Example: Basis Set Superposition Error (BSSE): Cr(CO)5 +CO¶
A study of the Basis Set Superposition Error (BSSE) in the formation of Cr(CO)6 . from CO and Cr(CO)5 .
For the BSSE calculation special chemical elements must be created to describe the ‘ghost’ atoms, which have zero nuclear charge and mass. They do have basis functions (and fit functions), however, and they are used to calculate the lowering of the energy of the system to which the ghost atoms are added, due to the enlargement of the basis by the ghost basis. The ghost atom has the same basis and fit set as the normal element but no nuclear charge and no frozen core. The BASIS key recognizes elements denoted with Gh.atom in the ATOMS key as being ghost atoms. If the basis file specifies a frozen core ADF will treat it as if no frozen core is present.
The following calculations are carried out:
- CO from C and O. This yields the bond energy of CO with respect to the (restricted) basic atoms.
- CO from the fragments CO (as calculated in 1) and the ghost atom Cr and 5 Carbon and 5 Oxygen ghost atoms. The ghost atomic fragments provide basis and fit functions but do not contribute charge or potential to the molecule. The bond energy of this calculation is the energy lowering of CO due to the additional basis functions. This is the BSSE for CO.
- Cr(CO)5 from Cr and 5 CO’s. This yields the (‘normal’) bond energy with respect to the given fragments.
- Cr(CO)5 from Cr(CO)5 as fragment (as calculated in 3) but with the CO basis functions added on the position of the 6th CO (‘ghost’ CO). The bond energy is the BSSE for Cr(CO)5 .
- Cr(CO)6 with Cr(CO)5 and CO as fragments. The bond energy is the one without BSSE. This bond energy can now be corrected by the sum of superposition contributions of calculations 2 and 4.
This series of calculations is carried out with basis set DZ.
Finally, the whole thing might be redone with basis set TZP, to show that the BSSE decreases with larger basis.
The calculations for the type DZ basis are contained in the sample script (with input- and output files). Those for type TZP bases can be set up easily and may be done as an exercise.
For the first series of calculations, with basis type DZ, the input files are discussed below and the relevant parts are echoed from the output files where the energy decomposition and the total bond energy are printed.
For the other series, using type TZP basis sets, only a summary of the results is given.
Computational details
The calculations in this example all use:
- Small core DZ basis set Frozen core level for the Chromium atom: 2p, for Carbon and Oxygen: 1s
- Numerical integration precision 4.0 (in Create runs 10.0, the default)
- Default settings for model parameters such as density functional (key XC) and for the remaining computational settings
For the BSSE calculations we first do the ‘normal’ calculations of CO and Cr(CO)5 , yielding the fragment files t21.CO and t21.CrCO5. The input files for these calculations are not shown here.
BSSE for CO
For the CO BSSE calculation the standard CO must have been computed first. In the BSSE run a Cr(CO)5 ghost fragment basis set is then added to the ‘normal’ CO input. Important is the use of the BASIS key. In this case the BASIS key is used for the generation of the ghost atoms, it should have the same definition for the atoms as will be used later for the Cr(CO)5 fragment. The FRAGMENTS key is used for the fragment CO. The energy change (the printed ‘bond energy’ in the output) is the BSSE.
The input file for the CO-BSSE run is:
title BSSE for CO due to Cr(CO)5 ghost
noprint sfo,frag,functions
atoms
Gh.Cr 0 0 0
Gh.C -1.86 0 0
Gh.C 1.86 0 0
Gh.C 0 1.86 0
Gh.C 0 -1.86 0
Gh.C 0 0 -1.86
Gh.O 3.03 0 0
Gh.O -3.03 0 0
Gh.O 0 3.03 0
Gh.O 0 -3.03 0
Gh.O 0 0 -3.03
C 0 0 1.86 f=CO
O 0 0 3.03 f=CO
end
Basis
Type DZ
Core Small
end
fragments
CO CO.t21
end
symmetry C(4V)
integration 4
endinput
In the output we find in the Bond Energy section:
hartree eV kcal/mol kJ/mol
-------------------- ----------- ---------- -----------
Pauli Repulsion
Kinetic (Delta T^0): 0.000000000000009 0.0000 0.00 0.00
Delta V^Pauli Coulomb: -0.000000000000007 0.0000 0.00 0.00
Delta V^Pauli LDA-XC: -0.000000000000003 0.0000 0.00 0.00
-------------------- ----------- ---------- -----------
Total Pauli Repulsion: -0.000000000000001 0.0000 0.00 0.00
(Total Pauli Repulsion =
Delta E^Pauli in BB paper)
Steric Interaction
Pauli Repulsion (Delta E^Pauli): -0.000000000000001 0.0000 0.00 0.00
Electrostatic Interaction: -0.000000000000017 0.0000 0.00 0.00
(Electrostatic Interaction =
Delta V_elstat in the BB paper)
-------------------- ----------- ---------- -----------
Total Steric Interaction: -0.000000000000018 0.0000 0.00 0.00
(Total Steric Interaction =
Delta E^0 in the BB paper)
Orbital Interactions
A1: -0.001838638722848 -0.0500 -1.15 -4.83
A2: 0.000000000000000 0.0000 0.00 0.00
B1: 0.000000000000000 0.0000 0.00 0.00
B2: 0.000000000000000 0.0000 0.00 0.00
E1: -0.002025936656647 -0.0551 -1.27 -5.32
-------------------- ----------- ---------- -----------
Total Orbital Interactions: -0.003864575379498 -0.1052 -2.43 -10.15
Alternative Decomposition Orb.Int.
