Results on Output
*****************

The (standard) output file contains information on the main
characteristics of the run, the SCF and geometry optimization results,
bonding energy and population analyses. Major parts of output can be
regulated with print switches, see section Controlling printed output.

By default the program produces quite a bit of output, for a large
part related to (Mulliken-type) population analyses of the molecule in
total, as well as of individual orbitals, both in terms of the
elementary basis functions and in terms of the SFOs, the symmetry-
adapted Fragment Orbitals.

The fragment-oriented approach of ADF is very suitable for a thorough
chemical analysis of molecular orbital properties and a conceptual
representation of results. New users are advised to spend time and get
familiar with the SFO-type analysis. It is a much more powerful tool
to understand the electronic structure of the molecule than the
classical atomic orbital populations.

The results for structure and reactivity, spectroscopic properties,
transport properties that are printed are meant to be self-
explanatory. See also the input options for each of these properties.


Electronic Configuration
========================

The direct results from the SCF are the orbital energies and
occupation numbers. This defines the electronic configuration: the
occupation numbers and HOMO and LUMO energies for instance show
whether or not the Aufbau principle is satisfied in the final
situation.

The energies of the Core Orbitals can be used to interpret for
instance XPS (X-ray Photoelectron Spectroscopy) data: from Koopman's
theorem these core orbital energies are an approximation to the core
ionization energies. This neglects the effect of relaxation upon the
ionization so that absolute energy values may not be very good;
relative values, however, should be fair and can therefore be used to
study (relative) chemical shifts.


Mulliken populations
====================

Mulliken populations are based on the elementary atomic basis
functions (bas). The individual BAS populations are printed together
with summaries of the populations in all basis functions with the same
angular moment quantum number on the same atom. A final summary is
obtained by adding all functions on each atom, yielding the atom-atom
populations. The atom-atom populations per l-value can be obtained if
the key EXTENDEDPOPAN is included. The atomic gross charges are
derived from the net and the overlap populations in the usual way. In
addition, a population analysis may be given of individual MOs (by
default this is suppressed). See the EPrint keys SCF (option mopop)
and orbpop.

Mulliken-type populations are computed and printed at various levels
of refinement (ranging from *per-basis function* to *per-fragment
type*, data for the whole molecule as well as for individual MOs), and
in two different representations, one based on the elementary basis
functions (bas), the other on SFOs (Symmetrized Fragment Orbitals).
This is potentially a very large amount of data. Precisely what is
printed by default, and how this can be modified so as to suppress
output or, alternatively, to get more information, is regulated by the
print keys (print, eprint).


Hirshfeld charges, Voronoi deformation density
==============================================

Mulliken populations can be summarized to yield atomic charges.
Alternative methods exist to deduce atom charges from the self-
consistent results of a molecular calculation. Several of those
alternatives are provided by ADF: Hirshfeld analysis, Voronoi
analysis, multipole derived charges, and charge model 5.

Of the methods applied in ADF to compute charges (Mulliken, Hirshfeld,
Voronoi) we recommend the Hirshfeld analysis [2],  [3] and the
analysis based on Voronoi *deformation* density (VDD) charges [1],
[4], see below. The fragments to which the Hirshfeld charges apply are
enumerated in the early geometry part of the output file, where for
each fragment the numbers of the atoms are given that belong to the
fragment. The sum of the Hirshfeld charges may not add up to the
analytical net total charge of the molecule. Any deviation from this
is caused by numerical integration precision (small effect) and the
neglect of long-distance terms that ADF uses to speed up the integral
evaluations. This approximation does not affect very much the energy
and molecular orbital properties, but it does show up in the sum-of-
charges somewhat more. It does not indicate an error (unless the
deviation is really large, say in the order of 1‰ of the total number
of electrons).

The Hirshfeld analysis produces a charge value per fragment, computed
as the integral of the SCF charge density over space, in each point
weighted by the relative fraction of the (initial) density of that
fragment in the total initial (sum-of-fragments) density:

   Q^\text{frag(i)} = \int \rho^\text{SCF} \rho^\text{initial frag(i)}
   / \sum_j \rho^\text{initial frag(j)}    \qquad (5.1.1)

The VDD method is based on the *deformation* density and a rigorous
partitioning of space into non-overlapping atomic areas, the so-called
Voronoi cells [1] [4] [5]. The Voronoi cell of an atom *A* is the
region in space closer to nucleus *A* than to any other nucleus (cf.
Wigner-Seitz cells in crystals). The VDD charge of an atom *A*
monitors the *flow* of charge into, or out of the atomic Voronoi cell
as a result of 'turning on' the chemical interactions between the
atoms. The VDD method summarizes the three-dimensional deformation
density on a per-atom basis. It is conceptually simple and affords a
transparent interpretation based on the plausible notion of charge
redistribution due to chemical bonding, i.e. the gain or loss of
charge in well-defined geometrical compartments of space. For the use
of VDD in analyses involving molecular fragments, see Ref. [6].

In the same fashion as for the Hirshfeld analysis, a summation over
all atoms is given which should yield zero (for a neutral molecule).
The deviation from zero is caused by numerical integration and by
neglect-of-long-distance-terms; the same remarks apply as for the
Hirshfeld analysis above.

The partitioning of space, using mid-way separation planes, is
inappropriate to produce useful absolute numbers when neighboring
atoms have very different sizes, for instance, Hydrogen and a heavy
metal. However, *changes* in the density analyzed in this way do give
a reasonable general insight in the effect of bonding on the location
of charge densities, in particular because the Voronoi data per atom
are split up in contributions within the atomic sphere and the rest of
its Voronoi cell.

For symmetric molecules one can do a (partial) symmetry partitioning
of the Voronoi deformation density if one includes

   PRINT VDDANALYSIS

This analysis is especially useful if all fragments are calculated in
the same symmetry as the full molecule, which is not always possible
or useful, for example, symmetry NOSYM does not provide any extra
information.

Hirshfeld and Voronoi charge analyses are printed at the end of the
SCF (of the last geometry, in case of an optimization).

The Hirshfeld analysis in ADF produces charges *per fragment*, so that
*atomic* charges are obtained only if single-atom fragments are used.
This limitation does not apply to Voronoi charges (data per atom).
Mulliken charges are given both per atom *and* per fragment.

In the printout of charges per fragment (as for the Hirshfeld
analysis), you have to be aware of the *ordering* of fragments. A
complete list of fragments is printed in the early GEOMETRY section of
standard output, where you also find which atom(s) correspond(s) to
which fragment. Note that even when you use single-atom fragments
only, the order of fragments is usually quite different from the order
of atoms in your input file. Typically (but not necessarily exactly in
each case), when you use single-atom fragments: consider the first
non-dummy atom in your ATOMS block. This defines the first atom
*type*. Then browse the ATOMS list until you find an atom of a
different type. This defines the second atom type, and so on. The
single-atom fragment list will often be such that you first get *all*
atoms of the first atom type, then all atoms of the second type, and
so on. Check the printed list-of-fragments always, to avoid mistakes
in assigning Hirshfeld charges to atoms (fragments).


Multipole derived charges
=========================

The multipole derived charges (MDC) analysis [7] uses the atomic
multipoles (obtained from the fitted density) up to some level X, and
reconstructs these multipoles exactly (up to level X) by distributing
charges over all atoms. This is achieved by using Lagrange multipliers
and a weight function to keep the multipoles local. Since the atomic
multipoles are reconstructed up to level X, the molecular multipoles
are represented also up to level X. The recommended level is to
reconstruct up to quadrupole: MDC-q charges. The SCF should have
converged for a meaningful MDC analysis.


Charges Derived from the Electrostatic Potential
================================================

Charges derived from the electrostatic potential (ESP) are available
for partial charge analysis. Two different charges are available in
ADF: ChElPG and RESP.

ChElPG is the acronym for 'Charges derived from the Electrostatic
Potential Grid'. They are defined by fitting point charges to the
electron density, on a a real-space atom-centric Lebedeev grid outside
the van der Waals region subject to the constraint that the ChElPG
charges must equal the total charge of the molecule. For input use the
keyword ESP-CHELPG.

RESP is the acronym for 'Restrained Electrostatic Potential Charges'.
RESP charges are the same as ChElPG charges but with a harmonic
restraint (0.05) to keep the charges and dipole moments as small as
possible. RESP charges are typically more physically reasonable and
robust than CHELPG charges. For input use the keyword ESP-RESP.

