Strong orthogonality condition

To build up vin the cluster function (1) we use the functions (PvA vA2---fvBi 9vs2 -- all of which satisfy the strong orthogonality condition in the sense of to (2), but do not satisfy the strong orthogonality needed for (1) We therefore consider the linear combination... 

The second term of (13) can be evaluated with the help of (8) and the knowledge that the RS can only involve, at most, two-electron operators. If any VRS contains only two-electron operators and the R are constrained by the strong orthogonality condition (11) then it is obvious that only the first two terms in the expansion (8) of Ax give rise to non-zero contributions to... 

In this case the spin and symmetry of the function eq. (2.104) coincide with the spin and symmetry of the wave functions of the ( /-system 4>)(. An assumption that the functions dn and1 satisfy the strong orthogonality condition of eq. (1.185) together with the variational principle yields a pair of the coupled equations for the functional multipliers ... 

The use of the Serber basis for spin functions combined with the strong orthogonality condition (100) ensures that separated pair functions (98)... 

Because of the strong orthogonality condition, two-electron functions are easier to construct, and to deal with. As mentioned above, a straightforward way of ensuring SO is to expand the geminals in mutually exclusive and orthogonal subspaces. This seems to be a very severe restriction, but, because of a famous statement that we call Arai s theorem, the fact is that the existence of an expansion of this kind and SO are completely equivalent. The theorem can be formulated as follows. 

One may observe that under the strong orthogonality condition (15), due to the summation over A, this product matrix vanishes ... 

The fact that these 1 DM y and 2 DM y2 have been calculated from the demonstrably antisymmetric many-body wave function (3) using the strong orthogonality conditions (1) is important for what follows in the context of density functional theory (DFT) to which we now turn. 

Alternative partitions of the whole system into groups according to Eq. (3.5) are possible. One of them is into K shell groups and a valence shell group, which one may call the K-V separation. It is in some respects better justified than the o—n separation and has been checked by ab initio calculations of small molecules 5>6>. One drawback of the K-V separation is that the strong orthogonality condition is not automatically satisfied for symmetry reasons, as in the o—n separation. However, this does not lead to serious difficulties. 

Then the so-called separated pair approximation is obtained.79 This approximation has been applied to several molecules.161-164 Further generalization of the method is possible by removing the strong orthogonality condition on the pair-functions. Such a completely generalized pair-function method has recently been applied to some very small diatomic systems.165... 

Ref. [14]). Hence, the pair energy is identified by the set T given in Eq. (10). Note from Eq. (18) that, for PWs defined by/i and/2 greater than the maximum angular momentum quantum number of the occupied orbitals, Zocc, the strong orthogonality conditions are automatically satisfied, and we have... 

McWeeny assumed that the group functions are completely arbitrary, i.e. they can correspond to highly correlated accurate wave functions of the individual groups. The only restriction, imposed upon the group functions was the strong orthogonality condition ... 

By the virtue of the strong orthogonality condition the two-particle density matrix of the total system could... 

This form of Qjk is not substantially simpler because we could not eliminate the offdiagonal terms. One may require also the so-called strong orthogonality condition ... 

Y.q 4>q 4>p)- The Strong orthogonality condition for the outgoing electron implies... 

With a strong orthogonality condition imposed on the scattered electron orbital, fulfilling the killer condition Po) = 0 the Auger amplitude resulting from the Fermi golden rule expression (Eq. 3.37) takes the form... 

With the single-channel and strong orthogonality condition imposed on the continuum electron orbital, a many-body factor is obtained that is only dependent on the characteristics of the residual bound states. A conventional MO analysis of the spectra is entailed only if there is just one large many-body term in the summation and if that term is close to 1. In that case one further reduces the MO factor in terms of, for example, local decomposition of symmetries and charges. One notes that for Auger a continuum orbital enters explicitly in the MO factor (as it does also in the photoionization case) the calculation of the MO factor will be discussed in a following section. 

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