Expansion Taylor

The idea in a Taylor expansion is to approximate the unknown function by a polynomial centred at an expansion point Xo, typically at or near the centre of the variable of interest. The coefficients of an Mh-order polynomial are determined by requiring that the first N derivatives match those of the unknown function at the expansion point. For a one-dimensional case this can be written as in eq. (16.91). 

For a many-dimensional function, the corresponding second-order expansion can be written as in eq. (16.92). 

Here g is a transposed vector (gradient) containing all the partial first derivatives, and H is the (Hessian) matrix containing the partial second derivatives. In many cases, the expansion point Xo is a stationary point for the real function, making the first derivative disappear, and the zeroth-order term can be removed by a shift of the origin. 

A Taylor expansion is an approximation to the real function by a polynomial terminated at order N. For a given (fixed) N, the Taylor expansion becomes a better approximation as the variable x approaches Xo. For a fixed point x at a given distance from Xo, the approximation can be improved by including more terms in the polynomial. Except for the case where the real function is a polynomial, however, the Taylor expansion will always be an approximation. Furthermore, as one moves away from the expansion point, the rate of convergence slows down, i.e. more and more terms are 

A specific example of a Taylor expansion is the molecular energy as a function of the nuclear coordinates. The real energy function is quite complex, l ut for describing a stable molecule at sufficiently low temperatures, only the functional form near the equilibrium geometry is required. Terminating the expansion at second order corresponds to modelling the nuclear motion by harmonic vibrations, while higher order terms introduce anharmonic corrections. 

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We deal witii the exponentials in (equation Al.4.102) and (equation Al.4.105) whose arguments are operators by using their Taylor expansion... 

The high-temperaUire expansion could also be derived as a Taylor expansion of the free energy in powers of X about X = 0 ... 

There are two approaches connnonly used to derive an analytical connection between g(i-) and u(r) the Percus-Yevick (PY) equation and the hypemetted chain (FfNC) equation. Both are derived from attempts to fomi fimctional Taylor expansions of different correlation fimctions. These auxiliary correlation functions include ... 

The PY equation is derived from a Taylor expansion of the direct correlation fimction, and has the fonn... 

Here E(t) denotes the applied optical field, and-e andm represent, respectively, the electronic charge and mass. The (angular) frequency oIq defines the resonance of the hamionic component of the response, and y represents a phenomenological damping rate for the oscillator. The nonlinear restoring force has been written in a Taylor expansion the temis + ) correspond to tlie corrections to the hamionic... 

According to the Porod law [28], the intensity in the tail of a scattering curve from an isotropic two-phase structure havmg sharp phase boundaries can be given by eqnation (B 1.9.81). In fact, this equation can also be derived from the deneral xpression of scattering (61.9.56). The derivation is as follows. If we assume qr= u and use the Taylor expansion at large q, we can rewrite (61.9.56) as... 

In Section III.D, we shall investigate when this happens. For the moment, imagine that we are at a point of degeneracy. To find out the topology of the adiabatic PES around this point, the diabatic potential matrix elements can be expressed by a hrst order Taylor expansion. 

The potential matrix elements are then obtained by making Taylor expansions around 00, using suitable zero-order diabatic potential energy functions,... 

I ask 3 Transform ( inherit ) local Taylor expansions from a upper hierarchy level to the next lower hierarchy level. 

For large systems comprising 36,000 atoms FAMUSAMM performs four times faster than SAMM and as fast as a cut-off scheme with a 10 A cut-off distance while completely avoiding truncation artifacts. Here, the speed-up with respect to SAMM is essentially achieved by the multiple-time-step extrapolation of local Taylor expansions in the outer distance classes. For this system FAMUSAMM executes by a factor of 60 faster than explicit evaluation of the Coulomb sum. The subsequent Section describes, as a sample application of FAMUSAMM, the study of a ligand-receptor unbinding process. 

Finite Difference Method To apply the finite difference method, we first spread grid points through the domain. Figure 3-49 shows a uniform mesh of n points (nonuniform meshes are possible, too). The unknown, here c(x), at a grid point x, is assigned the symbol Cj = c(Xi). The finite difference method can be derived easily by using a Taylor expansion of the solution about this point. Expressions for the derivatives are ... 

A good starting point for understanding finite-difference methods is the Taylor expansion about time t of the position at time t + At,... 

The most common integration algorithm used in the study of biomolecules is due to Verlet [11]. The Verlet integrator is based on two Taylor expansions, a forward expansion (t + At) and a backward expansion (t — At),... 

The Verlet algorithm is not self-starting. A lower order Taylor expansion [e.g., Eq. (13)] is often used to initiate the propagation. 

The goal of all minimization algorithms is to find a local minimum of a given function. They differ in how closely they try to mimic the way a drop of water or a small ball would roll down the slope, following the surface curvature, until it ends up at the bottom. Consider a Taylor expansion around a minimum point Xq of the general one-dimensional function F(X), which can be written as... 

A further improvement can be seen for the situation depicted in Eigure lb. Let ( )i, (r) denote the potential due to the charges in the cell about point b, evaluated at the point r. Let a be the center of the subcell containing q. Then (j), (r) can be approximated by a second-order Taylor expansion about a ... 

The free energy derivatives are also related to the coefficients in a Taylor expansion of the free energy with respect to X. In the case of linear coupling, we let Eba = XdJ-a — Up,)lkT in Eq. (9) we obtain... 

Fig. 5. Band gap as a function of nanotube radius calculated using empirical tight-binding Hamiltonian. Solid line gives estimate using Taylor expansion of graphene sheet results in eqn. (7).

M. Schoen. Taylor-expansion Monte Carlo simulations of classical fluids in the canonical and grand canonical ensembles. J Comput Phys 775 159-171, 1995. 

It is useful at this point if we examine the Taylor expansion for a general diatomic potential U R) about the equilibrium bond length R. ... 

If the position (r), velocity (v), acceleration (a) and time derivative of the acceleration (b) are known at time t, then these quantities can be obtained dX. t + 8t by a Taylor expansion ... 

If we start from the Taylor expansion about r(t) then... 

If we write a Taylor expansion of in normal coordinates about the equilibrium value Pe.eq... 

Estr is the energy function for stretching a bond between two atom types A and B. In its simplest form, it is written as a Taylor expansion around a namral , or equilibrium , bond length Rq- Tenninating the expansion at second order gives the expression... 

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