Download Term Symbol nomenclature and more Lecture notes Quantum Mechanics in PDF only on Docsity!
TERM SYMBOLS
Dr. Cyriac Mathew
Atomic Term Symbols In many-electron atoms the electron configuration is rather ambiguous. For example, consider the ground state electron configuration of a carbon atom, 1s^2 2s^2 2p^2. The two 2p electrons could be in any of the three 2p orbitals and have any spin consistent with Pauli Exclusion Principle. This may give rise to several atomic energy states because of the effects of inter-electron repulsions. Therefore, we need a more detailed designation of the electronic states of the atom. Such detailed designation of the electronic states of the atom is called term. The corresponding energy levels are represented using the term symbol. A term symbol tells us something about the angular momentum of the electrons in the atom. Spin-orbit coupling An electron is a charged particle, and hence its orbital angular momentum gives rise to a magnetic field just as an electric current in a loop gives rise to magnetic field in an electromagnet. Thus an electron with an orbital angular momentum possesses magnetic moment. The magnetic moment due to orbital
motion is given as l^ el , e is the
magnetogyric ratio of the electron and l is the orbital angular momentum. An electron also has a spin angular momentum. This intrinsic spinning motion also gives rise to magnetic moment. The spin magnetic moment is given by
s 2 e s .Where s is the spin angular
momentum. The spin magnetic moment is twice the value of the spin angular momentum. For electrons the magnetic moment is opposite to the angular momentum; see figure 1. Thus there are two sources for magnetic moment for the electrons. These two magnetic moments can interact and give rise to shifts in energies of the states of the atom. The interaction of these two magnetic moments is called spin-orbit coupling. The strength of the coupling, and its effect on the energy levels of the atom depend on the relative orientations of the spin and orbital magnetic moments, and hence on the relative orientations of the spin and orbital angular momenta. The total angular
NN
S (^) S
N
N
S
S
High J
Low J
S
N
l
l
s
s
l
l
s
s
l (^) : Orbital angular momentum
l : Orbital magnetic moment
s : spin angular momentum s spin magnetic moment: J: Total angular momentum Fig. 1
momentum of the electron is the vector sum of its spin and orbital momenta. Thus when the spin and orbital angular momenta are nearly parallel the total angular momentum is high; when the two angular momenta are opposed, the total angular momentum is low. When the total angular momentum (J) is high, the total energy also is high (unfavourable orientation of magnetic moments), and when the total angular momentum (J) is low, the total energy also is low (favourable orientation of magnetic moments), as shown in figure 1. The probable levels that may arise as a result of vector addition of the two angular momenta from a d^1 electron configuration is shown in fig. 2 When l = 0 the electron is having no orbital angular momentum and the total angular momentum is same as that of the spin angular momentum. Total electronic angular momentum When several electrons are present in an atom there are, generally, two ways in which the orbital and spin angular momenta add together or oppose each other. (i) Each spin may couple to its own orbital angular momentum as in a one-electron case. This type of coupling is called jj coupling. (ii) All the individual orbital angular momenta
( li
) couple to produce a total orbital angular
momentum ( L
). It is written as vector sums: i i
L (^) l
. Here the summation is over the
electrons in the atom. Similarly, the individual
spin angular momenta ( si
) couple together to produce a total spin angular momentum ( S
i i
S (^) s
. Here also the summation is over the electrons in the atom.
Now, the orbital and spin angular momenta couple to produce the total angular momentum ( J
This type of coupling is called the Russell-Saunders or L-S coupling. This coupling scheme is used when the spin-orbit coupling is weak, and generally used for atoms of low atomic number (z ≤ 30). (i) Total electronic orbital angular momentum When several electrons are present in an atom the total orbital angular momentum is obtained by the vector addition of individual orbital angular momenta.ie. in order to find out the
value of total orbital angular momentum we need an operator, L ˆ. It can be shown that L ˆ
commutes with the atomic Hamiltonian, H ˆ when spin-orbit coupling of individual electrons are
j = 3/
j = 5/
l = 2
s = 1/
s = 1/
l = 2
Fig. 2 The coupling of the spin and orbital angular momenta of a d electron (l =2) and s = ½ gives two possible values of j depending on the relative orientations of the spin and orbital angular momenta of the electron.
