Exponential Operators

It can be shown [ ] that the expansion of the exponential operators truncates exactly at the fourth power in T. As a result, the exact CC equations are quartic equations for the t y, etc amplitudes. The matrix elements... 

An alternative to split operator methods is to use iterative approaches. In these metiiods, one notes that the wavefiinction is fomially "tt(0) = exp(-i/7oi " ), and the action of the exponential operator is obtained by repetitive application of //on a function (i.e. on the computer, by repetitive applications of the sparse matrix... 

The problem is then reduced to the representation of the time-evolution operator [104,105]. For example, the Lanczos algorithm could be used to generate the eigenvalues of H, which can be used to set up the representation of the exponentiated operator. Again, the methods are based on matrix-vector operations, but now much larger steps are possible. 

M. Hochbruck and Ch. Lubich On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34 (1997) (to appear)... 

Coupled cluster is closely connected with Mpller-Plesset perturbation theory, as mentioned at the start of this section. The infinite Taylor expansion of the exponential operator (eq. (4.46)) ensures that the contributions from a given excitation level are included to infinite order. Perturbation theory indicates that doubles are the most important, they are the only contributors to MP2 and MP3. At fourth order, there are contributions from singles, doubles, triples and quadruples. The MP4 quadruples... 

Rotations are likewise unitary transformations, and we shall see that they can also be represented by an exponential operator. Let D(a) be a rotation about the z-axis, so that... 

Equation (4) is a three-term recursion for propagating a wave packet, and, assuming one starts out with some 4>(0) and (r) consistent with Eq. (1), then the iterations of Eq. (4) will generate the correct wave packet. The difficulty, of course, is that the action of the cosine operator in Eq. (4) is of the same difficulty as evaluating the action of the exponential operator in Eq. (1), requiring many evaluations of H on the current wave packet. Gray [8], for example, employed a short iterative Lanczos method [9] to evaluate the cosine operator. However, there is a numerical simplification if the representation of H is real. In this case, if we decompose the wave packet into real and imaginary parts. 

Equation 4.5 shows that the rate constant, k, is related to the activation energy, Ea, of the reaction by an inverse exponential operation. This means that the greater the activation energy, the smaller the rate constant, i.e., it is difficult to get the reactants to meet at high enough energies for the reaction to progress. The pre-exponential factor is a constant that includes information about how orientation of the reactant species to one another and the... 

However, these formal expressions are of no great help until we know how to operate with the exponential operator, which is very complicated because it involves the full iV-body problem with the interactions between the particles. In order to circumvent this difficulty, we shall use a resolvant technique 89 we define a resolvant operator (L — 2)-1, function of the complex variable z, and write ... 

The matrix elements of the operator (2.79) can be calculated by making use of Eq. (2.78) and of the usual formula for expansion of exponential operators. An alternative is to recognize, using Eq. (2.28), that f can be thought of as a rotation operator so that its matrix elements can be computed (Levine and Wulf-man, 1979) using the known results for the rotation matrices. 

The matrix elements of the operator (2.140) must, in general, be evaluated numerically.9 However, when N is large (a situation that is almost always encountered in actual spectra), the matrix elements of the exponential operator can be evaluated in closed form. Since the operator ft is a scalar, its matrix elements do not depend on J and one has... 

The matrix elements of the exponential operator are given by Eq. (2.141). Only the selection rule AJ = 0 remains for this operator. 

A source of error in the CASSCF(6e, 6o)-based methods is an incomplete treatment of the active-core relaxation. Although some effects of active-core relaxation are incorporated via the exponential operator in the CT calculations, this is incomplete due to the truncation of some operators in the ccaa class as explained in Section lll.C. Comparing CAS(10e, 80) with CAS(6e, 60) shows us the effects of the truncation. At the equilibrium structure (rNN = 1.15 A), we observe that the L-CTSD energy with CAS(6e, 60) is 3.5 mEh higher than that with CAS(10e, 80). For comparison, the MRMP energy with CAS(6e, 60) is 6.7 mEh higher that that with CAS(10e, 80). Thus the truncated ccaa operators... 

We have so far examined the performance of the canonical transformation theory when paired with a suitable multireference wavefunction, such as the CASSCF wavefunction. As we have argued, because the exponential operator describes dynamic correlation, this hybrid approach is the way in which the theory is intended to be used in general bonding situations. However, we can also examine the behavior of the single-reference version of the theory (i.e., using a Hartree-Fock reference). In this way, we can compare in detail with the related... 

A different way, developed extensively by Schwartz and his coworkeis, - is to use approximate quantum propagators, based on expansions of the exponential operators. These approximations have been tested for a number of systems, including comparison with the numerically exact results of Ref 38 for the rate in a double well potential, with satisfying results. 

Application of the Trotter factorization for the exponential operator appearing in Eq. (6) leads to the expression... 

If we now expand the exponential operators to second order we obtain for the operator in (4 6) the following expression ... 

A quantum-classical approximation for eq.(5) can be obtained by expanding the exponential operator in a power series of the reduced Planck constant h (or, equivalently, in a power series of (m/M)1/2 [11])... 

The logarithm is the inverse of the exponential operation. Thus, if y = ax we define x = loga y (read as log base a of y ). Again the common logarithms are base 10 (often written log ) and base e (often written In ) base 2 also appears occasionally. From the definition,... 

This expansion is valid to second order with respect to St. This is a convenient and practical method for computing the propagation of a wave packet. The computation consists of multiplying X t)) by three exponential operators. In the first step, the wave packet at time t in the coordinate representation is simply multiplied by the first exponential operator, because this operator is also expressed in coordinate space. In the second step, the wave packet is transformed into momentum space by a fast Fourier transform. The result is then multiplied by the middle exponential function containing the kinetic energy operator. In the third step, the wave packet is transformed back into coordinate space and multiplied by the remaining exponential operator, which again contains the potential. 

Now, the direct damping of the high frequency mode may be incorporated in this ACF by multiplying it by a damping exponential operator of the form exp —y°t. As a consequence, the SD, which is according to Eq. (4) the Fourier transform of the damped ACF, takes the form ... 

After using a circular permutation of the bra and kets that do not belong to the spaces where the exponential operators work and then, after simplification using the orthogonality property ( 0) = 8 it reads... 

Besides P, which showed not be confused with the momentum operator P of the H-bond bridge, is the Dyson time-ordering operator [57] acting on the Taylor expansion terms of the exponential operator in such a way so that the time arguments involved in the different integrals will be t > t > t". 

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