M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 44
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Since then, there has been atremendous progress, and today it is possible to calculate much largersystems with good accuracy. The quantum chemical codes as wellas the applied basis sets are continuously improving. While yearsago, calculation of molecules involving heavy metals, such as thelanthanides or actinides, was a formidable task, today the availableeffective core potentials make these calculations affordable and moreand more reliable. Computational studies of large biological macromolecules have also become an important tool in molecular biologyand biological chemistry [38].Here we focus mostly on one property, molecular geometry.
Withthe ever increasing accuracy of computations, it is important torealize that while calculations provide information on the equilibrium6.4. Quantum Chemical Calculations289geometry, the various experiments yield some effective geometries forthe molecule, averaged over molecular vibrations. Depending on themagnitude of these vibrations and their structural influence, the equilibrium and average structures may differ to various extents. Examples of rather extreme effects were mentioned in Section 3.7.6. This iswhy the following caveat has been issued: “For truly accurate comparison, experimental bond lengths [or, generally, geometries] should becompared with computed ones only following necessary corrections,bringing all information involved in the comparison to a commondenominator” [39].The application of computational chemistry is especially advantageous when structural differences rather than absolute values of thestructural parameters are sought.
Important systematic errors cancelto a large extent in the determination of structural differences in calculations as well as in experiments. The importance of small structuraldifferences in understanding various effects in series of substances hasbeen recognized [40].Small structural changes are especially important in molecularrecognition.
It has been noted, e.g., that “... subtle changes of molecular structure may result in severe changes of inclusion behavior of apotential host molecule due to the complicated interplay of weak intermolecular forces that govern host-guest complex formation” [41].Quantum chemical calculations have also proved to be importanttools in aiding the experimental determination of molecular geometryin that they can provide reliable constraints in the experimental analysis (see, e.g., the structure analysis of 2-nitrophenol [42] and of metalhalides that have a complex vapor composition [43, 44] or constitutevery floppy systems [45]).The difference is not merely practical, it is conceptual as well.
R.D.Levine [46] distinguished between physical and chemical shapes.According to him, the physical shape corresponds to a hard spacefilling model, whereas the chemical shape describes how molecularreactivity depends on the direction of approach and distance of theother reagent. In terms of geometry representations, the chemicalshape can be related to the average structures determined from theexperiments and the physical shape to the hypothetical equilibriumstructure.Quantum chemical calculations, of course, are the exclusive sourceof information for systems that are not amenable to experimental2906 Electronic Structure of Atoms and Moleculesstudy.
Such systems include transition states, highly reactive andunstable or even unknown species. Computations are used in drugdesign, materials science, surface science, just to mention a fewnew areas. Quantum chemical calculations have proved to be notonly complementary to experiments or be their alternatives, buthave opened up new research areas as well. Schafer has predictedthat chemical research will migrate each year from experiment tocomputation for the forseable future [47] and would level off at about50% each [48].6.5.
Influence of Environmental SymmetrySymmetry has a major role in two widely used and successfulapproaches of chemistry, viz., the crystal field and ligand field theoriesof coordination compounds. This topic has been thoroughly coveredin textbooks and monographs on coordination chemistry. Therefore, itis mentioned here only in passing.Hans Bethe showed that the degenerate electronic state of a cationis split by a crystal field into nonequivalent states [49]. The changeis determined entirely by the symmetry of the crystal lattice.
Bethe’soriginal work was concerned with ionic crystals, but his concept hasmore general applications. When an atom or an ion enters a ligandenvironment, the symmetry of the ligand arrangement will influence the electron density distribution of that atom or ion. The original spherical symmetry of the atomic orbitals will be lost, and thesymmetry of the ligand environment will be adopted. As a consequence of the usual decrease of symmetry, the degree of degeneracyof the orbitals decreases.The s electrons are already nondegenerate in the free atom, sotheir degeneracy does not change.
They will always belong to thetotally symmetric irreducible representation of the symmetry group.The p orbitals, however, are threefold degenerate, and the d orbitalsare fivefold degenerate. To determine their splitting in a certain pointgroup, we must use them, in principle, as bases for a representation ofthe group.
In practice, we can find in the character table of the pointgroup the irreducible representations to which the orbitals belong. Anorbital always belongs to the same irreducible representation as do6.5. Influence of Environmental Symmetry291Table 6-10. Splitting of Atomic Orbitals in Different Symmetry EnvironmentsspdOhTdD∞hD4dD4hC4C2a1ga1ga1a1ga1a1t1ut2u + ub2 + e1a2u + eua1 + ea1 + b1 + b2eg + t2ge + t2g + g + ⌬ga1 + e2 + e3a1g + b1g + b2g + ega1 + b1 + b2 + e2a1 + a2 + b1 + b2its subscripts. Some orbital splittings that accompany the decrease inenvironmental symmetry are shown in Table 6-10.As environmental symmetry decreases, the orbitals will becomesplit to an increasing extent.
In the C2v point group, for example,all atomic orbitals will be split into nondegenerate levels. This is notsurprising since the C2v character table contains only one-dimensionalirreducible representations. This result shows at once that there are nodegenerate energy levels in this point group. This has been stressed inChapter 4 in the discussion of irreducible representations.The symmetry of the ligand environment gives an important butlimited amount of information about orbital splitting. Both the octahedral and cubic ligand arrangements, for example, belong to the Ohpoint group, and we can tell that the d orbitals of the central atom willsplit into a doubly degenerate and a triply degenerate pair.
But nothingis revealed about the relative energies of these two sets of degenerateorbitals.The problem of relative energies is dealt with by crystal field theory.This theory examines the repulsive interaction between the ligandsand the central atom orbitals. Consider first an octahedral molecule(Figure 6-34), and compare the positions of one eg (e.g., dx2 −y2 ) andone t2g (e.g., dyz ) orbital. The others need not be considered, as they aredegenerate with, and thus have the same energy as, one of the eg or t2gorbitals.
The lobes of the dx2 −y2 orbital point towards the ligands. Theresulting electrostatic repulsion will destabilize this orbital, and itsenergies will increase accordingly. The dyz orbital, on the other hand,points in directions between the ligands. This is an energetically morefavorable position; hence, the energy of these orbitals will decrease.Examine now the cubic arrangement in Figure 6-35. It can beseen that the dyz orbital is in a more unfavorable situation relative to2926 Electronic Structure of Atoms and MoleculesFigure 6-34.
The orientation of the different symmetry d orbitals in an octahedralenvironment.the ligands than is the dx2 −y2 orbital, so their relative energies willbe reversed (see Figure 6-36). Some other typical orbital splittingsand the corresponding changes in the relative energies are shown inFigure 6-37.Prediction of Structural Changes. Crystal field theory is frequentlyapplied to account for and even predict structural and chemicalchanges.
A well-known example is the variation of first row transitionmetal ionic radii in an octahedral environment as illustrated inFigure 6-38 [50]. The dashed line connects the points for Ca, Mn, andZn, i.e., atoms with spherically symmetrical distribution of d electrons. Since the shielding of one d electron by another is imperfect,a contraction in the ionic radius is expected along this series. This initself would account only for a steady decrease in the radii, whereasthe ionic radii of all the other atoms are smaller than interpolationfrom the Ca–Mn–Zn curve would suggest. As is well known, theFigure 6-35.
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