Atomic spectra of many-electron atoms are very complex in nature—they contain multiple lines indicative of several transitions. The energy states of the atom corresponding to these transitions cannot be calculated as easily as in the case of hydrogen-like species with only one electron. The reason is obvious—one has to consider the mutual interaction between the electrons.
The wave mechanical procedure to tackle this situation is extremely lengthy and complicated. However, experimental spectroscopists have developed a procedure to assign the various energy states of an atom corresponding to observed spectral transitions without any reference to any theoretical description of electron motion. There is good correspondence between the procedures with experiment.
The Vector Atom Model:
Although the wave formalism abandons the picture of specific orbits for the electrons, the idea of angular momentum of an electron in an orbital is retained in this model. Next the orbital angular momentum and the spin angular momentum of the individual electrons in a multi-electron system may be combined according to the principle of vector addition to obtain the resultant spin and orbital angular momenta of the atomic system; these may be utilized to obtain different energy states of the atom.
This is the basis of the vector model of the atom. In this model, the different energy states of an atom are designated by particular term symbols derived from the resultant orbital and spin angular momenta of the extranuclear electrons. (A further interaction with the spin angular momentum of the nucleus will give rise to still finer splitting of the energy levels)
The interactions between the spin and orbital angular momenta of a many electron system may be considered in terms of the one-electron quantum numbers (n, l, ml and ms) of the individual electrons.
If we consider a two-electron system, for example, there may be three types of interaction:
(i) Orbit-orbit coupling, in which the orbital angular momenta of the two electrons couple;
(ii) Spin-spin coupling, in which the spin angular momenta of the two electrons couple; and
(iii) Spin-orbit coupling, in which the spin and orbital angular momenta of the same electron interact with one another.
The coupling of the spin of one electron with the orbital angular momentum of a different electron is small and can be ignored. There may be two ways to consider such interactions.
(i) Russel-Saunders Scheme (L-S Scheme):
In this scheme it is assumed that the strength of interaction varies as:
Spin – spin coupling > orbit-orbit coupling > spin-orbit coupling.
The individual orbital angular momenta of the electrons are first combined to give a total orbital angular momentum of the system. By analogy with the orbital angular momentum of a single electron (=√[l(l + 1) h]/2), this is expressed by means of a total orbital angular momentum quantum number L; the total orbital angular momentum is equal to √[L(L + 1) h]/2.
Similarly, the resultant spin angular momentum of the atom is expressed by a corresponding quantum number S as √[S(S + 1) h]/2.
The total angular momentum (spin + orbital) is now represented by a corresponding quantum number J and is equal to √[J(J + 1) h]/2.
This coupling scheme works fairly well for the lighter elements (say upto atomic number 30). As the radial function in heavier atoms decays more slowly to a larger distance in space, the interaction between the spin and orbital angular momenta of two electrons decreases and the L-S scheme becomes less appropriate.
(ii) j-j Coupling Scheme:
In this scheme, the spin and orbital angular momenta of individual electrons are first coupled to obtain a resultant “spin + orbital” angular momentum for each; the corresponding quantum number is denoted by j. A weaker coupling of the individual electron angular momenta now gives the total angular momentum for the atom. The relative importance of spin-orbit coupling increases in a gradual manner with atomic number and there is a smooth transition from the L-S to the j-j scheme.
Term Symbols (L-S Scheme):
First of all we recall the salient features of the single electron quantum numbers l, ml and ms.
Following are the features:
(i) The orbital quantum number l has integral values from 0 to (n – 1) for any principal quantum shell n and the orbital angular momentum is given by √[l(l +1) h]/2.
(ii) The orbital angular momentum has 2l + 1 possible directions in space (with reference to the direction of an external magnetic field) specified by ml, the magnetic quantum number: ml = l, (l – 1), … 0 …– (l – 1), – l.
(iii) The spin angular momentum is given by √[s(s + 1)h]/2 where s is the spin quantum number (½). The z-component of this vector (chosen arbitrarily) is given by ms (h/2) where ms is the spin magnetic quantum number; this can be +s or – s, i.e., +½ or –½.
Now we return to our case of a many electron atom and apply an analogous treatment.
The procedures for obtaining L, S and J will be described below.
