The hydrogen atom

The ground state is degenerate and the corresponding manifold of states are 

This wavefunction has been variationally determined for H with the optimal parameter values fj, = 0.6612 and ly = 0.3780. One can directly evaluate the amplitude 

The time of joining of the next monomer molecule to the already created polymeric chain, measured by very sensitive equipment, proves to be 10 sec, whereas computed from the Arrhenius equation gives a value near 10 ° years. 

The effect of tunneling enables us to take quite a new look at some physical, chemical and, particularly, biological processes in respect of the organization and behavior of biologically active systems, accompanying all natural processes—from the formation of planets to the most complex particularities of biosyntheses. The scales of these phenomena are inconceivable though it is presently possible to assert that tunneling effects play a very important role in many processes of vital activity. 

Consider now a more complex problem, which is very important for chemistry the motion of a charged particle in a spherically symmetric electric field. In this case a particle s 

The solution to this problem has an exceptionally important role in quantum mechanics and especially in quantum chemistry. Firstly, this is a problem that can be solved analytically (though some special mathematical functions must be used). Secondly, the solution is of great importance for chemistry where the electronic orbits arise from the solution moreover the theory of the chemical bond has been worked out using the results. Thirdly, an empirically modified hydrogen atom s orbits are widely used generally for heavier atoms because there are no other ways of achieving results. 

that the z-axis is formally distinguished from the other axes. It can be said that this axis is selected it is specially distinguished only geometrically, though it is always specially distinguished if there is an outer influence (electric, magnetic, etc.). It is said that z is the distinguished axis. 

EXAMPLE The lowest-frequency pure-rotational absorption line of occurs at 

The hydrogen atom consists of a proton and an electron. If e symbolizes the charge on the proton e - +1.6 x 10 C), then the electron s charge is -e. 

We shall assume the electron and proton to be point masses whose interaction is given by Coulomb s law. In discussing atoms and molecules, we shall usually be considering isolated systems, ignoring interatomic and intermolecular interactions 

The possibility of small deviations from Coulomb s law has been considered. Experiments have shown that if the Coulomb s-law force is written as being proportional to then s 10 . It can be shown that a deviation from Coulomb s law would imply a nonzero rest mass for the photon see A. S. Goldhaber and M. M. Nieto, Rev. Mod. Phys., 43,277 (1971). There is little or no evidence for a nonzero photon rest mass, and data indicate that any such mass must be less than 10 g L. Davis et al., Phys. Rev. Lett., 35,1402 (1975) R. Lakes, Phys. Rev. Lett., 80,1826 (1998). 

The force in (6.56) is central, and comparison with Eq. (6.4) gives dV r)/dr = Integration gives 

The hydrogen atom, consisting of a proton and only one electron, occupies a very important position in the development of quantum mechanics because the Schrddinger equation may be solved exactly for this system. This is true also for the hydrogen-like atomic ions He, Li, Be, etc., and simple one-electron molecular ions such as Hj. 

A vector quantity is indicated by bold italic type and its magnitude by italic type. 

The second term on the right-hand side is the coulombic potential energy for the attraction between charges —e and +e a distance r apart. The first term contains the reduced mass /r, equal to - - m ), for the system of an electron of mass and a proton of mass m.  

It also contains the laplacian which is here defined as 

The functions are known as the angular wave functions or, because they describe the distribution of p over the surface of a sphere of radius r, spherical harmonics. The quantum number n = l,2,3.oo and is the same as in the Bohr theory, is the azimuthal quantum number associated with the discrete orbital angular momentum values, and is 

The hydrogen atom, containing a single electron, has played a major role in the development of models of electronic structure. In 1913 Niels Bohr (1885-1962), a Danish physicist, offered a theoretical explanation of the atomic spectrum of hydrogen. His model was based largely on classical mechanics. In 1922 this model earned him the Nobel Prize in physics. By that time, Bohr had become director of the Institute of Theoretical Physics at Copenhagen. There he helped develop the new discipline of quantum mechanics, used by other scientists to construct a more sophisticated model for the hydrogen atom. 

like all the other individuals mentioned in this chapter, was not a chemist. His only real contact with chemistry came as an undergraduate at the University of Copenhagen. His chemistry teacher, Niels Bjerrum, who later became his close friend and sailing companion, recalled that Bohr set a record for broken glassware that lasted half a century. 

Before proceeding with the Bohr model, let us make three points  

Using this expression for AE and the equation En = —RH/n2, it is possible to relate the frequency of the light emitted to the quantum numbers, nhi and ni0, of the two states  

a giant of twentieth century physics, was respected by scientists and politicians alike. 

The hydrogen atom is the most important chemical application of the quantum theory. The solutions of SchrSdinger s equation—known as atomic orbitals—form the basis of our understanding, not only of atomic structure, but also of chemical bonding in molecules and solids. 

The im. functions are the associated Legendre polynomials of which a few are given in Table 1.1. They are independent of Z, the nuclear charge number, and therefore are the same for all one-electron atoms. 

The hydrogen atom is the most important testing ground for any quantum mechanical theory. Amazingly, it turns out that the Dirac equation describing a hydrogen-like system can be solved analytically. The results are in almost perfect agreement with the measurements. This success has been one of the main historical reasons for the quick acceptance of the Dirac equation by physicists. 

The potential energy of an electron (charge —e) in the field of an atomic nucleus of charge Ze is given by the Coulomb potential 

We divide the time-dependent Dirac equation by he and consider the Dirac operator 

In this expression we find a dimensionless combination of the physical constants e, h, and c, namely 

This constant is called the fine structure constant. Its numerical value is independent of the chosen system of units. The number Z of protons in the nucleus. 

like all the other individuals mentioned in this chapter, was not a chemist. His only real contact with chemistry came as an undergraduate at the University of Copen- 

TABLE 6.2 Wavelengths (nm) of Lines in the Atomic Spectrum of Hydrogen 

Ultraviolet (Lyman Series) Visible (Balmer Series) Infrared (Paschen Series) 

That is to say that the quantity p from classical mechanics is thus replaced in wave mechanics by an operator. Likewise the potential energy is also replaced by an operator, namely the multiplication of the function of the potential energy V by the wave function 9. 

In place of the classical equation (8) we can now also write for the wave equation  

In this H is thus now a symbol, not for a quantity but for an operation, to be applied to the wave function cp (see p. 123) H is called the HAMiLTONian operator. This operator H, therefore, has the same form as the function H in classical mechanics when the above-mentioned formal rules are borne in mind. 

In the calculation of the total energy W (p. 123) as well as other dynamical quantities which can be written classically as a function of p and q, the method of calculation in wave mechanics is to be found by means of the transformation to the corresponding operators according to the same rules. 

For the hydrogen atom we have simply V = —e2/r and the equation is then soluble, although even then some mathematical dexterity is needed. It appears that a solution is only possible for certain values of the total energy W thus the discrete energy states, which were assumed ad hoc in Bohr s theory, are produced automatically . These states are characterized by certain quantities which must be whole numbers, the quantum numbers, which thus also follow immediately from the mathematical behaviour of the equation. One finds for the wave function of the ground state of the hydrogen atom  

Equation (22.2) also shows that the total energy is made up of three contributions the first term in the equation is the contribution of the kinetic energy of the motion along the line of centers the second term is the kinetic energy associated with the rotation the third term is the potential energy, V(r). 

The hydrogen atom is a typical case of the central-field problem. As was shown in Fig. 19.5, the proton is at the center with a charge + e while the electron is at a distance r with a charge —e. The coulombic force acts along the line of centers and corresponds to a potential energy, V(r) = —e /4ncQr. 

We have replaced the rotational quantum number J by /, since this is the usual notation in atomic systems. The quantum number / is called the azimuthal quantum number and characterizes the total angular momentum of the atom, 

The quantum number m has the same interpretation as before it characterizes the z component of the angular momentum. 

In this situation, m is called the magnetic quantum number, for reasons that will be apparent later. 

