Orbitals of the hydrogen atom

At last we are ready to return to the periodic table. At last we are able to begin answering those who are wondering why about the special properties of the electron populations in Table 15-1. Let us reproduce Table 15-1 together with the numbers of orbitals of the hydrogen atom. The suggestion of a connection is irresistible, as seen in Table 15-II. 

According to the quantum mechanical description of the 1 s orbital of the hydrogen atom, what relation exists between the surface of a sphere centered about the nucleus and the location of an electron ... 

For any particular atom, the = 2 orbitals are larger than the Is orbital, the = 3 orbitals are larger than the = 2 orbitals, and so on. The electron density plots in Figure 7-20 show this trend for the first three s orbitals of the hydrogen atom. This plot also shows that the number of nodes increases as n increases. 

Electron density plots for the 1, 2, and 3 S atomic orbitals of the hydrogen atom. The vertical... 

Figure 3.10 Representations of the electron density ip2 of the Is orbital and the 2p orbital of the hydrogen atom. (b,e) Contour maps for the xe plane. (c,f) Surfaces of constant electron density. (a,d) Dot density diagrams the density of dots is proportional to the electron density. (Reproduced with permission from the Journal of Chemical Education 40, 256, 1963 and M. J. Winter, Chemical Bonding, 1994, Oxford University Press, Fig. 1.10 and Fig. 1.11.)...

The units we use in daily life, such as kilogram (or pound) and meter (or inch) are tailored to the human scale. In the world of quantum mechanics, however, these units would lead to inconvenient numbers. For example, the mass of the electron is 9.1095 X J0 31 kg and the radius of the first circular orbit of the hydrogen atom in Bohr s theory, the Bohr radius, is 5.2918 X 10 11 m. Atomic units, usually abbreviated as au, are introduced to eliminate the need to work with these awkward numbers, which result from the arbitrary units of our macroscopic world. The atomic unit of length is equal to the length of the Bohr radius, that is, 5.2918 X 10 n m, and is called the bohr. Thus 1 bohr = 5.2918 X 10"11 m. The atomic unit of mass is the rest mass of the electron, and the atomic unit of charge is the charge of an electron. Atomic units for these and some other quantities and their values in SI units are summarized in the accompanying table. 

By assigning various values to n, we can evaluate the corresponding energy of the electron in the orbits of the hydrogen atom. When this is done, we find the energies of several orbits as follows ... 

Figure 2.1 Electronic orbitals and the resulting emission spectrum in the hydrogen atom, (a) Bohr orbitals of the hydrogen atom and the resulting spectral series, (b) emission spectrum of atomic hydrogen. The spectrum in (b) is calibrated in terms of wavenumber (P), which is reciprocal wavelength. The Balmer series, which consists of those transitions terminating on the second orbital, give rise to emission lines in the visible region of the spectrum. ( 1990 John Wiley Sons, Inc. Reprinted from Brady, 1990, by permission of the publisher.)...

There are certain special cases, however, where Equation 9.1 does not provide the atomic number. The simplest example is the 2p orbital of the hydrogen atom, where the density... 

Similarly, C — H sigma bonds in the C2H6 molecule are formed by the end to end overlap of sp3 hybrid orbitals of the carbon atoms with the Is orbitals of the hydrogen atoms. The C—Co bond is formed by the end to end overlap of the sp3 hybrid orbitals of the C atoms. So in the C2H6 molecule there are six C — H o bonds and one C—Co bond making seven o bonds in total. 

DSO term is very small. However, as we will see latter, the DSO contribution is significant for certain coupling constants and cannot be discarded. Note that our criterion to estimate the order of magnitude of the individual terms is based on an electron in a Bohr orbit of the hydrogen atom. On some occasions this estimate may not give a good indication of the actual magnitude. 

One-center expansion was first applied to whole molecules by Desclaux Pyykko in relativistic and nonrelativistic Hartree-Fock calculations for the series CH4 to PbH4 [81] and then in the Dirac-Fock calculations of CuH, AgH and AuH [82] and other molecules [83]. A large bond length contraction due to the relativistic effects was estimated. However, the accuracy of such calculations is limited in practice because the orbitals of the hydrogen atom are reexpanded on a heavy nucleus in the entire coordinate space. It is notable that the RFCP and one-center expansion approaches were considered earlier as alternatives to each other [84, 85]. 

The spatial orientations of the atomic orbitals of the hydrogen atom are very important in the consideration of the interaction of orbitals of different atoms in the production of chemical bonds. 

Neither of the Is orbitals of the hydrogen atoms of the water molecule, taken separately, transform within the group of irreducible representations deduced for that molecule. The two Is orbitals must be taken together as one or the other of two group orbitals. A more formal treatment of the group orbitals which two Is orbitals may form is dealt with in Chapter 3. 

Whenever there are two or more identical atoms linked to a central atom, their wave functions must be combined in such a way as to demonstrate their indistinguishability. This is achieved by making linear combinations of the wave functions of the ligand atoms. For the two Is orbitals of the hydrogen atoms in the water molecule their wave functions may be combined to give ... 

To reinforce the meanings of diagrams such as those shown in Figure 3.1, it is helpful to plot the atomic and molecular wave functions along an axis which coincides with the molecular axis of H2. For this exercise the wave function for the Is orbital of the hydrogen atom may be written... 

The VSEPR treatment is best approached by considering the ion as made up from three ions F + H+ + F. The central proton possesses no electrons until the ligand fluoride ions supply two each. The two pairs of electrons repel each other to give the observed linear configuration of the three atoms. The two pairs of electrons would occupy the Is and 2s orbitals of the hydrogen atom and, what with a considerable amount of interelectronic repulsion, would not lead to stability. 

Table 6.2 Symmetries of the valence orbitals of the carbon atom and the group orbitals of the hydrogen atoms of the methane molecule...

The radial function for the 3s atomic orbital of the hydrogen atom has the form ... 

Figure 4.5 The radial probability density function for the 1s atomic orbital of the hydrogen atom...

If the relativistic effects are sufficiently large and therefore cannot be accounted for as corrections, then as a rule one has to utilize relativistic wave functions and the relativistic Hamiltonian, usually in the form of the so-called relativistic Breit operator. In the case of an N-electron atom the latter may be written as follows (in atomic units, in which the absolute value of electron charge e, its mass m and Planck constant h are equal to one, whereas the unit of length is equal to the radius of the first Bohr orbit of the hydrogen atom) ... 

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