The H Molecule

To understand the bonding in a hydrogen molecule, we have to see what happens when two hydrogen atoms are close enough for their atomic orbitals to interact. We now have two protons and two nuclei, and even with this small a molecule we cannot expect theory to give us complete solutions. We need a description of the electron distribution over the whole molecule—a molecular orbital. The way the problem is handled is to accept that a first approximation has the two atoms remaining more or less unchanged, so that the description of the molecule will resemble the sum of the two isolated atoms. Thus we combine the two atomic orbitals in a linear combination expressed in Equation 1.1, where the function which describes the new electron distribution, the molecular orbital, is called a and (j and p2 are the atomic Is wave functions on atoms 1 and 2. 

The coefficients, c , and c2, are a measure of the contribution which the atomic orbital is making to the molecular orbital. They are of course equal in magnitude in this case, since the two atoms are the same, but they may be positive or negative. To obtain the electron distribution, we square the function in Equation 1.1, which is written in two ways in Equation 1.2. 

The mathematical description of the effect of overlap on the electronic energy is complex, but some of the terminology is worth recognising, and will be used from time to time in the rest of this book. The energy E of 

The detailed form that a and (3 take is where the mathematical complexity appears. They come from the Schrodinger equation, and they are integrals over all coordinates, represented here simply by dx, in the form of Equations 1.6 and 1.7  

We can take away the sign changes in the wave function by plotting a2 along the intemuclear axis, as in Fig. 1.7. The solid lines are the plots for the molecular orbitals, and the dashed lines are plots, for comparison, of the undisturbed atomic orbitals o2. The electron population in the bonding orbital (Fig. 1,7a) can be seen to be slightly contracted relative to the sum of the squares of the atomic orbitals, and the electron population 

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The probability given by Eq. (2) is a function of an enormous number of variables. We can neither compute nor display such a function. The most with which we can deal are functions of the coordinates of one, two, three, or, at the outside, four molecules. It takes six variables to specify the positions of four molecules. Therefore, it is helpful to integrate over the positions of most of the molecules. The h molecule distribution function is given by... 

Describe the pholoeleclron spectrum of the H-, molecule and indicate what information can be gained about the H-,+ ion. 

Fig. IX-4.- Modes ot vibration of the H/> molecule The arrows indicate the direction of vibration ot each atom, for the normal mode whose characteristic temperature ip indicated. For similar information on a variety of molecules, see H. Spoilcr, Molekulspek-tren mid ihro An wend ungen attf chcmische Probleme, Springer, Berlin, 1935.

For molecules in which atoms A and B are identical, ft c n (see p e I4l for the H, molecule), and the contribution of the ionic structures is small. If B is more electronegative than A. then the energy of the contributing structure A B approaches more nearly that of ihe purely covalent structure A—B and resonance is enhanced. On the other hand, the energy of is so prohibitively high that this... 

The simplest model to describe a a as well as a a bond is the two-electron two-orbital model, which will be used frequently in the remainder of the text. The MO and the VB treatment of the H, molecule may serve as an example. In the... 

Molecules containing one to four electrons. We have already seen the H, molecule in which there are two electrons in the (r, orbital. Two bonding electrons constitute a chemical bond. The molecular orbital theory does not restrict itself to even numbers of bonding electrons, and so the bond order is given as one-half the difference between the number of bonding electrons and the number of antibonding electrons ... 

Since the function i/t in equation 3.11 cannot be determined, the approximate solution for the case of the H + molecule viz... 

A particnlarly clear example of Himd s case (c) coupling has been observed for the HeAr+ ion in its near-dissociation vibration rotation levels [58] it also occurs for the h molecule [59]. 

Symmetry-breaking arising from the presence of nuclear spin interactions, even in closed shell molecules, was discussed by Herzberg [267] and observed for the h molecule by Pique, Hartmann, Churassy and Bads [268]. As Bunker and Moss [265] point out, it will be most important for homonuclear open shell molecules with large... 

The H+ molecule is easily formed by electron bombardment of H2, being formed by the ion-molecule reaction... 

Since the accuracy of wavelength stabilization increases with decreasing molecular linewidth, spectroscopists have looked for particularly narrow lines that could be used for extremely well-stabilized lasers. It is very common to stabilize onto a hyperfine component of a visible transition in the h molecule... 

One striking difference between the pn orbitals of the cyclic polyene and those of the H molecules is that the symmetry labels are different. For example, the pir levels of cyclopropenyl transform as aS + e" (double primes since they are anti-83nmnetric with respect to reflection in the plane perpendicular, to the threefold rotation axis) but the 5 orbitals of H3 transform as a[ e (slnj primes since these orbitals are symmetric with respect to this symmetry operation). The breakdown into nondegenerate and degenerate orbitals (one of eadi) is the same in both cases. 

Section 7-5 Linear Combination of Atomic Orbitals The H+ Molecule-Ion... 

Since the accuracy of wavelength stabilization increases with decreasing molecular linewidth, spectroscopists have looked for particularly narrow lines that could be used for extremely well-stabilized lasers. It is very eommon to stabilize onto a hyperfine eomponent of a visible transition in the h molecule using Doppler-free saturated absorption inside [5.72] or outside [5.73] the laser resonator (Vol. 2, Seet. 2.3). The stabilization record was held for a long time by a HeNe laser at A, = 3.39 (xm that was stabilized onto a Doppler-free infrared transition in CH4 [5.74,5.75]. 

Two paths suggest themselves, illustrated in Scheme 1 the H-H bond in the gas could be broken, and the hydrogen atoms bound to some other molecule. Then, under the right set of conditions, the H-molecule bond is broken and the H-H bond is reformed. The other option is that the H-H bond could be left intact, but the molecule allowed to interact with another atom or molecule via a weak overlap of orbitals. This generally suggests a substrate or surface, and on the surface H atoms could be nearer to each other than Pauli repulsion would allow in the gas phase. 

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