5. Using HiPart¶
For the terminology used in this chapter we refer to the Theoretical background.
HiPart is actually not a single program but a collection of Python scripts that can be used from the command line to perform population analysis on Gaussian computations as a post-processing step. All scripts start with hi-, which makes it easy to get an overview of all available script. Just enter hi- on the command line and hit the TAB key twice:
toon@poony ~> hi-
hi-atomdb.py hi-multipoles.py
hi-bond-orders.py hi-net-overlap.py
hi-charges.py hi-overlap-matrices-orb.py
hi-dipoles.py hi-overlap-matrices.py
hi-esp-test.py hi-spin-charges.py
toon@poony ~> hi-
Note that the part toon@poony ~> is the shell prompt, which may look differently on your Linux box. You don’t have to copy it. It is always included in the examples below to show in which directory each command is executed, i.e. right before the > character, it shows the current directory.
5.1. Test molecules¶
All examples below illustrate the usage of HiPart through the analysis of the electronic structure of the alanine and alanine radical in the gas phase. The structure and atom numbering are depicted in the figure below:
All computations are carried out at the B3LYP/6-31G(d) level with Gaussian03. The Gaussian03 files for these test molecules example can be found with the source code. If you have not already done so, check out the latest version of the source code:
toon@poony ~> git clone git://molmod.ugent.be/git/hipart.git
These are the output files you should have now:
toon@poony ~> cd hipart/examples/001-usage
toon@poony ~/hipart/examples/001-usage> ls
alanine.com alanine.log alanine-rad.fchk
alanine.fchk alanine-rad.com alanine-rad.log
5.2. Running proper Gaussian jobs¶
In order to run a decent analysis of the wavefunction, one first needs a decent wavefunction to start with. Our alanine input file looks like:
toon@poony ~/hipart/examples/001-usage> cat alanine.com
%mem=500MB
%chk=alanine.chk
%nproc=1
#p B3LYP/6-31G(d) sp scf(fermi,tight) density(current)
Alanine
0 1
O 1.657642604 0.744201101 -0.489836837
O 1.102098787 -1.089074323 0.697793837
N -1.404760606 -1.091053462 -0.458202548
C -0.655372082 0.161523560 -0.415839794
C -1.204817386 1.230334427 0.557894814
C 0.774816114 -0.164020328 -0.012893818
H -0.627559422 0.594521751 -1.422939548
H -0.609640466 2.149682405 0.525945464
H -2.235463406 1.483421899 0.284034915
H -1.205447577 0.848510089 1.585349081
H -2.403964854 -0.897766300 -0.440132543
H -1.180805308 -1.635048404 0.373826645
H 2.530514260 0.486012629 -0.137289368
toon@poony ~/hipart/examples/001-usage>
The line %chk=gaussian.chk is mandatory to process the Gaussian output with HiPart. After the job is completed, one must convert the binary checkpoint file to a formatted checkpoint file as follows:
formchk alanine.chk alanine.fchk
The formatted checkpoint file contains the output of a Gaussian computation in a standard format with reasonable numerical precision. This file will be loaded by the HiPart scripts to perform the population analysis.
Although the basis functions and level of theory could be improved upon, there are also some technical details to take into account:
- The SCF option fermi is known to accelarate the rate of convergence in almost all cases, so it is certainly useful. Sometimes it also fixes cases where SCF convergence would otherwise fail completely.
- The SCF option tight increases the convergence criteria that Gaussian uses to stop the SCF cycle. It may be useful for the more delicate derived quantities such as high order atomic multipole moments.
- The argument density(current) is useful in post-HF computations, but I always add as a good habit. If it is not present in a post-HF computation, only the density matrix of the preceding HF computation is written to the checkpoint file, and not the post-HF density matrix.
5.3. Preparing the atomic database¶
Both the Hirshfeld scheme and Iterative Hirshfeld scheme rely on a database of pro-atoms. These are spherically averaged atomic densities obtained from single-atom computations. The program hi-atomdb.py generates such a database using the Gaussian program. Make sure the g03 or g09 binary is in the path before executing hi-atomdb.py.
The online help is as follows:
toon@poony ~/hipart/examples/001-usage> hi-atomdb.py --help
Usage: hi-atomdb.py [options] executable lot atoms
hi-atomdb.py computes a database of pro-atomic densities.
The following arguments are mandatory:
* executable -- the name of the Gaussian binary (g03 or g09)
* lot -- The level of theory to be used in Gaussian input notation.
* atoms -- The atoms to be computed. One can specify ranges, e.g 1,2-5'
(avoid whitespace)
It is recommended to run this script in a directory that is initially empty and
that will contain nothing but the generated atom database. This script will
generate quite a few files and subdirectories.
Examples:
hi-atomdb.py MP2/Aug-CC-pVDZ 1-10,17
hi-atomdb.py HF/3-21G 1,6,7,8 -l 110
Options:
-h, --help show this help message and exit
-l LEBEDEV, --lebedev=LEBEDEV
The number of grid points for the spherical averaging.
[default=350]. Select from: 6, 14, 26, 38, 50, 74, 86,
110, 146, 170, 194, 230, 266, 302, 350, 434, 590, 770,
974, 1202, 1454, 1730, 2030, 2354, 2702, 3074, 3470,
3890, 4334, 4802, 5294, 5810
--rlow=RLOW The smallest radius for the density profile (in
angstroms). [default=2e-05]
--rhigh=RHIGH The largest radius for the density profile (in
angstroms). [default=20.0]
--num-steps=NUM_STEPS
The number of steps in density profile. [default=100]
--max-ion=MAX_ION The maximum ionization to consider. [default=2]
--qc Specify the qc convergence scheme in Gaussian input.
[default=False]
--no-work Do not save intermediate results in work directory for
later reuse.
--no-random Do not randomly rotate angular grids.
-q, --quiet Do not write any screen output.
The default settings are good enough for most levels of theory and basis sets. hi-atomdb.py prints some progress information on screen. For our example, we need pro-atoms Hydrogen, Carbon, Nitrogen and Oxygen. The database of pro-atoms is generated as follows. (The level of theory is put between quotation marks to avoid the interpretation of the round brackets by the shell.)
toon@poony ~/hipart/examples/001-usage> mkdir atoms
toon@poony ~/hipart/examples/001-usage> cd atoms
toon@poony ~/hipart/examples/001-usage/atoms> hi-atomdb.py g03 'B3LYP/6-31G(d)' 1,6-8
---TIME--- ---------------------------------LOG---------------------------------
0.01 Computing atomic database with program Gaussian (g03,qc=False)
Creating input files: 0%.........33%.........66%.........100%
Atomic computations: 0%.........33%.........66%.........100%
0.05 Could not execute any job. Is g03 in the PATH?
This is what one gets when the g03 binary is not in the path. You probably need to load a g03.profile first.
toon@poony ~/hipart/examples/001-usage/atoms> . ~/g03.profile
toon@poony ~/hipart/examples/001-usage/atoms> hi-atomdb.py g03 'B3LYP/6-31G(d)' 1,6-8
---TIME--- ---------------------------------LOG---------------------------------
0.01 Computing atomic database with program Gaussian (g03,qc=False)
Creating input files: 0%.........33%.........66%.........100%
Atomic computations: 0%.........33%.........66%.........100%
139.45 Selecting ground states.
Density profiles: 0%.........50%.........100%
141.29 Total charge error: 1 H -2 -5.17837e-09
141.29 Total charge error: 1 H -1 2.39259e-09
141.29 Total charge error: 1 H +0 -9.13307e-10
141.29 Total charge error: 6 C -2 7.36471e-09
141.29 Total charge error: 6 C -1 2.21719e-09
141.29 Total charge error: 6 C +0 3.42406e-09
141.29 Total charge error: 6 C +1 -3.71426e-09
141.29 Total charge error: 6 C +2 -1.65750e-09
141.29 Total charge error: 7 N -2 -7.98666e-09
141.29 Total charge error: 7 N -1 -8.17507e-09
141.29 Total charge error: 7 N +0 -7.18856e-09
141.29 Total charge error: 7 N +1 -5.77149e-09
141.29 Total charge error: 7 N +2 -1.27853e-08
141.29 Total charge error: 8 O -2 -3.74776e-09
141.29 Total charge error: 8 O -1 -2.72562e-09
141.29 Total charge error: 8 O +0 -7.53968e-09
141.29 Total charge error: 8 O +1 -1.21337e-08
141.29 Total charge error: 8 O +2 -1.03187e-08
The script hi-atomdb.py consists of three major phases: (i) setup of the atomic input files for Gaussian, (ii) Gaussian computations on every atomic input, and (iii) derivation of the spherically averaged atomic densities. In the end a check is performed by integrating the total charge based on the spherically averaged densities. When too poor grids are used, it will be obvious from the errors in this last check. In this example the errors are small enough.
The choice of angular grid is not that important and can be chosen very large because the computations are fast enough anyway. The radial grid settings are more delicate. The radial grid is always logarithmic, i.e. equidistant on a logarithmic scale. The same radial grid will be used by all other HiPart programs that use this database. If for some reason large radial grids are required later, they have to be defined at this point. In case of Lithium, heavy atoms or large basis sets, one may want to tune the radial grid.
Once the program is finished, the following files are generated:
toon@poony ~/hipart/examples/001-usage/atoms> find | sort
./001H
./001H/neg1
./001H/neg1/gs
./001H/neg1/mult1
./001H/neg1/mult1/gaussian.com
./001H/neg1/mult1/gaussian.fchk
./001H/neg1/mult1/gaussian.log
./001H/neg1/mult1/grid.bin
./001H/neg1/mult1/grid_moldens.bin
...
./008O/pos2
./008O/pos2/gs
./008O/pos2/mult1
./008O/pos2/mult1/gaussian.com
./008O/pos2/mult1/gaussian.fchk
./008O/pos2/mult1/gaussian.log
./008O/pos2/mult3
./008O/pos2/mult3/gaussian.com
./008O/pos2/mult3/gaussian.fchk
./008O/pos2/mult3/gaussian.log
./008O/pos2/mult3/grid.bin
./008O/pos2/mult3/grid_moldens.bin
./chieta_au.txt
./chieta_ev.txt
./densities.txt
./energies.txt
For each atom-charge combination, all reasonable spin multiplicities are computed and the spin state with the lowest energy is selected. One can run hi-atomdb.py a second time with more atoms to extend the database. (Existing computations will be reused, but make sure the same basis and level of theory are used.)
Only the file densities.txt will be used later. In our example it contains the following data:
toon@poony ~/hipart/examples/001-usage/atoms> cat densities.txt
RLogIntGrid(3.7794522678425048e-05,3.7794522678425039e+01,100)
1 -2 1.4143823633854540e+00 1.4143823558521540e+00 1.4143823458935731e+00 1.4143823327289020e+00 1.4143823153259989e+00 1.4143822923203782e+00 1.4143822619083162e+00 1.4143822217053421e+00 1.4143821685593945e+00 1.4143820983035729e+00 1.4143820054295204e+00 1.4143818826555179e+00 1.4143817203555378e+00 1.4143815058045723e+00 1.4143812221808971e+00 1.4143808472472639e+00 1.4143803516073346e+00 1.4143796964009863e+00 1.4143788302575910e+00 1.4143776852684826e+00 1.4143761716628074e+00 1.4143741707696700e+00 1.4143715257145022e+00 1.4143680291209051e+00 1.4143634068538322e+00 1.4143572965304845e+00 1.4143492191155476e+00 1.4143385413757426e+00 1.4143244262547785e+00 1.4143057672845942e+00 1.4142811019039809e+00 1.4142484969104010e+00 1.4142053971037181e+00 1.4141484253233438e+00 1.4140731183212463e+00 1.4139735779720974e+00 1.4138420108406164e+00 1.4136681206466741e+00 1.4134383071130172e+00 1.4131346103329181e+00 1.4127333212955944e+00 1.4122031555628598e+00 1.4115028572428892e+00 1.4105780633929996e+00 1.4093572142725119e+00 1.4077462430316434e+00 1.4056217223560754e+00 1.4028220926795534e+00 1.3991365624275409e+00 1.3942912852697547e+00 1.3879325363192647e+00 1.3796069202319592e+00 1.3687392967959122e+00 1.3546103254307318e+00 1.3363376083650640e+00 1.3128676836714381e+00 1.2829907732045698e+00 1.2453958453984941e+00 1.1987884046751693e+00 1.1420929304417837e+00 1.0747476032877994e+00 9.9705959349788709e-01 9.1051850926034983e-01 8.1788157724819321e-01 7.2281332506545704e-01 6.2899585270542147e-01 5.3898995633071201e-01 4.5357060060082066e-01 3.7230832675937753e-01 2.9531055761567043e-01 2.2464199169476168e-01 1.6379487595041783e-01 1.1557573385930459e-01 8.0442075228479057e-02 5.6554660787928542e-02 4.1068398398227318e-02 3.1276867298832979e-02 2.4799019199622244e-02 1.9494757495560549e-02 1.4065837857584454e-02 8.6669135656273506e-03 4.3162270843275941e-03 1.6531390806556058e-03 4.5695610040091459e-04 8.3079436011078244e-05 8.7173880468681315e-06 4.4262155332295921e-07 8.6093787721969106e-09 4.7103176702608016e-11 4.8186228590891602e-14 5.3721814276498844e-18 3.1975574999064668e-23 3.9558204409997410e-30 2.9229404148632462e-39 2.4809876602580675e-51 2.7366215925662594e-67 2.2001791217421095e-88
1 -1 3.1195446156691437e-01 3.1195446059202026e-01 3.1195445930327270e-01 3.1195445759962498e-01 3.1195445534750033e-01 3.1195445237032565e-01 3.1195444843467685e-01 3.1195444323197835e-01 3.1195443635432102e-01 3.1195442726246142e-01 3.1195441524356093e-01 3.1195439935528341e-01 3.1195437835191542e-01 3.1195435058670384e-01 3.1195431388274319e-01 3.1195426536228255e-01 3.1195420122111889e-01 3.1195411643033155e-01 3.1195400434199055e-01 3.1195385616796434e-01 3.1195366029090488e-01 3.1195340135345834e-01 3.1195305905426157e-01 3.1195260655638368e-01 3.1195200838347387e-01 3.1195121763872119e-01 3.1195017232879407e-01 3.1194879050473595e-01 3.1194696383937925e-01 3.1194454913848768e-01 3.1194135712152504e-01 3.1193713759489677e-01 3.1193155985958482e-01 3.1192418682465600e-01 3.1191444081040859e-01 3.1190155838327871e-01 3.1188453072217731e-01 3.1186202491174619e-01 3.1183228011595465e-01 3.1179297070837014e-01 3.1174102600624981e-01 3.1167239313467832e-01 3.1158172557789560e-01 3.1146197500082778e-01 3.1130385781884568e-01 3.1109516073637089e-01 3.1081984127681872e-01 3.1045687087225260e-01 3.0997876094411009e-01 3.0934970974145198e-01 3.0852331536219912e-01 3.0743982859780838e-01 3.0602298494070307e-01 3.0417658456844399e-01 3.0178121978003997e-01 2.9869192709128423e-01 2.9473810540189876e-01 2.8972778299539004e-01 2.8345908455700275e-01 2.7574209397539629e-01 2.6643327954943941e-01 2.5548074451330016e-01 2.4297031713178169e-01 2.2915044454987263e-01 2.1440420483479944e-01 1.9914421414850864e-01 1.8364844327702498e-01 1.6792878566772168e-01 1.5176987786450274e-01 1.3498739970035270e-01 1.1772877173324872e-01 1.0052868308332318e-01 8.4060635338824227e-02 6.8819695467937078e-02 5.4981155276814776e-02 4.2515835456432345e-02 3.1446946106182573e-02 2.1979296809859258e-02 1.4355956981332456e-02 8.6388422829198376e-03 4.6674498608139650e-03 2.1645514216707719e-03 8.0620277387347767e-04 2.2121489094853613e-04 4.0172103819653579e-05 4.2147808482508385e-06 2.1400289504650849e-07 4.1625444797133146e-09 2.2773892672659307e-11 2.3297536918956619e-14 2.5973934629460171e-18 1.5459855664785757e-23 1.9125977579903914e-30 1.4132110816420360e-39 1.1995315529064438e-51 1.3231278821866738e-67 1.0637635651526014e-88
1 +0 3.0105828732899365e-01 3.0105828616196056e-01 3.0105828461920942e-01 3.0105828257978939e-01 3.0105827988378986e-01 3.0105827631984106e-01 3.0105827160851040e-01 3.0105826538040942e-01 3.0105825714722412e-01 3.0105824626343969e-01 3.0105823187571845e-01 3.0105821285600071e-01 3.0105818771305831e-01 3.0105815447557416e-01 3.0105811053758702e-01 3.0105805245417849e-01 3.0105797567136888e-01 3.0105787416907925e-01 3.0105773998914315e-01 3.0105756261137079e-01 3.0105732812875174e-01 3.0105701815715663e-01 3.0105660839404769e-01 3.0105606671324081e-01 3.0105535064644029e-01 3.0105440405411366e-01 3.0105315272497674e-01 3.0105149855933933e-01 3.0104931188090112e-01 3.0104642127517928e-01 3.0104260015974676e-01 3.0103754903640645e-01 3.0103087203936385e-01 3.0102204595027537e-01 3.0101037926769642e-01 3.0099495815114186e-01 3.0097457505300423e-01 3.0094763453238965e-01 3.0091202902298586e-01 3.0086497508794563e-01 3.0080279780048702e-01 3.0072064717636493e-01 3.0061212587449376e-01 3.0046880149853633e-01 3.0027956964787050e-01 3.0002982539136280e-01 2.9970039138900145e-01 2.9926614138997110e-01 2.9869425034787361e-01 2.9794200097125179e-01 2.9695408861095829e-01 2.9565940486677272e-01 2.9396736631013559e-01 2.9176402073298813e-01 2.8890845478736998e-01 2.8523049779385673e-01 2.8053140883406785e-01 2.7459012231506902e-01 2.6717850052999892e-01 2.5808930929942975e-01 2.4717910118476480e-01 2.3442307199352422e-01 2.1996872431689579e-01 2.0416112506535677e-01 1.8750300568698472e-01 1.7052546299687843e-01 1.5359751475998137e-01 1.3678544778696830e-01 1.1991339302221654e-01 1.0286220989822926e-01 8.5888399208535804e-02 6.9644949074992926e-02 5.4871229787039850e-02 4.2043536874906057e-02 3.1249676170048167e-02 2.2332105206289865e-02 1.5138868235657441e-02 9.6227251115645323e-03 5.7089316932365328e-03 3.1561421121873573e-03 1.6043175544854901e-03 7.1939497317967740e-04 2.6439014081167252e-04 7.2290297237494547e-05 1.3120349344360046e-05 1.3764974372946876e-06 6.9890689030574948e-08 1.3594353281319771e-09 7.4376704939374899e-12 7.6086862010886486e-15 8.4827644523379875e-19 5.0489968478928918e-24 6.2463067319420374e-31 4.6153718710779052e-40 3.9175210693382699e-52 4.3211713300379237e-68 3.4741196837902158e-89
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8 -1 2.9067784836865189e+02 2.9067770092814828e+02 2.9067750602095447e+02 2.9067724836594545e+02 2.9067690776263834e+02 2.9067645750764655e+02 2.9067586230147651e+02 2.9067507548169311e+02 2.9067403536577683e+02 2.9067266041739458e+02 2.9067084285787462e+02 2.9066844022336494e+02 2.9066526420812590e+02 2.9066106592339003e+02 2.9065551642333571e+02 2.9064818098409535e+02 2.9063848514145127e+02 2.9062566986337066e+02 2.9060873241070800e+02 2.9058634836796455e+02 2.9055676893804133e+02 2.9051768580951824e+02 2.9046605363070222e+02 2.9039785726714712e+02 2.9030780749970904e+02 2.9018894461207276e+02 2.9003212451767024e+02 2.8982535703605180e+02 2.8955296147615280e+02 2.8919450250146866e+02 2.8872347248372074e+02 2.8810570071590683e+02 2.8729750410372793e+02 2.8624366250670437e+02 2.8487542496268986e+02 2.8310895524051068e+02 2.8084492716990957e+02 2.7797037495248128e+02 2.7436430482919917e+02 2.6990872639643817e+02 2.6450612780623533e+02 2.5810215391467648e+02 2.5070754275656304e+02 2.4240666427041938e+02 2.3333522160794809e+02 2.2361603935074010e+02 2.1326975783683608e+02 2.0216099943368380e+02 1.9005607193982684e+02 1.7679504028839807e+02 1.6245063786593391e+02 1.4732710165508018e+02 1.3180734156115142e+02 1.1618886381253162e+02 1.0063597620883169e+02 8.5289676700397308e+01 7.0434799558537534e+01 5.6524394571377471e+01 4.4005044443681655e+01 3.3140381780125644e+01 2.4014467987689848e+01 1.6646226031388171e+01 1.1023916369638444e+01 7.0239116957206384e+00 4.3728893761583789e+00 2.7326707537369166e+00 1.7966833341378670e+00 1.3116272577029493e+00 1.0710706544391539e+00 9.2807581309777221e-01 8.0228371596304882e-01 6.6576594303069436e-01 5.2203017939035257e-01 3.8607967162303525e-01 2.6949711251794306e-01 1.7633918259737413e-01 1.0692877801853798e-01 6.0109325049147261e-02 3.1863701014722454e-02 1.5964700656913414e-02 7.1490985999124535e-03 2.5519352601333818e-03 6.3232136990670946e-04 9.3756872651218107e-05 6.9814821421956915e-06 2.0835246591391618e-07 1.8517029907910477e-09 3.3112769754947961e-12 7.0987128317425243e-16 9.2042519796535902e-21 2.9224872609989573e-27 6.8783521880659645e-36 2.4728019802069429e-47 1.6829923667135499e-62 1.3724339821295620e-82
8 +0 2.9110060808140298e+02 2.9110046049981258e+02 2.9110026540611017e+02 2.9110000750454918e+02 2.9109966657531493e+02 2.9109921588947032e+02 2.9109862011374258e+02 2.9109783254104241e+02 2.9109679142982679e+02 2.9109541516573842e+02 2.9109359586696780e+02 2.9109119093333436e+02 2.9108801187889975e+02 2.9108380957671710e+02 2.9107825476617137e+02 2.9107091230736160e+02 2.9106120718628381e+02 2.9104837964445181e+02 2.9103142598296296e+02 2.9100902051857344e+02 2.9097941278004828e+02 2.9094029224590605e+02 2.9088861064820003e+02 2.9082034900665349e+02 2.9073021303434354e+02 2.9061123634385501e+02 2.9045426607891250e+02 2.9024730055140145e+02 2.8997464400600387e+02 2.8961584144887695e+02 2.8914435970933704e+02 2.8852599507790370e+02 2.8771702213950931e+02 2.8666216703041368e+02 2.8529261151598774e+02 2.8352443667654575e+02 2.8125821726752861e+02 2.7838087294751483e+02 2.7477128414678663e+02 2.7031133249403894e+02 2.6490339122979054e+02 2.5849302394432226e+02 2.5109094195420110e+02 2.4278155243810170e+02 2.3370064002741151e+02 2.2397109268528149e+02 2.1361350371359546e+02 2.0249227411431482e+02 1.9037336151510934e+02 1.7709652128833969e+02 1.6273439109624940e+02 1.4759134755410034e+02 1.3205053516368650e+02 1.1640962031771782e+02 1.0083294255726378e+02 8.5461541335274092e+01 7.0580548550196639e+01 5.6643677437226344e+01 4.4098388107110679e+01 3.3209195093926269e+01 2.4061033143289077e+01 1.6673797761649848e+01 1.1036728670082342e+01 7.0267958093420040e+00 4.3705270333494770e+00 2.7287669699421153e+00 1.7932758679462919e+00 1.3087731983326487e+00 1.0672509191852708e+00 9.2123999731103279e-01 7.9086982448955834e-01 6.4917765938789429e-01 5.0065290756566327e-01 3.6118069413586912e-01 2.4317581772832622e-01 1.5131557125066447e-01 8.5812751930716746e-02 4.4384979935850881e-02 2.1502173813030941e-02 9.9722859107889313e-03 4.2480230049096934e-03 1.4777558611186014e-03 3.6136365067849005e-04 5.3135586151602928e-05 3.9319683416069288e-06 1.1676987223127105e-07 1.0338043256209246e-09 1.8431904080668780e-12 3.9423694524039376e-16 5.1027071884895113e-21 1.6180018167349790e-27 3.8042041144308692e-36 1.3665589932317662e-47 9.2952790534996630e-63 7.5766157728776280e-83
8 +1 2.9179597503409656e+02 2.9179582717437830e+02 2.9179563171300936e+02 2.9179537332541452e+02 2.9179503175367620e+02 2.9179458021848399e+02 2.9179398331997288e+02 2.9179319426303812e+02 2.9179215118977430e+02 2.9179077233201519e+02 2.9178894960463941e+02 2.9178654013872017e+02 2.9178335509309818e+02 2.9177914487131238e+02 2.9177357959220598e+02 2.9176622329578004e+02 2.9175649988429382e+02 2.9174364816730400e+02 2.9172666255410309e+02 2.9170421486266576e+02 2.9167455132209244e+02 2.9163535705519263e+02 2.9158357804696715e+02 2.9151518773891917e+02 2.9142488185941261e+02 2.9130568087971733e+02 2.9114841467230065e+02 2.9094105889266376e+02 2.9066788813914889e+02 2.9030840875371570e+02 2.8983603736291099e+02 2.8921650545544315e+02 2.8840600462653953e+02 2.8734915584888313e+02 2.8597700951713534e+02 2.8420548576699292e+02 2.8193496739992139e+02 2.7905215360616972e+02 2.7543568503267716e+02 2.7096720325442260e+02 2.6554887220218103e+02 2.5912611800719270e+02 2.5170962703541451e+02 2.4338391077690872e+02 2.3428494659070739e+02 2.2453578144578995e+02 2.1415695860459496e+02 2.0301250560143933e+02 1.9086778072964796e+02 1.7756204066391729e+02 1.6316784305020792e+02 1.4798992243807135e+02 1.3241193810695771e+02 1.1673191006662977e+02 1.0111427877182172e+02 8.5700182346721505e+01 7.0775264448741652e+01 5.6794315741316979e+01 4.4206159267605173e+01 3.3276635039715913e+01 2.4092006373216130e+01 1.6673725849253376e+01 1.1012690392422378e+01 6.9870085794834340e+00 4.3232115104447066e+00 2.6807846646018834e+00 1.7489091885731172e+00 1.2691728063157426e+00 1.0311668468115249e+00 8.8675108770057653e-01 7.5675089203311940e-01 6.1505474650543590e-01 4.6660060569909256e-01 3.2774975278867313e-01 2.1177820920220608e-01 1.2413056377805219e-01 6.4781997406589048e-02 3.0031470221961537e-02 1.2795938412451566e-02 5.2550041900662706e-03 2.0501328738887739e-03 6.7732916850441393e-04 1.6069209236304675e-04 2.3134557592055853e-05 1.6837870824036508e-06 4.9344452237971610e-08 4.3227581246440734e-10 7.6434321966153233e-13 1.6243024412645667e-16 2.0918796649099306e-21 6.6075735487066188e-28 1.5489746249418217e-36 5.5517366211414671e-48 3.7697785262198526e-63 3.0687409939055806e-83
8 +2 2.9301885885210442e+02 2.9301871044949837e+02 2.9301851427046392e+02 2.9301825493416180e+02 2.9301791210829316e+02 2.9301745891522199e+02 2.9301685982511162e+02 2.9301606787103503e+02 2.9301502096797014e+02 2.9301363704752396e+02 2.9301180762771997e+02 2.9300938931506255e+02 2.9300619257502041e+02 2.9300196689468663e+02 2.9299638118166126e+02 2.9298899787519423e+02 2.9297923876221319e+02 2.9296633985726038e+02 2.9294929187713882e+02 2.9292676176263461e+02 2.9289698930262819e+02 2.9285765111878851e+02 2.9280568197981660e+02 2.9273704053783479e+02 2.9264640303656319e+02 2.9252676430482876e+02 2.9236892051762777e+02 2.9216080313261239e+02 2.9188662892542470e+02 2.9152582884599292e+02 2.9105172166284956e+02 2.9042991260676706e+02 2.8961643167673316e+02 2.8855569524070393e+02 2.8717849838555770e+02 2.8540044894446942e+02 2.8312155804376079e+02 2.8022809930371074e+02 2.7659825266566463e+02 2.7211320196585069e+02 2.6667471773322757e+02 2.6022797758305728e+02 2.5278364288547706e+02 2.4442645444340829e+02 2.3529279146838252e+02 2.2550603750650163e+02 2.1508667107613442e+02 2.0389804100695085e+02 1.9170440995329631e+02 1.7834415102307281e+02 1.6388974775564003e+02 1.4864668504287513e+02 1.3299964405254579e+02 1.1724733298439524e+02 1.0155438327419554e+02 8.6062097615923022e+01 7.1057008091464894e+01 5.6995767010554893e+01 4.4329522076719627e+01 3.3326379564590589e+01 2.4074977667204852e+01 1.6599721167870300e+01 1.0894826634347261e+01 6.8409619978963150e+00 4.1652369091602859e+00 2.5251064735617033e+00 1.6052359681115400e+00 1.1415545151387685e+00 9.1941628757238869e-01 7.8961186460877730e-01 6.7385455943361183e-01 5.4615079363502539e-01 4.1019572615481403e-01 2.8165680391106945e-01 1.7476196027539526e-01 9.6263563089708057e-02 4.6038065614865585e-02 1.8991000467281961e-02 7.0159295955474462e-03 2.4940394980997949e-03 8.6818765330877532e-04 2.6591245899969880e-04 5.9916095773780312e-05 8.2850757100331942e-06 5.8290650121157214e-07 1.6602313756727717e-08 1.4205944874853597e-10 2.4644021061818120e-13 5.1578994561065978e-17 6.5632913155990273e-22 2.0537467067371380e-28 4.7795228241976242e-37 1.7034478610591646e-48 1.1517087252145078e-63 9.3444593934639259e-84
The first line describes the radial grid used for the atomic database. In this
case it is a logarithmic grid with lowest grid point at
bohr, highest grid point at
bohr, and 100 points in total. All
subsequent lines are the averaged densities of the atom-charge states at these
radial grids points. Densities with a value lower than 1e-100 bohr^-3 are
omitted to save space and time.
5.4. Partitioning tools¶
All other scripts besides hi-atomdb.py have the same usage description:
toon@poony ~/hipart/examples/001-usage/atoms> hi-some-script.py --help
Usage: hi-some-script.py [options] gaussian.fchk scheme [scheme parameters]
...
In the first subsection the usage will be discussed extensively for the script hi-charges.py, but this discussion also applies to all subsequent scripts. In every subsection, one of the possible partitioning schemes is used, but remind that each script works with all four partitioning schemes.
5.4.1. Atomic charges¶
Effective atomic charges are computed with hi-charges.py. The online help is as follows:
toon@poony ~/hipart/examples/001-usage/atoms> cd ..
toon@poony ~/hipart/examples/001-usage> hi-charges.py --help
Usage: hi-charges.py [options] gaussian.fchk scheme [scheme parameters]
hi-charges.py computes effective atomic charges.
The effective atomic charges are the monopole terms in the multipole expansion
of each atomic contribution to the density plus the monopole of the nucleus. The
atomic densities are obtained from the 'scheme' specified at the command line.
The file gaussian.fchk is a formatted checkpoint file from a Gaussian
computation. To obtain this file, add the following line on top of a Gaussian
com-file (before running the job)
%chk=gaussian.chk
After the Gaussian computation transform this binary checkpoint file into
a text file with the ``formchk`` program of the Gaussian software suite:
formchk gaussian.chk gaussian.fchk
Partitioning schemes:
* Becke's Smooth Voronoi Partitioning
scheme = becke
scheme parameters = [k] [r_low r_high steps]
The parameter k is optional and defaults to 3. It is the number of
iterations in the definition of the weight function in Becke's paper.
