Subject: Summary: Off-diagonal elements of the CI matrix - part 2

From: "Gustavo Moura" <[email-masked]>
Date: Thu, 20 Aug 1998 13:28:46 -0400

Dear CCL readers,
	after my last weeks summary I have received two more answers to the
question that I posted two weeks ago. Below you can find the two new answers
that I have received. My thanks to Darko Babic and James J. P. Stewart for
answering my question.
	Sincerely yours,
			Gustavo Moura
		   [email-masked]

Answer 1:

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	Hi!

	I thought you'll get many answers on this question and I didn't want
waste my time.  Probably many others thought the same, or it's the vacations
 ...

	There is no UNIQUE way of permuting orbitals in one sequence to match
the other one, but EACH way always includes either even or odd number of
transpositions.  This is all you have to know.  If there is an odd number of
permutations you must multiply the integral by -1, otherwise by +1.

	To determine the parity of the permutation, one may, for example,
count the number of inversions in it.  For each element in the permutation
count the number of preceding elements that are 'greater', and sum these
numbers up.  For example: |dbac> should be permuted to |abcd>.  Start with
the second element since there is NO element before the first one.  "b" is
the second element and it has "d" before it, which is 'greater' than "b",
so we start with 1.  "a" has two elements before: "d" and "b" and both are
'greater' than "a", so we add 2.  "c" has three elements before but only "d"
is 'greater' than "c", so we add 1 and the final result is 1+2+1=4=even.

	If the two sequences are not identical, say |abcd> and |bdaq>, you
can easily decide the ideal position of the different orbitals ("q" is to
be where "c" is), with some meaningless arbitrariness when they differ in
two orbitals.

	The above way of determining the permutation parity is not the only
one, but probably the most frequently used.

				Best wishes,
								Darko Babic
							Institute "Rudjer
Boskovic"
							HR-10001 Zagreb, POB
1016
								  Croatia

Answer 2:

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Dear Dr Moura,

The problem with determinants is also found in semiempirical
configuration interaction.  In the MOPAC 93 Manual, I describe
a procedure for bringing two (spin) determinants into maximum
coincidence, and working out the phase factor.

This problem is also given as FORTRAN code in the MECI function
of MOPAC.  However, it is not trivial...

Jimmy Stewart