Tensor contracted

So Hirata, Tensor contraction engine abstraction and automated parallel implementation of configuration-interaction, coupled-cluster, and many-body perturbation theories. J. Phys. Chem. A 107, 4940 (2003). 

This procedure reduces Eq. (631) to Eqs. (627) and (628), which are special cases obtained by tensor contraction. 

JJ For discussion of the Tensor Contraction Engine see http //www.csc.lsu.edu/gb/TCE/ and the recent publication of S. Hirata entitled Symbolic Algebra in Quantum Chemistry ... 

Tensors, from the same or different fields, can be combined by outer multiplication, denoted by juxtaposing indices with order preserved on the resultant tensor.33 It is possible that an index is present both in the covariant and contravariant index sets then with the repeated index summation convention, both are eliminated and a tensor of lower rank results. The elimination of pairs of indices is known as contraction, and outer multiplication followed by contraction is inner multiplication.33 In multiplication between tensors, contractions cannot take place entirely within one normal product (i.e., the generalized time-independent Wick theorem see Section IV) hence such tensors are called irreducible. 

The mode specific vibrational root mean squared displacement vector is thus replaced by the mean square displacement tensor of that atom in that specific mode, it has units of A. The tensor is represented in Eq. (A2.85) by a matrix, in more compact notation it is written, for a specific vibrational mode and atom Bi. The colon-operator ( ) or tensor contraction operation is found by taking the trace (Tr) of the product of the two tensors, see Eq. (2.50). Further we write the total mean square displacement tensor of an atom as the sum over all the individual vibrational contributions. 

As a disadvantage, we see particularly the very complex structure of the working equations resulting from these formalisms. This obstacle could be overcome by some automatized implementation tools like the Tensor Contraction Engine (TCE) [35], however, at least some of the methods have also very big computational demands, since solutions for many sectors of the Fock space are required. 

P. Sadayappan, G. Baumgartner, D.E. Bernholdt, R.J. Harrison, S. Hirata, M. Nooijen, R.M. Pitzer and J. Ramanujam The Tensor Contraction Engine, http //www.cse.ohio-state.edu / gb/TCE... 

When we switch to the quantum mechanical treatment in the next section the electronic and nuclear densities will be expressed by the corresponding wave functions and the tensor contraction finds its counterpart in the formalism of product Hilbert spaces. 

As it was mentioned before the general hyperfine Hamiltonian is formulated as a summation of rank-A tensor contractions according to... 

The force constant matrix for atomic displacements is denoted (t,t ), T( t) is the third-rank internal strain tensor, c is a fourth rank elastic constant tensor. Tensor contraction is understood throughout. Thus the restoring force on an atom is... 

The following important rules of tensor analysis should be mentioned Addition and subtraction is only defined for tensors of the same rank and of the same transformation properties (co-/contravariance). For example, adding a matrix and a vector is not valid. Multiplication (also called tensor contraction) is only defined for pairs of indices, where one index is co- and the other is contravariant. As another example, odxi is a valid tensor contraction, but x x is not. 

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