Orbital symmetiy

The best way to understand how orbital symmetiy affects pericyclic reactions is to look at some examples, bet s look first at a group of polyene rearrangements called electrocyclu reactions. An electrocyclic reaction is a pericyclic proce.ss that involves the cyclization of a conjugated polyene. One n bond is broken, the other 7T bonds change position, a new rr bond is formed, and a cyclic compound results. For example, a conjugated triene can be converted into a cyclohe.xa-dicne, and a conjugated diene can be converted into a cyclobutene. 

R. B. Woodward and R. Hoffmann, The Conseivation of Orbital Symmetiy. Verlag Chemie GmbH, Academic Press, Aschaffenburg, 1970. 

No two instructors teach organic chemistry exactly the same way. This book covers all the fundamental topics in detail, building each new concept on those that come before. Many topics may be given more or less emphasis at the discretion of the instructor. Examples of these topics are C NMR spectroscopy, ultraviolet spectroscopy, conservation of orbital symmetiy, amino acids and proteins, nucleic acids, and the special topics chapters, lipids and synthetic polymers. 

A complication arises for functions of d or higher symmetiy. There are five real d orbitals, which transform as xy, xz, yz, x —y, and z, that are called pure d functions. The orbital commonly referred to as z is actually... 

Frontier orbital theory also provides the basic framework for analysis of the effect that the symmetiy of orbitals has upon reactivity. One of the basic tenets of MO theory is that the symmetries of two orbitals must match to permit a strong interaction between them. This symmetry requirement, when used in the context of frontier orbital theory, can be a very powerful tool for predicting reactivity. As an example, let us examine the approach of an allyl cation and an ethylene molecule and ask whether the following reaction is likely to occur. 

The cyclobutene-butadiene interconversion can serve as an example of the reasoning employed in construction of an orbital correlation diagram. For this reaction, the four n orbitals of butadiene are converted smoothly into the two n and two a orbitals of the ground state of cyclobutene. The analysis is done as shown in Fig. 11.3. The n orbitals of butadiene are ip2, 3, and ij/. For cyclobutene, the four orbitals are a, iz, a, and n. Each of the orbitals is classified with respect to the symmetiy elements that are maintained in the course of the transformation. The relevant symmetry features depend on the structure of the reacting system. The most common elements of symmetiy to be considered are planes of symmetiy and rotation axes. An orbital is classified as symmetric (5) if it is unchanged by reflection in a plane of symmetiy or by rotation about an axis of symmetiy. If the orbital changes sign (phase) at each lobe as a result of the symmetry operation, it is called antisymmetric (A). Proper MOs must be either symmetric or antisymmetric. If an orbital is not sufficiently symmetric to be either S or A, it must be adapted by eombination with other orbitals to meet this requirement. 

Correlation diagrams can be constructed in an analogous fashion for the disrotatory and conrotatory modes for interconversion of hexatriene and cyclohexadiene. They lead to the prediction that the disrotatory mode is an allowed process whereas the conrotatory reaction is forbidden. This is in agreement with the experimental results on this reaction. Other electrocyclizations can be analyzed by the same method. Substituted derivatives of polyenes obey the orbital symmetry rules, even in cases in which the substitution pattern does not correspond in symmetiy to the orbital system. It is the symmetry of the participating orbitals, not of the molecule as a whole, that is crucial to the analysis. 

In particular, when all three of these indices are zero, the GTO has spherical symmetry, and is called an s-type GTO. When exactly one of the indices is one, the function has axial symmetiy about a single Cartesian axis and is called a p-type GTO. There are three possible choices for which index is one, corresponding to the pjc, p, and Pz orbitals. 

Various schemes exist to try to reduce the number of CSFs in the expansion in a rational way. Symmetry can reduce the scope of the problem enormously. In die TMM problem, many of die CSFs having partially occupied orbitals correspond to an electronic state symmetiy other than that of the totally symmetric irreducible representation, and dius make no contribution to the closed-shell singlet wave function (if symmetry is not used before the fact, die calculation itself will determine the coefdcients of non-contributing CSFs to be zero, but no advantage in efdciency will have been gained). Since this application of group dieoiy involves no approximations, it is one of the best ways to speed up a CAS calculation. 

