Designation of Spectroscopic States

A.1 INTRODUCTION

The recipes briefly summarized here are used in arriving at atomic and molecular state (energy) descriptions. The reader is referred to standard textbooks of quantum mechanics and spectroscopy, some of which are given in the Additional Reading, for the mathematical formulations of these principles.

A.2 ATOMS

The state of the electron in the hydrogen atom or hydrogen-like ion (e.g., He+ and Li2+, each containing only one electron) is specified by the three quantum numbers n, i, and mi. The principal quantum number n can have the values 1, 2, 3,...; it defines the shell of the atom occupied by the electron, and therefore the total energy of the single bound electron (neglecting relativistic effects) is specified completely by n. The azimuthal or orbital angular momentum quantum number i can have the values 0, 1, 2 ...(n — 1); it determines the total angular momentum of the electron, which is quantized and can be represented by a vector having magnitudes (h/2'n )[i(i + 1)]1/2 (where h is Planck's constant). The states (or atomic orbitals) with i = 0, 1, 2, 3,... are designated, respectively, s, p, d, f, . . . orbitals, so that the electron in the lowest electronic state of the hydrogen atom (n = 1, i = 0) is called a 1s electron. The quantum number mi is the magnetic quantum number (so named because it is related to the effects of a magnetic field on atomic spectra), and can have the values —i, — (i — 1),..., 0,... (i — 1), i; it determines the allowed components of the angular momentum in a definite z direction, which are also quantized and can only have values of m\h/2^.

In addition to the three quantum numbers given, another factor, electron spin, must be considered to explain the fine structure observed in atomic spectra. This fourth parameter is also a natural consequence of quantum mechanics if relativistic effects are included in the detailed wave mechanical treatment. In addition to orbital angular momentum described by the quantum number /, an electron has an intrinsic magnetic moment as if (classically) it were spinning about its own axis in addition to orbiting the nucleus. Associated with this moment is a quantum number s, such that the total spin angular momentum vector has a magnitude (h/2^)[s(s + 1)]1/2; however, s can have only the single value of 1/2. Analogous to the orbital angular momentum, the z component of the spin angular momentum is also quantized with allowed values of magnitude msh/2w, where ms = ±1/2 (so that there are only two possible orientations of the spin angular momentum). The electron spin and orbital angular momentum magnetic moments interact, so that / and s are coupled to give a total electronic angular momentum vector of magnitude h/2^[j(j + 1)]1/2 where j = / ± s = / ± 1/2.

With polyelectronic atoms, the complexities of the mathematics involved in describing the system as a result of electrostatic interactions among the electrons as well as between the electrons and the nucleus are such that only approximate solutions of the quantum mechanical treatment are possible. The concept of hydrogen-like orbitals is retained with the principal quantum number n designating the electronic shells of the atom, but now the energy of the state is a function of the total angular momentum of the electrons as well as the quantum number n. The number of electrons in a given orbital is governed by the Pauli exclusion principle, which states that no two bound electrons can exist in the same quantum state; that is, no two electrons in the atom can have the same values of n, /, m/, and ms. Thus, an atomic orbital characterized by a specific set of n, /, and m/ values can be occupied by two electrons only if their spins are opposed with ms = +1/2 and —1/2. Subject to this constraint, the orbitals are filled in the order of increasing energy (the aufbau principle). If two or more orbitals have the same energy (i.e., are degenerate), then the electrons go into different orbitals as much as possible, thus reducing electron-electron repulsion by keeping them apart, with spins parallel and in the same direction rather than opposed (Hund's rule).

For light atoms of the type we shall be concerned with here (nuclear charge < 40), the orbital angular momenta of all the electrons strongly couple together to give a total, or resultant, orbital angular momentum. This is designated by the quantum number L, which is obtained by the combination of the moments of the individual electrons. Similarly, the spin angular momenta combine to give a resultant spin, designated by the quantum number S. It follows that the total orbital angular momentum in the z direction is characterized by a quantum number ML = £m/, and the total spin angular momentum in the z direction has associated with it a quantum number MS = Xms.

Allowed values of Ml are -L, -(L - 1),..., 0,..., (L - 1), L (i.e., |Ml| < L) and allowed values of Ms are -S, -(S - 1), ...,0, ..., (S - 1), S (or |MS| < S). Closed (completely filled) shells and subshells have to a good approximation zero net resultant orbital and spin angular momenta, so that the combination need be carried out over only the electrons in partially filled subshells. Examples of this process are given later. The L and S momenta then couple together (Russell-Saunders coupling) to form a total atomic angular momentum J. Possible values of J are

J = L + S, L + S - 1, L + S - 2, ..., |L - S + 1|, |L - S| (A-1)

where |L - S| is equal to L - S, if L > S, and equals S - L if S > , L (i.e., J can only have positive values). The complete term symbol, which designates the electronic state of the atom, is written

Frequently, n is omitted. The quantity (2S + 1) is the multiplicity of the atom (called singlet, doublet, triplet, etc., respectively, for values of 1, 2, 3,...); it gives the total number of possible spin orientations. Often the value of J is omitted from the term symbol, particularly if the energy differences among the different J states are so small that they are not a factor in photochemical considerations. Analogous to hydrogen or hydrogen-like atoms, the atomic states of L = 0,1,2,3, are designated, respectively, S, P, D, F, states.