Three additional parameters can be provided of the file rs.bin is not yet
present in the work directory. The first two, r_low and r_high, are the
first and the last point on the logarithmic radial grid in angstrom. The
third, steps, is the number of grid points on the radial grid. The default
is 2.0e-5, 20.0 and 100, respectively.
Becke, A. D. J. Chem. Phys. 1988, 88, 2547-2553.
http://dx.doi.org/10.1063/1.454033
* Hirshfeld Partitioning
scheme = hirsh
scheme parameters = densities.txt
The file densities.txt is generated with the script hi-atomdb.py. It
contains spherically averaged densities of individual atoms. Make sure all
the atoms present in the molecule of interest are included in the file
densities.txt
Hirshfeld, F. L. Theor. Chim. Acta 1977, 44, 129-138.
http://dx.doi.org/10.1007/BF00549096
* Hirshfeld-I Partitioning
scheme = hirshi
scheme parameters = densities.txt
The file densities.txt is generated with the script hi-atomdb.py. It
contains spherically averaged densities of individual atoms. Make sure all
the atoms present in the molecule of interest are included in the file
densities.txt
Bultinck, P.; Van Alsenoy, C.; Ayers, P. W.; Dorca, R. C. J. Chem. Phys.
2007, 126, 144111.
http://dx.doi.org/10.1063/1.2715563
* Iterative Stockholder Partitioning
scheme = isa
scheme parameters = [r_low r_high steps]
Three additional parameters can be provided of the file rs.bin is not yet
present in the work directory. The first two, r_low and r_high, are the
first and the last point on the logarithmic radial grid in angstrom. The
third, steps, is the number of grid points on the radial grid. The default
is 2.0e-5, 20.0 and 100, respectively.
Lillestolen, T. C.; Wheatley, R. J. Chem. Commun. 2008, 5909-5911.
http://dx.doi.org/10.1039/b812691g
Options:
-h, --help show this help message and exit
-l LEBEDEV, --lebedev=LEBEDEV
The number of grid points for the atomic grids.
[default=110]. Select from: 6, 14, 26, 38, 50, 74, 86,
110, 146, 170, 194, 230, 266, 302, 350, 434, 590, 770,
974, 1202, 1454, 1730, 2030, 2354, 2702, 3074, 3470,
3890, 4334, 4802, 5294, 5810
-c, --clean Remove the workdir before and after the computation.
--no-work Do not save intermediate results in work directory for
later reuse.
--no-output Do not write any output to text files.
-q, --quiet Do not write any screen output.
--no-fix-total-charge
Do not correct the total charge.
--no-random Do not randomly rotate angular grids.
--save-mem Try to be less memory hungry at the expense of a
little efficiency.
Specific options for the iterative partitioning schemes:
-t THRESHOLD, --threshold=THRESHOLD
When the maximum change in the charges drops below
this threshold value, the iteration stops.
[default=0.0001]
--max-iter=MAX_ITER
Maximum number of iterations in self-consistent
procedures. [default=500]
We will first compute the Becke charges, as the Becke scheme is the simplest of the four partitioning methods. The screen output gives some progress information of the various parts of the program. This is screen output in the case of the alanine molecule:
toon@poony ~/hipart/examples/001-usage> hi-charges.py alanine.fchk becke
---TIME--- ---------------------------------LOG---------------------------------
0.03 BEGIN Loading Electronic structure
0.04 Data read from: alanine.fchk ()
0.04 Restricted: True
0.04 Orbitals present: True
0.04 Spin density present: False
0.04 Number of alpha electrons: 24
0.04 Number of beta electrons: 24
0.04 Number of electrons: 48
0.04 Total charge: 0
0.04 Number of atoms: 13
0.04 Chemical formula: O2NC3H7
0.04 END Loading Electronic structure
0.04 BEGIN Net and overlap populations
0.04 BEGIN Atomic grids
Computing/Loading atomic grids and distances: 0%.........5%......
...11%.........17%.........23%.........29%.........35%.........41
%.........47%.........53%.........59%.........65%.........71%....
.....76%.........82%.........88%.........94%........100%
0.21 END Atomic grids
0.21 BEGIN Molecular density on atomic grids
Computing/Loading densities: 0%.........76%..100%
0.21 END Molecular density on atomic grids
0.21 BEGIN Atomic charges
0.21 Written alanine.hipart/becke_charges.txt
0.21 END Atomic charges
0.21 BEGIN Atomic weights on other atoms' grids.
0.21 BEGIN Defining atomic weight functions (own atomic grid)
0.21 Trying to load weight functions
0.21 END Defining atomic weight functions (own atomic grid)
0.21 BEGIN Becke's Smooth Voronoi Partitioning
Computing/Loading cell functions: 0%.........76%..100%
0.24 END Becke's Smooth Voronoi Partitioning
Computing off-diagonal atom weights: 0%.........5%.........11%...
......17%.........23%.........29%.........35%.........41%........
.47%.........53%.........59%.........65%.........71%.........76%.
........82%.........88%.........94%........100%
0.25 END Atomic weights on other atoms' grids.
Integrating over products of stockholder weights: 0%.........10%...
......21%.........32%.........43%.........54%.........65%.........7
6%.........87%.........98%100%
0.33 Written alanine.hipart/becke_net_overlap.txt
0.33 END Net and overlap populations
The entire screen output is conceived as a call graph that shows in which part of Hipart the program is currently active. The order of the routines is determined by an internal dependency mechanism that allows many different workflows through the code. The first part of the output is a summary of the electronic structure stored in the file alanine.fchk. From then on the actual computation is carried out.
All output is stored in a subdirectory of the current directory whose name is based on the filename of the formatted checkpoint file. E.g. if the formatted checkpoint file is alanine.fchk, then the output directory is alanine.hipart. In this example the following output files can be found in alanine.hipart:
toon@poony ~/hipart/examples/001-usage> ls alanine.hipart/
becke_charges.txt work
All output that depends on the choice of the partitioning scheme is prefixed with the corresponding key, e.g. in this case we have becke_charges.txt. The work directory contains cached binary intermediate results that will be reused when another HiPart script (or the same script with different options) is executed afterwards. It can always be removed manually, or with the --clean option it is automatically removed at the end of the script. In this example the work directory contains the following files:
toon@poony ~/hipart/examples/001-usage> ls alanine.hipart/work/
atom00000_becke_atweights.bin atom00007_becke_atweights.bin
atom00000.bin atom00007.bin
atom00000_cell_functions.bin atom00007_cell_functions.bin
atom00000_moldens.bin atom00007_moldens.bin
atom00001_becke_atweights.bin atom00008_becke_atweights.bin
atom00001.bin atom00008.bin
atom00001_cell_functions.bin atom00008_cell_functions.bin
atom00001_moldens.bin atom00008_moldens.bin
atom00002_becke_atweights.bin atom00009_becke_atweights.bin
atom00002.bin atom00009.bin
atom00002_cell_functions.bin atom00009_cell_functions.bin
atom00002_moldens.bin atom00009_moldens.bin
atom00003_becke_atweights.bin atom00010_becke_atweights.bin
atom00003.bin atom00010.bin
atom00003_cell_functions.bin atom00010_cell_functions.bin
atom00003_moldens.bin atom00010_moldens.bin
atom00004_becke_atweights.bin atom00011_becke_atweights.bin
atom00004.bin atom00011.bin
atom00004_cell_functions.bin atom00011_cell_functions.bin
atom00004_moldens.bin atom00011_moldens.bin
atom00005_becke_atweights.bin atom00012_becke_atweights.bin
atom00005.bin atom00012.bin
atom00005_cell_functions.bin atom00012_cell_functions.bin
atom00005_moldens.bin atom00012_moldens.bin
atom00006_becke_atweights.bin becke_charges.bin
atom00006.bin becke_populations.bin
atom00006_cell_functions.bin context
atom00006_moldens.bin
Certain choices (grids and some other options) affect the contents of the files in the work directory. For example, when different grids are used in a second run, the work directory is no longer usable. In the following example we try to use a different Lebedev grid, which causes an error message:
toon@poony ~/hipart/examples/001-usage> hi-charges.py alanine.fchk becke -l14
---TIME--- ---------------------------------LOG---------------------------------
0.03 BEGIN Loading Electronic structure
0.05 Data read from: alanine.fchk ()
0.05 Restricted: True
0.05 Orbitals present: True
0.05 Spin density present: False
0.05 Number of alpha electrons: 24
0.05 Number of beta electrons: 24
0.05 Number of electrons: 48
0.05 Total charge: 0
0.05 Number of atoms: 13
0.05 Chemical formula: O2NC3H7
0.05 END Loading Electronic structure
The existing work directory (alanine.hipart/work) contains incompatible data.
Try using the --clean option once.
The following mismatch was detected in the work directory:
'ALebedevIntGrid(110)' (found in work) versus 'ALebedevIntGrid(14)' (current script) for property 'agrid'
Either use the --clean option, or stick to the options used in the first execution of a HiPart script.
The output file becke_charges.txt has the following contents:
toon@poony ~/hipart/examples/001-usage> cat alanine.hipart/becke_charges.txt
number of atoms: 13
i Z Charge
--------------------------------
0 O 8 -0.211830150924
1 O 8 -0.226117942774
2 N 7 -0.452908032027
3 C 6 -0.175834726603
4 C 6 -0.370816060791
5 C 6 -0.045089923958
6 H 1 0.195365605367
7 H 1 0.161545954455
8 H 1 0.146588570392
9 H 1 0.159295929951
10 H 1 0.232914891450
11 H 1 0.263608058943
12 H 1 0.323277826520
--------------------------------
It is easily processed with other programs in a follow-up analysis. Note that the same data are also present in binary format in the file work/becke_charges.bin. The module hipart.io contains Python routines to read and write most HiPart output files in your own Python scripts.
Although the numbers in the output file are printed with 13 decimals, one must realize that precision is not the same as accuracy. The accuracy of these numbers depends on the choice of the radial and angular grids. The accuracy is also inherently limited by the choices made in the Gaussian input file and the precision of the numbers in the formatted checkpoint file.
With the default options, even a second run of the program (after removing the work directory) will result in slightly different numbers:
toon@poony ~/hipart/examples/001-usage> rm -r alanine.hipart
toon@poony ~/hipart/examples/001-usage> hi-charges.py alanine.fchk becke
<output omitted>
toon@poony ~/hipart/examples/001-usage> cat alanine.hipart/becke_charges.txt
number of atoms: 13
i Z Charge
--------------------------------
0 O 8 -0.212286001098
1 O 8 -0.225962908984
2 N 7 -0.453499377490
3 C 6 -0.175833206860
4 C 6 -0.371150449115
5 C 6 -0.044682232259
6 H 1 0.195448605472
7 H 1 0.161683191955
8 H 1 0.146702787517
9 H 1 0.159442914700
10 H 1 0.233037455526
11 H 1 0.263694328503
12 H 1 0.323404892132
--------------------------------
This is due to the random rotations applied to the angular grids used for the atom-centered numerical integrations. These randomly rotated angular grids have several advantages:
- It improves the accuracy due to compensation of errors.
- It removes directionional preference in the grids and.
- It allows simple estimates of the accuracy by simply rerunning the same analysis twice.
Read the section Tuning integration grids for more details on numerical errors and how to control them.
One can always disable the random rotation of the angular grids with the --no-random option. This does not give more reproducible results because the charges systematically depend on the orientation of the molecule when --no-random is used.
For the sake of completeness, these are the commands to compute the charges on the same molecule with the three other partitioning schemes:
toon@poony ~/hipart/examples/001-usage> hi-charges.py alanine.fchk hirsh atoms/densities.txt
toon@poony ~/hipart/examples/001-usage> hi-charges.py alanine.fchk hirshi atoms/densities.txt
toon@poony ~/hipart/examples/001-usage> hi-charges.py alanine.fchk isa
5.4.2. Atomic dipoles¶
Atomic dipoles are compute with the program hi-dipoles.py. Like most HiPart scripts it takes the same arguments and options as the script hi-charges.py, which are discussed in the previous section. The online help starts as follows:
toon@poony ~/hipart/examples/001-usage> hi-dipoles.py --help
Usage: hi-dipoles.py [options] gaussian.fchk scheme [scheme parameters]
hi-dipoles.py computes atomic charges and dipoles.
These atomic charges and dipoles are the monopole and dipole terms in the
multipole expansion of each atomic contribution to the density. The atomic
densities are obtained from the 'scheme' specified at the command line.
...
The screen output is also very similar. Depending on the previously executed scripts, e.g. hi-charges.py, some intermediate results can be loaded from the work directory and do not have to be computed again.
The following example computes the atomic Iterative Hirshfeld dipoles in the alanine molecule:
toon@poony ~/hipart/examples/001-usage> hi-dipoles.py alanine.fchk hirshi atoms/densities.txt
---TIME--- ---------------------------------LOG---------------------------------
0.02 BEGIN Loading Electronic structure
0.05 Data read from: alanine.fchk ()
0.05 Restricted: True
0.05 Orbitals present: True
0.05 Spin density present: False
0.05 Number of alpha electrons: 24
0.05 Number of beta electrons: 24
0.05 Number of electrons: 48
0.05 Total charge: 0
0.05 Number of atoms: 13
0.05 Chemical formula: O2NC3H7
0.05 END Loading Electronic structure
0.05 BEGIN Atomic dipoles
0.05 BEGIN Atomic grids
Computing/Loading atomic grids and distances: 0%.........5%......
...11%.........17%.........23%.........29%.........35%.........41
%.........47%.........53%.........59%.........65%.........71%....
.....76%.........82%.........88%.........94%........100%
0.23 END Atomic grids
0.23 BEGIN Molecular density on atomic grids
Computing/Loading densities: 0%.........76%..100%
0.24 END Molecular density on atomic grids
0.24 BEGIN Defining atomic weight functions (own atomic grid)
0.24 Trying to load weight functions
0.24 Could not load all weight functions from workdir. Computing them.
0.24 BEGIN Iterative Hirshfeld
0.24 Iteration Max change Total charge
0.35 0 2.75474e-01 8.08195e-04
0.46 1 1.74579e-01 8.27390e-04
0.57 2 1.22188e-01 8.17670e-04
0.67 3 8.35207e-02 8.05282e-04
0.78 4 5.68051e-02 7.94794e-04
0.89 5 3.95991e-02 7.86665e-04
1.00 6 2.83158e-02 7.80534e-04
1.10 7 2.03801e-02 7.75934e-04
1.21 8 1.47532e-02 7.72484e-04
1.32 9 1.16319e-02 7.69893e-04
1.43 10 9.22139e-03 7.67926e-04
1.53 11 7.31924e-03 7.66432e-04
1.64 12 5.81607e-03 7.65289e-04
1.75 13 4.62651e-03 7.64410e-04
1.86 14 3.68384e-03 7.63730e-04
1.96 15 2.93585e-03 7.63203e-04
2.07 16 2.34162e-03 7.62792e-04
2.18 17 1.86901e-03 7.62470e-04
2.29 18 1.49276e-03 7.62217e-04
2.39 19 1.19293e-03 7.62018e-04
2.50 20 9.53810e-04 7.61860e-04
2.61 21 7.62963e-04 7.61736e-04
2.72 22 6.10542e-04 7.61637e-04
2.82 23 4.88738e-04 7.61558e-04
2.93 24 3.91352e-04 7.61495e-04
3.04 25 3.13453e-04 7.61446e-04
3.15 26 2.51116e-04 7.61406e-04
3.25 27 2.01215e-04 7.61374e-04
3.36 28 1.61257e-04 7.61348e-04
3.47 29 1.29253e-04 7.61328e-04
3.58 30 1.03613e-04 7.61312e-04
3.68 31 8.30677e-05 7.61299e-04
3.68 END Iterative Hirshfeld
3.79 Writing results to workdir
3.79 END Defining atomic weight functions (own atomic grid)
Computing dipoles: 0%.........76%..100%
3.80 Written alanine.hipart/hirshi_dipoles.txt
3.80 END Atomic dipoles
The output is stored in the file alanine.hipart/hirshi_dipoles.txt.
toon@poony ~/hipart/examples/001-usage> cat alanine.hipart/hirshi_dipoles.txt
number of atoms: 13
i Z Dipole-X Dipole-Y Dipole-Z Dipole-norm
-------------------------------------------------------------------------------
0 O 8 -0.070971228298 0.058939792138 -0.049805346138 0.104839815181
1 O 8 0.003427569749 -0.080037505223 0.049155449234 0.093989407202
2 N 7 0.032989636016 0.113658772017 0.007558083216 0.118590712801
3 C 6 -0.146086700416 -0.059768791465 0.048156232761 0.165023196022
4 C 6 -0.002296149335 -0.025989999686 -0.034734865594 0.043442643488
5 C 6 0.072468606622 0.028445758103 -0.004440958158 0.077978088006
6 H 1 -0.017722304183 0.043438610385 -0.095282537525 0.106206190473
7 H 1 0.040027317026 0.073894901988 -0.002654305078 0.084081436617
8 H 1 -0.077314096654 0.019857082661 -0.023393180432 0.083180611707
9 H 1 -0.003145374821 -0.026898862075 0.079983515109 0.084444093061
10 H 1 -0.064465604534 0.011013617024 -0.003193093485 0.065477551679
11 H 1 -0.000769543878 -0.041022742610 0.048994172037 0.063905293229
12 H 1 0.062268957875 -0.000049451488 0.011842353616 0.063385068427
5.4.3. Testing charges and dipoles on the ESP grid¶
It is well known that some charge population schemes are better in reproducing the electrostatic potential around a molecule than others. [Bultinck2009] Since this often a desirable property, the script hi-esp-test.py can be used to verify the quality of the atomic charges and/or dipoles obtained with a partitioning scheme.
hi-esp-test.py performs two tests:
- It computes the molecular dipole moment based on (i) charges, (ii) dipoles, and (iii) charges with dipoles derived from the population analysis. These molecular dipole approximations are compared with the molecular dipole vector reported in the formatted checkpoint file.
- A second test is based on an Electrostatic Potential (ESP) Fitting cost function for the charges and the dipoles. With this cost function the script computes how well (i) the charges, (ii) the dipoles, and (iii) the charges with the dipoles reproduce the ESP around the molecule.
Again, the script is executed in the same style as all other scripts. See the documentation of hi-charges.py for more details. The example below tests the charges and dipoles obtained with a Iterative Hirshfeld partitioning:
toon@poony ~/hipart/examples/001-usage> hi-esp-test.py alanine.fchk hirshi atoms/densities.txt
---TIME--- ---------------------------------LOG---------------------------------
0.02 BEGIN Loading Electronic structure
0.05 Data read from: alanine.fchk ()
0.05 Restricted: True
0.05 Orbitals present: True
0.05 Spin density present: False
0.05 Number of alpha electrons: 24
0.05 Number of beta electrons: 24
0.05 Number of electrons: 48
0.05 Total charge: 0
0.05 Number of atoms: 13
0.05 Chemical formula: O2NC3H7
0.05 END Loading Electronic structure
0.05 BEGIN Testing charges and dipoles on ESP grid.
0.05 BEGIN Atomic charges
0.05 BEGIN Atomic grids
Computing/Loading atomic grids and distances: 0%.........5%....
.....11%.........17%.........23%.........29%.........35%.......
..41%.........47%.........53%.........59%.........65%.........7
1%.........76%.........82%.........88%.........94%........100%
0.23 END Atomic grids
0.23 BEGIN Molecular density on atomic grids
Computing/Loading densities: 0%.........76%..100%
0.23 END Molecular density on atomic grids
0.23 BEGIN Defining atomic weight functions (own atomic grid)
0.23 Trying to load weight functions
0.24 END Defining atomic weight functions (own atomic grid)
Computing charges: 0%.........76%..100%
0.24 Written alanine.hipart/hirshi_charges.txt
0.24 END Atomic charges
0.24 BEGIN Atomic dipoles
0.24 Written alanine.hipart/hirshi_dipoles.txt
0.24 END Atomic dipoles
0.24 BEGIN Computing the ESP cost function
0.24 BEGIN Molecular density on the molecular grid
0.24 BEGIN Molecular grid
0.24 BEGIN Estimating noble gas core radii
0.24 END Estimating noble gas core radii
Constructing molecular grid: 0%.........33%.........66%......
...100%
1.20 END Molecular grid
1.70 END Molecular density on the molecular grid
1.70 BEGIN Molecular potential on the molecular grid
1.70 This may take a minute. Hang on.
41.90 END Molecular potential on the molecular grid
42.11 Written alanine.hipart/mol_esp_cost.txt
42.11 END Computing the ESP cost function
42.11 Written alanine.hipart/hirshi_esp_test.txt
42.11 END Testing charges and dipoles on ESP grid.
This script computes the charges and dipoles with the given scheme if they are not present yet. Then the matrix representation of the cost function is constructed and stored in the file mol_esp_cost.txt. The results of the test are written in hirsh_esp_test.txt. The output in this example is:
toon@poony ~/hipart/examples/001-usage> cat alanine.hipart/hirshi_esp_test.txt
Reproduction of the molecular dipole
-------------------------------------------------------------------------------
Dipole-X Dipole-Y Dipole-Z Dipole-norm
-------------------------------------------------------------------------------
charges (q) -0.054069592658 0.643203611818 0.166578561018 0.666620449804
dipoles (p) -0.171588914832 0.115481181768 0.032185519564 0.209319293676
q and p -0.225658507489 0.758684793585 0.198764080581 0.816107552807
total density -0.226635821000 0.757958121000 0.199290781000 0.815831553657
-------------------------------------------------------------------------------
Reproduction of the external molecular ESP
-------------------------------------------------------------
RMSD RMS CORRELATION
-------------------------------------------------------------
charges (q) 1.67208e-03 6.38350e-03 0.98
dipoles (p) 6.43865e-03 1.75708e-03 0.55
q and p 5.20147e-04 7.21791e-03 1.00
total density 7.22795e-03
-------------------------------------------------------------
As can be seen in the first section, the charges with the dipoles are able to reproduce the QM dipole moment from Gaussian up to some numerical error. This error can be controlled to some extent by tuning the grids. In principle, the correspondence should be exact. The dipole is reasonably well reproduced by just using the effective charges.
The second section of the output compares the QM ESP on the grid points is with the ESP generated by either the charges, the dipoles or the charges with dipoles. The first column is the root means square deviation over all grid points. The second column contains the root means square value of the ESP over all grid points. The third column contains the correlation coefficient between the approximate and QM ESP data. One can compute the relative errors by dividing an RMSD value by the RMS value of the total density. In this example the relative error (over all grid points) of the ESP predicted by the charges is 23%. Charges combined with dipoles give a fairly accurate description of the ESP around the molecule, with a relative error of 7%.
In both tests it is clear that the atomic dipoles contribute little to the ESP. This can be seen in two places: (i) the norm of the contribution to the molecular dipole of the atomic dipoles is small compared to the norm of the total molecular dipole, and (ii) the RMS value of the ESP contributions of the dipoles is relatively small.
5.4.4. Atomic multipole expansions¶
The multipole expansion of each atom, up to the hexadecapole, is computed with the script hi-multipoles.py. This is an example with the Iterative Stockholder Analysis:
toon@poony ~/hipart/examples/001-usage> hi-multipoles.py alanine.fchk isa
---TIME--- ---------------------------------LOG---------------------------------
0.02 BEGIN Loading Electronic structure
0.05 Data read from: alanine.fchk ()
0.05 Restricted: True
0.05 Orbitals present: True
0.05 Spin density present: False
0.05 Number of alpha electrons: 24
0.05 Number of beta electrons: 24
0.05 Number of electrons: 48
0.05 Total charge: 0
0.05 Number of atoms: 13
0.05 Chemical formula: O2NC3H7
0.05 END Loading Electronic structure
0.05 BEGIN Atomic multipoles (up to hexadecapols)
0.05 BEGIN Atomic grids
Computing/Loading atomic grids and distances: 0%.........5%......
...11%.........17%.........23%.........29%.........35%.........41
%.........47%.........53%.........59%.........65%.........71%....
.....76%.........82%.........88%.........94%........100%
0.23 END Atomic grids
0.23 BEGIN Molecular density on atomic grids
Computing/Loading densities: 0%.........76%..100%
0.23 END Molecular density on atomic grids
0.23 BEGIN Defining atomic weight functions (own atomic grid)
0.23 Trying to load weight functions
0.23 Could not load all weight functions from workdir. Computing them.
0.23 BEGIN Iterative Stockholder Analysis
0.23 Generating initial guess for the pro-atoms
0.24 Iteration Max change Total charge
0.34 0 6.31430e-01 1.91522e-04
0.45 1 6.20743e-02 2.32913e-04
0.56 2 4.84612e-02 2.85275e-04
0.67 3 3.93967e-02 3.26740e-04
0.78 4 3.23312e-02 3.34675e-04
0.89 5 2.69850e-02 3.19180e-04
1.00 6 2.29579e-02 3.03944e-04
1.11 7 1.98811e-02 2.59226e-04
1.21 8 1.74820e-02 2.28101e-04
1.32 9 1.55679e-02 2.36177e-04
1.43 10 1.40073e-02 2.22602e-04
1.54 11 1.27156e-02 1.84933e-04
1.65 12 1.16288e-02 2.21515e-04
1.76 13 1.06967e-02 2.37141e-04
1.86 14 9.89575e-03 2.02795e-04
1.97 15 9.19332e-03 2.24397e-04
2.08 16 8.57554e-03 2.08130e-04
2.19 17 8.02923e-03 2.12105e-04
2.30 18 7.53919e-03 1.99942e-04
2.41 19 7.10116e-03 2.01548e-04
2.51 20 6.70453e-03 1.96309e-04
2.62 21 6.34498e-03 1.98584e-04
2.73 22 6.01677e-03 1.90360e-04
2.84 23 5.71739e-03 1.90062e-04
2.95 24 5.44179e-03 1.91603e-04
3.05 25 5.18792e-03 1.89557e-04
3.16 26 4.95361e-03 1.87692e-04
3.27 27 4.73642e-03 1.83569e-04
3.38 28 4.53477e-03 1.85664e-04
3.48 29 4.34662e-03 1.86175e-04
3.59 30 4.17122e-03 1.85042e-04
3.70 31 4.00725e-03 1.81619e-04
3.81 32 3.85368e-03 1.82164e-04
3.92 33 3.70926e-03 1.78690e-04
4.02 34 3.57372e-03 1.77825e-04
4.13 35 3.44585e-03 1.76630e-04
4.24 36 3.32527e-03 1.75548e-04
4.35 37 3.21135e-03 1.74528e-04
4.45 38 3.10356e-03 1.73569e-04
4.56 39 3.00144e-03 1.72667e-04
4.67 40 2.90457e-03 1.71822e-04
4.78 41 2.81257e-03 1.71004e-04
4.88 42 2.72510e-03 1.70271e-04
4.99 43 2.64183e-03 1.69561e-04
5.10 44 2.56250e-03 1.68894e-04
5.21 45 2.48683e-03 1.68286e-04
5.31 46 2.41459e-03 1.67737e-04
5.42 47 2.34557e-03 1.67193e-04
5.53 48 2.27955e-03 1.66727e-04
5.64 49 2.21637e-03 1.66284e-04
5.74 50 2.15584e-03 1.65874e-04
5.85 51 2.09782e-03 1.65495e-04
5.96 52 2.04215e-03 1.65144e-04
6.07 53 1.98871e-03 1.64819e-04
6.17 54 1.93737e-03 1.64518e-04
6.28 55 1.88802e-03 1.64239e-04
6.39 56 1.84054e-03 1.63981e-04
6.50 57 1.79485e-03 1.63740e-04
6.60 58 1.75084e-03 1.63516e-04
6.71 59 1.70843e-03 1.63307e-04
6.82 60 1.66755e-03 1.63112e-04
6.92 61 1.62811e-03 1.62928e-04
7.03 62 1.59004e-03 1.62756e-04
7.14 63 1.55329e-03 1.62593e-04
7.25 64 1.51778e-03 1.62439e-04
7.35 65 1.48346e-03 1.62293e-04
7.46 66 1.45027e-03 1.62154e-04
7.57 67 1.41817e-03 1.62021e-04
7.68 68 1.38710e-03 1.61894e-04
7.78 69 1.35701e-03 1.61771e-04
7.89 70 1.32788e-03 1.61653e-04
8.00 71 1.29965e-03 1.61539e-04
8.10 72 1.27228e-03 1.61429e-04
8.21 73 1.24575e-03 1.61321e-04
8.32 74 1.22001e-03 1.61216e-04
8.43 75 1.19504e-03 1.61114e-04
8.53 76 1.17079e-03 1.61013e-04
8.64 77 1.14726e-03 1.60915e-04
8.75 78 1.12439e-03 1.60819e-04
8.86 79 1.10218e-03 1.60724e-04
8.96 80 1.08059e-03 1.60630e-04
9.07 81 1.05960e-03 1.60538e-04
9.18 82 1.03920e-03 1.60447e-04
9.29 83 1.01934e-03 1.60357e-04
9.39 84 1.00003e-03 1.60268e-04
9.50 85 9.81226e-04 1.60180e-04
9.61 86 9.62924e-04 1.60093e-04
9.72 87 9.45102e-04 1.60007e-04
9.82 88 9.27744e-04 1.59922e-04
9.93 89 9.10833e-04 1.59837e-04
10.04 90 8.94353e-04 1.59753e-04
10.15 91 8.78291e-04 1.59670e-04
10.25 92 8.62631e-04 1.59588e-04
10.36 93 8.47360e-04 1.59507e-04
10.47 94 8.32466e-04 1.59426e-04
10.58 95 8.17935e-04 1.59346e-04
10.68 96 8.03757e-04 1.59266e-04
10.79 97 7.89919e-04 1.59188e-04
10.90 98 7.76411e-04 1.59110e-04
11.01 99 7.63223e-04 1.59032e-04
11.11 100 7.50343e-04 1.58956e-04
11.22 101 7.37763e-04 1.58880e-04
11.33 102 7.25473e-04 1.58804e-04
11.44 103 7.13465e-04 1.58728e-04
11.56 104 7.01729e-04 1.58672e-04
11.66 105 6.90258e-04 1.58583e-04
11.77 106 6.79043e-04 1.58510e-04
11.88 107 6.68078e-04 1.58438e-04
11.99 108 6.57354e-04 1.58367e-04
12.09 109 6.46865e-04 1.58297e-04
12.20 110 6.36604e-04 1.58221e-04
12.31 111 6.26565e-04 1.58187e-04
12.42 112 6.16738e-04 1.57854e-04
12.52 113 6.07135e-04 1.53037e-04
12.63 114 5.97923e-04 1.72083e-04
12.74 115 5.87986e-04 1.57762e-04
12.85 116 5.79656e-04 1.58387e-04
12.95 117 5.70607e-04 1.58262e-04
13.06 118 5.61937e-04 1.58129e-04
13.17 119 5.53471e-04 1.58162e-04
13.27 120 5.45174e-04 1.57445e-04
13.38 121 5.37071e-04 1.57976e-04
13.49 122 5.29069e-04 1.57798e-04
13.60 123 5.21273e-04 1.57666e-04
13.70 124 5.13614e-04 1.57729e-04
13.81 125 5.06098e-04 1.56772e-04
13.92 126 4.98764e-04 1.57592e-04
14.03 127 4.91493e-04 1.56801e-04
14.14 128 4.84450e-04 1.57496e-04
14.24 129 4.77463e-04 1.57246e-04
14.35 130 4.70667e-04 1.57523e-04
14.46 131 4.63965e-04 1.56942e-04
14.57 132 4.57426e-04 1.56614e-04
14.67 133 4.50984e-04 1.57115e-04
14.78 134 4.44639e-04 1.57246e-04
14.89 135 4.38442e-04 1.56923e-04
15.00 136 4.32367e-04 1.56931e-04
15.10 137 4.26382e-04 1.56673e-04
15.21 138 4.20519e-04 1.57176e-04
15.32 139 4.14734e-04 1.56946e-04
15.43 140 4.09093e-04 1.56913e-04
15.54 141 4.03532e-04 1.56819e-04
15.65 142 3.98073e-04 1.56762e-04
15.75 143 3.92710e-04 1.56707e-04
15.86 144 3.87442e-04 1.56652e-04
15.97 145 3.82266e-04 1.56597e-04
16.08 146 3.77181e-04 1.56543e-04
16.18 147 3.72183e-04 1.56488e-04
16.29 148 3.67272e-04 1.56434e-04
16.40 149 3.62444e-04 1.56380e-04
16.51 150 3.57699e-04 1.56326e-04
16.61 151 3.53035e-04 1.56273e-04
16.72 152 3.48449e-04 1.56220e-04
16.83 153 3.43941e-04 1.56167e-04
16.94 154 3.39507e-04 1.56115e-04
17.04 155 3.35148e-04 1.56063e-04
17.16 156 3.30861e-04 1.56011e-04
17.26 157 3.26644e-04 1.55960e-04
17.36 158 3.22497e-04 1.55909e-04
17.47 159 3.18417e-04 1.55858e-04
17.57 160 3.14403e-04 1.55808e-04
17.68 161 3.10455e-04 1.55758e-04
17.78 162 3.06569e-04 1.55709e-04
17.89 163 3.02747e-04 1.55660e-04
18.00 164 2.98985e-04 1.55611e-04
18.10 165 2.95282e-04 1.55563e-04
18.20 166 2.91639e-04 1.55515e-04
18.31 167 2.88052e-04 1.55467e-04
18.41 168 2.84522e-04 1.55419e-04
18.52 169 2.81046e-04 1.55370e-04
18.62 170 2.77617e-04 1.55314e-04
18.73 171 2.74206e-04 1.55224e-04
18.83 172 2.70803e-04 1.55099e-04
18.94 173 2.67498e-04 1.55028e-04
19.04 174 2.64278e-04 1.54980e-04
19.15 175 2.61126e-04 1.54934e-04
19.25 176 2.58029e-04 1.54880e-04
19.36 177 2.54981e-04 1.54825e-04
19.46 178 2.51978e-04 1.54677e-04
19.57 179 2.49020e-04 1.54637e-04
19.67 180 2.46098e-04 1.54613e-04
19.78 181 2.43217e-04 1.54572e-04
19.88 182 2.40377e-04 1.54532e-04
19.98 183 2.37575e-04 1.54487e-04
20.09 184 2.34812e-04 1.54440e-04
20.20 185 2.32088e-04 1.54393e-04
20.30 186 2.29403e-04 1.54347e-04
20.40 187 2.26757e-04 1.54302e-04
20.51 188 2.24149e-04 1.54258e-04
20.61 189 2.21578e-04 1.54215e-04
20.72 190 2.19043e-04 1.54173e-04
20.83 191 2.16545e-04 1.54132e-04
20.93 192 2.14081e-04 1.54091e-04
21.03 193 2.11653e-04 1.54052e-04
21.14 194 2.09258e-04 1.54013e-04
21.24 195 2.06897e-04 1.53974e-04
21.35 196 2.04568e-04 1.53936e-04
21.45 197 2.02272e-04 1.53899e-04
21.55 198 2.00007e-04 1.53862e-04
21.66 199 1.97774e-04 1.53825e-04
21.76 200 1.95571e-04 1.53211e-04
21.87 201 1.93624e-04 1.53830e-04
21.97 202 1.91346e-04 1.53620e-04
22.07 203 1.89239e-04 1.53571e-04
22.18 204 1.87143e-04 1.53539e-04
22.28 205 1.85075e-04 1.53502e-04
22.39 206 1.83038e-04 1.53468e-04
22.50 207 1.81030e-04 1.53435e-04
22.61 208 1.79050e-04 1.53402e-04
22.71 209 1.77097e-04 1.53371e-04
22.82 210 1.75171e-04 1.53340e-04
22.93 211 1.73270e-04 1.53309e-04
23.03 212 1.71396e-04 1.53279e-04
23.14 213 1.69546e-04 1.53249e-04
23.25 214 1.67721e-04 1.53220e-04
23.36 215 1.65920e-04 1.53190e-04
23.46 216 1.64142e-04 1.53161e-04
23.57 217 1.62388e-04 1.53132e-04
23.68 218 1.60657e-04 1.53103e-04
23.79 219 1.58948e-04 1.53074e-04
23.89 220 1.57261e-04 1.53044e-04
24.00 221 1.55596e-04 1.53015e-04
24.11 222 1.53952e-04 1.52986e-04
24.22 223 1.52330e-04 1.52957e-04
24.32 224 1.50728e-04 1.52928e-04
24.43 225 1.49147e-04 1.52899e-04
24.54 226 1.47585e-04 1.52870e-04
24.64 227 1.46043e-04 1.52841e-04
24.75 228 1.44521e-04 1.52813e-04
24.86 229 1.43018e-04 1.52784e-04
24.97 230 1.41533e-04 1.52756e-04
25.07 231 1.40068e-04 1.52728e-04
25.18 232 1.38620e-04 1.52700e-04
25.29 233 1.37190e-04 1.52672e-04
25.39 234 1.35778e-04 1.52644e-04
25.50 235 1.34383e-04 1.52616e-04
25.61 236 1.33006e-04 1.52589e-04
25.71 237 1.31645e-04 1.52562e-04
25.82 238 1.30301e-04 1.52535e-04
25.93 239 1.28974e-04 1.52508e-04
26.03 240 1.27662e-04 1.52481e-04
26.14 241 1.26366e-04 1.52455e-04
26.25 242 1.25086e-04 1.52428e-04
26.36 243 1.23822e-04 1.52402e-04
26.46 244 1.22572e-04 1.52376e-04
26.57 245 1.21338e-04 1.52350e-04
26.68 246 1.20118e-04 1.52325e-04
26.78 247 1.18913e-04 1.52299e-04
26.89 248 1.17722e-04 1.52274e-04
27.00 249 1.16545e-04 1.52249e-04
27.11 250 1.15382e-04 1.52224e-04
27.21 251 1.14233e-04 1.52199e-04
27.32 252 1.13097e-04 1.52175e-04
27.43 253 1.11974e-04 1.52150e-04
27.53 254 1.10865e-04 1.52125e-04
27.64 255 1.09769e-04 1.52100e-04
27.75 256 1.08685e-04 1.52072e-04
27.85 257 1.07614e-04 1.52039e-04
27.96 258 1.06555e-04 1.51996e-04
28.07 259 1.05509e-04 1.51946e-04
28.17 260 1.04474e-04 1.51895e-04
28.28 261 1.03452e-04 1.51848e-04
28.39 262 1.02441e-04 1.51809e-04
28.50 263 1.01441e-04 1.51772e-04
28.60 264 1.00453e-04 1.51738e-04
28.71 265 9.94765e-05 1.51705e-04
28.71 END Iterative Stockholder Analysis
28.81 Writing results to workdir
28.82 END Defining atomic weight functions (own atomic grid)
Computing multipoles: 0%.........76%..100%
29.33 Written alanine.hipart/isa_multipoles.txt
29.33 END Atomic multipoles (up to hexadecapols)=
We often experience slow convergence with the ISA method. This may be inherent to the method. Any suggestions for improvement are always appreciated.
The output is written to alanine.hipart/isa_multipoles.txt and can also be read into your Python scripts with the routines in hipart.io.
toon@poony ~/hipart/examples/001-usage> cat alanine.hipart/isa_multipoles.txt
number of atoms: 13
number of fields: 25
Multipoles | (0,0) (1,0) (1,1+) (1,1-) (2,0) (2,1+) (2,1-) (2,2+) (2,2-) (3,0) (3,1+) (3,1-) (3,2+) (3,2-) (3,3+) (3,3-) (4,0) (4,1+) (4,1-) (4,2+) (4,2-) (4,3+) (4,3-) (4,4+) (4,4-)
----------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
0 O 8 | -0.592142491311 0.001756691643 -0.022195389767 -0.009958506655 -0.096074619322 -0.012204165528 -0.080959770781 0.084665172788 0.084489653241 0.159145877436 -0.021370210470 -0.044549296032 -0.086663485873 -0.023818154024 -0.136876599013 0.237362424122 -0.037329162671 0.026085390488 0.002983411820 0.127846086213 0.052762334620 -0.031369357410 -0.052063170688 -0.077951139944 -0.190580251156
1 O 8 | -0.544820972903 0.023233022104 0.011662324439 -0.048054564862 0.112052475725 0.016242669088 0.025199633606 -0.150028837482 -0.077194132363 0.134332091945 0.061410044195 -0.042738992271 -0.095693905175 -0.173719739902 -0.258349836774 0.066217878548 -0.027918088910 -0.060148268720 -0.013541963218 0.038512654429 0.068357659405 0.102636846624 0.055635325215 0.055007009959 -0.044823155468
2 N 7 | -0.863962016586 0.061811095124 0.011356681277 0.104428966703 -0.262942715151 0.143627936218 -0.278138792300 0.010915245609 0.149983135989 -0.100721861065 -0.171383688124 0.567446639622 -0.103982064111 0.116980291412 0.455912331230 -0.236204524129 0.117134316298 0.247900036744 -0.307921982246 0.283037885325 0.051612986709 -0.085578237990 -0.063620534199 -0.172776319780 -0.260088207083
3 C 6 | 0.396482257671 0.021305492206 -0.120291813857 -0.063759176835 -0.028108774212 0.002585440574 0.023952412692 -0.060260274454 0.054530493280 0.091424666622 -0.017153038089 -0.145719678212 -0.085930845937 0.017409849069 -0.243022872232 0.099673697519 0.086239691975 0.007534478010 0.008552775590 0.054424880149 0.029315806565 0.087523932480 0.202887509693 -0.006415824430 0.195670124882
4 C 6 | -0.382471197776 -0.006757574174 -0.003255473791 -0.013256342214 -0.009371044983 -0.009358671087 0.046178196240 -0.032913557932 -0.009256425447 -0.100616722685 -0.041934703371 0.192836914812 0.048375451395 -0.134016989547 0.227420373055 0.000414669217 -0.035876255898 -0.107605253991 -0.250196302175 -0.236802181540 -0.217822675451 0.190463100916 -0.115525109128 0.093270824362 -0.216770199293
5 C 6 | 0.637157404389 0.000854866740 0.100294189425 0.011144202835 0.034870209187 -0.070195303769 0.100647356337 -0.072508878932 0.048634815357 0.009896086599 0.087913042484 0.043422708709 0.073909255360 0.048811040787 0.085910988186 -0.112849628671 -0.030972169710 -0.069116948719 0.079616894868 -0.336643872468 -0.162383618204 -0.125119931297 0.143265804829 0.065026555257 0.227212363974
6 H 1 | 0.000259655854 -0.045219667604 -0.005899112669 0.015188588919 0.021571128433 -0.001612563022 -0.018519136687 -0.008668765200 0.020449862574 -0.000761830006 -0.001142537902 -0.004741314194 0.000894912272 0.007400369743 0.023363915412 -0.006332817813 -0.028355210627 -0.014719112200 0.037849096398 0.001287903218 0.015345710551 0.009145724857 0.026853059742 -0.005689451951 0.061178612421
7 H 1 | 0.098582951977 0.000173909024 0.013598806720 0.022660990593 -0.006909415859 -0.002754154119 -0.001870826902 -0.002691330036 0.010988461840 -0.012456224250 -0.014463285358 0.002573849779 0.008255332207 -0.009497865840 -0.015458673379 -0.003030991989 -0.017421974263 0.005448045237 -0.005110925296 0.036890953576 -0.005338485796 -0.018289624888 0.001782967664 -0.009222038568 -0.008702604483
8 H 1 | 0.089955660710 -0.005605665400 -0.023617366095 0.008566541582 -0.003905483623 0.009817375608 0.001082619117 0.009565588574 -0.009594065658 0.002074244440 0.003693578435 0.019260290334 -0.008740878475 0.000895650746 -0.015411683295 0.016638328005 -0.015296085868 0.026353344042 0.004746012365 -0.009845538104 0.051842080341 -0.010960712408 -0.009487764866 0.019113269010 0.016773996259
9 H 1 | 0.090954428556 0.024907775434 0.000527715491 -0.008086118909 0.012216146271 0.002188038643 -0.007617712288 -0.004206329005 -0.004246711318 0.010487046500 -0.006261866607 -0.018944109560 0.001573504204 -0.009607033210 0.018105707081 -0.003543602672 0.004164556456 0.001697055133 0.005986669240 0.000360581740 0.027979541361 -0.012816200181 -0.008957298568 0.008790712966 0.033509664228
10 H 1 | 0.315384819711 -0.006315649787 -0.010468987867 -0.002277645788 -0.005593858246 0.000606554369 -0.013533977148 -0.011522870483 0.007373969397 -0.011065363828 -0.007968013030 0.017448993453 -0.005384899602 -0.006862849737 0.005200761059 0.013604862882 0.024889710435 -0.006991951508 -0.006626458497 -0.017198418339 0.029589263248 -0.013230928107 -0.008598436322 0.028255719708 0.004012163548
11 H 1 | 0.326089354049 0.004752088078 0.004219675571 -0.012892057756 -0.010705019958 -0.000726444568 -0.003284978944 0.003341195094 0.015677034928 0.021415264647 -0.000681963703 0.010619561434 -0.009338282268 -0.002350513570 -0.000866476997 -0.007714671224 -0.033956886912 0.005033072024 -0.033546074016 -0.001194779316 0.001658351231 -0.014562643532 0.018132944523 0.001249203071 0.017870602272
12 H 1 | 0.428869626749 0.001674778712 0.011828079066 0.000366843257 -0.003897514266 -0.001074170207 -0.003417124512 0.003247533028 0.004857239188 0.005063346990 0.000594382004 0.007140417661 0.013739123816 0.004480411653 0.004416256613 -0.009541175053 -0.004337241709 -0.019476552965 0.005069779818 -0.001456973226 0.008535326396 0.013775097861 -0.006572479823 0.008115840646 -0.024621876730
5.4.5. Net and overlap populations¶
Net and overlap electron populations are computed with the script hi-net-overlap.py. This example computes the net and overlap populations using the Becke scheme:
toon@poony ~/hipart/examples/001-usage> hi-net-overlap.py alanine.fchk becke
---TIME--- ---------------------------------LOG---------------------------------
0.03 BEGIN Loading Electronic structure
0.05 Data read from: alanine.fchk ()
0.05 Restricted: True
0.05 Orbitals present: True
0.05 Spin density present: False
0.05 Number of alpha electrons: 24
0.05 Number of beta electrons: 24
0.05 Number of electrons: 48
0.05 Total charge: 0
0.05 Number of atoms: 13
0.05 Chemical formula: O2NC3H7
0.05 END Loading Electronic structure
0.05 BEGIN Net and overlap populations
0.05 BEGIN Atomic grids
Computing/Loading atomic grids and distances: 0%.........5%......
...11%.........17%.........23%.........29%.........35%.........41
%.........47%.........53%.........59%.........65%.........71%....
.....76%.........82%.........88%.........94%........100%
0.23 END Atomic grids
0.23 BEGIN Molecular density on atomic grids
Computing/Loading densities: 0%.........76%..100%
0.23 END Molecular density on atomic grids
0.23 BEGIN Atomic charges
0.23 Written alanine.hipart/becke_charges.txt
0.23 END Atomic charges
0.23 BEGIN Atomic weights on other atoms' grids.
0.23 BEGIN Defining atomic weight functions (own atomic grid)
0.23 Trying to load weight functions
0.24 END Defining atomic weight functions (own atomic grid)
0.24 BEGIN Becke's Smooth Voronoi Partitioning
Computing/Loading cell functions: 0%.........0%.........1%.....
....2%.........3%.........4%.........5%.........6%.........7%..
.......8%.........9%.........10%.........11%.........12%.......
..13%.........14%.........15%.........16%.........17%.........1
8%.........19%.........20%.........21%.........22%.........23%.
........24%.........25%.........26%.........27%.........28%....
.....29%.........30%.........31%.........32%.........33%.......
..34%.........35%.........36%.........37%.........38%.........3
9%.........40%.........41%.........42%.........43%.........44%.
........45%.........46%.........47%.........48%.........49%....