Kinetic: -0.056036605580477 -1.5248 -35.16 -147.12
Coulomb: 0.048666195764206 1.3243 30.54 127.77
XC: 0.003505834436773 0.0954 2.20 9.20
-------------------- ----------- ---------- -----------
Total Orbital Interactions: -0.003864575379498 -0.1052 -2.43 -10.15
Residu (E=Steric+OrbInt+Res): -0.000000000000003 0.0000 0.00 0.00
Total Bonding Energy: -0.003864575379519 -0.1052 -2.43 -10.15
Summary of Bonding Energy (energy terms are taken from the energy decomposition above)
======================================================================================
Electrostatic Energy: -0.000000000000017 0.0000 0.00 0.00
Kinetic Energy: -0.056036605580468 -1.5248 -35.16 -147.12
Coulomb (Steric+OrbInt) Energy: 0.048666195764196 1.3243 30.54 127.77
XC Energy: 0.003505834436770 0.0954 2.20 9.20
-------------------- ----------- ---------- -----------
Total Bonding Energy: -0.003864575379519 -0.1052 -2.43 -10.15
The BSSE for CO is computed as 2.43 kcal/mole
BSSE for Cr(CO)5
In similar fashion the BSSE is computed for Cr(CO)5 . In the BSSE run a ghost atoms C and O at the positions they will have in the Cr(CO)6 molecule are added to the normal Cr(CO)5 input:
title BSSE for Cr(CO)5 due to CO ghost
noprint sfo,frag,functions
atoms
Cr 0 0 0 f=CrCO5
C 1.86 0 0 f=CrCO5
C -1.86 0 0 f=CrCO5
C 0 1.86 0 f=CrCO5
C 0 -1.86 0 f=CrCO5
C 0 0 -1.86 f=CrCO5
O 3.03 0 0 f=CrCO5
O -3.03 0 0 f=CrCO5
O 0 3.03 0 f=CrCO5
O 0 -3.03 0 f=CrCO5
O 0 0 -3.03 f=CrCO5
Gh.C 0 0 1.86
Gh.O 0 0 3.03
end
Basis
Type DZ
Core Small
end
fragments
CrCO5 CrCO5.t21
end
symmetry C(4v)
integration 4
endinput
The Bond Energy result yields 1.93 kcal/mole for the BSSE.
Bond Energy calculation with BSSE correction
The bonding of CO to Cr(CO)5 is computed in the normal way: from fragments CO and Cr(CO)5 . The obtained value for the bond energy can then simply corrected for the two BSSE terms, 4.36 kcal/mole together.
Relevance of Core Functions
The two BSSE runs (#2 and #4 in the list above) can also be repeated, but now with the core orthogonalization functions omitted from the ghost bases. To to this one can not use the BASIS key, but one needs to explicitly ‘create’ the ghost atoms. This will not be done here, but only the results will be discussed. One may argue about whether these functions should be included in the ghost basis sets, but since they are very contracted around the ghost nuclei they are not expected to contribute significantly anyway and may then just as well be omitted. This has been explicitly verified by test examples. /p The Core Functions (the functions in the valence basis set that serve only for core-orthogonalization, for instance the 1S 5.40 in the Carbon basis set (see the $ADFHOME/atomicdata/DZ/C.1s basis set file) are removed from the Create data files used for the creation of the ghost atoms. This yields as BSSE values for CO and Cr(CO)5 respectively 2.33 and 1.87 kcal/mole (compare 2.43 and 1.93 kcal/mole for the case with Core Functions included). The net total effect of including/removing the Core Functions is therefore (2.43-2.33)+(1.93-1.87)=0.16 kcal/mole. This is an order of magnitude smaller than the BSSE effect itself.
BSSE and the size of the basis set
BSSE effects should diminish with larger bases and disappear in the limit of a perfect basis. This can be studied by comparing the BSSE for basis DZ, see above, with the BSSE for basis TZP. The procedure is completely similar to the one above and yields: For the BSSE terms: 0.7 kcal/mole for CO (compare: 2.4 kcal/mole for basis DZ), and 0.6 kcal/mole for Cr(CO)5 (1.9 for basis DZ) The total BSSE drops from 4.4 kcal/mole in basis DZ to 1.3 in basis TZP.
Reference
A systematic study with adf of the BSSE in metal-carbonyl complexes can be found in Rosa, A., et al., Basis Set Effects in Density Functional Calculations on the Metal-Ligand and Metal-Metal Bonds of Cr(CO)5-CO and (CO)5. Journal of Physical Chemistry, 1996, 100: p. 5690-5696.