L = 1 + 1, …. 1 – 1 = 2,……., 0 = 2, 1, 0
l 3 = 1, L’ = 0 L = 1 + 0 ,……………….1- 0 L = 1 The total L terms are :3,2,1,2,1,0,1. The corresponding states are: F, D, P, D, P, S, P
(ii) Total electronic spin angular momentum When there are several electrons to be considered we must assess their total spin angular momentum quantum number S (a non-negative integer or half-integer). Using the Clebsch- Gordan series we can decide the values of S;
S = s 1 + s 2 , s 1 + s 2 – 1, .….. s 1 (^) s 2
For example let us consider two electrons, each of them with spin s = ½. s 1 = ½, s 2 = ½
S = ½ + ½ , …….. 1 2 ^12 =1, ……0 = 1,
If there are three electrons the total spin angular momentum is obtained by coupling the third spin with each value of S. The spin multiplicity of a term is the value of 2S + 1. When S = 0 (as for a closed shell) the electrons are all paired and there is no net spin. Hence the spin multiplicity is 2×0 + 1 = 1 and the state is called a singlet state. If S = ½ , 2 × ½ + 1 = 2, a doublet state. When S =1 the state is a triplet state, 2×1 + 1 = 3, and so on. A singlet S term is written as, 1 S ( read as singlet S ), a doublet term as 2 S ( read as doublet S ). (iii) Total electronic angular momentum If there are several electrons outside a closed shell we have to consider the coupling of all the spin angular momenta and all the orbital angular momenta separately. For atoms of low atomic number the spin-orbit coupling is weak, and we are following the Russell-Saunders coupling scheme. We imagine that all the orbital angular momenta of the electrons couple to give the total orbital angular momentum L, and that all the spin angular momenta are similarly coupled to give total S. Now we imagine that the two kinds of angular momenta couple through the spin-orbit interaction to give a total angular momentum J. J is the total angular momentum quantum number. The permitted values of J are given by the Clebsh-Gordan series.
J = L + S, L + S – 1, ……… J S. For example if L = 2 and S = 1, then
J = 2 + 1, ………… 2 1 = 3, …….., 1 = 3,2,1( each value differ by 1 )
It is to be noticed that L, S and J are zero for completely filled sub-shells because, for every electron with a negative value of li , there is another electron with a corresponding positive value to cancel it; the same case exists for spin angular momentum quantum number si also. Thus we can ignore the electrons in completely filled sub-shells in finding the terms.
L is the total orbital angular momentum and its z component Lz can have 2L+1 values ranging from L to – L. These are represented by, ML = L, L – 1,…..,0,…. – L. Similarly MS can take 2S+1 values and MJ can take 2J +1 values. MS = S, S – 1, ………– S. and MJ = J, J – 1, … – J.
Now L = (^) ML (max)and S = MS (max)
Atomic Term symbol The atomic terms symbol is represented in the following way;
2 S 1 LJ
L is the total orbital angular momentum, (2S + 1) is the spin multiplicity and J is the total electronic angular momentum quantum number.eg 3 P 2 ; (read as triplet P two ). Two electrons in different sub-shells are called nonequivalent electrons. Nonequivalent electrons have different values for n or l or both, and we need not worry about any restrictions imposed by Pauli Exclusion Principle when we derive the terms. For example, the excited state of helium 1s^1 2s^1 , or configurations like 2s^1 2p^1 , 3p^1 3d^1 etc. Two electrons in the same sub-shell are called equivalent electrons. Equivalent electrons have the same value of n and l. Example ground state carbon 1s^2 2s^2 2p^2 , or configurations like 2p^3 , 3d^2 etc. The situation is complicated by the necessity to avoid giving two electrons the same four quantum numbers. Hence not all the terms derived for nonequivalent electrons are possible in the case of equivalent electrons. Term symbols for nonequivalent electrons As an example let us consider the excited state electron configuration 1s^1 2s^1 , of the He atom. Let us set up a table showing the possible ML and MS values, in the following manner. ML 0 0 0 0 MS (^1 0 0) − 1 Col. 1 2 3 4
There are four microstates in the table because there are two possible spins, ±½, for the electron in the 1s orbital and also for the electron in the 2s orbital. Since both the electrons are on different orbitals the Pauli Exclusion Principle need not be considered. ML is equal to zero for all the microstates in the table because the electrons are s-electrons, and they correspond to L = 0.
