Lecture Note for Oct. 9, 2002 (Wed.)

 

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:

 

       Valence correlation:

 

              Correlation among valence electrons

              Example:  3s2 in Mg

 

       Core-valence correlation:

 

Correlation between valence and core electrons

Example:  3s in Na

 

       Core correlation:

 

              Correlation among core electrons

              Example: 2p6 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 1s2 2s2 1S0

 

       CM wave function for Be:  1s2 2s2 1S0 + 1s2 2p2 1S0

 

       BE = (1s2 2s2 1S0) = single configuration wave function

 

       BE =  a(1s2 2s2 1S0) + b(1s2 2p2 1S0) = 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:

 

                           a2 + b2 = 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

 

              2s2 and 2p2 for the Be ground state

 

              3s2, 3p2, and 3d2 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

 

       (2s1/2)2 and (2p1/2)2 for Be-like ions, but not (2p3/2)2

 

      (3s1/2)2 and (3p1/2)2 for the Mg-like ions, but not (3p3/2)2, (3d3/2)2, and (3d5/2)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 4f5/2 with the configurations with 4d5/2 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:

 

              Eav = ヒJ (2J+1)EJ  / ヒJ (2J+1)

 

     Config. average wave functions are not  eigenfunctions of J2, 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 J2, appropriate input data must be given

 

     These input data consist of the Slater integrals—which are integrals of 1/r12

 

     Appropriate Slater integrals are also generated internally in the MCDF2002 code by specifying J and MJ

 

     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 3P and 1P 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 42 = 16 component Hamiltonian and wave functions

 

       For uranium, 492 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, 1s2

 

       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メ

 

We do not know how to extend the Coulomb interaction, 1/r12, to include relativity to all orders in Zメ

 

       Relativistic Coulomb interaction to the order mc2(Zメ)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, e2/rij

 

     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 i and j which are one-electron eigenvalues before electron-electron interaction is activated

 

       Although the HF equation contains diagonal Lagrange multipliers 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 ij are called the frequency-dependent Breit operator.  This should be used with care!!

 

       One can eliminate ij from Eq. (46) by taking the limit ij ≧ 0.  Such Breit operator is called the frequency-independent Breit operator.  Usually this is the one used.

 

       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 mc2

 

       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 (l), and angular (j) momenta] are coupled in the jj coupling

 

     Closed shells in the jj coupling are:

 

       (s1/2)2; (p1/2)2; (p3/2)4; (d3/2)4; (d5/2)6; (f5/2)6; (f7/2)8; etc

 

     Shorthand notations for relativistic orbitals:

 

       s = s1/2

 

       p* = p1/2

 

       p = p3/2

 

       d* = d3/2

 

       d = d5/2

       f* = f5/2

 

       f = f7/2, etc.

 

       * ≧ j = l – 1/2

 

       no * ≧ 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:              1s2 2s2 2p2 3P

 

       Relativistic:  (core) [ (2p*)2 + 2p*2p + (2p)2 ]

 

       (2p*)2 = closed shell, J = 0

 

       2p* 2p = open shell, J = 1, 2

 

       2p2 = open shell, J = 0, 2 (why?)


     Nonrelativistic terms for C

 

       1s2 2s2 2p23P0,1,2; 1D2; 1S0

 

       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, 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 j

 

     According to the Dirac solution, the binding energies of the 2s and 2p1/2 states of H are degenerate

 

     QED corrections make 2p1/2 lower than 2s

 

     The excitation energy, E(1s—2p1/2), of U91+ 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メ


     For heavy atoms, Zメ 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