Advanced charge density and bond order analysis¶

NBO analysis of EFG, NMR chemical shifts, NMR spin-spin coupling¶

For certain molecular properties it is possible to perform detailed analyses in terms of Natural Bond Orbitals (NBOs) and Natural Localized Molecular Orbitals (NLMOs). These features generally require a sequence of ADF and/or property code runs. An initial non-relativistic or scalar relativistic ADF run, followed by the generation of NBO and NLMO data, is required, and the resulting data files need to be present in subsequent property calculations, along with a keyword indicating that the NBO analysis is requested in the property module.

We have noted in the past some slight loss of numerical accuracy of the results after going through the various orbital transformations in the NBO - NLMO sequence. It is important that the user verifies in each case that the total contributions from the analysis are in agreement with the total calculated property, within the numerical integration accuracy limits. In order to assist the user with this, the analysis program always print the total analysis contributions, including small non-printed values.

Moreover, there appears to be a problem with the analysis of the Fock matrix in the NBO program in conjunction with ADF calculations. Therefore please do NOT use the Fock matrix second order perturbation theory analysis in NBO at this time. We will remove this disclaimer once the issue has been fixed. Applications of the NBO-NLMO property analysis codes have so far given no indication that the Fock matrix issue interferes with the analysis.

Important note: If properties are analyzed from within spin-orbit relativistic computations, the NBO/NLMO analysis is performed in terms of scalar (spin-free) relativistic orbitals, as detailed in the technical references. The results from these analyses are exact in the sense that they fully reproduce the final spin-orbit property result, and they allow to dissect the property in terms of more intuitive one component real scalar relativistic localized orbitals. Typically, the property analysis in a spin-orbit calculation involves contributions from unoccupied scalar NLMOs, whereas there are no such contributions if a non-relativistic or scalar relativistic property is analyzed.

Available properties for NBO analysis: EFG, NMR chemical shifts and NMR spin-spin coupling.

NBO analysis of EFG

EFGs: non-relativistic and scalar ZORA, in ADF/AORresponse. Requires initial ADF run with

AOresponse
 donothing
End

in order to generate orbitals that re equivalent to those generated in the subsequent ADF run where the EFG is calculated. Alternatively, simply calculate the EFG twice, once before the NBO generation step, and once afterward.

The next step (see below) is to create the NBOs and the required data files for the analysis. Afterward, in the second ADF run, use

Aoresponse
 efg NUC nbo
end

efg NUC nbo

Here NUC is the number of the nucleus at which the EFG is to be computed (ADF internal atom ordering). Example: efg 1 nbo.

The threshold for printing the EFG-NBO contributions to output can be adjusted via the Tresh options. The default is 0.05, which means that only orbitals with absolute value contribution larger than 5% of the total EFG are printed. To increase the number of contributions printed, specify a smaller threshold. For example:

Aoresponse
   efg 1 nbo Tresh 0.01
end

In addition to the optional NBO analysis, the EFG program in AOResponse prints a Mulliken type analysis of the EFG principal components, and an analysis in terms of canonical MOs.

WARNING: The ordering of the principal components is lowest to highest including the sign. That is, we have \(V11 \leq V22 \leq V33\). This does not conform to the usual convention of \(|V11| \leq |V22| \leq |V33|\). Please make sure you select the right component for your analysis.

Example job: $ADFHOME/examples/adf/AlCl3_efgnbo. For an explanation of the output and a general usage tutorial, see [327]. Further references and recommended citations, see [328].

NBO analysis of NMR Chemical shift

An implementation is currently available for spin-orbit ZORA computations. If scalar ZORA calculations are to be analyzed, provide the input keyword

FAKESO

in the NMR input (outside of the ‘nmr’ keyword). If this feature is requested one should restrict the calculation to a single shielding tensor per NMR run. It would be good practice to check the results against regular NMR calculations where the analysis feature is not requested. No ZORA scaling is applied in the analysis results. The data should be equivalent to a regular computation in the NMR input with

NMR
 u1k best
 calc all
END

Depending on whether scalar or spin-orbit calculations are to be analyzed, the sequence of calculations is different:

scalar:

1. ADF, scalar ZORA, symmetry NOSYM, closed shell
2. generate NBOs and required data files for analysis
3. NMR with FAKESO and analysis keywords, use TAPE21, TAPE10 from step 1.

spin-orbit:

