Transform Linear canonical

To summarize the theory dynamic correlations are described by the unitary operator exp A acting on a suitable reference funchon, where A consists of excitation operators of the form (4). We employ a cumulant decomposition to evaluate all expressions in the energy and amphtude equations. Since we are applying the cumulant decomposition after the first commutator (the term linear in the amplimdes), we call this theory linearized canonical transformation theory, by analogy with the coupled-cluster usage of the term. The key features of the hnearized CT theory are summarized and compared with other theories in Table II. 

In the current work, we consider primarily two theoretical models the linearized canonical transformation with doubles (L-CTD) and linearized canonical transformation with singles and doubles (L-CTSD) theories. These are defined by the choice of operators in A. The L-CTD theory contains only two-particle... 

The practical way of calculating 2 is different from that used in the derivation of (4.18). Since 2 is invariant with respect to canonical transformations, it is preferable to seek it in the initial coordinate system. Writing the linearized equation for deviations from the instanton solution 6Q,... 

Let us begin by studying the relative dynamics in the Hamiltonian case, that is, for deterministic driving. The shift (42) to the moving origin is a time-dependent canonical transformation [98]. It transforms the linearized Hamiltonian (33) into... 

They would become the stars of Prigoginian statistical mechanics. Their importance lies in the fact that, whenever it is possible to determine these variables by a canonical transformation of the initial phase space variables, one obtains a description with the following properties. The action variables / ( = 1,2,..., N, where N is the number of degrees of freedom of the system) are invariants of motion, whereas the angles a increase linearly in time, with frequencies generally action-dependent. The integration of the equations... 

Exercise. It was understood in (6.3) that the momenta pk are the actual velocities or linear combinations of them. This need not be true if general canonical transformations are allowed. Show, however, that regardless of the choice of variables there always exists an automorphism x- x having the properties (6.4), (6.5), (6.6), (6.7) and that therefore the proof still holds. 

As a result of this type of Hamiltonian transformation the linear electron-phonon interaction disappears, an effective interaction between the JT centers is created, but the Hamiltonian may become very complicated. The transfer of the mixing of the phonon and electron operators to the other Hamiltonian terms is the price for the accuracy of the canonical transformation. Of course, in this case also the problem can not be solved exactly and some approximations should be applied. 

Using the invariabihty of the trace, we perform the canonical transformation in expression (217) by means of quantum mechanical thermal averaging over Fermi operators with an accuracy to terms linear in the carrier concentration, and then averaging over the phonon bath, we obtain [165]... 

Second, canonical transformation methods may be employed to diagonalize the system-bath Hamiltonian partially by a transformation to new ( dressed ) coordinates. Such methods have been in wide use in solid-state physics for some time, and a large repertoire of transformations for different situations has been developed [101]. In the case of a linearly coupled harmonic bath, the natural transformation is to adopt coordinates in which the oscillators are displaced adiabatically as a function of the system coordinates. This approach, known in solid-state physics as the small-polaron transformation [102], has been used widely and successfully in many contexts. In particular, Harris and Silbey demonstrated that many important features of the spin-boson system can be captured analytically using a variationally optimized small-polaron transformation [45-47]. As we show below, the effectiveness of this technique can be broadened considerably when a collective bath coordinate is first included in the system directly. 

Q j = -f) where is a unit matrix of order n. From linear algebra, it is known that if 0 is an arbitrary nondegenerate skew-symmetric matrix, it can be reduced to the above-mentioned canonical form through a linear nondegenerate transformation of the basis. 

It is also possible to define real normal coordinates n(j) which describe running waves. In contrast to the real normal coordinates a ( ), the coordinates n( ) are not just linear combinations of Q(S) and Q ( ) but also of the momenta P(p and P (. They are coordinates which can be obtained by a canonical transformation, but we shall introduce them here in an elementary way as follows [2.6] ... 

Ham, F.C. (1968) Canonical transformation approach to the linear Jahn-Teller effect E-e. Phys. Rev., 166, 307. 

Barentzen, H. and Polansky, O.E. (1978) Canonical transformation approach to the linear Jahn—Teller effect E0s. J. Chem. Phys., 68, 4398 Barenteen, H. and Polansky, O.E. (1977) Variational approach to the linear Jahn—Teller effect E0e. Chem. Phys. Lett., 49, 121—124. 

