Subject: Three Electron Problem / ZINDO output

From: [email-masked] (Knut Asmis)
Date: Tue, 3 Sep 1996 13:50:41 +0200

Dear netters,

I encounter a problem when performing CI calculations on doublet
states with the ZINDO program. The problem is related to the so 
called "Three electron problem", which deals with the question:
"What are the wavefunctions describing the true spin states, 
e.g. eigenfunctions of the total spin operator S^2 and of the 
z-component spin operator Sz, for a system of three electrons and
three spaceorbitals".

My question doesn't concern the "three electron problem" itself,
but the way computational programs like ZINDO refer to Slater
determinants or in this case linear combinations of Slater
determinants. 

The ZINDO output generally relates the CI coefficients to the
appropriate Slater determinants (e.g. "antisymmetrized" electron
configurations) by a numbering scheme. For example the following 
scheme

(1)         1 2 3 4 5
            1 2 3 4

identifies the electron configuration, in which the spaceorbitals
1 to 4 are doubly occupied and the orbital 5 is occupied by a 
single electron with spin alpha.

What about the following case ?

(2)        1 2 3 4 6
           1 2 3 5

The Slater determinant, corresponding to an electron
configuration with three unpaired electrons in the spaceorbitals
4, 5 and 6 is not an eigenfunction of the total spin operator S^2.
Only the appropriate linear combination of Slater determinants is
an eigenfunctions of S^2. 

My question is: What linear combination of Slater determinants is
associated with the general numbering scheme (3)?

(3)      a c         
         b  

My answer would be: (3):   1/sqrt(2) * ( DET(aBc) - DET(Abc) )
 
In this case DET(aBc) refers to the Slater determinant, which is 
related to the electron configuration in which electron 1 
occupies orbital a and has spin alpha, electron 2 occupies 
orbital b and has spin beta (indicated by the capital letter),
and electron 3 occupies spaceorbital c and has spin alpha. 
DET(Abc) refers to the Slater determinant, which is related to
the electron configuration in which electron 1 occupies orbital a
and has spin beta, electron 2 occupies orbital b and has spin
alpha and electron 3 occupies spaceorbital c and has spin alpha.

Is this correct?

Thank you in advance for suggestions or answers,

Knut Asmis
Institut for Physical Chemistry
University of Fribourg
Switzerland

e-mail: [email-masked]