Diagonal representation

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... 

Local Minimum. Any Stationary Point on a Potential Energy Surface for which all elements in the diagonal representation of the Hessian are positive. 

Let Rao be the matrix representing the symmetry operation R in the representation rAG. Equation (9.64) means that there is a similarity transformation that transforms RAO to RAO, where RAO is in block-diagonal form, the blocks being the matrices R],R2,...,Rfc of the irreducible representations ri,r2,...,ri. Let rAO denote the block-diagonal representation equivalent to TAO and let A be the matrix of the similarity transformation that converts the matrices of I AO to TAO Rao = A, RaoA. We form the following linear combinations of the AOs ... 

To see the solutions to Eq. (3.13) more clearly we have plotted below its left-and right-hand sides as functions of p for a function in three variables and with Hessian eigenvalues -1, 2, and 3. The step length function has poles at the eigenvalues as can be seen from Eq. (3.13) in the diagonal representation... 

To develop a method to locate saddle points by selectively following one eigenvector we note that at a stationary point all n components of the gradient in the diagonal representation are zero ... 

Hence, in the diagonal representation the gradient and Hessian are identical except for opposite sign in the lowest mode.19 Therefore, a first-order saddle point of the function coincides with a minimum of the image and we may determine the transition state by minimizing the image function. 

To minimize the image function we use a second-order method since in each iteration the Hessian is needed anyway to identify the mode to be inverted (the image mode). Line search methods cannot be used since it is impossible to calculate the image function itself when carrying out the line search. However, the trust region RSO minimization requires only gradient and Hessian information and may therefore be used. In the diagonal representation the step Eq. (5.8) becomes... 

To compare the image and GE methods, notice that in the diagonal representation the quadratic model may be written... 

Thus, we obtain a special class of so-called diagonal representations of so(4) characterized by j0 = 0 (or jl = j2). This is analogous to the situation in angular momentum theory where only integral values of orbital angular momentum are possible. The half-odd-integral values are ruled out because of the particular realization of L in terms of coordinates and momenta. 

To evaluate the first- and second-order molecular properties, we choose the diagonal representation of the Hamiltonian. In this representation, the electronic energy, the electronic gradient, and the electronic Hessian of the electronic ground state 0) may be written in the following manner... 

It is simple to verify that Eq. (S.15) becomes an identity if one uses the diagonal representation of G E). 

An operational statement of the difference between a pure state and a statistical mixture can be made with respect to their diagonal representations. In such representation the pure state density matrix will have only one nonzero element on its diagonal, that will obviously take the value 1. A diagonal density matrix representing a statistical mixture must have at least two elements (whose sum is 1) on its diagonal. 

This follows from the fact that this relation has to hold for the diagonal representation and from the fact that the Trace operation does not depend on the representation used. 

To show that this remains true for mixed states consider the diagonal representation of the initial state, p t = 0) = T i/f )(V . This describes a mixed state in which the probability to be in the pure state is P . In the following time evolution each pure state evolves according to the time-dependent Schrodinger equation so that F (Z) = exp(— i/h H and therefore P also represents the probability to be in state > (/) at time t. It follows that p(Z) = 7 %(Z))(%(Z) so that... 

Like any valid density operator it has to be semi-positive (i.e. no negative eigenvalues) at all time. This is implied by the fact that these eigenvalues correspond to state probabilities in the diagonal representation. Indeed, if the overall density operator p satisfies this requirement it can be shown that so does CT = TrB/5. 

The model (17.19) remains the same, except that Hs is written in this non-diagonal representation as... 

Hence, the operators a in (67) also form a representation of the Weyl-Heisenberg algebra of the electric dipole photons. Employing this transformation (67) then gives the diagonal representation of the operator (63)... 

We saw that an appropriate choice of a local reference frame leads to the diagonal representation (148) of the vacuum polarization matrix (142). The use of the unitary transformation (147) allows the operator polarization matrix (142) to be cast into the form... 

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 ... 

Transform the diagonal representation back to the harmonic oscillator basis by using the inverse transformation of step 1. 

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