Basis set superposition error in the DFT

The basis set superposition error (BSSE, also known as counterpoise correction, see [Boys:1970]) can be estimated by performing subsystem calculations using ghost atoms which carry basis sets of the full system.

This tutorial describes how to use ghost atoms in DIRAC and how to avoid certain pitfalls when working with ghost centers. As an example we shall consider the helium dimer, having linear symmetry \(D_{\infty h}\) . For the counterpoise correction calculation one beryllium atom is replaced by a ghost atom without charge, but carrying the full beryllium basis. The symmetry is now reduced to \(C_{\infty v}\) . DIRAC can automatically detect and exploit the linear symmetry, thus providing computational savings. In the case of DFT there is, however, a complication: The ghost center needs not only the basis of a beryllium atom, but also the same numerical grid. This will force us to reduce the symmetry of the molecular calculation to \(C_{\infty v}\) . When carrying out the counterpoise correction we shall furthermore have to make sure that atoms are in the same place as in the molecular calculation. Below we see how to handle these issues, first with the use of xyz-files, next using mol-files.

Another matter is that conventional DFT based on local functionals is not a good choice for the description of a van der Waals complex like the helium dimer. We shall therefore combine short-range DFT with MP2.

Using mol-files

We start with the molecular calculation. Without consideration of counterpoise we would use

INTGRL
Helium dimer
...
C   1              A
     2.0     2
He1  0.0 0.0  0.0
He2  0.0 0.0  2.98
LARGE BASIS aug-cc-pVDZ
FINISH

However, DIRAC will not detect the full molecular symmetry \(D_{\infty h}\) , which is different from the counterpoinse system and so the same numerical grid can not be used. We therefore add a ghost center, with no basis, to break inversion symetry.

INTGRL
Helium dimer + ghost
...
C   2              A
     2.0     2
He1  0.0 0.0  0.0
He2  0.0 0.0  2.98
LARGE BASIS aug-cc-pVDZ
     0.0     1
Gh   0.0 0.0  10.0
LARGE 0
FINISH

The input file for our calculation combining MP2 with short-range LDA reads

**DIRAC                                                                                             
.TITLE                                                                                              
 Helium dimer                                                                                       
.WAVE FUNCTION                                                                                      
**INTEGRALS                                                                                         
*READIN                                                                                             
.UNCONT                                                                                             
.NORTSD
**HAMILTONIAN                                                                                       
.DFT                                                                                                
CAM p:alpha=0.0 p:beta=1.0 p:mu=0.40 x:slater=1.0 c:srvwn5=1.0                                      
**WAVE FUNCTIONS                                                                                    
.SCF                                                                                                
.MP2                                                                                                
*SCF                                                                                                
.CLOSED SHELL                                                                                       
4                                                                                                   
*MP2CAL                                                                                             
.VIRTHR                                                                                             
40.00                                                                                               
*END OF       

Note a final touch: We use the keyword NORTSD to avoid that our molecule is moved (placing origin at center of mass) during symmetry detection. We run the molecular calculation using:

pam --inp=MP2srLDA --mol=He2Gh --get=numerical_grid

Note that we save the file containing the numerical grid. Naming the ghost center ‘Gh’ assures that no grid is generated for it, as can be seen from the output:

Atom  Deg   Rmin        Rmax        Step size     #rp     #tp
=============================================================
He1     1   0.998E-05   0.157E+02   0.123E+00      99    8552
He2     1   0.998E-05   0.157E+02   0.123E+00      99    8552

We now turn to the counterpoise calculation. We replace one helium atom by a ghost atom, but this time we want a helium basis set for it. We therefore specify

INTGRL
Helium atom + ghost with basis
...
C   2              A
     2.0     1
He1  0.0 0.0  0.0
LARGE BASIS aug-cc-pVDZ
     0.0     1
Gh   0.0 0.0  2.98
LARGE BASIS Q=2.0 aug-cc-pVDZ
FINISH

DIRAC uses nuclear charge to find basis sets from libraries so we have to add “Q=2.0” for the ghost atom in the mol-file to provide this information. Note also that we again call the ghost atom ‘Gh’, but it will get a grid since we import it from the previous calculation. The corresponding menu file reads

**DIRAC                                                                                             
.TITLE                                                                                              
 Helium dimer                                                                                       
.WAVE FUNCTION                                                                                      
**INTEGRALS                                                                                         
*READIN                                                                                             
.UNCONT                                                                                             
.NORTSD
**GRID
.IMPORT
numerical_grid
**HAMILTONIAN                                                                                       
.DFT                                                                                                
CAM p:alpha=0.0 p:beta=1.0 p:mu=0.40 x:slater=1.0 c:srvwn5=1.0                                      
**WAVE FUNCTIONS                                                                                    
.SCF                                                                                                
.MP2                                                                                                
*SCF                                                                                                
.CLOSED SHELL                                                                                       
2                                                                                                   
*MP2CAL                                                                                             
.VIRTHR                                                                                             
40.00                                                                                               
*END OF       

We have adjusted the electron number to that of the atom and also ask for the import of the numerical grid. We run the calculation using:

pam --inp=MP2srLDA_cp --mol=HeGh --put=numerical_grid

Using xyz-files

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Symmetry recognition

If you leave the symmetry text in the molecule specification blank, DIRAC will try to evaluate the symmetry of the system automatically. Note that the symmetry recognition uses the atom types and coordinates but not the respective basis sets. This means that if you use ghost atoms with different basis sets, you should give the symmetry explicitly or check the symmetry recognized by DIRAC.