.....50%.........51%.........52%.........53%.........54%.......
..55%.........56%.........57%.........58%.........59%.........6
0%.........61%.........62%.........63%.........64%.........65%.
........66%.........67%.........68%.........69%.........70%....
.....71%.........71%.........72%.........73%.........74%.......
..75%.........76%.........77%.........78%.........79%.........8
0%.........81%.........82%.........83%.........84%.........85%.
........86%.........87%.........88%.........89%.........90%....
.....91%.........92%.........93%.........94%.........95%.......
..96%.........97%.........98%.........99%...100%
0.26 END Becke's Smooth Voronoi Partitioning
Computing off-diagonal atom weights: 0%.........5%.........11%...
......17%.........23%.........29%.........35%.........41%........
.47%.........53%.........59%.........65%.........71%.........76%.
........82%.........88%.........94%........100%
0.28 END Atomic weights on other atoms' grids.
Integrating over products of stockholder weights: 0%.........10%...
......21%.........32%.........43%.........54%.........65%.........7
6%.........87%.........98%100%
0.39 Written alanine.hipart/becke_net_overlap.txt
0.39 END Net and overlap populations
The output, alanine.hipart/becke_net_overlap.txt, contains a symmetric matrix with net and overlap populations. The atomic net populations are put on the diagonal, while the overlap populations are off-diagonal elements.
toon@poony ~/hipart/examples/001-usage> cat alanine.hipart/becke_net_overlap.txt
number of atoms: 13
Net | 0 O 1 O 2 N 3 C 4 C 5 C 6 H 7 H 8 H 9 H 10 H 11 H 12 H
----------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
0 O 8 | 7.760657682242 0.004242209550 0.000001240048 0.010327257936 0.001834326183 0.309650111420 0.003109667757 0.002858032885 0.000000000011 0.000000394254 0.000000000000 0.000000000000 0.118508248899
1 O 8 | 0.004242209550 7.811157777014 0.005643292500 0.003006786245 0.000244595025 0.389855643654 0.000000000055 0.000000026510 0.000000000000 0.000654548302 0.000000000001 0.007559009497 0.003813753419
2 N 7 | 0.000001240048 0.005643292500 6.901120381146 0.280042871987 0.010402128647 0.011471177554 0.003354277926 0.000000000000 0.000194072808 0.000681230346 0.122697913691 0.118827303804 0.000000000000
3 C 6 | 0.010327257936 0.003006786245 0.280042871987 5.285573432105 0.257360791966 0.229060476396 0.102629643859 0.001278590028 0.002733387801 0.001496012821 0.002433145392 0.001410038502 0.000000000005
4 C 6 | 0.001834326183 0.000244595025 0.010402128647 0.257360791966 5.719188449572 0.014583814747 0.003074753540 0.121224128595 0.122678163539 0.120107972870 0.001932061337 0.000120815421 0.000000002202
5 C 6 | 0.309650111420 0.389855643654 0.011471177554 0.229060476396 0.014583814747 5.082712117170 0.001695019621 0.000533904745 0.000000000000 0.001214625061 0.000000000000 0.000494474224 0.003421510823
6 H 1 | 0.003109667757 0.000000000055 0.003354277926 0.102629643859 0.003074753540 0.001695019621 0.690658404811 0.000010697813 0.000019103080 0.000000000000 0.000001525013 0.000000000000 0.000000000000
7 H 1 | 0.002858032885 0.000000026510 0.000000000000 0.001278590028 0.121224128595 0.000533904745 0.000010697813 0.711605116204 0.000498869875 0.000332477157 0.000000000000 0.000000000000 0.000000000016
8 H 1 | 0.000000000011 0.000000000000 0.000194072808 0.002733387801 0.122678163539 0.000000000000 0.000019103080 0.000498869875 0.726250143111 0.000417149043 0.000596297273 0.000000000023 0.000000000000
9 H 1 | 0.000000394254 0.000654548302 0.000681230346 0.001496012821 0.120107972870 0.001214625061 0.000000000000 0.000332477157 0.000417149043 0.715608280305 0.000000711335 0.000064714409 0.000000000000
10 H 1 | 0.000000000000 0.000000000001 0.122697913691 0.002433145392 0.001932061337 0.000000000000 0.000001525013 0.000000000000 0.000596297273 0.000000711335 0.638898285494 0.000432616253 0.000000000000
11 H 1 | 0.000000000000 0.007559009497 0.118827303804 0.001410038502 0.000120815421 0.000494474224 0.000000000000 0.000000000000 0.000000000023 0.000064714409 0.000432616253 0.607611760446 0.000000000000
12 H 1 | 0.118508248899 0.003813753419 0.000000000000 0.000000000005 0.000000002202 0.003421510823 0.000000000000 0.000000000016 0.000000000000 0.000000000000 0.000000000000 0.000000000000 0.550908894051
5.4.6. Bond orders, valences and free valences¶
The computation of bond orders, valences and free valences is supported for both SCF computations and correlated methods, either closed or open shell. We show the computation on the alanine radical is it has non-zero free valences. This time we use the Hirshfeld partitioning method.
toon@poony ~/hipart/examples/001-usage> hi-bond-orders.py alanine-rad.fchk hirsh atoms/densities.txt
---TIME--- ---------------------------------LOG---------------------------------
0.02 BEGIN Loading Electronic structure
0.06 Data read from: alanine-rad.fchk ()
0.06 Restricted: False
0.06 Orbitals present: True
0.06 Spin density present: True
0.06 Number of alpha electrons: 24
0.06 Number of beta electrons: 23
0.06 Number of electrons: 47
0.06 Total charge: 0
0.06 Number of atoms: 12
0.06 Chemical formula: O2NC3H6
0.06 END Loading Electronic structure
0.06 BEGIN Bond orders and valences
0.06 BEGIN Atomic charges
0.06 BEGIN Atomic grids
Computing/Loading atomic grids and distances: 0%.........6%....
.....13%.........20%.........27%.........34%.........41%.......
..48%.........55%.........62%.........69%.........76%.........8
3%.........90%.........97%...100%
0.45 END Atomic grids
0.45 BEGIN Molecular density on atomic grids
Computing/Loading densities: 0%.........83%.100%
3.89 END Molecular density on atomic grids
3.89 BEGIN Defining atomic weight functions (own atomic grid)
3.89 Trying to load weight functions
3.89 Could not load all weight functions from workdir. Computing them.
3.89 BEGIN Conventional Hirshfeld (with neutral pro-atoms)
3.89 END Conventional Hirshfeld (with neutral pro-atoms)
3.98 Writing results to workdir
3.98 END Defining atomic weight functions (own atomic grid)
Computing charges: 0%.........83%.100%
3.99 Written alanine-rad.hipart/hirsh_charges.txt
3.99 END Atomic charges
3.99 BEGIN Atomic overlap matrices (contracted Gaussians)
Computing matrices: 0%.........83%.100%
9.91 Written alanine-rad.hipart/hirsh_overlap_matrices.txt
9.91 END Atomic overlap matrices (contracted Gaussians)
Computing bond orders: 0%.........12%.........25%.........38%......
...51%.........64%.........76%.........89%.......100%
10.14 Written alanine-rad.hipart/hirsh_bond_orders.txt
10.14 Written alanine-rad.hipart/hirsh_valences.txt
10.14 Written alanine-rad.hipart/hirsh_free_valences.txt
10.14 END Bond orders and valences
There are three output files:
toon@poony ~/hipart/examples/001-usage> cat alanine-rad.hipart/hirsh_bond_orders.txt
number of atoms: 12
Bond order | 0 O 1 O 2 N 3 C 4 C 5 C 6 H 7 H 8 H 9 H 10 H 11 H
----------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
0 O 8 | 0.000000000000 0.534631488015 0.060919358595 0.145663459036 0.036932587808 1.995808397598 0.019231451199 0.028485628653 0.003443182171 0.004963593822 0.006995364889 0.004537735097
1 O 8 | 0.534631488015 0.000000000000 0.185672119473 0.147787798591 0.019526066492 1.756074283462 0.016029051946 0.004156837608 0.003507343856 0.001997818127 0.011856734820 0.021802632150
2 N 7 | 0.060919358595 0.185672119473 0.000000000000 1.189610812775 0.118292945985 0.151005387974 0.100697272791 0.016204194751 0.014288383032 0.016771731003 1.047803312038 1.037093750023
3 C 6 | 0.145663459036 0.147787798591 1.189610812775 0.000000000000 1.043720806760 0.978862943094 0.880786936070 0.089253931327 0.091608666005 0.089891027745 0.094871057580 0.092787948071
4 C 6 | 0.036932587808 0.019526066492 0.118292945985 1.043720806760 0.000000000000 0.109940225834 0.090345507715 0.945522668753 0.962633286697 0.956084911932 0.011300539385 0.021182631040
5 C 6 | 1.995808397598 1.756074283462 0.151005387974 0.978862943094 0.109940225834 0.000000000000 0.095903046680 0.017045952134 0.012837690915 0.013835547789 0.014080044170 0.014193869460
6 H 1 | 0.019231451199 0.016029051946 0.100697272791 0.880786936070 0.090345507715 0.095903046680 0.000000000000 0.009856646732 0.011758520217 0.011788172938 0.014608166511 0.012451378912
7 H 1 | 0.028485628653 0.004156837608 0.016204194751 0.089253931327 0.945522668753 0.017045952134 0.009856646732 0.000000000000 0.080447636827 0.080837681225 0.001457460059 0.002148019336
8 H 1 | 0.003443182171 0.003507343856 0.014288383032 0.091608666005 0.962633286697 0.012837690915 0.011758520217 0.080447636827 0.000000000000 0.082064181450 0.003226333710 0.002247858928
9 H 1 | 0.004963593822 0.001997818127 0.016771731003 0.089891027745 0.956084911932 0.013835547789 0.011788172938 0.080837681225 0.082064181450 0.000000000000 0.001392981918 0.008884055620
10 H 1 | 0.006995364889 0.011856734820 1.047803312038 0.094871057580 0.011300539385 0.014080044170 0.014608166511 0.001457460059 0.003226333710 0.001392981918 0.000000000000 0.078776390871
11 H 1 | 0.004537735097 0.021802632150 1.037093750023 0.092787948071 0.021182631040 0.014193869460 0.012451378912 0.002148019336 0.002247858928 0.008884055620 0.078776390871 0.000000000000
toon@poony ~/hipart/examples/001-usage> cat alanine-rad.hipart/hirsh_valences.txt
number of atoms: 12
i Z Valences
--------------------------------
0 O 8 2.929965288620
1 O 8 3.014517554824
2 N 7 3.950283201335
3 C 6 4.845036817629
4 C 6 4.315171579586
5 C 6 5.163057369578
6 H 1 1.263313618624
7 H 1 1.275578396275
8 H 1 1.268272597174
9 H 1 1.268972818152
10 H 1 1.286526859695
11 H 1 1.296006218835
--------------------------------
toon@poony ~/hipart/examples/001-usage> cat alanine-rad.hipart/hirsh_free_valences.txt
number of atoms: 12
i Z Free valences
--------------------------------
0 O 8 0.088353041738
1 O 8 0.311475380286
2 N 7 0.011923932894
3 C 6 0.000191430575
4 C 6 -0.000310598815
5 C 6 0.003469980470
6 H 1 -0.000142533086
7 H 1 0.000161738869
8 H 1 0.000209513365
9 H 1 0.000461114583
10 H 1 0.000158473743
11 H 1 -0.000100050674
--------------------------------
The bond orders are written as off-diagonal elements in the first output file. The diagonal elements are always zero. The atomic valence and free valence are written in the two following output files, respectively.
5.4.7. Spin charges¶
The computation of spin charges only makes sense in the case of open shell computations. The screen output must contain the line Spin density present: True in the beginning. If not, the reported spin charges are always zero.
This example computes Hirshfeld spin charges in the alanine radical:
toon@poony ~/hipart/examples/001-usage> hi-spin-charges.py alanine-rad.fchk hirsh atoms/densities.txt
---TIME--- ---------------------------------LOG---------------------------------
0.02 BEGIN Loading Electronic structure
0.06 Data read from: alanine-rad.fchk ()
0.06 Restricted: False
0.06 Orbitals present: True
0.06 Spin density present: True
0.06 Number of alpha electrons: 24
0.06 Number of beta electrons: 23
0.06 Number of electrons: 47
0.06 Total charge: 0
0.06 Number of atoms: 12
0.06 Chemical formula: O2NC3H6
0.06 END Loading Electronic structure
0.06 BEGIN Atomic spin charges
0.06 BEGIN Atomic grids
Computing/Loading atomic grids and distances: 0%.........6%......
...13%.........20%.........27%.........34%.........41%.........48
%.........55%.........62%.........69%.........76%.........83%....
.....90%.........97%...100%
0.22 END Atomic grids
0.22 BEGIN Molecular spin density on atomic grids
Computing/Loading spin densities: 0%.........83%.100%
3.68 END Molecular spin density on atomic grids
3.68 BEGIN Defining atomic weight functions (own atomic grid)
3.68 Trying to load weight functions
3.68 END Defining atomic weight functions (own atomic grid)
Computing spin charges: 0%.........83%.100%
3.68 Written alanine-rad.hipart/hirsh_spin_charges.txt
3.68 END Atomic spin charges
The output looks like:
toon@poony ~/hipart/examples/001-usage> cat alanine-rad.hipart/hirsh_spin_charges.txt
number of atoms: 12
i Z Spin charge
--------------------------------
0 O 8 0.296354798239
1 O 8 0.571212878771
2 N 7 0.107194498509
3 C 6 -0.001845675962
4 C 6 0.002960695340
5 C 6 0.011314770995
6 H 1 -0.000230137092
7 H 1 0.002340260889
8 H 1 0.000613031523
9 H 1 -0.000033836171
10 H 1 0.007597792743
11 H 1 0.002837236327
--------------------------------
The spin charge is not evenly distributed over the two oxygen atoms due to the polarization by the rest of the molecule.
5.4.8. Atomic overlap matrices¶
The atomic overlap matrices are used, amongst other things, for the computation of the bond orders. If you are just interested in these matrices, use the command hi-overlap-matrices.py or hi-overlap-matrices-orb.py to compute them.
- hi-overlap-matrices.py computes the atomic overlap matrices in the (original) basis of the contracted Gaussians. The output is written to <name>.hipart/<scheme>_overlap_matrices.txt
- hi-overlap-matrices-orb.py computes the atomic overlap matrices in the basis of the orbitals. For a restricted wavefunction there is only one output: <name>.hipart/<scheme>_alpha_overlap_matrices.txt. In the unrestricted case there are two additional outputs: <name>.hipart/<scheme>_beta_overlap_matrices.txt and <name>.hipart/<scheme>_natural_overlap_matrices.txt.
The Hirshfeld atomic overlap matrices in the original basis of the alanine radical are computed as follows.
toon@poony ~/hipart/examples/001-usage> hi-overlap-matrices.py alanine-rad.fchk hirsh atoms/densities.txt
---TIME--- ---------------------------------LOG---------------------------------
0.02 BEGIN Loading Electronic structure
0.06 Data read from: alanine-rad.fchk ()
0.06 Restricted: False
0.06 Orbitals present: True
0.06 Spin density present: True
0.06 Number of alpha electrons: 24
0.06 Number of beta electrons: 23
0.06 Number of electrons: 47
0.06 Total charge: 0
0.06 Number of atoms: 12
0.06 Chemical formula: O2NC3H6
0.06 END Loading Electronic structure
0.06 BEGIN Atomic overlap matrices (contracted Gaussians)
0.06 BEGIN Atomic grids
Computing/Loading atomic grids and distances: 0%.........6%......
...13%.........20%.........27%.........34%.........41%.........48
%.........55%.........62%.........69%.........76%.........83%....