The largest value of MS is 1 and therefore, S = MS (max)= 1. All the values of MS = 1, 0, −
correspond to L = 0 and S = 1. When L = 0 the term is S and the spin multiplicity is 2S+1 = 2×1+1 = 3. Therefore the term is 3 S. Thus the microstates of columns 1, 2 (or 3), and 4 give rise to 3 S term. Now column 3 corresponds to ML =0 and MS = 0, and since there is only one entry, L = 0 and S = 0. It represents a 1 S term. For the 3 S term L = 0 and S = 1. Hence,
J = L+S, …… L S = 3+0, … 3- 0 = 3. The term symbol is 1 F 3
Taking L =2 and S =1we have 3 D term. The corresponding term symbols are
J = L+S, …… L S = 2+1, …2-1 = 3,..,1 = 3,2,1. (^3) D 3 , 3 D2, (^3) D 1
Taking L = 1and S = 0 the term is 1 D and the corresponding term symbol is 1 D 1 For L = 1 and S =1 we have a 3 P term. The corresponding term symbols are, 3 P 2 , 3 P 1 , 3 P 0 When L = 1 and S = 0 we have a 1 P term and the corresponding term symbol is 1 P 1 Thus the total terms for this electron configuration are: (^3) F, 1 F, 3 D, 1 D, 3 P and 1 P.
Example 4 : Write down the term symbol for the ground state configuration of F and Na. a. Fluorine The ground state configuration of fluorine is [He]2s^2 2p^5 or [Ne]2p−1. We treat this as a p^1 configuration. For a p electron l =1and since there is only one electron, L = 1. For a single electron s = ½ and hence S = ½. Therefore 2S+1 = 2. J values are: J = 1 + ½, …… 1 - ½ = 3 2 , …, ½ = 3 2 ,½. Hence the term symbols are, (^2) P 3 (^) 2 and (^2) P (^12)
b. Sodium The ground state electron configuration of Na is [Ne]3s^1. For an s electron l = 0 and hence L = 0. For a single electron s =½ and hence S = ½. Therefore, 2S+1 = 2. Since L =0 and S = ½, J = ½. The term symbol is 2 S½ Exercise 1 (a) Write down term symbols for the configurations (i) 3d^10 4s^2 (ii) 3d^1 4s^2 (iii) 1s^2 2s^1 (b) What values of J may occur in the following terms? (i) 1 S (ii) 2 P (iii) 3 D (iv) 4 F Configurations with equivalent electrons Electrons in the same sub-shell are equivalent electrons. They have the same n and l values. As an example we shall consider the ground state carbon atom. The electron configuration is 1s^2 2s^2 2p^2. Since we need not consider completely filled sub-shells we focus on the electron configuration 2p^2 or in general np^2. Since the two 2p electrons do not differ in their n or l values, only the terms that are consistent with Pauli Exclusion Principle need be considered. We are going to assign two electrons to two of the six possible spin-orbitals (2pxα, 2pxβ, 2pyα, 2pyβ, 2pzα, 2pzβ).The number of distinct ways (microstates) of assigning N electrons to G
spin-orbitals belonging to the same sub-shell (equivalent orbitals) is given by;
G
N G N
For the 2p^2 configuration N = 2 and G = 6. (For p orbitals G = 6, for d orbitals G = 10 etc.). The
number of distinct ways are:
. We write down all these spin combinations in the
form of a table.
Sl. No
ml ML
MS
L
( ML (max))
S
( MS (max) )
Term +1 0 − 1
(^1 0 0 0 0 1) S 2 2 0
2 0 1 D
1 1 3 P
Table 1 For the spin combination given in 1st^ row ML and MS are zero and hence M and S also are zero. Therefore this arrangement corresponds to 1 S term. For the spin combinations given in rows 2 to 6 the maximum value of ML is 2. Hence all these five combinations correspond to a D term. Also, for all the five combinations S = 0, and hence represents 1 D term. For the spin combinations given in rows 7 to 15 the maximum value of ML is 1.Hence it corresponds to a P term. Now there are three values for MS and the maximum is 1. Therefore S =
- Hence all these 9 combinations correspond to 3 P term. Therefore, the corresponding term symbols are 1 S 0 , 1 D 2 , 3 P 0 , 3 P 1 and 3 P 2 .If we compare this with an arrangement of two non-equivalent electrons, such as 2p^1 3p^1 where Pauli Exclusion Principle need not be considered, the possible terms will be 3 D, 3 P, 3 S, 1 D,^1 P and 1 S instead of 1 S, (^1) D and 3 P.
