Operator space diagonalization

Consequently, Eqs. (43) and (59) are identical, for applications in a 3D parameter space, except that the vector product in the former is expressed as a commutator in the latter. Both are computed as diagonal elements of combinations of strictly off-diagonal operators and both give gauge independent results. Equally, however, both are subject to the limitations with respect to the choice of surface for the final integration that are discussed in the sentence following Eq. (43). 

Both Simons and Yeager employ the 3-block basis operators as the secondary operator space, retaining only portions of the diagonal matrix elements thereof. When the correlation coefficients are calculated by RSPT and the 5-block operators (i.e., a al,a ala and a ala ala ) are Schmidt orthogonalized to the simple electron removal operators (the 1-block), the matrix vanishes through first order. Therefore, the 5-block basis operators do not contribute until fourth order [since (37) is bilinear in. 4 ]. Differences between the approaches of Yeager and Simons are described more fully and tested numerically in Section III.A. 

The EOM results for BH are well converged even when the primary operator space is restricted to simple ionization basis operators (the dimension of App is 15), yielding an IP that is 0.2 eV above the best Cl case (SDT, relaxed). Since there are no shake-up basis operators in the P-space, this calculation does not contain any effects due to the 5-block basis opeators or any off-diagonal couplings. (Only the diagonal terms in Aqq through first order are retained.)... 

There can be seen that, not only the Hamilton operator could be written as a diagonal matrix, but the elements of the EH space too. It is only necessary to take into account the isomorphism between the extended wavefunction form and a diagonal structure, which can be defined employing a diagonal operator instead of the vector one used in writing equation (II) ... 

Note that the metric operator fi commutes with all diagonal operators in the space Y. 

In the above discussion of relaxation to equilibrium, the density matrix was implicitly cast in the energy representation. However, the density operator can be cast in a variety of representations other than the energy representation. Two of the most connnonly used are the coordinate representation and the Wigner phase space representation. In addition, there is the diagonal representation of the density operator in this representation, the most general fomi of p takes the fomi... 

The Hamiltonian again has the basic form of Eq. (63). The system is described by the nuclear coordinates, Q, which are relative to a suitable nuclear configuration Q. In conbast to Section in.C, this may be any point in configmation space. As a diabatic representation has been assumed, the kinetic energy operator matrix, T, is diagonal with elements... 

These six matrices can be verified to multiply just as the symmetry operations do thus they form another three-dimensional representation of the group. We see that in the Ti basis the matrices are block diagonal. This means that the space spanned by the Tj functions, which is the same space as the Sj span, forms a reducible representation that can be decomposed into a one dimensional space and a two dimensional space (via formation of the Ti functions). Note that the characters (traces) of the matrices are not changed by the change in bases. 

Suppose that B is an operator defined in configuration space so that its matrix is diagonal and of the form... 

When spin-orbit coupling is introduced the symmetry states in the double group CJ are found from the direct products of the orbital and spin components. Linear combinations of the C"V eigenfunctions are then taken which transform correctly in C when spin is explicitly included, and the space-spin combinations are formed according to Ballhausen (39) so as to be diagonal under the rotation operation Cf. For an odd-electron system the Kramers doublets transform as e ( /2)a, n =1, 3, 5,... whilst for even electron systems the degenerate levels transform as e na, n = 1, 2, 3,. For d1 systems the first term in H naturally vanishes and the orbital functions are at once invested with spin to construct the C functions. 

The quantum-mechanical state is represented in abstract Hilbert space on the basis of eigenfunctions of the position operator, by F(q, t). If the eigenvectors of an abstract quantum-mechanical operator are used as a basis, the operator itself is represented by a diagonal square matrix. In wave-mechanical formalism the position and momentum matrices reduce to multiplication by qi and (h/2ni)(d/dqi) respectively. The corresponding expectation values are... 

The previous argument is valid for all observables, each represented by a characteristic operator X with experimental uncertainty AX. The problem is to identify an elementary cell within the energy shell, to be consistent with the macroscopic operators. This cell would constitute a linear sub-space over the Hilbert space in which all operators commute with the Hamiltonian. In principle each operator may be diagonalized by unitary transformation and only those elements within a narrow range along the diagonal that represents the minimum uncertainties would differ perceptibly from zero. 

In the previous discussion the semiclassical separation of particles and antiparticles employed projection operators and the associated subspaces of the Hilbert space. By suitable choices of bases such a separation can also be constructed with the help of unitary operators rotating the Hamiltonian into a block-diagonal form. Such a procedure is closely analogous to the Foldy-Wouthuysen transformation that provides a similar separation in a non-relati-vistic limit. A (unitary) semiclassical Foldy-Wouthuysen transformation Usc rotates the Dirac-Hamiltonian Hd into... 

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