(a) The Resultant Orbital Quantum Number, L:
This gives the resultant orbital angular momentum of the system and is obtained by vectorial addition of individual l values. The different L values, 0, 1, 2, 3, 4, etc. are represented by a letter symbol (like s, p, d, … for l values) ―
But how should one combine the individual l-values? The angular momentum vectors related to l are, in general, not parallel and so direct addition of such vectors is very difficult. But the individual ml values represent the component of the orbital angular momentum vectors (mlh/2) oriented in the same direction (conventionally the z-axis) and so their vector addition can be done simply by algebraic summation. Such algebraic summation for all the electrons in the system gives the resultant orbital angular momentum along the z-axis, which may be represented by ML (h/2). ML = (Σml) may be called the resultant orbital magnetic quantum number.
Just like ml, there are (2L + 1) values for ML: L, (L – 1), .… 0 … – (L – 1), – L.
Different possible distribution of electrons among different ml values gives different ML values (Σml) and hence L-values. These correspond to different terms or energy states for the system.
(b) The Resultant Spin Quantum Number, S:
Similar to L, we may introduce a resultant spin quantum number S which gives the resultant spin angular momentum as √[S(S + 1)h]/2. Algebraic summation of ms values of individual electrons gives a resultant spin magnetic quantum number Ms = (Σms).
Ms = (Σms) may have 2S + 1 values: S, (S – 1), … 0 … – (S – 1), – S. The quantity (2S + 1) is known as the spin multiplicity of a state.
Note that S is used for the resultant spin quantum number as well as the symbol for a state with L = 0.
(c) The Inner Quantum Number, J:
This expresses the total angular momentum as √ [J(J +1) h]/2. J can have the values L + S, L + S – 1, … |L – S|. The modulus notation indicates that only the magnitude of (L – S) is involved. Like j for a single electron, J can be either positive or zero. So two cases may arise –
L > S ― there are 2S + 1 possible values of J.
L < S ― there are 2L + 1 possible values of J.
The complete term symbol (or state symbol) for an energy state is written using (2S + 1), L and J as ―
Instead of the numerical value of L, we use the corresponding symbols: S (L = 0), P (L =1) etc. Thus, 3P0 (triplet-P-zero) stands for a system in which L = I, 2S + 1 = 3 and J = 0. Terms which only differ in J values differ slightly in energy. The energy state is therefore often expressed by 25 + 1L with the J-values omitted. The different J values now give different components of that state.
Now we shall work out the ground-state terms for a few species:
Hydrogen Atom (1s1):
l = 0; hence L = 0. This corresponds to an 5 state. Now for the single electron, S = ½, 2S +1 = 2, and J = ½. So the ground state term symbol is 2S1/2.
Helium Atom (1s2):
For both electrons l = 0. So L = 0. Since the electrons are spin-paired, ms = + ½ and –½ and S = 0. 2S + 1 = 1, J = 0. So the state symbol is 1S0.
For any ns2 or np2 configuration or the like, i.e., a closed subshell (or a closed shell), MS and ML will be zero. The spins will cancel one another, so also will the ‘+’ and ‘–’ values of ml. So for lithium and beryllium, the 1s shell does not contribute anything to the term symbol. Li (1s22s1) has the same term symbol as hydrogen (2S1/2) and Be (1s22s2) has the same term symbol as helium (1S0).
Here also, only the p electron need be considered. I = 1, so L = 1, i.e., a P state is indicated. S = ½ gives 2S + 1 = 2, a doublet state. J may be L + S or L – S, i.e., 3/2 or ½ . So the state symbols are 2P3/2 or 2P1/2. The ground state term will be 2P1/2.
Let us consider only the two p-electrons. For each, l = 1 and ml = + 1, 0 or – 1. Let us, for convenience, label the electrons as 1 and 2. Then the possible values of ml for electron(1) and electron(2) are –
The possible values of ML (= ml(1) + ml(2)) are then 2, 1, 0, 0, –1, 0, –1, –2. These can be arranged and the different L values (hence the term symbols) derived therefrom as―
For ML = + 2 or – 2, ml (1) = ml (2) = 1 or ml (1) = ml (2) = – 1. Thus, n, I, and ml for the two electrons become identical; hence the Pauli principle requires that the spins of the two electrons must be opposite, that is MS = +½ + (– ½) = 0. So S = 0 and 2S + 1 = 1; it is a singlet state. L + S = 2 and L – S = 2, so that J = 2. The complete term symbol is thus 1D2.
For each value of ML, the two electrons may have spins (½, ½), (½ – ½), (–½ –½), giving ML = +1, 0, or –1. We have 9 possible combinations of ML and MS.
When MS = +1, 0 and –1, S = 1 and 2S + 1 = 3. The term symbol is thus 3P. Next we find that L + S = 1 + 1 = 2 and L – S = 1 – 1 = 0. So J may have values 2, 1, 0. The complete term symbols will be 3P2, 3P1 and 3P0.
This corresponds to ML = 0 and MS = 0, and hence S = 0. The complete symbol will be 1S0.
In working out the term symbols for the carbon atom, we have ultimately considered the various possible distributions of the p-electrons among the ml and ms values obeying the exclusion principle. Each such combination of ml and ms for the two electrons is called a microstate.
In Table 4.7, these permitted combinations (15 all total) are shown with the resultant ML and MS value for each:
These 15 microstates can be grouped into three sets – (i) a set of five with ML = 2, 1, 0, –1, – 2, all having MS = 0; these correspond to the 1D2 term. MJ = 2, 1, 0, –1, –2, i.e., J = 2. (ii) a set of nine microstates corresponding to the 3p terms, that is. ML = 1, 0 or – 1 and MS = 1, 0 or – 1. These can be subdivided into five with J = 2 (3P2), three with J = 1 (3P1) and one with J = 0 (3P0). (iii) a single combination with ML = 0 and MS = 0, corresponding to the term 1S0. However, there are three combinations (serial no 2, 12 and 13) for ML = 0, MS = 0 and there is no way to decide which ML – MS combination should be assigned to which state. It is not meaningful either.
The total number of microstates can be ascertained very easily. The first electron can be placed in any of the three p-orbitals with either of two spin quantum numbers; this can be done in six different ways (these may be called six spin orbitals). The second electron can be placed in the particular p-orbital already occupied by the first electron with an opposite spin only; in the other two p-orbitals it may have either spin. So this electron may be placed in the system in five different manners. Combining, there are 6 x 5 = 30 different ways. But since the electrons are indistinguishable, the actual number will be 30 ÷ 2 = 15. The number of microstates N arising from ‘e’ number of electrons filling a set of orbitals of same n and l values are –
Where, Nl is the number of ml – ms combinations for a single electron in the orbital set [= 2(2l + 1)]. Nl = 6 for np orbitals, 10 for nd orbitals etc.
Also, each term with quantum numbers L and S consists of (2L + 1). (2S + 1) number of ML – MS pairs or microstates.
We consider only the 2p3 configuration and label the electrons as 1, 2 and 3. The possible values of ml are 1, 0, – 1 for each. This gives the possible L-values as 3, 2, 1, 0. But L = 3 means ML should also have the value 3 (or –3). This will occur only if all three electrons belong together to ml = + 1 (or – 1). This is not permitted since ms may have only two values +½ or –½ and there may be at best two electrons for each ml. So L cannot be 3 and there cannot be any F term.
L = 2 indicates a D-term. ML = 2, 1, 0, – 1, – 2. ML can be equal to 2 only when two of the electrons are spin-paired. So MS can be only +½ or – ½. This gives S = ½ and 2S + 1 = 2.L + S = 2 + ½; L – S = 2 – ½; so the allowed values of J are 5/2 and 3/2. So the complete term symbols for the D-term will be 2D5/2 and 2D3/2. From L = 2 and S = ½, there will be (2S + 1) (2L + 1) i.e., 10 microstates for the D-terms. Each J-value has 2J+ 1 microstates. (6 and 4 respectively).
L = 1 corresponds to a P-term. Here ML = 1, 0, – 1. For a combination of three electrons this again gives MS = ½ or – ½ and S = ½. 2S + 1 = 2 and J values are 3/2 and 1/2. The term symbols are 3P3/2 and 3P1/2 (six microstates).
L = 0 means ML = 0. But all three electrons cannot be together in ml = 0 as that would violate the Pauli principle. So they must arise from combinations like ml(1) = 1, ml(2) = –1 and ml(3) = 0. Since the ml values are different, ms may be same. So the possible values of MS are 3/2. ½, –1/2, –3/2. This gives S = 3/2 and J = 3/2. The complete term symbol is 4S3/2 (four microstates).
Oxygen and Fluorine:
Discussion of the 2p4 and 2p5 configurations becomes easier when we introduce a useful simplification known as the hole formalism. According to this, the 2p4 configuration may be looked upon as a 2p6 configuration plus two positrons to annihilate two electrons. We know that positrons differ from electrons only with respect to their charge. So two positrons will give rise to the same states as given by two electrons. The filled p6 configuration does not contribute anything to the terms and so we find that the p4 configuration will have the same terms as the p2 configuration. Similarly, the p5 configuration will have the same terms as the p1 configuration. In general, for any subshell which is more than half-filled, p6–n or d10–n configurations will give rise to the same terms as pn or dn configurations respectively.
