Operator antisymmetrizer

A word about notation is in order, regarding Eq. (37). Previously (cf. Eq. (26)), P and P were defined to act upon primed and unprimed coordinates of n-electron kernels. Where tensors are involved, such as in Eq. (37), P represents signed permutations over the row indices, (i.e., the first set of indices) and P denotes signed permutations over column indices. Thus, for example, when P 2 acts on A1A2 in Eq. (37), this operation antisymmetrizes the indices q and 2 appearing in Eq. (38). The column indices (ji and 72) of this product are already antisymmetric, having inherited this property from A2. 

This orientation of the molecule reveals that methane possesses three twofold symmetry axes, one each along the x, y, and z axes. Because of this molecular symmetry, the proper molecular orbitals of methane must possess symmetry with respect to these same axes. There are two possibilities the orbital may be unchanged by 180° rotation about the axis (symmetric), or it may be transformed into an orbital of identical shape but opposite sign by the symmetry operation (antisymmetric). The carbon 2s-orbital is symmetric with respect to each axis, but the three 2p-orbitals are each antisymmetric to two of the axes and symmetric with respect to one. The combinations which give rise to molecular orbitals that meet these symmetry requirements are shown in Fig. 1.11. 

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 ... 

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 ... 

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... 

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. 

Various types of antisymmetric wavefunction can be obtained by applying different functions of the T operators to fi o. and the unknown coefficients together with the energy can be determined from the projection equations... 

The 1 operator is the identity, while Py generates all possible permutations of two electron coordinates, Pyi all possible permutations of three electron coordinates etc. It may be shown that the antisymmetrizing operator A commutes with H, and that A acting twice gives the same as A acting once, multiplied by the square root of N factorial. 

In addition to the Schrodinger equation we have the antisymmetry requirement (Eq. II.2) connected with the Pauli principle and, by means of the antisymmetrization operator (Eq. 11.16), the Hartree product (Eq. 11.37) is then transformed into a Slater determinant ... 

When dealing with systems described by antisymmetrical states, the creation and annihilation operators are defined in such a way that the occupation numbers can never be greater than unity. Thus we have a creation operator af defined by... 

In formulating the second-quantized description of a system of noninteracting fermions, we shall, therefore, have to introduce distinct creation and annihilation operators for particle and antiparticle. Furthermore, since all the fermions that have been discovered thus far obey the Pauli Exclusion principle we shall have to make sure that the formalism describes a many particle system in terms of properly antisymmetrized amplitudes so that the particles obey Fermi-Dirac statistics. For definiteness, we shall in the present section consider only the negaton-positon system, and call the negaton the particle and the positon the antiparticle. 

It is required, in accordance with the Fermi character of particles and antiparticles, to be separately antisymmetric in the particle and antiparticle variables, which in turn requires that the operator b and d satisfy the following anticommutation rules ... 

The commutations (9-416)-( 9-419) guarantee that the state vectors are antisymmetric and that the occupation number operators N (p,s) and N+(q,t) can have only eigenvalues 0 and 1 (which is, of course, what is meant by the statement that particles and antiparticles separately obey Fermi-Dirac etatistios). In fact one readily verifies that... 

When the carbonyl groups are present, the transition state for syn attack is sta-bihzed by interactions between the in-phase combination of the NN lone pairs and the antisymmetric n orbital of the CO-X-CO bridge (100). Although the secondary effect (SOI) operates only during syn approach and contributes added stabilization to this transition state, the primary orbital interaction (see 103) between the HOMO of the cyclohexadiene moiety of 100 and the n orbital of the dienophile (NN, Fig. 16) is differentiated with respect to the direction of attack, i.e., syn or anti, of triazolinedione (NN, Fig. 16). 

In Section II, we explained that and Fq are respectively symmetric and antisymmetric under the operator. R271 in the double space. More generally, if the... 

The wave funetion obtained eorresponds to the Unrestricted Hartree-Fock scheme and beeomes equivalent to the RHF ease if the orbitals (t>a and (()p are the same. In this UHF form, the UHF wave funetion obeys the Pauli prineiple but is not an eigenfunction of the total spin operator and is thus a mixture of different spin multiplicities. In the present two-eleetron case, an alternative form of the wave funetion which has the same total energy, which is a pure singlet state, but whieh is no longer antisymmetric as required by thePauli principle, is ... 

In equation (8.32) the operator P is any one of the N operators, including the identity operator, that permute a given order of particles to another order. The summation is taken over all N permutation operators. The quantity dp is always - -1 for the symmetric wave function Ps, but for the antisymmetric wave function Wa, dpis-l-l(—l)if the permutation operator P involves the... 

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