Hydrogen (H) is the simplest and lightest atom in the periodic table. We drink it every day it is an essential component of water in fact, hydro-gen means water-generating, It has played a crucial role in many developments of modern physics. In this book we will model the hydrogen atom by a single quantum particle (the electron) moving in a spherically symmetric force field (created by the proton in the nucleus). There are certainly more sophisticated models available — for example, it is more precise to model the hydrogen atom as the mutual interaction of two particles, a proton and an electron — but our model is simple and quite accurate. 

The strongest, most easily discerned set of lines were called the principal spectrum. After the principal spectrum, there are two series of lines, the sharp spectrum and the diffuse spectrum. In addition, there was a fourth series of lines, the Bergmann or fimdamental spectrum. 

In the spectroscopy literature, a color is usually labeled by the corresponding wavelength of Ught (in angstroms A) or by the reciprocal of the wavelength (in cm ), called the wave number. One angstrom equals 10 meters, while one centimeter equals 10 meters, so to convert from wavelength to wave number one must multiply by a factor of 10  

As a concrete example, consider the strongest spectral line of hydrogen, corresponding to a wavelength of about 1200A. The corresponding wave number is 

The wave number is natural because it is proportional to the energy of a photon of the given frequency. More specifically, we have 

The study of the hydrogen atom also played an important role in the development of quantum theory. The Lyman, Balmer, and Paschen series of spectral lines observed in incandescent atomic hydrogen were found to obey the empirical equation 

Any description of the free atom begins with the Schroedinger wave equation for a single electron in the field of a positive point charge +Ze  

Hr = mM/(m + M) is the reduced mass of the electron and nucleus, — e is the charge on an electron, and h = 2wh is Planck s constant. For bound states, i.e. E 0, this equation has the well-known solution (194) 

For an equilibrium classical orbit, the attractive electrostatic force must equal the centrifugal force, or 

Combination of equations 3 and 4 gives the classical Bohr orbits 

The reciprocal of the Bohr radius is just the quantum-mechanical mean reciprocal radius of an electron with quantum number n  

The Hydrogen When the Schrodinger equation is solved for the hydrogen atom, it is found that 

In order to plot the complete wave functions, one would in general require a four-dimensional graph with coordinates for each of the three spatial dimensions (x. y, z or r, 6, i ) and a fourth value, the wave function. 

In order to circumvent this problem and also to make it easier to visualize the actual distribution of electrons within the atom, it is common to break down the wave function, S, into three parts, each of which is a function or but a single variable. It is most convenient to use polar coordinates, so one obtains 

The Radial Wave The radial functions for the first three orbitals1 in the hydrogen atom are  

Although the radial functions may appear formidable, the important aspects may be made apparent by grouping the constants. For a given atom, Z will be constant and may be combined with the other constants, resulting in considerable simplification  

We start by considering the hydrogen atom, the simplest possible system, in which one electron interacts with a nucleus of unit positive charge. Only two terms are required from the master equation (3.161) in chapter 3, namely, those describing the kinetic energy of the electron and the electron-nuclear Coulomb potential energy. In the space-fixed axes system and SI units these terms are 

The Schrodinger equation for the hydrogen atom may therefore be written 

It turns out that the solutions of (6.5) are much simpler if one transforms from cartesian to spherical polar coordinates, as defined in figure 6.1. The relationships between the two are 

X = r sin0 cos , Y = r sin0sin0, Z = rcos9, so that, in spherical polar coordinates, the Laplacian is given by 

The solutions of the Schrodinger equation in these spherical polar coordinates are described in many books. They can be factorised and have the following form  

When the Schrddinger equation is solved for the hydrogen atom, it is found that there are three characteristic quantum numbers n. (, and m, (as expected for a three-dimensional system)L The allowed values for these quantum numbers and their relation to the physical system will be discussed below, but for now they may be taken as a set of three inleyers specifying a particular situation. Each solution found for a dilTeiient set of ti, I, and m, is called an eU/enfmiction and represents an orbital in the hydrogen atom. 