1. ADF, scalar ZORA, closed shell
2. generate NBOs and required data files for analysis
3. delete TAPE21, TAPE10, TAPE15
4. ADF, spin-orbit ZORA, symmetry NOSYM, closed shell
5. NMR with analysis keywords, using TAPE21, TAPE10 from step 4

In the NMR run, in addition to the NMR keyword, provide the following

analysis
 print 0.01
 canonical
 nbo
 components
end

The optional canonical keyword can be used independently from the NBO analysis features. It enables an analysis of the shielding in terms of the canonical MOs. The components keyword is optional and enables an analysis not only of the isotropic shielding but also of each principal component of the tensor. The print keyword selects printout of contributions relative to the total diamagnetic, paramagnetic. In the example, only contributions greater than 1% are printed. Set to zero to print ALL contributions.

Example job: $ADFHOME/examples/adf/CH4_nmrnbo. References [329-331].

NBO analysis of NMR spin-spin coupling (J-coupling)

Non-relativistic, scalar ZORA, spin-orbit ZORA

The sequence of jobs is similar to those in the NMR section.

scalar or non-relativistic:

1. ADF, scalar ZORA or nonrel.
2. generate NBOs and required data files for analysis
3. CPL with analysis keyword, use TAPE21, TAPE10 from step 1.

spin-orbit:

1. ADF, scalar ZORA
2. generate NBOs and required data files for analysis
3. delete TAPE21, TAPE10, TAPE15
4. ADF, spin-orbit ZORA
5. CPL with analysis keyword, using TAPE21, TAPE10 from step 4

In the CPL run provide the following ‘contributions’ keyword to enable the analysis

nmrcoupling
   ... other options
   contributions 1E19 nbo
end

The numerical value selects a print threshold in SI units of T**2/J for the analysis. Increase the value to obtain less detail in the analysis. By default, ‘contributions’ triggers an analysis of the J-coupling in terms of canonical MOs. The nbo keyword enables in addition the NBO-NLMO analysis.

Please note that due to the history of how the program was developed the output from the scalar/nrel. analysis and from the spin-orbit calculations differs somewhat. The qualitative content is the same.

In scalar ZORA or non-relativistic CPL calculations without the SD term an orbital based analysis is only performed for the Fermi-contact mechanism. If you also need an analysis for the PSO and SD mechanisms but do not want to run a spin-orbit calculation with ADF please use the SD or NOSD keywords which will cause the spin-orbit branch of the CPL code to be used. In ZORA spin-orbit calculations the FC, SD, PSO, and cross terms are analyzed together by default. You can selectively switch them on or off in order to get individual mechanism analyses. The DSO mechanism is often negligible. An analysis tool for this mechanism has therefore not yet been developed.

Example job: $ADFHOME/examples/adf/CPL_CH3OH_NBO. References NMR spin-spin couplings with NBO analysis [331-334]:

Generation of NBOs

How to generate the NBOs, NLMOs, and the data files needed for these calculations (step 2 below is step 2 in the examples above):

1. run ADF with scalar ZORA or non-relativistic options, and keep TAPE21 and TAPE15.

# run adfnbo in WRITE  mode to create the gennbo input file FILE47
# and one of the required property analysis files, adfnbo.kf

$ADFBIN/adfnbo << eor
write
spherical
end input
eor

rm -f adfnbo.37 adfnbo.39 adfnbo.49 adfnbo.48
$ADFBIN/gennbo6 << FILE47

# run adfnbo in COPY mode to create the second property analysis
# file, adfnbo2.kf

$ADFBIN/adfnbo <<  eor
 spherical
 copy
end input
eor

# run adfnbo in READ mode: prepare locorb on TAPE21

$ADFBIN/adfnbo <<  eor
 spherical
 read
end input
eor

rm -f adfnbo.37 adfnbo.39 adfnbo.49 adfnbo.48

# keep the TAPE21 after this sequence in order to
# be able to plot the NBOs and NLMOs with adfview:

mv TAPE21 nbonlmo.t21

# clean up, keep adfnbo*.kf for any NBO property analyses.

Bader atomic properties (grid based method)¶

A fast Bader atomic property calculation (see Refs. [228,229]) is performed by specifying the BADER keyword in the input file. This calculation can be used for relatively big systems (hundreds of atoms). The default calculation produces atomic electron density populations, charges, density Laplacian, dipole and quadrupole moments, and Shannon and Renyi entropies. Starting from ADF2012 the default calculation also includes the calculation of critical points of the density and bond paths (see Ref. [363]), which can be visualized with the ADF-GUI. The following CP descriptors are available: density, laplacian and Hessian matrix, ellipticity, Jenkins’ metallicity index [456], reduced density gradient and DFT energy densities variations rates [457], and Abramov’s local energy components [458].

BADER {Energy} {Spacing=value} {Reactivity}

Spacing

Specifies spacing (distance between neighboring points) of the initial grid when searching for critical points, in Angstrom. The default spacing value is 0.5 Bohr (or approximately 0.26 Angstrom). It may be useful to specify a smaller value if the default results in some critical points being missed, which will result in a more accurate but slower calculation.

Energy

If the argument Energy is included, also calculate Bader atomic energies [453]. Note that this option to calculate Bader atomic energies is only valid in cases, where the key TOTALENERGY can be used, and that this option further implies that the key EXACTDENSITY is set.

Reactivity

If Reactivity is specified then certain reactivity descriptors [459] are calculated, such as atomic Fukui functions f+ and f- [483], the Koopmans dual descriptor [484, 460], and the atomic electronegativity[461], by integrating corresponding properties over the atomic basin. Besides, MO populations per basin, the localization and delocalization indices (LI/DI) [462],[463], and the matrix of the condensed linear response function (in the frozen orbital approximation)[464] are computed. When the Koopmans approximation is used (f+, f- and dual descriptor), degenerate or quasi-degenerate FMOs are taken into account based on [491].

The accuracy of this this calculation [228,229] can be determined by the standard method: the integration of the Laplacian of the electron density must vanish over the Bader atomic basins. The default output produces these integrations. The accuracy of the method can be improved by using larger integration grids. Usually the default grid suffices an average atomic integration accuracy of 10^-3 a.u. (differences of milliHartree in the energies energies). The convergence of the integration of the electron density Laplacian is not monotone but sinusoidal. So the integration of the Laplacian is not always closer to zero as a larger grid is used. So this type of Bader atomic property calculation might be considered as an approach where computational efficiency is critical and moderate accuracy is sufficient [228,229].

Relativistic QTAIM Bader properties and topology [454] are computed if both arguments Bader and RELATIVISTIC ZORA are included along with CORE NONE argument since relativistic QTAIM calculations can be only computed for all electron calculations currently. Relativistic effects on QTAIM properties and topology might be important for heavy atoms [454].

ADF2AIM¶

The ADF utility adf2aim (original name rdt21) developed by Xavi López, Engelber Sans and Carles Bo converts an ADF TAPE21 to WFN format (for Bader analysis)

The program rdt21 is now called adf2aim and is part of the ADF package.

The WFN file is an input file for the third party program Xaim (see http://www.quimica.urv.es/XAIM for details), which is a graphical user interface to programs that can perform the Bader analysis. Usage of adf2aim can be found in the Examples Document.

Aromaticity index with QTAIM¶

The cyclic delocalisation of mobile electrons in two or three dimensions is probably one of the key aspects that characterise aromatic compounds. However, aromaticity cannot be measured directly by any physical or chemical experiment because it is not a well-defined magnitude. One can define indicators of aromaticity, some of which are implemented in ADF. The matrix of localisation/delocalisation indices (LI-DI) and the following from it Iring (ring index) and MCI (multi center index) aromaticity indices (see Ref. [440]) can be computed for rings specified by the user. The rings are specified using the AROMATICITY input block, one line per ring. For example, for a benzene molecule where atoms 1-6 are carbon atoms the input would look like this:

AROMATICITY
  1 2 3 4 5 6
END

The program does not check whether the atoms in the AROMATICITY input block are connected so it is responsibility of the user to specify them correctly. Moreover, for the Iring index to be calculated correctly the atoms must be listed in the order they are connected in the ring.

Calculation of the aromaticity indices invokes the QTAIM analysis automatically and uses its results so all accuracy considerations for the BADER key above are also valid here. This feature requires that the calculation is performed without symmetry (Symmetry NOSYM).

Results of the calculation are printed to the output under the “Aromaticity Index” header. The first table presents the total number of electrons per atom and their breakdown in “localized” and “delocalized”. Note: the total number of electrons per atom \(N_{total} = N_{loc} + N_{deloc} / 2\) since delocalized electrons are counted twice. The second table presents the Iring and MCI ring indices for each ring specified in the AROMATICITY block. These results (and the original LiDi matrix from which they are computed) are stored in the Properties section in the t21 file.