Matrix Elements under Time Reversal

MfJIfM Matrix Elements under Time Reversal 

We have already examined the matrix elements of a one-particle operator. The integrals of the one-particle Hermitian operator / have the following relations  

the application of time reversal reduces the number of unique matrix elements by a factor of 2. 

Matrix elements of two-particle operators arise primarily either from the Coulomb or the Breit (or Gaunt) interaction. 

The Coulomb integral (pq rs) is essentially a matrix element over two charge densities. In the relativistic case the permutational symmetry of the integrals is reduced relative to the nonrelativistic case because the functions are in general complex. Thus, we have 

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MfJIfM Matrix Elements under Time Reversal... 

The s.o.c. operator is a one-electron operator which is even under time reversal, and non-totally symmetric in spin and orbit space. The trace of the spin-orbit coupling matrix for the t2g-shell thus vanishes. As a result the s.o.c. operator is found to transform as the MK = 0 component of a pure quasi-spin triplet (Cf. Eq. 26). Application of the selection rule in Eq. 28 shows that allowed matrix elements must involve a change of one unit in quasi-spin character, i.e. AQ = 1. Since 4S and 2D are both quasi-spin singlets while 2P is a quasispin triplet, s.o.c. interactions will be as follows ... 

The interaction occurs between half-filled shell states K) and IL) which possess different parities with respect to CL, ttk and ttl, respectively. This implies that necessarily the configurations are different, i.e., I A") lL), and also that the product of parities ttkttl has the value — 1. The hypothesis that Tt% is antisymmetric under time reversal is represented in equation (32) by setting thm = —1. It follows that the interaction matrix element (LlH lK) must vanish if these criteria are met. 

Further, if it is hypothesized that be symmetric under time reversal (thm = +1) then it follows from equation (32) that the off diagonal matrix elements must necessarily vanish. 

The diagonal matrix elements between half-filled shell states are now considered. If it is assumed that the interaction operator is symmetric under time reversal (also as in case 2), then thm = +1. The diagonal interaction elements are just the expectation value of in the closed shell, which is zero if is not totally symmetric under spatial operations (Another way of saying this is that (H%) vanishes if Mr K X K KMKr ), but obviously has time reversal parity +1. It now follows that if the above criteria are met then the diagonal matrix elements must vanish. 

The elements of the correlation function matrix C (t) have the following transformation properties under time reversal... 

Operators, Matrix Elements, and Wave Functions under Time-Reversal Symmetry... 

Transforming the Hamiltonian matrix elements into this basis and making use of the behavior of the functions under time reversal, we have... 

As we have shown elsewhere, the coefficients in this expression correspond to the elements of the bond-order matrix [4]. Because the JT Hamiltonian is Hermitian and invariant under time reversal, it is represented by a symmetric matrix in a real function space. The symmetries of this interaction matrix will therefore correspond to the symmetrized direct square of the degenerate irrep of the function space. This part of the direct square is represented by square brackets. The corresponding character is given by [8] ... 

It can be proven generally (19) that the matrix elements of operators invariant under time reversal (such as ) in the time-reversal invariant... 

The latter result (82) yields a quantum probability amplitude that, under Hermitian conjugation and time reversal, correctly equates to the corresponding amplitude for the time-inverse process of degenerate downconversion. To see this, we note that the matrix element for SHG invokes the tensor product Py (—2co co, ) p([/lC., where the brackets embracing two of the subscripts (jk) in the radiation tensor denote index symmetry, reflecting the equivalence of the two input photons. As shown previously [1], this allows the tensor product to be written without loss of generality as ( 2co co, co), entailing an index-symmetrized form of the molecular response tensor,... 

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