Advanced charge density and bond order analysis
***********************************************

In addition to Mulliken charge analysis, ADF calculates several atomic
charges that do not share the flaws of Mulliken (strong basis set
dependence). The multipole-derived charge analysis exactly reproduces
dipole and higher multipole moments of the molecule. Other charge
analysis methods (Voronoi deformation density and Hirshfeld) provide
atomic charges that agree well with chemical intuition. Nalewajski
bond orders can be calculated and show good agreement with
experimental trends and chemical intuition, even for transition metal
compounds.

Note that the amount of data can be controlled with the keys PRINT,
NoPrint, EPrint, and Debug.


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 is 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 is required. See also the section on MDC.

**ESP-CHELPG and ESP-RESP charges**

Charges derived from the Electrostatic Potential Grid (ChElPG) are
calculated with the following key:

   ESP-CHELPG

RESP (Restrained Electrostatic Potential Charge) is calculated with
the following key:

   ESP-RESP

See also the section on CHELPG and RESP.

**Charge model 5 (CM5)**

The calculation of CM5 charges is activated by the keyword

   CM5

See also the section on CM5.

**QTAIM**

QTAIM charges are calculated if "QTAIM" is enabled, and the
"AnalysisLevel" is at least "extended". See also the section on QTAIM.

**Charges for AMS**

Only one of the charges above can be returned to the AMS driver if
charges are needed at the driver level or explicitly requested with
"Properties%Charges" in the AMS driver input. The choice of which
charge analysis scheme is used for the returned charges is made with
the "AtomicChargesTypeForAMS" keyword. This key will also set the
required input for the chosen charge analysis scheme.

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

"AtomicChargesTypeForAMS"
   Type:
      Multiple Choice

   Default value:
      Mulliken

   Options:
      [Mulliken, Hirshfeld, CM5, Voronoi, MDC-M, MDC-D, MDC-Q, QTAIM,
      ESP-CHELPG, ESP-RESP]

   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
===========

There are several ways to calculate bond orders with ADF. See also the
section on bond order analysis for how bond orders are calculated.

**Bond Orders with atomic fragments**

If atomic fragments are used, 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

The bond order analysis is based on SFOs. The symmetry used in the
calculation should be "NOSYM".

By default, bond orders are computed using the Nalewajski-Mrozek [3]
[4] [37] [38] [39] method. There are 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
\DeltaP) (i.e. Nalewajski-Mrozek-3) are calculated and printed in the
output. For more information on alternative Nalewajski-Mrozek bond
order indices, see the section on bond order analysis.

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

   BondOrders
      Calculate Yes/No
      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.

   "Calculate"
      Type:
         Bool

      Default value:
         No

      Description:
         Whether or not the bond orders should be calculated. This is
         for internal use/debug purposes, except for multi-atomic
         fragments. In case of atomic fragments one should request the
         bond orders at the AMS driver level via the
         Properties%BondOrders input option. For multi-atomic
         fragments, however, one should not request the bond orders at
         the AMS driver level, but instead use this key.

   "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.

**Bond Orders with multi-atomic fragments**

The bond analysis may also be used 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.
Symmetry should be "NOSYM". In this case you may not specify the
Properties%BondOrders key in the AMS part of the input, because AMS
wants a bond analysis based on atomic fragments. However, you should
specify BondOrders%Calculate in the "Engine ADF" part of the input:

   $AMSBIN/ams << eor
   ...
   Engine ADF
      BondOrders
         Calculate Yes
         ...
      End
      Fragments
         ...
      End
      Symmetry NOSYM
   EndEngine
   eor

**ExtendedPopan**

Mayer bond orders and Mulliken atom-atom populations per l-value can
be enabled using the "ExtendedPopan" key. Symmetry can be used. The
atom-atom bond orders are printed, even if one uses multi-atomic
fragments.

   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


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 are calculated for each specific orbital
interaction between fragments, giving insight into 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.

Tip:

  See also the EDA-NOCV tutorial

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.

   \Delta P \Psi_i = \nu_i \Psi_i

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

   \Delta \rho (r) = \sum \Delta \rho_k (r) = \sum \nu_k \left(
   -\Psi^2_{-k} (r) + \Psi^2_{k} (r)   \right)

Here, k goes over the pairs of NOCVs.

