Extended Huckel start

Background

The basis theory is given here: [Hoffman1963].

The present development was further motivated by [Norman2012].

The particularity of the present approach is that it employs pre-calculated atomic fragments.

A start guess is generated by solving the following general eigenvalue problem

\begin{equation*} H\mathbf{c} = S\mathbf{c}\varepsilon \end{equation*}

where \(S\) is the overlap matrix in terms of pre-calculated atomic orbitals. The Hamiltonian matrix is defined as

\begin{equation*} H_{ij} = \left\{\begin{array}{lcl}\epsilon_i;&\quad&i=j\\KS_{ij};&\quad&i\ne j\end{array}\right. \end{equation*}

where \(K\) is the Wolfsberg-Helmholtz constant. The default value is 1.75, but it can be changed with the keyword .HUCPAR.

Example

alternate text

We consider a Kohn-Sham calculation of the closed-shell \(LuF_3\) molecule using the PBE functional. We start from the molecular input file LuF3.xyz

    4
 
Lu   -0.1027466346   0.0000000000   0.0000000000
F     0.3153699354   1.9200009200   0.0000000000
F     0.3153699354  -0.9600004600   1.6627695700
F     0.3153699354  -0.9600004600  -1.6627695700

and the menu file PBE.inp

**DIRAC
.WAVE FUNC
.ANALYZE
#.INPTES
**INTEGRALS
*READIN
.UNCONT
*TWOINT
.SCREEN
1.0D-16
**HAMILTONIAN
.DFTAUTO
PBE
**WAVE FUN
.SCF
*SCF
.CLOSED SHELL
98
**ANALYZE
.MULPOP
*MULPOP
.LABEL
SHELL
**MOLECULE
*BASIS
.DEFAULT
 cc-pVTZ
.SPECIAL
 Lu BASIS dyall.v3z
**END OF

This menu file includes specifications to use uncontracted integrals and specifies a screening threshhold for 2-electron integrals. The the method is specified; Here the Kohn-Sham DFT will be used with the PBE functional. The molecule is defined as having 98 closed shell electrons (for the fermion irrep without inversion symmetry). Further more, the Mulliken population analysis is specified to provide labels for individual orbitals and lastly the basis sets used are defined. Note that a special basis set is used for the Lu atom. The basis set must be specified in the menu file when using an xyz file rather than a mol file. Note that the keyword INPTES for testing the inputfile before the run is commented out. Observe that the caclulation will take more than an hour. Unfortunately, and curiously, this calculation does not converge

                                   SCF - CYCLE
                                   -----------

* Convergence on norm of error vector (gradient).
  Desired convergence:1.000D-07
  Allowed convergence:1.000D-06

* ERGVAL - convergence in total energy
* FCKVAL - convergence in maximum change in total Fock matrix
* EVCVAL - convergence in error vector (gradient)
--------------------------------------------------------------------------------------------------------------------------------
           Energy               ERGVAL    FCKVAL    EVCVAL      Conv.acc    CPU          Integrals   Time stamp
--------------------------------------------------------------------------------------------------------------------------------
It.    1    -7606.329622178      0.00D+00  0.00D+00  0.00D+00              28.97259600s   Scr. nuclei    Wed Dec 10
It.    2    -14547.31897887      6.94D+03 -4.12D+02  1.08D+02               2min32.266s   LL SL          Wed Dec 10
It.    3    -14073.36388927     -4.74D+02  1.23D+02  1.66D+02   DIIS   2    2min31.717s   LL SL          Wed Dec 10
It.    4    -14792.03577675      7.19D+02 -4.80D+01  4.86D+01   DIIS   3    2min35.039s   LL SL          Wed Dec 10
It.    5    -14695.41264973     -9.66D+01 -5.33D+01  3.35D+01   DIIS   4    2min31.924s   LL SL          Wed Dec 10
It.    6    -14619.54437781     -7.59D+01  6.02D+01  5.70D+01   DIIS   5    2min20.840s   LL SL          Wed Dec 10
It.    7    -14630.80789739      1.13D+01 -8.06D-01  5.57D+01   DIIS   6    2min21.446s   LL SL          Wed Dec 10
It.    8    -14632.21062186      1.40D+00 -9.11D-02  5.55D+01   DIIS   7    2min21.657s   LL SL          Wed Dec 10
It.    9    -14632.21078416      1.62D-04 -2.00D-03  5.55D+01   DIIS   8    2min19.026s   LL SL          Wed Dec 10
It.   10    -14630.85572005     -1.36D+00  1.07D-01  5.57D+01   DIIS   9    2min20.250s   LL SL          Wed Dec 10
It.   11    -14632.27345480      1.42D+00 -8.90D-02  5.56D+01   DIIS   9    2min19.892s   LL SL          Wed Dec 10
It.   12    -14675.22022580      4.29D+01 -2.22D+00  5.27D+01   DIIS   9    2min24.215s   LL SL          Wed Dec 10
It.   13    -14674.79019885     -4.30D-01 -3.70D-01  5.17D+01   DIIS   9    2min32.673s   LL SL          Wed Dec 10
It.   14    -14773.70533600      9.89D+01 -4.57D+00  4.40D+01   DIIS   9    2min40.921s   LL SL          Wed Dec 10
It.   15    -14030.36289440     -7.43D+02 -8.76D+01  4.18D+01   DIIS   9    2min31.891s   LL SL          Wed Dec 10
It.   16    -14534.80551844      5.04D+02  1.05D+02  7.86D+01   DIIS   9    2min46.992s   LL SL          Wed Dec 10
It.   17    -14535.64458223      8.39D-01  4.70D-02  7.88D+01   DIIS   9    2min35.034s   LL SL          Wed Dec 10
It.   18    -14523.96963624     -1.17D+01  7.36D-01  8.06D+01   DIIS   9    2min37.812s   LL SL          Wed Dec 10
It.   19    -14527.13835101      3.17D+00 -1.86D-01  8.00D+01   DIIS   9    2min39.389s   LL SL          Wed Dec 10
It.   20    -14532.42279875      5.28D+00 -3.49D-01  7.92D+01   DIIS   9    2min38.201s   LL SL          Wed Dec 10
It.   21    -14521.10965318     -1.13D+01  6.40D-01  8.09D+01   DIIS   9    2min23.733s   LL SL          Wed Dec 10
It.   22    -14471.27866690     -4.98D+01  3.20D+00  8.78D+01   DIIS   9    2min27.103s   LL SL          Wed Dec 10
It.   23    -14458.64743054     -1.26D+01  8.10D-01  8.95D+01   DIIS   9    2min26.239s   LL SL          Wed Dec 10
It.   24    -14247.57474297     -2.11D+02  1.42D+01  1.26D+02   DIIS   9    2min27.943s   LL SL          Wed Dec 10
It.   25    -13755.84951325     -4.92D+02 -1.34D+02  4.31D+01   DIIS   9    2min34.127s   LL SL          Wed Dec 10
It.   26    -14155.33995820      3.99D+02  1.40D+02  1.42D+02   DIIS   9    2min27.956s   LL SL          Wed Dec 10
It.   27    -14203.53818312      4.82D+01 -2.81D+00  1.34D+02   DIIS   9    2min26.276s   LL SL          Wed Dec 10
It.   28    -14372.59026376      1.69D+02 -1.02D+01  1.08D+02   DIIS   9    2min29.781s   LL SL          Wed Dec 10
It.   29    -14403.99910127      3.14D+01 -1.84D+00  1.04D+02   DIIS   9    2min27.335s   LL SL          Wed Dec 10
It.   30    -14398.26182697     -5.74D+00  4.01D-01  1.05D+02   DIIS   9    2min27.010s   LL SL          Wed Dec 10
It.   31    -14390.97015529     -7.29D+00  4.69D-01  1.06D+02   DIIS   9    2min22.080s   LL SL          Wed Dec 10
It.   32    -14392.09835723      1.13D+00 -3.80D-02  1.06D+02   DIIS   9    2min15.570s   LL SL          Wed Dec 10
It.   33    -14357.31922163     -3.48D+01  2.18D+00  1.11D+02   DIIS   9    2min20.425s   LL SL          Wed Dec 10
It.   34    -14403.41219753      4.61D+01 -3.07D+00  1.04D+02   DIIS   9    2min20.358s   LL SL          Wed Dec 10
It.   35    -13757.03956058     -6.46D+02 -1.24D+02  3.89D+01   DIIS   9    2min22.830s   LL SL          Wed Dec 10
It.   36    -14319.29356334      5.62D+02  1.29D+02  1.17D+02   DIIS   9    2min20.737s   LL SL          Wed Dec 10
It.   37    -14301.26184351     -1.80D+01  1.14D+00  1.20D+02   DIIS   9    2min18.481s   LL SL          Wed Dec 10
It.   38    -14321.83223258      2.06D+01 -1.36D+00  1.17D+02   DIIS   9    2min18.980s   LL SL          Wed Dec 10
It.   39    -14312.58970385     -9.24D+00  5.89D-01  1.18D+02   DIIS   9    2min18.591s   LL SL          Wed Dec 10
It.   40    -14303.93716589     -8.65D+00  5.49D-01  1.20D+02   DIIS   9    2min18.135s   LL SL          Wed Dec 10
It.   41    -14234.12176998     -6.98D+01  4.06D+00  1.30D+02   DIIS   9    2min19.220s   LL SL          Wed Dec 10
It.   42    -14262.78273430      2.87D+01 -1.69D+00  1.26D+02   DIIS   9    2min19.694s   LL SL          Wed Dec 10
It.   43    -14284.94272404      2.22D+01 -1.32D+00  1.22D+02   DIIS   9    2min22.317s   LL SL          Wed Dec 10
It.   44    -14283.69349005     -1.25D+00  1.21D-01  1.22D+02   DIIS   9    2min18.819s   LL SL          Wed Dec 10
It.   45    -13756.02930378     -5.28D+02 -1.32D+02  4.12D+01   DIIS   9    2min20.891s   LL SL          Wed Dec 10
It.   46    -14294.32431815      5.38D+02  1.31D+02  1.21D+02   DIIS   9    2min21.661s   LL SL          Wed Dec 10
It.   47    -14271.05944724     -2.33D+01  1.47D+00  1.25D+02   DIIS   9    2min19.840s   LL SL          Wed Dec 10
It.   48    -14111.67647533     -1.59D+02  9.25D+00  1.50D+02   DIIS   9    2min21.312s   LL SL          Wed Dec 10
It.   49    -14198.35642753      8.67D+01 -4.95D+00  1.36D+02   DIIS   9    2min20.919s   LL SL          Wed Dec 10
It.   50    -14205.82719171      7.47D+00 -3.60D-01  1.35D+02   DIIS   9    2min19.880s   LL SL          Wed Dec 10
--------------------------------------------------------------------------------------------------------------------------------
* No convergence after   50 iterations

