MacLaurin expansion

Swalin then uses a Maclaurin expansion of the Anhenius term to obtain the energy of formation of a cavity in the liquid which permits a small jump, to obtain the equation... 

We have shown that the number of linearly independent invariants of degree k under the permutation group H is equal to the coefficient of in the Maclaurin expansion of (1.27). This represents an important special case of a proposition by Th. Molien. ... 

Typically the experimental measurements are most accurate at small q. Inserting the MacLaurin expansion, Eq. (5.40), of IIo(p) into Eq. (5.63) leads to the MacLaurin expansion [167,187] for the isotropic Compton profile ... 

Hartree-Fock calculations of the three leading coefficients in the MacLaurin expansion, Eq. (5.40), have been made [187,232] for all atoms in the periodic table. The calculations [187] showed that 93% of rio(O) comes from the outermost s orbital, and that IIo(O) behaves as a measure of atomic size. Similarly, 95% of IIq(O) comes from the outermost s and p orbitals. The sign of IIq(O) depends on the relative number of electrons in the outermost s and p orbitals, which make negative and positive contributions, respectively. Clearly, the coefficients of the MacLaurin expansion are excellent probes of the valence orbitals. The curvature riQ(O) is a surprisingly powerful predictor of the global behavior of IIo(p). A positive IIq(O) indicates a type 11 momentum density, whereas a negative rio(O) indicates that IIo(O) is of either type 1 or 111 [187,230]. MacDougall has speculated on the connection between IIq(O) and superconductivity [233]. 

Calculations of IIq(O) are very sensitive to the basis set. The venerable Clementi-Roetti wavefunctions [234], often considered to be of Hartree-Fock quality, get the sign of IIq(O) wrong for the sihcon atom. Purely numerical, basis-set-free, calculations [232,235] have been performed to establish Hartree-Fock limits for the MacLaurin expansion coefficients of IIo(p). The effects of electron correlation on IIo(O), and in a few cases IIq(O), have been examined for the helium atom [236], the hydride anion [236], the isoelectronic series of the lithium [237], beryllium [238], and neon [239] atoms, the second-period atoms from boron to fluorine [127], the atoms from helium to neon [240], and the neon and argon atoms [241]. Electron correlation has only moderate effects on IIo(O). 

Choosing a coordinate system that diagonalizes the Hessian matrix, the MacLaurin expansion of the three-dimensional momentum density II( p) can be written as [241]... 

An alternative way of computing the coefficients [p , -r and q , -r comes from observing that the two power series representations in Eq. (114) are the truncated Maclaurin expansions. Thus,... 

Free energy will be defined in Section 4.6. If (r) varies slowly enough in the region where p(r) is significant, then one can use a Maclaurin expansion of 4>(r) around some suitably chosen origin r = 0 ... 

The P-B equation can be approximately linearized using the Maclaurin expansion of the sinh function and then neglecting higher-order terms [37. The P-B equation for ion distribution is thus simplified and yields... 

The coefficients, a, are often related to one another in a simple way which is determined by the nature of the function. An important method of expressing functions in a power series is the Taylor and Maclaurin expansions. In a Taylor expansion the function f(x) is expanded about a given point xq and the coefficients are related to the values of the derivatives of the function at x = xq. Thus, the Taylor expansion of f(x) is... 

The Maclaurin expansion is the special case in which the function is expanded about X = 0. Thus, the Maclaurin expansion of/(x) is... 

It should also be mentioned that, for L and A isotropic functions, truncated Maclaurin expansions of Eqs. 43 and 44 and use of Eq. 29 to change the independent variables from (s,v) to (a,p) results in the traditional forms for Fourier s law of heat condition and Newton s law of viscosity, viz. 

The small p behavior of Tlaip) is most easily treated with the MacLaurin expansion given by [19,54] ... 

Evaluations are made by series representations (8, 9). The Maclaurin expansion is... 

The parameter fi measures the strength of the Kronig-Penney barrier between neighboring square weUs. In this limit y (2me oo, whereas yb (2me VobVb ) / 0. Thus, substituting into (C9) and using the small argument Maclaurin expansions, sin yb yh and cos y b 1 we have... 

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