Ground State Term – Hund’s Rules:
Once the energies of various states for an electron configuration are labeled by respective term symbols, it becomes necessary to know the relative energies of these states. A number of rules have been framed for the purpose from analysis of spectroscopic data. These are collectively known as Hund’s rules.
Where Russel-Saunders coupling holds (Z < 30), the rules essentially state that:
1. The energy of the state’s decreases as the spin multiplicity (and hence S) increases. The ground state of an atom should therefore show highest spin multiplicity and hence possess the maximum number of unpaired spins in parallel orientation.
2. When the value of S is same for more than one state, the state with the highest value of L has the minimum energy.
3. For all states having same S and L, the following order of energy will hold:
(i) The subshell is less than half-filled – the component with the smallest value of J has the least energy.
(ii) The subshell is more than half-filled or half-filled ― the component with the highest value of J has the least energy.
For boron, we obtained two terms— 2P3/2 and 2P1/2. The ground state term will be 2P1/2 (less than half-filled, least value of J).
For carbon, the triplet state 3P will be most stable as it has the highest spin multiplicity. Since the electron configuration is less than half-filled, the 3P0 state, with lowest value of J, will be the ground state.
One can derive the ground state term for a given configuration using Hund’s rules, without deriving all the possible terms. Thus for carbon (2s22p2), we consider only the 2p electrons as the closed shell 2s2 contributes nothing. We have ―
Maximum ML = 1; L = 1, S = 1. Spin multiplicity = 2S + 1 = 3. L = 1 denotes a P state. Since we are dealing with a less than half-filled configuration, the lowest value of J will give the ground state, which is 3P0.
Derive the term for a hydrogen atom in the excited states (i) 2s1, (ii) 2p1.
(i) For an s-orbital, L = 0;
Here S = ½, 2S + 1 = 2. J = ½. The term will be 2S1/2.
(ii) L = 1, S = ½. So, J = ½, 3/2
The states will be 2P3/2 and 2P1/2.
The ground state term symbols of the first few elements are given below:
Examine whether the ground-state term 7S3 for Cr corresponds to the 3d44s2 configuration.
The closed-shell s2 configuration does not contribute to the term. For 3d4 configuration we get –
L = 2 + 1 + 0 + (– 1) = 2
S = 4 x ½ = 2.
2S + 1 = 5
Jmax = 4.
Hence the term would be 5D4 which does not match the term 7S3.
(For 3d54s1 configuration L = 0, 5 = 6 x ½ = 3 which gives the 7S3 term).
Terms, Levels and States:
A term symbol summarily gives the symbolic representation of the state of an atom. However, the various words used in this context are used to imply different meanings.
An atomic term conveys definite values of the total orbital angular momentum quantum number L and the total spin angular momentum quantum number S. The L value is represented by a code letter (S for L = 0, P for L = 1 etc.). S is shown by 2S + 1 (spin multiplicity) as a left superscript. J is not included in the term but the quantum numbers n and I may precede the term. A term is thus designated by the quantum numbers n, I, L and S.
Levels arise from splitting of terms according to different values of J (fine structure). A given value of L gives the levels of its term according to the values of J. A level is thus designated by the quantum numbers n, I, L, S and J. When L > S, the multiplicity (2S + 1) is the number of levels of the term. A level consists of a group of states with a common value of J.
In magnetic fields, levels can further split into states.
Show the terms and levels for (i) [Ne] 3s1 (ii) [Ne] 3p1 (iii) the 3D term of [Ne] 2p1 3p1.
(i) For 3s1, L = I = 0; S = s = ½; J = ½.
Term = 2S. There is only one level for J = ½ which we denote 2S1/2.
(ii) For 3p1 L = I = 1; S = s = ½; J = 3/2, 1/2
Term = 2P. There are two levels, 2P3/2 and 2P1/2.
(iii) For 3D term, L = 2, S = 1, So, J = 3, 2, 1. The levels are 3D3, 3D2, 3D1.
A spectroscopic term symbol sometimes contains an additional information about parity. The term parity in general refers to how a wave function changes when all the spatial coordinates undergo an inversion in space (for example we replace x, y, z by –x, –y and –z respectively).
