Lecture Note for

III-C. Electron Correlation (Nonrelativistic)

¡¤ The difference between the ¡°exact¡± solution
and the solution based on independent particle model (such as the Hartree-Fock wave functions) is called electron correlation

¡¤ Another name for electron correlation is many-body effects

¡¤ Many-body effects appear both in nonrelativistic and relativistic atomic structure

¡¤ For neutral and
lightly-charged ions, nonrelativistic correlation
effects are the most important corrections to the Hartree-Fock
solutions, followed by relativistic corrections. QED
effects are very small

¡¤ For inner shells of
heavy atoms and highly charged ions, relativity is the most important
corrections to the nonrelativistic HF solutions. QED corrections can be as important as
electron correlation

¡¤ The most difficult task in atomic structure
theory is to accurately account for electron correlation

¡¤ There are 3 types of electron correlation:

Correlation
among valence electrons

Example: 3s^{2} in Mg

Core-valence correlation:

Correlation between valence
and core electrons

Example: 3s in Na

Core correlation:

Correlation
among core electrons

Example:
2p^{6} in Ar

¡¤ Different approaches are used to account for
different types of electron correlation

¡¤ There is no magic
solution to account for all types of electron correlation, for to do so
is to solve the Schödinger equation exactly

¡¤ Two most successful theories for electron
correlation are many-body perturbation theory (MBPT)
and configuration mixing

¡¤ MBPT has been
adapted from nuclear physics

¡¤ MBPT can be applied if the initial state can
be expressed by a single Slater determinant

Example: Rare gas atoms, alkali atoms

¡¤ MBPT needs a ¡°complete¡± set of unperturbed,
basis set functions, discrete and continuum

Example: Hartree
functions (i.e., do not satisfy Pauli exclusion principle), or replace the Fock
part by a local potential, such as the Slater approx.

¡¤ Slater approx.
replaces the exchange (=Fock) terms in the Hartree-Fock equation by a function of the total charge
density [see Eqs. (42) and
(43) of the lecture note]

¡¤ There are different forms of the Slater
approx.

¡¤ The Slater approx. is better known as the
Kohn-Sham approx., a density functional theory

¡¤ The Kohn-Sham theory is popular in condensed
matter theory, but does not help atomic structure theory

¡¤ The Slater approx. was made popular by
Herman and Skillman, who published a book in early 1960s containing nonrelativistic ground-state wave functions and matching
local potentials for all atoms in the periodic table

¡¤ The Herman-Skillman potential and wave
functions are easy to use, and therefore suitable for mass production and
survey work, but not for high-precision predictions

¡¤ Since the Herman-Skillman potential is
local, it is easy to generate a complete set of wave functions, a product of
numerical radial functions and spherical harmonics

¡¤ For neutral atoms, 3rd or higher order
perturbation theory is needed for accurate predictions of atomic properties

¡¤ For highly-charged ions, 2nd order
perturbation theory is sufficient to generate reliable atomic data

¡¤ Configuration Mixing
(CM) is often called configuration interaction
(CI)

¡¤ Configuration Mixing
is the most popular, versatile method to account for electron
correlation

¡¤ MCDF2002 code uses relativistic CM
method. Users
of MCDF2002 must be familiar with the procedures to construct accurate wave
functions using the CM method

¡¤ As usual, there are different ways to
implement the CM method

¡¤ CM uses occupied configurations and
unoccupied correlation configurations to represent electron correlation

Example: Ground state of Be
is 1s^{2} 2s^{2} ^{1}S_{0}

CM wave
function for Be:
1s^{2} 2s^{2} ^{1}S_{0} + 1s^{2}
2p^{2} ^{1}S_{0}

*¥×*_{BE}
=* ¥÷(*1s^{2}
2s^{2} ^{1}S_{0}) = single configuration wave function

*¥×*_{BE}
=_{ } *a¥÷*(1s^{2} 2s^{2} ^{1}S_{0})
+ *b¥÷*(1s^{2} 2p^{2} ^{1}S_{0})
= multiconfiguration wave function