In the combined ETS-NOCV scheme, the orbital interaction term \Delta
E_\text{orb} is expressed in terms of NOCVs as [5] [6]:

   \Delta E_\text{orb} = \sum \Delta E_k^\text{orb} = \sum \nu_k
   \left( -F^\text{TS}_{-k} + F^{TS}_k \right)

Here, -F^\text{TS}_{-k} and F^\text{TS}_{k} are diagonal transition-
state Kohn-Sham matrix elements corresponding to NOCVs with
eigenvalues -\nu_k and \nu_k, respectively. The advantage of this
expression is that usually only a few complementary 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]:

   \Delta E_\text{orb} = \sum \nu_k \left( -F^\text{TS}_{-k} +
   F^\text{TS}_{k} \right) = \sum \nu_k \left( -T^\text{TS}_{-k} -
   V^\text{TS}_{-k} + T^\text{TS}_k + V^\text{TS}_{k} \right) = \sum
   \left( \Delta T_\text{orb} + \Delta V_\text{orb} \right)

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

**Remarks**

The ETS-NOCV analysis is often not very useful when atomic fragments
are used. No symmetry should be used in the final calculation, so use
the "Symmetry NOSYM" keyword if your molecule is symmetric. The
analysis is not completely implemented for meta-GGAs and meta-hybrids.
In the case of meta-GGAs 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.

ADF2012 introduced improvements to both the ETS and NOCV analyses 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.

In case of bonding between closed-shell molecular fragments, the set
of NOCVs 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. In
case of bonding between open-shell molecular fragments, two sets of
NOCVs 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 using the key FRAGOCCUPATIONS, or the
example of CH3I bonding between the CH3 radical and I using the key
UNRESTRICTEDFRAGMENTS.

**Usage**

In order to perform the ETS-NOCV analysis, include the block key
ETSNOCV:

   ETSNOCV
      {RHOKMIN rhokmin}
      {EKMIN ekmin}
      {ENOCV enocv}
      {TVANALYSIS}
   End

"ETSNOCV"
   The ETSNOCV keyword specifies thresholds for printing NOCV-related
   information. All three arguments are optional, and when all three
   are omitted, only the NOCVs corresponding to eigenvalues with
   absolute values satisfying (\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 SFOs.

   "EKMIN"
      The threshold for orbital interaction energy contributions
      corresponding to deformation density components originating from
      each NOCV pair.

   "ENOCV"
      The threshold for NOCV eigenvalues.

   "TVANALYSIS"
      Enable T/V decomposition of energy contributions.

"PRINT NOCVHirshfeld"
   The NOCVs can be integrated per fragment using the Hirshfeld
   partitioning scheme. Using these integrals, you can distinguish
   between inter- and intra-fragment NOCVs. An inter-fragment NOCV has
   a nonzero integrated 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 input file generator, called adfnbo, for the GENNBO program of
Prof. Weinhold's Natural Bond Orbital (NBO) package. In ADF2013 and
later, the NBO 6.0 version is supported https://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:
https://nbo6.chem.wisc.edu. These include the 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 version in the ADF distribution was updated to
the March 2017 version; see also https://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."

**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".
The usage is as follows:

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

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

   $AMSBIN/gennbo6 FILE47

In order to get, for example, the second order perturbation theory
analysis of the Fock matrix in the NBO basis, you should include the
keyword FOCK in the adfnbo run that is performed before gennbo6 is
called. Remark: it was noted that the gennbo6 results for heavy
elements related to the Fock matrix calculated with ADF do not seem to
make sense. That is why the key FOCK is not included in this adfnbo
call if AMSinput (GUI module) is used. The key Fock is not necessary
to calculate the NBOs or NLMOs.

The default for the so-called NBO key list that adfnbo puts on FILE47
is: BNDIDX NBONLMO=W AONBO=W AONLMO=W NLMOMO=W STERIC DIST. You can
also write a user-defined NBO keylist in FILE47 using the key
NBOkeylist in adfnbo, for example like

   $AMSBIN/adfnbo <<eor
      ADFfile ams.results/adf.rkf
      TAPE15file ams.results/TAPE15
      write
      fock
      spherical
      NBOkeylist BNDIDX NBONLMO=W AONBO=W AONLMO=W NLMOMO=W NAOMO=PVAL NRT STERIC CMO 3CBOND DIST
   eor

   $AMSBIN/gennbo6 FILE47

Note that here NRT is included. The calculation of NRT (natural
resonance theory) with gennbo6 can be very expensive for larger
systems (or may fail to converge) and/or gennbo6 may try to allocate
too much memory. Thus it is not recommended to include this key NRT in
the NBOkeylist for larger systems.


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 you verify 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 with this, the analysis program
always prints 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, 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 you 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/AOResponse. Requires an
initial ADF run with

   AOresponse
    donothing
   End

in order to generate orbitals that are 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: $AMSHOME/examples/adf/AlCl3_efgnbo. For an explanation of
the output and a general usage tutorial, see [7]. For further
references and recommended citations, see [8].

**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, you 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 adf.rkf (previously
   known as TAPE21), TAPE10 from step 1.

**spin-orbit**:

1. ADF, scalar ZORA, closed shell

2. generate NBOs and required data files for analysis

3. delete adf.rkf, TAPE10, TAPE15

4. ADF, spin-orbit ZORA, symmetry NOSYM, closed shell

5. NMR with analysis keywords, using adf.rkf, 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 and paramagnetic
terms. In the example, only contributions greater than 1% are printed.
Set to zero to print all contributions.

Example job: $AMSHOME/examples/adf/CH4_nmrnbo. References [40] [41]
[42].

**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 adf.rkf (previously known as
   TAPE21), TAPE10 from step 1.

**spin-orbit**:

1. ADF, scalar ZORA

2. generate NBOs and required data files for analysis

3. delete adf.rkf, TAPE10, TAPE15

4. ADF, spin-orbit ZORA

5. CPL with analysis keyword, using adf.rkf, 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 the NBO-
NLMO analysis in addition.

Please note that due to the history of how the program was developed,
the output from the scalar/non-relativistic 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: $AMSHOME/examples/adf/CPL_CH3OH_NBO. References NMR spin-
spin couplings with NBO analysis [42] [43] [44] [45]:

**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
   "adf.rkf" and TAPE15.

   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.


Local, atomic, and non-local properties
---------------------------------------

The QTAIM input block replaces the *Bader* keyword. The former *Bader
Reactivity* option now corresponds to the *ConceptualDFT* block (see
below).

The QTAIM functionality encompasses two different but related
features: a topological analysis of the electron density gradient
field [10], and a calculation of condensed atomic properties by
integrating various local functions over atomic basins [46] [47]. The
topological analysis is performed at all levels of the analysis
(normal, extended, or full) and it produces a list of the molecule's
electron density critical points and bond paths that can be visualized
by the AMSview GUI module. In addition, the following properties at
the critical points are calculated: electron density, its Laplacian
and Hessian matrix, ellipticity, Jenkins' metallicity index [12],
reduced density gradient, DFT energy density variation rates [13], and
Abramov's local energy components [14]. At the *Extended* analysis
level the following condensed atomic properties are calculated:
charges, density Laplacian, dipole and quadrupole moments, Shannon and
Renyi entropies, spatial distribution of the molecular orbitals over
atomic basins. If the *Energy* keyword is specified, the atomic
energies [11] are also calculated. At the *Full* level, the atom-atom
matrix elements of the localization (LI) and delocalization (DI)
indices [48] [49] are computed. In ADF2023, we also added the
(optional) calculation of the Source Function introduced by Gatti and
Bader [54] [55]. This function is evaluated for each bond or ring
critical point of the molecule, allowing a decomposition of the values
of the electron density at these points as a sum of atomic
contributions. Since this calculation requires the integration of the
Laplacian of the electron density, we strongly recommend a numerical
quality set (at least) to "Good" (or higher).

"QTAIM"
   Type:
      Block

   Description:
      This block is used to request a topological analysis of the
      gradient field of the electron density, also known as the
      Bader's analysis. If this block is specified without any sub-
      key, only local properties are calculated.

   "AnalysisLevel"
      Type:
         Multiple Choice

      Default value:
         Normal

      Options:
         [Normal, Extended, Full]

      Description:
         Set the level of the QTAIM analysis:
         Normal - topology analysis and properties at the density
         critical points,                       Extended - same as
         Normal plus condensed atomic descriptors, plus basin
         information on TAPE10                       Full - same as
         Extended plus non-local descriptors.

   "Source"
      Type:
         Bool

      Default value:
         No

      GUI name:
         Source Function

      Description:
         Calculate the Source Function at BCPs and RCPs.

   "AtomsToDo"
      Type:
         Integer List

      GUI name:
         Include atoms

      Description:
         List of atoms for which condensed descriptors are to be
         calculated. By default all atoms are included.

   "Spacing"
      Type:
         Float

      Default value:
         0.5

      Unit:
         Bohr

      Description:
         Specifies spacing of the initial Cartesian grid when
         searching for critical points. It may be useful to specify a
         smaller value than the default if some critical points are
         missed. This will result in a more accurate but slower
         calculation.

Note that the accuracy of condensed atomic properties can be estimated
by integrating the Laplacian of the electron density over a given
basin, which, ideally, must vanish. The accuracy of the method can be
improved by using a larger integration grid (NumericalQuality).
Usually, the default grid is sufficient to achieve the accuracy of
10^-3 a.u. (differences of milliHartree in the energies).
Unfortunately, the convergence of the electron density Laplacian
integral with the grid size is not monotonous. Therefore this type of
Bader atomic property calculation should be considered for
applications where computational efficiency is critical and moderate
accuracy is sufficient.

Note:

  The *Energy* keyword is now obsolete (AMS2023). The (virial) atomic
  energies are now automatically calculated at the *Extended* or
  *Full* level except for relativistic calculations. However, the
  other QTAIM properties *are* correct for such calculations (they
  depend on the sole electron density; see reference [15]). Please
  note that the atomic energies are correct only at equilibrium
  geometries (the virial theorem is used).


ADF2AIM
-------

The ADF utility adf2aim (original name rdt21), developed by Xavi
Lopez, Engelber Sans and Carles Bo, converts an ADF "adf.rkf"
(formerly known as 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 delocalization of mobile electrons in two or three
dimensions is probably one of the key aspects that characterize
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
localization/delocalization indices (LI-DI), together with the Iring
(ring index) and MCI (multi center index) aromaticity indices derived
from it (see Ref. [16]), can be computed for rings specified by the
user. The rings are specified using the AROMATICITY input block, one
line per ring.

   Aromaticity
      RingAtoms integer_list
   End

"Aromaticity"
   Type:
      Block

   Description:
      Calculate aromaticity indicators, i.e. the matrix of
      localization/delocalization indices (LI-DI), Iring (ring index)
      and MCI (multi center index) aromaticity indices.

   "RingAtoms"
      Type:
         Integer List

      Recurring:
         True

      Description:
         List of the atom numbers that form a ring. For each ring a
         separate line.

For example, for a benzene molecule where atoms 1-6 are carbon atoms,
the input would look like this:

   AROMATICITY
     ringatoms 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 the 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 QTAIM 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.


Conceptual DFT
==============


Global, atomic, and non-local descriptors
-----------------------------------------

The *ConceptualDFT* block replaces and extends the set of features
behind the *Bader Reactivity* keyword found in the ADF versions before
ADF2019. The frontier molecular orbital (FMO) approximation is used
for all calculations. Orbital degeneracy is taken into account
following Martinez [17]. All these descriptors are described in Ref.
[18]. Which descriptors are calculated depends on the *AnalysisLevel*
key:

* *Normal:* global conceptual DFT descriptors, i.e. electronic
  chemical potential, electronegativity (its opposite), hardness and
  softness, hyperhardness [19], electrophilicity index [20],
  dissociation energies (nucleofuge and electrofuge, see Ref. [21]),
  electrodonating and electroaccepting powers [22] and the net
  electrophilicity [23]. Also, global descriptors called *global dual
  descriptors* \Delta f+ and \Delta f- are calculated [24], which
  correspond to the spatial integral of the positive (electrophilic)
  or negative (nucleophilic) regions of the original dual descriptor,
  respectively. These descriptors allow for comparing total reactivity
  of molecules, in contrast with Fukui functions or the original dual
  descriptor that integrate over the whole space to 1 or 0 for all
  molecules.

* *Extended:* both global and atomic properties, including condensed
  (QTAIM) descriptors in both canonical and grand canonical ensembles.
  More precisely, the following properties are computed: condensed
  Fukui functions [26], dual descriptor [25] [27] and some of their
  composite functions [50] [51] [52], as well as condensed local
  electrophilicity and nucleophilicity [28].

* *Full:* all properties are calculated, including non-local ones,
  such as the condensed linear response function [29] and the softness
  kernel [30].

