Taylor’s expansion

If we cany out a Taylor s expansion of the potential energy about the equilibrium length of an isolated chemical bond, we get... 

The linearisation of the non-linear component and energy balance equations, based on the use of Taylor s expansion theorem, leads to two, simultaneous, first-order, linear differential equations with constant coefficients of the form... 

We will skip the algebraic details. The simple idea is that we can do long division of a function of the form in Eq. (3-30) and match the terms to a Taylor s expansion of the exponential function. If we do, we ll find that the (1/1) Pade approximation is equivalent to a third order Taylor... 

The terms, Enn(Q)and Hpq(Q) are introduced over the Taylor s expansion at the equilibrium geometry of fixed nuclear configuration Ro (crude level -clamped nuclei),... 

We now simplify the left-hand side by first making a Taylor s expansion of JM+dM around point x... 

This equation is simply the first nonzero term in a Taylor s expansion of z about y = 0.) The far right term in this expression is written to account for the fact that z is a function of pH. When the z of Eq. 8.38 is substituted into Eq. 8.8, we get... 

In an electrochemical system, Taylor s expansion shows that... 

When the second- and higher-order terms in the Taylor s expansion of the activation energies can be neglected, we have ... 

Here s a more algebraic version of the same argument. We regard f as a function of both X and r, and examine the behavior of x = /(x, r) near the bifurcation at X = X and r =. Taylor s expansion yields... 

We are especially interested in the mean of the ensemble x). We therefore average over Eq. 1.25 and make a Taylor s expansion of the right hand side around x). We obtain ... 

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