.....90%.........97%...100%
0.22 END Atomic grids
0.22 BEGIN Defining atomic weight functions (own atomic grid)
0.22 Trying to load weight functions
0.22 END Defining atomic weight functions (own atomic grid)
Computing matrices: 0%.........83%.100%
0.40 Written alanine-rad.hipart/hirsh_overlap_matrices.txt
0.40 END Atomic overlap matrices (contracted Gaussians)
The output file alanine-rad.hipart/hirsh_overlap_matrices.txt is rather large. Below we only show the overlap matrix for the first atom:
toon@poony ~/hipart/examples/001-usage> cat alanine-rad.hipart/hirsh_overlap_matrices.txt
number of orbitals: 102
number of atoms: 12
Atom 0 O 8
9.9778812931e-01 2.3103169663e-01 -1.0099846236e-03 1.4183348057e-03 7.6096094615e-07 1.6582482976e-01 -1.9601190240e-04 2.7537384597e-04 1.6979738547e-07 3.2978249648e-02 1.7813163656e-04 -3.1077414015e-07 3.2906157469e-02 -5.9992013037e-07 3.3052636046e-02 3.9862172347e-23 1.9045740371e-07 -6.0747055300e-08 8.5775712418e-07 1.2161463413e-07 2.2656583348e-03 -6.3340218194e-04 9.1236500470e-03 1.2963831723e-03 5.3813729614e-07 -6.6739795229e-06 -9.4802712895e-07 5.5213778400e-05 1.3507287852e-05 1.3766678414e-06 6.3640644356e-53 7.8532940352e-15 -3.9693738481e-14 3.5985349326e-14 8.9115815822e-16 1.2091189851e-05 -5.4115174806e-05 4.8968917437e-05 1.2189805771e-06 9.3845488446e-14 -1.4683226913e-13 -3.6353448139e-15 7.7202684214e-14 3.2960514142e-15 3.9144701219e-16 2.3021217387e-21 4.8516686061e-06 -1.8121307812e-05 8.6266080753e-06 -2.5767174000e-06 3.2861043633e-03 -1.1036114458e-02 5.2478266107e-03 -1.5701621760e-03 1.3485042388e-06 -1.1081521172e-06 3.3031268375e-07 3.1130059129e-07 -1.5747671127e-07 3.3472136914e-08 5.1639077551e-30 3.3163608059e-08 -1.6493617191e-07 -7.0067570839e-09 1.8196538964e-08 6.4381651244e-04 -2.8905273076e-03 -1.2437042509e-04 3.1902287159e-04 2.1362756490e-09 1.5437617332e-10 -4.0686040721e-10 9.6981677858e-12 -1.7007941153e-11 3.1988091073e-11 1.4438973705e-06 1.8581823556e-02 -2.2912884596e-02 3.2147457232e-02 2.8151471638e-06 4.7940284156e-02 -5.2442757810e-02 7.3578526137e-02 6.4055058315e-06 1.8400347231e-02 -4.3129424993e-02 -3.8102321225e-06 3.5589072843e-02 5.2903754604e-06 6.5256891105e-04 8.2092534768e-09 1.1450364746e-03 5.5918927931e-08 1.9154253868e-03 6.8755497500e-16 1.3693270154e-05 4.8593936907e-11 2.8489745354e-04 4.3802722749e-19 1.8839310577e-06 4.1763362304e-15 2.2694772066e-05
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1.8839310577e-06 2.1242940175e-05 2.0877514346e-05 -1.3344923304e-05 3.3548811211e-06 4.9267667638e-05 6.1807238611e-05 -2.9926611678e-05 1.3442760643e-05 4.3944740436e-05 -1.9876296098e-05 7.4825069543e-06 2.4853570834e-05 -4.6435525851e-06 2.2504132319e-05 3.7110452884e-09 1.1199227715e-06 8.2711182350e-07 1.6429868994e-06 4.9321447693e-07 1.0840191717e-05 1.0248438381e-05 2.1960684402e-05 6.7305747812e-06 2.6164936302e-06 3.3071476990e-06 9.1975621280e-07 5.7027735619e-06 2.0348438030e-06 2.3646863261e-06 1.4478009276e-11 5.5335187356e-08 -1.1505848876e-07 7.9883423476e-08 -1.7177000091e-09 2.6255885230e-06 -5.9286999719e-06 5.6931808580e-06 4.4087062320e-07 2.7718644922e-07 -2.8794458896e-07 6.7399458573e-09 1.4882131413e-07 -7.2823240032e-09 5.2855973812e-08 2.6173039828e-07 1.3081965108e-05 -1.8870676365e-05 7.4167455427e-06 -2.1425501949e-06 3.5118512472e-05 -5.5831056387e-05 2.5547597637e-05 -4.1903131113e-06 2.0729714993e-05 -1.0937258663e-05 2.5544459738e-06 8.2499642369e-06 -1.2534392278e-06 6.5564484045e-06 3.2550200630e-10 9.1865030170e-07 -2.0460256715e-06 -4.8530316929e-07 2.8728959375e-07 1.0709993721e-05 -2.6855351995e-05 -6.4151743885e-06 4.9530796083e-06 1.7536881062e-06 7.0315977540e-07 -4.0911599086e-07 2.5189642473e-07 -8.4032630996e-08 2.0941768559e-07 6.6188443198e-06 6.7348636946e-05 1.3589518713e-05 4.3805324088e-05 1.4209715384e-05 7.6458781277e-05 1.7416969919e-05 4.9944371247e-05 1.9092868119e-05 4.5352875835e-05 9.1148489673e-06 -5.5757704828e-07 6.3919185367e-05 7.5117863075e-06 5.4894646146e-05 1.8794211096e-06 2.0048101132e-05 5.2164784889e-07 1.2016981725e-05 4.6952270882e-10 9.9623114360e-07 8.0510448434e-08 5.3941201298e-06 2.2037360356e-10 5.1217991827e-07 1.3060617550e-08 1.9090879841e-06
4.1763362304e-15 1.8991467207e-11 4.0311146247e-11 -3.0556499551e-11 -2.0217111981e-11 7.3168113554e-09 1.9511494665e-08 -1.6867026454e-08 -1.0820174658e-08 4.0360189666e-10 -4.9779470864e-10 -3.3127048816e-10 2.5484954293e-10 2.5829731118e-10 1.3387997831e-10 7.3414421011e-13 1.8988766557e-09 3.7906862206e-09 1.5975267417e-09 -1.5523062153e-09 5.7106326789e-08 1.4244889539e-07 6.0381621462e-08 -4.7651549254e-08 2.4313424005e-08 1.5761857030e-08 -1.1528246964e-08 5.4974554720e-09 -4.6559466071e-09 5.0181798490e-09 2.4263019591e-12 1.2108029503e-08 -2.1007414650e-08 1.2592161655e-08 -1.2237721502e-08 9.0459046992e-08 -1.5309596120e-07 1.1896827344e-07 -9.7543148780e-08 4.3353680650e-08 -4.3072230243e-08 4.0819932958e-08 1.6290478039e-08 -2.4493774727e-08 1.4851480230e-08 1.6042586207e-09 5.3759075783e-08 -5.0266892599e-08 -3.0251307903e-08 -8.2482584380e-08 1.5630335302e-07 -1.4035453060e-07 -8.9935765463e-08 -2.5018456287e-07 3.8512998760e-08 3.0758295727e-08 7.3731955507e-08 2.4271435236e-08 3.5008846520e-08 8.5099810270e-08 1.2896891936e-13 2.3569455531e-09 -4.0466731120e-09 -4.5954593857e-09 -1.7847748472e-09 3.8672239755e-08 -7.1536247424e-08 -7.9791145621e-08 -3.0833384422e-08 2.0444762538e-09 3.6508612761e-09 1.3350816891e-09 2.4789628386e-09 1.6156690276e-09 4.4295989808e-10 7.3522896616e-10 7.6119524763e-08 9.6309105830e-08 -5.2594095537e-08 -8.2404488489e-08 1.8163657481e-07 2.1909579414e-07 -1.1492533258e-07 -2.1988509487e-07 9.5404209321e-08 -8.4986991860e-08 -1.1715982305e-07 4.5038635948e-08 5.8142084758e-08 7.2435710383e-08 1.3348335240e-10 2.1827178511e-08 8.9905503235e-12 8.6334023960e-09 6.2904918887e-13 4.2352662046e-09 1.1999123041e-09 5.0592227308e-08 1.4313719722e-11 1.3060617550e-08 7.3086892961e-09 1.1352799060e-07
2.2694772066e-05 2.1204704219e-04 1.6994583782e-04 -1.2957279133e-04 -8.1323910888e-05 4.4582050462e-04 4.4629050384e-04 -2.6273207591e-04 -3.0918716803e-04 3.4288466114e-04 -1.3883772624e-04 -1.4517185241e-04 2.4107473401e-04 1.0113441405e-04 2.5727824457e-04 4.1722689967e-08 1.1000993668e-05 5.8583469725e-06 1.7973537937e-05 -2.1014027308e-06 1.0907823294e-04 7.2485251571e-05 2.4049384230e-04 -5.1272994682e-05 2.1008029061e-05 2.4516002815e-05 -7.0126913560e-06 6.7354040843e-05 -1.6044685373e-05 1.9772782607e-05 1.3627771607e-11 1.8553264284e-07 -4.0194756103e-07 2.7227875478e-07 -1.1155786874e-07 1.3631577732e-05 -3.2240755709e-05 3.1656079029e-05 -1.0366442408e-05 9.7385932427e-07 -1.0517035995e-06 4.2994091360e-07 4.9071136449e-07 -2.9471889562e-07 1.7754338002e-07 4.6118388160e-07 3.9520269502e-05 -5.8334931925e-05 2.4657051919e-05 -2.9644238592e-05 1.7642447955e-04 -3.0426596655e-04 1.4099531922e-04 -1.8266116158e-04 5.7203126885e-05 -3.3606180221e-05 3.7810904826e-05 2.2119508925e-05 -1.5698059360e-05 2.7150810770e-05 7.4105987161e-10 5.0593379814e-06 -1.2009056346e-05 -2.5826648099e-06 -9.7491614940e-07 7.8602212271e-05 -2.0968063168e-04 -4.6096335808e-05 -2.8394272726e-05 9.4692007199e-06 3.2905196720e-06 9.4877694296e-07 1.1352210525e-06 2.3664124036e-07 9.2383272133e-07 3.9234733372e-05 4.5839330429e-04 -1.3209913815e-05 3.3147716929e-04 -2.4776845196e-04 5.8894533093e-04 1.5609781313e-05 4.2135514289e-04 -4.2414731382e-04 2.8932342648e-04 -1.2971514680e-05 -1.9140722769e-05 4.3194890863e-04 -1.8644054400e-04 4.1509634692e-04 1.4849247410e-06 4.9003551548e-05 2.4800843768e-06 7.9814811416e-05 1.0556320624e-09 5.3657997257e-06 1.3598897366e-06 7.6836960420e-05 2.5181415360e-10 1.9090879841e-06 1.1352799060e-07 2.0525299446e-05
Atom 1 O 8
...
5.5. Using HiPart as a library¶
Next to the command-line scripts illustrated in the previous section, HiPart can also be used as a library to write your own partitioning scripts. This is mostly useful when one has to perform rather repetitive work, or when some subsequent analysis is necessary. A few typical examples are shown below, although the possibilities are endless.
5.5.1. Batch processing of many fchk files¶
When a large number of Gaussian files must be processed to generate a database of atomic partial charges, the standard HiPart command-line scripts may not be suitable enough. The following custom script uses the HiPart library to compute Hirshfeld charges for all molecules in the previous section. The regular output and work files are suppressed and all results are written to a single file with a custom format. The script is included in the source tree in the directory examples/002-batch.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 | #!/usr/bin/env python
# -*- coding: utf-8 -*-
# HiPart is a program to analyze the electronic structure of molecules with
# fuzzy-atom partitioning methods.
# Copyright (C) 2007 - 2012 Toon Verstraelen <Toon.Verstraelen@UGent.be>
#
# This file is part of HiPart.
#
# HiPart is free software; you can redistribute it and/or
# modify it under the terms of the GNU General Public License
# as published by the Free Software Foundation; either version 3
# of the License, or (at your option) any later version.
#
# HiPart is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program; if not, see <http://www.gnu.org/licenses/>
#
#--
#!/usr/bin/env python
def main():
import glob
from molmod import angstrom
from hipart.atoms import AtomTable
from hipart.context import Context, Options
from hipart.schemes import HirshfeldScheme
# load the atom table from the first example
atom_table = AtomTable("../001-usage/atoms/densities.txt")
# select formatted checkpoint files from the first example
fns_fchk = glob.glob("../001-usage/*.fchk")
f = file("all.txt", "w")
# loop over all formatted checkpoint files
for fn_fchk in fns_fchk:
# setup a HiPart computation
options = Options(lebedev=266, do_work=False, do_output=False)
context = Context(fn_fchk, options)
scheme = HirshfeldScheme(context, atom_table)
# the following line does the computation
scheme.do_charges()
# print output to the file all.txt
m = scheme.molecule
print >> f, "XXX", m.size
for i in xrange(m.size):
print >> f, "%3i % 15.10f % 15.10f % 15.10f % 9.5f" % (
m.numbers[i], m.coordinates[i,0]/angstrom,
m.coordinates[i,1]/angstrom, m.coordinates[i,2]/angstrom,
scheme.charges[i]
)
f.close()
if __name__ == "__main__":
main()
|
The script is executed as follows:
toon@poony ~/hipart/examples/001-usage> cd ../002-batch
toon@poony ~/hipart/examples/002-batch> ./batch.py
The output file, all.txt, reads:
toon@poony ~/hipart/examples/002-batch> cat all.txt
XXX 13
8 1.6576426043 0.7442011013 -0.4898368374 -0.16996
8 1.1020987870 -1.0890743230 0.6977938371 -0.27558
7 -1.4047606065 -1.0910534617 -0.4582025482 -0.22138
6 -0.6553720816 0.1615235597 -0.4158397944 0.03542
6 -1.2048173855 1.2303344269 0.5578948140 -0.08058
6 0.7748161142 -0.1640203278 -0.0128938185 0.21813
1 -0.6275594218 0.5945217510 -1.4229395476 0.03376
1 -0.6096404655 2.1496824048 0.5259454633 0.03120
1 -2.2354634061 1.4834218989 0.2840349146 0.03120
1 -1.2054475774 0.8485100890 1.5853490811 0.02841
1 -2.4039648542 -0.8977663004 -0.4401325436 0.09646
1 -1.1808053082 -1.6350484044 0.3738266447 0.09148
1 2.5305142603 0.4860126298 -0.1372893682 0.18144
XXX 12
8 -1.5492981588 1.0659318744 0.0758484510 -0.22111
8 -1.4032848053 -1.0506244235 -0.2250982449 -0.14769
7 1.0892470147 -1.3210053537 0.0163836069 -0.16458
6 0.6573065841 0.0175080319 0.3899234851 0.03854
6 1.4014571966 1.1941383090 -0.2499169713 -0.07890
6 -0.8312103705 0.0584386208 0.0757614040 0.19371
1 0.7371478941 0.0945698786 1.4825857195 0.04989
1 0.9191461021 2.1375698401 0.0290968359 0.03963
1 2.4461106491 1.2221187655 0.0775005591 0.03058
1 1.3827823213 1.1176697205 -1.3435700943 0.03189
1 1.9071124462 -1.6446422178 0.5219844772 0.11557
1 1.2383147103 -1.4232178471 -0.9828918999 0.11248
5.5.2. Tuning integration grids¶
5.5.2.1. Method 1¶
One can not know a priori which combination of radial and angular integration grid gives a good trade-off between accuracy of the output of interest, e.g. effective atomic charges, and the computational cost. The program below tests all combinations of a set of radial and angular grids on the alanine wavefunction from the previous section, and measures for each combination the error on the Hirshfeld charges and the computational cost in seconds. The error is estimated as the standard deviation on the effective charges over 40 runs of the same computation. Due to the random orientation in the angular grids, the results vary each run for purely numerical reasons. The script concludes with a summary of the configurations that are Pareto-optimal with respect to a minimal error and a minimal computational cost. The script is included in the source tree in the directory examples/003-grid-tuning.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 | #!/usr/bin/env python
