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
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