Technical remarks and Terminology
*********************************

A few words about ADF as regards its technical setup and the names and
abbreviations used in this manual. References to these will be made in
the discussion of output and print switches.


Density functional theory
=========================

The underlying theory of the ADF engine is based on the Kohn-Sham
approach to Density-Functional Theory (DFT). Kohn-Sham DFT is an
important first-principles computational method to predict chemical
properties accurately and to analyze and interpret these in convenient
and simple chemical terms.

The reasons for its popularity and success are easy to understand. In
the first place, the DFT approach is in principle exact. In
particular, the Kohn-Sham method implies a one-electron picture of the
many-electron systems but yields in principle the exact electron
density (and related properties) and the total energy. The exact
exchange-correlation (XC) functional is unknown, but the currently
available XC functionals provide in most cases already a 'chemical'
accuracy of a few kcal/mol for binding energies. Moreover, the quest
for more accurate ones based on a more detailed understanding of their
essential properties is continuing.

In the past decades, computational chemistry has evolved from a
curiosity of theoreticians into a mainstream tool used by all types of
chemists, physicists, and engineers who have an interest in research
and development. In that time Density Functional Theory has become the
dominant method for modeling chemistry at the molecular level.

In the second place, it preserves at all levels of approximation the
appealing one-electron molecular orbital (MO) view on chemical
reactions and properties. The computed orbitals are suitable for the
typical MO-theoretical analyses and interpretations. The KS method
effectively incorporates all correlation effects.

In the third place, it is a relatively efficient computational method,
and its fundamental scaling properties do not deteriorate when
methodological precision is increased, in particular, when a more
accurate XC functional is applied. Recent research paves the way to
implementations that scale only linearly with the system size. This
brings within reach the treatment by fundamental quantum chemical
methods of systems with hundreds, maybe even thousands of atoms.

DFT gives superior accuracy to Hartree-Fock theory and semi-empirical
approaches, and it is well suited for molecules containing metal
atoms. In contrast to conventional ab initio methods (MP2, CI, CC), it
enables accurate treatment of systems with several hundreds of atoms
(or several thousands with QM/MM).

Text is mostly taken from: *Chemistry with ADF*, G. te Velde, F.M.
Bickelhaupt, E.J. Baerends, C. Fonseca Guerra, S.J.A. van Gisbergen,
J.G. Snijders, T. Ziegler, Journal of Computational Chemistry 22, 931
(2001)


The Kohn-Sham MO model
======================

The basic postulate in Kohn-Sham DFT is that we can apply a one-
electron formulation to the system of N interacting electrons by
introducing a suitable local potential V_\text{XC} (r), in addition to
any external potentials V_\text{ext} (r) and the Coulomb potential of
the electron cloud V_\text{C} (r), and solving:

   [T +  V_\text{ext} (r) + V_\text{C} (r) + V_\text{XC} (r)]  \phi_i
   (r) = \epsilon_i \phi_i (r)

Here T is the kinetic energy operator. The potential V_\text{XC} (r)
is the functional derivative with respect to the density \rho of
E_\text{XC} [\rho], the exchange-correlation energy functional. The
one-electron molecular orbitals (MOs) \phi_i  with corresponding
orbital energies \epsilon_i  define the exact electronic charge
density and give, in principle, access to all properties because these
are expressible as a functional of the density, in particular the
energy. Moreover, they provide an intuitively appealing view of the
system as being built from independent-electron orbitals with all
ensuing interpretations. The exact form of the exact energy density
E_\text{XC} (r), representing and incorporating all exchange and
correlation (XC) effects is unknown. From general principles one can
formulate conditions on what E_\text{XC} (r) should look like, and
several, more and more advanced expressions have been advocated for it
in the literature. Their application to real systems has been
impressively successful, and it seems likely that a further increase
of accuracy is a matter of time.


Basis functions and orbitals
============================

Let us make a clear distinction between (basis) *functions* and
*orbitals*, even where these phrases are sometimes mixed up in the
traditional terminology. Orbitals are always specific combinations of
the basis functions. Orbitals are related to the computed
eigenfunctions of some Fock operator or Hamiltonian occurring in the
run or in a related preceding calculation. Functions are merely the
elementary mathematical entities in which the orbitals are expressed.
A Slater-Type Orbital (STO), for instance, is a function, not an
orbital.

The physical meaning of one-electron orbitals in DFT has often been
questioned. We believe that they are useful quantities for
interpretation, just like the HF orbitals. For a discussion see ref.
[1].

**See also**

* ADF-GUI tutorial: basis set effects


Cartesian function sets, spurious components
--------------------------------------------

ADF employs Slater-type exponential basis functions centered on the
atoms. Such a function consists of an exponential part exp(-ar) and a
polynomial prefactor r^kr x^kx y^ky z^kz . A function *set* is
characterized by its radial behavior (the exponential part and the
power of r, kr) and by its angular momentum quantum number *l*. The
functions in such a set consist of all possible combinations x^kx y^ky
z^kz , such that kx+ky+kz=l. These are denoted the *Cartesian*
spherical harmonics.

The Cartesian function sets are very suitable for computational
manipulations, but they have a drawback. By inspection it is easily
verified that a *d*-set consists of 6 Cartesian functions, while there
can of course be only 5 true *d*-type functions among them: one
(linear combination) of them is in fact an *s*-type function (x^2 +y^2
+z^2 ). Similarly, there are 10 *f*-type Cartesian functions, 3 of
which are in fact *p*-functions. And so on. In ADF all such lower-*l*
(combinations of) functions are projected out of the basis and not
employed. As a consequence the basis set *size* in the sense of the
number of degrees of freedom and hence the number of possible
eigenfunctions of the Fock operator is smaller than the number of
expansion coefficients that refer to the primitive (Cartesian) basis
functions.

The abbreviation BAS is used for references to the elementary
Cartesian basis functions.


Frozen core: Core Orbitals and Core Functions
---------------------------------------------

To speed up calculations the innermost atomic shells are kept frozen.
The frozen Core Orbitals (CO), which are solutions of a large-basis
all-electron calculation on the isolated atom, are expressed in an
auxiliary set of (Slater-type) basis functions corbas, distinct from
the valence set. The core basis set and the COs expansion coefficients
are stored in the basis set files.

Orthogonality of the valence Molecular Orbitals (MO) to the COs is
achieved with the help of so-called Core Functions (CF). These
functions are included in the valence set but they are not additional
degrees of freedom. Each of the normal valence functions is combined
with a linear combination of all CFs in the molecule in such a way
that the transformed function (cbas) is orthogonal to all frozen COs
in the molecule. There are exactly as many CFs as COs, so the
orthogonality condition for all valence basis functions amounts to the
solution of a linear system where the number of conditions equals the
number of parameters.

This aspect once more increases the discrepancy between the number of
expansion coefficients of an MO and the number of MOs: the expansion
coefficients in the most elementary bas representation run over all
bas functions, including the CFs among them. In some places,
expansions may be alternatively given in the core-orthogonalized BAS
functions (CBAS), in which case the CFs do not count anymore: they are
included implicitly in the CBAS functions.


Symmetry
--------

The Overlap and Fock matrices become block-diagonal by using symmetry-
adapted combinations of the (C)BAS functions, such that each such
function transforms under the symmetry operators as one of the
subspecies of the irreducible representations (irrep) of the symmetry
group. Symmetry-adapted functions are denoted (C)SBAS.

For a given irrep and subspecies not all elementary basis functions
can participate in the symmetry-adapted combinations. For instance,
for an atom in a reflection plane a basis function that is anti-
symmetric with respect to the reflection cannot be part of any
symmetric combination of functions. In particular, for higher
symmetries the number of BAS functions that are relevant for a
subspecies may be considerably smaller than the total number of BAS
functions. This is used to cut down expansion lengths, both as used
internally in the computation and construction of the Fock matrix, and
in printed output. The printed expansion coefficients (in the bas
representation) refer only to the participating BAS functions. A
defining list of them is printed at an early stage of the run for each
of the subspecies.


Orthonormal basis
-----------------

It is often computationally convenient to use an orthonormal basis.
This is constructed from the CSBAS basis by a Löwdin orthogonalization
procedure. The resulting symmetry-adapted orthonormal basis is denoted
low.

The MOs are computed by diagonalization of the Fock matrix in the LOW
representation. The resulting eigenvectors are easily transformed back
to any other representation whenever suitable, for instance, to the
primitive Cartesian bas representation (including the CFs).


Fragments
---------

Except in Create mode, where a *basic atom* is constructed, the system
is built up from fragments and the corresponding fragment files are
attached to the run. The ADF engine reads from the files the fragment
MOs and these are used as (compound) *basis functions* for the
molecular calculation. The fragment MOs are called Fragment Orbitals
(FO).

FOs belong, of course, to one of the symmetry representations of the
fragment, but not necessarily to a symmetry representation of the new
molecule. The FOs are therefore combined into symmetry-adapted
combinations, SFOs, to serve as a symmetry-adapted basis in the
molecule. These combinations may involve one or more FOs from the same
fragment and/or from different fragments. In the latter case the
fragments must be symmetry related by one of the operators of the
molecule. Symmetry related fragments must of course be identical,
apart from their spatial location: they must be of the same fragment
*type*.

FOs are naturally orthogonal to the Core Orbitals of their own
fragment, but not necessarily to COs of other fragments. By a suitable
combination of the SFOs with all CFs in the molecule we obtain the
core-orthogonalized symmetry-adapted CSFOs.

The CSFOs can be transformed to an orthonormal basis by a Löwdin
transformation. The resulting basis is called low, as above.


Summary of functions and orbitals
---------------------------------

In Create mode the (conceptual) approach is:

BAS → (core-orthogonalization) → CBAS → (symmetry) → CSBAS →
(orthonormality) → LOW → (Fock diagonalization) → MO

In Fragment mode:

FO (=MO from fragment file) → (symmetry) → SFO → (core-orth.) → CSFO →
(orthonormality) → LOW → (Fock diagonalization) → MO


Acronyms
--------

"BAS"
   Elementary Cartesian basis functions, consisting of a radial part
   (exponential factor and power of r) and an angular part (Cartesian
   spherical harmonic). The complete BAS set contains spurious
   lower-*l* combinations; these combinations are projected out and
   not used in the calculation. The BAS set contains also Core
   Functions.

"SBAS"
   Symmetry-adapted combination of BAS functions.

"CF"
   Core Function, part of the bas set. The CFs do not represent
   degrees of freedom in the basis set but serve only to ensure
   orthogonalization of the valence space to all frozen Core Orbitals.

"CBAS"
   Core-orthogonalized elementary basis functions: the true valence
   (not-CF) BAS functions transformed by adding a suitable combination
   of the CFs. The total number of CBAS + the total number of CFs
   equals the total number of BAS.

"CSBAS"
   Symmetry-adapted combination of cbas functions.

"CO"
   Frozen Core Orbitals, expressed as linear combinations of an
   auxiliary corbas basis set. The corbas set plays no role in the
   further discussion. The corbas functions are *not* the CFs.

   The number of COs equals the number of CFs.

"LOW"
   Löwdin orthonormalized symmetry-adapted core-orthogonalized basis.
   In Create mode they are derived directly from the BAS functions, in
   Fragment mode from the Fragment Orbitals, which are themselves of
   course expressible in the BAS set.

"FO"
   Fragment Orbital: the MO of a fragment calculation, now used as a
   *basis function* in the molecule of which the fragment is part.

"SFO"
   Symmetry-adapted combination of FOs.

"CSFO"
   Core-orthogonalized SFO.


Coulomb potential evaluation, density fitting
=============================================

Using Slater-type basis functions yields awkward multi-center
integrals in the evaluation of the Coulomb potential. We therefore
first need to find an approximate density-representation for which the
Coulomb integral can be evaluated efficiently. This procedure is
commonly referred to as density fitting. The default density fitting
procedure in ADF is described in Ref. [2].

One of the most important properties of a molecule is its energy, or
its bonding energy with respect to the constituent fragments. The
approximate nature of the fitting procedure introduces two types of
errors. The first is that, since the Coulomb potential is only
approximated, the SCF solution itself, i.e. the set of self-consistent
Molecular Orbitals and their energy eigenvalues may be slightly wrong,
yielding an error in the charge density and hence in the energy. Since
the energy is to first order stable with respect to changes in the MO
coefficients, this error in the energy can be assumed very small. The
second type of error derives from the computation of the energy from
the (self-consistent) charge density, via the Coulomb potential. Let

   \rho \equiv \rho_{exact} (r) = \rho_{fit} (r) + \delta (r)

and

   V_{fit} (r) = \int \rho_{fit} (r')/|r-r'| dr

For the Coulomb energy of the charge density we have

   2E_{Coul} & = \iint \rho(r) \rho(r')/|r-r'| dr dr' = \int \rho (r)
   V_{fit} (r) dr + \iint \rho(r) \delta(r)/|r-r'| dr dr' \\
   & = \int V_{fit} (r) (\rho(r)+\delta(r)) dr + \iint \delta(r)
   \delta(r')/|r-r'| dr dr'

from which we see that the fit error is corrected to first order (by
adding the fit deficiency \delta(r) to the exact charge density when
integrating against the fit potential) and that only a second order
term remains that cannot be evaluated, the last term in the right-
hand-side of the last equation.

A fair impression of the fit quality and the importance of the second
order error term is obtained by checking

* the size of the first order correction term \int V_{fit} (r)
  \delta(r) dr and

* the norm of the deficiency function, \int \delta^2(r) dr.

Both are printed in standard output, at the end of the SCF procedure's
computational report. They are usually very small, which gives some
confidence that the second order fit error can be ignored.


Three-step build-up of the bonding
==================================

The approach of ADF is based on fragments. This applies not only in
the analysis at the end of the computation but also in the set-up of
the ADF engine. The computation of the molecule from its constituent
fragments takes place in three steps, and these are reflected in the
analysis of bond energy components.

First, the (free, unrelaxed) fragments are placed at their positions
in the molecule. This implies an *electrostatic* interaction: for each
fragment the Coulomb interaction of its undisturbed charge density
with the fields of the other fragments.

Next, the Pauli exclusion principle is applied. Even without
considering self-consistency the one-electron orbitals of the combined
fragments cannot represent a correct one-determinant wave function
because the orbitals of different fragments are not orthogonal to one
another. The ADF engine performs an orthonormalization of the occupied
Fragment Orbitals to obtain an antisymmetrized product. This implies a
change in the total molecular charge density from the sum-of-fragments
to what is called the sum-of-*orthogonalized*-fragments. The
corresponding (repulsive) energy term is evaluated separately and is
called *Exchange* repulsion, or alternatively *Pauli* repulsion. The
phrase *orthogonal(ized) fragments*, if you find it elsewhere in this
manual or in the source code of ADF, refers to this aspect. The sum of
Pauli repulsion and electrostatic interaction is called the *steric
interaction*.

The third phase is the relaxation to self-consistency, with of course
the ensuing contributions to the bond energy.


Transition State procedure
==========================

This phrase stands for an analysis method described in ref. [4] and
has no relation to transition states in chemical reactions. An
extensive discussion of bond energy analysis by ADF is given in ref.
[5] [6].

