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Basis-Sätze (basis sets)

Molekülorbitale werden im Allgemeinen durch eine Linearkombination von Atomorbitalen dargestellt (LCAO-MO). Dabei werden die Atomorbitale durch eine Summe von Gaussfunktionen gebildet (grafische Darstellung).
Der einfachste Basissatz wird als STO-3G (Slater Type Orbitals) bezeichnet. Dabei werden die AO's durch 3 Gaussfunktionen gebildet. Minimaler Basissatz bedeutet, dass nur so viele AO's verwendet werden, um die Elektronen des neutralen Atoms unterzubringen.
Die entsprechende Befehlsgruppe im Input-File lautet: 

  $BASIS GBASIS=STO NGAUSS=3 $END

Die in GAMESS eingebauten Basissätze sind:

$BASIS group
   
This group allows certain standard basis sets to be easily given. If this group is omitted, the basis set must be given instead in the $DATA group (externer Basissatz) 

GBASIS = Name of the Gaussian basis set.

      
= MINI - Huzinaga's 3 gaussian minimal basis set.
               
Available H-Rn.
      
= MIDI - Huzinaga's 21 split valence basis set 
                Available H-Xe.
      
= STO  - Pople's STO-NG minimal basis set.
                Available H-Xe, for N=2,3,4,5,6 
      
= N21  - Pople's N-21G split valence basis set. 
               
Available H-Xe, for NGAUSS=3. 
                Available H-Ar, for NGAUSS=6.
      
= N31  - Pople's N-31G split valence basis set.
               
Available H-Ne,P-Cl for NGAUSS=4.
               
Available H-He,C-F for NGAUSS=5.
               
Available H-Ar, for NGAUSS=6.
               
For Ga-Kr, N31 selects the BC basis.
      
= N311 - Pople's "triple split" N-311G basis set.
               
Available H-Ne, for NGAUSS=6.
    
           Selecting N311 implies MC for Na-Ar.
      
= DZV  - "double zeta valence" basis set.
               
a synonym for DH for H,Li,Be-Ne,Al-Cl.
               
a synonym for BC for Ga-Kr.
     
 = DH   - Dunning/Hay "double zeta" basis set.
               
(3s)/[2s] for H.
               
(9s,4p)/[3s,2p] for Li.
               
(9s,5p)/[3s,2p] for Be-Ne.
               
(11s,7p)/[6s,4p] for Al-Cl.
   
   = BC   - Binning/Curtiss "double zeta" basis set.
               
(14s,11p,5d/[6s,4p,1d] for Ga-Kr.
      
= TZV  - "triple zeta valence" basis set.
               
(5s)/[3s] for H.
               
(10s,3p)/[4s,3p] for Li.
               
(10s,6p)/[5s,3p] for Be-Ne.
               
a synonym for MC for Na-Ar.
               
(14s,9p)/[8s,4p] for K-Ca.
               
(14s,11p,6d)/[10s,8p,3d] for Sc-Zn.
  
    = MC   - McLean/Chandler "triple split" basis.
               
(12s,9p)/[6s,5p] for Na-Ar.
               
Selecting MC implies 6-311G for H-Ne. 

       * * * the next two are ECP bases only * * * 

GBASIS = SBK  - Stevens/Basch/Krauss/Jasien/Cundari
               
valence basis set, for Li-Rn.  This choice
               
implies an unscaled -31G basis for H-He.
      
= HW   - Hay/Wadt valence basis.
               
This is a -21 split, available Na-Xe,
               
except for the transition metals.
               
This implies a 3-21G basis for H-Ne. 

        * * * semiempirical basis sets * * * 

The elements for which these exist can be found
in the 'further information' section of this
manual.  If you pick one of these, all other data
in this group is ignored. 

GBASIS = MNDO - selects MNDO model hamiltonian
      
= AM1  - selects AM1 model hamiltonian

      
= PM3  - selects PM3 model hamiltonian  

NGAUSS = the number of Gaussians (N).   This parameter
         pertains only to GBASIS=STO, N21, N31, or N311. 

NDFUNC = number of heavy atom polarization functions to
        
be used.  These are usually d functions, except
 
       for MINI/MIDI.  The term "heavy" means Na on up
        
when GBASIS=STO, HW, or N21, and from Li on up
        
otherwise.  The value may not exceed 3.  The
        
variable POLAR selects the actual exponents to
        
be used, see also SPLIT2 and SPLIT3. (default=0) 

NFFUNC = number of heavy atom f type polarization
        
functions to be used on Li-Cl.  This may only
        
be input as 0 or 1.  (default=0) 

NPFUNC = number of light atom, p type polarization
        
functions to be used on H-He.  This may not
        
exceed 3, see also POLAR.  (default=0) 

DIFFSP = flag to add diffuse sp (L) shell to heavy atoms.
        
Heavy means Li-F, Na-Cl, Ga-Br, In-I, Tl-At.
        
The default is .FALSE. 

DIFFS  = flag to add diffuse s shell to hydrogens.
        
The default is .FALSE. 

POLAR  = exponent of polarization functions
      
= POPLE     (default for all other cases)
      
= POPN311   (default for GBASIS=N311, MC)
      
= DUNNING   (default for GBASIS=DH, DZV)
      
= HUZINAGA  (default for GBASIS=MINI, MIDI)
      
= HONDO7    (default for GBASIS=TZV) 

SPLIT2 = an array of splitting factors used when NDFUNC or NPFUNC is 2.
        
Default=2.0,0.5 
SPLIT3 = an array of splitting factors used when NDFUNC or NPFUNC is 3. 
        
Default=4.00,1.00,0.25           ==========================================================
  
The splitting factors are from the Pople school, and are probably too far apart. See for example the Binning and Curtiss paper. For example, the SPLIT2 value will usually cause an INCREASE over the 1d energy at the HF level for
hydrocarbons. The actual exponents used for polarization functions, as well as for diffuse sp or s shells, are described in the 'Further References' section of this manual.  This section also describes the sp part of the basis set chosen by
GBASIS fully, with all references cited.  
Note that GAMESS always punches a full $DATA group.  Thus, if $BASIS does not quite cover the basis you want, you can obtain this full $DATA group from EXETYP=CHECK, and then change polarization exponents, add Rydbergs, etc.

Erläuterungen:

split valence basis set: In these bases the AO's are split into two parts: an inner, compact orbital and an outer, expanded orbital. The coefficients of these two kinds of orbitals can be varied independently during the construction of the MO. 


STO-3-21G
means that the core orbitals are represented by three Gaussians, whereas the inner and outer valence orbitals consist of two and one Gaussian, respectively.

STO-3-311G means a triply split basis, with an inner orbital represented by 3 Gaussian, and middle and outer orbitals represented as single Gaussians.  
Basis sets that similarly split the core orbitals are called
double zeta, DZ, and triple zeta, TZ.
Further improvement of basis function is achieved by adding d-orbitals to all heavy (non-hydrogen) atoms. This allows a shift of the center of an orbital away from the position of the nucleus. For example, a p-orbital on carbon can be
polarized away from the nucleus by mixing into it a d-orbital of lower symmetry.

One obvious place where this can improve results is in the modeling of small rings. Compounds of second-row elements also are more accurate described by the inclusion of polarization. The presence of polarization is indicated in the Pople notation by appending an asterisk to the set designator. Thus,
STO-3-21G* implies the previously described split valence basis with polarisation added. An alternative description of this kind of basis is DZP, Double Zeta Polarization. A second asterisk, as in STO-6-31G**, implies the addition of a set of p-orbitals to each hydrogen to provide for their polarization. 
A further improvement is achieved if the basis is augmented with so-called
diffuse functions. These are intended to inprove the basis set at large distances from the nuclei, thus better describing the barely bound electron in anions or neutral molecules with unshared pairs. Typically, the augmentation takes the form of a single set of very diffuse (exponents from 0.1 to 0.01) s and p orbitals. The presence of diffuse functions is symbolyzed by the addition of a plus sign, +, to the basis set designator: STO-6-31+G. Again, a second plus implies diffuse functions added to hydrogens.   

