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
\[H\mathbf{c} = S\mathbf{c}\varepsilon\]
where \(S\) is the overlap matrix in terms of pre-calculated atomic orbitals. The Hamiltonian matrix is defined as
\[\begin{split}H_{ij} = \left\{\begin{array}{lcl}\epsilon_i;&\quad&i=j\\KS_{ij};&\quad&i\ne j\end{array}\right.\end{split}\]
where \(K\) is the Wolfsberg-Helmholtz constant. The default value is 1.75, but it can be changed with the keyword .HUCPAR.
Example¶
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
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
In addition to 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). 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