NUMERICALQUALITY good is recommended. These charges can be used in the
3D-RISM and SM12 solvation models.


Charge model 5
==============

The charge model 5 (CM5) [8] [9] uses the Hirshfeld analysis in
combination with a parametrization to yield atomic charges that can
accurately reproduce dipole moments obtained from experimental
results. CM5 charges can be used in the SM12 solvation model. For
input use the keyword CM5.


Bond order analysis
===================

The bond order analysis (see Bond orders). produces the output in
which the bond order values are printed for each pair of atoms for
which the Nalewajski-Mrozek bond order value is larger than the
threshold that can be specified with the keyword BONDORDER. For
convenience the printed bond orders are accompanied by the
corresponding inter-atomic distance.  In the Nalewajski-Mrozek
approach [11] [12] [16] [17] [18] [19] the bond order indices b_AB
are calculated based on  the one- and two-center valence indices

   b_{AB} = V_{AB} + W^{AB}_A V_A + W^{AB}_B V_B

with the weighting factors for one-center indices given by

   W^{XY}_X = \frac{V^\text{COV}_{XY}}{\sum_Z V^\text{COV}_{XZ}}

Unlike other definitions of covalent bond orders, the Nalewajski-
Mrozek valence indices comprise both, covalent and ionic
contributions. There exist three alternative sets of the Nalewajski-
Mrozek valence indices [11] [12] [16] [17] [18] [19] [10]. The bond
order indices calculated from each set of the valence indices differ
slightly due to arbitrariness in the way of splitting the one-center
terms between bonds. More detailed description of alternative valence
indices and their physical meaning is summarized in [11]; see also
original papers [12] [16] [17] [18] [19].

By default the bond order indices based on the valence indices
obtained from partitioning of  Tr(P\Delta P) are printed in the ADF
output. Note that in this version the covalent two-center part (also
printed in the output) is equal to the Gopinathan-Jug [12] bond order.
The default values are:

   & V_A = V^\text{ion}_A + V^\text{COV}_A \\ & V^\text{ion}_A =
   \sum_{a \in A} \left(  P^\alpha_{aa} \Delta P^\alpha_{aa}  +
   P^\beta_{aa} \Delta P^\beta_{aa}  \right) \\ & V^\text{COV}_A = 2
   \sum_{a \in A} \sum_{a' \in A, a<a`} \left(  P^\alpha_{aa'} \Delta
   P^\alpha_{a'a}  +  P^\beta_{aa'} \Delta P^\beta_{a'a}  \right) \\ &
   V^\text{COV}_{AB} = 2 \sum_{a \in A} \sum_{b \in B} \left(
   P^\alpha_{ab} \Delta P^\alpha_{ba}  +  P^\beta_{ab} \Delta
   P^\beta_{ba}  \right)

To produce the values from all alternative versions of Nalewajski-
Mrozek valence indices, accompanied by the Gopinathan-Jug [12] and
Mayer [10] bond orders, see the keyword BONDORDER.

The Mayer [10] bond orders can also be calculated (see
BondOrders%PrintAll). The two implementations of calculating the Mayer
bond orders differ slightly if one uses frozen cores. They should
agree exactly in all electron calculations.


Dipole moment, Quadrupole moment, Electrostatic potential
=========================================================

Dipole moment. Note that in an ion the value of the dipole moment
depends on the choice of the origin, as follows from elementary
electrostatic theory.

Quadrupole moment. Note that the value of the quadrupole moment often
depends on the choice of the origin, as follows from elementary
electrostatic theory.

Electrostatic potential at the nuclei: the Coulomb potential of the
molecule at the nuclear positions, where the contribution from the
nucleus itself is omitted.


Fragments and Basis Functions
=============================

"SFOs"
   SFOs: the Symmetry combinations of Fragment Orbitals. The SFOs are
   the basic conceptual entities for the analysis of MOs and other
   results. Note: The FO *coefficients* that expand the SFOs are
   normalized in the sense that they add up (squared) to unity. The
   resulting SFO *function* is not necessarily a normalized function.
   The FOs are normalized, so it depends on the *overlap* between the
   FOs what the self-overlap and hence the norm of the SFO is. Also
   printed are, for each subspecies in each irrep separately, the
   indices of the elementary basis functions from which the FOs and
   hence the SFOs are built up. (The overlap matrix of SFOs is printed
   much later, in the SFO Populations section after everything (SCF,
   Geometry) has cycled to convergence).

"Elementary basis functions (BAS)"
   First the lists of function *sets*, defined by radial behavior and
   the angular quantum number, are printed for all atom types on which
   the functions are centered. Thereafter follows the complete BAS
   list where the function sets have been expanded over all atoms (the
   *sets* are printed only for the atom *types*) and also over all
   Cartesian harmonics (6, not 5 *d*-functions, et cetera). In this
   printout the numbering can be found to which the SFO survey above
   refers.


MO analysis
===========

"MOs expanded in SFOs"
   This gives a useful characterization of the character of the self-
   consistent molecular orbitals. Additional information is supplied
   by the SFO population analysis, see below. The definition of the
   SFOs in terms of the Fragment MOs has been given in an earlier part
   of output (section build). The SFO occupation numbers that applied
   in the fragments are printed. This allows a determination of the
   orbital interactions represented in a MO. Be aware that the
   bonding/antibonding nature of a SFO combination in a MO is
   determined by the relative signs of the coefficients *and* by the
   overlap of the SFOs. This overlap *may be negative!* Note also that
   SFOs are generally *not* normalized functions. The SFO overlap
   matrix is printed later, in the SFO-populations part below.

"SFO population analysis"
   For each irrep:

   * Overlap matrix of the SFOs. Diagonal elements are not equal to
     1.0 if the SFO is a linear combination of two or more Fragment
     Orbitals. The Fragment Orbitals themselves are normalized so the
     diagonal elements of the SFO overlap matrix give information
     about the overlap of the Fragment Orbitals that were combined to
     build the SFO.

   * Populations on a per-fragment basis for a selected set of MOs
     (see EPrint, subkey *OrbPop*). This part is by default *not*
     printed, see EPRINT subkey *SFO*.

   * SFO contributions per MO: populations for each of the selected
     MOs. In these data the MO occupation numbers are not included, so
     that also useful information about the virtual MOs is obtained.
     The printout is in matrix form, with the MOs as columns. In each
     printed matrix a row (corresponding to a particular SFO) is
     omitted if all populations of that SFO are very small in all of
     the MOs that are represented in that matrix. See eprint, subkey
     *orbpop*. Note that this method to define SFO populations (for
     orbitals) is very similar to the classical Mulliken type
     analysis, in particular regarding the aspect that *gross*
     populations are obtained as the diagonal (*net*) populations plus
     half of the related off-diagonal (overlap) populations.
     Occasionally this may result in negative (!) values for the
     population of certain SFOs, or in percentages higher than 100%.
     If you have such results and wonder if they can be right, work
     out one of the offending cases by hand, using the printed SFO
     overlap matrix and the printed expansion of the MOs in SFOs to
     compute 'by hand' the population matrix of the pertaining MO. To
     avoid doing large calculations it is usually sufficient to take
     only the few largest MO expansion coefficients; this should at
     least qualitatively give the correct outcomes.

   * Total SFO gross populations in a symmetry representation: from a
     summation over all MOs (not only those analyzed in the previous
     section of output) in the symmetry representation under
     consideration. In the gross populations the MO occupation numbers
     have been included.

   * (Per spin): A full list of all MOs (combining all symmetry
     representations), ordered by energy, with their most significant
     SFO populations. Since there might be several significant SFO
     populations for a particular MO, and an SFO may actually be a
     linear combination of several (symmetry-related) Fragment
     Orbitals, this table could get quite extensive. In order to
     confine each SFO population specification to one line of output,
     the SFOs are indicated by the characteristics of the first term
     (Fragment Orbital) of its expansion in Fragment Orbitals. So, if
     you see the SFO given as the '2 P:x on the first Carbon
     fragment', it may actually refer to the symmetry combination of,
     for instance, 2P:x and 2P:y orbitals on the first, second and
     third Carbon fragments. A full definition of all SFOs in terms of
     the constituting Fragment Orbitals is given in an early part of
     the output.


