The eigenvalue equation for the Hamiltonian H_{0} of a particle with potential energy -Ze^{2}/r is
.
Common eigenfunctions of H_{0}, L^{2}, and L_{z} are of the form y_{nlm}(r)=R_{nl}(r)Y_{lm}(q,f).
.
If E>0, then a solution exists for any value of E and l.
If E<0, then solutions exist only for discrete values of E, .
n=1 | ground state | ionization potential | |
n=2 | resonance level | resonance potential |
For hydrogen E_{I}=13.595eV, E_{R}=10.196eV.
K | L | M | N | O |
1s | 2s | 3s | 4s | 5s |
2p | 3p | 4p | 5p | |
3d | 4d | 5d | ||
4f | 5f | |||
5g |
The probability per unit time of a spontaneous transition from a stationary state |a> to a stationary state |b>, accompanied by the emission of a photon of energy =E_{a}-E_{b} is
,
with x_{ab}=<a|x|b>.
The selection rules are Dl=±1, Dm_{l}=0, ±1, Dn is not restricted.
.
R=109677.581cm^{-1} for hydrogen.
n'=1 | n=2,3,4,… | Lyman series |
n'=2 | n=3,4,5,… | Balmer series |
n'=3 | n=4,5,6,… | Paschen series |
n'=4 | n=5,6,7,… | Bracket series |
n'=5 | n=6,7,8,… | Pfund series |
I=neutral, II=singly ionized, III=doubly ionized, … .
(Example: Be_{I}, Be_{II}, Be_{III}, … .)
H=H_{0}+H_{f}.
.
First order energy shifts:
.
Dipole selection rules: Dj=0, ± 1, Dm_{j}=0, ± 1.
The Hamiltonian H_{0} for a multi-electron atom is
.
H_{0}y=Ey.
We often use the central field approximation. We assume that each electron moves in a centrally symmetric field created by the nucleus and all the other electrons. Let U(r_{i}) denote the potential energy of electron i in the centrally symmetric field of all the other electrons.
.
H=H_{0c}+H'. We treat the non-central part of the potential as a perturbation of H_{0c}. This perturbation is referred to as electron-electron correlations.
To solve H_{0c}y=Ey for the eigenfunctions and eigenvalues of H_{0c}, we assume that y is of the form y=y_{1}(r_{1})y_{2}(r_{2})… . Then the eigenvalue equation can be separated into equations of the form
.
For each electron we therefore have an equation of the form
.
The solutions are of the form y(r)=R(r)Y_{lm}(q,f)c(x), where R(r) is a solution to the radial equation
.
We have N independent equations. The central field approximation is therefore called an independent electron approximation.
To calculate U(r_{i}) due to the centrally symmetric field produced by all the other electrons, we must know the wave functions of all electrons.
Procedure:
The wave function y thus found is the optimum wavefunction that can result from a variational calculation with a trial function of the form y=y_{1}(r_{1})y_{2}(r_{2})….
In the central field approximation 2(2l_{1}+1)2(2l_{2}+1)2(2l_{3}+1) states, differing by the values of the quantum numbers m_{l} and m_{s} correspond to each configuration n_{1}l_{1}, n_{2}l_{2}, n_{3}l_{3},… . The non-central part of the electrostatic interaction between the electrons and the spin-orbit interaction lead to splitting of the level n_{1}l_{1}, n_{2}l_{2}, n_{3}l_{3},… into a number of sublevels.
We assume that the non-central part of the electrostatic interaction is much bigger than the spin-orbit interaction. (This is usually true for light multi-electron atoms.) The electrostatic interaction leads to a splitting of the level corresponding to a given electron configuration into a number of sublevels characterized by different values of the total orbital angular momentum of the electrons, L, and the total spin, S. The operator for the electrostatic interaction commutes with L=l_{1}+l_{2}+l_{2}+… and S=s_{1}+s_{2}+s_{3}+… . (See Cohen-Tannoudji, page1000). To each term LS belong (2L+1)(2S+1) states, differing by the values of M_{L} and M_{S}. The spin-orbit interaction leads to a splitting of the term LS into a number of components corresponding to different values of the total angular momentum J. But it does not completely remove the degeneracy. Each J component is degenerate with a multiplicity of 2J+1.
.
In the LS coupling scheme, a term is designated by ^{2S+1}L_{J}. 2S+1 is called the multiplicity of the term.
Which terms corresponding to a given configuration have the lowest energy?
Hund's Rule (established empirically)
np, n'p | [^{1}S] | n''p | ----> | ^{2}P |
np, n'p | [^{1}P] | n''p | ----> | ^{2}S, ^{2}P, ^{2}D |
np, n'p | [^{1}D] | n''p | ----> | ^{2}P, ^{2}D, ^{2}F |
np, n'p | [^{3}S] | n''p | ----> | ^{2}P, ^{4}P |
np, n'p | [^{3}P] | n''p | ----> | ^{2}S, ^{2}P, ^{2}D, ^{4}S, ^{4}P, ^{4}D |
np, n'p | [^{3}D] | n''p | ----> | ^{2}P, ^{2}D, ^{2}F, ^{4}P, ^{4}D, ^{4}F |
In brief form this is written as
^{2}S P D F | ^{4}S P D F |
2 6 4 2 | 1 3 2 1 |
The number under the term symbol indicates the number of identical terms.
Among the values of L and S obtained from the general rules for addition of angular momenta are those which correspond to states forbidden by the Pauli principle.
For each electron the following values are
possible:
m_{l}=1,0,-1, m_{s}=1/2,-1/2
Combining the different values of m_{l} and m_{s}, we
obtain the following possible states:
m_{l}=1 | m_{s}=1/2 | (1^{+}) |
m_{l}=0 | m_{s}=1/2 | (0^{+}) |
m_{l}=-1 | m_{s}=1/2 | (-1^{+}) |
m_{l}=1 | m_{s}=-1/2 | (1^{-}) |
m_{l}=0 | m_{s}=-1/2 | (0^{-}) |
m_{l}=-1 | m_{s}=-1/2 | (-1^{-}) |
In each of these states there cannot be more than one electron.
The following states with non-negative values of M_{L} and M_{s} are possible.
State | M_{L} | M_{s} |
(1^{+})(0^{+}) | 1 | 1 |
(1^{+})(-1^{+}) | 0 | 1 |
(1^{+})(1^{-}) | 2 | 0 |
(1^{+})(0^{-}) | 1 | 0 |
(1^{+})(-1^{-}) | 0 | 0 |
(0^{+})(1^{-}) | 1 | 0 |
(0^{+})(0^{-}) | 0 | 0 |
(-1^{+})(1^{-}) | 0 | 0 |
State with negative values of M_{L} and M_{s} need not be written out.
Thus only three terms are possible, ^{1}D, ^{3}P, and ^{1}S for the configuration np^{2}.
The statistical weight is the total number of terms pertaining to a given configuration.
For a configuration with no equivalent electrons the statistical weight is
.
For the configuration l^{n}, the statistical weight is determined by the number of possible combinations which can be formed from the quantum numbers m_{l}, m_{s} taking into account the Pauli principle. Let N_{0}=2(2l+1). The statistical weight of l^{n} is
.
We assume that the spin-orbit interaction considerably exceeds the non-central part of the electrostatic interaction. (This is usually true for heavy multi-electron atoms.) Then the energy depends primarily on how the orbital angular momentum l_{i} and the spin angular momentum s_{i} of each electron sum up into the total angular momentum j_{i} of each electron. j_{1}=l_{1}+s_{1}, j_{2}=l_{2}+s_{2}, … . The electrostatic interaction now leads to a splitting, depending on how the vectors j_{i} sum up into the total angular momentum J. In the jj coupling scheme the state of each electron is described by four quantum numbers, n, l, j, m_{j}, where j=l ±1/2.
1s_{1/2}, 2s_{1/2}, 2p_{1/2}, 2p_{3/2}, …
If we neglect the electrostatic energy completely, then the energy of each electron is determined by the quantum numbers nlj. The level corresponding to each j is 2j+1 fold degenerate.
^{}
_{possible terms} | ||
np_{1/2} , np_{1/2} | m_{1}=1/2, m_{2}=-1/2 | (1/2,1/2)_{0} |
np_{1/2} , np_{3/2} | m_{1}= ±1/2, m_{2}= ±1/2, ±3/2 | (1/2,3/2)_{2,1} |
np_{3/2} , np_{3/2} | m_{1}=3/2, m_{2}=1/2 m_{1}=3/2, m_{2}=-1/2 m_{1}=3/2, m_{2}=-3/2 m_{1}=1/2, m_{2}=-1/2 |
(3/2,3/2)_{2,0} |