Many-electron atoms defined

The true value of tk for a many-electron atom or a molecule is unknown. If we could set it equal ( expand it) to a linear combination of an infinite number of basis functions, each defined in a space of infinite dimensions, we could carry out an exact calculation of (k. Such a set of basis functions would be a complete set. 

In the case of many-electron atoms the total electronic orbital angular momentum is defined as the vector sum over n electrons... 

In order to understand how electrons of many-electron atoms arrange themselves into the available orbitals it is necessary to define a fourth quantum number ... 

More generally, for many-electron atoms we define U as the energy required to transfer an electron from one atom to another, so that... 

In the theory of many-electron atoms, the particle-hole representation is normally used to describe atoms with filled shells. To the ground state of such systems there corresponds a single determinant, composed of one-electron wave functions defined in a certain approximation. This determinant is now defined as the vacuum state. In the case of atoms with unfilled shells, this representation can be used for the atomic core consisting only of filled shells. Then, the excitation of electrons from these shells will be described as the creation of particle-hole pairs. 

Two examples of polyatomic calculations, on H20 and NH3, are outlined and explained in detail. In both cases the analysis starts from an assumed molecular structure of known symmetry. The transformation properties of the atomic orbitals on each atomic centre, under the symmetry operations of the group, are examined next. The atomic orbitals are defined as Is, 2s, 2pxi 2py and 2pz. Nothing can be more explicit - these are the occupied atomic orbitals of a many-electron atom. This configuration violates the exclusion principle9. Although the quantum numbers may not be needed,... 

In 1972 T. L. Allen used Monte Carlo for FSGO method by least squares solution of the Schrodinger equation for many electron atoms and molecules. The least squares solution of the Schrodinger equation was introduced by D. H. Weinstein in 1934, and developed by others. Let us define the local energy, for our system of interest as... 

These Hartree orbitals resemble the atomic orbitals of hydrogen in many ways. Their angular dependence is identical to that of the hydrogen orbitals, so quantum numbers and m are associated with each atomic orbital. The radial dependence of the orbitals in many-electron atoms differs from that of one-electron orbitals because the effective field differs from the Coulomb potential, but a principal quantum number n can still be defined. The lowest energy orbital is a Is orbital and has no radial nodes, the next lowest s orbital is a 2s orbital and has one radial node, and so forth. Each electron in an atom has associated with it a set of four quantum numbers (n, , m, mfj. The first three quantum numbers describe its spatial distribution and the fourth specifies its spin state. The allowed quantum numbers follow the same pattern as those for the hydrogen atom. However, the number of states associated with each combination of (n, , m) is twice as large because of the two values for m. ... 

Let us consider a many-electron atom of nuclear charge Z confined by a hard prolate spheroidal cavity. In this study the nuclear position will correspond to one of the foci as shown in Figure 4. In terms of prolate spheroidal coordinates, the nuclear position then corresponds to one of the foci for a family of confocal orthogonal prolate spheroids and hyperboloids defined, respectively, by the variables f and rj as [73] ... 

The family of confocal ellipsoids and hyberboloids represented by the prolate spheroidal coordinates allows us now to treat the case of a many-electron atom spatially limited by an open surface in half-space. A special case of the family of hyperboloids corresponds to an infinite plane defined by jj = 0 according to Equations (35) and (36). We now treat the specific case of an atom whose nuclear position is located at the focus a distance D from the plane as shown in Figure 4. 

Whereas, for non-Coulombic potentials, one can define nr and , n is then no longer related simply to the binding energy. Indeed, for a complex, many-electron atom, it is not at all obvious how one should set about quantising the system, since there is no guarantee that the orbits of individual electrons will close.6 In fact, conservation of the angular momentum for individual electrons is, at best, only an approximation. It would hold exactly for central fields. Even then, the same simple, precise relationship between n and as for H is not to be expected for many-electron atoms. As we shall see, the very meaning of n (the principal or most important quantum number) becomes less clear-cut for many-electron systems. In a nutshell the n and quantum numbers of... 

In this respect, the highly-correlated, many-electron atom is similar to the situation encountered in nuclear physics, where there are many dynamical variables and there exists no simple semiclassical limit. Random matrix theory can be a useful framework, but the word chaos is not easy to define in this context. 

The power of quantum mechanics is revealed by experimental confirmation of the predicted spectroscopic properties of atomic hydrogen. The reasonable expectation of successfully extending the method to many-electron atoms and molecules has been thwarted by mathematical complexity. It has never been possible to solve the wave equation for the motion of more than one particle. The most complex chemical system that has been solved (numerically) is for the single electron in the field of two protons, clamped in place, to define the molecular ion hJ. In order to apply the methods of quantum mechanics to any atom or molecule, apart from H and H, it is necessary to apply approximation methods or introduce additional assumptions based on chemical intuition. 

A wavefunction, ip, is a solution to the Schrodinger equation. For atoms, wavefunctions describe the energy and probabihty of location of the electrons in any region around the proton nucleus. The simplest wavefunctions are found for the hydrogen atom. Each of the solutions contains three integer terms called quantum numbers. They are n, the principal quantum number, I, the orbital angular momentum quantum number and mi, the magnetic quantum number. These simplest wavefunctions do not include the electron spin quantum number, m, which is introduced in more complete descriptions of atoms. Quantum numbers define the state of a system. More complex wavefunctions arise when many-electron atoms or molecules are considered. 

For each electron in a many-electron atom, the centrifugal potential was proportional to an inverse squared distance, namely the distance of the electron from the subspace defined by the other N—1 electrons and the nucleus. In the units of this chapter, namely 4/( )—1) ... 

A free radical (often simply called a radical) may be defined as a species that contains one or more unpaired electrons. Note that this definition includes certain stable inorganic molecules such as NO and NO2, as well as many individual atoms, such as Na and Cl. As with carbocations and carbanions, simple alkyl radicals are very reactive. Their lifetimes are extremely short in solution, but they can be kept for relatively long periods frozen within the crystal lattices of other molecules. Many spectral measurements have been made on radicals trapped in this manner. Even under these conditions, the methyl radical decomposes with a half-life of 10-15 min in a methanol lattice at 77 K. Since the lifetime of a radical depends not only on its inherent stabihty, but also on the conditions under which it is generated, the terms persistent and stable are usually used for the different senses. A stable radical is inherently stable a persistent radical has a relatively long lifetime under the conditions at which it is generated, though it may not be very stable. 

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