Advanced charge density and bond order analysis¶

Charges, Populations¶

Mulliken populations

See the input key EPRINT. See also the section on Mulliken populations.

Hirshfeld charges, Voronoi deformation density (VDD)

No special input key required, except for ‘PRINT VDDANALYSIS’ for a symmetry decomposed VDD analysis. See also the section on Hirshfeld charges, Voronoi deformation density.

Multipole derived charges

No special input key required. See also the section on MDC.

Charge model 5 (CM5)

Charges calculated with CM5 activated by keyword

CM5

See also the section on CM5.

Only one of the charges above can be returned to the AMS driver, in case charges are needed at the driver level or explicitly requested with Properties%Charges in the AMS driver input. The choice for which charge analysis scheme is used for the returned charges in made with the AtomicChargesTypeForAMS keyword.

AtomicChargesTypeForAMS [Mulliken | Hirshfeld | CM5 | Voronoi | MDC-M | MDC-D | MDC-Q | 
                         QTAIM]

AtomicChargesTypeForAMS

Type

Multiple Choice

Default value

Mulliken

Options

[Mulliken, Hirshfeld, CM5, Voronoi, MDC-M, MDC-D, MDC-Q, QTAIM]

GUI name

Atomic charges for AMS

Description

Type of atomic charges to be used by AMS. Note that some of these atomic charges are computed and printed by default in ADF. Hirshfeld charges are available only for default atomic fragments.

Bond orders¶

The calculation of bond orders can be requested at the AMS driver level (see the Properties section of the AMS driver manual):

$AMSBIN/ams << eor
...
Properties
   BondOrders Yes
End

Engine ADF
   ...
EndEngine
eor

By default, bond orders are computed using the Nalewajski-Mrozek 3 4 37 38 39 method. There exist three alternative definitions of the valence and bond order indices within the Nalewajski-Mrozek approach. By default the values obtained from partitioning of Tr(P \(\Delta\) P) (i.e. Nalewajski-Mrozek-3) are calculated and printed in the output. For more information on alternative Nalewajski-Mrozek bond order indices see (see also: section on bond order analysis).

In the Engine ADF part of the input, you can specify the following options:

BondOrders
   PrintAll Yes/No
   PrintTolerance float
   TypeForAMS [Nalewajski-Mrozek-1 | Nalewajski-Mrozek-2 | Nalewajski-Mrozek-3 | Mayer | 
               Gopinathan-Jug]
End

BondOrders

Type

Block

Description

Options for the calculation of bond orders. Note: the calculation of bond orders should be requested via the Properties%BondOrders input option in the AMS driver input.

PrintAll

Type

Bool

Default value

No

Description

If ‘Yes’, all five types of bond orders (i.e. Nalewajski-Mrozek-1,2 & 3, Mayer and Gopinathan-Jug) will be printed to the output. Otherwise only the Nalewajski-Mrozek-3 and the type requested in BondOrders%TypeForAMS will be printed.

PrintTolerance

Type

Float

Default value

0.2

Description

Only bond orders larger than this threshold will be printed in the output (this treshold applies only to the printing in the ‘BOND-ORDER ANALYSIS’ section of the ADF output.

TypeForAMS

Type

Multiple Choice

Default value

Nalewajski-Mrozek-3

Options

[Nalewajski-Mrozek-1, Nalewajski-Mrozek-2, Nalewajski-Mrozek-3, Mayer, Gopinathan-Jug]

GUI name

Bond order type for AMS

Description

The type of bond order that will be saved, printed and used by AMS. Nalewajski-Mrozek-1,2: bond orders calculated from two-electron valence indices based on partitioning of tr(Delta_P^2) using 3-index set or 4-index set respectively. Nalewajski-Mrozek-3: bond-orders calculated from valence indices based on partitioning of tr(P*Delta_P). Inter-atomic bond orders are not defined with non-atomic fragments.

The bond order analysis is based on SFOs. The symmetry used in the calculation should be NOSYM. The bond analysis may be used also for multi-atomic fragments. The fragment-fragment bond orders are printed in such a case. Note that in the present implementation all fragment types should be different.

Mayer bond orders and Mulliken atom-atom populations per l-value can be enabled using the ExtendedPopan key:

ExtendedPopan Yes/No

ExtendedPopan

Type

Bool

Default value

No

GUI name

: Extended population analysis

Description

Calculate the Mayer bond orders and Mulliken atom-atom populations per l-value

See also the section on bond order analysis.