Please note that atomic electronegativities [31] are calculated
separately with an optional *Electronegativity* keyword. In contrast
with other properties, it requires all-electron basis sets and
triggers the *TotalEnergy* keyword.

"ConceptualDFT"
   Type:
      Block

   Description:
      Conceptual DFT Properties

   "AnalysisLevel"
      Type:
         Multiple Choice

      Default value:
         Normal

      Options:
         [Normal, Extended, Full]

      Description:
         Set the level of the ConceptualDFT analysis:
         Normal - global descriptors only,
         Extended - both global and condensed (QTAIM) local
         descriptors,                      Full - all descriptors
         including non local ones.

   "Electronegativity"
      Type:
         Bool

      Default value:
         No

      GUI name:
         Atomic electronegativities

      Description:
         Calculate atomic electronegativities. Requires an all-
         electron calculation (no frozen core), triggers the
         TotalEnergy and increases the [AnalysisLevel] to at least
         Extended.

   "AtomsToDo"
      Type:
         Integer List

      GUI name:
         Include atoms

      Description:
         Define a subset of atoms for which properties are calculated.
         If the [Domains] block is present then this list specifies
         which atoms are used to define the domains bounding box.

For more information about *Conceptual DFT*, we suggest reading these
excellent reviews: Refs. [33] and [32].


Domains of the dual descriptor
------------------------------

In a paper [34], Tognetti et al. proposed a real-space partitioning
scheme of the dual descriptor (DD) into chemically meaningful
*reactivity domains*. Basically, a predominantly electrophilic
(nucleophilic) DD domain constitutes a region of space where this
function remains positive (negative). They can be visualized using the
GUI and quantitative information can be obtained using the additional
*Domains* key block.

Note:

  This analysis requires that the calculation is performed without
  symmetry (Symmetry NOSYM).

"ConceptualDFT"
   Type:
      Block

   Description:
      Conceptual DFT Properties

   "Domains"
      Type:
         Block

      Description:
         Calculate integrated properties for the domains (same sign)
         of the dual descriptor.

      "Border"
         Type:
            Float

         Default value:
            7.0

         Unit:
            Bohr

         Description:
            Set the extent of the Cartesian grid. Extent is the
            distance between a face of the grid's bounding box and the
            most outlying atom in the corresponding direction. If the
            [AtomsToDo] key is present, the bounding box is created
            around the specified atoms.

      "Display"
         Type:
            Float

         Default value:
            0.005

         Description:
            Domains for which the integrated DD value is smaller (in
            magnitude) than the specified value are omitted from the
            printed output.

      "Enabled"
         Type:
            Bool

         Default value:
            No

         GUI name:
            Properties of reactivity domains

         Description:
            Calculate properties of reactivity domains.

      "Ensemble"
         Type:
            Multiple Choice

         Default value:
            Canonical

         Options:
            [Canonical, GrandCanonical]

         Description:
            Statistical ensemble for DD domains. Canonical: DD values
            are calculated using the statistical canonical ensemble.
            GrandCanonical: DD values are calculated using the
            statistical grand canonical ensemble. The grand canonical
            DD corresponds to (S^2 f(2) - (gamma/eta^3) f^0), where
            f(2) is the canonical DD, gamma and eta - the hyper-
            hardness and hardness of the chemical system,
            respectively, and f^0 is the mean Fukui function. This
            statistical ensemble is a natural choice when comparing
            two chemical systems with a different number of electrons.

      "Radius"
         Type:
            Float

         Default value:
            0.0

         Description:
            This option adds a sphere around each nucleus, excluding
            all points inside it. This can help to separate domains
            around an atom or to exclude core electrons. Be careful
            when using this option. In particular, the radius of the
            sphere should exceed two or three times the [Spacing]
            value to be effective. By default, no spheres are added.

      "Spacing"
         Type:
            Float

         Default value:
            0.1

         Unit:
            Bohr

         Description:
            Specifies spacing (distance between neighboring points) of
            the rectangular Cartesian grid used when searching for DD
            domains. It may be useful to specify a smaller value (or
            increase the size of the grid, see [Border] key) if a
            substantial part of the electronic density is accounted
            for.

      "Threshold"
         Type:
            Float

         Default value:
            0.001

         Description:
            Arbitrary value of dual descriptor used to separate DD
            domains (values below this threshold are ignored).


adf2damqt: DAMQT interface
==========================

Interface (adf2damqt) to the third-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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