¡¤ Mixing coefficients *a* and *b*
are determined by the variational method, i.e., by
minimizing the total energy

¡¤ In general, the coefficient *a* for the
occupied configuration will be much larger than the coefficient *b* for
the correlation configuration

¡¤ The total wave function is normalized to
unity:

*a*^{2}
+ *b*^{2} = 1

¡¤ Correlation configurations must have the
same total parity and angular momentum (*J* for relativistic wave
functions, *L* and *S* for nonrelativistic
wave functions) as the occupied configurations

¡¤ In principle, there are infinitely many
candidates for correlation configurations.

¡¤ However, the most important ones are those
with the same principal quantum numbers and *j*¡¯s as the occupied configuration

2s^{2}
and 2p^{2} for the Be ground state

3s^{2},
3p^{2}, and 3d^{2} for the Mg ground state

¡¤ These configurations are hydrogenically
degenerate in energy, i.e., the sum of their hydrogenic
energies is the same. They belong to a Coulomb complex

¡¤ For highly-charged ions

(2s_{1/2})^{2}
and (2p_{1/2})^{2} for Be-like ions, but not (2p_{3/2})^{2}

(3s_{1/2})^{2}
and (3p_{1/2})^{2 }for the Mg-like ions, but not (3p_{3/2})^{2},
(3d_{3/2})^{2}, and (3d_{5/2})^{2}

¡¤ For neutral atoms, the nonrelativistic
mixing of configurations with 4f with the configurations with 4s, 4p, and 4d is
weak

¡¤ For highly-charged ions, the relativistic
mixing of configurations with 4f_{5/2} with the configurations with 4d_{5/2}
is strong

¡¤ In the CM method, we can either make both the
mixing coefficients and the radial functions self consistent, or just the
mixing coefficients while keeping the radial functions frozen

¡¤ The former is known as the CM with relaxed orbitals

¡¤ The latter is known as the CM with frozen orbitals

¡¤ The CM with relaxed orbitals
produces compact and superior wave functions, in principle

¡¤ In practice, however, wave functions with
too many correlation configurations may have numerical convergence problems

¡¤ To avoid numerical convergence problems, it
is common to generate radial functions using a local potential from the Slater
approx. or configuration average radial
functions, and keep them fixed while the configuration mixing coefficients are
determined by variational method

¡¤ Configuration
average means the total energy is a weighted sum of all possible *J*¡¯s
that can be constructed from the configurations included in a wave function:

E_{av} = ¥Ò* _{J}* (2

¡¤ Config. average wave functions are
not eigenfunctions
of *J*^{2}, but their radial functions converge easily

¡¤ In most Hartree-Fock wave function codes (both relativistic and nonrelativistic), config. average total energy is automatically calculated by the
codes

¡¤ To make a total wave function to be an eigenfunction of *J*^{2}, appropriate
input data must be given

¡¤ These input data consist of the Slater
integrals—which are integrals of 1/*r*_{12}

¡¤ Appropriate Slater integrals are also
generated internally in the MCDF2002 code by specifying *J* and *M _{J}*

¡¤ When a configuration leads to a unique
angular quantum number, such as the rare gas atoms and alkali atoms,
configuration average total energy is the same as the *J*-specific total
energy

¡¤ For the valence correlation, multiconfiguration wave functions with relaxed, *J*-specific
radial functions can be used

¡¤ However, for the core-valence and core
correlation, *J*-specific, relaxed radial functions do not converge easily
because the radial size of the valence and core orbitals
is very different, preventing them to have significant overlap and large matrix
elements

¡¤ It is very important to have good working
experience in choosing correlation configurations appropriate to individual
cases

¡¤ To get good energy levels, an extensive list
of correlation configurations (often > 1,000) must be used.

¡¤ For most collision cross sections,
properties of valence orbitals play a decisive role,
and it is important to use a small number of well-chosen relaxed radial orbitals to represent the valence correlation

¡¤ For accurate results, term-dependent (= *J* specific) radial functions
must be used

Example: 2s2p ^{3}P and ^{1}P
radial functions for Be are very different!!

