Algebra permutations

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. 

Permutations are unitary operators as seen in Eq. (5.27). This tells us how to take the Hermitian conjugate of an element of the group algebra. 

For many purposes it is convenient to eliminate the restriction (1.1) by the following purely algebraic device. One allows each xa to range from — oo to +oo, but agrees that all s sets ti,t2,... ts that are the same apart from a permutation correspond to one and the same state. In addition one extends the definition of Qs xl9 t2,. .., t5) to the whole s-dimensional space by stipulating that it is a symmetric function of its variables. The normalization condition (1.2) may then be written... 

The same result could have been obtained more succinctly by starting with Eq. 11.40 and carrying out the manipulations algebraically (as in Eq. 11.38). The algebraic calculations depend on the fact that the matrices for Ix, Iy, and lz are traceless and on the cyclic permutation relations IrIs = ilt among Ix, Iy, and I2. 

According to these definitions of constitutional chemistry and stereochemistry, they are two disjoint aspects of chemistry. It follows that their treatment by qualitative mathematics must differ profoundly. The logical structure of constitutional chemistry is represented by the algebra of be- and r-matrices. The logical structure of stereochemistry is essentially permutation group theoretical in nature. 

Similarly, the external particle lines in the rightmost diagram must be permuted in its algebraic expression ... 

When the permutation operators are expanded, these expressions are identical to those given in Eq. [145] derived earlier using Wick s theorem and some complicated algebra. 

Careful inspection, however, reveals that the diagrams are equivalent because one can be produced from the other by permutation of the hole or particle lines on the T2 fragment. (This equivalence can also be proven algebraically, and the reader is encouraged to carry this analysis out independently.)... 

The permutation operators appear in order to maintain the antisymmetry of the algebraic expressions, as explained earlier. Note that the factors of lA appearing in the second and third terms result from both a pair of equivalent lines and a pair of equivalent vertices in each of the corresponding diagrams. 

Rule 4. Generate the algebraic expressions corresponding to the non-canonical diagrams directly from the algebraic expressions for the canonical diagrams by applying permutation operators. 

To maintain full antisymmetry of an amplitude, the algebraic expression for a diagram should be preceded by a permutation operator permuting the open lines in all distinct ways, 2P(— )pP. 

The choice of translationally invariant electronic coordinates makes He(te) trivially invariant under permutations of the original electronic coordinates and independent of any particular choice of translationally invariant nuclear coordinates. Similarly Hn(tn) is independent of any particular choice of translationally invariant electronic coordinates and can be shown, after some algebra, to be invariant under any permutation of the original coordinates of identical nuclei. The interaction operator Hen(tn,te) is obviously invariant under a permutation of the original electronic coordinates and, again after a little algebra, can be shown to be invariant under any permutation of the original coordinates of identical nuclei. 

---