Pseudopotenciales

Ver pág. 65 de ref. [11].

Sólo consideran los electrones de valencia moviéndose en el potencial generado por el núcleo y los electrones del core (core congelado)[16].:

Pasan del Hamiltoniano:

$$H = \sum_i^{n_e}\left( -\frac{1}{2}\nabla^2_i - \frac{Z}{r_i} \right) + \sum_{i<j}^{n_e} \frac{1}{r_{ij}}$$ (1.26)

al de pseudo-potenciales:

$$H^{ps} = \sum_i^{n_v}\left( -\frac{1}{2}\nabla^2_i + V^{ps}_i \right) + \sum_{i<j}^{n_v} \frac{1}{r_{ij}}$$ (1.27)

A partir de estos dos hamiltonianos construyen los operadores de Fock correspondientes, separando la parte de valencia del primero, que es a la que ajustan el de pseudo-potenciales para obtener $V^{ps}$.

Se suelen utilizar unos pseudo-potenciales sin nodos en la parte interna del core, dado esos nodos complican mucho el calculo, y lo que se pretende es que los resultados (pseudo-orbitales) de la capa de valencia si sean los más próximos a los HF en la zona de valencia.

El nombre de pseudo-potenciales viene del cambio de orbitales de valencia con nodos a pseudo-orbitales sin nodos.

Hay varias aproximaciones, la más utilizada es la forma semi-local[17] :

$$V^{ps}= -\frac{Z}{r} + \sum_l V_l(r) P_l$$ (1.28)

donde $V_l(r)$ es una función de $r$ y $P_l$ representa el proyector sobre los armónicos esféricos de simetría l.

o, la no-local:

$$V^{ps}= -\frac{Z}{r} + \sum_{p,q} C_{pq}\vert f_p><f_q\vert$$ (1.29)

siendo $f_j$ un conjunto de funciones gaussianas.

Basis Set Library Information on: Hay-Wadt MB (n+1) ECP      

              Hay-Wadt (n+1) ECP Minimal Basis Sets                            
              -----------------------------------------                        
Elements     Contraction                   References                          
 K - Ca: (5s,5p)    -> [2s,1p]     P. J. Hay and W. R. Wadt, J. Chem. Phys.    
Sc - Cu: (5s,5p,5d) -> [2s,1p,1d]  82, 299 (1985).                             
Rb - Sr: (5s,6p)    -> [2s,1p]                                                 
 Y - Ag: (5s,6p,5d) -> [2s,1p,1d]                                              
Cs - Ba: (5s,6p)    -> [2s,1p]                                                 
La - Au: (5s,6p,5d) -> [2s,1p,1d]                                              
**                                                                             
The Hay-Wadt (n+1) effective core potentials include an extra shell of         
electrons beyond what is traditionally available in effective core potentials. 
For example, on potassium the 3s and 3p electrons are not subsumed into the    
core as they would be in other ECP's.  This set is derived from the exponents  
and contraction coefficients given in the Hay-Wadt paper and were obtained     
directly from P. J. Hay.  Any discrepancy between the numbers contained in     
these basis sets and the JCP paper are attributable to the file obtained from  
Jeff at LANL.  The elements beyond Kr include the 1-electron Darwin and        
mass-velocity relativitics corrections in their definitions.

Elements supported

K Ca Sc Ti V Cr Mn Fe Co Ni Cu Rb Sr Y Zr Nb Mo Tc Rh Rh Pd Ag Cs Ba La Hf 
Ta W Re Os Ir Pt Au 


Basis Set Library Information on: Hay-Wadt VDZ (n+1) ECP     

              Hay-Wadt (n+1) ECP VDZ Basis Sets                                
              -----------------------------------------                        
BASIS="Hay-Wadt VDZ (n+1) ECP"
 PT  0
 S   4  1.00
       2.54700000     -1.47391750
       1.61400000      1.91157190
       0.51670000      0.39223190
       0.26510000      0.00000000
 S   4  1.00
       2.54700000      1.43881660
       1.61400000     -2.09118210
       0.51670000     -1.09213150
       0.26510000      1.34265960
 S   1  1.00
       0.05800000      1.00000000
 P   3  1.00
       2.91100000     -0.52474380
       1.83600000      0.96718840
       0.59820000      0.54386320
 P   2  1.00
       0.60480000     -0.10614380
       0.09960000      1.03831020
 P   1  1.00
       0.02900000      1.00000000
 D   2  1.00
       1.24300000      0.55981500
       0.42710000      0.55110900
 D   1  1.00
       0.13700000      1.00000000
 ****

Effective Core Potentials
-------------------------
PT-ECP   4   60
g potential         
 5
1      728.93940560     -0.16192680
2      320.65678000  -1320.28738520
2       52.86801740   -298.31781350
2       12.02801280    -87.58370650
2        3.52389130     -8.14932740
s-g potential       
 6
0      409.44373580      2.73342180
1      274.54192310     59.70243290
2      127.56585700    891.45895500
2       32.90366310    368.44676560
2        5.05938800    238.02630900
2        4.15065560   -107.05564540
p-g potential       
 4
0      466.17288920      1.88785680
1      120.78882590     76.01386290
2       36.41187910    343.55111160
2        5.69854080    119.49117860
d-g potential       
 5
0      249.56507630      2.93436780
1      126.66785850     59.33065710
2       63.14305860    452.44451940
2       17.90594700    210.47694790
2        4.22393730     58.62541120
f-g potential       
 5
0      121.81587990      3.95342530
1       60.87570300     53.85551820
2       31.47671470    247.43051330
2        9.88117510    127.81879760
2        2.73198740     15.37720460



Basis Set Library Information on: SBKJC VDZ ECP              

          Stevens/Basch/Krauss/Jasien/Cundari -21G ECP Basis                   
          --------------------------------------------------                   
Elements     Contraction                   References                          
 H - He: (3s)       -> [2s]        Taken from 3-21G basis.                     
Li - Ar: (4s,4p)    -> [2s,2p]     W.J. Stevens, H. Basch, M. Krauss, J. Chem. 
                                   Phys. 81, 6026 (1984).                      
 K - Ca: (5s,5p)    -> [3s,3p]     W.J. Stevens, M. Krauss, H. Basch, P.G.     
Sc - Zn: (8s,8p,6d) -> [4s,4p,3d]  Jasien, Can. J. Chem. 70, 612 (1992).       
Ce - Lu:                           T.R. Cundari and W.J. Stevens, J. Chem.     
                                   Chem. Phys. 98, 5555 (1993).                
La     : (9s,9p,5d) -> [4s,4p,3d]                                              
Hf - Hg: (7s,7p,5d) -> [4s,4p,3d]                                              
**                                                                             
The Stevens/Basch/Krauss/Jasien/Cundari -21G relativistic ECP's and their      
associated basis sets are designed to replace all but the outermost electrons  
in an atom.  For example, for K the 1s, 2s, 2p, 3s and 3p are considered core. 

Elements supported

H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn 
Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Rh Rh Pd Ag Cd In Sn Sb Te I Xe Cs Ba La 
Ce Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn