Subject: Re: CCL:CC versus CI

From: Allen Adler <[email-masked]>
Date: Fri, 30 Oct 1998 01:08:37 -0600

Let me apologize in advance.

I'm walking into this discussion in the middle and understand
neither the terminology, the acronymns nor the context. However,
I registered some recognition upon reading Steven Creve's remark:

>Truncated CCSD does *not* generate all excitations. By making use of the
>hausdorff comutator expansion, one can show that the exp-series terminates
>exactly after a certain number of orders.

in regard to the expression CCSD (exp(T1+T2)). Being a mathematician,
I usually run across the Campbell-Baker-Hausdorff formula in connection
with Lie groups and Lie algebras. So now I'm trying to guess whether
(1) Steven Creve is saying, in effect, that T1 and T2 are operators that
    generate a Lie algebra which is nilpotent.
(2) David Heisterberg was, in effect, guessing that the algebra was
    either abelian or 2-step nilpotent.

Anyway, now I'm curious to know what it actually is that is being
discussed, on the off chance that I might understand it. Specifically,
(a) what are T1,T2?
(b) what is the definition of the expression CCSD (exp(T1+T2))?
(c) where is the result on the nilpotence of the Lie algebra proved?
(d) what do all the acronyms stand for?
(e) in what readily accessible work is all this explained? I may be able to
    get my hands on a copy of the 1989 edition of Ostland and Sczabo.
    Is it there?

Naively,
Allan Adler
[email-masked]

---

From: Ron Shepard <[email-masked]>
Date: Fri, 30 Oct 1998 17:19:07 -0500

>> Suppose we perform a CI calculation with all determinants that are
>> normally generated by CCSD (exp(T1+T2)). Would the obtained result be
>> worse or better than the CCSD value?
>> 
>> Any comments are welcome?

This problem is not phrased exactly correctly.  The set of determinants
generated by exp(T1) itself is the full set of determinants, so if a CI
calculation were done in this full space the answer would be the full-CI
energy, which is the best that can be done in that finite basis.  Adding
the T2 operator is redundant in this context, since the determinants that
it generates are also generated by the T1^2 terms in the exponential. 

>    You ask interesting questions, Steven, and this one I think I can
>answer.
>
>    Both CISD and CCSD represent the correlated wavefunction |corr> as a
>sum over configuration state functions.  The difference between them is
>that CI exploits the variational prinicple to minimise the energy, thus 
>
>  CI:     <E> = <corr|H|corr>
>
>ensuring a strict upper bound to the total energy, while CC solves the
>Schroedinger equation within a subspace.  As a consequence, the computed 
>
>  CC:     H|corr> = E|corr>
>
>energy may be greater than or less than the true energy.  Use of the
>cluster operator to parameterise |corr> results in the implicit
>estimation of contributions from higher excitations ( quadruples, etc )
>so the energy in practice tends to be quite accurate.  

This is correct, but there are other important differences in the CI and
CC energy expressions too.  It is not that CC represents the wave
function as an expansion in CSFs, it is that the CC represents the wave
function as a nonlinear contraction of CSFs.  So another important
difference between CC and CI is that in CI the individual CSFs each have
their own coefficient, but in CC each individual CSF has a coefficient
that is constructed, in a nonlinear way, as sums of products of the T1
and T2 coefficients.

Consider for example the contribution of a particular 4-fold excitation
CSF in the two cases.  In CI, this CSF would have its own coefficient,
which would be optimized variationally.  But in CC, this coefficient
would be determined as a sum of T2*T2+T2*T1*T1+T1*T1*T1*T1 terms.  In the
CI case there would be a lot of independent parameters, which means an
expensive calculation, while in the CC case the number of parameters is
limited, which means a relatively cheap calculation.

Then, there is the philisophical question of exactly what is in the CC
wave function.  If you think of the CC wave function as exp(T), then you
might think that there are all possible excitations.  But if you look at
the way the parameters T1 and T2 are computed, then you will see that
there are only 4-fold excitations included.  Since the energy is computed
from only T1 and T2, and since they are determined only from the 4-fold
excitations, you might also say that the energy itself depends only on
4-fold excitations.  Sound like a paradox?  I believe that you just have
to meditate on this for a while before you can make peace with it. :-)

$.02 -Ron Shepard