Orbital optimization

The basis orbitals can be variationally optimized using the orbital optimization method [23]. As an illustration of the orbital optimization, let us explain using a methane molecule of which input file is Methane_OO.dat. The following keywords in this file are set as follows:

    <Definition.of.Atomic.Species
     H   H4.0-s41p41        H_TM
     C   C4.5-s41p41        C_TM_PCC
    Definition.of.Atomic.Species>

    orbitalOpt.Method          species     # Off|Unrestricted|Restricted
    orbitalOpt.InitCoes     Symmetrical    # Symmetrical|Free
    orbitalOpt.initPrefactor   0.1         # default=0.1
    orbitalOpt.scf.maxIter      25         # default=12
    orbitalOpt.MD.maxIter       10         # default=5
    orbitalOpt.per.MDIter       20         # default=1000000
    orbitalOpt.criterion      1.0e-6       # default=1.0e-4 (Hartree/borh)^2
    Num.CntOrb.Atoms             2         # default=1
    <Atoms.Cont.Orbitals
     1
     2
    Atoms.Cont.Orbitals>

Then, we execute OpenMX as:

    % ./openmx Methane.dat
  

When the execution is completed normally, you can find the history of orbital optimization in the file 'met_oo.out' as:

   ***********************************************************
   ***********************************************************
            History of orbital optimization   MD= 1          
   *********     Gradient Norm ((Hartree/borh)^2)     ********
                 Required criterion=  0.000001000000                  
   ***********************************************************

      iter=   1  Gradient Norm=  0.081251614657  Uele= -2.750500719281
      iter=   2  Gradient Norm=  0.018543400953  Uele= -2.933260690003
      iter=   3  Gradient Norm=  0.005918002913  Uele= -2.966113950591
      iter=   4  Gradient Norm=  0.001553729359  Uele= -3.010077558163
      iter=   5  Gradient Norm=  0.000356946294  Uele= -3.012729963043
      iter=   6  Gradient Norm=  0.000119196944  Uele= -3.024577717351
      iter=   7  Gradient Norm=  0.000042934968  Uele= -3.024772396249
      iter=   8  Gradient Norm=  0.000031243105  Uele= -3.026624698820
      iter=   9  Gradient Norm=  0.000020515771  Uele= -3.026569330230
      iter=  10  Gradient Norm=  0.000015126154  Uele= -3.026833093004

In most cases, ten iterative steps are enough to achieve a sufficient convergence. The comparison between the primitive basis orbitals and the optimized orbitals in the total energy is given by

    Primitive basis orbitals
       Utot  =      -8.032594073571 (Hartree) 

    Optimized orbitals by the orbital optimization 
       Utot  =      -8.150139929748 (Hartree)

Figure: The total energy for a carbon dimer C$_2$, a methane molecule CH$_4$, carbon and silicon in the diamond structure, a ethane molecule C$_2$H$_6$, and a hexafluoro ethane molecule C$_2$F$_6$ as a function of the number of primitive and optimized orbitals. The total energy and the number of orbitals are defined as those per atom for C$_2$, carbon and silicon in the diamond, and as those per molecule for CH$_4$, C$_2$H$_6$, and C$_2$F$_6$.

We see that the small but accurate basis set orbitals can be generated by the orbital optimization. In Fig. 14 we show the convergence properties of total energies for a carbon dimer C$_2$, a methane molecule CH$_4$, and the diamond as a function of the number of unoptimized and optimized orbitals. We see that a remarkable convergent results are obtained using the optimized orbitals for all systems. In this illustration of a methane molecule, the optimized radial orbitals are output to files, C_1.pao and H_2.pao. These output files, C_1.pao and H_2.pao, could be an input data for pseudo-atomic orbitals as it is. This means that it is possible to perform a pre-optimization of basis orbitals for systems you are interested in. The pre-optimization could be performed for smaller but chemically similar systems.

The following three options are available for the keyword 'orbitalOpt.Method', the unrestricted optimization 'Unrestricted', the restricted optimization 'Restricted', and Orbital optimization restricted to species 'Species'.

• Unrestricted

  The radial functions of basis orbitals are optimized without any constraint. Thus, all the radial functions could differ from each other, which could depend on the following indices, atomic number, angular moment quantum number, magnetic quantum number, and orbital multiplicity.

• Restricted

  The radial functions of basis orbitals are optimized with a constraint that the radial wave function $R$ is independent on the magnetic quantum number. We prefer 'Restricted' to 'Unrestricted', since the restricted optimization guarantees the rotational invariance of the total energy.

• Species

  Basis orbitals in atoms with the same species name, that you define in 'Definition.of.Atomic.Species', are optimized as the same orbitals. If you want to assign the same orbitals to atoms with almost the same chemical environment, and optimize these orbitals, this scheme could be quite convenient.