Scalar expansion functions

Let us first consider two scalar potentials (r, t) and < (r, t), which differ by more than a mere TD function, and two vector potentials A(r, t) and A (r, t), which are also different. Their Taylor expansions demand that there must exist at least one time derivative of the potentials differing from zero, in the case of scalar as well as vector potentials. In other words, there must exist a minimal nonnegative integer, say k for the scalar potential and l for the vector potential, such that... 

If V(r, x) were a known function, this linear expansion could be used to determine how the velocity varies for short intervals of time and in any arbitrary short spatial direction dx. In a Taylor-series expansion of a scalar field, it is often conventional to post-multiply by the dx. Since the gradient of a scalar field is a vector and because the inner product of two vectors is commutative, the order of the product is unimportant. However, because of the tensor structure of the gradient of a vector field, the pre-multiply is essential. 

For actual calculations one needs the values of the scalar products entering Eqs. (4.184)-(4.188). In Section III.B.8.C we obtain their representations in terms of the moments Qt and St of the functions /0 and /j, respectively. The procedures of asymptotic expansion of Qt and St are given in Section... 

In the previous section we used quaternions to construct a convenient parameterization of the hybridization manifold, using the fact that it can be supplied by the 50(4) group structure. However, the strictly local HOs allow for the quaternion representation for themselves. Indeed, the quaternion was previously characterized as an entity comprising a scalar and a 3-vector part h = (h0, h) = (s, v). This notation reflects the symmetry properties of the quaternion under spatial rotation its first component ho = s does not change under spatial rotation i.e. is a scalar, whereas the vector part h — v — (hx,hy,hz) expectedly transforms as a 3-vector. These are precisely the features which can be easily found by the strictly local HOs the coefficient of the s-orbital in the HO s expansion over AOs does not change under the spatial rotation of the molecule, whereas the coefficients at the p-functions transform as if they were the components of a 3-dimensional vector. Thus each of the HOs located at a heavy atom and assigned to the m-th bond can be presented as a quaternion ... 

The orthogonal characteristic polynomials or eigenpolynomials Qn(u) play one of the central roles in spectral analysis since they form a basis due to the completeness relation (163). They can be computed either via the Lanczos recursion (84) or from the power series representation (114). The latter method generates the expansion coefficients q , -r through the recursion (117). Alternatively, these coefficients can be deduced from the Lanczos recursion (97) for the rth derivative Q /r(0) since we have qni r = (l/r )Q r(0) as in Eq. (122). The polynomial set Qn(u) is the basis comprised of scalar functions in the Lanczos vector space C from Eq. (135). In Eq. (135), the definition (142) of the inner product implies that the polynomials Qn(u) and Qm(u) are orthogonal to each other (for n= m) with respect to the complex weight function dk, as per (166). The completeness (163) of the set Q (u) enables expansion of every function f(u) e C in a series in terms of the... 

Procedures 2 and 3 are very convenient when looking for a quick test on the efficiency of adding or not a new CETO set to the expansion (4.4), while optimizing the non-linear parameters. All the procedure becomes simpler when only one function is added, because in this case one has m=l. The (nxm) matrices appear to be column vectors (nxl) and the square (mxm) a and x matrices become (1x1), scalars that is 0=1 and x=(l -t.). ... 

In their original theory, Maier and Saupe supposed that the molecular interactions responsible for the nematic state are anisotropic van der Waals interactions (discussed in Section 2.3), in which case mms should be temperature-independent. However, it is now recognized that shape anisotropy is also important, even for small-molecule thermotropic nematics. By making mms temperature-dependent, the Maier-Saupe potential can, in principle, accommodate both energetic and entropic effects. In fact, if the function sin(u, u) in the purely entropic Onsager potential Eq. (2-5) is approximated by the expansion 1 — V2 cos (u, u)+. . ., then to lowest order the Maier-Saupe potential (2-7) is obtained with C/ms — Uo bT/S, where we have defined the dimensionless Maier-Saupe energy constant by Uus = ums/ksT, Thus, the Maier-Saupe potential can be used as an approximation to describe orientational order in either lyotropic (solvent-based) or thermotropic nematics. For a thermotropic melt, the Maier-Saupe theory predicts a first-order transition from the isotropic to the nematic phase when mms/ bT = U s — t i.MS = 4.55, and at this transition the scalar order parameter S jumps from zero to 0.43. S increases toward unity with further increases in Uus- The spinodal point at which the isotropic phase is unstable to even small orientational perturbations occurs atU — = 5 for the Maier-... 

