Antisymmetry operations

Figure 4-12. Antisymmetry operations. First row antirotation axes 2,4, 6 Second row antimirror rotation axes 2, 4, 6 Third row antirotation axes combined with ordinary mirror planes 2m, 4 m, 6 m Fourth row ordinary rotationaxes combined with antimirror planes 2 m, 4 m, 6 m Fifth row 1 m, 3 m, after Shubnikov [16], Reproduced with permission from Nauka Publishing Co., Moscow.

The essential points can be summarized as follows. The introduction of the intra-monomer antisymmetry can be done at different levels of the theory. The simplest formulation is just to use I Aab o unperturbed wave function, introducing a truncated expansion of the antisymmetry operator. This means to pass from rigorous to approximate formulations. 

A b is the additional antisymmetry operator we have aheady introduced. Unfortunately IAab I o > is not an eigenfunction of H as asked by the PT. Two ways of overcoming this difficulty are possible. One could abandon the natural partitioning [8.36] and define another... 

A general molecular wavefunction can be written in terms of a Hartree product of orbitals with an antisymmetry operator d as... 

In these equations, J and M are quantum numbers associated with the angular momentum operators and J, respectively. The number II = 0, 1 is a parity quantum number that specifies the symmetry or antisymmetry of the column vector with respect to the inversion of the nuclei through G. Note that the same parity quantum number II appears for and Also, the... 

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 summary, proper spin eigenfunetions must be eonstmeted from antisymmetrie (i.e., determinental) wavefunetions as demonstrated above beeause the total and total Sz remain valid symmetry operators for many-eleetron systems. Doing so results in the spin-adapted wavefunetions being expressed as eombinations of determinants with eoeffieients determined via spin angular momentum teehniques as demonstrated above. In... 

The opposite of a creation operator is an annihilation operator a which removes orbital i from the wave function it is acting on. The a-a product of operators removes orbital j and creates orbital i, i.e. replaces the occupied orbital j with an unoccupied orbital i. The antisymmetry of the wave function is built into the operators as they obey the following anti-commutation relationships. 

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]

Three distinct sets of linear mappings for the partial 3-positivity matrices in Eqs. (31)-(36) are important (i) the contraction mappings, which relate the lifted metric matrices to the 2-positive matrices in Eqs. (27)-(29) (ii) the linear interconversion mappings from rearranging creation and annihilation operators to interrelate the lifted metric matrices and (iii) antisymmetry (or symmetry) conditions, which enforce the permutation of the creation operators for fermions (or bosons). Note that the correct permutation of the annihilation operators is automatically enforced from the permutation of the creation operators in (iii) by the Hermiticity of the matrices. 

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... 

This kind of wavefunction is called a Hartree Product, and it is not physically realistic. In the first place, it is an independent-electron model, and we know electrons repel each other. Secondly, it does not satisfy the antisymmetry principle due to Pauli which states that the sign of the wavefunction must be inverted under the operation of switching the coordinates of any two electrons, or... 

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. 

The phase factor r(n) is introduced in order to endow the antisymmetry of many-electron wave functions in the Fock space, as we soon will see. The definition that ai operating on an occupation number vector gives zero if spin... 

One particular advantage of Slater determinants constructed from orthonormsd spin-orbitals is that matrix elements between determinants over operators such as H sure very simple. Only three distinct cases arise, as is well known and treated elsewhere. It is perhaps not surprising that the simplest matrix element formulas should be obtained from the treatment that exploits symmetry the least, as only the fermion antisymmetry has been accounted for in the determinants. As more symmetry is introduced, the formulas become more complicated. On the other hand, the symmetry reduces the dimension of the problem more and more, because selection rules eliminate more terms. We consider here the spin adaptation of Slater determinants. 

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