Chapter 7

It may lead to a change in some of the bond lengths which may be readily absorbed into by changing the reference geometry. Thus a non-degenerate state is stable with respect to a distortion which lowers the symmetry. More interesting is the case for degenerate , because an energy lowering by distortion is possible.
The first order Jahn-Teller theorem is categorized by two terms in the potential energy expansion:

The first term gives the direction of energy curvature with respect to the distortion coordinate, q. The second term gives the slope of this curvature.
Lets take a very simple system H₃-. A Walsh diagram for
distortion from D_3h is shown below:



Chapter VII. Molecular Orbitals and Geometric Perturbation

	In this chapter we will do basically two things - examine in some detail the form of the molecular orbitals for an AH₂ molecule and secondly find out how to predict the form and energetic consequences of molecular orbitals when a molecule is distorted via geometric perturbation theory A. Linear AH₂Molecules The orbitals of D_h AH₂molecules are extremely easy to build up in an orbital interaction diagram:

		The resultant MOs from intermolecular perturbation theory are:



	B. The C_2v AH₂ Molecule Likewise we get (now there is a three orbital pattern to worry about)without too much more difficulty:


		Notice that we have set up the phases so that:



		A contour plot of the MOs for H₂O are shown below:



	C. Orbital Correlations Let us now figure out how the MOs of D_h AH₂ correlate to (evolve into) those of the C_2v geometry. This is shown below were it is important to remember the way the AO overlap runs for:


		This is called an orbital correlation diagram:



	D. Electronegativity considerations We have put the electronegativity of A to be similar to that of H for each of these cases. Clearly for H₂O this is not correct, nor is it for H₂Be. Three general cases are shown below:



For D_h AH₂ one can easily construct interaction diagrams for each case. However, it is more illustrative to use electronegativity perturbation theory:



		We will use this result a little later. Notice that the form of the MOs does not change at all.



	Likewise for the C_2v AH₂ system:



	Here the interactions can be described as: case A 3a₁ µ z - () + [s] 2a₁ µ s - () - [z] 1a₁ µ + (s) + (z) case B 3a₁ µ z - () + [s] 2a₁ µ - (s) + (z) 1a₁ µ s + () + [z] case C 3a₁ µ - (s) - (z) 2a₁ µ z + () - [s] 1a₁ µ s + () + [z]. There is a kind of avoided crossing between the 1a₁ and 2a₁ molecular orbitals in terms of their composition on going from case A to case B. Likewise, an analogous crossing occurs between 3a₁ and 2a₁ for going from case B to case C. These are indicated by the dashed lines. E. Geometrical Perturbations Suppose that a molecule undergoes a geometrical distortion. We can examine how the molecular orbitals of one geometry are related to those of another by considering the structural distortion as a perturbation. The molecular geometries before and after distortion are described as the unperturbed and perturbed structures, respectively. Given any two geometries of a molecule, either one may in principle be considered as the perturbed one. In practice, however, it leads to a substan



		tially simpler analysis if the molecular geometry of the lower symmetry is chosen as the perturbed one. Group theory tells us in a simple, straightforward way how to construct the molecular orbitals of molecules in a highly symmetric geometry. Geometrical perturbation theory can be used to see how the shapes and energies of the molecular orbitals change upon going to a less symmetric (perhaps the experimental) geometry. The relevant perturbation theory formulae are:


			The changes in the overlap and resonance integrals are defined as


			Let us first figure out what happens to the overlap for each MO during the D_h to C_2v distortion:

So is there mixing between 1s_u and 2s_u after the distortion takes place?

= -S + S 0.

This is a general phenomena, when two orbitals have the same symmetry in the unperturbed molecule then they are orthogonal and so there is, of course, little overlap that is turned on. But what happens when two MOs have different symmetry in the unperturbed structure and upon geometrical distortion now have the same symmetry?



			The overlap turned on between the one component of p_u and 1s_g along with 2s_g is not zero. We have chosen phases so that in each case it is positive. The geometric perturbation construction of this distortion is therefore given by:



			What may seem as ambiguous in this diagram is the behavior of 2a₁. There are two competing interactions. We have drawn it so that it is stabilized. Is this always true? Why don't the two effects cancel? We have previously looked at the way electronegativity works in D_h. From this we have:





				Thus in all cases:


			F. Walsh Diagrams These are calculated orbital energies of a molecule versus a distortion coordinate. An example for H₂S is shown below:



		A major function of a Walsh diagram is to account for the structural regularity observed for a series of related molecules with the same number of valence electrons, and to see how molecules change structure with the number of electrons or spin state. Walsh's original rule for predicting molecular shapes may simply be stated as follows: A molecule adopts the structure that best stabilizes the HOMO. If the HOMO is unperturbed by the structural change under consideration, the occupied MO lying closest to it governs the geometrical preference. Let us illustrate Walsh's rule by examining the shapes of AH₂ molecules based upon the Figure . The HOMO of a four- electron AH₂ molecule is , and this orbital is destabilized on bending so that BeH₂ is linear. The 2a₁ orbital of AH₂ lies lower in a bent structure than 1p_uwhile b₁ of AH₂ is ener- getically unaffected by the D_·h Æ C_2v distortion. Conse- quently, the shape of AH₂ molecules with five to eight elec- trons is governed by the energetics of 2a₁. Thus BH₂, CH₂, NH₂, and H₂O all adopt a C_2v structure. Typical Bond Angles in AH₂ Molecules molecule electronic configuration H-A-H bond angle BeH₂ ()²()² 180° BH₂ (1a₁)²(1b₂)² (2a₁)¹ 127° CH₂^a (1a₁)²(1b₂)² (2a₁)¹(b₁)¹ 134° CH₂^b (1a₁)²(1b₂)² (2a₁)² 102° NH₂ (1a₁)²(1b₂)² (2a₁)²(b₁)¹ 103° OH₂ (1a₁)²(1b₂)² (2a₁)²(b₁)² 104° MgH₂ ()²()² 180° AlH₂ (1a₁)²(1b₂)² (2a₁)¹ 119° SiH₂^a (1a₁)²(1b₂)² (2a₁)¹(b₁)¹ 118° SiH₂^b (1a₁)²(1b₂)² (2a₁)² 93° PH₂ (1a₁)²(1b₂)² (2a₁)²(b₁)¹ 92° SH₂ (1a₁)²(1b₂)² (2a₁)²(b₁)² 92° ^aFor the ³B₁ state. ^bFor the ¹A₁ state.

G. Jahn-Teller Distortions

The ideas of Jahn and Teller have had a strong influence on the way both molecular and solid state structures are viewed in electronic terms. Their initial ideas were centered around the geometric stability of molecules and ions in solids described by degenerate electronic states, but the approach has been taken further by others. The Jahn-Teller theorem (we shall see later that strictly this should be called the first order Jahn-Teller theorem) is often stated in the following way. An orbitally degenerate electronic state of a non-linear molecule is unstable with respect to a distortion which removes the degeneracy. The theory Jahn and Teller derived in fact specified the symmetry species of the 'Jahn-Teller active' vibration and thus the possible structures of the distorted molecule or ion

The energy of the electronic ground state () can be expanded as a function of some distortion coordinate q,

1. FIRST-ORDER JAHN-TELLER DISTORTION

Consider initially the first order term,

Since the Hamiltonian operator must be totally symmetric, for this term to be nonzero the symmetry representation of q, G_q, must be contained in the symmetric direct product of G_Yi (=G), i.e., G_YiG_Yi(=G_iG_i). In all point groups direct products of non-degenerate representations lead to the totally symmetric representation. A totally symmetric vibration does not change the point group of the molecule.



	the Walsh diagram for the equilateral triangle to linear or isosceles triangle (i.e., C_2v ¨ D_3h Æ D_·h) distortion in a simple three-center system. It predicts that the two-electron molecule H₃⁺ should be an equilateral triangle while a four-electron one, H₃^-, should be linear or an isosceles triangle. In a D_3h structure, the HOMO of H₃^- is doubly degenerate and half-filled. The degeneracy is lifted by the D_3h Æ D_h or D_3h Æ C_2v distortions since it stabilizes one component of the e' set in each of the two directions. Throughout the D_3h Æ D_·h distortion the overlap of le'_x with le'_z or la'_l vanishes, so that the stabilization of the le'_x level is caused by a first-order energy change which results from decreasing the extent of antibonding interactions in the le'_x orbital. The same is true for the distortion to the C_2v structure. This instability of D_3h H₃^- is an example of a first-order Jahn-Teller distortion. Now let us examine this from an electronic state perspective. There are two electrons in the 1e' set for the D_3h H₃^- molecule so the possible electronic states are ³A₂', ¹A₁' and ¹E'. Consider first the two A states. The symmetric direct product of a₁' or a₂' leads to a₁', so that both ³A₂' and ¹A₁' states are stable with respect to a symmetry-lowering distortion. The symmetric direct product of e' leads to a₁' + e', so a vibration of e' symmetry may lower the energy of the molecule.

D_3h E 2C₃ 3C₂ s_h 2S₃ 3s_v A'₁ 1 1 1 1 1 1 A'₂ 1 1 -1 1 1 -1 E' 2 -1 0 2 -1 0 A"₁ 1 1 1 -1 -1 -1 A"₂ 1 1 -1 -1 -1 1 E" 2 -1 0 -2 1 0
		So what is a distortion of e' symmetry? The Spectroscopy core course - CHM 6314 - covers this in great depth. What I will do is give you the results for common structures. These are complied together in Appendix I. Be sure to download a copy. The e' modes for D_3h H₃ are:





		The e' modes in fact create the D_3h Æ D_h and D_3h Æ C_2v distortions. The potential energy surface for H₃^- (and many other molecules with a C₃ axis present in them and this electronic state problem) is shown below in two different views. This is called a "monkey saddle" potential energy surface.