ETS-NOCV: Natural Orbitals for Chemical Valence¶

With the ETS-NOCV charge and energy decomposition scheme the deformation density is partitioned into the different components (\(\sigma\), \(\pi\), \(\delta\)) of the chemical bond. The energy contributions to the total bond energy is calculated for each specific orbital interactions between fragments, giving insight in the orbital interactions also for non-symmetric molecules. The ETS-NOCV analysis offers a compact quantitative picture of the chemical bond, which is also qualitatively attractive to chemists.

The Natural Orbitals for Chemical Valence (NOCV) approach has been derived from the Nalewajski-Mrozek valence theory 3 4. From the mathematical point of view, each NOCV \(\psi\)_i is defined as an eigenvector of the deformation density matrix in the basis of fragment orbitals.

Thus, the deformation density \(\Delta\)\(\rho\) can be expressed in the NOCV representation as a sum of pairs of complimentary eigenfunctions (\(\psi_{-k}\), \(\psi_k\)) corresponding to eigenvalues -\(\nu\)_k and \(\nu\)_k with the same absolute value but opposite signs:

here, k goes over the pairs of NOCV’s.

In the combined ETS-NOCV scheme the orbital interaction term \(\Delta E_\text{orb}\) is expressed in terms of NOCV’s as 5 6:

here, \(-F^\text{TS}_{-k}\) and \(F^\text{TS}_{k}\) are diagonal transition-state Kohn-Sham matrix elements corresponding to NOCV’s with eigenvalues -\(\nu\)_k and \(\nu\)_k , respectively. The advantage of this expression is that usually only a few complimentary NOCV pairs significantly contribute to the total \(\Delta E_\text{orb}\). Another advantage of this approach is that not only can each \(\Delta \rho_k (r)\) be visualized but there is also a well defined bonding energy contribution \(\Delta E_k^\text{orb}\) corresponding to it.

Further decomposition of \(\Delta E_{k}^\text{orb}\) into \(\Delta T_{k}^\text{orb}\) and \(\Delta V_{k}^\text{orb}\) components is possible as the Fock matrix can be decomposed into kinetic and potential energy matrices 53:

where \(T^\text{TS}_{-k}\) and \(V^\text{TS}_{-k}\) are diagonal elements of T and V matrices corresponding to \(F^\text{TS}\) matrix.

Remarks

The ETS-NOCV analysis is often not very useful when atomic fragments are used. No symmetry must be used in the final calculation, thus, use a Symmetry NOSYM keyword if your molecule is symmetric. The analysis is not completely implemented for meta-GGA’s and meta-hybrids. In case of meta-GGA’s and meta-hybrids the so called ‘MetaGGA correction’ or ‘MetaHybrid correction’ part of the orbital interaction term is not taken into account when calculating energy contributions.

Improvements in ADF2012 to both the ETS and NOCV analysis with hybrids. ETS: Now the exact exchange contribution to the Pauli term is isolated and the contributions to the orbital term are divided among orbital symmetries. NOCV: The exact exchange contribution to the Fock operator is included when calculating energy contributions. These changes do not apply to meta-hybrids.

Usage

In order to perform the ETS-NOCV analysis, the following two keywords must be specified at the same time:

ETSNOCV
   {RHOKMIN rhokmin}
   {EKMIN ekmin}
   {ENOCV enocv}
   {TVANALYSIS}
End
PRINT {ETSLOWDIN | ETSLOWDIN-Unrestricted}

ETSNOCV

The ETSNOCV keyword specifies thresholds for printing of NOCV-related information. All three arguments are optional and when all three are omitted only the NOCV’s corresponding to eigenvalues abs \((\nu_k) \geq 0.05\) are included in the analysis.

RHOKMIN

The threshold for population analysis of each deformation density contribution in terms of individual SFO’s.

EKMIN

The threshold for orbital interaction energy contributions corresponding to deformation density components originating from each NOCV-pairs.

ENOCV

The threshold for NOCV-eigenvalues.

TVANALYSIS

Enable T/V decomposition of energy contributions.

PRINT {ETSLOWDIN | ETSLOWDIN-Unrestricted}

Only one of the two PRINT options is supposed to be used to activate printing of ETS-NOCV results. The choice depends on the bonding situation.

