Lecture Note for Sept.25, 2002 (Wed.)

III. Nonrelativistic
Atomic Structure Theory for Complex Atoms

¡¤ Schrödinger eq.
for H solved in 1925

¡¤ Dirac eq. for H solved in 1928

¡¤ What do we know
about the wave functions of other atoms now?

¡¤ ¡°Almost exact¡± nonrelativistic
wave functions for bound states of He is known

¡¤ No relativistic wave functions for bound and

¡¤ Hamiltonian for He
contains *e*^{2}/*r*_{12}.

¡¤ Without this Coulomb
interaction term, the ¡°exact¡± solution for He is the product of two hydrogenic wave functions.

¡¤ With the Coulomb
interaction term, we do not know how to find the ¡°exact¡± solution

III-A. Hylleraas
Wave Functions for He

¡¤ Hylleraas wave
function is nonrelativistic

¡¤ Use powers of *r*_{12} in
addition to powers of *r*_{1} and *r*_{2}

¡¤ Example

*¥×*(1,2) = *r*_{12}{*r*_{1}*r*_{2}
exp[–¥æ(*r*_{1}+¥ç*r*_{2})] *Y*_{00}(1) *Y _{lm}*(2)

+
*r*_{1}*r*_{2}exp[–¥æ(¥ç*r*_{1}+*r*_{2})]
*Y _{lm}*(1)

¡¤ In practice, many powers of *r*_{1},
*r*_{2}, and *r*_{12} are used

¡¤ Hylleraas proposed
this type of wave function in 1929

¡¤ Pekeris used over
1,000 terms and obtained a very accurate ground-state wave function of He
(1957)

¡¤ The Hylleraas
method has been extended to Li-like systems successfully

¡¤ Main difficulty in extending the Hylleraas method to many-electron atoms is the
interdependent integration limits for *r _{ij}*
and

¡¤ So far, no one succeeded in using the Hylleraas method for a relativistic wave function of He

¡¤ The fundamental difficulty in solving the
Schrödinger equation for atoms with 2 or more electrons is the fact that *e*^{2}/*r*_{ij}
in the Hamiltonian cannot be treated as a perturbation to other terms in the
Hamiltonian

¡¤ To obtain the total energy for He as accurate as that from the best Hylleraas
wave function, more than 20th order perturbation theory must be used!

III-B. Hartree-Fock Method for Complex Atoms

¡¤ In addition to solving the Schrödinger
equation, Pauli exclusion principle
must be satisfied

¡¤ Pauli exclusion
prinsiple ¡æ electron exchange

*¥×*(1,2) = – *¥×*(2,1)

¡¤ Hartree¡¯s original
self-consistent field (SCF) theory did not satisfy Pauli
exclusion principle, and it was not based on the variational principle

¡¤ Hartree function
is the product of one-electron functions, referred to as orbitals

¡¤ Fock (1931?)
introduced antisymmetrized wave functions

¡¤ Slater (1931?) introduced the Slater
determinant to represent the Hartree-Fock (HF) wave
functions

¡¤ HF wave functions consist of one-electron orbitals, and each orbital is of the form:

*¥÷** _{nlm}*(

¡¤ Steps to derive the Hartree-Fock equation

(1) Construct Slater determinants with
appropriate one-electron orbitals

(2) Construct the total wave function *¥×*
by combining the determinants with Clebsch-Gordan
coefficients to represent desired ** L** and

(3) Derive the expression for the total
energy *E* = expectation value of the Hamiltonian using *¥×*
constructed in step (2)

(4) Carry out the integration of angular
variables in the expression for *E*

(5) The remaining expression is a function
of the radial orbitals, *P _{nl}*(

(6) Integration of the angular variables
results in various coefficients, known as the Slater
coefficients, which are functions of one-electron quantum numbers and the
total angular momentum quantum numbers, *L* and *S* [*I*(*ab*) and *R ^{k}*(

(7) One-electron integral *I*(*ab*) is associated with the one-electron operators in
the Hamiltonian [Eq. (35)]

(8) Two-electron integral *R ^{k}*(

(9) There are 3 kinds of two-electron
integrals:

*F ^{k}*(

(10) Apply the variational
principle:

*¥äE* = 0

(11) This will lead to a set of coupled
differential equations for radial functions *P _{nl}*(

