# Principles of Quantum Mechanics as Applied to Chemistry and Chemical Physics

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number n, so we may drop the subscript l. A comparison with equation (6.57) shows that the total energy is equal to one-half of the average potential energy En 12hVin (6:80) Since the total energy is the sum of the kinetic energy T and the potential energy V, we also have the expression Tn ÿEn Z 2e92 2a0 n2 (6:81) The relationship En ÿTn (Vn=2) is an example of the quantum-mechani- cal virial theorem. 6.5 Spectra The theoretical results for the hydrogen-like atom may be related to experimen- tally measured spectra. Observed spectral lines arise from transitions of the atom from one electronic energy level to another. The frequency í of any given spectral line is given by the Planck relation í (E2 ÿ E1)=h where E1 is the lower energy level and E2 the higher one. In an absorption spectrum, the atom absorbs a photon of frequency í and undergoes a transition from a lower to a higher energy level (E1 ! E2). In an emission spectrum, the process is reversed; the transition is from a higher to a lower energy level (E2 ! E1) and a photon is emitted. A spectral line is usually expressed as a wave number ~í, defined as the reciprocal of the wavelength º ~í \ufffd 1 º í c jE2 ÿ E1j hc (6:82) The hydrogen-like atomic energy levels are given in equation (6.48). If n1 and n2 are the principal quantum numbers of the energy levels E1 and E2, respectively, then the wave number of the spectral line is 6.5 Spectra 187 ~í RZ2 1 n21 ÿ 1 n22 \ufffd \ufffd , n2 . n1 (6:83) where the Rydberg constant R is given by R ìe9 4 4ð"3c (6:84) The value of the Rydberg constant varies from one hydrogen-like atom to another because the reduced mass ì is a factor. It is not appropriate here to replace the reduced mass ì by the electronic mass me because the errors caused by this substitution are larger than the uncertainties in the experimental data. The measured values of the Rydberg constants for the atoms 1H, 4He, 7Li2, and 9 Be3 are listed in Table 6.3. Following the custom of the field of spectroscopy, we express the wave numbers in the unit cmÿ1 rather than the SI unit mÿ1. Also listed in Table 6.3 is the extrapolated value of R for infinite nuclear mass. The calculated values from equation (6.84) are in agreement with the experimental values within the known number of significant figures for the fundamental constants me, e9, and " and the nuclear masses mN . The measured values of R have more significant figures than any of the quantities in equation (6.84) except the speed of light c. The spectrum of hydrogen (Z 1) is divided into a number of series of spectral lines, each series having a particular value for n1. As many as six different series have been observed: n1 1, Lyman series ultraviolet n1 2, Balmer series visible n1 3, Paschen series infrared n1 4, Brackett series infrared n1 5, Pfund series far infrared n1 6, Humphreys series very far infrared Table 6.3. Rydberg constant for hydrogen-like atoms Atom R (cmÿ1) 1H 109 677.58 2H (D) 109 707.42 4He 109 722.26 7Li2 109 728.72 9Be3 109 730.62 1 109 737.31 188 The hydrogen atom Thus, transitions from the lowest energy level n1 1 to the higher energy levels n2 2, 3, 4, . . . give the Lyman series, transitions from n1 2 to n2 3, 4, 5, . . . give the Balmer series, and so forth. An energy level diagram for the hydrogen atom is shown in Figure 6.6. The transitions corresponding to the spectral lines in the various series are shown as vertical lines between the energy levels. n 5 ¥ n 5 6 n 5 5 n 5 4 n 5 3 n 5 2 n 5 1 Pfund series Brackett seriesPaschen series Balmer series Lyman series 212 210 28 26 24 22 0 214 E ne rg y (e V ) Continuum Figure 6.6 Energy levels for the hydrogen atom. 6.5 Spectra 189 A typical series of spectral lines is shown schematically in Figure 6.7. The line at the lowest value of the wave number ~í corresponds to the transition n1 ! (n2 n1 1), the next line to n1 ! (n2 n1 2), and so forth. These spectral lines are situated closer and closer together as n2 increases and converge to the series limit, corresponding to n2 1. According to equation (6.83), the series limit is given by ~í R=n21 (6:85) Beyond the series limit is a continuous spectrum corresponding to transitions from the energy level n1 to the continuous range of positive energies for the atom. The reduced mass of the hydrogen isotope 2H, known as deuterium, slightly differs from that of ordinary hydrogen 1H. Accordingly, the Rydberg constants for hydrogen and for deuterium differ slightly as well. Since naturally occurring hydrogen contains about 0.02% deuterium, each observed spectral line in hydrogen is actually a doublet of closely spaced lines, the one for deuterium much weaker in intensity than the other. This effect of nuclear mass on spectral lines was used by Urey (1932) to prove the existence of deuterium. Pseudo-Zeeman effect The influence of an external magnetic field on the spectrum of an atom is known as the Zeeman effect. The magnetic field interacts with the magnetic moments within the atom and causes the atomic spectral lines to split into a number of closely spaced lines. In addition to a magnetic moment due to its orbital motion, an electron also possesses a magnetic moment due to an intrinsic angular momentum called spin. The concept of spin is discussed in Chapter 7. In the discussion here, we consider only the interaction of the external magnetic field with the magnetic moment due to the electronic orbital motion and neglect the effects of electron spin. Thus, the following analysis ¥í~ Figure 6.7 A typical series of spectral lines for a hydrogen-like atom shown in terms of the wave number ~í. 190 The hydrogen atom does not give results that correspond to actual observations. For this reason, we refer to this treatment as the pseudo-Zeeman effect. When a magnetic field B is applied to a hydrogen-like atom with magnetic moment M, the resulting potential energy V is given by the classical expression V ÿM : B ìB " L : B (6:86) where equation (5.81) has been introduced. If the z-axis is selected to be parallel to the vector B, then we have V ìB BLz=" (6:87) If we replace the z-component of the classical angular momentum in equation (6.87) by its quantum-mechanical operator, then the Hamiltonian operator H^B for the hydrogen-like atom in a magnetic field B becomes H^ B H^ ìB B " L^z (6:88) where H^ is the Hamiltonian operator (6.14) for the atom in the absence of the magnetic field. Since the atomic orbitals \u142nlm in equation (6.56) are simultan- eous eigenfunctions of H^ , L^2, and L^z, they are also eigenfunctions of the operator H^ B. Accordingly, we have H^ B\u142nlm H^ ìB B " L^z \ufffd \ufffd \u142nlm (En mìB B)\u142nlm (6:89) where En is given by (6.48) and equation (6.15c) has been used. Thus, the energy levels of a hydrogen-like atom in an external magnetic field depend on the quantum numbers n and m and are given by Enm ÿ Z 2e92 2aìn2 mìB B, n 1, 2, . . . ; m 0, \ufffd1, . . . , \ufffd(nÿ 1) (6:90) This dependence on m is the reason why m is called the magnetic quantum number. The degenerate energy levels for the hydrogen atom in the absence of an external magnetic field are split by the magnetic field into a series of closely spaced levels, some of which are non-degenerate while others are still degenerate. For example, the energy level E3 for n 3 is nine-fold degenerate in the absence of a magnetic field. In the magnetic field, this energy level is split into five levels: E3 (triply degenerate), E3 ìB B (doubly degenerate), E3 ÿ ìB B (doubly degenerate), E3 2ìB B (non-degenerate),