Walsh’s rule

The most widely used qualitative model for the explanation of the shapes of molecules is the Valence Shell Electron Pair Repulsion (VSEPR) model of Gillespie and Nyholm (25). The orbital correlation diagrams of Walsh (26) are also used for simple systems for which the qualitative form of the MOs may be deduced from symmetry considerations. Attempts have been made to prove that these two approaches are equivalent (27). But this is impossible since Walsh s Rules refer explicitly to (and only have meaning within) the MO model while the VSEPR method does not refer to (is not confined by) any explicitly-stated model of molecular electronic structure. Thus, any proof that the two approaches are equivalent can only prove, at best, that the two are equivalent at the MO level i.e. that Walsh s Rules are contained in the VSEPR model. Of course, the transformation to localised orbitals of an MO determinant provides a convenient picture of VSEPR rules but the VSEPR method itself depends not on the independent-particle model but on the possibility of separating the total electronic structure of a molecule into more or less autonomous electron pairs which interact as separate entities (28). The localised MO description is merely the simplest such separation the general case is our Eq. (6)... 

Problem 9-15. Could the same predictions be made from a simple electron repulsion argument If n pairs of electrons must be accommodated in the valence-shell molecular orbitals, then assume simply that they will be as far apart as possible. Up to four electrons will push each other as far apart as possible, to create linear geometry more than four must be distributed more densely, so that the angle between the substituents will be less that 180°. Does this simple hypothesis explain everything that Walsh s rules do Is there any advantage to using Walsh s correlation diagram analysis ... 

Before entering into a discussion of the assignments and analysis of the observed band structure it is necessary to return to a consideration of the molecular geometry. Recall that Walsh s rules predicted that the electron occupation of the 71 orbital would render the Si state nonplanar. Such a conformation would have two distinct equilibrium configurations that are related to each other... 

A discussion of Walsh s rules and the first-row hydrides has been given by Wasserman now that quantitative calculations have been carried out.439 These rules have also been reviewed by Buenker11 and further studied by Davidson et a/.405... 

C. AB2 Molecules.—In the case of AB2 molecules, the number of electrons in the system is now 30—40 for some of the molecules containing second-row atoms consequently these species present a much greater computational problem if high accuracy is to be achieved. Nevertheless, much useful information can be obtained from wavefunctions at the SCF level, or including limited Cl, and the past four years have seen a large amount of effort in this area. An extensive discussion of Walsh s rules, including their application to AB2 molecules, has already been referred to.11... 

According to the number of valence electrons, Walsh s rules predict bent structures for SiH2(6), SiH(Halogen) (12) and Si(Halogen)2 (18 electrons) silylenes. 

The intermediate of the 0+ + N2 reaction, N20+, has 15 valence electrons, and thus it is assumed to have a linear configuration (N-N-0+) in accordance with Walsh s rules [97] for a 15 valence electron triatomic. Experimentally the ground state of N20+ is known to be 2n, and its first excited state, lying at 3.45 eV, to be 2 + [174], A second excited state with no symmetry assigned had been reported at 7.19 eV [6]. McLean and Yoshimine at IBM had done accurate LCAO-MO-SCF calculations on the parent system N20 itself [57]. The bond lengths of ground state N20+ and N20 are quite similar [174],... 

Since one expects, for bound molecules, that the chemical potential p will be negative, the fact that J(Vp)2jp dr is a positive quantity shows from equation(120) that deviations from relation (84) of the simplest density treatment can, in principle, be of either sign, depending on the relative magnitudes of the chemical potential and density gradient corrections. We shall discuss numerical values for the deviation from equation (84) in Section 16 below, in the light of equation (120). But before doing this it is of interest to re-examine the theoretical basis of Walsh s rules. 

Basis for the Derivation of Walsh s Rules.—As already remarked (see also ref. 52), Walsh s rules governing molecular shapes were based, at least implicitly, on the assumption that the total energy was the sum of orbital energies. Though this... 

The status of Walsh s rules for molecular shapes still needs further fundamental clarification, though their overall usefulness can hardly be in doubt. However, the simplest density description, as shown by March and Plaskett18 for neutral atoms, and by Ruedenberg48 for neutral molecules, does indeed relate the total energy to the sum of orbital energies, with a multiplying factor of 3/2, both for atoms and for molecules at equilibrium. Corrections to this relation undoubtedly involve the chemical potential.50... 

Stenkamp, L. Z., and E. R. Davidson (1973). An ICSCF investigation of Walsh s Rules. Theor. Chim. Acta 30, 283-314. 

Bonds to Hydrogen.—NH and NH2 Species. Walsh s rules have been applied to NH2, and the computed geometry is in agreement with that predicted by more accurate calculations.21 The use of an argon laser has allowed the photoelectronic spectra of NH2 and NH- to be measured and the electron affinities to be determined EA(NH2) = 0.779 0.037 eV and Ea(NH) - 0.38 0.03 eV.22... 

Ab initio SCF calculations on the ground state of the LiBOj molecule show that the bent structure has a lower energy than the linear, in agreement with Walsh s rules. Only calculations including polarization functions produced this result. 

This approach was first applied to relatively simple cases many years ago by Walsh4 and led to certain generalizations, called Walsh s rules, relating the shapes of triatomic molecules to their electronic structures. The basic approach is to calculate, or estimate, the energies of molecular orbitals for two limiting structures, say, Unear and bent (to 90°) for an AB2 molecule,... 

Walsh s rules predict that the excited states may be bent and, in particular, Renner-Teller interaction will lift the degeneracy of the state. The... 

Just as in the diatomics with 10 valence electrons (VE), the 10 VE system HCN is one of the most stable species, occupying all A-B bonding MOs, and it has therefore received considerable attention in a number of experimental studies. It is linear in its ground state according to Walsh s rules. " The isomer HNC was first observed by Milligan and Jacox about 20 years ago and later an emission line observed at 90 665 MHz in interstellar space s was... 

Both HSO and SOH are bent (Fig. 7, Table XIII), in agreement with Walsh s rules the in-plane component of the Tig-type MO is able to mix with the hydrogen AOs upon bending, becomes more stable in this manner and is doubly occupied in both A and A" states. The SO separation is larger in the HOS component, as discussed earlier. It is also interesting that the transition energy to the first excited state is quite different in both isomers, around 1.6 eV in HSO and only 0.6 eV in HOS, which makes possible a clear identification of which isomer is present under experimental conditions. It is also seen that the barrier to linearity is much smaller in the energetically preferred system HOS. 

The forbidden band system (3400-4000 A) arises from a <- A spin-forbidden transition. (This B state is the triplet counterpart to the aforementioned Bi state.) In this transition, an electron jumps from the 6sli orbital to the 2b 1 orbital, orbitals that favor smaller and larger bond angles, respectively (Fig. 7.3). Thus Walsh s rules predict that SO2 in the Bi state will exhibit a larger bond angle than in the ground state, as indeed it does (126.1° as compared to 119.5°). 

Results such as these have tended to restrict use of the EH method to qualitative predictions of conformation in molecules too large to be conveniently treated by more accurate methods. However, just as the simple Huckel method underwent various refinements (such as the cjo technique) to patch up certain inadequacies, so has the EH method been rehned. Such refinements have been shown to give marked improvement in numerical predictions of various properties. The EH method has been overtaken in popularity by a host of more sophisticated computational methods. (See Chapter 11.) However, it is still sometimes used as a first step in such methods as a way to produce a starting set of approximate MOs. The EHMO method also continues to be important as the computational equivalent of qualitative MO theory (Chapter 14), which continues to play an important role in theoretical treatments of inorganic and organic chemistries (as, for example, in Walsh s Rules and in Woodward-Hoffmann Rules). 

In the F2 + ICl system, no product mass peaks at m/e 56 (C137f) or m/e (Cl Jp) were observed and no m/e 20 (HF) peak was observed for the F2 + HI reaction. These mass spectral results indicate that the trihalogen and pseudo-trihalogen species formed involve bonding schemes such as ClIF and HIF with the more electropositive atom in the center of the molecule as predicted by Walsh s rules (60,61). 

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