motion of an electron in the field of the nuclei and of a cloud 

Let us consider the form of the MOs of H2. Since electrons repel one another but are attracted by nuclei, we would expect the two electrons in H2 to correlate their motions in such a way that each tends to be near a different nucleus. When electron 1 is near nucleus A (Fig. la), electron 2 will be near nucleus B. The average distance rj2 between electron 1 and electron 2 will then be similar to the distance r 2 between electron 1 and nucleus B the attraction between electron 1 and nucleus B will then be balanced by the interelectronic repulsion. Electron 1 will then be moving in a field very similar to that for a single hydrogen atom consequently, the MO it occupies must resemble, in the neighborhood of nucleus A, an AO of atom A. Likewise, in the neighborhood of nucleus B, the MO must resemble an AO of nucleus B. This suggests that the MO ij/ may be represented approximately as a linear combination of the two AOs and (j)  

If we reflect the molecule in the plane of symmetry indicated above, we inter- 

We can arrive at this result in a different way. Consider what happens when two hydrogen atoms approach so that their Is AOs overlap (Fig. 1.10b). From the analogy between waves and wave functions, one might expect an interference effect analogous to that of the experiment indicated in Fig. 1.2. In other words, the orbitals should combine. When they do so, we may get constructive interference, the orbitals adding [c equation (1.13)], or we may get destructive interference, the orbitals subtracting [ct equation (1.14)]. The first situation is indicated in Fig. 1.10(c, d) the electron density is given by 

The extra electron density between the nuclei is clearly determined by the extent to which the AOs and (j) overlap in space, the extra density in some volume element dx being given [see equation (1.15)] by 20a Ab The total extent of this overlap cloud is given by summing the contributions by different volume elements dr by definition, this sum is equal to twice 

---

The hydrogen molecule molecular orbital and valence bond treatments... 

Dipole-dipole forces among HCl molecules Hydrogen bonds between water molecules Eondon forces between hydrogen molecules Molecular orbitals in the hydrogen molecule Molecular orbitals in a helium molecule ... 

As described previously for the hydrogen molecule, molecular orbital (MO) theory takes the atomic orbitals of the atoms, and mathematically combines the wave functions that represent these atomic orbitals (using an approach known as the linear combination of atomic orbitals). This combination produces new molecular orbitals that describe the regions of space occupied by the bonding electrons. The number of new molecular orbitals formed is the same as the number of atomic orbitals combined. The wave functions that represent the new molecular orbitals can be used to calculate the energy of an electron in those molecular orbitals. 

We will now describe the bonding in the hydrogen molecule using this model. The first step is to obtain the hydrogen molecule s orbitals, a process that is greatly simplified if we assume that the molecular orbitals can be constructed from the hydrogen I5 atomic orbitals. 

Tie hydrogen molecule is such a small problem that all of the integrals can be written out in uU. This is rarely the case in molecular orbital calculations. Nevertheless, the same irinciples are used to determine the energy of a polyelectronic molecular system. For an ([-electron system, the Hamiltonian takes the following general form ... 

We shall examine the simplest possible molecular orbital problem, calculation of the bond energy and bond length of the hydrogen molecule ion Hj. Although of no practical significance, is of theoretical importance because the complete quantum mechanical calculation of its bond energy can be canied out by both exact and approximate methods. This pemiits comparison of the exact quantum mechanical solution with the solution obtained by various approximate techniques so that a judgment can be made as to the efficacy of the approximate methods. Exact quantum mechanical calculations cannot be carried out on more complicated molecular systems, hence the importance of the one exact molecular solution we do have. We wish to have a three-way comparison i) exact theoretical, ii) experimental, and iii) approximate theoretical. 

Approximate Theoretical. The simplest molecular orbital problem is that of the hydrogen molecule ion (Pig KJ-3), is a preliminary example of all molecular orbital problems to come, w hich, although they may be very complicated, are elaborations on this simple example. 

Even though the problem of the hydrogen molecule H2 is mathematically more difficult than, it was the first molecular orbital calculation to appear in the literature (Heitler and London, 1927). In contrast to Hj, we no longer have an exact result to refer to, nor shall we have an exact energy for any problem to be encountered from this point on. We do, however, have many reliable results from experimental thermochemistry and spectroscopy. 

