INDEX permutation symmetry

The notation < i j k 1> introduced above gives the two-electron integrals for the g(r,r ) operator in the so-called Dirac notation, in which the i and k indices label the spin-orbitals that refer to the coordinates r and the j and 1 indices label the spin-orbitals referring to coordinates r. The r and r denote r,0,( ),a and r, 0, ( ), a (with a and a being the a or P spin functions). The fact that r and r are integrated and hence represent dummy variables introduces index permutational symmetry into this list of integrals. For example,... 

Besides the elementary properties of index permutational symmetry considered in eq. (7), and intrinsic point group symmetry of a given tensor accounted for in eqs. (8)-(14), much more powerful group-theoretical tools [6] can be developed to speed up coupled Hartree-Fock (CHF) calculations [7-11] of hyperpolarizabilities, which are nowadays almost routinely periformed in a number of studies dealing with non linear response of molecular systems [12-35], in particular at the self-consistent-field (SCF) level of accuracy. 

The algebraic equations and efficient computational sequences were derived by smith and reported by us [33] for CCSD-, CCSDT-, and CCSDTQ-R12, their excited-state analogues via the equation-of-motion (EOM) formalisms (EOM-CC-R12 up to EOM-CCSDTQ-R12), and the so-called A equations for the analytical gradients and response properties, again up to A-CCSDTQ-R12. The full CCSD-, CCSDT-, and CCSDTQ-R12 methods [34,35] were implemented by smith into efficient computer codes that took advantage of spin, spatial, and index-permutation symmetries. 

Kleinman symmetry (index permutation symmetry). Far from resonances of the medium where dispersion is negligible, the susceptibilities become to a good approximation invariant with respect to permutation of all Cartesian indices (without simultaneous permutation of the frequency arguments). This property is called Kleinman symmetry (Kleinman, 1962). It is important in the discussion of the exchange of power between electromagnetic waves in an NLO medium. In many cases approximate validity of Kleinman symmetry can be used effectively to reduce the number of independent tensor components of an NLO susceptibility. 

In the case of permutation isomers with some indistinguishable ligands and/or with skeletal symmetry there are different but equivalent ligand permutations leading to the reference isomer of the configuration. Among the numerically ordered equivalent ligand index permutations that one is chosen for the descriptor which contains the lowest number of transpositions ). 

For special choices of the involved frequencies with respect to the resonances of the material the permutation symmetry of the nonlinearity can be further developed. These permutation symmetries are normally referred to as Kleinman relations [7]. If none of the frequencies is in resonance with a transition frequency of the material and consequently no absorption occurs, then also the spatial index i and the frequency o>4 of the nonlinear polarization can be permuted. 

It is to be noticed the all linear transformations which are obtained by contracting all but one index of a given tensor, yield vectors which all have the structure of a gradient. All that is needed is therefore a general gradient formulation which allows for all kinds of spin combinations and which does not assume hermitean symmetry or permutation symmetry in the two-electron integrals. 

Intrinsic permutation symmetry was already used in (18)-(21) so that only one of the equivalent terms occurs in the equation. Far from resonances of the medium and in the limit /2, wi, W2,... o -+ 0 the permutation symmetry of (25) can be extended to include the first Cartesian index p, and the induced frequency fl. This property, if at least valid to a good approximation, is called overall permutation symmetry. 

Other important systems are uniaxial isotropic systems, because the widely studied poled polymers belong to this symmetry class (o°w or Co ). of such systems has seven non-vanishing components of which four are independent, = X k, x9k = xSk = X lx and x9k- For the SHG susceptibility X -2u> o),u)) the number of independent components reduces to three because of intrinsic permutation symmetry in the second and third index. If the uniaxial system is created by poling of an isotropic system by an external electric field, e.g. a poled polymer or liquid, then to first order in the applied field, z, the number of independent components of x -2a> w, w) is only two (Kielich, 1968). It is thus equal to the number of independent components of x -2a) a),o),0) because of (28). 

The number of independent components may be further reduced by intrinsic or Kleinman symmetry (cf. p. 131). Intrinsic permutation symmetry, /3 , = holds for the second-order polarizability /3 ,(-2w Kleinman symmetry, i.e. permutation symmetry in all Cartesian indices (cf. p. 131), generally holds only in the limit w—>0. 

There is an even more compact way of defining the Kramers single-replacement operators, Xfq, which also has the advantage of displaying the permutational symmetry. To do this, we introduce two auxiliary operators. One is the bar-reversal operator, Kp, which is the time-reversal operator for a spinor with index p. The effect of this operator is... 

Although the order of frequencies in coqOJ — (o ) has no special meaning itself, owing to the intrinsic permutation symmetry, we will keep it fixed (Raman convention), using co and -co as second and third arguments. Besides the intrinsic permutation symmetry, the medium macroscopic symmetry imposes further restrictions on the tensor index of nonvanishing, independent components A very important result is that the nth-order susceptibility vanishes for even n in media showing inversion symmetry and contributes the low-est-order nonlinearity. For isotropic media, it can be shown that vanishes if some Cartesian index appears an odd number of times in the subscript. 

The general problem is now clear the quantities i /,. p are tensor components, with respect to the group U(m), and we want to find linear combinations of these components that will display particular symmetries under electron permutations and hence under index permutations. Each set of symmetrized products, with a particular index symmetry, will provide a basis for constructing spin-free CFs (as in Section 7.6) for states of given spin multiplicity and in this way the full-CI secular equations will be reduced into the desired block form, each block corresponding to an irreducible representation of U(m). It is therefore necessary to study both groups U(m), which describes possible orbital transformations, and which provides a route (via the Young tableaux of Chapter 4) to the construction of rank-N tensors of particular symmetry type with reject to index permutations. 

Show that by suitably combining two functions of the set of six found in Problem 10.4 it is possible to obtain a ffinction that behaves like d and that it is possible to find another combination that is invariant under all rotations (i.e. is of S type). [Note This problem and the previous one illustrate the classification of 2-electron wavefunctions, forming a 2-electron tensor space , with respect to their behaviour under transformations induced by rotations. In Chapter 10 the emphasis has been on transformations of the full linear group (or one of its subgroups) but in both cases symmetry with respect to index permutations is of importance in reducing the representation carried by the tensor space into its irreducible components.]... 

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