Diagonal representability problem

It must be reemphasized that the exact nature of [( ] is not necessary to the physical solution of our problem. Because the normal-coordinate approach merely represents a linear transformation of the real coordinates, the motion of the polymer represented by all the qls will be identical to the motion of the polymer represented by all the jc/s. Our problem thus becomes the rather simple one of finding a diagonal representation of the (z + 1) x (z + 1) matrix [A]. This rather well known result (a similar form applies in the treatment of a vibrating string, among others) is derived in the appendix at the end of this chapter, and is merely stated here ... 

Finally, we consider the complete molecular Hamiltonian which contains not only temis depending on the electron spin, but also temis depending on the nuclear spin / (see chapter 7 of [1]). This Hamiltonian conmiutes with the components of Pgiven in (equation Al.4,1). The diagonalization of the matrix representation of the complete molecular Hamiltonian proceeds as described in section Al.4,1.1. The theory of rotational synnnetry is an extensive subject and we have only scratched the surface here. A relatively new book, which is concemed with molecules, is by Zare [6] (see [7] for the solutions to all the problems in [6] and a list of the errors). This book describes, for example, the method for obtaining the fimctioiis ... 

The next step might be to perform a configuration interaction calculation, in order to get a more accurate representation of the excited states. We touched on this for dihydrogen in an earlier chapter. To do this, we take linear combinations of the 10 states given above, and solve a 10 x 10 matrix eigenvalue problem to find the expansion coefficients. The diagonal elements of the Hamiltonian matrix are given above (equation 8.7), and it turns out that there is a simplification. 

Section II discusses the real wave packet propagation method we have found useful for the description of several three- and four-atom problems. As with many other wave packet or time-dependent quantum mechanical methods, as well as iterative diagonalization procedures for time-independent problems, repeated actions of a Hamiltonian matrix on a vector represent the major computational bottleneck of the method. Section III discusses relevant issues concerning the efficient numerical representation of the wave packet and the action of the Hamiltonian matrix on a vector in four-atom dynamics problems. Similar considerations apply to problems with fewer or more atoms. Problems involving four or more atoms can be computationally very taxing. Modern (parallel) computer architectures can be exploited to reduce the physical time to solution and Section IV discusses some parallel algorithms we have developed. Section V presents our concluding remarks. 

The evaluation of the action of the Hamiltonian matrix on a vector is the central computational bottleneck. (The action of the absorption matrix, A, is generally a simple diagonal damping operation near the relevant grid edges.) Section IIIA discusses a useful representation for four-atom systems. Section IIIB outlines one aspect of how the action of the kinetic energy operator is evaluated that may prove of general interest and also is of relevance for problems that require parallelization. Section IIIC discusses initial conditions and hnal state analysis and Section HID outlines some relevant equations for the construction of cross sections and rate constants for four-atom problems of the type AB + CD ABC + D. 

An example of the application of Eq. (47) is provided by the group < 3v whose symmetry operations are defined by Eqs. (18). If the same arbitrary function,

symmetry operation can be worked out, as shown in the last column of Table 13. With the use of the projection operator defined by Eq. (47) and the character table (Table 6), it is found (problem 16) that the coordinate z is totally symmetric (representation Ai). However, it is the sum xy + zx that is preserved in the doubly degenerate representation, E. It should not be surprising that the functions xy and zx are projected as the sum, because it was the sum of the diagonal elements (the trace) of the irreducible representation that was employed in each case in the... 

There are also some unexpected problems, related to the fact that the stationarity conditions do not discriminate between ground and excited states, between pure states and ensemble states, and not even between fermions and bosons. The IBQ give only information about the nondiagonal elements of y and the Xk, whereas for the diagonal elements other sources of information must be used. These elements are essentially determined by the requirement of w-representability. This can be imposed exactly to the leading order of perturbation theory. Some information on the diagonal elements is obtained from the lCSE,t, though in a very expensive and hence not recommended way. The best way to take care of -representability is probably via a unitary Fock-space transformation of the reference function, because this transformation preserves the -representability. 

The ACSE has important connections to other approaches to electronic structure including (i) variational methods that calculate the 2-RDM directly [36-39] and (ii) wavefunction methods that employ a two-body unitary transformation including canonical diagonalization [22, 29, 30], the effective valence Hamiltonian method [31, 32], and unitary coupled cluster [33-35]. A 2-RDM that is representable by an ensemble of V-particle states is said to be ensemble V-representable, while a 2-RDM that is representable by a single V-particle state is said to be pure V-representable. The variational method, within the accuracy of the V-representabihty conditions, constrains the 2-RDM to be ensemble N-representable while the ACSE, within the accuracy of 3-RDM reconstruction, constrains the 2-RDM to be pure V-representable. The ACSE and variational methods, therefore, may be viewed as complementary methods that provide approximate solutions to, respectively, the pure and ensemble V-representabihty problems. 

Constraints on the diagonal element of the density matrix can be useful in the context of the density matrix optimization problem, Eq. (8). As Weinhold and Wilson [23] stressed, the A-representability constraints on the diagonal elements of the density matrix have conceptually appealing probabilistic interpretations this is not true for most of the other known A-representability constraints. 

A complementary approach to the parabolic barrier problem is obtained by considering the Hamiltonian equivalent representation of the GLE. If the potential is parabolic, then the Hamiltonian may be diagonalized" using a normal mode transformation. One rewrites the Hamiltonian using mass weighted coordinates q Vmd. An orthogonal transformation matrix... 

The triangular molecule has [21,22] two normal vibrational coordinates that span two irreducible representations A. In order to diagonalize the nuclear kinetic and potential energy we solve the classical problem of normal vibrations. In this way one obtains two A coordinates... 

---