Cartesian expansion

The superscript S denotes the spherical expansion and the range of the nuclear coordinate now only comprises the spatial region of the nucleus. This point needs some clarification concerning the assumption that AV = 0. The Cartesian expansion of the term is written as... 

Obviously, the spherical and Cartesian expressions for cannot be identical since the Cartesian expansion parameters a and /3 are not confined to specific spatial regions or have to fulfill an inequality such as r > r. A more explicit analysis which can be found in [28] states the relation between these two expansions as... 

From these expressions, the fiill set of Hermite-to-Cartesian expansion coefficients may be generated and the overlap distribution may then be expanded in Hermite Gaussians according to (9.5.1). 

NexL at each abscissa, we calculate the modihed Hermite polynomials (9.11.30) using the recurrence relations (9.11.36). The resulting polynomial values are then eontraeted with the Hermite-to-Cartesian expansion coefficients, yielding the one-dimensional d artesian integrals (9.11.43)-(9.11.45). The expansion coefficients, which may be obtained from the two-term recurrence relations (9.5.15)-(9.5.17), are the same for all the abscissae. The final Cartesian one-electron Coulomb integral is obtained by carrying out the summation (9.11.42). 

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 algorithm outlined above is a level 1 cell multipole or Cartesian multipole algorithm [28]. A number of modifications are possible. Accuracy can be raised by using higher order expansions, which unfortunately are more expensive. The cost can be alleviated by... 

Appendix B Expansion of Cartesian Gaussian Basis Functions Using Spherical Harmonics... 

APPENDIX B EXPANSION OF CARTESIAN GAUSSIAN BASIS FUNCTIONS USING SPHERICAL HARMONICS... 

This expansion can be continued beyond the terms which are second order in deviations of from (i), but only at the cost of calculating the corresponding higher order ah initio Cartesian derivatives (which is not sufficiently efficient at present57). 

The expansion of Fourier components of the dipole interaction tensor in the vicinity of the minimum point at the boundary of the first Brillouin zone, with the Cartesian axes Ox and Oy respectively chosen along bi and b2 (see Fig. 2.9b), has the form... 

In an alternative formulation of the Redfield theory, one expresses the density operator by expansion in a suitable operator basis set and formulates the equation of motion directly in terms of the expectation values of the operators (18,20,50). Consider a system of two nuclear spins with the spin quantum number of 1/2,1, and N, interacting with each other through the scalar J-coupling and dipolar interaction. In an isotropic liquid, the former interaction gives rise to J-split doublets, while the dipolar interaction acts as a relaxation mechanism. For the discussion of such a system, the appropriate sixteen-dimensional basis set can for example consist of the unit operator, E, the operators corresponding to the Cartesian components of the two spins, Ix, ly, Iz, Nx, Ny, Nz and the products of the components of I and the components of N (49). These sixteen operators span the Liouville space for our two-spin system. If we concentrate on the longitudinal relaxation (the relaxation connected to the distribution of populations), the Redfield theory predicts the relaxation to follow a set of three coupled differential equations ... 

We begin by defining a set of N atoms and writing their Cartesian coordinates as a single vector with 3N components, r = (r, ..., ro,N). If locating the atoms at ro is a local minimum in the energy of the atoms, then it is convenient to define new coordinates x = r ro. The Taylor expansion of the atom s energy about the minimum at ro is, to second order,... 

The tensor defined by Eq. (2.133) is a Cartesian representation of the constrained mobihty tensor K = (C which may also be represented by the expansion... 

This may be confirmed by expanding an arbitary contravariant Cartesian vector (with a raised bead index) in a basis of a and m vectors and confirming that one recovers the original vector if such an expansion vector is left-multiplied by the RHS of Eq. (2.149). 

The main result of this section is the derivation of a class of alternative expressions for the Cartesian drift velocity that is based on an expansion of the divergence of the diffusivity of as a sum... 

In this section, we first give an elementary derivation of Eq. (2.182) for the covariant divergence of and then derive an expansion for the resulting Cartesian divergence of in order to recover the more explicit expression for... 

For most molecules, the small momentum expansion of the momentum density requires the full 3x3 Hessian matrix A of n( p) at p = 0. In Cartesian coordinates, this matrix has elements... 

The Hy-CI function used for molecular systems is based on the MO theory, in which molecular orbitals are many-center linear combinations of one-center Cartesian Gaussians. These combinations are the solutions of Hartree-Fock equations. An alternative way is to employ directly in Cl and Hylleraas-CI expansions simple one-center basis functions instead of producing first the molecular orbitals. This is a subject of the valence bond theory (VB). This type of approach, called Hy-CIVB, has been proposed by Cencek et al. (Cencek et.al. 1991). In the full-CI or full-Hy-CI limit (all possible CSF-s generated from the given one-center basis set), MO and VB wave functions become identical each term in a MO-expansion is simply a linear combination of all terms from a VB-expansion. Due to the non-orthogonality of one-center functions the mathematical formalism of the VB theory for many-electron systems is rather cumbersome. However, for two-electron systems this drawback is not important and, moreover, the VB function seems in this case more natural. 

For molecules with inversion symmetry, like H2, the expansion parameter X must be even, Eqs. 4.15 through 4.17. (It also must be non-negative.) In order to relate the expansion coefficient Axl to the Cartesian dipole components calculated in a body-fixed frame, we choose the unit separation vector, R, to be parallel to the z-axis, hence M = 0, Yw = [(2L + 1)/4k] /2, and... 

Consider the two-dimensional stresses on the faces of a cartesian control volume as illustrated in Fig. 2.25. The differential control-volume dimensions are dx and dy, with the dz = 1. Assuming differential dimensions and that the stress state is continuous and differentiable, the spatial variation in the stress state can be expressed in terms of first-order Taylor series expansions. 

The Onk and Q k are operators which are respectively linear combinations of spherical harmonics and expansions in terms of Cartesian coordinates, 1 = 2 for d-orbitals, 3 for /-orbitals. The parameters B k and A k are, of course, specialized forms of the general form given in equation (2), but including the evaluation of the relevant radial integrals. 

The determination of these normal frequencies, and the forms of the normal vibrations, thus becomes the primary problem in correlating the structure and internal forces of the molecule with the observed vibrational spectrum. It is the complexity of this problem for large molecules which has hindered the kind of detailed solution that can be achieved with small molecules. In the general case, a solution of the equations of motion in normal coordinates is required. Let the Cartesian displacement coordinates of the N nuclei of a molecule be designated by qlt q2,... qsN. The potential energy of the oscillating system is not accurately known in the absence of a solution to the quantum mechanical problem of the electronic energies, but for small displacements it can be quite well approximated by a power series expansion in the displacements ... 

For the space group 225 (Fmim or Ojj) write down the coset expansion of the little group G(k) on T. Hence write down an expression for the small representations. State the point group of the k vector P(k) at the symmetry points L(/4 V-i Vi) and X(q a 2a). Work out also the Cartesian coordinates of L and X. Finally, list the space-group representations at L and X. 

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