Hydrogen orbitals used

Traditional hydrogenic orbitals used in atomic and molecular physics as expansion bases belong to the nlm) representation, which in configuration space corresponds to separation in polar coordinates, and in momentum space to a separation in spherical coordinates on the (Fock s) hypersphere [1], The tilm) basis will be called spherical in the following. Stark states npm) have also been used for atoms in fields and correspond to separation in parabolic coordinates an ordinary space and in cylindrical coordinates on (for their use for expanding molecular orbitals see ref. [2]). A third basis, to be termed Zeeman states and denoted nXm) has been introduced more recently by Labarthe [3] and has found increasing applications [4]. 

Find the radius of the sphere defining the Is hydrogen orbital using the 95% probability definition. 

The orbitals used for methane, for example, are four Is Slater orbitals of hydrogen and one 2s and three 2p Slater orbitals of carbon, leading to an 8 x 8 secular matrix. Slater orbitals are systematic approximations to atomic orbitals that are widely used in computer applications. We will investigate Slater orbitals in more detail in later chapters. 

It is easy to see that the full shape of the orbital is better represented by the sum of these two Gaussians, especially at the tail of the cur ve where chemical bonding takes place, than it is by one Gaussian. When we run an STO-2G ab initio calculation on the hydrogen atom using the GAUSSIAN stored parameters rather than supplying oirr own, the input file is... 

According to this model, a covalent bond consists of a pair of electrons of opposed spin within an orbital. For example, a hydrogen atom forms a covalent bond by accepting an electron from another atom to complete its Is orbital. Using orbital diagrams, we could write... 

The HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied MO) levels for hydrogen donors used in coal liquefaction are not yet well known, but the principles involved can be illustrated with the group transfer reaction between molecular hydrogen, a (4n+2)e donor with n=0, and naphthalene, a (4m)e acceptor with m=l ... 

The combination of boron and hydrogen orbitals in the three-center bond can be shown in a molecular orbital diagram as in Figure 13.2. Using this approach to bonding, the structures of some of the... 

The relationship between alternative separable solutions of the Coulomb problem in momentum space is exploited in order to obtain hydrogenic orbitals which are of interest for Sturmian expansions of use in atomic and molecular structure calculations and for the description of atoms in fields. In view of their usefulness in problems where a direction in space is privileged, as when atoms are in an electric or magnetic field, we refer to these sets as to the Stark and Zeeman bases, as an alternative to the usual spherical basis, set. Fock s projection onto the surface of a sphere in the four dimensional hyperspace allows us to establish the connections of the momentum space wave functions with hyperspherical harmonics. Its generalization to higher spaces permits to build up multielectronic and multicenter orbitals. 

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 molecule of hydrogen fluoride, HF, belongs to the Coov point group. The hydrogen atom uses its 1 s atomic orbital to make bonding and antibonding combinations with the 2pz orbital of the fluorine atom, the z... 

Figure 6.14 The orbitals of two BH2 groups and of two hydrogen atoms used in the formation of the MOs of the bridge bonding in diborane...

The two metal atoms make use of hybrid orbitals as indicated, and each of these overlaps a hydrogen orbital that is centered midway between the metals but substantially above the metal-metal axis. For a dimeric species, such as diborane, this is repeated with a second set of orbitals and hydrogen atom symmetrically placed on the opposite side of the metal-metal axis. Examination of this model shows that each of the terminal bonds is a normal 2-electron-2-center bond. Only the bonds between the metal centers are nonconventional. An extensive review of these systems has been given by Lipscomb (69). 

Abstract—The spectroscopic phenomena of strong hydrogen bonds (band shifts, band broadening, intensity increase) are attributed to the interaction of the n electrons with the hydrogen atom, using the latter s distorted p-orbit. The analogy of the spectra phenomena in the hydrogen-chelated and metal-chelated coinpounds is stressed. 

Fig. 11-3.1. A set of vectors v1( va, V , and v4 representing the four o hybrid orbitals used by carbon to bond the four hydrogens in methane.

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