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:
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 (
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.
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. When comparing the plot above to Norlis results for the same system, one should consider the following:
(1) The plot above spans a larger region along both axes compared to Norli's plot.
(2) The basis sets used differ (STO-3G vs. 6-311++G(2p,2d))
This may account for the fact that the correlation converges above -1 a.u., but a more comparable calculations could possibly shed some more light on the issue.
| 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 [ ]:
for(k=0; k<nElectron, k++)
for(a=nElectron, k<nStates, k++)