Kernels Tensor product

Note that the most negative eigenvalue of the tensor-producted operator within the square brackets is simply 4ra-1(—2) = —22n 1. Because of this we may immediately place a lower bound that any kernel function may take as... 

The tensor product of two or more kernels is also a kernel function ... 

We denote the tensor of such elements as Dp, which is the tensor representation of the kernel Dp in a basis of p-electron direct products of the spin orbitals 4>j) [46]. The convention introduced in Eq. (8), that the number of indices implicitly specifies the tensor rank, is followed wherever tensors are used in this chapter. 

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. 

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