Quantum Numbers of Multielectron Atoms

Absorbance is a dimensionless quantity. An absorbance of 1.0 corresponds to 90% absorption at a given wavelength, an absorbance of 2.0 corresponds to 99% absorption, and so on. 

Although the quantity most commonly used to describe absorbed light is the wavelength, energy and frequency are also used. In addition, the wavenumber, the numbo of waves per centimeter (a quantity proportional to the energy), is frequently used, especially in reference to infrared light. For reference, the relations between these quantities are given by the equations 

Although the quantum numbers and energies of individual electrons can be described in fairly simple terms, interactions between electrons complicate this picture. Some of these interactions were discussed in Section 2.2.3 as a result of repulsions between electrons (characterized by energy 11 ), electrons tend to occupy separate orbitals as a result of exchange energy (Ilg), electrons in separate orbitals tend to have parallel spins. 

Independently, each of the 2p electrons could have any of six possible m , combinations  

Because the quantum numbers m and provide information about the magnetic fields generated by electrons due to their orbital and spin, respectively, we need to determine how many possible combinations of m and /n valnes there are for a pi configuration to assess the different possible interactions between these fields. These combinations allow determination of the corresponding values of M and Ms. For shorthand, we will 

The 2p electrons are not independent of each other, however the orbital angular momenta (characterized by mi values) and the spin angular momenta (characterized by m, values) of the 2p electrons interact in a manner called Russell-Saunders coupling or LS coupling. The interactions produce atomic states called microstates that can be described by new quantum numbers  

Total orbital angular momentum Total spin angular momentum 

We need to determine how many possible combinations of m/ and m, values there are for a configuration. Once these combinations are known, we can determine the corresponding values of Ml and Ms For shorthand, we will designate the ms value of each electron by a superscript +, representing = -I-5, or -, representing m, = - 2 For example, an electron having mi  

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Quantum Numbers of Multielectron Atoms 409 TABLE 11.3 Examples of Atomic States (Free-lon Terms) and Quantum Numbers... 

L, Ml, J, Mj, S, quantum numbers of multielectron atom /, Mj nuclear quantum numbers... 

For reasons we will discuss later, a fourth quantum number is required to completely describe a specific electron in a multielectron atom. The fourth quantum number is given the symbol ms. Each electron in an atom has a set of four quantum numbers n, l, mi, and ms. We will now discuss the quantum numbers of electrons as they are used in atoms beyond hydrogen. 

The quantum numbers that describe states of multielectron atoms are defined as follows ... 

To this point in the discussion of multielectron atoms, the spin and orbital angular momenta have been treated separately. In addition, the spin and orbital angular momenta couple with each other, a phenomenon known as spin-orbit coupling. In multielectron atoms, the S and L quantum numbers combine into the total angular momentum quantum number J. The quantum number J may have the following values ... 

In addition to the conditions for the electronic structures of multielectron atoms established by the monoelectronic wave functions and their relative energies mentioned above, other restrictions should also be considered. One of them is the Pauli principle stating that no two electrons can have the same quantum numbers. Thus one orbital can contain a maximum of two electrons provided they have different spin quantum numbers. Other practical rules or restrictions refer to the influence of interelectronic interactions on the electronic structures established by Hund s rules. The electrons with the same n and / values will occupy first orbitals with different nti and the same rris (paired spins). 

In Section 8-10 and in Chapter 24, we will see that orbital energies of multielectron atoms also depend on the quantum numbers and m. ... 

Now the parity of a multielectron, ionic, or atomic wave function is given by TT ( — l)/fc, where lk (the angular-momentum quantum number)... 

The Pauli Exclusion Principle states that no two electrons of any single atom may simultaneously occupy a slate described by only a single set of quantum numbers. Five such numbers arc needed to describe fully the quantum-mechanical conditions of an electron. For j-j coupling this set is generally ti. I., v. j. iij. and for l.-S it is /t. /. j. u(. nr,. From die coupling of the angular momentum associated with the latter sets a full description of the multielectron stale, described by it, L. S, J. Mis determined. 

The three quantum numbers n, l, and wi/ discussed in Section 5.7 define the energy, shape, and spatial orientation of orbitals, but they don t quite tell the whole story. When the line spectra of many multielectron atoms are studied in detail, it turns out that some lines actually occur as very closely spaced pairs. (You can see this pairing if you look closely at the visible spectrum of sodium in Figure 5.6.) Thus, there are more energy levels than simple quantum mechanics predicts, and a fourth quantum number is required. Denoted ms, this fourth quantum number is related to a property called electron spin. 

The importance of the spin quantum number comes when electrons occupy specific orbitals in multielectron atoms. According to the Pauli exclusion principle,... 

Which two of the four quantum numbers determine the energy level of an orbital in a multielectron atom ... 

In a hydrogen atom, the orbital energy is determined exclusively by the principal quantum number n—all the different values of / and mi are degenerate. In a multielectron atom, however, this degeneracy is partially broken the energy increases as / increases for the same value of n. 

In this volume, principal consideration is given to the lighter elements, so that the Russell-Saunders (549) vector model of the atom is used. In this model a multielectron atom is assumed to have the quantum numbers n, L = lif Ml, 8 = siy (or n, L, J = L + S, Mj). This implies stronger and Si-Sj coupling than U-Si coupling. It follows from Pauli s principle that for a closed shell =... 

Values of the magnetic quantum number, mp, depend on the value off. When = 0, mp can only take one value (0) when = 1, mp has three possible values (+1, 0, or -1). There are five possible values of mp when - 2 and seven when C = 3. In more familiar terms, there is only one sort of s orbital there are three sorts of p orbitals, five sorts of d orbitals, and seven sorts of f orbitals. All three p orbitals are degenerate as arc all five d orbitals and all seven f orbitals (for both single-electron and multielectron atoms). We shall see how to represent these orbitals later. 

A multielectron atom can exist in several electronic states, called microstates, which are characterized by the way the electrons are distributed among the atomic orbitals. The number of microstates for a free atom with a valence shell consisting of a set of degenerate orbitals with orbital angular momentum quantum number I housing n electrons is given by ... 

In a free multielectron atom or ion, the spin and orbital angular moments of the electrons couple to give a total angular momentum represented in the Russell-Saunders scheme by the quantum number J. Since J arises from vectorial addition of L (the total orbital quantum number) and 5 (total spin quantum number), it may take integral (or half-integral... 

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