Bond energy analysis
====================

The bond energy and its decomposition in conceptually useful terms:
Pauli (exchange) repulsion, total steric repulsion, orbital
interactions (partitioned into the contributions from the distinct
irreducible representations), and corrections for some approximations
(fitting and Transition State analysis procedure). For a discussion of
bonding energy decompositions and applications see e.g. refs. [13]
[14] [15] [20] [21] [22] [23] [24] [25] [26].

The program prints the bonding energy (not in a Create or Frequencies
run) and its decomposition in terms that are useful for chemical
interpretation. The *total* energy is not computed. The bonding energy
is defined relative to the fragments. When *basic atoms* are employed
as fragments one should realize that these do not represent the atomic
ground state since they are computed as spin-restricted and
spherically symmetric objects, with possibly fractional occupation
numbers. The correct multiplet state is not computed. To obtain the
bonding energy with respect to isolated atoms you should therefore add
atomic correction terms to account for spin polarization and the
multiplet state. See also the SLATERDETERMINANTS key and the
discussion on multiplet states.

The spin polarization energy can be computed by running the single
atom unrestricted, using as fragment the corresponding (restricted)
basic atom. The true multiplet state is not necessarily obtained in
this way.

For the comparison of computed bonding energies with experimental data
one should furthermore be aware of any aspects that are not
represented in the computational formalism, such as zero-point motions
and environment (solvent) effects.

In a Geometry Optimization or Transition State search, the program may
print a bonding energy evaluation at each geometry (depending on print
switches). A  test-energy value is written in the log file. This is
*not* the bonding energy, although the difference is usually small.
The test-energy printed in the log file is the energy expression from
which the energy gradients are computed. The true bonding energy
contains in addition a few (small) correction terms that are mostly
related to the fit incompleteness. These correction terms are usually
very small.

If Electric Fields are used in the computation (homogeneous and/or
point charges), the printed Bonding Energy is the energy of the
molecule in the field minus the energy of the fragments in the same
field. The energy terms due to the field are also printed separately
so that one can subtract them from the total bonding energy to obtain
the energy-change without field-terms.

[1] G. te Velde, *Numerical integration and other methodological
    aspects of bandstructure calculations*, in *Chemistry*. 1990,
    Vrije Universiteit: Amsterdam.

[2] F.L. Hirshfeld, *Bonded-atom fragments for describing molecular
    charge densities*, Theoretica Chimica Acta 44, 129 (1977)

[3] K.B. Wiberg and P.R. Rablen, *Comparison of atomic charges derived
    via different procedures*, Journal of Computational Chemistry 14,
    1504 (1993)

[4] F.M. Bickelhaupt, N.J.R. van Eikema Hommes, C. Fonseca Guerra and
    E.J. Baerends, *The Carbon-Lithium Electron Pair Bond in*
    (CH3Li)_n (n = 1, 2, 4), Organometallics 15, 2923 (1996)

[5] C. Fonseca Guerra, J.-W. Handgraaf, E. J. Baerends and F. M.
    Bickelhaupt, *Voronoi Deformation Density (VDD) charges.
    Assessment of the Mulliken, Bader, Hirshfeld, Weinhold and VDD
    methods for Charge Analysis*, Journal of Computational Chemistry
    25, 189 (2004)

[6] C. Fonseca Guerra, F.M. Bickelhaupt, J.G. Snijders and E.J.
    Baerends, *The Nature of the Hydrogen Bond in DNA Base Pairs: The
    Role of Charge Transfer and Resonance Assistance*, Chemistry - A
    European Journal 5, 3581 (1999)

[7] M. Swart, P.Th. van Duijnen and J.G. Snijders, *A charge analysis
    derived from an atomic multipole expansion*, Journal of
    Computational Chemistry 22, 79 (2001)

[8] A.V. Marenich, S.V. Jerome, C.J. Cramer, D.G. Truhlar, *Charge
    Model 5: An Extension of Hirshfeld Population Analysis for the
    Accurate Description of Molecular Interactions in Gaseous and
    Condensed Phases*, Journal of Chemical Theory and Computation 8,
    527 (2012)

[9] C.A. Peeples and G. Schreckenbach, *Implementation of the SM12
    Solvation Model into ADF and Comparison with COSMO*, Journal of
    Chemical Theory and Computation 12, 4033 (2016)

[10] I. Mayer, *Charge, bond order and valence in the ab initio SCF
     theory*, Chemical Physics Letters 97, 270 (1983)

[11] A. Michalak, R.L. De Kock and T. Ziegler, *Bond Multiplicity in
     Transition-Metal Complexes: Applications of Two-Electron Valence
     Indices*, Journal of Physical Chemistry A 112, 7256 (2008)

[12] M.S. Gopinathan and K. Jug, *Valency. I. A quantum chemical
     definition and properties*, Theoretica Chimica Acta 1983 63, 497
     (1983)

[13] T. Ziegler and A. Rauk, *On the calculation of Bonding Energies
     by the Hartree Fock Slater method. I. The Transition State
     Method*, Theoretica Chimica Acta 46, 1 (1977)

[14] T. Ziegler and A. Rauk, *A theoretical study of the ethylene-
     metal bond in complexes between copper(1+), silver(1+), gold(1+),
     platinum(0) or platinum(2+) and ethylene, based on the Hartree-
     Fock-Slater transition-state method*, Inorganic Chemistry 18,
     1558 (1979)

[15] F.M. Bickelhaupt, N.M. Nibbering, E.M. van Wezenbeek and E.J.
     Baerends, *The Central Bond in the Three CN* Dimers NC_CN, CN-CN,
     and CN-NC: Electron Pair Bonding and Pauli Repulsion Effects*,
     Journal of Physical Chemistry 96, 4864 (1992)

[16] R.F. Nalewajski and J. Mrozek, *Modified valence indices from the
     two-particle density matrix*, International Journal of Quantum
     Chemistry 51, 187 (1994)

[17] R.F. Nalewajski, J. Mrozek and A. Michalak, *Two-electron valence
     indices from the Kohn-Sham orbitals*, International Journal of
     Quantum Chemistry 61, 589 (1997)

[18] R.F. Nalewajski, J. Mrozek and A. Michalak, *Exploring Bonding
     Patterns of Molecular Systems Using Quantum Mechanical Bond
     Multiplicities*, Polish Journal of Chemistry 72, 1779 (1998)

[19] R.F. Nalewajski, J. Mrozek and G. Mazur, *Quantum chemical
     valence indices from the one-determinantal difference approach*,
     Canadian Journal of Chemistry 74, 1121 (1996)

[20] K. Kitaura and K. Morokuma, *A new energy decomposition scheme
     for molecular interactions within the Hartree-Fock
     approximation*, International Journal of Quantum Chemistry 10,
     325 (1976)

[21] T. Ziegler and A. Rauk, *Carbon monoxide, carbon monosulfide,
     molecular nitrogen, phosphorus trifluoride, and methyl isocyanide
     as sigma donors and pi acceptors. A theoretical study by the
     Hartree-Fock-Slater transition-state method*, Inorganic Chemistry
     18, 1755 (1979)

[22] H. Fujimoto, J. Osamura and T. Minato, *Orbital interaction and
     chemical bonds. Exchange repulsion and rehybridization in
     chemical reactions*, Journal of the American Chemical Society
     100, 2954 (1978)

[23] S. Wolfe, D.J. Mitchell and M.-H. Whangbo, *On the role of steric
     effects in the perturbational molecular orbital method of
     conformational analysis*, Journal of the American Chemical
     Society 100, 1936 (1978)

[24] A.J. Stone and R.W. Erskine, *Intermolecular self-consistent-
     field perturbation theory for organic reactions. I. Theory and
     implementation; nucleophilic attack on carbonyl compounds*,
     Journal of the American Chemical Society 102, 7185 (1980)

[25] F. Bernardi, A. Bottoni, A. Mangini and G. Tonachini,
     *Quantitative orbital analysis of ab initio SCF=MO computations :
     Part II. Conformational preferences in H2N---OH and H2N---SH*,
     Journal of Molecular Structure: THEOCHEM 86, 163 (1981)

[26] P.J. van den Hoek and E.J. Baerends, *Chemical bonding at metal-
     semiconductor interfaces*, Applied Surface Science 41/42, 236
     (1989)
