Cr(CO)5+CO: Basis Set Superposition Error
Sample directory: adf/BSSE_CrCO6/
A study of the Basis Set Superposition Error (BSSE) in the formation of Cr(CO)6. from CO and Cr(CO)5.
The basis set superposition error (BSSE) can be calculated with the help of the option to create Alternative Chemical Elements. or Ghost atoms.
An alternative chemical element is an element with a special feature, not corresponding to one of the predefined chemical elements. It may have, for instance, a different effective nuclear charge or a 'special' atomic mass.
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 (there must be no core description in the Create data file for the ghost atom!).
The following calculations are carried out:
- 1. CO from C and O. This yields the bond energy of CO with respect to the (restricted) basic atoms.
- 2. 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. - 3. Cr(CO)5 from Cr and 5 CO's.
This yields the ('normal') bond energy with respect to the given fragments. - 4. 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. - 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.
Next, the two BSSE runs (#2 and #4 in the list above) are repeated, but now with the core orthogonalization functions omitted from the ghost bases. 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 is explicitly verified in the current example by demonstrating that the BSSE is not significantly affected by omitting these functions.
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:
- 1. Frozen core level for the Chromium atom: 2p (for Carbon and Oxygen: 1s);
- 2. Numerical integration precision 4.0 (in Create runs 10.0, the default);
- 3.
Default settings for model parameters such as density
functional (key XC)
and for the remaining computational settings
Basis DZ, including Core Functions
Creation of ghost atoms
Ghost atoms must be created like normal chemical elements. The adf database does not provide the ghost database files. They are easily constructed from the normal database file of the pertaining chemical element: only the frozen core references have to be adapted such that the ghost atom will not have a frozen core. This affects the sections 'core' and 'description' in the database file (see the User's Guide).
For the creation of the Carbon ghost atom with basis DZ the database file is:
Carbon (II, ghost) BASIS 1S 5.40 2S 1.24 2S 1.98 2P 0.96 2P 2.20 END CORE 0 0 0 0 END DESCRIPTION END FIT 1S 10.80 2S 11.59 2S 7.59 2S 4.97 3S 4.79 3S 3.35 3S 2.34 3S 1.64 2P 8.34 2P 5.14 3P 4.67 3P 3.10 3P 2.06 3D 5.88 3D 3.84 3D 2.51 3D 1.64 4F 5.40 4F 3.55 5G 4.50 END FITCOEFFICIENTS / END
Observe that there are four integers zero after the keyword core, indicating that there are no s-, p-, d-, or f-type frozen core shells. Specification of any frozen core shells would imply the insertion of (core) electrons around the ghost atoms in the calculation.
Consequently, the data block directly below core is empty: no Slater-type functions are required to describe any frozen core orbitals.
Finally, the description data block is empty: no expansion coefficients that would describe the frozen core orbitals in terms of the Slater-type expansion functions.
All other data (apart from the title, which is just a label) in the Create data file are unchanged. The ghost file has the same Basis set, the same Fit set as for a normal atom. The values of the fit coefficients are irrelevant and could as well be put zero altogether: in the scf part of the create run on the ghost atom the fit coefficients will be set to zero after the first cycle since there is no charge density to be fitted.
Then the corresponding Create run is carried out.
$ADFBIN/adf -n1 << eor Create Gh.C q=0 m=0 file=in.ghost end input eor mv TAPE21 t21.C_ghost
The options 'q=' and 'm=' specify the nuclear charge and atomic mass respectively. Both are zero for a ghost atom: it is not a physical object, only the center for a set of functions.
In the same fashion the Oxygen and Chromium ghost atoms are created. The inputs for these are not shown here.
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. 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
fragments
Gh.Cr t21.Cr_ghost
Gh.C t21.C_ghost
Gh.O t21.O_ghost
CO t21.CO
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): 0.000000000000025 0.0000 0.00 0.00
Delta V^Pauli Coulomb: -0.000000000000021 0.0000 0.00 0.00
Delta V^Pauli LDA-XC: -0.000000000000007 0.0000 0.00 0.00
-------------------- ----------- ---------- -----------
Total Pauli Repulsion: -0.000000000000004 0.0000 0.00 0.00
(Total Pauli Repulsion =
Delta E^Pauli in BB paper)
Steric Interaction
Pauli Repulsion (Delta E^Pauli): -0.000000000000004 0.0000 0.00 0.00
Electrostatic Interaction: 0.000000000000057 0.0000 0.00 0.00
(Electrostatic Interaction =
Delta V_elstat in the BB paper)
-------------------- ----------- ---------- -----------
Total Steric Interaction: 0.000000000000054 0.0000 0.00 0.00
(Total Steric Interaction =
Delta E^0 in the BB paper)
Orbital Interactions
A1: -0.001838637105191 -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: -0.002025936206895 -0.0551 -1.27 -5.32
-------------------- ----------- ---------- -----------
Total Orbital Interactions: -0.003864573312086 -0.1052 -2.43 -10.15
Alternative Decomposition Orb.Int.
Kinetic: -0.056036607909737 -1.5248 -35.16 -147.12
Coulomb: 0.048666200804427 1.3243 30.54 127.77
XC: 0.003505833793224 0.0954 2.20 9.20
-------------------- ----------- ---------- -----------
Total Orbital Interactions: -0.003864573312086 -0.1052 -2.43 -10.15
Residu (E=Steric+OrbInt+Res): -0.000000000000002 0.0000 0.00 0.00
Total Bonding Energy: -0.003864573312034 -0.1052 -2.43 -10.15
Summary of Bonding Energy (energy terms are taken from the energy decomposition above)
======================================================================================
Electrostatic Energy: 0.000000000000057 0.0000 0.00 0.00
Kinetic Energy: -0.056036607909712 -1.5248 -35.16 -147.12
Coulomb (Steric+OrbInt) Energy: 0.048666200804404 1.3243 30.54 127.77
XC Energy: 0.003505833793217 0.0954 2.20 9.20
-------------------- ----------- ---------- -----------
Total Bonding Energy: -0.003864573312034 -0.1052 -2.43 -10.15
The BSSE for CO is computed as 2.42 kcal/mole
BSSE for Cr(CO)5
In similar fashion the BSSE is computed for Cr(CO)5. In the BSSE run a ghost CO is 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 fragments CrCO5 t21.CrCO5 Gh.C t21.C_ghost Gh.O t21.O_ghost 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 (not included in the sample): from fragments CO and Cr(CO)5. The obtained value for the bond energy is then simply corrected for the two BSSE terms, 4.35 kcal/mole together.
Relevance of Core Functions
The whole procedure explained above is repeated with now 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) 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 1.88 kcal/mole (compare 2.42 and 1.93 kcal/mole for the case with Core Functions included). The net total effect of including/removing the Core Functions is therefore (2.42-2.33)+(1.93-1.88)=0.14 kcal/mole. This is an order of magnitude smaller than the BSSE effect itself.
In the last calculation a PRINT instruction is inserted in the input file to let the program output the symmetry group representations, character table and multiplication table. This information is printed after the lists of basis and fit sets.
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, using basis sets with Core Functions included: 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)
Without Core Functions the numbers are similar.
The total BSSE drops from 4.3 kcal/mole in basis DZ to 1.3 in basis TZP (if Core Functions are included in the Create runs for the ghosts), and changes very slightly when the Core Functions are omitted.
Create runs for the ghosts), and changes very slightly when the Core Functions are omitted.
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