Because we are principally Interested in the prohtihiliiy of finding electrons at various points in space, we shall be more concerned with the squares of the radial functions than with the functions themselves. It i.s the square of the wave ftinction 

The interaction between an electron and a nucleus in a hydrogen atom gives rise to a potential energy that can be described by the relationship -e2/r. Therefore, using the Hamiltonian operator and postulate IV, the wave equation can be written as 

Rearranging the equation and representing the potential energy as V gives 

The difficulty in solving this equation is that when the Laplacian is written in terms of Cartesian coordinates we find that r is a function of x, y, and z, 

The wave equation is a second-order partial differential equation in three variables. The usual technique for solving such an equation is to use a procedure known as the separation of variables. However, with r expressed as the square root of the sum of the squares of the three variables, it is impossible 

The three quantum numbers that arise as mathematical restraints on the differential equations (boundary conditions) can be summarized as follows  

For more discussion of nuclear motion in diatomic molecules, see Section 13.2. For the rotational energies of polyatomic molecules, see Townes and Schawlow, chaps. 2-4. 

The lowest-frequency pure-rotational absorption line of C S occurs at 48991.0 MHz. Find the bond distance in C S. 

The lowest-frequency rotational absorption is the J = 0- 1 line. Equations (1.4), (6.52), and (6.51) give 

A few scientists have speculated that the proton and electron charges might not be exactly equal in magnitude. Experiments show that the magnitudes of the electron and proton charges are equal to within one part in 10. See G. Bressi et al., Phys. Rev. A, 

In this chapter we shall consider atoms or ions which have a single electron. These include the hydrogen atom, He and LP . The solutions of Schrodinger s wave equation for such systems provide important information on the way in which an electron moves around the nucleus. The actual trajectory followed by an electron cannot be known in detail because of the operation of the uncertainty principle, but the wavefunctions obtained from the Schrodinger equation provide probability dis- 

The wavefunctions that we shall discuss form the basis of our understanding of atomic structure in general, because the concepts introduced can be extended to many-electron atoms. They will also prove useful when we come to discuss chemical bonding. 

Now we are ready to apply the method of wave mechanics to study the electronic structure of the atoms. At the beginning of this chapter, we concentrate on the hydrogen atom, which consists of one proton and one electron. After treating the hydrogen atom, we will proceed to the other atoms in the Periodic Table. 

The discrimination of emission frequencies leads to the concept of discrete energy levels within the atom that may be occupied by electrons. Detailed analysis of the wavelengths of the lines in the emission spectrum of the hydrogen atom led to the formulation of the onpirical Rydberg equation  

The Rydberg constant, R, has an experimentally observed value of 1.096776 X 10 m for the hydrogen atom. Because it is the frequency, V, of the radiation which is proportional to the energy, equation (1.16) can be transformed into one expressing the frequencies of lines by using the relationship given by equation (1.2)  

The other term in the equation, c, is the velocity of light. The terms in 

Bohr interpreted spectral lines in the hydrogen spectrum in terms of electronic transitions within the hydrogen atom. The Bohr equation (1.15) expresses the idea that Ej - E, represents the difference in energy between the two levels, A , and may be written in the form  

Equation (1.20) may be regarded as being the difference between the two equations  

Each photon of blue light carries a larger quantity of energy than a photon of red light. 

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Fischer-Hepp rearrangement The nitros-amines of aromatic secondary amines when treated with hydrochloric acid give nuclear substituted nitrosoamines. Among the benzene derivatives, if the para position is free the -NO group displaces the hydrogen atom there in naphthalene derivatives it enters the 1-position ... 

Covalent. Formed by most of the non-metals and transition metals. This class includes such diverse compounds as methane, CH4 and iron carbonyl hydride, H2Fe(CO)4. In many compounds the hydrogen atoms act as bridges. Where there are more than one hydride sites there is often hydrogen exchange between the sites. Hydrogens may be inside metal clusters. 

They are formed by heating dibasic acids or their anhydrides with ammonia. The hydrogen atom of the NH group is acidic and can be replaced by a metal. Mild hydrolysis breaks the ring to give the half amide of the acid. See succinimide and phthalimide. 

