Example the hydrogen molecule

First we briefly discuss the simplest possible 1-electron 2-centre system, the hydrogen molecule ion H2, in order to Ulustrate the nature of the problem. The one-electron eigenvalue equation is now (using lower-case 

Here ra and are the distances of the electron from the two nuclei a and b. Equation (1.3.1) was first solved, with high accuracy, by Burrau (1927), who transformed to confocal elliptic coordinates in which the equation is separable, the solution becoming a product of three factors tlrat may be obtained by solution of three separate differential equations. Solutions in this way is not generally possible, and we pass directly to the construction of simpler approximations. 

We note that if the electron is close to nucleus a then the l/rg term in (1.3.2) will be so large that l/rt may be neglected by comparison the Hamiltonian operator is then quite close to that for one electron near nucleus a alone, i.e. for a single hydrogen atom. If we use a(r) for the Is orbital centred on nucleus a and 6(r) for that centred on nucleus b, it is therefore reasonable to expect that 

since the two nuclei are identical, we expect that 0(r)p will be completely symmetrical across a plane cutting the system into two identical halves there is no reason to expect a higher probability of finding the electron at one end of the molecule than the other. But the required symmetry of 

The normal molecule H2, in which two electrons move in the field of the two nuclei, may now be discussed in essentially the same way as the helium atom in Section 1.2. In the absence of electron interaction, an approximate spatial wavefunction for the state of lowest energy would be 

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We can extend the Lewis symbols introduced in Section 2.2 to describe covalent bonding by using a line (—) to represent a shared pair of electrons. For example, the hydrogen molecule formed when two H- atoms share an electron pair (H=H) is represented by the symbol H—H. A fluorine atom has seven valence electrons and needs one more to complete its octet. It can achieve an octet by accepting a share in an electron supplied by another atom, such as another fluorine atom ... 

Now let us look at the coupled fermions. Fillaux considers, as an example, the hydrogen molecule and argues that a similar pattern should take place for the oscillators in KHCO3. The vibrational ground-state wave functions are... 

As usual, let us consider the simplest example the hydrogen molecule. The normalized Hartree-Fock determinant,... 

There are many compounds in existence which have a considerable positive enthalpy of formation. They are not made by direct union of the constituent elements in their standard states, but by some process in which the necessary energy is provided indirectly. Many known covalent hydrides (Chapter 5) are made by indirect methods (for example from other hydrides) or by supplying energy (in the form of heat or an electric discharge) to the direct reaction to dissociate the hydrogen molecules and also possibly vaporise the other element. Other known endothermic compounds include nitrogen oxide and ethyne (acetylene) all these compounds have considerable kinetic stability. 

An acid was once defined simply as a substance which produces hydrogen ions, or protons. However, the simple proton, H , is never found under ordinary conditions, and this definition required amendment. Bronsted and, independently, Lowry, therefore redefined an acid as a susbstance able to donate protons to other molecules or ions, and a base as a substance capable of accepting such protons. If we consider hydrogen chloride, HCl, as an example, the HCl molecule is essentially covalent, and hydrogen chloride (gas or liquid) contains no protons. But anhydrous hydrogen chloride in benzene will react with anhydrous ammonia ... 

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. 

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 simplest example of covalent bonding is the hydrogen molecule. The proximity of the two nuclei creates a new electron orbital, shared by the two atoms, into which the two electrons go (Fig. 4.5). This sharing of electrons leads to a reduction in energy, and a stable bond, as Fig. 4.6 shows. The energy of a covalent bond is well described by the empirical equation... 

The number and kinds of atoms in a molecule can also be shown in a molecular formula. For example, the water molecule is symbolized H20. In this molecular formula, H means hydrogen atom, O means oxygen atom, and the subscript 2 following H indicates there are two hydrogen atoms bound to the single oxygen atom. The molecular formula of ammonia, NH3, indicates that one molecule of ammonia contains one atom of nitrogen (N) and... 

The presence of intermolecular forces also accounts for the variation in the compression factor. Thus, for gases under conditions of pressure and temperature such that Z > 1, the repulsions are more important than the attractions. Their molar volumes are greater than expected for an ideal gas because repulsions tend to drive the molecules apart. For example, a hydrogen molecule has so few electrons that the its molecules are only very weakly attracted to one another. For gases under conditions of pressure and temperature such that Z < 1, the attractions are more important than the repulsions, and the molar volume is smaller than for an ideal gas because attractions tend to draw molecules together. To improve our model of a gas, we need to add to it that the molecules of a real gas exert attractive and repulsive forces on one another. 

This resembles Eq. (1.1), but here each j/ represents a wave equation for an imaginary canonical form and each c is the amount contributed to the total picture by that form. For example, a wave function can be written for each of the following canonical forms of the hydrogen molecule " ... 

In some cases the excited state is entirely dissociative (Fig. 7.3), that is, there is no distance where attraction outweighs repulsion, and the bond must cleave. An example is the hydrogen molecule, where a ct 0 promotion always results in cleavage. 

As a final comment, it is interesting to note that this FS(K) study of the hydrogen molecule offers a new and simple illustration of the behavior of sophisticated Hartree-Fock schemes like UHF, PHF and EHF. Furthermore, it provides a very efficient numerical example of instabilities in the standard Hartree-Fock method. It is important to see that the UHF, PHF and EHF schemes all correct the wrong RHF behavior and lead to the correct dissociation limit. However, the UHF and PHF schemes only correct the wave function for large enough interatomic distances and the effect of projection in the PHF scheme even results in a spurious minimum. The EHF scheme is thus the only one which shows a lowering of the energy with respect to RHF for all interatomic distances. 

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