		So the "polytopal" rearrangement of H₃^- is very complicated and looks like:



	2. SECOND-ORDER JAHN-TELLER DISTORTION The distortion term for this component is given by:


	This term is always negative (i.e., stabilizing) since represents the ground electronic state. One can see that the stabilization occurs as a result of mixing an excited state () into the electronic ground state. The sign of is thus set by the relative magnitudes of these two terms. The energy gap expression appears in the denominator, so that states lying close to the ground state will be the most important for the energy stabilization. If the lowest lying one of these is of the correct symmetry for to be nonzero (G_q = G_i G_j), then the lead term in this expansion



		can become important and may become negative. As a result the system will now distort away from the symmetrical structure along the coordinate (G_q) whose symmetry is set by this symmetry prescription. The second order Jahn-Teller effect couples electronic states which arise from different electron configurations. Because the symmetry species of the states and are determined by the symmetry of the orbitals which are occupied, the problem often reduces to an orbital rather than state picture. Thus if the HOMO and LUMO are non-degenerate, then the symmetry species of q is given by G_q = G_HOMO G_LUMO. This makes the method easy to apply. Let us take an eight electron AH₂ molecule as an example at the linear, geometry. There are four electrons in the HOMO, 1p_u, and the LUMO is . There is a distortion coordinate then of symmetry G_q = = . The p_u mode is given by:


		So provided that there is a small 1p_u - energy gap, then there will be a sizable driving force to distort from D_h Æ C_2v.



		Here is some real data for 8 valence electron molecules: AH₂ angle AH₂ angle OH₂ 104.5° NH₂^- 104° SH₂ 92.1° OH₂ 104.5° SeH₂ 90.6° FH₂⁺ 118.1° TeH₂ 90.3°
	There are a number of trends here:





	In going down a column in the Periodic Table the AH bond length increases with decreasing the electronegativity of A (e.g., r_AH = 0.956, 1.328, 1.460 and 1.653Å for H₂0, H₂S, H₂Se and H₂Te, respectively).This is also a reflection of the fact that, with increasing the principal quantum number n, the np atomic orbital of A has maxima at larger distances from the nucleus. The overlap of O 2p, S 3p, etc. with H 1s is fairly constant at their optimal distances. However, the ns orbital becomes increasingly more contracted than np as one proceeds down the sixth column. This leads to a smaller overlap between the ns AO and the H 1s combination. Thus, the antibonding in



	The PE spectrum of water is quite different:



	Three ionizations are observed with adiabatic ionizations potentials of 12.6. 13.8 and 17.0 eV. These correspond to the b₁, 2a₁ and 1b₂ molecular orbitals, respectively. The ionization required for the 1a₁ MO is at 32.2 eV. With He(I) radiation one can obtain a photoelectron spectrum with ionization potentials less than 21 eV. Note that the 2a₁ and b₁ (n_s and n) nonbonding levels are separated by 1.2 eV or about 28 kcal/mol; they are in no way close to being degenerate. Analysis of the vibrational progression for the b₁ ionization has lead to a determination that in the ²B₁ state the O-H bond lengths increase by 0.08Å and the H-O-H bond angle increases by 4.4°. These small changes are fully consistent with the identification of b₁ being a purely nonbonding molecular orbital. The 2a₁ MO is also nonbonding, but referring back to the Walsh diagram, removal of an electron from this orbital should cause the H-O-H bond angle to increase. The intricate fine structure associated with this ionization has been used to propose a linear geometry for the ion with a very small potential required to bend it.The bond angles in BH₂, ³B₁ CH₂ and ²A₁ NH₂ are 127°, 134° and 144°, respectively. These compounds have the electronic configuration ···(1a₁)¹(b₁)^x, where x = 0, 1 and 2 (in the latter case this is isoelectronic to the ²A₁ ion of H₂O⁺). The



		occupation of the b₁ MO should play a small role in setting the geometry. Given the discussion previously about the effect of the electronegativity on the geometry and slope of 2a₁, it is clear that the bond angle in the ²A₁ state of H₂O⁺ should be greater than 144°. The 1b₂ MO is O-H bonding via a p AO at O. Removal of an electron from this MO should cause the O-H bond distance to increase and the H-O-H bond angle to decrease, the 1b₂ orbital rises in energy on bending). This appears to be the case from the analysis of the vibrational fine structure. The O-H distance is thought to increase by ~0.2Å and the H-O-H bond angle to decrease by ~18°. It is clear that the gross, as well as, fine details associated with the photoelectron spectrum of H₂O are fully consistent with the delocalized picture of bonding. The ionization potentials for the group 18 dihydrides are shown below:



	The VB approach can be used but we must make sure that the bond orbitals have the full symmetry that the molecule (really the Hamiltonian) has. To do this one needs to take linear combinations of all equivalent bond orbitals. Starting with the lone pairs:


	The combinations when simplified become nothing more than the LCAO ones. The O-H bonds can similarly be combined to produce: