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Review of Quantum Numbers
Electrons in an atom reside in shells characterized by a particular value of n, the Principal Quantum Number. Within each shell an electron can occupy an orbital which is further characterized by an Orbital Quantum Number, \(l\), where \(l\) can take all values in the range:
\[l = 0, 1, 2, 3, ... , (n-1),\]
traditionally termed s, p, d, f, etc. orbitals.
Each orbital has a characteristic shape reflecting the motion of the electron in that particular orbital, this motion being characterized by an angular momentum that reflects the angular velocity of the electron moving in its orbital. A quantum mechanics approach to determining the energy of electrons in an element or ion is based on the results obtained by solving the Schrödinger Wave Equation for the H-atom. The various solutions for the different energy states are characterized by the three quantum numbers, n, l and m_{l}.
- m_{l} is a subset of l, where the allowable values are: m_{l} = l, l-1, l-2, ..... 1, 0, -1, ....... , -(l-2), -(l-1), -l.
- There are thus (2l +1) values of m_{l} for each l value, i.e. one s orbital (l = 0), three p orbitals (l = 1), five d orbitals (l = 2), etc.
- There is a fourth quantum number, m_{s}, that identifies the orientation of the spin of one electron relative to those of other electrons in the system. A single electron in free space has a fundamental property associated with it called spin, arising from the spinning of an asymmetrical charge distribution about its own axis. Like an electron moving in its orbital around a nucleus, the electron spinning about its axis has associated with its motion a well defined angular momentum. The value of m_{s} is either + ½ or - ½.
In summary then, each electron in an orbital is characterized by four quantum numbers (Table 1).
symbol | description | range of values |
---|---|---|
n | Principal Quantum Number - largely governs size of orbital and its energy | 1,2,3 etc |
l | Azimuthal/Orbital Quantum Number - largely determines shape of subshell 0 for s orbital, 1 for p orbital etc | (0 ≤ l ≤ n-1) for n = 3 then l = 0, 1, 2 (s, p, d) |
m_{l} | Magnetic Quantum Number - orientation of subshell's shape for example p_{x} with p_{y} and p_{z} | l ≥ m_{l} ≥ -l for l = 2, then m_{l} = 2, 1, 0, -1, -2 |
m_{s} | Spin Quantum Number | either + ½ or - ½ for single electron |
Russell Saunders coupling
The ways in which the angular momenta associated with the orbital and spin motions in many-electron-atoms can be combined together are many and varied. In spite of this seeming complexity, the results are frequently readily determined for simple atom systems and are used to characterize the electronic states of atoms. The interactions that can occur are of three types.
- spin-spin coupling
- orbit-orbit coupling
- spin-orbit coupling
There are two principal coupling schemes used:
- Russell-Saunders (or L - S) coupling
- and jj coupling.
In the Russell Saunders scheme (named after Henry Norris Russell, 1877-1957 a Princeton Astronomer and Frederick Albert Saunders, 1875-1963 a Harvard Physicist and published in Astrophysics Journal, 61, 38, 1925) it is assumed that:
spin-spin coupling > orbit-orbit coupling > spin-orbit coupling.
This is found to give a good approximation for first row transition series where spin-orbit (J) coupling can generally be ignored, however for elements with atomic number greater than thirty, spin-orbit coupling becomes more significant and the j-j coupling scheme is used.
Spin-Spin Coupling
S - the resultant spin quantum number for a system of electrons. The overall spin S arises from adding the individual m_{s} together and is as a result of coupling of spin quantum numbers for the separate electrons.
Orbit-Orbit Coupling
L - the total orbital angular momentum quantum number defines the energy state for a system of electrons. These states or term letters are represented as follows:
L | 0 | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|
S | P | D | F | G | H |
Spin-Orbit Coupling
Coupling occurs between the resultant spin and orbital momenta of an electron which gives rise to J the total angular momentum quantum number. Multiplicity occurs when several levels are close together and is given by the formula (2S+1). The Russell Saunders term symbol that results from these considerations is given by:
\[ \large ^{(2S+1)}L\]
Configuration
S= + ½, hence (2S+1) = 2
L=2 and the Ground Term is written as ^{2}D
The Russell Saunders term symbols for the other free ion configurations are given in the Table below.
Configuration | # of quantum states | # of energy levels | Ground Term | Excited Terms |
---|---|---|---|---|
d^{1},d^{9} | 10 | 1 | ^{2}D | - |
d^{2},d^{8} | 45 | 5 | ^{3}F | ^{3}P, ^{1}G,^{1}D,^{1}S |
d^{3},d^{7} | 120 | 8 | ^{4}F | ^{4}P, ^{2}H, ^{2}G, ^{2}F, 2 x ^{2}D, ^{2}P |
d^{4},d^{6} | 210 | 16 | ^{5}D | ^{3}H, ^{3}G, 2 x ^{3}F, ^{3}D, 2 x ^{3}P, ^{1}I, 2 x ^{1}G, ^{1}F, 2 x ^{1}D, 2 x ^{1}S |
d^{5} | 252 | 16 | ^{6}S | ^{4}G, ^{4}F, ^{4}D, ^{4}P, ^{2}I, ^{2}H, 2 x ^{2}G, 2 x ^{2}F, 3 x ^{2}D, ^{2}P, ^{2}S |
Note that d^{n} gives the same terms as d^{10}^{-n}
Hund's Rules
The Ground Terms are deduced by using Hund's Rules. The two rules are:
- The Ground Term will have the maximum multiplicity
- If there is more than 1 Term with maximum multipicity, then the Ground Term will have the largest value of L.