# -*- coding: utf-8 -*-
# HiPart is a program to analyze the electronic structure of molecules with
# fuzzy-atom partitioning methods.
# Copyright (C) 2007 - 2012 Toon Verstraelen <Toon.Verstraelen@UGent.be>
#
# This file is part of HiPart.
#
# HiPart is free software; you can redistribute it and/or
# modify it under the terms of the GNU General Public License
# as published by the Free Software Foundation; either version 3
# of the License, or (at your option) any later version.
#
# HiPart is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program; if not, see <http://www.gnu.org/licenses/>
#
#--
#!/usr/bin/env python
def run_atom_dbs():
import os
from hipart.atomdb import run_atomdb, Options
from molmod.periodic import periodic
# atom numbers of interest
atom_numbers = [1, 6, 7, 8]
# auxiliary function to setup an atomic database with a given radial grid
# size
def my_run_atomdb(size):
directory = "atoms%03i" % size
if os.path.isfile(os.path.join(directory, "densities.txt")):
return
lot = 'B3LYP/6-31g(d)'
options = Options(num_steps=size, do_work=False)
run_atomdb("g03", lot, atom_numbers, options, directory)
# array with different sizes for the radial grid
sizes = [50, 100, 200, 400]
# first run with the least radial grid points
my_run_atomdb(sizes[0])
# then make symlinks to reuse the atomic computations
for size in sizes[1:]:
directory = "atoms%03i" % size
if not os.path.isdir(directory):
os.mkdir(directory)
for number in atom_numbers:
name = "%03i%s" % (number, periodic[number].symbol)
source = os.path.join("../atoms050", name)
link_name = os.path.join(directory, name)
os.symlink(source, link_name)
# run atomdb in other directories
for size in sizes[1:]:
my_run_atomdb(size)
# load the tables with atomic density profiles
from hipart.atoms import AtomTable
atom_tables = {}
for size in sizes:
densities_fn = os.path.join("atoms%03i" % size, "densities.txt")
atom_tables[size] = AtomTable(densities_fn)
return atom_tables
class Config(object):
def __init__(self, size, lebedev, error, cost):
self.size = size
self.lebedev = lebedev
self.error = error
self.cost = cost
self.optimal = False
def __str__(self):
return "size = %3i lebedev = %3i error = %.2e cost = %.2e" % \
(self.size, self.lebedev, self.error, self.cost)
def estimate_errors(atom_tables, fn_fchk):
from hipart.context import Context, Options
from hipart.schemes import HirshfeldScheme
from hipart.log import log
import numpy, time
configs = []
# The loops below run 40 jobs for each grid configuration, i.e. combination
# of radial and angular grid. Due to the random rotations of the angular
# integration grids, the resulting charges will differ in each run. The
# standard devation on each charge over the 40 runs is computed, and then
# the error over all atoms is averaged. This error is used as 'the' error on
# the charges. The cost is the cpu time consumed for computing this error.
for size, atom_table in sorted(atom_tables.iteritems()):
for lebedev in 26, 50, 110, 170, 266, 434:
log.begin("Config size=%i lebedev=%i" % (size, lebedev))
start = time.clock()
all_charges = []
for counter in xrange(40):
log.begin("Sample counter=%i" % counter)
# Make a new working environment for every sample
options = Options(lebedev=lebedev, do_work=False,
do_output=False)
context = Context(fn_fchk, options)
scheme = HirshfeldScheme(context, atom_table)
scheme.do_charges()
all_charges.append(scheme.charges)
del scheme
log.end()
all_charges = numpy.array(all_charges)
error = all_charges.std(axis=0).mean()
cost = time.clock() - start
log("error=%.2e cost=%.2e" % (error, cost))
configs.append(Config(size, lebedev, error, cost))
log.end()
log.begin("All configs")
for config in configs:
log(str(config))
log.end()
return configs
def pareto_front(configs):
from hipart.log import log
# take a copy of the list, so that we can modify locally.
configs = list(configs)
# eliminate all configs that are not pareto-optimal
configs.sort(key=(lambda c: c.error))
i = 0
while i < len(configs):
ref_cost = configs[i].cost
j = len(configs) -1
while j > i:
if configs[j].cost >= configs[i].cost:
del configs[j]
j -= 1
i += 1
# print the pareto front.
log.begin("Pareto front")
for config in configs:
log(str(config))
config.optimal = True
log.end()
def write_table(configs):
# This is an ulgy routine, and it is not so important as an illustration of
# using HiPart as a library. It is usefull to write documentation though.
# 1) Arrange everything in a convenient data format for making a table
config_map = {}
lebedevs = set([])
sizes = set([])
for config in configs:
config_map[(config.size, config.lebedev)] = config
lebedevs.add(config.lebedev)
sizes.add(config.size)
lebedevs = sorted(lebedevs)
sizes = sorted(sizes)
# 2) Write the table header
print "="*9, ("="*10 + " ")*(len(lebedevs)*2)
print "lebedev ", " ".join(("l=%i" % l).center(21) for l in lebedevs)
print "-"*9, ("-"*21 + " ")*(len(lebedevs))
for s in sizes:
words = []
for l in lebedevs:
c = config_map[(s,l)]
if c.optimal:
word = "**%7.1e** " % c.error
word += ("**%.1f**" % c.cost).center(10)
else:
word = " %7.1e " % c.error
word += ("%.1f" % c.cost).center(10)
word = word.replace("e+0", "e+").replace("e-0", "e-")
words.append(word)
print "%9s" % ("s=%i" % s), " ".join(words)
print "="*9, ("="*10 + " ")*(len(lebedevs)*2)
if __name__ == '__main__':
# 1) Generate databases of spherically averaged atomic densities at
# different grid resolutions.
atom_tables = run_atom_dbs()
# 2) Use each radial grid to estimate the error on the Hirshfeld-I charges.
configs = estimate_errors(atom_tables, "../001-usage/alanine.fchk")
# 3) Find the pareto front
pareto_front(configs)
# 4) Write a table in reStructuredText.
write_table(configs)
|
The script is executed as follows:
toon@poony ~/hipart/examples/002-batch> cd ../003-grid-tuning
toon@poony ~/hipart/examples/003-grid-tuning> ./tune.py
The Pareto-optimal choices are printed in the end of the screen output:
---TIME--- ---------------------------------LOG---------------------------------
<lots of output omitted>
10938.71 BEGIN Pareto front
10938.71 size = 400 lebedev = 434 error = 3.65e-06 cost = 2.37e+03
10938.71 size = 200 lebedev = 434 error = 5.58e-06 cost = 1.18e+03
10938.71 size = 100 lebedev = 434 error = 8.50e-06 cost = 5.84e+02
10938.71 size = 100 lebedev = 266 error = 2.10e-05 cost = 3.62e+02
10938.71 size = 100 lebedev = 170 error = 3.72e-05 cost = 2.33e+02
10938.71 size = 100 lebedev = 110 error = 7.81e-05 cost = 1.54e+02
10938.71 size = 200 lebedev = 50 error = 2.10e-04 cost = 1.49e+02
10938.71 size = 100 lebedev = 50 error = 2.98e-04 cost = 7.47e+01
10938.71 size = 100 lebedev = 26 error = 1.26e-03 cost = 4.29e+01
10938.71 size = 50 lebedev = 26 error = 3.18e-03 cost = 2.19e+01
10938.71 END Pareto front
All configurations that are beaten both in error and computing time by some other configuration are not present in the Pareto front. The output also shows for each improvement in error (or computing time) what price one has to pay in computing time (or error). The (error,time) pair of each combination is included in the table below, and the Pareto-optimal cases are printed in bold.
| lebedev | l=26 | l=50 | l=110 | l=170 | l=266 | l=434 | ||||||
| s=50 | 3.2e-3 | 21.9 | 3.8e-3 | 37.8 | 7.1e-3 | 77.3 | 1.9e-3 | 117.3 | 3.1e-3 | 180.8 | 1.3e-3 | 292.1 |
| s=100 | 1.3e-3 | 42.9 | 3.0e-4 | 74.7 | 7.8e-5 | 153.9 | 3.7e-5 | 232.9 | 2.1e-5 | 361.5 | 8.5e-6 | 584.4 |
| s=200 | 9.6e-4 | 85.3 | 2.1e-4 | 149.0 | 5.1e-5 | 307.9 | 2.9e-5 | 466.9 | 1.2e-5 | 726.5 | 5.6e-6 | 1177.7 |
| s=400 | 6.1e-4 | 170.0 | 1.5e-4 | 297.8 | 3.7e-5 | 616.7 | 2.1e-5 | 936.4 | 9.6e-6 | 1460.4 | 3.7e-6 | 2366.7 |
The choice of the angular grid seems to be the most determinant for the accuracy (as soon as one uses 100 radial grid points). This is not surprising because the angular grids are randomly rotated, while the radial grids are always the same. Nevertheless, this exercise gives us at least some idea of the accuracy of the effective charges.
The combination (s=100,l=110) is the default choice in HiPart, and results in an error of about 1e-4 for the effective partial charges. (This error is also used as the standard convergence criterion for the iterative procedures.) When too small grids are used (e.g. s=50,l=26) the iterative procedures (Iterative Hirshfeld and Iterative Stockholder Analysis) will only converge with relaxed thresholds.
5.5.2.2. Method 2¶
An alternative method to test the quality of the charges is to compare them with charges obtained with larger grids. The script examples/004-grid-tuning/tune.py performs such an analysis and is very similar to the script for Method 1. It computes charges for ethanol, formaldehyde and oxalic acid for many combinations of angular and radial grids, estimates the error by comparing with the best grid, and finally prints out the Pareto-optimal choices.
The script is executed as follows:
toon@poony ~/hipart/examples/003-grid-tuning> cd ../004-grid-tuning
toon@poony ~/hipart/examples/004-grid-tuning> ./tune.py
The Pareto-optimal choices are printed in the end of the screen output:
---TIME--- ---------------------------------LOG---------------------------------
<lots of output omitted>
11898.77 BEGIN Pareto front
11898.77 size = 1600 lebedev = 2702 error = 0.00e+00 cost = 1.40e+03
11898.77 size = 800 lebedev = 2702 error = 2.09e-08 cost = 6.89e+02
11898.77 size = 400 lebedev = 2702 error = 1.12e-07 cost = 3.44e+02
11898.77 size = 400 lebedev = 1454 error = 1.49e-07 cost = 1.86e+02
11898.77 size = 400 lebedev = 770 error = 1.41e-06 cost = 9.77e+01
11898.77 size = 200 lebedev = 1454 error = 1.61e-06 cost = 9.22e+01
11898.77 size = 200 lebedev = 770 error = 2.05e-06 cost = 4.86e+01
11898.77 size = 200 lebedev = 434 error = 1.15e-05 cost = 2.76e+01
11898.77 size = 200 lebedev = 230 error = 1.56e-05 cost = 1.50e+01
11898.77 size = 100 lebedev = 434 error = 2.95e-05 cost = 1.37e+01
11898.77 size = 100 lebedev = 230 error = 3.63e-05 cost = 7.45e+00
11898.77 size = 100 lebedev = 110 error = 8.92e-05 cost = 3.74e+00
11898.77 size = 100 lebedev = 50 error = 2.96e-04 cost = 1.89e+00
11898.77 size = 50 lebedev = 110 error = 5.48e-04 cost = 1.88e+00
11898.77 size = 50 lebedev = 50 error = 6.13e-04 cost = 9.70e-01
11898.77 size = 50 lebedev = 26 error = 1.13e-03 cost = 5.90e-01
11898.77 END Pareto front
The (error,time) pair of each combination is included in the table below, and the Pareto-optimal cases are printed in bold.
| lebedev | l=26 | l=50 | l=110 | l=230 | l=434 | l=770 | l=1454 | l=2702 | ||||||||
| s=50 | 1.1e-3 | 0.6 | 6.1e-4 | 1.0 | 5.5e-4 | 1.9 | 1.1e-3 | 3.8 | 5.0e-4 | 6.9 | 1.1e-3 | 12.2 | 1.2e-3 | 22.9 | 3.7e-4 | 42.5 |
| s=100 | 1.2e-3 | 1.1 | 3.0e-4 | 1.9 | 8.9e-5 | 3.7 | 3.6e-5 | 7.5 | 3.0e-5 | 13.7 | 2.7e-5 | 24.2 | 2.7e-5 | 45.8 | 2.7e-5 | 85.2 |
| s=200 | 1.3e-3 | 2.3 | 1.9e-4 | 3.8 | 4.1e-5 | 7.5 | 1.6e-5 | 15.0 | 1.1e-5 | 27.6 | 2.0e-6 | 48.6 | 1.6e-6 | 92.2 | 1.7e-6 | 171.7 |
| s=400 | 8.5e-4 | 4.5 | 2.1e-4 | 7.5 | 6.3e-5 | 15.0 | 1.2e-5 | 30.1 | 5.1e-6 | 55.6 | 1.4e-6 | 97.7 | 1.5e-7 | 185.7 | 1.1e-7 | 343.8 |
| s=800 | 5.6e-4 | 9.1 | 2.0e-4 | 15.0 | 2.5e-5 | 30.1 | 1.3e-5 | 61.1 | 6.0e-6 | 111.7 | 3.8e-7 | 196.3 | 1.2e-7 | 371.3 | 2.1e-8 | 688.8 |
| s=1600 | 2.9e-4 | 18.2 | 1.1e-4 | 30.2 | 1.9e-5 | 60.5 | 5.2e-6 | 120.9 | 2.1e-6 | 224.2 | 8.1e-7 | 396.9 | 1.1e-7 | 745.4 | 0.0e+0 | 1401.3 |
The same script can be used to estimate the errors when the --no-random option is used. The table with (error,time) pairs becomes:
| lebedev | l=26 | l=50 | l=110 | l=230 | l=434 | l=770 | l=1454 | l=2702 | ||||||||
| s=50 | 2.5e-3 | 0.5 | 1.1e-3 | 0.8 | 2.8e-4 | 1.8 | 6.8e-4 | 3.7 | 4.8e-4 | 6.9 | 3.1e-4 | 12.2 | 7.6e-4 | 23.0 | 4.1e-4 | 42.9 |
| s=100 | 2.5e-3 | 0.9 | 8.5e-4 | 1.6 | 1.0e-4 | 3.5 | 3.8e-5 | 7.2 | 2.8e-5 | 13.7 | 3.0e-5 | 24.2 | 3.0e-5 | 46.2 | 2.7e-5 | 86.1 |
| s=200 | 2.5e-3 | 1.7 | 8.2e-4 | 3.2 | 1.1e-4 | 7.0 | 3.7e-5 | 14.6 | 8.4e-6 | 27.4 | 3.4e-6 | 48.7 | 2.1e-6 | 92.9 | 1.7e-6 | 173.1 |
| s=400 | 2.5e-3 | 3.4 | 8.2e-4 | 6.4 | 1.1e-4 | 14.0 | 3.7e-5 | 29.2 | 8.9e-6 | 55.3 | 4.0e-6 | 98.0 | 4.3e-7 | 186.9 | 1.2e-7 | 347.0 |
| s=800 | 2.5e-3 | 6.8 | 8.2e-4 | 12.8 | 1.1e-4 | 28.1 | 3.7e-5 | 58.8 | 9.0e-6 | 110.8 | 4.0e-6 | 196.7 | 4.2e-7 | 375.4 | 6.8e-9 | 697.6 |
| s=1600 | 2.5e-3 | 13.6 | 8.2e-4 | 25.7 | 1.1e-4 | 56.4 | 3.7e-5 | 118.0 | 9.0e-6 | 222.2 | 4.0e-6 | 399.5 | 4.2e-7 | 751.8 | 0.0e+0 | 1414.3 |
The error estimates in the first table are very comparable to the numbers in method 1, which confirms the validity of the assumptions made in method 1.
The error is apparently not always lower when randomly rotated angular grids are used. It is mainly advantageous when the radial grids have much more grid points than the angular grids (lower left of the table), but these cases are not Pareto-optimal. When Pareto-optimal grids configurations are used, the effect of random rotations is small, but sometimes still useful, e.g. for the case (s=200,l=230). When there are more angular grids points than radial grid points (upper right of the table), the randomly rotated angular grids are counterproductive. With very fine angular grids, the advantage of random rotations generally diminishes.
Note that the error with randomly-rotated grids is truly random, i.e. it is different every time the computations are done from scratch. In case of non-randomly-rotated grids, the error is systematic and will be exactly the same when a computation is done twice.
