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 free states of He and nonrelativistic wave functions for free states of He

 

     Hamiltonian for He contains e2/r12.

 

     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 r12 in addition to powers of r1 and r2

 

     Example

 

       (1,2) = r12{r1r2 exp[–(r1+r2)] Y00(1) Ylm(2)

      

       + r1r2exp[–(r1+r2)] Ylm(1) Y00(2)}

 

     In practice, many powers of r1, r2, and r12 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 rij and ij

 

     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 e2/rij 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 electron exchange

 

              (1,2) = – (2,1)

 

     Hartrees 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(r,,) = r–1Pnl(r)Ylm(,)

 

     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 S

      

(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, Pnl(r)  [See Eq. (34) of the lecture note]

 

       (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 Rk(abcd) in Eq. (34)]

 

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

 

       (8)  Two-electron integral Rk(abcd) is associated with the Coulomb interaction e2/r12

 

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

 

       Fk(abab);     Gk(abba);     Rk(abcd)

      

(10)  Apply the variational principle:

      

                     E = 0

 

       (11)  This will lead to a set of coupled differential equations for radial functions Pnl(r) [Eq. (36)]

 

       (12)  Combine the equations for Pnl(r) with the orthonormality conditions for the radial functions using Lagrange multipliers

 

              drPnl(r)Pnl(r) = nn

 

       (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 ZYnl

 

     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, ML, and MS, the resulting wave function is called a term-dependent HF wave function

 

     When L, S, ML, and MS are not specified, it is called an average configuration HF wave function

 

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

 

       Examples:

 

       1s for H

 

       1s22s22p2 for C

 

1s22s22p6 for Ne

 

     For H and Ne, L and S are unique, but for C, 2p2 can lead to 3P, 1D, and 1S

 

     Question:  Why are 1P, 3D, and 3S absent?

 

     Superscripts on the left are the spin multiplicity, 2S+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 = 0

 

       L and l are used for orbital angular momentum and L, l = 8

 

       M and m are used for angular momentum projection and L, l = 9

 

     Total angular momentum J is used to distinguish different states with the same L and S

 

     When coupling 3 or more electrons, L, S, and J may not be enough to distinguish all allowed states.  Then the seniority quantum number is introduced: 

Example:      3d3


     Steps to solve HF equations

 

       (1)  Choose trial radial functions for Pnl

 

       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, Ynl and Xnl in Eq. (36)

 

       (4)  Solve Eq. (36) with the effective potentials and get a new set of Pnl

 

(5)  Compare the new set of Pnl with the set used in step (3)

 

       (6)  If the new set of Pnl is not close to the old set of Pnl within the desired accuracy, then construct a new set of effective potentials with the new set of Pnl

 

        (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 Xnl 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, Oxford Univ.

 

       (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