Benchmarking the CCSolver

In this notebook I will do some Coupled-Cluster benchmark calculations using a Slater Determinant produced from a Restricted Hartree-Fock (RHF) calculation on an STO-3G basis set for the hydrogen gas (H2) and a water molecule (H2O).

The water molecule

We have previously done some calculations on H2O using RHF and STO-3G to determine geometry and ground state energy. The results corresponded rather nice to the ones presented in the literature (see Report from FYS4411 - Gøran Brekke Svaland and Audun Skau Hansen).

We now use the geometry and energy presented in the following document to compare our Coupled Cluster Singles Doubles (CCSD) solver to the corresponding calculations performed by NWChem: http://institute.loni.org/NWChem2012/documents/tce-session.pdf.

Prior to the CCSD calculation the different algorithms starts out with the following parameters:

HF-energy (NWChem) : -74.962663062148

HF-energy (CCSolve): -74.962677575226

HF-limit : -76.067 (http://chemistry.illinoisstate.edu/standard/che460/handouts/460performance.pdf)

The results from each iteration is presented in the table below. The rightmost coulomn contains the results obtained by CCSolve.

Iter Correlation(NWChem) Correlation (CCsolve) Rel.Error
1 -0.0358672469179 -0.035897626185118 0.0008469919
2 -0.0454068882657 -0.04544804888198 0.000906484
3 -0.0483870059027 -0.048432634274069 0.0009429881
4 -0.0494370597647 -0.049484831256215 0.0009663093
5 -0.0498391184890 -0.049888079084351 0.0009823728
6 -0.0500021724029 -0.050051825093841 0.0009930107
7 -0.0500711904756 -0.050121251581979 0.0009997986
8 -0.0501014381364 -0.050151741527804 0.0010040309
9 -0.0501150974135 -0.05016554465157 0.0010066276
10 -0.0501214303300 -0.050171962746426 0.0010081998
11 -0.0501244348663 -0.050175017473014 0.0010091407
12 -0.0501258887096 -0.050176500709333 0.0010096978
13 -0.0501266039080 -0.050177233008851 0.0010100246
14 -0.0501269605251 -0.050177599511074 0.0010102146
15 -0.0501271402835 -0.050177784948453 0.0010103242
16 -0.0501272316751 -0.050177879584002 0.0010103871
17 -0.0501272784536 -0.05017792820593 0.0010104229
18 -0.0501273025229 -0.050177953317881 0.0010104433
19 -0.0501273149581 -0.050177966340254 0.0010104547
20 -0.0501273214031 (converged) (none)

CCSolve obtains a lower ground state energy already at the onset from the SCF-procedure. This is also reflected in both the first iteration of the CCSD procedure and the final converged correlation energy.

Revisiting the Hydrogen Molecule

We spent some time dealing with this simple diatomic system in FYS4411, and found our code to produce some nice results. (See report from FYS4411 - Gøran Brekke Svaland and Audun Skau Hansen). As the RHF procedure forces the electrons into spin-orbitals, the method is not well suited for calculating the energy of H2 as the bondlength increase beyond the point of electron interaction.

In our new calculation we wish to perform a CCSD or (CCD) for H2 to investigate how the energy compares to RHF as we increase the bondlength. The results are shown in the plot below.


In [7]:
from IPython.display import Image
h2 = Image(filename='H2bondlength.png')
h2


Out[7]:

As the plot above shows, we see that we get the expected behaviour from the CCSD solver as we increase the distance between the nucelis.

General result comparison

System Total energy (a.u.) Correlation (a.u.) Parameters Comparison
H2 -1.13728 -0.0205616 Bond length: 1.4, STO-3G, RHF+CCSD HF-limit: -1.132 (2), Exact correlation -0.03969 (1)
O2 -147.696 -0.145003 Bond lenght: 2.287, STO-3G, RHF+CCSD HF-limit: -, Exact correlation -0.37 (1)
O -73.7092 -0.047385 STO-3G, RHF+CCSD HF-limit-74.729 (2), Correlation: -0.262 (2)
Be -14.4037 -0.0517703 STO-3G, RHF+CCSD General HF: -14.67 (3)
Be -14.5199 -0.00484722 Hydrogenlike (4 functions), RHF+CCSD General HF: -14.67 (3)
He -2.84228 -0.00868762 Hydrogenlike (4 functions), RHF+CCSD General HF: -2.904 (3)

(1) Results from Thijssen, p84, obtained from variational calculus

(2) results from 460performance.pdf (see text above)

(3) Source given in "Report from FYS4411 - Gøran Brekke Svaland and Audun Skau Hansen"


In [ ]: