Column antisymmetrizer

The Young operator Y in (1) antisymmetrizes with respect to permutations of sites in the same column in its tableau. The monomial yi, therefore, cannot be symmetrical with respect to any two sites in the same column, i.e., it cannot contain the same power of A for any two such sites. The powers of A for the sites in a given column, therefore, must all be different. The lowest possible choice consistent with this is that they be 0, 1, 2,. .., p, for a column of length /t. Thus, y can be chosen to be independent of A for sites in the first row of the tableau, and to contain A for sites in the second row, A2 for those in the third, etc. The total order is therefore... 

Also, the fermion anticommutation rules interrelate the ROMs with the HRDMs they render these matrices antisymmetric with respect to odd permutations of the row or column indices and, finally, they interrelate them with two other families of matrices the G-matrices and the comelation matrices. 

Although a formal solution of the A-representability problem for the 2-RDM and 2-HRDM (and higher-order matrices) was reported [1], this solution is not feasible, at least in a practical sense [90], Hence, in the case of the 2-RDM and 2-HRDM, only a set of necessary A-representability conditions is known. Thus these latter matrices must be Hermitian, Positive semidefinite (D- and Q-conditions [16, 17, 91]), and antisymmetric under permutation of indices within a given row/column. These second-order matrices must contract into the first-order ones according to the following relations ... 

For the a/l-block of the 2-RDM the decomposition was generated ad hoc [70]. This is because this block is not antisymmetric under permutation of the orbital indices within the row or column subsets of indices and thus the unitary decomposition reported by Coleman cannot be applied. Hence the ad hoc decomposition is given here by... 

It must be emphasized that the a S-block of the 2-RDM may also be decomposed into two subblocks, the singlet and the triplet one. This clearly enhances the computational efficiency of the purification of the 2-RDM, since these subblocks may be corrected separately. On the other hand, since the singlet and triplet subblocks are symmetric and antisymmetric, respectively, under permutation of the orbital indices within the row or column subsets of labels, it would be possible to use the unitarily invariant decomposition of Coleman [73] and that of Sun et al. [102] to correct both the N- and S -representability defects of these subblocks. However, this would be formally equivalent to use the decomposition given by Eqs. (94)-(97), since this latter decomposition implicitly presents the former ones as particular cases [77]. 

In order to get significant results, the initial data must be formed by a set of clearly non-A -representable second-order matrices, which would generate upon contraction a closely ensemble A -representable 1-RDM. It therefore seemed reasonable to choose as initial data the approximate 2-RDMs built by application of the independent pair model within the framework of the spin-adapted reduced Hamiltonian (SRH) theory [37 5]. This choice is adequate because these matrices, which are positive semidefinite, Hermitian, and antisymmetric with respect to the permutation of two row/column indices, are not A -representable, since the 2-HRDMs derived from them are not positive semidefinite. Moreover, the 1-RDMs derived from these 2-RDMs, although positive semidefinite, are neither ensemble A -representable nor 5-representable. That is, the correction of the N- and 5-representability defects of these sets of matrices (approximated 2-RDM, 2-HRDM, and 1-RDM) is a suitable test for the two purification procedures. Attention has been focused only on correcting the N- and 5-representability of the a S-block of these matrices, since the I-MZ purification procedure deals with a different decomposition of this block. 

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. 

Interchange of two rows of the Slater determinant changes the sign of the wave function, which is therefore antisymmetric with respect to interchange of electrons. When two rows or columns are identical, the determinant is zero. The Slater determinant wave function therefore obeys the Pauli exclusion principle for fermions. 

The second of these is the column antisymmetrizer and is symbolized by M. As might be expected, for the 3,2 tableau the column antisymmetrizer is the product of the antisymmetrizer for each column and is... 

We have so far said little about the nature ofthe space function, S. Earlier we implied that it might be an orbital product, but this was not really necessary in our general work analyzing the effects of the antisymmetrizer and the spin eigenfunction. We shall now be specific and assume that S is a product of orbitals. There are many ways that a product of orbitals could be arranged, and, indeed, there are many of these for which the application of the would produce zero. The partition corresponding to the spin eigenfunction had at most two rows, and we have seen that the appropriate ones for the spatial functions have at most two columns. Let us illustrate these considerations with a system of five electrons in a doublet state, and assume that we have five different (linearly independent) orbitals, which we label a, b,c,d, and e. We can draw two tableaux, one with the particle labels and one with the orbital labels. 

