Hydrogen atom, atomic orbital orbitals

Suggested Extra Reading- Appendix B The Hydrogen Atom Orbitals]... 

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... 

So called Hydrogenic atomic orbitals (exact solutions for the hydrogen atom) have radial nodes (values of the distance r where the orbital s value goes to zero) that make them somewhat inconvenient for computation. Results are not sensitive to these nodes and most simple calculations use Slater atomic orbitals of the form... 

The electronic Hamiltonian commutes with both the square of the angular momentum operator r and its z-component and so the three operators have simultaneous eigenfunctions. Solution of the electronic Schrddinger problem gives the well-known hydrogenic atomic orbitals... 

The hydrogen atom orbitals give us the numbers 2, 8, 18, and 32—the numbers we find separating the specially stable electron populations of the inert gases. It was necessary to multiply n2 by two—an important factor that could not have been anticipated. Furthermore, it will be necessary to find an explanation for the occurrence of eight-electron differences both at neon and at argon and eighteen-electron differences both at krypton and at xenon. 

In NH and NFS, three p orbitals are involved in the bonding [see representation (30)]. Figure 16-10 shows the spatial arrangement implied by assuming persistence of the hydrogen atom orbitals after bonding. We expect, then, that ant-... 

The solutions of the angular dependent part are the spherical harmonics, Y, known to most chemists as the mathematical expressions describing shapes of (hydrogenic) atomic orbitals. It is noted that Y is defined only in terms of a central field and not for atoms in molecules. 

For the hydrogen atom, orbital energy depends only on the value of n. For example, all four n = 2 orbitals have the same energy. All nine n = 3 orbitals have tbe same energy. What must the value of /be for each of these orbitals ... 

There is a similar effect in the length of C-H bonds, but this is less dramatic, primarily because the hydrogen atomic orbital involved (1 ) is considerably smaller than any of the hybrid orbitals we are considering. Nevertheless, C-H bonds involving sp-hybridized carbon are shorter than those involving i /t -hybridized carbon, and those with -hybridized carbon are the longest. 

The hydrogen atom orbitals are functions of three variables the coordinates of the electron. Their physical interpretation is that the square of the amplitude of the wave function at any point is proportional to the probability of finding a particle at that point. Mathematically, the electron density distribution is equal to the square of the absolute value of the wave function ... 

The basic concept of molecular orbital theory is that molecular orbitals may be constructed from a set of contributing atomic orbitals such that the molecular wave functions consist of linear combinations of atomic orbitals (LCAO).. in the case of the combination of two Is hydrogen atomic orbitals to give two molecular orbitals, the two linear combinations are written below so that atomic wave functions are represented by j/ and molecular wave functions by <)> ... 

In our QM systems, we have temporarily restricted ourselves to systems of one electron. If, in addition, our system were to have only one nucleus, then we would not need to guess wave functions, but instead we could solve Eq. (4.16) exactly. The eigenfunctions that are determined in that instance are the familiar hydrogenic atomic orbitals. Is, 2s, 2p, 3s, 3p, 3d, etc., whose properties and derivation are discussed in detail in standard texts on quantum mechanics. For the moment, we will not investigate the mathematical representation of these hydrogenic atomic orbitals in any detail, but we will simply posit that, as functions, they may be useful in the construction of more complicated molecular orbitals. In particular, just as in Eq. (4.10) we constructed a guess wave function as a linear combination of exact wave functions, so here we will construct a guess wave function as a linear combination of atomic wave functions (p, i.e.,... 

Slater-type orbitals were introduced in Section 5.2 (Eq. (5.2)) as the basis functions used in extended Huckel theory. As noted in that discussion, STOs have a number of attractive features primarily associated with the degree to which they closely resemble hydrogenic atomic orbitals. In ab initio HF theory, however, they suffer from a fairly significant limitation. There is no analytical solution available for the general four-index integral (Eq. 

Fig. 1. Hydrogen atomic orbitals (a) for isolated atoms (h) with one direction defined (c) with three directions defined.

Figure 2.22 A plot of the radial function for a 3s hydrogen atomic orbital...

There are several discrete atomic orbitals available to the electron of a hydrogen atom. These orbitals differ in energy, size, and shape, and exact mathematical descriptions for each are possible. Following is a qualitative description of the nature of some of the hydrogen atomic orbitals. 

Figure 1.1 Hydrogen atomic orbital functions, (a) Is (b) 2p (c) 3d. The edges drawn are artificial, because orbitals have no edges but merely decrease in magnitude as distance from the nucleus increases. The important features of the orbitals are the nodal planes indicated, and the algebraic signs of the orbital functions, positive in the shaded regions and negative in the unshaded regions.

The hydrogen atomic orbitals would not do us a great deal of good if orbitals of other atoms were radically different, since in that case different pictures would be required for each atom. But the feature of the hydrogen atom problem that determines the most important characteristics of the hydrogen atom orbitals is the spherical symmetry. Since all the atoms are spherically symmetric, the atomic orbitals of all atoms are similar, the main difference being in their radial dependence, that is, in how rapidly they approach zero as one moves away from the nucleus. Because the radial dependence is of minimal importance in qualitative... 

A complete review of the characteristics of various types of basis sets has been given recently by Schaefer.44 The radial form of STO s is similar to the (nodeless) hydrogenic atomic orbitals, rn -1e - r where n is the principal quantum number and C is a variable exponent.45 Their angular dependence is described by multiplication by a spherical harmonic The use of (jTO s in molecular calculations was... 

In the most commonly utilized approximation, the many-electron wave functions are written in terms of products of one-electron wave functions similar to the solutions obtained for the hydrogen atom. These one-electron functions used to construct the many-electron wave function are called atomic orbitals. They are also called hydrogen-like orbitals since they are one-electron orbitals and also because their shape is similar to that of the hydrogen atom orbitals. 

Analogously to all similar UHF wave functions describing dissociation of a closed shell molecule Into two odd-electron fragments, the UHF/1 wave fimctions is characterized by a pair of molecular orbitals of opposite spins, which are more and more localized on the boron and on the hydrogen, respectively, as the intemuclear distance R increases. (In the limit R- >, one of these orbitals becomes a pure boron, another a pure hydrogen atomic orbital). 

Fig. 1.1 Overlap of hydrogen atom orbitals to form (a) a bonding orbital, and (b) an antibonding orbital, with energies as shown in (c).

Mulliken proposed a different trial function than that used in the valence bond approach. In general, this function is a normalized sum of hydrogen-atom orbitals and is written in the form... 

So far we have talked about the shapes of the hydrogen atomic orbitals but not about their energies. For the hydrogen atom the energy of a particular orbital is determined by its value of n. Thus all orbitals with the same value of n have the same energy—they are said to be degenerate. This feature... 

---