ETSLOWDIN

If one is interested in a description of bonding between closed-shell molecular fragments, then ‘PRINT ETSLOWDIN’ keyword must be used. In such a case one set of NOCV’s originating from the total deformation density matrix \(\Delta P = ( \Delta P_\alpha + \Delta P_\beta)\) will be printed out. See the example of carbene bonding between closed shell CH2 and Cr(CO)5.

ETSLOWDIN-Unrestricted

If, however, one is interested in a description of bonding between open-shell molecular fragments then the ‘PRINT ETSLOWDIN-Unrestricted’ keyword must be used. In this case two sets of NOCV’s originating from \(\Delta P_\alpha\) and \(\Delta P_\beta\) will be printed out. See the example of CH3-CH3 bonding between two CH3 radicals with opposite spins. This option must also be used when one wants to analyze bonding in a molecule with unpaired electrons.

PRINT NOCVHirshfeld

The NOCVs can be integrated per fragment using the Hirshfeld partitioning scheme. Using these integrals, one can distinguish between inter- and intra-fragment NOCVs. An inter-fragment NOCV has non-zero integral Hirshfeld \(\Delta Q\) value and corresponds to a charge transfer between fragments. An intra-fragment NOCV has a small \(\Delta Q\) value and corresponds to polarization of the fragments. When ‘PRINT NOCVHirshfeld’ is specified, the \(\int{\Delta \rho} = -\Delta Q\) values per NOCV are printed to the output file in the “Hirshfeld partitioning” table at the end of the ETS-NOCV section.

Adfnbo, gennbo: NBO analysis¶

• (PDF) NBO manual

Dr. Autschbach, SCM, and Prof. Weinhold have collaborated to prepare a simple in put file generator, called adfnbo, for the GENNBO program of Prof. Weinholds Natural Bond Orbital (NBO) package. In ADF2013 and later the NBO 6.0 version is supported http://nbo6.chem.wisc.edu.

The GENNBO executable is included in the ADF distribution and can be enabled via the license file for all those who buy an NBO license from SCM ( info@scm.com). An extensive documentation of GENNBO is part of the NBO manual. The application of ADFNBO to frozen-core basis sets needs to be further tested. Usage can be found below and in the Examples Document.

Next a brief summary of the capabilities of GENNBO is given (by Prof. Weinhold). GENNBO implements most capabilities of the full NBO 6.0 program suite as described on the NBO website: http://nbo6.chem.wisc.edu These include determination of natural atomic orbitals (NAOs), bond orbitals (NBOs), and localized MOs (NLMOs), as well as the associated NPA (atomic charges and orbital populations) and NRT (resonance structures, weightings, bond orders) valence descriptors, for a wide variety of uncorrelated and correlated (variational, perturbative, or density functional) theoretical levels. GENNBO-supported options include all keywords except those explicitly requiring interactive communication with the host electronic structure system (viz., $DEL deletions, NEDA, NCS, NJC). The GENNBO program typically sits conveniently on the PC desktop, ready to analyze (or re-analyze at will, with altered options) the final results of a complex ADF calculation performed on a remote cluster.

GENNBO “communicates” with the original ADF calculation through an archive file (JOB.47 file, preserving all necessary details of the final density) that is initially generated by ADF and subsequently becomes the input file for GENNBO. The .47 file contains a standard $NBO … $END keylist that can be edited with a standard word processor or text editor to include chosen NBO keyword options, just as though they might have appeared in the original input stream of an interactive ADFNBO run. The stand-alone GENNBO program therefore allows many alternative NBO analysis options to be explored at leisure, without costly re-calculation of the wave function.

In ADF2018 the NBO 6.0 version6 in the ADF distribution is updated to the March 2017 version, see also http://nbo6.chem.wisc.edu: “The 14-Mar-2017 distribution includes a number of other features of interest to general NBO users: The default NAO search algorithm has been modified to better preserve core-valence separation in rare cases where numerical near-degeneracies can lead to unphysical core-valence mixing. The numerical effects are generally negligible except where they become necessary. The “OLDNAO” keyword restores the legacy NAO algorithm for comparison purposes.”

This new NAO method may change the NBO analysis of some properties (EFG, NMR) quite substantially, especially individual contributions.

