5.2.1. Extra information about basis sets and integrals

5.2.1.1. Basis set format

The Inline option for a basis set will read the basis set as defined by the following pseudo code.

Read Charge, lAng
Do iAng = 0, lAng
   Read nPrim, nContr
   Read (Exp(iPrim),iPrim=1,nPrim)
   Do iPrim=1,nPrim
      Read (Coeff(iPrim,iContr),iContr=1,nContr)
   End Do
End Do

where Charge is the nuclear charge, lAng is the highest angular momentum quantum number, nPrim is the number of primitive functions (exponents) for a given shell, and nContr is the number of contracted functions for a given shell.

The following is an example of a DPZ basis set for carbon. Normally, however, the basis set will be read from a library file following the specified label (like, e.g., C.DZP…4s2p1d.), and not be inserted inline at the input file.

Basis set                                -- Start defining a basis set
C.DZP.Someone.9s5p1d.4s2p1d. / inline    -- Definition in input stream
       6.0              2                -- charge, max l-quantum no.
    9    4                               -- no. of prim. and contr. s-functions
4232.61                                  -- s-exponents
634.882
146.097
42.4974
14.1892
1.9666
5.1477
0.4962
0.1533
  .002029   .0       .0       .0         -- s-contraction matrix
  .015535   .0       .0       .0
  .075411   .0       .0       .0
  .257121   .0       .0       .0
  .596555   .0       .0       .0
  .242517   .0       .0       .0
  .0       1.0       .0       .0
  .0        .0      1.0       .0
  .0        .0       .0      1.0
    5    2                               -- no. of prim. and contr. p-functions
18.1557                                  -- p-exponents
3.98640
1.14290
0.3594
0.1146
 .018534   .0                            -- p-contraction matrix
 .115442   .0
 .386206   .0
 .640089   .0
 .0       1.0
    1    1                               -- no. of prim. and contr. d-functions
   .75                                   -- d-exponents
  1.0                                    -- d-contraction matrix
C1 0.00000 0.00000 0.00000               -- atom-label, Cartesian coordinates
C2 1.00000 0.00000 0.00000               -- atom-label, Cartesian coordinates
End Of Basis                             -- end of basis set definition

5.2.1.2. The basis set label and the ECP libraries

The label within the ECP library is given as input in the line following the keyword BASIS SET. The label defines either the valence basis set and core potential which is assigned to a frozen-core atom or the embedding potential which is assigned to an environmental froze-ion. Here, all the comments made about this label in the section The basis set label and the basis set library for all-electron basis sets stand, except for the following changes:

  1. The identifier type must be ECP or PP.
  2. The identifier aux specifies the kind of the potential. It is used, for instance, to choose between non-relativistic, Cowan–Griffin, or no-pair Douglas–Kroll relativistic core potentials (i.e. Pt.ECP.Casarrubios.13s10p9d5f.1s2p2d1f.16e-NR-AIMP. or Pt.ECP.Casarrubios.13s10p9d5f.1s2p2d1f.16e-CG-AIMP. or Pt.ECP.Rakowitz.13s10p9d6f.5s4p4d2f.18e-NP-AIMP.) and to pick up one among all the embedding potentials available for a given ion (i.e. F.ECP.Lopez-Moraza.0s.0s.0e-AIMP-KMgF3. or F.ECP.Lopez-Moraza.0s.0s.0e-AIMP-CsCaF3.).
  3. The identifier contracted is used here in order to produce the actual basis set out of the basis set included in the ECP library, which is a minimal basis set (in general contraction form) augmented with some polarization, diffuse, … function. It indicates the number of s, p, … contracted functions in the actual basis set, the result being always a many-primitive contracted function followed by a number of primitives. As an example, At.ECP.Barandiaran.13s12p8d5f.3s4p3d2f.17e-CG-AIMP. will generate a (13,1,1/12,1,1,1/8,1,1/5,1) formal contraction pattern which is in this case a (13,1,1/12,1,1,1/7,1,1/5,1) real pattern. Other contraction patters should be input “Inline”.
  4. The user is suggested to read carefully section 5.2.2 of the tutorials and examples manual before using the ECP utilities.

5.2.1.3. One-Electron Integral Labels

The storage of one-electron integrals on disk is facilitated by the one-electron integral I/O facility. The internal structure of the one-electron file and the management is something which the user normally do not need to worry about. However, for the general input section of the FFPT, the user need to know the name and structure of the internal labels which the one-electron integral I/O facility associates with each type of one-electron integral. The labels are listed and explained here below for reference. The component index is also utilized by the one-electron integral I/O facility to discriminate the various components of the one-electron integrals of a certain type, for example, the dipole moment integrals have three components (1=x-component, 2=y-component, 3=z-component). The component index is enumerated as a canonical index over the powers of the Cartesian components of the operator (e.g. multipole moment, velocity, electric field, etc.). The order is defined by following pseudo code,

Do ix = nOrder, 0, -1
   Do iy = nOrder-ix, 0, -1
      iz = nOrder-ix-iy
   End Do
End Do,

where nOrder is the total order of the operator, for example, nOrder=2 for the electric field gradient and the quadrupole moment operator.

Label Explanation
Mltpl nn the nnth order Cartesian multipole moments.
MltplS the overlap matrix used in the semi-empirical NDDO method.
Kinetic the kinetic energy integrals.
Attract the electron attraction integrals.
AttractS the electron attraction integrals used in the semi-empirical NDDO method.
PrjInt the projection integrals used in ECP calculations.
M1Int the M1 integrals used in ECP calculations.
M2Int the M2 integrals used in ECP calculations.
SROInt the spectrally resolved operator integrals used in ECP calculations.
XFdInt the external electric field integrals.
MassVel the mass-velocity integrals.
Darwin the Darwin one-electron contact integrals.
Velocity the velocity integrals.
EF0nnnnn the electric potential at center nnnnn.
EF1nnnnn the electric field at center nnnnn.
EF2nnnnn the electric field gradient at center nnnnn.
AngMom the angular momentum integrals.
DMS the diamagnetic shielding integrals.
Wellnnnn the nnnnth set of spherical well integrals.
OneHam the one-electron Hamiltonian.
AMProd the hermitized product of angular momentum integrals.
AMFI the atomic mean field integrals.