Slater, coordinate

We 6hall first consider the isotope effects to be expected from an oversimplified model of the activated complex namely, all frequencies in the activated complex are identical with those in the normal molecule except the C—C stretching frequency, which becomes a translation. The temperature independent factorvL fvL+ will, first, be calculated from the Slater coordinate. In addition we assume that the isotope shift in the C—C stretching frequency is again given by the diatomic reduced mass relationship. 

This same quantum correction can be used with the Slater coordinate for the temperature independent factor. This would give kjnk3 = 1.045 at 400°K, and would not change the conclusions to be reached. 

For both types of orbitals, the coordinates r, 0, and (j) refer to the position of the electron relative to a set of axes attached to the center on which the basis orbital is located. Although Slater-type orbitals (STOs) are preferred on fundamental grounds (e.g., as demonstrated in Appendices A and B, the hydrogen atom orbitals are of this form and the exact solution of the many-electron Schrodinger equation can be shown to be of this form (in each of its coordinates) near the nuclear centers), STOs are used primarily for atomic and linear-molecule calculations because the multi-center integrals < XaXbl g I XcXd > (each... 

As mentioned in Chapter 5, one can think of the expansion of an unknown MO in terms of basis functions as describing the MO function in the coordinate system of the basis functions. The multi-determinant wave function (4.1) can similarly be considered as describing the total wave function in a coordinate system of Slater determinants. The basis set determines the size of the one-electron basis (and thus limits the description of the one-electron functions, the MOs), while the number of determinants included determines the size of the many-electron basis (and thus limits the description of electron correlation). 

The variational problem may again be formulated as a secular equation, where the coordinate axes are many-electron functions (Slater determinants), <, which are orthogonal (Section 4.2). 

The total wavefunction, , is an antisymmetrized product of the one-electron functions i/q (a Slater determinant). The i/tj are called one-electron functions since they depend on the coordinates of only one electron this approximation is embedded in all MO methods. The effects that are missing when this approximation is used go under the general name of electron correlation. 

The orbital phase continuity conditions stem from the intrinsic property of electrons. Electrons are fermions, and are described by wavefnnctions antisymmetric (change plus and minus signs) with respect to an interchange of the coordinates of an pair of particles. The antisymmetry principle is a more fnndamental principle than Pauli s exclusion principle. Slater determinants are antisymmetric, which is why the overlap integral between t(a c) given above has a negative... 

An operator, depending of an arbitrary number of electron coordinates, has an easily expressible set of matrix elements, using two Slater determinants D(j) and D(k). 

Suppose a r-electron operator to be written as S2(r), with the r-dimensional vector r representing the coordinates of the canonically ordered r (matrix element between two Slater determinants can te written as ... 

Calculation of the Slater-Condon, spin-orbit coupling and ligand field parameters. The luminescence of CsMgBr3 Eu2+ is crucially dependent on the local coordination geometry of the Eu2+ dopant. Besides, a geometry change occurs in the excited state 4f 5d1 (see Table 2), leading to shifts... 

Here Slater functions (a3 2/7r 1/2) exp( —a r — ra ) with the atom being centered at position vectors ra. The overlap between these functions is given by S. After an FT and integrating over momentum coordinates of one particle, the EMD of H2 molecule within VB and MO theory are derived as... 

J. Slater. Methods of self-coordinated field for molecules and solids. — M 1978, 662 p. 

Iosio Kato in 1957. [92] Unfortunately, any trial wave function composed of Slater determinants has smooth first and higher derivatives with respect to the interelec-tronic coordinates. Thus, even though such expansions are insightful and preserve the concept of orbitals to some extent, from a mathematical point of view they are expected to be slowly convergent. 

The coefficient of (1 / /2) is simply a normalization factor. This expression builds in a physical description of electron exchange implicitly it changes sign if two electrons are exchanged. This expression has other advantages. For example, it does not distinguish between electrons and it disappears if two electrons have the same coordinates or if two of the one-electron wave functions are the same. This means that the Slater determinant satisfies... 

The deformation functions, however, must also describe density accumulation in the bond regions, which in the one-center formalism is represented by the atom-centered terms. They must be more diffuse, with a different radial dependence. Since the electron density is a sum over the products of atomic orbitals, an argument can be made for using a radial dependence derived from the atomic orbital functions. The radial dependence is based on that of hydrogenic orbitals, which are valid for the one-electron atom. They have Slater-type radial functions, equal to exponentials multiplied by r1 times a polynomial of degree n — l — 1 in the radial coordinate r. As an example, the 2s and 2p hydrogenic orbitals are given by... 

The multipole formalism described by Stewart (1976) deviates from Eq. (3.35) in several respects. It is a deformation density formalism in which the deformation from the IAM density is described by multipole functions with Slater-type radial dependence, without the K-type expansion and contraction of the valence shell. While Eq. (3.35) is commonly applied using local atomic coordinate systems to facilitate the introduction of chemical constraints (chapter 4), Stewart s formalism has been encoded using a single crystal-coordinate system. 

Eastman A, Richon VM. Mechanisms of cellular resistance to platinum coordination complexes. In (McBrien DCH, Slater TF, eds) Biochemical Mechanisms of Platinum Anticancer Drugs 1986 IRL Oxford, UK pp. 91-119. 

The electronic Schrodinger equation is still intractable and further approximations are required. The most obvious is to insist that electrons move independently of each other. In practice, individual electrons are confined to functions termed molecular orbitals, each of which is determined by assuming that the electron is moving within an average field of all the other electrons. The total wavefunction is written in the form of a single determinant (a so-called Slater determinant). This means that it is antisymmetric upon interchange of electron coordinates. ... 

To determine the image, the first step is to determine the distribution of tunneling current as a function of the position of the apex atom. We set the center of the coordinate system at the nucleus of the sample atom. The tunneling matrix element as a function of the position r of the nucleus of the apex atom can be evaluated by applying the derivative rule to the Slater wavefunctions. The tunneling conductance as a function of r, g(r), is proportional to the square of the tunneling matrix element ... 

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