Electron polar coordinates

This section contains a brief review of the molecular version of Marcus theory, as developed by Warshel [81]. The free energy surface for an electron transfer reaction is shown schematically in Eigure 1, where R represents the reactants and A, P represents the products D and A , and the reaction coordinate X is the degree of polarization of the solvent. The subscript o for R and P denotes the equilibrium values of R and P, while P is the Eranck-Condon state on the P-surface. The activation free energy, AG, can be calculated from Marcus theory by Eq. (4). This relation is based on the assumption that the free energy is a parabolic function of the polarization coordinate. Eor self-exchange transfer reactions, we need only X to calculate AG, because AG° = 0. Moreover, we can write... 

The spacial distribution of electron density in an atom is described by means of atomic orbitals Vr(r, 6, (p) such that for a given orbital xp the function xj/ dv gives the probability of finding the electron in an element of volume dv at a point having the polar coordinates r, 6, 0. Each orbital can be expressed as a product of two functions, i e. 0, [Pg.1285]

Atoms are spherical objects, and what we do is to write the electronic Schrddinger equation in spherical polar coordinates, to mirror the symmetry of the problem. 

Polar Coordinates. Yes, this is a type of factor space. We might use polar coordinates when we are mapping electron densities, seasonal population variations, or anytime it makes our work more convenient or allows us to better visualize or understand our data. 

Figure 4. The following data were used leak radius r0 (40 micron), leak conductance F = 1 cc. sec.-1, gas pressure (air) p (20 torr), diffusion coefficient (26) for ions Di — 2 sq. cm. sec.-"1, for electrons De = 2000 sq. cm. sec.-1 (at 20 torr air). The flow velocities were assumed independent of 6 and (spherical polar coordinates). The spherically symmetrical flow pattern, which obviously overestimates the flow velocity for large values of 6 was chosen because of its simplicity. The velocity of the radially directed flow at a distance r is v = F/2irr2. The time re-...

Here the eigenfunctions referred to are those for an electron in an atom, and r, 0 and

polar coordinates of the electron, the nucleus being at the origin of the coordinate system. 

The Tetrahedral Carbon Atom.—We have thus derived the result that an atom in which only s and p eigenfunctions contribute to bond formation and in which the quantization in polar coordinates is broken can form one, two, three, or four equivalent bonds, which are directed toward the corners of a regular tetrahedron (Fig. 4). This calculation provides the quantum mechanical justification of the chemist s tetrahedral carbon atom, present in diamond and all aliphatic carbon compounds, and for the tetrahedral quadrivalent nitrogen atom, the tetrahedral phosphorus atom, as in phosphonium compounds, the tetrahedral boron atom in B2H6 (involving single-electron bonds), and many other such atoms. 

The radial functions, R depend only upon the distance, r, of the electron from the nucleus while the angular functions, (6,(p) called spherical harmonics, depend only upon the polar coordinates, 6 and Examples of these purely angular functions are shown in Fig. 3-11. 

In the REC model, the ligand is modelled through an effective point charge situated in the axis described by the lanthanide-coordinated atom axis, at a distance R, which is smaller than the real metal-ligand distance (Figure 2.6). To account for the effect of covalent electron sharing, a radial displacement vector (Dr) is defined, in which the polar coordinate R is varied. At the same time, the charge value (q) is scanned in order to achieve the minimum deviation between calculated and experimental data, whereas 9 and cp remain constant. 

In order to solve the wave equation for the hydrogen atom, it is necessary to transform the Laplacian into polar coordinates. That transformation allows the distance of the electron from the nucleus to be expressed in terms of r, 9, and (p, which in turn allows the separation of variables technique to be used. Examination of Eq. (2.40) shows that the first and third terms in the Hamiltonian are exactly like the two terms in the operator for the hydrogen atom. Likewise, the second and fourth terms are also equivalent to those for a hydrogen atom. However, the last term, e2/r12, is the troublesome part of the Hamiltonian. In fact, even after polar coordinates are employed, that term prevents the separation of variables from being accomplished. Not being able to separate the variables to obtain three simpler equations prevents an exact solution of Eq. (2.40) from being carried out. 

The well balanced electronic and coordinative unsaturation of their Ru(II) center accounts for the high performance and the excellent tolerance of these complexes toward an array of polar functional groups. This discovery has triggered extensive follow up work and carbenes 1 now belong to the most popular metathesis catalysts which set the standards in this field [3]. Many elegant applications to the synthesis of complex target molecules and structurally diverse natural products highlight their truely remarkable scope. 

Iti wave mechanics yt(r, 0, wave-function is yi(r, 0, y) is to be found in u small volume dx centred at the point whose polar coordinates arc (r, 0,9 ). To make the total probability unity we must have... 

Figure 7.6 The change in the polar coordinate 0 of an electron due to rotation through an angle a around the z-axis.

For compactness, the subscript M for the electronic density parameters has been omitted in Eq. (8.49). The polar coordinate system has the z axis of the local Cartesian coordinate system as the polar axis, and the vector RMP is referred to this local coordinate system. 

Within the Born-Oppenheimer approximation, the nuclei are at rest and have zero momentum. So the electron momentum density is an intrinsically one-center function that can be expressed usefully in spherical polar coordinates and expanded as follows [162,163]... 

Here, ma is the mass of the nucleus a, Zae2 is its charge, and Va2 is the Laplacian with respect to the three cartesian coordinates of this nucleus (this operator Va2 is given in spherical polar coordinates in Appendix A) rj a is the distance between the jth electron and the a1 1 nucleus, rj k is the distance between the j and k electrons, me is the electron s mass, and Ra>b is the distance from nucleus a to nucleus b. 

Equation (9.38), if restricted to two particles, is identical in form to the radial component of the electronic Schrodinger equation for the hydrogen atom expressed in polar coordinates about the system s center of mass. In the case of the hydrogen atom, solution of the equation is facilitated by the simplicity of the two-particle system. In rotational spectroscopy of polyatomic molecules, the kinetic energy operator is considerably more complex in its construction. For purposes of discussion, we will confine ourselves to two examples that are relatively simple, presented without derivation, and then offer some generalizations therefrom. More advanced treatises on rotational spectroscopy are available to readers hungering for more. 

The hydrogen atom has a single electron confined to the neighbourhood of the nucleus by a potential field V, given by — e jr. The solution of the appropriate Schrodinger equation becomes possible if the equation is expressed in polar coordinates r, 0 and independent equations each containing only one variable ... 

Having decided to use AOs (or combinations of them) for yrA and pB> we will now look at the form these take. They are approximate solutions to the Schrodinger equation for the atom in question. The Schrodinger equation for many-electron atoms is usually solved approximately by writing the total electronic wavefunction as the product of one-electron functions (these are the AOs). Each AO 4>i is a function of the polar coordinates r, 0, and single electron and can be written as... 

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