Operators complex antisymmetric

Following Ref. [5] the T1 condition is obtained by considering an operator A = Y ij gij,kaiajak, where the gij k are arbitrary real or complex coefhcients totally antisymmetric in the three indices. (We view g as a vector of dimension (0, where r is the size of the one-electron basis.) The contractions (t / A+A t /) and (t / AA+ t /) both involve the 3-RDM, but with opposite sign, and so the nonnegativity of (tk 4 4 -f AA I ) for all three-index functions g provides a representability condition involving only the 1-RDM and 2-RDM. In exphcit form the condition is of semidefinite form, 0 T, where the Hermitian matrix T is... 

The T2 condition, slightly strengthened from Ref. [5], but in the real case aheady contained in Ref. [4], is obtained by considering operators A = Sij,kPi OjOk and B = hiai for arbitrary real or complex coefhcients gij k and hi, with gij antisymmetric in (/, fe). (So this g may be compressed to... 

One of the pedagogically unfortunate aspects of quantum mechanics is the complexity that arises in the interaction of electron spin with the Pauli exclusion principle as soon as there are more than two electrons. In general, since the ESE does not even contain any spin operators, the total spin operator must commute with it, and, thus, the total spin of a system of any size is conserved at this level of approximation. The corresponding solution to the ESE must reflect this. In addition, the total electronic wave function must also be antisymmetric in the interchange of any pair of space-spin coordinates, and the interaction of these two requirements has a subtle influence on the energies that has no counterpart in classical systems. 

The matrix elements in both BRe and Bim will, for both uniform and exterior complex scaling, be built from terms which, when the matrix representation of the rotated operator is constructed, are multiplied with complex constants. This will make the matrices BRe and BIm complex, but they will still be symmetric and antisymmetric, respectively, with respect to transposition, i.e.,... 

It is common to use an alternative notation for the electron-electron interaction integrals (rr ss ) = (rs r s ), i.e., with the complex conjugate spin orbitals, belonging to the operators, on the left. One can introduce the compact notation (rs tu) = (rs fn) — rs vt) for an antisymmetric interaction integral and write... 

This can be readily understood by recalling (see equation (A4) that P is essentially a nondiagonal antisymmetric momentum matrix its elements are those of a differential operator, and close scrutiny shows that eigenvectors change drastically at the ridge from those of the transition state (an intermediate complex) to those of the separated collision partners. It is easy to show that asymptotically P matrix elements decay as p l, and tend to overlaps of vibrational wave-functions of diatoms. 

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