This function is antisymmetric with respect to the interchange of any pair of orbitals. The same pertains to structure 1 of Table 11.11 with respect to either ofthe columns. Thus the dominant structure is very much two atoms. 

As we have pointed out many times previously, the columns of the standard tableaux functions are antisymmetrized, and the orbitals in a column may be replaced by any linear combination of them with no more than a change of an unimportant overall constant. In this case, consider a linear combination that has two hybrid orbitals that point directly at the H atoms in accord with Pauling s principle of maximum overlap. Using the parameter

[Pg.180]

The antisymmetrized orbital product A i( )2ct>3 is represented by the short hand I 4> l4>24>3 I and is referred to as a Slater determinant. The origin of this notation can be made clear by noting that (1/a/N ) times the determinant of a matrix whose rows are labeled by the index i of the spin-orbital (f>i and whose columns are labeled by the index j of the electron at rj is equal to the above function A (f>i< )2<1>3 = (1/V3 ) det(cf>i (rj)). The general structure of such Slater determinants is illustrated below ... 

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... 

The l/v/4 factor ensures that the wavefunction is normalized, i.e. that IT)2 integrated over all space = 1 [11]). This Slater determinant ensures that there are no more than two electrons in each spatial orbital, since for each spatial orbital there are only two 1-electron spin functions, and it ensures that F is antisymmetric since switching two electrons amounts to exchanging two rows of the determinant, and this changes its sign (Section 4.3.3). Note that instead of assigning the electrons successively to row 1, row 2, etc., we could have placed them in column 1, column 2, etc. of... 

Owing to the indistinguishability of electrons, the wavefunction of a molecule s electron-cloud must be antisymmetric in the coordinates of the electrons. Hence, in the orbital-approximation, the wavefunction of a molecule (whose state corresponds to a set of complete electronic shells) can be expressed as a Slater-determinant, each column or row of which is written in terms of a single spin-orbital 8>. As pointed out, however, by Fock 9> and Dirac 10>, and later stressed by Lennard-Jones n> and Pople 12 the orbitals of a Slater-determinant are not uniquely determined, mathematically. 

This is called a Slater determinant, in honor of physicist J. C. Slater. Since a determinant changes its sign upon the exchange of any two rows or columns, any wavefunction written in determinantal form must be antisymmetric with respect to electron exchange. If we take the determinantal function in eq. (2.2.36) and exchange its two rows ... 

Such a function vanishes because any determinant with two identical rows or columns vanishes. In other words, any system having both electrons in the Is orbital with a spin cannot exist. Now we see there is an alternative way of saying the Pauli Exclusion Principle A wavefunction for a system with two or more electrons must be antisymmetric with respect to the interchange of labels of any two electrons. 

In this set the functions can be classified into two types in the right column the spatial multiplier is symmetric with respect to transpositions of the spatial coordinates and the spin multiplier is antisymmetric with respect to transpositions of the spin coordinates in the left column the spatial multiplier is antisymmetric with respect to transpositions of the spatial coordinates and the spin multipliers are symmetric with respect to transpositions of the spin coordinates. Because in the second case the spatial (antisymmetric) multiplier is the same for all three spin-functions, the energy of these three states will be the same i.e. triply degenerate - a triplet. The state with the antisymmetric spin multiplier is compatible with several different spatial wave functions, which probably produces a different value of energy when averaging the Hamiltonian, thus producing several spin-singlet states. From this example one may derive two conclusions (i) the spin of the many electronic wave function is important not by itself (the Hamiltonian is spin-independent), but as an indicator of the symmetry properties of the wave function (ii) the symmetry properties of the spatial and spin multipliers are complementary - if the spatial part is symmetric with respect to permutations the spin multiplier is antisymmetric and vice versa. 

---