Coordination Chemistry Reviews

185–186 (1999) 81 – 98

Metal ion selectivity and molecular modeling

Peter Comba *
Anorganisch-Chemisches Institut, Uni6ersität Heidelberg, Im Neuenheimer Feld 270,
69120 Heidelberg, Germany
Received 28 October 1998
Contents

Abstract. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2. Steric effects and beyond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.1. Preorganization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.2. Molecular mechanics—general aspects and limitations . . . . . . . . . . . . . . . . . . . . 83
2.3. Force fields: the problem of transferability. . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.4. Preorganization: metal ion versus ligand selectivity . . . . . . . . . . . . . . . . . . . . . . 86
3. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.1. Quantitative assessment of the degree of ligand preorganization . . . . . . . . . . . . . . 86
3.2. Chelate ring size effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.3. Bonding cavity shapes and sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.4. Quantitative structure stability relationships . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4. Conclusions and possible extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Abstract

The evaluation of ligand molecules that are able to coordinate selectively specific metal
ions is a difficult task. Possible applications range from metal refinement and detoxification
of industrial waste to medical applications and the desire to understand biological processes.
Molecular mechanics modeling was used to predict highly preorganized ligand systems and
the molecular mechanics design, followed by the synthesis of size selective macrocyclic
donors was used extensively in this area. So far, only few truly successful studies have
emerged, and some of these are reviewed. The limitations of the models used are highlighted,
the reasons for expected pitfalls are discussed and possible models to solve some of the
remaining problems are analyzed. © 1999 Elsevier Science S.A. All rights reserved.
Keywords: Metal ion selectivity; Molecular mechanics; Force fields; Complex stability; QSAR

* Tel.: +49-6221-548453; fax: +49-6221-546617.

E-mail address: comba@akcomba.oci.uni-heidelberg.de (P. Comba)

0010-8545/99/$ - see front matter © 1999 Elsevier Science S.A. All rights reserved.
PII: S 0 0 1 0 - 8 5 4 5 ( 9 8 ) 0 0 2 4 9 - 5
82 P. Comba / Coordination Chemistry Re6iews 185–186 (1999) 81–98

1. Introduction

Ligands that are able to selectively build strong coordination compounds with
specific metal ions are of importance in many areas [1–7]. Metal ion selective
ligands are used in medicine, for the treatment of metal intoxication [8–12], the
complexation of paramagnetic metal ions used in magnetic resonance imaging
(MRI) [13 – 17] and for complexation of radio isotopes used for imaging and
therapy of tumors [18 – 22]; in environmental sciences, for waste water treatment
and for the quantitative analysis of soil, water and air [23,24]; in industry, for
separation and recycling in hydrometallurgic processes [25,26]; in fundamental
research, to understand the selectivity in metal ion transport through the cell wall
[3,27], in binding to siderophores, ionophores [3,28] and proteins [8,29], and to
understand the effects of ligand systems on reduction potentials of transition metal
coordination compounds, that is, the stabilization of uncommon oxidation states of
transition metal ions by coordination to specific ligand systems. Thus, there have
been and still are enormous efforts for the rational design of metal ion selective
ligands. All approaches used in this area are based on a thorough understanding of
complex stability (or complex stability differences), as in Eqs. (1)–(3):
K
Mnaq+ +Laq X MLnaq+ (1)
DG°c
K=e − (2)
RT
DG°c =DH°c −TDS°c (3)
The complexation reaction involves desolvation of the metal ion and of the ligands,
the complexation process and solvation of the complex. Electronic effects (metal
ion–donor atom bonding), steric effects (e.g. preorganization of the ligand, size-
fitting), entropic terms (e.g. chelate and macrocycle effects), solvent dependencies
(i.e. solvation and ion-pairing) are the basis of complex stability. These are
requirements that, so far, have not been met altogether in a general approach for
the accurate and, in terms of computational expense, acceptably fast calculation of
metal ion selectivities. There are a number of concepts that are used in the
interpretation and prediction of complex stabilities and in related areas (e.g. the
prediction of reduction potentials [30 –38], the correlation of reduction potentials
with ligand field strengths [39 – 41] and NMR chemical shifts [42]). Some of these
approaches are purely qualitative, others allow (semi-)quantitative predictions but
all are, for the reasons given above, limited in their applicability. Electronic factors
(bond strengths) are often interpreted on the basis of Pearson’s HSAB principle
[43], the Irving – Williams series [44 – 47] and general ligand field effects [32,39–
41,46,47]. The approaches to quantity steric effects range from the concepts of cone
angles [48 – 50] and seat – ligand – fitting [51–53], the VSEPR [54,55] and ligand–lig-
and repulsion models [56 – 58] to the computation of ligand bonding cavities
[1,59–61] and force field calculations in general [62]. Molecular mechanics has often
been used in the area of the prediction of metal ion discrimination [1,6,7,59–72].
However, in many of these examples the straight relation to the computation of
 P. Comba / Coordination Chemistry Re6iews 185–186 (1999) 81–98 83

