Antisymmetric/antisymmetries operators

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 ... 

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 ... 

The requirement of overall exchange antisymmetry of the /V-clcct.ron wavefunction [Pg.36]

The symbols used for the representations are those proposed by Mulliken. The A representations are those which are symmetric with respect to the C2 operation, and the Bs are antisymmetric to that operation. The subscript 1 indicates that a representation is symmetric with respect to the ov operation, the subscript 2 indicating antisymmetry to it. No other indications are required, since the characters in the o column are decided by another rule of group theory. This rule is the product of any two columns of a character table must also be a column in that table. It may be seen that the product of the C2 characters and those of gv give the contents of the The representations deduced above must be described as irreducible representations This is because they... 

In the case of conrotatory mode, the symmetry is preserved with respeo to C2 axis of rotation. On 180° rotation along this axis, F goes to H. and H2 to H, and the new configuration is indistinguishable from the original. An orbital symmetric with respect to rotation is called a and antisymmetric as b. On the other hand, in the case of disrotatory moot-the elements of symmetry are described with respect to a mirror plane. Tilt symmetry and antisymmetry of an orbital with respect to a mirror plant of reflection is denoted by a and a" respectively (Section 2.9). The natun of each MO of cyclobutene with respect to these two operations is shov. n in the Table 8.4 for cyclobutene and butadiene. 

When constructing many-electron wave functions it is necessary to ensure their antisymmetry under permutation of any pair of coordinates. Having introduced the concepts of the CFP and unit tensors, Racah [22, 23] laid the foundations of the tensorial approach to the problem of constructing antisymmetric wave functions and finding matrix elements of operators corresponding to physical quantities. 

Since the reference system s consists of noninteracting particles, the results of Section 6.2 and the antisymmetry requirement show that the ground-state wave function i/fj 0 of the reference system is the antisymmetrized product (Slater determinant) of the lowest-energy Kohn-Sham spin-orbitals mP of the reference system, where the spatial part flf (r, ) of each spin-orbital is an eigenfunction of the one-electron operator P ... 

Each of the MOs in Fig. 10-3 is symmetric or antisymmetric for some of the operations that apply to a tetrahedron. 02 is symmetric for rotations about the z axis by 27t/3, and also for reflection through the xz plane. This same reflection plane is a symmetry plane for 03 and 04, but neither of these MOs shows symmetry or antisymmetry for rotation about the z axis. Each MO contains one p AO and, perforce, has the symmetry of that AO. We shall refer to these as p-type MOs. Note that hydrogen Is AOs lying in the nodal plane of a p AO do not mix with that p AO in formation of MOs. This results from zero interaction elements in H, which, in turn, results from zero overlap elements in S. Note also that the MO 02 is the MO that we anticipated earlier on the basis of inspection of the matrix H. Because of phase agreements between the hydrogen Is AOs and the adjacent lobes of the p AOs, 0j, 02, and 03 are C-H bonding MOs. 

The function (P is an eigenfunction of the operator F. The simplest perturbation expansion that one may employ is the RS (polarization) method (extended to the case of two perturbation operators) discussed in Sect. 2. However, as explained in that section, the RS method is not adequate, except for large intermonomer separations. The underlying reason is that the wave functions in this approach do not completely fulfill the Pauli exclusion principle, Le. the wave functions are not fully antisymmetric with respect to exchanges of electrons [the antisymmetry is satisfied for exchanges within monomers but not between them]. As described in Sect. 2, the antisymmetry requirement can be imposed by acting on the wave functions with the N-electron antisymmetrization operator. This (anti)symmetrization can be performed in many ways and leads to various versions of SAPT. The simplest of them, the SRS method, has been implemented in the many-electron context [24]. 

In second quantization, the Pauli antisymmetry principle is incorporated through the algebraic properties of the creation and annihilation operators as discussed in Chapter 1. We note that, in density-functional theory (which bypasses the construction of the wave function and concentrates on the electron density), the fulfilment of the A -representability condition on the density represents a less trivial problem. A density is said to be N-representable if it can be derived from an antisymmetric wave function for N particles [1]. 

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