Both compounds are soluble in water and are readily hydrolysed to sulphamic acid, HjN S03" andammonia the hydrogen atoms are in each case replaceable by metals to form salts. Many derivatives of sulphamide and cyclic derivatives of sulphimide are known. 

The hydrogen atom attached to each carbon atom in the hexagon has been omitted by convention. 

The hydrogen atoms have been omitted for the sake of simplification. 

Despite its success in reproducing the hydrogen atom spectmm, the Bolir model of the atom rapidly encountered difficulties. Advances in the resolution obtained in spectroscopic experiments had shown that the spectral features of the hydrogen atom are actually composed of several closely spaced lines these are not accounted for by quantum jumps between Bolir s allowed orbits. However, by modifying the Bolir model to... 

It is admittedly inconsistent to begin a section on many-particle quantiun mechanics by discussing a problem that can be treated as a single particle. Flowever, the hydrogen atom and atomic ions in which only one... 

Applications of quantum mechanics to chemistry invariably deal with systems (atoms and molecules) that contain more than one particle. Apart from the hydrogen atom, the stationary-state energies caimot be calculated exactly, and compromises must be made in order to estimate them. Perhaps the most useful and widely used approximation in chemistry is the independent-particle approximation, which can take several fomis. Conuiion to all of these is the assumption that the Hamiltonian operator for a system consisting of n particles is approximated by tlie sum... 

The miderstanding of the quantum mechanics of atoms was pioneered by Bohr, in his theory of the hydrogen atom. This combined the classical ideas on planetary motion—applicable to the atom because of the fomial similarity of tlie gravitational potential to tlie Coulomb potential between an electron and nucleus—with the quantum ideas that had recently been introduced by Planck and Einstein. This led eventually to the fomial theory of quaiitum mechanics, first discovered by Heisenberg, and most conveniently expressed by Schrodinger in the wave equation that bears his name. 

The simplest system exliibiting a nuclear hyperfme interaction is the hydrogen atom with a coupling constant of 1420 MHz. If different isotopes of the same element exhibit hyperfme couplings, their ratio is detemiined by the ratio of the nuclear g-values. Small deviations from this ratio may occur for the Femii contact interaction, since the electron spin probes the inner stmcture of the nucleus if it is in an s orbital. However, this so-called hyperfme anomaly is usually smaller than 1 %. 

Figure Bl.23.9. Scattering intensity of 4 keV Ne versus azimuthal angle 8 for a Ni 110] surface in the clean (1 X 1), (1 X 2)-H missing row, and (2 x l)-0 missing row phases. The hydrogen atoms are not shown. The oxygen atoms are shown as small open circles. 0-Ni and Ni-Ni denote the directions along which O and Ni atoms, respectively, shadow the Ni scattering centre.

Figure C2.14.2. The hydrogen bond in water. The oxygen lone pairs (shaded blobs) are the donors, and the hydrogen atoms the acceptors [ 177, 178].

Next, we consider one pair of it electrons and one pair of cj elections. The cj electrons may originate from a CH or from a CC bond. Let us consider the loop enclosed by the three anchors formed when the electron pair comes from a C-H bond. There are only three possible pairing options. The hydrogen-atom originally bonded to carbon atom 1, is shifted in one product to carbon atom 2,... 

It is interesting to note that this is the first time that in the present framework the quantization is formed by two quantum numbers a number n to be termed the principal quantum number and a number , to be termed the secondary quantum number. This case is reminiscent of the two quantum numbers that characterize the hydrogen atom. 

The hydroxyl hydrogen exchanges but the hydrogen atoms of the CH3 (methyl) group do not. 

Arsenic (but not antimony) forms a second hydride. This is extremely unstable, decomposing at very low temperatures. Replacement of the hydrogen atoms by methyl groups gives the more stable substance tetramethyldiarsane, cacodyl, (CH3)2As -AsfCHj), a truly foul-smelhng liquid. 

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