The number of terms that arise from an np^4 configuration also will be the same as that of np^2 configuration. The following table gives the terms that would arise from different configurations. (Any sub-shell that contains n electrons will give exactly the same term symbols as the same sub-shell when it is n electrons short of being full). Table 2 shows the terms that may arise from equivalent and non-equivalent electron configurations.
The classical explanation of Hund’s rule is that electrons with the same spin tend to keep out of each other’s way, thereby minimizing the Coulombic repulsion between them. The term that has the greatest number of parallel spins (highest value of S) will therefore be lowest in energy. But this traditional explanation turns out to be wrong in most cases. For example consider the 3 S (1s^1 2s^1 ) term and the 1 S (1s^1 2s^1 ) term of Helium. The 3 S term is found to be lower in energy than the 1 S term. But calculations using accurate wave functions have shown that the average distance between the two electrons is slightly less for 3 S term than 1 S term. Hence the traditional explanation could not be applied here. The reason for the 3 S term lies below in energy than the 1 S term is because of a substantially greater electron-nucleus attraction in 3 S term as compared with 1 S term. The following explanation is given for this: the ‘repulsion’ between electrons of like spins makes the average angle between the radius vectors of two electrons large for the 3 S term than for the 1 S term. This reduces the screening of the nucleus and allows the electrons to get closer to the nucleus in the 3 S term, making electron nucleus attraction grater for 3 S term which results in energy lowering compared to 1 S term.
Terms, Levels and States
The electrostatic interaction between electrons give rise to different terms while the spin-orbit (magnetic) interaction gives rise to different levels. The interaction with external magnetic field gives rise to different states. Different states are represented by MJ values. A summary of the different types of interaction that are responsible for the various kinds of splitting of energy levels in atoms is shown in Fig 3. Only some terms, levels and states are shown as an example. This kind splitting is applicable in lighter atoms only. In heavy atoms magnetic interaction may dominate electrostatic (charge-charge) interaction. For the ground state electron configuration of carbon the term symbols are 3 P 0 , 3 P 1 , 3 P 2 , 1 D 2 and 1 S 0. Fig 4 shows the different terms, levels and states that are possible for this arrangement. Each level is 2J+ degenerate and the splitting of levels (MJ values) occur in the presence of an external magnetic field only.
Configuration
Electrostatic interaction
G F D P^ S
Spin correlation
(^1) P (^3) S (^1) S
Magnetic interaction (spin-orbit)
(^3) P 1 3 P 0 1 S 0
Interaction with external magnetic field
TERMS
LEVELS
(+J, ........, -J)^ STATES Fig 3 :
(^3) P
(^1) S 1 S 0
(^3) P 0
(^3) P 1
(^3) P 2
0
0
1 0
2 1 -1^0
Terms Levels^ States^ MJ
(^1) D
0
Observed atomic energy levels in ground state carbon
10194 cm-
11454 cm-
(The separation of the levels for 3 P term is very small and is exaggerated for clarity)
(^1) D 2
Fig 4
Example 5 Using Hund’s rule deduce the lowest energy level of an excited state of beryllium atom whose electron configuration is 1s^2 2s^1 3s^1 The term symbols for the configuration are, 3 S 1 and 1 S 0. From Hund’s first rule 3 S 1 is the lowest energy level. Example 6 Using Hund’d rule select the ground state term from the following (a) 3 P, 1 P, 3 F, 1 G (b) 4 P, 4 G, 6 S, 2 I Ans (a) 3 F (b) 6 S Example 7 Explain the fine structure of sodium D-line. The atomic Hamiltonian does not include electron spin. But in reality the existence of spin introduces an additional term (usually small), to the Hamiltonian. This term is called spin- orbit interaction. This spin-orbit interaction splits the atomic terms into levels. When we include the spin-orbit interaction, the energy of an atomic level depends on its total angular momentum J. Thus each atomic term is split into levels, each level having a different value of J. For example the 1s^2 2s^2 2p^6 3p^1 configuration of sodium has the single term 2 P, which is composed of the two
configuration 3p^1 gives rise to 2 P3/2 and 2 P1/2. Similarly with other excited states. Consider transitions (1) and (2). For these transitions; ∆L = +1 (S to P) ∆S = 0 (^2 S to 2 P) ∆J = 0 for (1) and +1 for (2)
1s^1 ( (^2) S1/2)
2s^1 ( (^2) S1/2)
3s^1 ( 2 S1/2)
3p^1 (^2 P3/2)
3p^1 (^2 P1/2)
2p^1 (^2 P3/2)
2p^1 (^2 P1/2)
3d^1 (^2 D5/2) 3d^1 (^2 D3/2)
Fig. 6 The energy levels of a hydrogen atom showing possible transitions which is responsible for the fine structure of the spectrum
(1)
(2)
The Zeeman Effect
In 1896, Zeeman observed that application of an external magnetic field caused a splitting of atomic spectral lines. Electrons possess magnetic moments due to orbital and spin angular momenta. In the presence of an external magnetic field the magnetic moment of the electrons interact with the external field resulting shifts in energy level which causes the apparent split in spectral lines.