Usage

ADF needs to write some data to file, which is done by including FULLFOCK, AOMAT2FILE, SAVE TAPE15, and SYMMETRY NOSYM in the input file for ADF. An all electron basis set is needed. Use the key ‘spherical’ in the adfnbo input. A file named FILE47 is generated by adfnbo which is an input file for the general NBO program gennbo6. Thus the usage is like

$AMSBIN/ams <<eor
...
Engine ADF
   Basis
     ...
     Core None
   End
   FullFock
   AOMat2File
   Save TAPE15
   Symmetry Nosym
EndEngine
eor

$AMSBIN/adfnbo <<eor
   write
   fock
   spherical
   end input
eor

$AMSBIN/gennbo6 FILE47

Remark: if the key TESTJOB in included in the input for adfnbo, NRT will be included in FILE47. The calculation of NRT (natural resonance theory) can be very expensive for larger systems (or may fail to converge). Thus it is not recommended to include the key TESTJOB in the input for adfnbo for larger systems.

AOresponse
 donothing
End

Aoresponse
 efg NUC nbo
end

Aoresponse
   efg 1 nbo Tresh 0.01
end

FAKESO

NMR
 u1k best
 calc all
END

analysis
 print 0.01
 canonical
 nbo
 components
end

nmrcoupling
   ... other options
   contributions 1E19 nbo
end

AMS_JOBNAME=Scalar $AMSBIN/ams <<eor
  ...
eor

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

$AMSBIN/adfnbo << eor
 ADFfile Scalar.results/adf.rkf
 TAPE15file Scalar.results/TAPE15
 write
 spherical
eor

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

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

$AMSBIN/adfnbo <<  eor
 ADFfile Scalar.results/adf.rkf
 TAPE15file Scalar.results/TAPE15
 spherical
 copy
eor

# run adfnbo in READ mode: prepare locorb on TAPE21

$AMSBIN/adfnbo <<  eor
 ADFfile scalar.results/adf.rkf
 TAPE15file scalar.results/TAPE15
 spherical
 read
eor

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

# keep the adf.rkf after this sequence in order to
# be able to plot the NBOs and NLMOs with amsview

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

QTAIM: Atoms in Molecules¶

One can calculate local and atomic properties using a real-space partition of the electronic density. It is based on the quantum theory of atoms in molecules (QTAIM) developed by Richard Bader 9. Another possibility for Bader analysis is to use the adf2aim utility with a third party program such as Xaim.

AROMATICITY
  1 2 3 4 5 6
END

Conceptual DFT¶

adf2damqt: DAMQT interface¶

Interface (adf2damqt) to the 3rd party DAMQT 2.0 package 35, which can analyze the electron density in molecules, and related quantities. DAMQT includes the method of deformed atoms in molecules (DAM). The DAMQT 2.0 package is described in Ref. 35. In the ADF calculation TAPE15 and adf.rkf need to be saved. The executable adf2damqt can be run with up to three optional arguments. If no argument is supplied, “ADF” is chosen as default root name (fname) for files generated by the interface, and files containing electron density matrix (fname.den) and molecular orbitals (fname.SLorba and, eventually, fname.SLorbb) will be created in a format suitable to be read by DAMQT.

$AMSBIN/ams << eor
...
Engine ADF
   ...
   SAVE TAPE15
EndEngine
eor
mv ams.results/adf.rkf TAPE21
mv ams.results/TAPE15 TAPE15
$AMSBIN/adf2damqt {fname {SPIN} {NOORBITALS}}

fname

If a specific name is desired for the files (fname), it must be supplied as first optional argument, provided that the name does not coincide with any of the two additional options SPIN and NOORBITALS.

SPIN

SPIN: for storing spin density matrix in fname.den file (instead of total electron density, which is the default).

NOORBITALS

NOORBITALS: to prevent generation of files with molecular orbitals (by default orbitals are generated).

SPIN and NOORBITALS are case insensitive and can be given in any order (but always after optional fname when required).

FOD: fractional orbital density¶

Following the analysis method by Grimme and Hansen (Ref. 36) a fractional occupation number weighted electron density (FOD) can be plotted with AMSview. The scalar field is obtained by finite-temperature DFT calculations with pre-defined electronic temperature \(T_{el}\), typically \(T_{el}\) =5000 K, and for hybrids \(T_{el}\) =20000*(amount of HF exchange)+5000, see Ref. 36. This analysis tool can be used to determine whether static electron-correlation effects are important.

OCCUPATIONS ELECTRONICTEMPERATURE=5000

The NFOD (integrated FOD) is written in the output of ADF. The FOD can be visualized with AMSview. The contour surface value in AMSview should be set to 0.005 (\(e/(bohr)^3\)).

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