Antisymmetric orbitals

Withdrawal of electrons from orbitals antisymmetric between hydrogen atoms (generally hyperconjugative effects) should lead to a negative change in the coupling constants. ... 

The above argument about the structure of N2O4 is based upon the assumption32 that the odd electron of N02 occupies a orbital symmetric in the plane of the nuclei) rather than a r orbital (antisymmetric). The alternative assumption has been proposed 33 with it the N—N bond would be described as a t bond without an associated a... 

Ethylene, the first member of the series, is already familiar. Two adjacent p orbitals interacting will yield a bonding orbital, symmetric with respect to reflection in the mirror plane lying midway between the two carbons and perpendicular to the C—C axis, and an antibonding orbital antisymmetric with respect to that mirror plane. Figure 10.18 illustrates the interaction and the resulting orbitals. 

Section 4-3-3 g for gerade, orbitals symmetric to inversion, and u for ungerade, orbitals antisymmetric to inversion (those whose signs change on inversion). The g or m notation describes the symmetry of the orbitals without a judgment as to their relative eneigies. 

Conversely, since totai bas this character when orbital antisymmetrical (different electronic orbital quantum numbers), the spin combinations must be symmetrical. There are three of them and the triplet ortho states result. 

Screening length of orbital Symmetric orbital Antisymmetric orbital Symmetric p-type orbital Antisymmetric p-type orbital Angle... 

A species closely related to SF, ASF3", has been detected in Y-irradiated polycrystalline ASF3 (37). The ESR parameters for this radical are consistent with a planar T-shaped structure we feel, however, that the unpaired electron occupies a totally symmetric orbital, rather than an orbital antisymmetric with respect to the molecular plane as suggested by Subramanian and Rogers (37). 

The so-ealled Slater-Condon rules express the matrix elements of any one-eleetron (F) plus two-eleetron (G) additive operator between pairs of antisymmetrized spin-orbital produets that have been arranged (by permuting spin-orbital ordering) to be in so-ealled maximal eoineidenee. Onee in this order, the matrix elements between two sueh Slater determinants (labelled >and are summarized as follows ... 

HMO theory is named after its developer, Erich Huckel (1896-1980), who published his theory in 1930 [9] partly in order to explain the unusual stability of benzene and other aromatic compounds. Given that digital computers had not yet been invented and that all Hiickel s calculations had to be done by hand, HMO theory necessarily includes many approximations. The first is that only the jr-molecular orbitals of the molecule are considered. This implies that the entire molecular structure is planar (because then a plane of symmetry separates the r-orbitals, which are antisymmetric with respect to this plane, from all others). It also means that only one atomic orbital must be considered for each atom in the r-system (the p-orbital that is antisymmetric with respect to the plane of the molecule) and none at all for atoms (such as hydrogen) that are not involved in the r-system. Huckel then used the technique known as linear combination of atomic orbitals (LCAO) to build these atomic orbitals up into molecular orbitals. This is illustrated in Figure 7-18 for ethylene. 

For Iran sition metals th c splittin g of th c d orbitals in a ligand field is most readily done using HHT. In all other sem i-ctn pirical meth -ods, the orbital energies depend on the electron occupation. HyperCh em s m oiccii lar orbital calcii latiori s give orbital cri ergy spacings that differ from simple crystal field theory prediction s. The total molecular wavcfunction is an antisymmetrized product of the occupied molecular orbitals. The virtual set of orbitals arc the residue of SCT calculations, in that they are deemed least suitable to describe the molecular wavefunction, ... 

To see how and under what conditions stability is enhanced or diminished, we need to consider the symmetry of the orbital (9-32), Flectrons in the antisymmetric orbital r r have a 7ero probability of occurring at the node in u where U] = rj. Electron mutual avoidance of the node due to spin correlation reduces the total energy of the system because it reduces electron repulsion energy due to charge... 

In summary, the Hariree-Fock equation for antisymmetr ized orbitals is written... 

In particular, within the orbital model of eleetronie strueture (whieh is developed more systematieally in Seetion 6), one ean not eonstruet trial waveflmetions whieh are simple spin-orbital produets (i.e., an orbital multiplied by an a or P spin funetion for eaeh eleetron) sueh as lsalsP2sa2sP2pia2poa. Sueh spin-orbital produet funetions must be made permutationally antisymmetrie if the N-eleetron trial funetion is to be properly antisymmetrie. This ean be aeeomplished for any sueh produet wavefunetion by applying the following antisymmetrizer operator ... 

It should be noted that the effeet of A on any spin-orbital produet is to produee a funetion that is a sum of N terms. In eaeh of these terms the same spin-orbitals appear, but the order in whieh they appear differs from term to term. Thus antisymmetrization does not alter the overall orbital oeeupaney it simply "serambles" any knowledge of whieh eleetron is in whieh spin-orbital. 

The orbitals of an atom are labeled by 1 and m quantum numbers the orbitals belonging to a given energy and 1 value are 21+1- fold degenerate. The many-eleetron Hamiltonian, H, of an atom and the antisymmetrizer operator A = (V l/N )Zp Sp P eommute with total =Zi (i), as in the linear-moleeule ease. The additional symmetry present in the spherieal atom refleets itself in the faet that Lx, and Ly now also eommute with H and A. However, sinee does not eommute with Lx or Ly, new quantum... 

In this formulation, the electron density is expressed as a linear combination of basis functions similar in mathematical form to HF orbitals. A determinant is then formed from these functions, called Kohn-Sham orbitals. It is the electron density from this determinant of orbitals that is used to compute the energy. This procedure is necessary because Fermion systems can only have electron densities that arise from an antisymmetric wave function. There has been some debate over the interpretation of Kohn-Sham orbitals. It is certain that they are not mathematically equivalent to either HF orbitals or natural orbitals from correlated calculations. However, Kohn-Sham orbitals do describe the behavior of electrons in a molecule, just as the other orbitals mentioned do. DFT orbital eigenvalues do not match the energies obtained from photoelectron spectroscopy experiments as well as HF orbital energies do. The questions still being debated are how to assign similarities and how to physically interpret the differences. 

The fact that an electron has an intrinsic spin comes out of a relativistic formulation of quantum mechanics. Even though the Schrodinger equation does not predict it, wave functions that are antisymmetric and have two electrons per orbital are used for nonreiativistic calculations. This is necessary in order to obtain results that are in any way reasonable. 

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