IV. Relativistic Atomic Structure Theory for
Complex Atoms

¡¤ Relativistic theory for many-electron atoms
contain many theoretical problems

No
relativistic Hamiltonian for many-electron atoms

No
relativistic Hamiltonian for the Lamb shift, even for H atom

Rigorous
relativistic theory for He should use 4^{2} = 16 component Hamiltonian
and wave functions

For uranium, 4^{92} component wave functions?

Negative energy
band introduces dilemma known as the Brown-Ravenhall ¡°disease¡±

Relativistic
ground state for He should have infinitely many degenerate states with one
electron in the positive energy band and another in the negative energy band,
while its total energy is the same as the ground state with both electrons
bound, 1s^{2}

Real He atom does not disintegrate into electrons in the positive
and negative continuum bands

Relativistic
effects in atoms and molecules are classified by the powers of *Z¥á*

We do not know how to extend
the Coulomb interaction, 1/*r*_{12}, to include relativity to all
orders in *Z¥á*

Relativistic
Coulomb interaction to the order *mc*^{2}(*Z¥á*)^{4}
is given by the Breit interaction [see Eq. (46) of lecture note]. Higher order terms are not known
analytically

IV-A. Dirac-Fock Method

¡¤ In analogy to the nonrelativistic
Hartree-Fock method, we can construct relativistic Hartree-Fock method

Use the Dirac Hamiltonian for one-electron Hamiltonian [see Eq. (48)], after subtracting the rest mass [– 1 in Eq. (45)]

Use
extended nucleus [point nucleus is an option]. Analytic relativistic solutions for H
exist only for a point nucleus

Use the Breit operator for relativistic correction to
electron-electron interaction, *e*^{2}/*r _{ij}*

¡¤ Breit Operator [Eq. (46)]

Came from a
2nd-order perturbation theory, not a real operator like the Dirac
Hamiltonian

The first
term in [¡¦] is the magnetic interaction term

The second
term in [¡¦] is the retardation term

Since the Breit operator came from a 2nd-order perturbation theory,
it contains *¥å _{i}* and

Although
the HF equation contains diagonal Lagrange multipliers *¥å _{i}*
that look like energy eigenvalues, one-electron
energies in a many-electron atom are not physically meaningful!!

This is
another dilemma. Nevertheless, the Breit operator with *¥ø _{ij}*
are called the frequency-dependent Breit operator.
This should be used with care!!

One can
eliminate *¥ø _{ij}* from Eq. (46) by taking the limit

Exercise:
Derive the frequency-independent Breit
operator

¡¤ Since the Breit
operator is based on a perturbation theory, it is not included in the
Hamiltonian, and hence not subject to variational
principle

¡¤ In the course of deriving the Breit operator, it was assumed that the energy between the
two interacting electrons was far less than mc^{2}

This
condition is violated if one of the electrons is in the negative energy state

This is why
the Breit operator should not be subjected to a variational procedure that lets one of the electrons be in
the negative energy band

¡¤ If one wants to include
the Breit operator in the relativistic Hamiltonian
and apply the variational principle, then the eigenfunctions of the Hamiltonian
should exclude the negative energy solutions

¡¤ Projection operators are used to exclude
solutions belonging to a certain class.
Such a projection operator can be defined formally, but is very difficult
to find one that can be used in calculating wave functions

¡¤ Since an electron with a negative energy is
equivalent to a positron, a Hamiltonian with a projection operator to leave out
the negative energy states are also called a ¡°no
virtual pair (of *e*^{–} and *e*^{+})¡± Hamiltonian

¡¤ The relativistic HF equation, better known
as the Dirac-Fock equation, is derived essentially in
the same manner as that used for deriving the HF equation [see Eq. (48)]

¡¤ The Dirac-Fock
equation consists of a set of coupled, first order integro-differential
equations linking large and small component radial functions in two-dimensional
matrices

¡¤ One-electron angular momenta [spin (** s**), orbital (

¡¤ Closed shells in the *jj* coupling are:

(s_{1/2})^{2};
(p_{1/2})^{2}; (p_{3/2})^{4}; (d_{3/2})^{4};
(d_{5/2})^{6}; (f_{5/2})^{6}; (f_{7/2})^{8};
etc

¡¤ Shorthand notations for
relativistic orbitals:

s = s_{1/2}

p* = p_{1/2}

p = p_{3/2}

d* = d_{3/2}

d = d_{5/2}

f* = f_{5/2}

f = f_{7/2}, etc.