We therefore try the extended Hückel method for a better start guess. The input file PBEhuc.inp reads

**DIRAC
.WAVE FUNC
.ANALYZE
#.INPTES
**INTEGRALS
*READIN
.UNCONT
*TWOINT
.SCREEN
1.0D-16
**HAMILTONIAN
.DFTAUTO
PBE
**WAVE FUN
.SCF
*SCF
.CLOSED SHELL
98
.AD HOC
AFLUXX
1..40
AFFXXX
1..5
.OVLSEL
**ANALYZE
.MULPOP
*MULPOP
.LABEL
SHELL
**MOLECULE
*BASIS
.DEFAULT
 cc-pVTZ
.SPECIAL
 Lu BASIS dyall.v3z
*SYMMETRY
.NOSYM
**END OF

Using the keyword .AD HOC file names for the Lu and F atomic types are given, followed by orbital strings specifying what atomic orbitals to select (see Specification of orbital strings for the syntax). These files can be generated following the procedure outlined in the tutorial Atomic start for “Preparing the atomic start”. Note, that the calculation is specified to be performed without the use of symmetry. The calculation now converges in 20 iterations

                                   SCF - CYCLE
                                   -----------

* Convergence on norm of error vector (gradient).
  Desired convergence:1.000D-07
  Allowed convergence:1.000D-06

* ERGVAL - convergence in total energy
* FCKVAL - convergence in maximum change in total Fock matrix
* EVCVAL - convergence in error vector (gradient)
--------------------------------------------------------------------------------------------------------------------------------
           Energy               ERGVAL    FCKVAL    EVCVAL      Conv.acc    CPU          Integrals   Time stamp
--------------------------------------------------------------------------------------------------------------------------------
It.    1    -9105.410351646      1.00D+20  0.00D+00  0.00D+00               0.56891400s   Atom. Huckel   Fri Dec 12
It.    2    -14880.04552091      5.77D+03  0.00D+00  0.00D+00               4min13.235s   LL SL          Fri Dec 12
It.    3    -14864.30290550     -1.57D+01 -4.18D+00  8.86D+00               4min31.750s   LL SL          Fri Dec 12
It.    4    -14674.59364082     -1.90D+02 -3.17D+01  2.78D+01   DIIS   2    4min34.358s   LL SL          Fri Dec 12
It.    5    -14870.34991888      1.96D+02  3.26D+01  1.29D+01   DIIS   3    4min33.188s   LL SL          Fri Dec 12
It.    6    -14878.63912566      8.29D+00 -2.37D+00  4.95D+00   DIIS   4    4min32.808s   LL SL          Fri Dec 12
It.    7    -14880.25355557      1.61D+00 -3.10D+00  3.52D+00   DIIS   5    4min33.803s   LL SL          Fri Dec 12
It.    8    -14880.91452088      6.61D-01  1.42D+00  5.62D-01   DIIS   6    4min31.141s   LL SL          Fri Dec 12
It.    9    -14880.94191467      2.74D-02  2.03D-01  2.62D-01   DIIS   7    4min30.183s   LL SL          Fri Dec 12
It.   10    -14880.94481063      2.90D-03  2.55D-02  1.83D-01   DIIS   8    4min29.212s   LL SL          Fri Dec 12
It.   11    -14880.94590393      1.09D-03  1.83D-02  1.22D-01   DIIS   9    4min30.443s   LL SL          Fri Dec 12
It.   12    -14880.94690911      1.01D-03  5.01D-02  2.80D-02   DIIS   9    4min29.728s   LL SL          Fri Dec 12
It.   13    -14880.94696438      5.53D-05  1.38D-02  2.95D-03   DIIS   9    4min26.465s   LL SL          Fri Dec 12
It.   14    -14880.94696474      3.63D-07 -1.21D-03  1.86D-04   DIIS   9    4min22.408s   LL SL          Fri Dec 12
It.   15    -14880.94696474      2.89D-09  8.38D-05  1.03D-04   DIIS   9    4min20.725s   LL SL          Fri Dec 12
It.   16    -14880.94696475      1.82D-09 -7.01D-05  5.88D-05   DIIS   9    4min17.819s   LL SL          Fri Dec 12
It.   17    -14880.94696475     -1.75D-10  2.56D-05  2.90D-06   DIIS   9    4min15.351s   LL SL          Fri Dec 12
It.   18    -14880.94696475      9.82D-11 -1.01D-06  5.15D-07   DIIS   9    4min 7.462s   LL SL          Fri Dec 12
It.   19    -14880.94696475      3.51D-10 -1.01D-07  1.38D-07   DIIS   9    4min 3.581s   LL SL          Fri Dec 12
It.   20    -14880.94696475     -1.91D-10 -5.55D-08  5.12D-08   DIIS   9    3min 3.805s   LL SL          Fri Dec 12
--------------------------------------------------------------------------------------------------------------------------------
* Convergence after   20 iterations.
* Average elapsed time per iteration: 
      No 2-ints    :    0.56250000s
      LL SL        :    4min20.660s


                                   TOTAL ENERGY
                                   ------------

   Electronic energy                        :    -15435.865958186127

   Other contributions to the total energy
   Nuclear repulsion energy                 :       554.918175409854
   SS Coulombic correction                  :         0.000818030697

   Sum of all contributions to the energy
   Total energy                             :    -14880.946964745575