Exponential transformations

The orbital rotation is given by a unitary matrix U, which can be written as an exponential transformation. 

Quadralically Convergent or Second-Order SCF. As mentioned in Section 3.6, the variational procedure can be formulated in terms of an exponential transformation of the MOs, with the (independent) variational parameters contained in an X matrix. Note that the X variables are preferred over the MO coefficients in eq. (3.48) for optimization, since the latter are not independent (the MOs must be orthonormal). The exponential may be written as a series expansion, and the energy expanded in terms of the X variables describing the occupied-virtual mixing of the orbitals. 

Equation (7.30) shows that the fundamental information on recombination kinetics is contained in the solution of the scavenger-free case, from which the recombination kinetics with a scavenger may be obtained via an exponential transformation. The scavenger reaction probability is now given by... 

D. A. Mazziotti, Exactness of wave functions from two-body exponential transformations in many-body quantum theory. Phys. Rev. A 69, 012507 (2004). 

The resulting model is a MINLP with linear constraints and nonlinear objective function. The objective function terms can beconvexified using the exponential transformation with the exception of the condenser-cold utility expressions. If we replace the LMTDs with 2-3 times the ATmln, then, the whole objective can be convexified Floudas and Paules (1988). This implies that its global solution can be attained with the OA/ER or the v2-GBD algorithms. 

Previous results show that the EJR method is very promising to solve the convergence problem in MCSCF computations. The only snag is the number of integral transformations to be done in the exact formulation. However, this problem can be obviated if instead of the "exact" EJR algorithm outlined before, some kind of accumulated EJR is used /I/, where the integrals are only transformed after the full set of rotations is done, in the way this problem is treated in the exponential transformations. Preliminary work in this direction shows that the efficiency of the method is kept while the amount of time in the intregral transformations is lowered. 

For one-sided specifications, a or yj > fc, another exponential transformation can be used to change the discrete step function into a sigmoid curve, dj = exp[-exp-(co -I- c y- )... 

The Hermitian h, utilized by Westhaus and co-workers in some [103-115] of their numerical computations, is not really canonical as conditions other than (5.11) are applied to fully specify their exponential transformation operator. Suzuki [116] obtains a recursive scheme for a Hermitian h which he claims to be H. This is not the case, however, because his method is based on multiple transformations, which as mentioned above, produce a Hermitian h that differs [37] from H(-. 

It is worth noting the mutual connections between the commutation relations such as Equation (3.1) and exponential transformations, for example, in the context of Flausdorf s relations. Note that the coordinate transformation is a standard method for the analysis of boundary value problems (see e.g. [25]). An important type of commutation relation is naturally connected with... 

The discussion of orbital relaxation in the preceding section provides us with a clear way of understanding the shortcomings of the TDA. Recall that any Slater determinant formed from a particular set of basis functions may be obtained from any other Slater determinant that lies in the same basis by means of the exponential transformation... 

Using the observed agreement with the variational eigenvalues as the criterion of goodness, these results imply that for general use in this approach an exponential transformation of type 2 may be superior to that of type 1. Its utilization for the calculations described below on hydrogen was found to yield results equally encouraging to those described above. 

To calculate the front velocity for (7.21) we reduce the system to a Hamilton-Jacobi equation. To do so, we employ the hyperbolic scaling (4.33), the exponential transformation (4.35), the large scale limit e 0, and the definitions in (4.38) to obtain the Hamilton-Jacobi equation... 

While the exponential transformation is appealing by virtue of its computational simplicity, it is desirable to reformulate the problem in a non-orthogonal basis for reasons that will become apparent presently. Following Shavitt et al., we define a new set of parameters X as... 

There are many successful ways to increase the efficiency of the Monte Carlo method, in particular by decreasing the dispersion, and these are described in several textbooks. Numerous variance reduction techniques, such as splitting/Russian roulette, weight windows, and the exponential transformation method have been proposed to improve the efficiency of Monte Carlo transport calculations. 

Baker, R.S., Larsen, E.W., "A Local Exponential Transform Method for Global Variance Reduction in Monte Carlo Transport Problem", Karlsruhe, 1993, Vol.2. p.725. 

Approximately 20,000 neutron histories have been computed on the AVIDAC and a partial svimmary of the results obtained is presented in Tables VII and VIII. The fraction s/a is the parameter of the exponential transformation in units of the minimum total cross section in the energy range from zero to the initial energy. The lower entry in parentheses is the total number of neutron histories computed to obtain the result in each case. 

Now we introduce curvy steps. An intuitive interpretation of what is done here is to expand the Taylor series of the exponential transformation to higher orders, such that the step directions are no longer straight lines, but instead they are curved. Invoking the Baker-(Campbell-)Hausdorff lemma (see, e.g.. Ref. 123), the unitary transformation of the density matrix can be written as... 

Despite the formal solution (1.158) was achieved, it still suffers from a lack in analytical shape since the IT-Lambert poses an implicit functional character. In order to improve such implicit solution one can consider in Eq. (1.157) the same binomial to exponential transformation for initial substrate [SJ as previously performed for the instantaneous free substrate [5](0, see Eqs. (1.151) and (1.153), viz. ... 

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