metal ion selectivity is not obvious and limitations and pitfalls are not clearly
discussed. In this report I will mainly concentrate on force field based methods,
stress the problems, critically discuss a few (partly) successful examples and try
to indicate possible approaches that might in future lead to methods that allow
the prediction of metal ion discrimination.

2. Steric effects and beyond

2.1. Preorganization

The coordination of ligand molecules to a metal ion induces strain by the

metal center to the ligand molecule and strain by the ligand to the metal center
[41]. The loss of steric energy is compensated by the bonding energy that results
from metal ion – donor atom bond formation (Eq. (4)):
DG°c =(DH°ML +DUstrain) − TDSc (4)
In a fully preorganized ligand the structure of the metal-free ligand is identical
to that of the coordinated ligand [73,74], else DUstrain (the difference of strain
energy between the metal-free and the coordinated ligand) is positive (note that
it is possible to include steric strain that involves the metal ion–donor atom
stretching mode; this does, however, not fully include the metal ion/donor atom
pair dependent electronic part of DH°ML). It follows that preorganization refers
to a specific metal ion/ligand pair, and it involves the size and shape of the
bonding cavity of the ligand (that is, the ligand preferences) and the preferences
of the metal ion. Size- and shape-selective ligands need to be highly preorganized
and the energy cost for their structural reorganization needs to be large, else
they do not discriminate enough. That is, the ligand needs to be highly preorga-
nized and rigid. The main flexibility of ligand molecules is torsional freedom that
involves single bonds. That is, flexible ligands may easily adapt to metal ions of
variable size. This may be restricted by multiple bonds, small ring systems and
sterically demanding substituents [41]. This is the general basis of molecular
mechanics with the aim of a rational design of metal ion selective ligand sys-
tems. Note again, that, usually, this does not account for DH°ML, and it does not
fully account for − TDSc in Eq. (4).

2.2. Molecular mechanics—general aspects and limitations

The general assumption of molecular mechanics is that the positions of all

atoms of a molecule are determined by forces between each atom and all the
others. A set of potential energy functions describes the bonds, angles, torsional
angles, van der Waals and electrostatic interactions and some other terms. Struc-
tures are optimized by minimizing the resulting total strain energy. Thus, the
results of a molecular mechanics calculation are an optimized structure and the
corresponding minimized strain energy (depending on the minimization
84 P. Comba / Coordination Chemistry Re6iews 185–186 (1999) 81–98