For a poly-electron atom the magnetic moment associated with the orbital motion of the
electrons is given by L (^2) e
e L m
and that due to spin motion is given by S (^2) e
e S m
; where
e is the charge of the electron, me is the mass of the electron considered as a point mass, L is the total orbital angular momentum and S is the total spin. The magnetic moment due to total
angular momentum J is given by (^2) e e
eg J gJ m
; where g is Landé g-factor and β e is the
Bohr magneton (which has a value of 9.274 × 10 -24^ JT-1). The value of g is given by ( 1) ( 1) ( 1) 1 2 ( 1)
J J S S L L
g J J
. It depends on the state of the electron in the atom and in
general, g lies between 0 and 2. When an external magnetic field is applied to the atom, say in the z direction, the
component of the magnetic moment along z-direction, μ z , will interact with the applied field. μ z
is given by −β eg MJ. Now the interaction energy E is given by
E = − μ z B = β eg MJB
where B is the strength of the applied magnetic field, MJ is the components of the total angular momentum in the z-direction. MJ has (2J + 1) values ranging from + J to – J. Therefore the interaction energy also will have (2J + 1) different values. These energy states are degenerate in the absence of the external field. Thus in the presence of an external magnetic field a particular energy level splits into (2J +1) different energy states. The splitting of the MJ energy levels in the presence of an external magnetic field is called the Zeeman Effect. The splitting is proportional to the strength of the applied field and is very small in magnitude. For example in an applied field of one tesla (SI unit of magnetic field strength, 1T = 10,000 gauss) the splitting is of the order of 0.5 cm−^1. Normal Zeeman Effect For singlet states the total spin magnetic moment (S) is zero. The magnetic moment of the electron system is then due to the orbital motion alone. Hence there will not be any coupling between spin and orbit motion and hence MJ is equal to ML and g become 1. The interaction energy is now, E =
β e MLB. This type of splitting of energy
levels due to applied field when S = 0 is called Normal Zeeman Effect. As an example let us consider the spectral lines that may arise due to the transition 1 P → 1 S, in the presence of an external magnetic field. A 1 S term has neither orbital nor spin angular momentum, so it is unaffected by the external magnetic field.
B = 0 B > 0
ML
0
0
(^1) P
(^1) S
Normal Zeeman Effect
Fig. 6
instead of being coupled to each other, they couple to the electrostatic field produced by the two nuclear charges of the diatomic molecule. This situation is shown in Figure 7 and is referred to as the Hund’s case. The direction of the electrostatic field of the two nuclei is taken as the inter-nuclear axis (taken as the z-axis). The vector L is strongly coupled to the electrostatic field so that it precess about the inter-nuclear axis. As a result, the magnitude of L is not defined. Only the component
of L along the inter-nuclear axis is defined, which is (^) ml ML . ( ml is for individual
electrons and to calculate ML we simply add algebraically the ml ’ s of individual electrons).
Here is a quantum number taking values, 0, 1, 2, 3 …… All electronic states with > 0 are doubly degenerate. Classically, the degeneracy is ascribed to the electrons being orbiting, clockwise or anticlockwise around the inter-nuclear axis represented by ± ml. The value of is similar to the value of L in the case of atoms. The electronic terms corresponding to different values of are , , , …..
Symbol (^) …
The different types of molecular orbitals are σ, π, δ … We use the symbol λ to represent the angular momentum of an electron in a molecule; ml = λ. Depending upon the type of MO
the value of λ varies as shown in Table 3.
MOs ml λ (^ ml )^ AOs from which MOs are formed σ (^0 0) s, pz, d z 2 π ±^1 1 px, py, dxz, dyz δ ± (^2 2) dxy , d x (^) (^2) (^) y 2 Table 3 The coupling of S to inter-nuclear axis is caused by magnetic field along the axis due to the orbital motion of the electrons, and the electrostatic field has no effect on S. The component of S along z-axis can be taken as and the quantum number (beta) is analogues to Ms in atoms. is the component of S along z-axis (the symbol is , but to avoid confusion is used). can have values, S, S – 1, ….. – S. It can be computed from the Clebsch-Gordan series. For states > 0, there are (2S + 1) components corresponding to the values that can take. ie the multiplicity of the level is (2S + 1). The component of the total (orbital + electron spin) angular momentum along the inter- nuclear axis is given by , where (omega) is the absolute value of (.
. is analogous to the quantum number J in atoms. It is actually the quantum number for the z-component of the total electronic angular momentum and therefore can take on negative values. It is the value of ( and not the value of is written as the subscript on the term symbol. For eg., consider the 3 term. Since the term is , = 1. Now (2S + 1) = 3 or S = 1. Hence the values of are S, S – 1, …. S = 1, 0, 1. = S, (S), …..+ (S) = 1+1, 1+11, 11 = 2, 1, 0. Therefore the term symbols are, 3 2 , 3 1 , 3 0.
A 4 term has four levels, (^12) 4 (^12) 4 (^32) 4 (^52)
. This arises in the following way:
Spin multiplicity (2S + 1) = 4, therefore S = 32. Hence the values of B are: 32 ,^12 ,^ ^12 ,^32.
= S, (S), …..+ (S) = 1 32 ,1 12 ,1 ( 12 ),1 ( 32 ) 52 , 32 , 12 ,^12
For a state, there is no orbital angular momentum ( = 0). Hence, a state has only one component irrespective of the multiplicity. The quantum numbers and are not defined for state. A filled molecular orbital has both and equal to zero and gives rise to only a 1 term. It always corresponds to a single non- degenerate energy level. Example is the ground state electronic configuration of hydrogen molecule. For atoms, the electronic energy states may be classified entirely by the use of L, S, and J. In diatomic molecules, the corresponding quantum numbers , , and are not quite sufficient. We must also use the symmetry properties of the electronic wave function. For homopolar diatomic molecules, the states are labeled as g or u , which indicates the wavefunction is symmetric or anti-symmetric respectively to inversion through a centre of symmetry. The other symmetry property concerns the symmetry of electron wave function with respect to reflection across any plane (V) containing the inert-nuclear axis. If the wave function is unchanged by this reflection (ie. symmetric), the state is labeled +, and if it changes in sign by this reflection (ie. Anti-symmetric), the state is labeled – , as in 3 g+^ or 2 g. This symbolism is normally used for states only. Similar to atomic term symbol molecular term symbols also are also represented by writing spin multiplicity as left superscript and total angular momentum as right subscript to the code letter for .ie.
u g
g u
Parity - behaviour under inversion. Figure 9
(-) v(+) anti-symmetric with respect to plane
symmetric with respect to plane
v
Figure 10
Examples ( i ) H 2 + The electron configuration is 1σg^1. Since there is only one electron, S = s = ½. Spin multiplicity is 2. For σ electron λ = 0 and hence, Λ = 0. The term symbol is 2 Σg+. ( ii ) B 2 + The electron configuration is, 1σg^2 1σu^22 σg^22 σu^21 πu^1. Only one electron need to be considered. For that electron λ = Λ = 1 (π-electron) and S = s = ½. Hence the term is Π. The spin multiplicity is 2. Therefore the term symbol is 2 Π (or 2 Πu).