These are expressed in terms of scalar products between the unit axis system vectors on sites 1 and 2 (on different molecules) and the unit vector 6. from site 1 to 2. The S functions that can have nonzero coefficients in the intermolecular potential depend on the symmetry of the site. This table includes the first few terms that would appear in the expansion of the atom-atom potential for linear molecules. The second set illustrate the types of additional functions that can occur for sites with other than symmetry. These additional terms happen to be those required to describe the anisotropy of the repulsion between the N atom in pyridine (with Zj in the direction of the conventional lone pair on the nitrogen and yj perpendicular to the ring) and the H atom in methanol (with Z2 along the O—H bond and X2 in the COH plane, with C in the direction of positive X2). The important S functions reflect the different symmetries of the two molecules.Note that coefficients of S functions with values of k of opposite sign are always related so that purely real combinations of S functions appear in the intermolecular potential. 

Equation A.6.1 arises when the Maxwell-Stefan equations are solved for the case of steady-state, one-dimensional mass transfer, as discussed in Chapter 8. The matrices [A ] and [O] are as defined in Chapter 8, is the molar density of the mixture and a scalar, and (Ax) is a column matrix of mole fraction differences. All matrices in Eq. A.6.1 are of order n — 1 where n is the number of components in the mixture. For the purposes of this discussion we shall assume that the matrices [/ ] and [O] have already been calculated. The matrix function [0][exp[] - [7]] denoted by [2], can be computed using Sylvester s expansion formula (see, however, below) so the immediate problem is the calculation of the column matrix (7) from... 

Some functions of matrices are defined as expansions. If /(x) is a function of a scalar variable x that has the expansion... 

We therefore adapt the locally quadratic Hamiltonian treatment of Gaussian wave packets, pioneered by Heller [18], to a system with an induced adiabatic vector potential. The locally quadratic theory replaces the anharmonic time-independent nuclear Hamiltonian by a time-dependent Hamiltonian which is taken to be of second order about the instantaneous center of the wave packet. Since the nuclear wave packet continually evolves under an effective harmonic Hamiltonian, an initially Gaussian wave form remains Gaussian. The treatment yields equations of motion for the wave function parameters that can be solved numerically [36-38]. The locally quadratic Hamiltonian includes a second order expansion of the scalar potential, consisting of the last three terms in Eq. (2.18), which we write as... 

At this stage, we need some graphical definitions. With each circle we associate a function of position (and orientation) of one particle in this expansion the function is Zi(i). White circles are labeled, while black circles are unlabeled and an integral with respect to all coordinates is implied. With each bond we associate a function of the coordinates of two particles (both position and orientation) in this expansion the function is = exp [ — Other examples would be the scalar separation or... 

A different approach is chosen when the screening of nuclear potential due to the electrons is incorporated in /z . Transformation to the eigenspinor basis is then only possible after the DHF equation is solved which makes it more difficult to isolate the spin-orbit coupling parts of the Hamiltonian. Still, it is also in this case possible to define a scalar relativistic formalism if the so-called restricted kinetic balance scheme is used to relate the upper and lower component expansion sets. The modified Dirac formalism of Dyall [24] formalizes this procedure and makes it possible to identify and eliminate the spin-orbit coupling terms in the selfconsistent field calculations. The resulting 4-spinors remain complex functions, but the matrix elements of the DCB Hamiltonian exhibit the non-relativistic symmetry and algebra. 

The functional basis that supplies the global picture is connected through the expansion coefficients an to the spatial grid. This provides the ability to define the scalar product of two functions. If... 

In addition to the electronically adiabatic representation described by (4) and (5) or, equivalently (57) and (58), other representations can be defined in which the adiabatic electronic wave function basis set used in expansions (4) or (58) is replaced by some other set of functions of the electronic coordinates rel or r. Let us in what follows assume that we have separated the motion of the center of mass G of the system and adopted the Jacobi mass-scaled vectors R and r defined after (52), and in terms of which the adiabatic electronic wave functions are i] l,ad(r q) and the corresponding nuclear wave function coefficients are Xnd (R). The symbol q(R) refers to the set of scalar nuclear position coordinates defined after (56). Let iKil d(r q) label that alternate electronic basis set, which is allowed to be parametrically dependent on q, and for which we will use the designation diabatic. We now proceed to define such a set. LetXn(R) be the nuclear wave function coefficients associated with those diabatic electronic wave functions. As a result, we may rewrite (58) as... 

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