* ¡æ *j* = *l*
– 1/2

no * ¡æ *j* = *l* + 1/2

¡¤ Conversion between the *LS*
coupling and *jj*
coupling is unique only when there are two electrons. For three or more electrons, careful
matching is required

¡¤ Multiconfiguration
Dirac-Fock wave functions are necessary even for
simple atomic structures that require single configuration Hartree-Fock
wave functions

Example: Carbon

NR: 1s^{2}
2s^{2} 2p^{2} ^{3}P

Relativistic: (core) [ (2p*)^{2}
+ 2p*2p + (2p)^{2} ]

(2p*)^{2} = closed shell, *J* = 0

2p* 2p =
open shell, *J* = 1, 2

2p^{2}
= open shell, *J *= 0, 2 (why?)

¡¤ Nonrelativistic
terms for C

1s^{2}
2s^{2} 2p^{2} ¡æ ^{3}P_{0,1,2};
^{1}D_{2}; ^{1}S_{0}

Total
number of *J* states is the same in the *jj* and *LS* couplings

¡¤ Exercise: How many nonrelativistic orbitals in
Hg? How many relativistic orbitals in Hg?

¡¤ To calculate accurate
excitation/ionization energies, take the difference between total energies
before and after excitation/ionization

¡¤ As an approximation,
orbital energies, *¥å _{i}*, may be used
as ionization energies for inner-shell electrons. This is known as the Brillouin
theorem. The Brillouin theorem
should not be used for the outermost electron.

IV-B. QED Corrections

¡¤ Quantum electrodynamic (QED) corrections refer to physical effects
not covered by the Dirac equation. There are many effects, but they are
collectively referred to as the Lamb shift

¡¤ Change in the total
energy due to the finite size of the nucleus is not a QED effect by the above
definition. However, since the
exact solution of the Dirac equation for the H atom
is known only for a point nucleus, the finite nuclear size correction is
included in QED corrections. This
is numerically the largest correction,
particularly for heavy atoms

¡¤ Two largest QED
corrections are:

Self energy correction: Shift in the binding energy of an
electron due to the exchange of virtual photons with the nucleus

Reduces the
binding energy, and the largest QED correction

Vacuum polarization correction: Shift in the binding energy of an
electron due to the creation and annihilation of virtual *e*^{–} *e*^{+}
pairs in the vacuum

Increases the
binding energy—most of the time, the second largest correction

¡¤ QED corrections apply to all atoms, molecules, at all levels

¡¤ QED corrections are
most important for heavy atoms, inner shells, and low ** l** and

¡¤ According to the Dirac solution, the binding energies of the 2s and 2p_{1/2}
states of H are degenerate

¡¤ QED corrections make 2p_{1/2}
lower than 2s

¡¤ The excitation energy, *E*(1s—2p_{1/2}), of U^{91+} is shifted
by more than 70 eV due to the Lamb shift

¡¤ Many QED corrections
are calculated internally by the MCDF2002 code and included in the total energy

¡¤ For light atoms, QED
corrections are often calculated using a perturbation expansion in powers of *Z¥á*

¡¤ For heavy atoms, *Z¥á*
is not very small and perturbation expansions cannot be used. Non-perturbative
QED theory is an active field of current research

¡¤ There are different
methods to estimate QED corrections.
The MCDF2002 code offers the most reliable estimates, particularly for
heavy atoms. It is important that
the same methods for QED corrections be used before
taking the difference between total energies to calculate excitation/ionization
energies