(iii)N 2 +
The electron configuration is 2σg^22 σu^21 πu^43 σg^1. All orbitals except 3σg are filled. Only one electron is present in the 3σg orbital. Hence, Λ = 0 and S = s = ½. Therefore the term symbol is (^2) Σg+.
( iv ) O 2 (or B 2 ) The electron configuration for O 2 is 2σg^22 σu^23 σg^21 πu^41 πg^2. All orbitals up to 1πg are filled. There are two electrons in the πg orbital. For π-electrons λ = 1 (and ml = ±1). The two electrons in the1πg orbital can be arranged in the following way (Table 5). π+ and π− are the two different angular momenta of the electron. Ψ 1 , Ψ 2 , … are the spin-orbit wave functions
π+ ⇵ ↓ ↑ ↑ ↓ π− ⇵ ↑ ↑ ↓ ↓ ML +2 −2 0 0 0 0 MS 0 0 0 +1 0 −
term 1 ∆
1 g
^ ^3
g
^
Table 5 In the table, ML values +2 and −2 represent a term with Λ = 2. MS values for both the values are zero and hence S = 0. Therefore this arrangement represents a 1 Δ term. When ML and MS are zero it represents a 1 Σg term. For the remaining three arrangements ML = 0, but the values MS are +1, 0, −1, and it corresponds to S = 1. ie. Λ = 0 and S = 1. Hence a 3 Σg term. Now we have to assign +/− to Σ terms. The O 2 molecule has a π^2 configuration, and anti-symmetric wave functions can be formed either by combining symmetric spatial functions with anti-symmetric spin functions or vice versa. All the six combinations are shown below in table 6. The spatial part of Ψ 1 and Ψ 2 are symmetric and this being a ∆ term we do not assign +/− to it. For Ψ 3 the spatial function is symmetric and hence we assign + sign. Ψ 4 to Ψ 6 are anti-symmetric so it combine with symmetric spin function. We assign ‘–‘ for these three terms.
Spatial (orbital) function Spin function Symmetry of spin function
Symmetry of orbital function
Ψ 1 π+ π+ α(1)β(2) - β(1) α(2) Anti-symmetric symmetric 1
Ψ 2 π− π− α(1)β(2) - β(1) α(2) Anti-symmetric^ symmetric
Ψ 3 (π+ π− )+ (π− π+) α(1)β(2) - β(1) α(2) Anti-symmetric^ symmetric^1 Σ+
Ψ 4 (π+ π− ) − (π− π+) α(1)^ α^ (2)^ symmetric^
Anti- symmetric
Ψ 5 (π+ π− ) − (π− π+) α(1)β(2) +^ β(1)^ α(2)^ symmetric^ Anti-^3 Σ−
symmetric
Ψ 6 (π+ π− ) − (π− π+) β(1)β(2) symmetric
Anti- symmetric
Table 6
Thus the term symbols for the ground state of O 2 are: 1 Δ, (^1) ^ g and (^3) g^ .
Exercise : Write the term symbol for (i) Li 2 +^ with the electron configuration 2σg^2 , (ii) F 2 and (iii) H 2
Electronic spectra of diatomic molecules.
Let us take the simplest molecule H 2 as example. For H 2 molecule the configuration is 1σg^2 , hence the term symbol is 1 Σg+. We can also imagine a large number of singlet excited states and let us consider some lower energy levels in which only one electron has been raised from the ground state into some higher molecular orbitals (ie. singly excited states). We can ignore any promotion into any anti-bonding states since this would lead to the formation of an unstable molecule leading to immediate dissociation of the molecule. Thus we may consider, for example, three states; 1sσg^1 2sσg^1 , 1sσg^1 2pσg^1 , 1sσg^1 2pπu^1 (a) 1sσg^1 2sσg^1 Here both the electrons are σ electrons. Hence Λ = λ 1 + λ 2 = 0. We are considering only singlet states, so S = 0. Since both the constituent orbitals are gerade and symmetrical, the
over all state will be the same. Therefore we have 1 g^ state.
(b) 1sσg^1 2pσg^1 Since both the electrons are σ- electrons we have a 1 Σ state. But the overall state is now odd. This can be understood in the following way. Imagine that one of the electrons is coming from a hydrogen atom in the gerade 1s state, and the other electron from a hydrogen atom in the ungerade 2p state. Combination of
g 1s gerade
u 2p ungerade