We may introduce a quantum number P which may have values +1 for even parity and –1 for odd parity ―
Since atoms are symmetrical, its associated wave functions must either remain unchanged or change signs under inversion. Orbitals of odd l value are said to have odd parity (p, f etc.). An odd number of p or f electrons thus lead to an odd parity. An even number of p and f electrons leads to an even parity of the term. Thus we simply add the l values of the electrons to see whether a function is odd or even.
For even parity (P = + 1), no indication appears on the term. Odd parity (P = –1) is sometimes indicated by writing ‘0’ as a right superscript in the term. Thus, the term symbol for CI (ground state) is written 2Po3/2.
Selection Rules for Spectral Transitions:
Not all possible spectral transitions between different states of an atom are actually observed. At first some empirical selection rules were proposed to account for such “unseen” transitions which were declared forbidden. Subsequently, these and other related selection rules have been largely rationalized from various theoretical approaches.
A photon has a spin of unity. For left and right circularly polarized light, ms = +1 and –1 respectively. When an atom absorbs a photon, the angular momentum of the photon is transferred to the electrons of the atom; when a photon is emitted in a transition, its angular momentum must come from a change in angular momentum of the electron undergoing transition.
Summarily, interaction of the electric vector of the radiation with instantaneous electric dipoles in the atom (or molecule) results in oscillating electric field component of the light, giving rise to electric dipole transitions. Oscillating magnetic field components of light will similarly give rise to magnetic dipole transitions which are usually much weaker. Since angular momentum has to be conserved in emission or absorption, Δl should be equal to ±1 for the atom.
The selection rules set the criterion for allowed transitions. The criterion referred to in the selection rules may vary in different cases. For the simplest one electron species like hydrogen-like atoms, the selection rules summarily state that for allowed transitions
Δl = ±1 and Δml = 0, ±1.
There is no restriction on an (the principal quantum number).
Thus a d electron (l = 2) can make radiative transition to an orbital with I = 1 (p-orbital or l = 3 (f-orbital). It cannot undergo a transition to an s-orbital (l = 0) or another d-orbital (l = 2). Similarly, a 4s electron can make radiative transition to np orbitals only.
We note that an allowed transition is accompanied by a change in parity. The parity of a polyelectronic atom is even if Σli is even over all electrons; parity is odd if Σli is odd. When only one electron is promoted from the ground configurations, Δl = ± 1 results in a change in parity.
For many electron atoms, the states of the atom and spectral transitions are stated using term symbols. However, the selection rules are applicable when Russel-Saunders coupling is valid.
The following criterions are used to identify allowed transitions:
Let us consider the principal series of sodium corresponding to electronic transitions from various p-excited states to the 3s ground electronic state.
Ground state (3s1) – l = 0, s = ½ give L = 0, S = ½ and J = ½. Term symbol – 2S1/2
Excited states (np1) – I = +1, s = ½ give L = 1, S = ½ and J = (1 + ½), (1 + ½ –1) = 3/2, 1/2
Term symbols – 2P3/2, 2P1/2 (2P1/2 will be lower in energy).
The electronic spectra should consist of emission lines for transitions between the 2P states and the 2S state. We consider the selection rules one by one – ΔS = 0; ΔL = ±1 (P ↔ S); ΔJ = ±1 (3/2 ↔ 1/2); 0 (1/2 ↔ 1/2); Δl = ±1. Thus all the selection rules are satisfied.
The golden-yellow “D-lines” of sodium around 589 nm (∼17000 cm–1) are actually doublets at 589.16 nm and 589.76 nm. The difference of 0.6 nm (17 cm–1) corresponds to splitting between the 2P3/2 and 2P1/2 states.
When the sodium atom (or any atom with S = 0) is placed in a magnetic field parallel to the z-axis assigned to the atom, the multiplets will be split into 2L + 1 components according to the ML values ― Ml = Σmi = L, L– 1, – L (in this case +1, 0, –1). The allowed transition will now obey the additional rule –
ΔML = 0, +1.
This is observed in normal Zeeman Effect.
The Zeemann effect becomes complicated for atoms with S ≠ 0.
The applicability of the selection rules depends largely on the coupling of spin and orbital angular momenta of the electrons. The L-S coupling scheme or the Russel-Saunders term symbols are useful guides for lighter atoms. For heavier atoms, spin-orbit interaction of individual electrons become more dominating (j-j coupling) and we find breakdown of the above selection rules. For example, transitions between singlet and triplet states (ΔS = ±1) gain prominence in heavier atoms. Such “forbidden transitions” usually show weaker intensity.
The relative intensity of allowed transitions depends upon ΔL and ΔJ: