Taylor series function

Equation (63) was obtained after performing a Taylor series functional... 

By performing a Taylor series functional expansion of one of the functional derivatives in Eq. (3), around the other one, and keeping only terms up to first order, one may rewrite this equation in the form... 

The basic idea of NMA is to expand the potential energy function U(x) in a Taylor series expansion around a point Xq where the gradient of the potential vanishes ([Case 1996]). If third and higher-order derivatives are ignored, the dynamics of the system can be described in terms of the normal mode directions and frequencies Qj and Ui which satisfy ... 

When discussing derivative methods it is useful to write the function as a Taylor series expansion about the point jc. ... 

Although Eq. (10.50) is still plagued by remnants of the Taylor series expansion about the equilibrium point in the form of the factor (dn/dc2)o, we are now in a position to evaluate the latter quantity explicitly. Equation (8.87) gives an expression for the equilibrium osmotic pressure as a function of concentration n = RT(c2/M + Bc2 + ) Therefore... 

The value functions appearing in equation 3 may be expanded in Taylor series about x and, because the concentration changes effected by a single stage are relatively small, only the first nonvanishing term is retained. When the value of is replaced by its material balance equivalent, ie, equation 4 ... 

Taylor Series The Taylor series for a function of two variables, expanded about the point (xq, yo), is... 

Sufficient conditions are that any local move away from the optimal point ti gives rise to an increase in the objective function. Expand F in a Taylor series locally around the candidate point ti up to second-order terms ... 

All fluid properties are functions of space and time, namely p(x, y, z, t), p(x, y, z, t), T(x, y, z, t), and u(x, y, z, t) for the density, pressure, temperature, and velocity vector, respectively. The element under consideration is so small that fluid properties at the faces can be expressed accurately by the first two terms of a Taylor series expansion. For example, the pressure at the E and W faces, which are both at a distance l/26x from the element center, is expressed as... 

Suppose a system is a function of its components (equation 2.7-12). Expand the function / about the mean values of its arguments in a multi-variable Taylor series (equation 2.7 -13). The mean of Q, which is the expectation of equation... 

In the lowest approximation the molecular vibrations may be described as those of a harmonic oscillator. These can be derived by expanding the energy as a function of the nuclear coordinates in a Taylor series around the equilibrium geometry. For a diatomic molecule this is the intemuclear distance R. 

The Rational Function Optimization (RFO) expands the function in terms of a rational approximation instead of a straight second-order Taylor series (eq. (14.3)). 

The simplest form of approximation to a continuous function is some polynomial. Continuous functions may be approximated in order to provide a simpler form than the original function. Truncated power series representations (such as the Taylor series) are one class of polynomial approximations. 

Now, since the random variable — m /jj has finite mean (=0) and variance (= 1), both its characteristic function and the logarithm of its characteristic function have finite first and second derivatives. It follows that In Mi1 mi)/ffl(i ) can be expanded in a Taylor series with remainder42 as follows43... 

If D( ) is a linear displacement operator that converts a function [Pg.392]

Taylor series 260 torque, correlation functions 28 transfer time, rotational relaxation 51 transitions dipole moment 30 forbidden 30 non-adiabatic 130 translational velocity v 6... 

In the next section we derive the Taylor expansion of the coupled cluster cubic response function in its frequency arguments and the equations for the required expansions of the cluster amplitude and Lagrangian multiplier responses. For the experimentally important isotropic averages 7, 7i and yx we give explicit expressions for the A and higher-order coefficients in terms of the coefficients of the Taylor series. In Sec. 4 we present an application of the developed approach to the second hyperpolarizability of the methane molecule. We test the convergence of the hyperpolarizabilities with respect to the order of the expansion and investigate the sensitivity of the coefficients to basis sets and correlation treatment. The results are compared with dispersion coefficients derived by least square fits to experimental hyperpolarizability data or to pointwise calculated hyperpolarizabilities of other ab inito studies. 

Table 2 Convergence of the Taylor series and the series of diagonal Fade approximants (CCSD response, t-aug-cc-pVDZ basis). The inifinite order results were calculated using the implementation for the frequency-dependent response function.

Consider some point (aq, bo, .) within the region of definition of the functions F,G,... and suppose that the functions can be represented by an multidimensional Taylor series about this point. Truncating the series after... 

The complete formula for the Taylor series expansion attached to a n-variable function f (x) in the neighbourhood of the point x, possess the following peculiar simple structure when using NSS s ... 

In general, the angular frequency (o(k) is a function of k, so that the angular frequencies in the composite wave (jc, t), as well as the wave numbers, vary from one plane wave to another. If (o k) is a slowly varying function of k and the values of k are confined to a small range AA , then (x) k) may be expanded in a Taylor series in k about some point ko within the interval AA ... 

To evaluate L ip) from equation (F.2), we first factor out in the generating function and expand the remaining exponential function in a Taylor series... 

Thus, A (p) are just the coefficients in a Taylor series expansion of the function... 

With a few minor modifications, the Gauss-Newton method presented in Chapter 4 can be used to obtain the unknown parameters. If we consider Taylor series expansion of the penalty function around the current estimate of the parameter we have,... 

To handle the time delay, we do not simply expand the exponential function as a Taylor series. We use the so-called Pade approximation, which puts the function as a ratio of two polynomials. The simplest is the first order (1/1) Pade approximation ... 

One idea (not that we really do that) is to apply the Taylor series expansion on the exponential function of A, and evaluate the state transition matrix with... 

As foe molecule executes small-amplitude vibrations with respect to the equilibrium intenmclear distance, it is appropriate to develop the potential function in a Taylor series about that position. Thus,... 

Although the equilibrium configuration of a molecule can usually be specified, at ordinary temperatures, all of the atoms undergo oscillatory motions. The forces between the atoms in the molecule are described by a Taylor series of the intramolecular potential function in the internal coordinates. This function can then be written in the form... 

In this section, we take an approach that is characteristic of conventional perturbation theories, which involves an expansion of a desired quantity in a series with respect to a small parameter. To see how this works, we start with (2.8). The problem of expressing ln(exp (—tX)) as a power series is well known in probability theory and statistics. Here, we will not provide the detailed derivation of this expression, which relies on the expansions of the exponential and logarithmic functions in Taylor series. Instead, the reader is referred to the seminal paper of Zwanzig [3], or one of many books on probability theory - see for instance [7], The basic idea of the derivation consists of inserting... 

The potential energy V of the elastomer is presumed to be given as a function of the atomic coordinates x (lwell-defined equilibrium shape, there must be equilibrium positions x for all atoms that are part of the continuous network. Expand the potential in a Taylor series about the equilibrium positions, and set the potential to zero at equilibrium, to obtain... 

This integral of the electronic dipole moment operator is a function of a nuclear coordinate Q. The integral may be expanded in a Taylor series with respect to Q (equation 4) and... 

Even for a diatomic molecule the nuclear Schrodinger equation is generally so complicated that it can only be solved numerically. However, often one is not interested in all the solutions but only in the ground state and a few of the lower excited states. In this case the harmonic approximation can be employed. For this purpose the potential energy function is expanded into a Taylor series about the equilibrium separation, and terms up to second order are kept. For a diatomic molecule this results in ... 

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