The energy associated with a change in charge density, say the
relaxation to self-consistency from the sum-of-orthogonal-fragments,
can be computed by subtracting final and initial energies, each
obtained from the corresponding charge density. For purposes of
analysis the change in energy dE can be reformulated as

   dE = \int dr \left( \left(\rho_\text{final}(r) -
   \rho_\text{initial}(r) \right)
   \int_{\rho_\text{initial}}^{\rho_\text{final}} d\rho F[\rho (r)]
   \right) \quad (1.2.8)

F(\rho) is the Fock operator belonging to the charge density \rho

By expressing the density difference \rho_final  - \rho_initial  as a
sum over contributions from the different irreducible representations
\Gamma of the molecular symmetry group, one obtains an expression that
allows the bond energy to be decomposed into components corresponding
to these symmetry representations:

   dE = \sum_\Gamma \int dr \left( \Delta \rho^\Gamma (r)
   \int_{\rho_\text{initial}}^{\rho_\text{final}} d\rho F[\rho (r)]
   \right) \quad (1.2.9)

The integral of the Fock operator over the charge density is now
approximated by a weighted summation (in fact, a Simpson integration):

   \int_{\rho_\text{initial}}^{\rho_\text{final}} d\rho F[\rho (r)]
   \approx  \frac{1}{6} F[\rho_\text{initial}] + \frac{2}{3}
   F[\rho_\text{average}] + \frac{1}{6} F[\rho_\text{final}] \quad
   (1.2.10)

where

   \rho_\text{average} = \frac{1}{2} (\rho_\text{initial} +
   \rho_\text{final}) \quad (1.2.11)

The term with the Fock operator due to the average charge density has
given rise to the phrase *transition state*. To avoid confusion we
will often refer to it as the transition *field*.

The approximate integral (1.2.10) involves two errors. The first,
rather obvious, is the approximation of the exact integral in (1.2.9)
by the weighted sum in (1.2.10). Except in pathological cases this
approximation is highly accurate.

The second error comes from the fact that the Coulomb and XC
potentials in the Fock operator are computed from the *fit* density.
This is only an approximation to the true density, while in the
original bond-energy expression (energy due to the final density minus
energy due to the initial density) no potentials occur and the *exact*
charge density can be used. As mentioned before, these fit-related
errors are usually small. For the XC potential the true density can be
used if one includes the keyword EXACTDENSITY.

All such errors in the total bonding energy are easily corrected by
comparing the summation over the \Gammas with the correct value for
the total bonding interaction term. The difference is simply added to
the total bond energy, so no true error remains. We only have a
(correction) term that cannot be split in contributions from the
distinct symmetry representations. In the printed bond energy analysis
such small corrections are 'distributed' over the other terms by
scaling the other terms such that their sum is the correct total
value.

[1] F.M. Bickelhaupt and E.J. Baerends, *Kohn-Sham DFT: Predicting and
    Understanding Chemistry*, in *Reviews in Computational Chemistry*,
    D.B. Boyd and K.B. Lipkowitz, Editors. 2000, Wiley-VCH: New York.
    p. 1-86.

[2] M. Franchini, P.H.T. Philipsen, E. van Lenthe, L. Visscher,
    *Accurate Coulomb Potentials for Periodic and Molecular Systems
    through Density Fitting*, Journal of Chemical Theory and
    Computation 10, 1994 (2014)

[3] E.J. Baerends, D.E. Ellis and P. Ros, *Self-consistent molecular
    Hartree-Fock-Slater calculations I. The computational procedure*,
    Chemical Physics 2, 41 (1973)

[4] 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)

[5] P.J. van den Hoek, A.W. Kleyn and E.J. Baerends, *What is the
    origin of the repulsive wall in atom-atom potentials*, Comments
    Atomic and Molecular Physics 23, 93 (1989)

[6] E.J. Baerends, *Pauli repulsion effects in scattering from and
    catalysis by surface*, in *Cluster models for surface and bulk
    phenomena*, ISBN13: 9780306441028, G. Puccchiori, P.S. Bagus and
    F. Parmigiani, Editors. 1992, Springer: New-York. p. 189-207.