algorithm used, vibrational frequencies may also be obtained). These general

principles and the inherent limitations, specifically in the area of coordination
compounds, have been discussed in an increasing number of books and review
articles [62,75 – 83]. Important factors are that (i) molecular mechanics is a fully
interpolative method, that is, the results depend on the data base to which the
force field (functional form and parameterization) has been fitted; (ii) for appli-
cations that involve the computation of thermodynamic parameters (complex
stabilities) force fields that are fitted to experimental structures—and this usually
is the case in the area of transition metal ion compounds—are not appropriate
[62] (reasons for the fact that, nevertheless, fortunately the steepness of the
potential energy surfaces generally is rather accurate have been discussed in
detail [41,62,84]); (iii) strain energies are relative quantities; thus, only strain
energy differences between isomers may be used for thorough interpretations
(note, that this may be a serious problem in the area of metal ion selectivities);
(iv) environmental effects (solvation and ion pairing) and entropy terms are
often neglected in molecular mechanics studies; note, that there are possibilities
to optimize solvated and ion-paired compounds and to compute entropies
[62,84–92], and the correlation of strain energies with (computed) solvation ener-
gies and entropies has been discussed [32,33].
In addition to these general approximations and limitations there are two
points which are especially important for applications in the area of metal ion
recognition: (i) metal ion selectivity is based on the difference of the complexa-
tion free energies D(DG°c ) between two or more metal ions and a common ligand
(Eq. (4)), where the difference between the strain energies D(Ustrain) (degree of
preorganization) is only one term. That is, electronic effects (metal ion–donor
atom bonding) are of importance, and these are generally not easy to quantify;
(ii) the result of an optimization of a structure by strain energy minimization
depends on the starting structure; while there are methods to scan the conforma-
tional space and while some force fields allow flexibility in terms of the coordi-
nation geometry (points-on-a-sphere models in particular), coordination modes
and coordination numbers must usually be defined in the starting model. This is
a serious restriction and a relevant example is EDTA which may bind as a 5- or
6-coordinating ligand, and 6-, 7- and 8-coordinate [M(EDTA)(OH2)n ]m + com-
plexes have been observed) [62,93].

2.3. Force fields: the problem of transferability

The enthalpy DH MLc of complex formation may be correlated with the corre-
sponding strain energy terms (Eqs. (5) and (6)):

M +nL X MLn (5)

DH ML
c :U total MLn Maq Laq
strain =U strain −U strain −nU strain (6)
 P. Comba / Coordination Chemistry Re6iews 185–186 (1999) 81–98 85

This approach has been used successfully to verify the influence of the chelate ring
size for a series of nickel(II) amine complexes (Table 1). The good agreement
between the observed and computed stability sequences indicates that, in this case,
the differences as a function of the chelate ring size are dominated by steric strain
[63].
It emerges that Eq. (6) may be used to compute the steric energy contribution of
a complexation reaction. Note, that all data in Table 1 are on nickel(II) amines
with the same number of donor groups in each pair. Therefore, the bonding energy
term (see Eq. (4)) is constant (see also below). However, this approach requires
transferability of the force field between metal-free and coordinated ligands.
Obviously, this approximation may not be valuable in general: there is some
electron transfer between the donor and the metal center [62,94]. The recent
development of a new force field that distinguishes between metal-free and coordi-

Table 1
Experimentally determined and calculated stability constants of high spin nickel(II) amines with five-
and six-membered chelates [63]

Complex U (kJ mol−1) −DU (kJ mol−1)a DH (kJ mol−1) −D(DH) (kJ mol−1)
MM MM obs obs

Ni(en) 0.27 −2.15

Ni(tn) 0.73 0.37 −1.86 0.29
Ni(en)2 0.80 −4.37
Ni(tn)2 1.71 0.73 −3.59 0.79
Ni(en)3 1.09 −6.69
Ni(tn)3 3.14 1.78 −5.09 1.60
Ni(dien) 1.45 −2.84
Ni(dpt) 1.98 0.35 −2.53 0.31
Ni(dien)2 2.84 −6.05
Ni(dpt)2 5.10 1.91 −4.21 1.84
Ni(2,2,2-tet) 2.26 −3.35
Ni(2,3,2-tet) 1.75 −0.60 −4.28 −0.93

a
Corrected for strain energy differences of the free ligands
86 P. Comba / Coordination Chemistry Re6iews 185–186 (1999) 81–98

nated ligands takes account of this problem [95]. The observed effects are rather
small, and this may explain why studies on metal ion selectivity with variants of Eq.
(6) that assumed full transferability of the parameters did not lead to undue
inaccuracies [62,68,70,71,94,95].

2.4. Preorganization: metal ion 6ersus ligand selecti6ity

Molecular mechanics is a valuable tool to quantify the preorganization of a

ligand with respect to a specific metal complex and therefore, to design highly
preorganized ligands (see examples given below). This is because, based on a high
quality force field, molecular mechanics allows to design a ligand with a given
donor set that has an optimum fit (preferred coordination geometry). However,
maximum stability is not necessarily equal to optimized discrimination. Selectivity
means that the stability for one metal ion needs to be optimized while the stability
for all the others is minimized. The problem with respect to the metal–donor
bonding energy terms has been discussed above. In terms of the steric energy the
problem to solve is that the design of a metal ion selective ligand must involve a
high degree of preorganization for a specific metal ion and also a high degree of
‘disorganization’ or ‘mismatch’ for other metal ions. The latter is not an easy task
since this reverts the ligand design from a one- to a multidimensional problem and,
indeed, this aspect has not been addressed in detail so far.
Quite often, therefore, reports on the molecular mechanics design of metal ion
selective ligands involve the strain energy optimized structures of a series of ligand
derivatives and the corresponding metal complexes—sometimes involving a single
metal ion. Clearly, this type of study is not directly related to the design of metal
ion selective ligands but rather to the design of ‘ligand selectivity by a metal ion or
by a few metal ions’. In practice this is a much less interesting problem to solve and
it leads to much less revealing and less applicable results.

3. Applications

3.1. Quantitati6e assessment of the degree of ligand preorganization

The degree of preorganization of a metal-free ligand must be defined with respect

to the structure of the ligand in the corresponding coordination compound with a
specific metal ion. Optimization of the degree of preorganization, on the basis of a
constant donor set and topology of a ligand molecule, may help to optimize
complex stability. Oligothiamacrocylic ligands are especially valuable examples here
since these are known to prefer, in contrast to the corresponding oxa- and
aza-macrocycles, exodentate conformations [96–101]. The degree of preorganiza-
tion (or complementarity) of the metal-free ligand involves the size and shape of the
ligand cavity. The amount of reorganization that is necessary for the coordination
of the ligand molecule to a specific metal ion must involve the structure and/or
strain energy of the metal-free ligand and the ligand structure and/or ligand-based
 P. Comba / Coordination Chemistry Re6iews 185–186 (1999) 81–98 87

strain energy when it is coordinated to the metal ion. Thus, these structural and
strain energy differences are useful for the comparison with experimental (thermo-
dynamic) parameters. The general approach for obtaining this information includes
the experimentally determined or computed structures of the metal-free ligand and
of the metal complex, and the comparison of the structures and/or strain energies
of the two geometries of the ligand molecules, that is, for the coordinated ligand the
strain energy has to be computed after removal of the metal ion. Note that this
general approach may allow to obtain a meaningful idea of the degree of preorga-
nization of a particular ligand with a series of metal ions (metal ion selectivity) or
of a series of ligands with a specific metal ion (optimization of a ligand system or
‘ligand selectivity’, see above).
The structural reorganization of a ligand was considered to be a two step process
that involves a conformational ligand reorganization term DUconf from the most
stable conformation of the metal-free ligand to that of the coordinated ligand
(‘coordinating conformer’, reorganization prior to the complexation reaction), and
a structural reorganization term DUcomp that occurs during the complexation
reaction and is a measure of the complementarity of the ligand molecule (see Chart
1, Eq. (7)) [68].
(Chart 1)

DUreorg =DUconf +DUcomp (7)

Note that the mechanism of complexation by a macrocyclic ligand is, depending on
the type of ligand and metal ion involved, a step-wise process that does not
necessarily involve the ‘coordinating conformer’ [102,103]. Thus, the splitting of the
ligand structural reorganization into two terms that occur before and during the
coordination is arbitrary and, in terms of the complexation mechanism, not
necessarily reasonable. However, it may allow to separate specific factors of
importance for the ligand design, such as the rigidity (related to the conformational
reorganization) and the cavity size and shape of a ligand (related to the complemen-
tarity). Binding sites of alkylated bidentate ether ligands have been correlated with
this method to the corresponding stability constants [68].
A strain energy-based and a structural reorganization parameter were used to
assess the preorganization of structurally reinforced tetrathiamacrocyclic ligands
[69]. Both parameters are based on the experimentally determined and/or computed
structures of the metal-free and the coordinated ligand molecules. The strain energy
ratio EL/EC (EL =Ustrain (metal-free ligand); EC = Ustrain (coordinated ligand)) was
used as a parameter that is related to the thermodynamics of the reorganization
process; the sum of the absolute values of the differences of the intramolecular
donor–donor distances 6n = 1 Ddn (see Chart 2) was used as a structural parameter
related to the ligand reorganization.
88 P. Comba / Coordination Chemistry Re6iews 185–186 (1999) 81–98

(Chart 2)

A fully preorganized ligand has EL/EC = 1 and 6n = 1 Ddn = 0, generally EL/EC B 1
and 6n = 1 Ddn \0. These are fully empirical correlations and there is no justifica-
tion for using the strain energy ratio EL/EC instead of the strain energy difference
EC −EL for the assessment of the preorganization (other terms, such as (EC − EL)/
(EC + EL) were used with similar results [69].
3.2. Chelate ring size effects

For both amine [7,63] and (ether) oxygen donors [7,104] there is a selectivity of
five-membered chelate rings for relatively large metal ions while six-membered
chelate rings prefer relatively small metal ions. A simple geometric model is often
used to interpret these observations [6,7,63,65,105]: for a six-membered chelate ring
three appropriate corners of a cyclohexane molecule (chair conformation) are
replaced by a metal ion M and two amine nitrogen (or ether oxygen) donors N,
leading to a putative metal-1,3-diaminopropane ([M(tn)]n + ) chelate with M–N
distances of 1.54 Å and an N – M – N angle of 109.5°. In a similar model a
metal–1,2-diaminoethan ([M(en)]n + ) chelate leads to M–N distances of 2.50 Å and
an N–M – N angle of 69° (Chart 3).
(Chart 3)

This crude model neglects the variation in plasticity (or rigidity) and the variable
preferences of metal ions. The metal–donor distances, the donor–metal–donor
angles, and the angles around the donor atoms are coupled, that is, the optimum
structure largely depends on the relative steepness of the three potentials involved
(metal donor stretching and the two angle bending terms; potentials involving the
carbohydrate backbone usually are much stiffer) [41,62,79,84]. The fact that, with
a given metal ion, metal – amine, metal–1,2-diaminoethane and metal–1,3-di-
aminopropane compounds often have similar metal–nitrogen bond distances and
angles involving the metal center and the donor atoms [41,76,84,106,107] indicates
that the directionality exerted by the metal center and the donor atoms are of some
importance (the directionality of the metal center–donor bonds is comparatively
weak, however (plasticity of transition metal ions), and this was discussed in some
detail [108]). Also, the angular dependence involving the donor groups was dis-
cussed as one of the primary factors in the determination of complex stability of
crown ether ligands [66]. A well-balanced force field must take all these factors into
 P. Comba / Coordination Chemistry Re6iews 185–186 (1999) 81–98 89

account (note that a force field that is only based on experimental structures, i.e. a
force field that was not validated with thermodynamic data, might not fulfill these
requirements, see above).

