Taylor expansion representation

Here and throughout, x = sign(x). These are small-time expansions, but they are not Taylor expansions, as the appearance of nonanalytic terms indicates. From the parity and symmetry rules, the odd coefficients are block-adiagonal, and the even coefficients are block-diagonal in the grouped representation. Equating the coefficients of x / x in Eqs. (28) and (29), it follows that... 

In order to compare our approach with other approaches dealing with adiabatic corrections we perform simple model calculations for adiabatic corrections to ground state energy. We start with adiabatic Hamiltonian (32). We now perform the following approximation. We limit ourselves to finite orders of Taylor expansion of the operators H and H g We shall use similar approximation as in [25]. The diagrammatic representation of our approximate Hamiltonian will be... 

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. 

The ordering operator Tc places the operators in the Taylor expansion of the Tc exponent to the left with a later-in-time variable on the contour y. The operators are taken in the interaction representation on the Keldysh contour. 

An alternative representation is the stochastic series expansion (SSE) [41], a generalization of Handscomb s algorithm [42] for the Heisenberg model. It starts from a Taylor expansion of the partition function in orders of / ... 

All of these representations are based on the Taylor series of U( (R)) in the neighborhood of R = Re via powers of the chosen variable = (R). The only problem is to choose a convenient variable-function = (R). Let us assume that we And this (R), and that it works adequately, but at the same time the problem of convergence will arise (Beckel and Engelke, 1968 Beckel, 1976). And finally we must truncate the expansion, and this truncation may result in artifacts, i.e., we have some very difficult problems which are the corollaries of the adopted Taylor expansion. 

The relevant second-order Taylor expansion of the molecular electronic energy in powers of displacements of the canonical state parameters, [d/V, dl/(r)], is determined by the relevant principal derivatives of the energy representation ... 

Therefore, the two leading terms of the Taylor expansions for the centroid and Kubo correlation functions are the same, the difference between them beginning with the third term (i.e., at order t ). The latter term can be taken as an example of how to evaluate the leading correction term to the centroid correlation function (and thereby demonstrate that the centroid correlation function is a well-defined approximation to the Kubo correlation function). The Gaussian representation of operators in phase space [Eq. (2.61)] proves to be useful but not essential in this analysis. 

For class-1 states, a simple harmonic representation of U leads to a complete set of eigenfunctions ( ) this harmonic oscillator basis set is used to diagonalize equation (6). In this case, it is sufficient to construct U( 4>k) using a standard approach involving mass fluctuation (or nuclear ) coordinates and the corresponding electronic state dependent Hessian. The higher terms in the Taylor expansion define anharmonic contributions to the transition moments. These diabatic states are confining and only one stationary point in -space would be found for each... 

Here gg, Fg, and Gg are, respectively, the first (gradient), second (force constants), and third energy derivatives evaluated at Xg. The square brackets indicate that the three-dimensional array of third derivatives is contracted with the vector of coordinate changes to yield a square matrix. If Xg is a stationary point, i.e., a minimum, then the usual theory of small vibrations applies the gradient term vanishes, and truncation after the second-order term leads to separable, harmonic normal modes of vibration. However, on the MEP, the gradient term generally is not zero. The second relevant expansion is the Taylor series representation of the path (of the solution to Eq. [8]) in the arc length parameter, s, about the same point, Xg ... 

In summary, this iterative procedure of solution involves two approximations (1) neglecting all nonlinear terms in the Taylor expansion of fora first-estimate solution, and (2) retaining only the lowest nonlinear term in further iterations. As for the Taylor representation itself in step 1 above, the validity of these approximations underlying the iterative method must be carefully examined for every form of holonomic constraint In particular, two related points must be considered ... 

Linearization and iteration The nonlinear system of equations, Eq. 57, is linearized and solved for a first estimate solution of [7], as discussed in connection with Eq. [39]. The solution is then inserted in the retained quadratic terms, and the linear system is solved for an improved estimate of the I7). This iterative procedure is repeated until the 7 converge within a desired tolerance. For the bond-stretch constraint, there is just one nonlinear (quadratic) term in its Taylor expansion (see later, Eq. [95]), and the linearization and iteration procedure is a fairly good approximation, justified even for relatively large corrections. For the bond-angle and torsional constraints, with infinite series Taylor representations, tighter limits are imposed on the allowable constraint... 

Another possibility is to adapt to In A a method developed by Clarke and Glew (32) for the representation of In Kp as a function of temperature, where Kp is an equilibrium constant. In this method, AHridH at temperature T) is expressed as a Taylor expansion round its value at some reference temperature 6. 

The representation of E q) by the average phase shift exp[i27c / ] leads to a useful Taylor expansion... 

Critical dynamic analysis using renormalisation techniques were presented by Walgraef et aL, 1982. A method based on the systematic study of the Taylor expansion of the stochastic potential was applied for reaction-diffusion systems exhibiting Hopf bifurcation (Fraikin Lemarchand, 1985). The Poissonian representation technique of Gardiner Chaturvedi (1977) is also an efficient procedure for evaluating the parameters of fluctuations. A more rigorous derivation of reaction-diffusion equations with fluctuations were given by De Masi et al (1985). 

It should be stressed that for multidimensional curve crossing problems the low-order Taylor expansions (8), (9) and (19) are justified only in the diabatic electronic representation. In the adiabatic representation, curve crossings generally lead to rapid variations of potential-energy functions and transition dipole moments, rendering a low-order Taylor expansion of these functions in terms of nuclear coordinates meaningless. 

Immediately we face the problem of interpreting the square-root operator on the right-hand side in Eq. [46]. Using, for example, a Taylor expansion would lead to an equation containing all powers of the derivative operator and thus to a nonlocal theory. Such theories are very difficult to handle, and they present an unattractive version of the Schrodinger equation with space and time coordinates appearing in an unsymmetrical form. In the interest of mathematical simplicity, we return to Eq. [40], making the transformation to a quantum mechanical operator representation ... 

The diabatic representation can now be used to characterize the topography of the PESs at the vicinity of conical intersections. We first expand the diabatic potential energy matrix elements as Taylor expansions around a reference geometry Jlo... 

The terms of the electron-hypervibrational Hamiltonian up to the second order of Taylor expansion in the general representation are presented here in details. The... 

Later, in Figure 3-22, we give a graphical representation of the principles of the truncated Taylor series expansion for a one-parameter function. 

Short-time expansion. The Taylor series expansion of the correlation function is a useful representation at small times,... 

Typically, these methods arrive at the same finite difference representation for a given problem. However, we feel that Taylor-series expansions are easy to illustrate and we will therefore use them here in the derivation of finite difference equations. We encourage the student of polymer processing to look up the other techniques in the literature, for instance, integral methods and polynomial fitting from Tannehill, Anderson and Pletcher [26] or from Milne [16] and finite volume approach from Patankar [18], Versteeg and Malalasekera [27] or from Roache [20]. 

---