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

Lorenzo Lodi l.lodi at ucl.ac.uk
Thu Sep 14 11:42:21 BST 2006

I inquired some days ago about the difference between dipole moments 
calculated as expectation values (XP) or as energy derivatives (ED). As 
I think this may be of interest for other people, I'd like to briefly 
summarize my conclusions, at the same time ask a couple of questions.
There is a some literature on the subject, starting from the paper 
(kindly pointed out to me by P. Knowles)

-- G.H.F. Diercksen, B.O. Roos, A.J. Sadlej, Legitimate Calculation of 
First-Order Molecular Properties in the Case of Limited CI Functions. 
Dipole Moments., Chem. Phys. 59, 29 (1981)

A couple of more recent papers discussing this issue in more details are:

-- J. Lipinski, On the consequences of the violation of the 
Hellmann-Feynman theorem in calculations of electric properties of 
molecules, Chem. Phys. Lett. 363, 313 (2002)
-- M. Ernzerhof, C.M. Marian, S.D. Peyerimhoff, On the calculation of 
first-order properties: expectation value versus energy derivative 
approach, Int. J. Quant. Chem. 43, 659 (1992)

For exact wavefunctions the dipole moment can defined either as the 
expectation value of \mu or as the energy derivative E'(\lambda) for the 
perturbed hamiltonian H=H0 + \lambda \mu
The two way of calculation lead exactly to the same numerical value.
In the practical case of approximate wavefunctions, the two methods 
don't necessarily give the same result. They do for some classes of 
approximate wavefunctions (Hartree-Fock, CASSCF, Full-CI) but not for 
many others (truncated CI, perturbation theory, coupled cluster).
In the cases where they differ there are some some reasons to favour the 
ED method (this is all the more true for second-order properties like 
electron polarizability, where the XP method may lead to wrong 
symmetries between the components) but, at the same time, there is no 
garantee that either the XP or the ED value is closer to the exact value.
Ideally the most sensible strategy I guess would be to converge the 
wavefunction to a level where the difference between XP and ED doesn't 
matter, but this may not be achievable in practice.

I quote some results to give an idea of the level of disagreement. I did 
some sample computations on water (r1=r2=0.95782 ang, theta=104.485) 
with a large aug-cc-pCV6Z basis set and looked at the differences XP-FF 
for the RHF, CCSD, CASSCF, MRCI, ACPF, AQCC and RSPT2 methods 
(CAS::OCC,5,2,2,0; CLOSED, 1,0,0,0; CORE,0,0,0,0 ; \lambda=+/- 0.00075).
I assumed that the value given by Molpro is calculated via XP (can 
anyone confirm this? It's quite important for me to know exactly how 
Molpro computes dipoles with the various methods)
Here are the differences \mu(XP) - \mu(FF) (in a.u.):
RHF     0.00002
CCSD    0.00001
MCSCF   0.00000
MRCI    0.00521
ACPF    0.00249
AQCC    0.00320
RSPT2   0.00204

So all the multi-reference methods show a variance at the level of ~3e-3 
a.u. with this CAS.
Any comment on the whole matter would be warmly welcomed!

Lorenzo Lodi

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