As a specific example, consider the oxygen atom with eight electrons in an electronic configuration 1s22s22p4. The 1s shell and the 2s subshell are completely filled, hence only the four 2p (/ = 1) electrons need be considered. The state of lowest energy, hence the most stable, is called the ground state. According to Hund's rules, this state is the state of maximum multiplicity; for a given multiplicity, the state of maximum L; and for given L and S, the state of minimum J if the partially filled subshell is less than half-occupied, or conversely, the state of maximum J if the partially filled subshell is more than half-occupied. Thus the ground state of oxygen is given by the four electrons occupying the three 2p orbitals in the following manner:

The quantum numbers for the four electrons in this unfilled p subshell are as follows:

Efecfron n

For this configuration Ms is Xms = 1/2 — 1/2 + 1/2 + 1/2 = 1, which is the maximum possible for 4 electrons in 3 orbitals to be consistent with the Pauli exclusion principle, and therefore S = 1 (since |MS| < S) and the multiplicity (2S + 1) =3. Similarly, ML = Xmi = 1, which is also the maximum possible value; hence L must also be 1, since |ML| < L. The quantities S and L are maximum allowed values, and therefore they give the state of lowest energy which is a 23P (a "triplet-P") state. Possible J values are 2, 1, and 0; since the subshell is more than half-filled, maximum J gives the most stable state, and therefore the complete term symbol for the oxygen atom in its lowest energy state is 23P2.

Electronic excitation may result in change of J (generally unimportant in photochemistry, an exception being the rather large separations between the halogen 2P3/2 and 2Pi/2 states), transition to another orbital within the subshell, or transition to a higher energy subshell. For example, two possible excited states of the oxygen atom within the same subshell (same n = 2 and i = 1) are as follows:

The term symbols are 21D2 and 21So, respectively, the former being of lower energy because of maximum L.

Similarly, it can readily be verified using the same procedure that the total momenta for nitrogen with seven electrons (1s22s22p3) in the ground-state configuration is S = 3/2 (multiplicity = 4), L = 0, and J = 3/2; hence the term symbol is 24S3/2. A possible excited state is 22D3/2.

It should be pointed out that some excited electronic configurations are excluded by the Pauli exclusion principle for atoms with electrons with the same values of n and I (called equivalent electrons), because simply interchanging the order in the designation does not lead to different states.

A.3 DIATOMIC MOLECULES

For diatomic (or linear polyatomic) molecules, it is possible to designate the electronic states by a set of quantum numbers analogous to atoms. Thus, electrons with atomic orbital angular momentum quantum number I can have quantized molecular values X from 0 to /, and these combine in a manner analogous to polyelectronic atoms to give a quantum number A representing the total orbital angular momentum along the internuclear axis. It is possible for A to have values 0, 1,2,..., L (where L is the quantum number for the resultant orbital angular momentum for all the electrons in the molecule), and these are designated, respectively, X, n, A,... states, analogous to s, p, d,..., for the hydrogen atom and to S, P, D,... for polyelectronic atoms. Similarly, the total molecular spin quantum number is S, and the component of S along the internuclear axis is X with possible values S, S — 1,..., 0, —(S — 1), — S. The sum of these two quantum numbers is a quantum number analogous to /, ft = |A + X|, and the total term symbol for a specific diatomic or linear polyatomic electronic state is

In addition to these quantum numbers A, S, and ft, there are two other properties of homonuclear diatomic species dealing with the symmetry of its wave function:

1. If inversion of all electrons through a center of symmetry of the molecule leads to no change in sign of the electronic wave function, then the state is e^e«, or gerade (g); if a change of sign results, the state is odd, or ungerade (u). The symbol g or u is included in the molecular term symbol as a second subscript to A.

2. If reflection in a plane of symmetry passing through the nuclei (containing the internuclear axis of symmetry) leads to no change in sign of the electronic wave function; then the state is poszizVe (+); if it changes sign, the state is «eg^ízVe (-) . The symbol + or - is written as a second superscript to A. (This element of symmetry applies only to A = 0 or X states.)

Molecular oxygen provides a practical atmospheric photochemical example for illustrating these state designation factors. The electronic configuration for ground-state 02 (16 electrons) is

K1s)2 K1s)2 K2s)2 K2s)2 K2Pz)2 K2PJ2 K2py)2 (^pj1 (^g2py)1

The (o-g1s), and so on designate molecular orbitals, the asterisk (*) refers to an antibonding orbital (i.e., the density of electrons between the two oxygen atoms in this type of orbital is less than that for the two free atoms, and therefore there is a repulsive force between them), and the superscript gives the number of electrons in the molecular orbital. Since there are two unpaired electrons in the equivalent (degenerate) (wg2px) and (wg2py) orbitals, by Hund's rule of maximum multiplicity the total spin is 1 and the ground state is a triplet. The quantum number A can be 2 or 0; however, if A = 2, both electrons would have the same X = ±1 value, which is forbidden by the Pauli exclusion principle for electrons in equivalent orbitals also with the same spins; therefore only the A = 0, or X, state is allowed for S = 1. Figure A-1 shows the wave functions for one of the two unpaired electrons—the one in the (wg2px) orbital—for the lowest energy configuration, from which it is seen that interchange through the center of symmetry does not lead to a change in sign, whereas interchange through the plane of symmetry does. The ground state of oxygen is therefore gerade and negative, and the term symbol is 3Xg .

While the ground electronic state of 02 is the 3Xg state, there are two other possible states (of somewhat higher energies) for the electrons in these same two equivalent orbitals in which the total spin is zero (multiplicity = 1). In this case, both electrons can now have either the same X values (each +1 or g1), leading to A = 2 and a 1A state, or different X values (+1 and g1) giving a net angular momentum component A = 0 and hence a 1X state. From the same symmetry considerations based on Figure A-1 that were used to obtain the term symbol for the ground state, it follows that the term symbols for these two electronically excited states are 1Ag and 1X+. On the other hand, if an electron in a lower energy orbital [such as the (wu2py) state] is excited to the

FIGURE A-1 Schematic drawing of the (wg2px) orbital of O2 in the X-Z plane, the plane of the page. [The (wg2py) orbital, not shown, is in the y—z plane perpendicular to the plane of the page.] Inversion through the center of symmetry C from P to P' does not change sign, whereas inversion through a plane of symmetry passing through the two nuclei such as the y—z plane (from P to P'') does lead to a change in sign.

FIGURE A-1 Schematic drawing of the (wg2px) orbital of O2 in the X-Z plane, the plane of the page. [The (wg2py) orbital, not shown, is in the y—z plane perpendicular to the plane of the page.] Inversion through the center of symmetry C from P to P' does not change sign, whereas inversion through a plane of symmetry passing through the two nuclei such as the y—z plane (from P to P'') does lead to a change in sign.

(wg2py) orbital, then two nonequivalent orbitals, (wu2py) and (w*2px), are singly occupied; the Pauli exclusion principle no longer applies, since the two unpaired electrons occupy separate orbitals, and six states are possible: 1S+,1S—,1AU,3S+,3S—, and 3AU.

A.4 POLYATOMIC MOLECULES

Although the concepts developed for atoms and diatomic species can be extended to polyatomic molecules, the designation of electronic states for more complex molecules can become quite complex if the maximum spectral information is to be retained. However, many of the more subtle points that are especially important for fine-structure spectroscopic characterizations in most cases simply do not affect understanding of photochemical transformations involving complex species, and therefore for our purposes we can get by with relatively simple terms involving orbitals for only a single optical electron. The optical electron is the one electron promoted in the light absorption process.

Of importance for these orbitals are their symmetry and multiplicity characteristics, plus their involvement in bonding within the molecule. These may be of the following types involving both bonding and antibonding characteristics:

1. a and o-*orèzia/s: these are associated with two atoms and involve two tightly bound electrons

2. w and w*orèzia/s: for example, the contribution of p electrons to the double bond in ethylene, or the conjugated electrons in benzene

3. n orèzia/s: these are nonbonding orbitals occupied by lone pair electrons in heteroatomic molecules

Excited states are then designated by the initial and final orbitals associated with the transition of the optical electron. Thus, the first excited state of ethylene is formed by promoting an electron from a bonding w orbital to an antibonding w* orbital, leading to two unpaired electrons. Hund's rule of maximum multiplicity again suggests the spins of these electrons should be in the same direction, so that S = 1 and the multiplicity is 3. The designation of this state is 3(w, w*). For a molecule involving a carbonyl group, r'

an additional low-lying state is the 3(n, w*) state formed by exciting an electron from the nonbonding n orbital to the antibonding w* orbital. The state of lower energy, whether the 3(n, w*) or the 3(w, w*) state, will depend somewhat on substituent (R and R') groups and on the physical environment of the molecule.

Additional Reading

Harmony, M. D., torodwcfiOK fo Mo/ec«/ar Energies awd Specira. Holt, Rinehart, & Winston, New York, 1972.

Herzberg, G., Atomic Specira awd Afom/c Sfractare, 2nd ed. Dover, New York, 1974. Herzberg, G., Specira o/D/atom/c Mo/ec«/es. Van Nostrand, Princeton, NJ, 1950. Herzberg, G., E/ecfrorac Specira awd E/ecfrornc Sfractare o/Po/yafom/c Mo/ec«/es. Van Nostrand, Princeton, NJ, 1966.

Karplus, M., and Porter, R. N., Afoms arcd Mo/ec«/es. W. A. Benjamin, New York, 1970. McQuarrie, D. A., and Simon, J. D., P^ys/ca/ C^em/sfry: A Mo/ec«/ar Approach. University Science Books, Sausalito, CA, 1997.

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