When two molecular orbitals of the same symmetiy have similar energies, they interact to lower the energy of the lower orbital and raise the energy of the higher. For example, in the homonuclear diatomics, the (Jg 2s) and Ug 2p) orbitals both have rig symmetry (symmetric to infinite rotation and inversion) these orbitals interact to lower the energy of the ag 2s) and to raise the eneigy of the [Pg.124]

In summary, the free-ion term is split into Eg and "Tig by a field of Of, symmetry, and further split on distortion to D411 symmetiy. The labels of the states resulting from the free-ion term (Figure 11-10) are in reverse order to the labels on the orbitals for example, the b g atomic orbital is of highest energy, whereas the Big state originating from the free-ion term is of lowest energy. 

Draw an orbital energy-level diagram for diberyllium, Be2. Label the molecular orbitals with their symmetiy labels and represent the electrons as arrows in boxes. The spectrum of Be2 has been observed at low temperature. Would you have predicted the existence of Be2(g) from your diagram What is the normal form of beryllium at room temperature and atmospheric pressure ... 

In low-spin d octahedral complexes, e. g. [Cr(NO)(NH3)5], the depopulation of the t2g orbitals does not change the bond lengths associated with the metal-nitrosyl moiety greatly [41], The M-N-O bond angle is symmetiy imposed at 180°, the M-N(O) bond is 0.9 A shorter than the cis bonds to the amine ligands and the tram influence is 0.09 A. In these complexes, the multiple bmid character associated with and dy is retained, and the odd electron resides in the nmilxHidtng d orbital. 

Consider first transitions to another Ag wavefunction. in which case we need the (xoduct AgiiAg. Now AgAg = Ag. and the only character that returns Ag when multiplied by Ag is A itself. No component of the dipole operator belongs to. species Ag, so no Ag <— Ag transitions are allowed. (Note such transitions are transitions from an orbital occupied in the ground state to an excited-state orbital of the same symmetry.) The other possibility is a transition from an orbital of one symmetry (A or Bg) to the other in that case, the excited-state wavefunction will have symmetiy of AuBg = Bu from the two singly occupied orbitals in the excited state. The symmetry of the transition dipole, then, is Ag/tB = /tBu, and the only species that yields Ag when multiplied by Bu is Bu itself. The jc and > components of the dipole operator belongs to species By, so the.se transitions are allowed. 

Fig. 8.25. Symmetiy of the MOs results from the mutual arrangement of those AOs of both atoms, whidt have the largest LCAO coefficients. Panels (a) through (d) show the a type bonds, panels (e) through (g) show the n type bonds, and panels (h) and (i) show the S type bonds. The cr bond orbitals have no nodal plane (containing the nuclei), the n orbitals have one such plane, and the S ones have two such planes.

There will be a lot of atomic orbitals, and therefore also an astronomic number of integrals to compute (infinite for the infinite crystal), and there is nothing that can be done about that On the other hand, if we begin such a hopeless task, the value of any integral would repeat ar infinite number of times. This indicates a chance to simplify the problem. Indeed, we have not yet used the translational symmetiy of the system. 

Note, that e.g., if one of the Is orbitals had the opposite sign, then the function /(r) would nothave the symmetiy of the equilateral triangle, although it would be invariant too with respect to some of the operations of the equilateral triangle. 

We know how to q)ply the symmetiy operations on molecular orbitals (p. e20) and transforming them to other functions. 

The no-pair DCB Hamiltonian (6) is used as a starting point for variational or many-body relativistic calculations [9], The procedure is similar to the nonrelativistic case, with the Hartree-Fock orbitals replaced by the four-component Dirac-Fock-Breit PFB) functions. The spherical symmetiy of atoms leads to the separation of the one-electron equation into radial and spin-angular parts [10], The radial four-spinor has the so-called large component the upper two places and the small component in the lower... 

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