For computational purposes it is convenient to work with canonical MOs, i.e. those which make the matrix of Lagrange multipliers diagonal, and which are eigenfunctions of the Fock operator at convergence (eq, (3.41)). This corresponds to a specific choice of a unitary transformation of the occupied MOs. Once the SCF procedure has converged, however, we may chose other sets of orbitals by forming linear combinations of the canonical MOs. The total wave function, and thus all observable properties, are independent of such a rotation of the MOs. 

If A has repeated eigenvalues (multiple roots of the characteristic polynomial), the result, again from introductory linear algebra, is the Jordan canonical form. Briefly, the transformation matrix P now needs a set of generalized eigenvectors, and the transformed matrix J = P 1 AP is made of Jordan blocks for each of the repeated eigenvalues. For example, if matrix A has three repealed eigenvalues A,j, the transformed matrix should appear as... 

In the following discussion we assume that, in the system of Equations (7.6)-(7.8), all lower bounds lj = 0, and all upper bounds Uj = +< >, that is, that the bounds become 0. This simplifies the exposition. The simplex method is readily extended to general bounds [see Dantzig (1998)]. Assume that the first m columns of the linear system (7.7) form a basis matrix B. Multiplying each column of (7.7) by B-1 yields a transformed (but equivalent) system in which the coefficients of the variables ( x,. . . , xm) are an identity matrix. Such a system is called canonical and has the form shown in Table 7.1. 

This cubic can he transformed to canonical form (with the quadratic term missing) hy means of the linear transformation... 

As an example of the application of matrix methods to MO theory, consider the transformation between the set of delocalized canonical Hartree-Fock MOs energy-localized MOs linear combinations of the canonical MOs ... 

We now have to determine a linear map that will transform the observed gamut of colors to the canonical gamut of colors. This map will simply be a diagonal 3x3 matrix of the form... 

If there are n0 open channels at energy E, there are n linearly independent degenerate solutions of the Schrodinger equation. Each solution is characterized by a vector of coefficients aips, for i = 0,1, defined by the asymptotic form of the multichannel wave function in Eq. (8.1). The rectangular column matrix a consists of the two n0 x n0 coefficient matrices ao, < i Any nonsingular linear combination of the column vectors of a produces a physically equivalent set of solutions. When multiplied on the right by the inverse of the original matrix a0, the transformed a-matrix takes the canonical form... 

As seen in Equation 8.10, there is a linear dependence between the input variables or controlled factors that create a nonunique solution for the regression coefficients if calculated by the usual polynomials. To avoid this problem, Scheffe [3] introduced the canonical form of the polynomials. By simple transformation of the terms of the standard polynomial, one obtains the respective canonical forms. The most commonly used mixture polynomials are as follows ... 

Vogin et al. [235] have created a program for the computer design of a free radical reaction mechanism in the gas phase, in agreement with the rules formulated in Sect. 2.5.3. An algorithm has been devised to transform by the computer the formula of a compound, written in the linear notation described in Sect. 6.2.1 [182], into a canonical notation. Thus, the system both preserves the flexibility of a simple natural language and gains the sophistication of a canonical notation. 

Let us start from a linearly independent set = Ot, 2,..., complex functions the set has the additional property that the overlap matrix A = <0 fl>) is nonsingular, i.e., that A 0. Let y be the similarity transformation, which brings A to classical canonical form k with the eigenvalues on the diagonal and Os and Is on the line above the diagonal ... 

The linear terms can be removed from the canonical model by shifting the origin of the zx. .. zk system to the stationary point by the transformation... 

Thus the spectrum which arises when Eq. (8) is Fourier transformed consists of a set of -functions at the energies corresponding to the stationary states of the ion (which via the theorem of Koopmans) are the one-electron eigenvalues of the Hartree-Fock equations). The valence bond description of photoelectron spectroscopy provides a novel perspective of the origin of the canonical molecular orbitals of a molecule. Tlie CMOs are seen to arise as a linear combination of LMOs (which can be considered as imcorrelated VB pairs) and coefficients in this combination are the probability amplitudes for a hole to be found in the various LMOs of the molecule. 

---