# CCSD and CISD size-consistency problem

I have performed a Gaussian16 calculation to explore the size-consistency problem for the CISD method. The system I am studying is the water dimer and the results are not what I expected at first. I have computed the energy of the dimer for both models (single point calculation) where the two monomers are 100 Å away and the energy of the monomer.

I wanted to compute the difference between those energies for both methods as two times the energy of the monomer and the energy of the dimer. If a method is size-consistent, the difference should be zero. These are my results:

$$\begin{array}{lrr} \hline \text{Quantity} & \text{CCSD} & \text{CISD} \\ \hline \text{Monomer energy}/\pu{Ha} & -75.013 & -75.012 \\ \text{Dimer energy}/\pu{Ha} & -150.030 & -150.026 \\ \text{Abs. difference}/\pu{Ha} & 0.004 & 0.002 \\ \hline \end{array}$$

I have been reading many articles and papers and what I expected is that CISD is not size-consistent for a system of more than two electrons while CCSD is size-consistent.

However, the value of the difference is larger in the case of CCSD, what makes me think that it is telling me that CCSD is less size-consistent for my problem than CISD. Am I wrong? What do these values mean?

This is the code I used in Gaussian for the distanced monomers:

CISD

#p CISD SP

water dimer CISD SP

0  1
O
H  1  r2
H  1  r3  2  a3
O  2  r4  1  a4  3  d4
H  4  r5  2  a5  1  d5
H  4  r6  2  a6  1  d6
Variables:
r2= 0.9700
r3= 0.9700
a3= 108.00
r4= 100.0000
a4= 175.00
d4= 179.97
r5= 0.9700
a5= 110.00
d5= 300.00
r6= 0.9700
a6= 110.00
d6=  50.00


CCSD

#p CCSD SP

water dimer CCSD SP

0  1
O
H  1  r2
H  1  r3  2  a3
O  2  r4  1  a4  3  d4
H  4  r5  2  a5  1  d5
H  4  r6  2  a6  1  d6
Variables:
r2= 0.9700
r3= 0.9700
a3= 108.00
r4= 100.0000
a4= 175.00
d4= 179.97
r5= 0.9700
a5= 110.00
d5= 300.00
r6= 0.9700
a6= 110.00
d6=  50.00


Your CISD and CCSD energies are practically the same, since you have not defined a basis set. It appears Gaussian still defaults to STO-3G in this case, although it is a minimal basis set, which usually do not yield even qualitatively correct results even at the SCF level of theory.

To be reliable, a post-HF calculation should employ basis sets of at least triple- to quadruple-zeta quality, preferably with extrapolation to the complete basis set limit.

Repeating the calculations in a non-minimal basis set gives me a CISD/cc-pVDZ energy of -152.44051461, and a CCSD/cc-pVDZ energy of -152.47534355; so you already see clearly that the CCSD energy is lower than the CISD one. Going to larger basis sets will result in a better capture of correlation energy (although the Hartree-Fock energy will also go down), and the difference between CCSD and CISD may still increase.

CISD is not size-consistent for more than two electrons, and CCSD is size-consistent and size-extensive for any number of electrons.

With CCSD, if you calculate the $$\ce{Li2}$$ energy for a bond length of 1000 Å, you should get the same thing as you would get by calculating the energy of a $$\ce{Li}$$ atom and multiplying it by two, and this would work for all "dimers" as long as the bond length is large enough, but will not always work with CISD.

In your case, the "dimer" is not of two atoms, but is of two molecules. I suppose an atom and a molecule are not much different, in that they are both just many-electron systems (ignoring cases like the H atom or its cation), though there is the difference that one's geometry is typically treated as just a point, versus the other one having a more complicated geometry (the impact of this should get smaller as the distance between the two "monomers" gets larger though).

Here is what I would recommend:

• Try again after switching from 100 Å to 1000 Å.
• Tell us your basis set and give us the XYZ or ZMAT file so that we can double-check to see that we're getting the same problem (perhaps also tell us if you're using RHF or UHF and RCCSD or UCCSD, etc.).

Also: why is your CCSD energy exactly the same as your CISD energy? Something must be wrong, and simply fixing this may fix your problem.

• Hi, thank you for your answer. I have corrected the values of the energies as well as included the gaussian input files I used to run the calculations. – Paul Logan Apr 12 at 9:12

The issue might be numerical as well. At theory level, CCSD could be rigorously prove to be size extensive. However, in practice, when one solves CC residue equation numerically, there is some convergence threshold that affect the accuracy of the result. Math is always simple and graceful but numerics can be complicated and nesty