Taylor-Series Expansions

The following day Maria Mayer visited me to find out how I was coming along with the summary of the spectroscopy program. I Informed her that I had temporarily set that aside and was using what data we had to estimate chemical exchange factors. Marla Mayer knew all the difficulties in the calculation of partition function ratios from Incomplete spectroscopic data. 

She had been through it in connection with the heavy water program. She was excited by my approach and said that I could now finish the whole matter by taking out the classical contribution to Q,/Q. If we now added (- l/u.)Au. to my Taylor 

For most molecules the classical partition function Is an adequate approximation for the translation and rotation in 

ACS Symposium Series American Chemical Society Washington, DC, 1975. 

We next show that (Q/Q ). depenis only on the symmetry number ratio and a mass factor lndependent of chemical composition. Write the Hamiltonian of the molecule in Cartesian coordinates. Then the classical partition function 

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Evidently, this fomuila is not exact if fand vdo not connnute. However for short times it is a good approximation, as can be verified by comparing temis in Taylor series expansions of the middle and right-hand expressions in (A3,11,125). This approximation is intrinsically unitary, which means that scattering infomiation obtained from this calculation automatically conserves flux. 

The molecular dipole moment (not the transition dipole moment) is given as a Taylor series expansion about the equilibrium position... 

Raman scattering has been discussed by many authors. As in the case of IR vibrational spectroscopy, the interaction is between the electromagnetic field and a dipole moment, however in this case the dipole moment is induced by the field itself The induced dipole is pj j = a E, where a is the polarizability. It can be expressed in a Taylor series expansion in coordinate isplacement... 

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. ... 

There are many algorithms for integrating the equations of motion using finite difference methods, several of which are commonly used in molecular dynamics calculations. All algorithms assume that the positions and dynamic properties (velocities, accelerations, etc.) can be approximated as Taylor series expansions ... 

Table 7.1 presents us with something of a dilemma. We would obviously desire to explore i much of the phase space as possible but this may be compromised by the need for a sma time step. One possible approach is to use a multiple time step method. The underlyir rationale is that certain interactions evolve more rapidly with rime than other interaction The twin-range method (Section 6.7.1) is a crude type of multiple time step approach, i that interactions involving atoms between the lower and upper cutoff distance remai constant and change only when the neighbour list is updated. However, this approac can lead to an accumulation of numerical errors in calculated properties. A more soph sticated approach is to approximate the forces due to these atoms using a Taylor seri< expansion [Streett et al. 1978] ... 

Truncating this series after the first derivative and integrating provides the basis for the hermodynamic integration approach. Moreover, if the Taylor series expansion is continued intil it converges then Equation (11.45) is equivalent to the thermodynamic perturbation brmula, so providing a link between the two approaches. In practice, it is always necessary... 

Following the procedure described in Chatper 2, Section 2.5 the Taylor series expansion of the field unknowns at a time level equal to + oAt, where 0 < a < 1, are obtained as... 

The selection of a time increment dependent on parameter a (i.e. carrying out Taylor series expansion at a level between successive time steps of n and n+Y) enhances the flexibility of the temporal discretizations by allowing the introduction of various amounts of smoothing in different problems. The first-order time derivatives are found from the governing equations as... 

This kind of matr ix is called a Hessian matrix. The derivatives give the cmvatme of V(x[,X2) in a two-dimensional space because there are two masses, even though both masses are constrained to move on the -axis. As we have already seen, these derivatives are pari of the Taylor series expansion... 

The simple harmonie motion of a diatomie moleeule was treated in Chapter 1, and will not be repeated here. Instead, emphasis is plaeed on polyatomie moleeules whose eleetronie energy s dependenee on the 3N Cartesian eoordinates of its N atoms ean be written (approximately) in terms of a Taylor series expansion about a stable loeal minimum. We therefore assume that the moleeule of interest exists in an eleetronie state for whieh the geometry being eonsidered is stable (i.e., not subjeet to spontaneous geometrieal distortion). 

The Taylor series expansion of the eleetronie energy is written as ... 

The molecular quantities can be best understood as a Taylor series expansion. For example, the energy of the molecule E would be the sum of the energy without an electric field present, Eq, and corrections for the dipole, polarizability, hyperpolarizability, and the like ... 

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... 

Aris and Amundson (1958) solved the coupled, time-dependent material and energy balances, linearizing the equations about the operating point by a Taylor series expansion. This made the solution possible by the method of characteristic equations. The solution yielded two equations, one the slope condition and the other recognized by Gilles and Hofmann (1961) as the condition that sets the limits to avoid rate oscillation. This is called the... 

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... 

Recall that equations 9.86 and 9.100 have been both derived using only the first-order terms in the Taylor series expansion of our basic kinetic equation (equation 9.77). It is easy to show that if instead all terms through second-order in 6x and 6t are retained, the continuity equation ( 9.86) remains invariant but the momentum equation ( 9.100) requires correction terms [wolf86c]. The LHS of equation 9.100, to second order in (ia (5 << 1, is given by... 

The content of Eq. (3-81) is sometimes expressed in a somewhat different way by writing the (formal) Taylor series expansion of in the form... 

Lindstrom and Bates argue that a Taylor series expansion of (Eq. 3.4) around the expectation of the random effects bi = 0 may be poor. Instead, they consider linearizing (Eq. 3.4) in the random effects about some value bf closer to bi than its expectation 0. 

In particular, retaining the hrst two terms of the Taylor series expansion about bi = bf of b(Pi, Xi) and the leading term of Ri (Pi, ) i, it follows that... 

Finally, we show how to relate the modified Schrodinger equation evolution X(m) to the usual evolution T (t) [14]. Consider the modified Schrodinger equation, Eq. (12). We approximate f H) in this equation with a first-order Taylor series expansion. 

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 ... 

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

SO that the series u and U2 converge for all finite values of To see what happens to u and M2 as 00, we consider the Taylor series expansion of e ... 

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