[molpro-user] Computation of the dipole moment with CCSD(T)

[Tatiana Korona]

Dear Lorenzo (and Peter),

Analytical CCSD first-order properties (but not gradients) were coded by 
me in 2001, see the testjob co_cordip.test and the paper 
Mol.Phys.100,1723,2002, there is also a few lines in the manual about 
this. CCSD(T) properties are not coded, here I agree with Peter.

For the CCSD case the main part of the correlation effect comes from the 
unrelaxed part, or, as it is written in the output, the "expectation" 
part, although the latter name is not entirely correct, see below. The 
unrelaxed dipole moment is calculated from the unsymmetrical formula, 
which looks like a kind of the "expectation value" expression, and this is 
why it is called so:

<left_eigenvector|{\bar DM}right_eigenvector>, where

{\bar DM}=exp(-T)DM exp(T),

and DM is a dipole moment operator (or any other one-electron operator). 
For the ground state CCSD we have right_eigenvector=HF (Hartree-Fock 
determinant), left_eigenvector=(1+Lambda)HF, where Lambda is the 
excitation operator of the same form as T. The relaxation part accounts 
for the orbital contributions, resulting from the orbital changes under 
perturbation, and usually is not important. Actually, if it IS important, 
your CCSD dipole moment is most probably far away from the true value and 
you need to look for a higher-correlated method. I suggest to read a 
chapter of J. Gauss in "Modern Methods and Algorithms of Quantum 
Chemistry", John von Neumann Institute for Computing, year 2000, for a 
short and clear introduction into the CC properties (available on-line).

CCSD first-order properties can be calculated by my program by using

{ccsd
expec,relax,dm}

but the relaxation part is coded for the all-electron case only (core,0).

Both relaxed and unrelaxed CCSD values can be reproduced by finite field 
calculations. For an example of a finite field unrelaxed calculation see 
the same testjob (assuming you use the molpro version 2006.1). Note that 
you need tigher thresholds for integrals, energy etc. and that it is 
better to use the 4-point formula instead of 2-point.

Returning to the terminology: THE TRUE expectation value expression can be 
found in any textbook of QCh:

                          <Psi| DM Psi> 
expectation_value_of_DM= ------------
                           <Psi|Psi>

and specifically for the coupled cluster case we have Psi=exp(T) HF, ALSO 
for the bra:

                          <HF| exp(T^dagger) DM exp(T) HF>
expectation_value_of_DM= -------------------------------
                           <HF|exp(T^dagger) exp(T) HF>

where T^dagger is the Hermitian adjoint of T.

By the way, I have coded this expression for CCSD recently, using 
approximations introduced by Jeziorski and Moszynski in the 
IJQC48,191,1993, see the testjob h2o_xccprop.test. Since the code is quite 
new, there is nothing about it in the manual.

Last but not least, it is debatable that the unrelaxed coupled cluster 
properties are less accurate than the relaxed ones. For the coupled 
cluster level there is usually only a small difference between them. One 
can argue that one should rather use unrelaxed properties, since dynamic 
properties (e.g. frequency-dependent polarizabilities) have unphysical 
poles originating from Hartree-Fock, if RELAXED second-order properties 
are computed.

Best wishes,

Tatiana


Dr. Tatiana Korona http://tiger.chem.uw.edu.pl/staff/tania/index.html
Quantum Chemistry Laboratory
University of Warsaw
Pasteura 1, PL-02-093 Warsaw, POLAND

`The man who makes no mistakes does not usually make anything.'
                                        Edward John Phelps (1822-1900)

On Fri, 8 Sep 2006, Peter Knowles wrote:

> I guess you need to read the literature to understand that ccsd and ccsd(t) 
> do not satisfy the Hellmann-Feynman theorem in the usual sense, and that for 
> methods in this class the energy derivative is more accurate than an 
> expectation-value formulation.
>
> Section 42.1 of the Molpro manual states that analytic energy derivatives are 
> available for qcisd and qcisd(t). Unfortunately CCSD and CCSD(T) are not yet 
> coded, so if you want the dipole with these methods you have to use finite 
> differences.  testjobs/co_qcidip.test gives an example.
>
> Peter
>
> Lorenzo Lodi wrote:
>> I am computing the dipole moment of water with various methods and I'd now 
>> like to use the CCSD(T) method.
>> 
>> At the CCSD level I can compute the dipole with
>> ---
>> RHF
>> CCSD ; CORE, 0,0,0,0
>> EXPEC, DM
>> ---
>> and I understand that the value given as "orbitally relaxed CCSD dipole 
>> moment" is the same value as the expectation value of \mu on the CCSD 
>> wavefunction. (Incidentally, the GEXPEC directive does not work for me, is 
>> this behaviour normal?)
>> 
>> Now, for the CCSD(T) dipole.
>> Looking at some old posts it was suggested to calculate it using the DIP 
>> keyword with a small field and then taking the derivative of the energy at 
>> zero field. Now, my objection is as follows: the Hellmann-Feynmann theorem, 
>> on which this finite-field approach is based, holds for
>> 1) the exact wavefunction
>> and
>> 2) a wavefunction which is variationally optimised in all its parameters
>> The coupled cluster method is, of course, non-variational so I don't expect 
>> the finite-field approach to work well in this case.
>> In fact, I verified this comparing the CCSD dipole obtained by finite-field 
>> and as given by molpro (equilibrium geometry, cc-pV6Z basis) and I got a 
>> difference of ~0.003 a.u., which is too large for the level of accuracy I 
>> am looking for.
>> 
>> My conclusion would be that there is no accurate way to compute the dipole 
>> with the CCSD(T) method (in fact, as there is no CCSD(T) wavefunction, this 
>> may not be anything new...).
>> 
>> Could anyone confirm these comments and/or give any suggestion about how to 
>> proceed in this direction?
>> 
>> Thank you.
>> 
>> Lorenzo Lodi
>> 
>
> -- 
> Prof. Peter J. Knowles
> School of Chemistry, Cardiff University, Main Building, Park Place, Cardiff 
> CF10 3AT, UK
> Telephone: +44 29208 79182 Fax: +44 2920874030
> Email KnowlesPJ at Cardiff.ac.uk  WWW 
> http://www.cardiff.ac.uk/chemy/staff/knowles.html
>