(12) Combine the equations for *P _{nl}*(

*¡òdrP _{nl}*(

(13) Hartree-Fock equation
is a set of coupled, second order, inhomogeneous, integro-differential
equations [Eqs. (36)—(41)]

¡¤ Eq. (36) without
the right-hand-side is called the Hartree equation

¡¤ Hartree equation
is a homogenous equation, similar to the Schrödinger equation for H, but the
nuclear charge *Z* replaced by the screened nuclear charge *Z* – *Y _{nl}*

¡¤ HF theory is a typical independent particle model

¡¤ The Schrödinger Hamiltonian does not contain
spin-dependent operators

¡¤ When 2-dimensional spinors
for spin-up and spin-down are attached to the one-electron orbital, they are
called spin-orbitals

¡¤ Without a spin-dependent operator in the
Hamiltonian, HF wave functions for spin-up and spin-down cases should be the
same

¡¤ When orbitals with
different spin are allowed to be different (for instance, by introducing a
spin-dependent term in the Hamiltonian), such wave functions are called unrestricted HF wave functions

¡¤ In contrast, when orbitals
with different spin are assumed to be the same, such wave functions are called restricted HF wave functions

¡¤ When Slater determinants are combined to
form an eigenfunction of *L,*** S**,

¡¤ When ** L**,

¡¤ Combination of occupied orbitals
and their electron occupation numbers are called electron
configuration:

Examples:

1s for H

1s^{2}2s^{2}2p^{2}
for C

1s^{2}2s^{2}2p^{6}
for Ne

¡¤ For H and Ne, ** L**
and

¡¤ Question: Why are ^{1}P, ^{3}D,
and ^{3}S absent?

¡¤ Superscripts on the left are the spin multiplicity, 2*S*+1

¡¤ Capital letters denote *L*

All letters
in the alphabet, except for A, B, C, E, and J, are used

¡¤ Inconsistency:

*S* and *s* are
used for spin and ** L**,

*L*
and *l* are used for orbital angular momentum and
** L**,

M and m are
used for angular momentum projection and ** L**,

¡¤ Total angular momentum ** J** is
used to distinguish different states with the same

¡¤ When coupling 3 or more electrons, ** L**,

Example: 3d^{3}

¡¤ Steps to solve HF
equations

(1) Choose trial radial functions for *P _{nl}*

Examples: Thomas-Fermi functions; screened hydrogenic functions, solutions from previous calculations;
solutions from other theoretical models

(2) Radial functions can be a numerical
table as a function of *r*, or an analytic expression such as Legendre polynomials with screened nuclear charges

(3) Calculate the ¡°effective potential¡± terms,
*Y _{nl}*

(4) Solve Eq. (36)
with the effective potentials and get a new set of *P _{nl}*

(5) Compare the new set of *P _{nl}* with the set used in step (3)

(6) If the new set of *P _{nl}*
is not close to the old set of

(7)
Repeat steps (4)—(6) until the old and new sets of radial functions
agree within the desired convergence threshold

¡¤ The Hartree
equation is easy to solve, but the HF equation with *X _{nl}*
does not always converge to the desired accuracy

¡¤ Now, most atomic structure codes generate
numerical radial functions

¡¤ Most molecular structure codes generate
radial functions expressed in terms of analytic functions, known as the basis functions

¡¤ MCDF2002 code generates numerical radial
functions

¡¤ There are 4 popular atomic structure codes:

Nonrelativistic HF wave functions

(1) Charlotte Froese-Fischer
[Ref. 9], Vanderbilt Univ.

(2) Robert Cowan (very popular among experimental
spectroscopists), Los Alamos National Lab.

Relativistic
HF (= Dirac-Fock) wave functions

(3) Ian Grant,

(4) Jean-Paul Desclaux,
Grenoble

¡¤ Original authors of all these codes are
retired now! However, they are all
still involved in improving and modernizing their codes

¡¤ New directions for atomic structure codes

(1) Numerical solutions have convergence
problems, but fast

(2) Basis function method always converges,
but the accuracy of resulting wave functions strongly depends on the choice of
basis functions, and requires large memory and cpu time

(3) Computers are becoming faster, and
equipped with large memory

(4) New Challenge: Splines can be
used as the basis functions, and offers opportunity to combine the advantages
of the numerical solutions and the basis function method

(5) Developing a new atomic structure code
is a major commitment

III-C. Electron Correlation

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

¡¤ Another name for the electron correlation is
many-body effect