Die Zusammensetzung der obigen Basissätze sieht man am besten, wenn man sich den Output für ein einfaches Molekül, mit dem jeweiligen Basissatz berechnet, ansieht.

Beispiele: H, STO-6-311G
           H, STO-6-311G NPFUNC=3 DIFFS=.TRUE.    

Weitere Ausführungen:

Basis sets

You can see this by looking at the provided file '3-21G basis set for H, C and N'.  Open this in a separate browser window, so that you can read these notes at the same time.

Polarisation basis function

 

Man kann den Basissatz auch explizit eingeben. Dazu ein Beispiel:

 $DATA
H2O
CNV 2

O 8.0    0.000000   0.000000  -0.113552
S 8
1    0.153300D+05   0.508000D-03
2    0.229900D+04   0.392900D-02
3    0.522400D+03   0.202430D-01
4    0.147300D+03   0.791810D-01
5    0.475500D+02   0.230687D+00
6    0.167600D+02   0.433118D+00
7    0.620700D+01   0.350260D+00
8    0.688200D+00  -0.815400D-02
S 8
1    0.153300D+05  -0.115000D-03
2    0.229900D+04  -0.895000D-03
3    0.522400D+03  -0.463600D-02
4    0.147300D+03  -0.187240D-01
5    0.475500D+02  -0.584630D-01
6    0.167600D+02  -0.136463D+00
7    0.620700D+01  -0.175740D+00
8    0.688200D+00   0.603418D+00
S 1
1    0.175200D+01   1.0
S 1
1    0.238400D+00   1.0
P 3
1    0.344600D+02   0.159280D-01
2    0.774900D+01   0.997400D-01
3    0.228000D+01   0.310492D+00
P 1
1    0.715600D+00   1.0
P 1
1    0.214000D+00   1.0
D 1
1    0.231400D+01   1.0
D 1
1    0.645000D+00   1.0
F 1
1    0.142800D+01   1.0

H   1.0   0.000000   -0.753006   0.454207
S 3
1    0.338700D+02   0.606800D-02
2    0.509500D+01   0.453080D-01
3    0.115900D+01   0.202822D+00
S 1
1    0.325800D+00   1.0
S 1
1    0.102700D+00   1.0
P 1
1    0.140700D+01   1.0
P 1
1    0.388000D+00   1.0
D 1
1    0.105700D+01   1.0

 $END

Erläuterungen anhand des Basissatzes für ein H-Atom:
Nach den Koordinaten des Atoms (kartesische Koordinaten) erfolgt in der nächsten Zeile die Angabe des AO-Typs: S und die Zahl der Gaussfunktionen mit denen das AO dargestellt wird: 3. Es folgen die 3 Gaussfunktionen (Nummer, Exponent, Faktor der e-Funktion).
Es folgt ein weiteres s-AO bestehend aus 1 Gaussfunktion (diffuses s-AO; GTO: Gaussian Type Orbital)) und noch ein s-AO (noch diffuser).
Dann folgen 2 p-AO's und ein d-AO.

Bei den auf obige Weise eingegebenen Basissätzen muss man darauf achten, dass der Basissatz aus einer Summe von Funktionen bestehen muss, die unabhängig von einander sein müssen.

Solche Basissätze kann man sich aus dem Internet herunterladen:
http://www.emsl.pnl.gov:2080/forms/basisform.html

Man kann sich z.B. einige Maximalbasissätze für einige Atome herunterladen und durch Weglassen einzelner Funktionen (von hinten beginnend) sich geeignete Basissätze "zurechtstutzen". Wenn man z. B. den d-aug-cc-pVQZ Basissatz für das H-Atom verwendet, erhält man 75 Basisfunktionen. Lässt man den letzten Typ weg (2 Zeilen), nur noch 65, - 2 Typen: 59, -3: 56, -4: 55, -5: 45, -6: 39. Beispiel

Man kann aber auch selbst Atomorbitale entwerfen und sie mit GAMESS optimieren.

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