A very important difference between H2 and molecular orbital calculations is electron correlation. Election correlation is the term used to describe interactions between elections in the same molecule. In the hydrogen molecule ion, there is only one election, so there can be no election correlation. The designators given to the calculations in Table 10-1 indicate first an electron correlation method and second a basis set, for example, MP2/6-31 G(d,p) designates a Moeller-Plesset electron coiTclation extension beyond the Hartiee-Fock limit canied out with a 6-31G(d,p) basis set. 

Asimple example is the formation of the hydrogen molecule from two hydrogen atoms. Here the original atomic energy levels are degenerate (they have equal energy), but as the two atoms approach each other, they interact to form two non degenerate molecular orbitals, the lowest of which is doubly occupied. 

The most important molecular- orbitals are the so-called frontier molecular- orbitals. These are the highest (energy) occupied molecular- orbital (HOMO), and lowest (energy) unoccupied molecular- orbital (LUMO). The following picture shows the LUMO surface for the hydrogen molecule, H2. The LUMO consists of two separate surfaces, a red... 

Coulson, C. A., and Fischer, I., Phil. Mag. 40, 386, Notes on the molecular orbital treatment of the hydrogen molecule."... 

Wallis, R. F., J. Chem. Phys. 23, 1256, "Molecular orbitals for the hydrogen molecule ground state." In-out effect for H2. 

To get the molecular orbital of the hydrogen molecule, the orbital equations of the two atoms are combined. When the orbital equations are added together, the result is a bonding molecular orbital that extends over both atoms. Subtracting the orbital equations of the atoms produces an antibonding molecular orbital. This process is called the linear combination of atomic orbitals or LCAO. 

Wave functions for the orbitals of molecules are calculated by linear combinations of all wave functions of all atoms involved. The total number of orbitals remains unaltered, i.e. the total number of contributing atomic orbitals must be equal to the number of molecular orbitals. Furthermore, certain conditions have to be obeyed in the calculation these include linear independence of the molecular orbital functions and normalization. In the following we will designate wave functions of atoms by % and wave functions of molecules by y/. We obtain the wave functions of an H2 molecule by linear combination of the Is functions X and of the two hydrogen atoms ... 

Figure 1.10 Energy diagram for the hydrogen molecule. Combination of two atomic orbitals, is, gives two molecular orbitals, 0iec and iF moiec- The energy of Iffnoiec is lower than that of the separate atomic orbitals, and in the lowest electronic state of molecular hydrogen it contains both electrons.

In order to describe the hydrogen molecule by quantum mechanical methods, it is necessary to make use of the principles given in Chapter 2. It was shown that a wave function provided the starting point for application of the methods that permitted the calculation of values for the dynamical variables. It is with a wave function that we must again begin our treatment of the H2 molecule by the molecular orbital method. But what wave function do we need The answer is that we need a wave function for the H2 molecule, and that wave function is constructed from the atomic wave functions. The technique used to construct molecular wave functions is known as the linear combination of atomic orbitals (abbreviated as LCAO-MO). The linear combination of atomic orbitals can be written mathematically as... 

In this expression, the term hydrogen atoms A and B. The term interaction with the electrons interchanged. However, the term A2 represents both electrons 1 and 2 interacting with nucleus A. That means the structure described by the wave function is ionic, HA HB+. In an analogous way, the term B1 Bj2 represents both electrons interacting with nucleus B, which corresponds to the structure HA+ Hb . Therefore, what we have devised for a molecular wave function actually describes the hydrogen molecule as a "hybrid" (a valence bond term that is applied incorrectly) of... 

Molecular orbitals are generated by combining atomic orbitals. The number of molecular orbitals formed is always equal to the number of atomic orbitals that combine. So, if two atomic orbitals combine, then two molecular orbitals will be formed. This is the case when two hydrogen Is atomic orbitals combine to produce two molecular orbitals in a hydrogen molecule (H ). 

In the MO approach molecular orbitals are expressed as a linear combination of atomic orbitals (LCAO) atomic orbitals (AO), in return, are determined from the approximate numerical solution of the electronic Schrodinger equation for each of the parent atoms in the molecule. This is the reason why hydrogen-atom-like wavefunctions continue to be so important in quantum mechanics. Mathematically, MO-LCAO means that the wave-functions of the molecule containing N atoms can be expressed as... 

---