Taylor series transformations

Aspects of the Jahn-Teller symmetry argument will be relevant in later sections. Suppose that the electronic states aie n-fold degenerate, with symmetry at some symmetiical nuclear configuration Qq. The fundamental question concerns the symmetry of the nuclear coordinates that can split the degeneracy linearly in Q — Qo, in other words those that appeal linearly in Taylor series for the matrix elements A H B). Since the bras (/1 and kets B) both transform as and H are totally symmetric, it would appear at first sight that the Jahn-Teller active modes must have symmetry Fg = F x F. There... 

By inverse Fourier transformation of eq. 1 and expansion of both sides in a Taylor series we obtain ... 

Laplace transform is only applicable to linear systems. Hence, we have to linearize nonlinear equations before we can go on. The procedure of linearization is based on a first order Taylor series expansion. 

This problem was solved approximately in 1947 (B2, K9, Jl), wherein it was suggested that the transformed function of y be expanded in a linear Taylor series to provide... 

Note that as long as the unitary transformation is exact, i.e., as long as the Taylor series is infinite and converges, the exact energy eigenvalues of the 4-component untransformed Hamiltonian are obtained. 

This effectively states that the probability of the final state (left-hand side) is equal to that of all the initial states transforming to the final state (with probability P). Chandrasekhar expanded out the infinitesimal velocity and time changes of these quantities as Taylor series and used the Langevin equation to relate 5u and 5f. He showed that if the probability of changing velocity and position is given by a Gaussian distribution, then the probability, W(u, r, t) that a Brownian particle has a velocity u at a position r and at time t is... 

This equation gives the dynamics of the quantum-classical system in terms of phase space variables (R, P) for the bath and the Wigner transform variables (r,p) for the quantum subsystem. This equation cannot be simulated easily but can be used when a representation in a discrete basis is not appropriate. It is easy to recover a classical description of the entire system by expanding the potential energy terms in a Taylor series to linear order in r. Such classical approximations, in conjunction with quantum equilibrium sampling, are often used to estimate quantum correlation functions and expectation values. Classical evolution in this full Wigner representation is exact for harmonic systems since the Taylor expansion truncates. 

Three successive volume elements have concentrations C/ i, c, and cl+1, (Fig. A3.1), and the time increment is At. It is not difficult to show, by Taylor series expansion, that (A3.1) transforms into... 

The Taylor series for the potentials (10) and (11) must be consistent with the corresponding ones for the electric and magnetic fields. To this end let us consider the gauge transformation... 

All the formal manipulations performed so far are exact, and the nuclear evolution is still described at the full quantum level. To proceed to a computable expression [16-21], we now change bath subsystem variables to mean, Rk = [Rk + Rk)/% and difference, Zk = Rk Rk, coordinates (with similar transformation for the bath momenta, Pk = (Pfc + Pk)l and Yk = Pk — Pk) and Taylor series expand the phase in (39) in the difference variables. Truncating this expansion to linear order we obtain the following approximate expression for the correlation function... 

Now we can show the explicit relation with experiment. What is usually measured in spectroscopic or scattering experiments is the spectral density function /(to), which is the Fourier transform of some correlation function. For example, the absorption intensity in infrared spectroscopy is given by the Fourier transform of the time-dependent dipole-dipole correlation function <[/x(r), ju,(0)]>. If one expands the observables, i.e., the dipole operator in the case of infrared spectroscopy, as a Taylor series in the molecular displacement coordinates, the absorption or scattering intensity corresponding to the phonon branch r at wave vector q can be written as (Kobashi, 1978)... 

To avoid this, we use domain perturbation theory (see Section E) to transform from the exact boundary conditions applied at rs = R + sf to asymptotically equivalent boundary conditions applied at the spherical surface rs = R(t). For example, instead of a condition on ur at r = R(t) + ef from the kinematic condition, we can obtain an asymptotically equivalent condition at r = R by means of the Taylor series approximation... 

Because the shape function h is unknown, we transform the boundary conditions to the undisturbed interface position at z7 = 0 by using the domain perturbation technique, which was introduced in previous chapters. Hence we can express eu[ at z = eh in terms of its value at z = 0 by using a Taylor series approximation ... 

Manipulation of symbolic expressions and numerics (e.g., differentiation integration Taylor series Laplace transforms ordinary differential equations systems of linear equations, polynomials, and sets vectors matrices and tensors)... 

Within any decoupling scheme there are only a few restrictions on the choice of the transformations U. First, they have to be unitary and analytic (holomorphic) functions on a suitable domain of the one-electron Hilbert space V, since any parametrization has necessarily to be expanded in a Taylor series around W = 0 for the sake of comparability but also for later application in nested decoupling procedures (see chapter 12). Second, they have to permit a decomposition of in even terms of well-defined order in a given expansion parameter of the Hamiltonian (such as 1/c or V). It is thus possible to parametrize U without loss of generality by a power-series ansatz in terms of an antihermitean operator W, where unitarity of the resulting power series is the only constraint. In the next section this most general parametrization of U is discussed. 

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