Molecular orbitals antisymmetric

Ross, I. G., Trans. Faraday Soc. 48, 973, Calculations of the energy levels of acetylene by the method of antisymmetric molecular orbitals, including a—7r-interaction."... 

The electronic structure of this compd detd by the method of antisymmetrical molecular orbitals (ASMO) has been reported by Paoloni (Ref 4)... 

In contrast to 3, which can bind one electrophile at each Fe atom as in 6, the doubly bridged anion 4 is best suited to bind only one bridging group as in 7. In effect, the two electron pairs of 4 used to bind to an electrophile can be described by symmetric and antisymmetric molecular orbitals centered at the metal atoms and directed to the vacant bridging position (9). Hence, a wide variety of acceptor groups... 

Interpretation in terms of thermal population of symmetrical and antisymmetrical molecular orbitals, McLachlan calculations of spin densities. ... 

FIGURE 1.9 Twofold axis (C2) symmetric and antisymmetric molecular orbitals. 

As above, the active orbitals are called ([) and ( )a on one chromophore (A) and ( )j and ([), on the other chromophore (B) (Eigure 14.4). We assume that the two chro-mophores are identical, and related by symmetry operations. The ground state wave fnnction contains symmetric and antisymmetric molecular orbitals (MO) of the type ([) <[)j and <[)j <[), . Index i and a belong to center A, and j and b to center B. We inclnde only the relevant orbitals and write the ground state wave fnnction as... 

Appendix 11 The Coefficients of All Antisymmetric Molecular Orbitals of the Tetracene Derivative C20H14... 

The coefficients a, b, c of antisymmetric molecular orbitals of the tetracene derivative C2oH,4 satisfy the same snm rules that hold for antisymmetric molecular orbitals of hexatriene in Table AlO.l in Appendix 10 that the magnitudes of the coefficients of symmetric and antisymmetric eigenvectors have the same magnitudes and only differ in their signs. 

In the monoalkylcyclopentadienyl radical, the degeneracy of the symmetric and antisymmetric molecular orbitals of the n system is broken by electron release from the substituent which destabilizes the symmetric MO, which becomes singly occupied, as indicated by the large hyperfine couping constants at Hi and Hg 4 and the small values at Hz.s- ... 

The terminal 2p orbitals of an antisymmetric molecular orbital have the positively signed lobes on the opposite sides of the molecule. Thus, the orbitals must rotate in the same direction to form a a bond. The rotation is conrotatory. 

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, ... 

According to one classification (15,16), symmetrical dinuclear PMDs can be divided into two classes, A and B, with respect to the symmetry of the frontier molecular orbital (MO). Thus, the lowest unoccupied MO (LUMO) of class-A dyes is antisymmetrical and the highest occupied MO (HOMO) is symmetrical, and the TT-system contains an odd number of TT-electron pairs. On the other hand, the frontier MO symmetry of class-B dyes is the opposite, and the molecule has an even number of TT-electron pairs. 

Fig. 2. Simplified molecular orbital diagram for a low spia octahedral complex, such as [Co(NH3 )g, where A = energy difference a, e, and t may be antisymmetric (subscript ungerade) or centrosymmetric (subscript, gerade) symmetry orbitals. See text.

The simplest antisymmetric function that is a combination of molecular orbitals is a determinant. Before forming it, however, we need to account for a factor we ve neglected so far electron spin. Electrons can have spin up i+Vi) or down (-V2). Equation 20 assumes that each molecular orbital holds only one electron. However, most calculations are closed shell calculations, using doubly occupied orbitals, holding two electrons of opposite spin. For the moment, we will limit our discussion to this case. 

The problem has now become how to solve for the set of molecular orbital expansion coefficients, c. . Hartree-Fock theory takes advantage of the variational principle, which says that for the ground state of any antisymmetric normalized function of the electronic coordinates, which we will denote H, then the expectation value for the energy corresponding to E will always be greater than the energy for the exact wave function ... 

Figure B A qualitative molecular orbital diagram for ferrocene. The subscripts g and u refer to the parity of the orbitals g (German gerade, even) indicates that the orbital (or orbital combination) is symmetric with respect to inversion, whereas the subscript u (ungerade, odd) indicates that it is antisymmetric with respect to inversion. Only orbitals with the same parity can combine.

This is first illustrated for the two nonbonding -type orbitals n, and n2 of para-benzyne and pyrazine (Fig. 31). These nonbonding orbitals are derived from outer (2s, 2p) sp2 type hybrids which have not been used in any bonding interaction. Although the overlap between n, and n2 is zero each one overlaps with the central CC bond orbitals. All told, there will arise two distinct molecular orbitals in which nj and n2 enter as combinations (symmetric or antisymmetric) and which have different energies, because of selective interactions with the central bonds. 

The FMO of the diene having substituent X at the 5-positions is comprised of three molecular orbitals, namely, jt-HOMO of the diene part, a-orbital of carbon framework, and the nonbonding (n) orbital of X (Scheme 3). The FMO of the diene for Diels-Alder reactions should mainly consist of n-HOMO. The jt-HOMO is antisymmetric with respect to reflection in the plane containing C5 carbon and its substitu-... 

The AB supermolecule is described by a single determinant wave function formulated in terms of doubly occupied molecular orbitals with no orthonormality constraints. For a system with 2N = 2Na +2Nb electrons the SCF-MI wave function expressed in terms of the antisymmetrizer operator A is... 

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