A simple graphical method for determining just the ground term alone for the free-ions uses a "fill in the boxes" arrangement.
d^{n} | 2 | 1 | 0 | -1 | -2 | L | S | Ground Term |
---|---|---|---|---|---|---|---|---|
d^{1} | ↑ | 2 | 1/2 | ^{2}D | ||||
d^{2} | ↑ | ↑ | 3 | 1 | ^{3}F | |||
d^{3} | ↑ | ↑ | ↑ | 3 | 3/2 | ^{4}F | ||
d^{4} | ↑ | ↑ | ↑ | ↑ | 2 | 2 | ^{5}D | |
d^{5} | ↑ | ↑ | ↑ | ↑ | ↑ | 0 | 5/2 | ^{6}S |
d^{6} | ↑↓ | ↑ | ↑ | ↑ | ↑ | 2 | 2 | ^{5}D |
d^{7} | ↑↓ | ↑↓ | ↑ | ↑ | ↑ | 3 | 3/2 | ^{4}F |
d^{8} | ↑↓ | ↑↓ | ↑↓ | ↑ | ↑ | 3 | 1 | ^{3}F |
d^{9} | ↑↓ | ↑↓ | ↑↓ | ↑↓ | ↑ | 2 | 1/2 | ^{2}D |
To calculate S, simply sum the unpaired electrons using a value of ½ for each. To calculate L, use the labels for each column to determine the value of L for that box, then add all the individual box values together.
Configuration
For a d^{7} configuration, then:
- in the +2 box are 2 electrons, so L for that box is 2*2= 4
- in the +1 box are 2 electrons, so L for that box is 1*2= 2
- in the 0 box is 1 electron, L is 0
- in the -1 box is 1 electron, L is -1*1= -1
- in the -2 box is 1 electron, L is -2*1= -2
Total value of L is therefore +4 +2 +0 -1 -2 or L=3.
Note that for 5 electrons with 1 electron in each box then the total value of L is 0. This is why L for a d^{1} configuration is the same as for a d^{6}.
The other thing to note is the idea of the "hole" approach. A d^{1} configuration can be treated as similar to a d^{9} configuration. In the first case there is 1 electron and in the latter there is an absence of an electron i.e., a hole.
The overall result shown in the Table above is that:
- 4 configurations (d^{1}, d^{4}, d^{6}, d^{9}) give rise to D ground terms,
- 4 configurations (d^{2}, d^{3}, d^{7}, d^{8}) give rise to F ground terms
- and the d5 configuration gives an S ground term.
The Crystal Field Splitting of Russell-Saunders terms
The effect of a crystal field on the different orbitals (s, p, d, etc.) will result in splitting into subsets of different energies, depending on whether they are in an octahedral or tetrahedral environment. The magnitude of the d orbital splitting is generally represented as a fraction of Δ_{oct} or 10Dq.
The ground term energies for free ions are also affected by the influence of a crystal field and an analogy is made between orbitals and ground terms that are related due to the angular parts of their electron distribution. The effect of a crystal field on different orbitals in an octahedral field environment will cause the d orbitals to split to give t_{2g} and e_{g} subsets and the D ground term states into T_{2g} and E_{g}, (where upper case is used to denote states and lower case orbitals). f orbitals are split to give subsets known as t_{1g}, t_{2g} and a_{2g}. By analogy, the F ground term when split by a crystal field will give states known as T_{1g}, T_{2g}, and A_{2g}.
Note that it is important to recognize that the F ground term here refers to states arising from d orbitals and not f orbitals and depending on whether it is in an octahedral or tetrahedral environment the lowest term can be either A_{2g} or T_{1g}.
Russell-Saunders Terms | Crystal Field Components |
---|---|
S (1) | A_{1g} |
P (3) | T_{1g} |
D (5) | E_{g} , T_{2g} |
F (7) | A_{2g} , T_{1g} , T_{2g} |
G (9) | A_{1g} , E_{g} , T_{1g} , T_{2g} |
H (11) | E_{g} , 2 x T_{1g} , T_{2g} |
I (13) | A_{1g} , A_{2g} , E_{g} , T_{1g} , 2 x T_{2g} |
Note that, for simplicity, spin multiplicities are not included in the table since they remain the same for each term. The table above shows that the Mulliken symmetry labels, developed for atomic and molecular orbitals, have been applied to these states but for this purpose they are written in CAPITAL LETTERS.
Mulliken Symbol for atomic and molecular orbitals | Explanation |
---|---|
a | Non-degenerate orbital; symmetric to principal C_{n} |
b | Non-degenerate orbital; unsymmetric to principal C_{n} |
e | Doubly degenerate orbital |
t | Triply degenerate orbital |
(subscript) g | Symmetric with respect to center of inversion |
(subscript) u | Unsymmetric with respect to center of inversion |
(subscript) 1 | Symmetric with respect to C_{2} perp. to principal C_{n} |
(subscript) 2 | Unsymmetric with respect to C_{2} perp. to principal C_{n} |
(superscript) ' | Symmetric with respect to s_{h} |
(superscript) " | Unsymmetric with respect to s_{h} |
For splitting in a tetrahedral crystal field the components are similar, except that the symmetry label g (gerade) is absent. The ground term for first-row transition metal ions is either D, F or S which in high spin octahedral fields gives rise to A, E or T states. This means that the states are either non-degenerate, doubly degenerate or triply degenerate.