Review: Simple spectra (one-electron atoms)

The eigenvalue equation for the Hamiltonian H₀ of a particle with potential energy -Ze²/r is

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Common eigenfunctions of H₀, L², and L_z are of the form y_nlm(r)=R_nl(r)Y_lm(q,f).

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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, .

Definitions:

n=1	ground state		ionization potential
n=2	resonance level		resonance potential

For hydrogen E_I=13.595eV, E_R=10.196eV.

Definition of shells:

Electric dipole transitions:

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

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with x_ab=<a|x|b>.

The selection rules are Dl=±1, Dm_l=0, ±1, Dn is not restricted.

Series regularities:

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R=109677.581cm^-1 for hydrogen.

Notation for ionized atoms:

I=neutral, II=singly ionized, III=doubly ionized, … .
(Example: Be_I, Be_II, Be_III, … .)

Fine Structure:

H=H₀+H_f.

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First order energy shifts:

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Dipole selection rules: Dj=0, ± 1,  Dm_j=0, ± 1.

Spectra of multi-electron atoms

The Hamiltonian H₀ for a multi-electron atom is

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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.

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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_0cy=Ey for the eigenfunctions and eigenvalues of H_0c, we assume that y is of the form y=y₁(r₁)y₂(r₂)… .  Then the eigenvalue equation can be separated into equations of the form

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For each electron we therefore have an equation of the form

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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

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We have N independent equations.  The central field approximation is therefore called an independent electron approximation.

The self-consistent field 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:

• Use a trial function of the form y=y₁(r₁)y₂(r₂)….
  • Assume some reasonable wave functions for the single electron wave functions, for example hydrogenic wave functions.
• Calculate U(r).
• Use U(r) in the eigenvalue equation for the Hamiltonian for each electron to calculate new wave functions for each electron using the variational method.
• Use these new wave functions to calculate U(r).
• Repeat the last two steps until self consistency is reached to a high order of accuracy.

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=y₁(r₁)y₂(r₂)….

Definitions:

• State:
  • The state of an atom is the condition of motion of all the electrons.  To specify a state, one must list four quantum numbers for each electron, for example n,l,m_l,m_s, or n,l,j,m_j.
• Level:
  • A level is a collection of states having the same energy in the absence of an external electric and magnetic field.
• Sublevel:
  • External fields split energy levels into several sublevels.
• Configuration:
  • A configuration is an enumeration of the values of n and l for all electrons of an atom.
• Equivalent orbitals:
  • Equivalent orbitals are orbitals with the same quantum numbers n and l.
• Coupling scheme:
  • A coupling scheme is a recipe for adding angular momenta.

In the central field approximation 2(2l₁+1)2(2l₂+1)2(2l₃+1) states, differing by the values of the quantum numbers m_l and m_s correspond to each configuration n₁l₁, n₂l₂, n₃l₃,… .  The non-central part of the electrostatic interaction between the electrons and the spin-orbit interaction lead to splitting of the level n₁l₁, n₂l₂, n₃l₃,… into a number of sublevels.

LS coupling

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=l₁+l₂+l₂+… and S=s₁+s₂+s₃+… .  (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.

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In the LS coupling scheme, a term is designated by ^2S+1L_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)

• The level with the largest multiplicity has the lowest energy.
• For a given multiplicity, the level with the largest value of L has the lowest energy.
• For less than half-filled shells:
  • The component with the smallest value of J has the lowest energy (normal order).
    Examples of less than half-filled shells: np², nd²
• For more than half-filled shells:
  • The component with the largest value of J has the lowest energy (inverted order).
    Examples of more than half-filled shells: np⁴, nd⁸
• When the number of electrons is 2l+1, i.e. when the shell is half filled, there is no multiplet splitting.

Example:

Finding the terms of a multi-electron configuration:

• Non-equivalent electrons:
  L=L₁+L₂, L₁+L₂-1, …, |L₁-L₂|
  S=S₁+S₂, S₁+S₂-1, …, |S₁-S₂|
  Addition is first carried out for two electrons, then the third is added, then the fourth, and so on.

  Examples:

  • Configuration np, n'p
    L=0,1,2, S=0,1
    The terms ¹S,¹P, ¹D, ³S, ³P, and ³D, are possible.
  • Configuration np, n'p, n''p

     

    np, n'p	[1S]	n''p	---->	2P
    np, n'p	[1P]	n''p	---->	2S, 2P, 2D
    np, n'p	[1D]	n''p	---->	2P, 2D, 2F
    np, n'p	[3S]	n''p	---->	2P, 4P
    np, n'p	[3P]	n''p	---->	2S, 2P, 2D, 4S, 4P, 4D
    np, n'p	[3D]	n''p	---->	2P, 2D, 2F, 4P, 4D, 4F

    In brief form this is written as

    2S P D F	4S P D F
    2 6 4 2	1 3 2 1

    The number under the term symbol indicates the number of identical terms.

• Equivalent electrons:

  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.

  Examples:

  • Configuration np²

    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:

    ml=1	ms=1/2	(1+)
    ml=0	ms=1/2	(0+)
    ml=-1	ms=1/2	(-1+)
    ml=1	ms=-1/2	(1-)
    ml=0	ms=-1/2	(0-)
    ml=-1	ms=-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	ML	Ms
    (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.

    • The presence of the M_L=2, M_s=0 term shows that a ¹D term is among the possible terms. To this term we must further assign states with M_L=1, M_s=0 and M_L=0, M_s=0.
    • Among the states left is a state with M_L=1, M_S=1. This and and states with  M_L=1, M_S=0, and M_L=0, M_S=1, and M_L=0, M_S=0 yield the ³P term.
    • The only remaining state has M_L=0, M_S=0. It corresponds to the ¹S term.

    Thus only three terms are possible, ¹D, ³P, and ¹S for the configuration np².

  • The possible terms for the configuration np³ are ⁴S, ²P, and ²D.
  • The possible terms for the configuration np⁴ are ¹D, ³P, and ¹S, the same terms as for the configuration np².

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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

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For the configuration lⁿ, 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 N₀=2(2l+1).  The statistical weight of lⁿ is

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jj coupling

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. j₁=l₁+s₁, j₂=l₂+s₂, … .  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.

Notation for each electron:

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.

Example:

• Configuration np, n'p

• Configuration np²

  		possible terms
  np1/2 , np1/2	m1=1/2, m2=-1/2	(1/2,1/2)0
  np1/2 , np3/2	m1= ±1/2, m2= ±1/2, ±3/2	(1/2,3/2)2,1
  np3/2 , np3/2	m1=3/2, m2=1/2 m1=3/2, m2=-1/2 m1=3/2, m2=-3/2 m1=1/2, m2=-1/2	(3/2,3/2)2,0