3.3. Bonding ca6ity shapes and sizes

The calculation of the hole size of macrocyclic ligands is by far the widest area
of application of molecular mechanics toward the design of metal ion selective
ligands. A ligand with a hole that entirely fits a specific metal ion, that is, a highly
preorganized ligand, leads to maximum stability. Rigidity of the macrocycle pre-
vents the ligand from adapting to a variety of metal centers and therefore enhances
the selectivity. From the fact that metal ions do not merely need to fit into the hole,
but that they need to be fixed (bound) to the cavity, it emerges that the ligand
cavity must have a specific size and shape, that is, the number and type of donors,
their angular orientation around and the distance to the metal center and the
directionality of the donor lone pairs with respect to the metal–donor vectors are
of importance [41,72,84]. The rigidity of the ligand also refers to the size and the
shape of the cavity.
A number of primarily molecular mechanics based methods for the computation
of ligand cavity sizes have been reported (note that cavity shapes and sizes are not
necessarily restricted to macrocyclic ligands). These have been discussed in detail
and controversially and corresponding results have been reviewed extensively
[1,7,59–62,75,109 – 112]. Molecular mechanics based approaches generally compute
the ligand-based strain energy as a function of the metal–donor distance. Based on
the often used harmonic potential (Eq. (8)) there are three main methods to scan
the potential energy surface as a function of the M–L distance:
U ML
strain =1/2k
ML ML
(r −r°) (8)
“ variation of r°;
“ restraints, i.e. fixing of r ML by (exceedingly) large force constants;
“ constraints, i.e. fixing of r ML mathematically (this is only possible when mini-
mization techniques are used that include second derivatives).
Constraining bond distances (or other internal parameters) with Lagrange multipli-
ers probably is the most elegant method since it is mathematically precise and does
not lead to artefacts. Also, it may be used to compute ligand cavities without metal
ion dependent energy terms. This is a requirement for the computation of metal ion
selectivities [62,72,75].
Ligand systems used for the selective binding of metal ions are only rarely
symmetrical. The asymmetry may be based on the ligand backbone and/or the
donor set. It is astonishing that only recently it was pointed out that erroneous
results are expected when all metal –donor distances are varied uniformly for
asymmetrical ligands [62,113]. From the methods that have been proposed to shrink
and blow-up cavities asymmetrically, the most elegant is to use Lagrange multipli-
ers on the sum of a selected number of bonds (that is on all metal–donor distances;
sum constraints) [72,114]. This allows to scan a bonding cavity under the condition
90 P. Comba / Coordination Chemistry Re6iews 185–186 (1999) 81–98

Fig. 1. Strain energy Estrain as a function of the average M – N bond distance rav for [M(bispidine)]n +
(see Chart 4 for ligand structure). sym, symmetrical variation of the hole size; asym, asymmetrical
variation of the hole size [115]; sum, variation with sum constraints [71].

that each bond that is included in the constraint may react individually to the stress
imposed by enforcing a certain cavity size. Similar methods may be used to scan the
shape of a cavity by constraining sets of valence or torsional angles but this has not
been done so far. An example of the computation of the cavity size and shape is
given in Fig. 1 (see Chart 4 for the structure of the corresponding bispidine type
ligand). The important result is that the optimum cavity size (1.78, 1.81, 1.84 Å)
and the relative energy cost upon shrinking and blowing-up the ligand (steepness of
the total energy curve) depend to some extend on the approach used (the force field
was the same for the computation of the three curves) [72].
(Chart 4)
 P. Comba / Coordination Chemistry Re6iews 185–186 (1999) 81–98 91

The lowest energy curve in Fig. 1 fulfills all the requirements for the computation
of the shape and size of a ligand cavity: it is metal-ion independent (all metal
dependent terms are removed: a points-on-a-sphere model is used, i.e. angle
bending potentials around the metal center are replaced by 1,3-nonbonded interac-
tions and metal – donor potentials are set to zero) and sum-constraints are used for
metal–donor bonds, i.e. elongation and compression of all four bonds are decou-
pled. Hence, the plotted strain energy of this curve is the strain induced by a metal
ion to the ligand without any metal ion present. The important questions are what
the significance of such a curve is and how its accuracy may be checked. The naive
answer to the first question is that the bispidine ligand (Chart 4) prefers very small
metal ions (M – L 1.8 Å, see Fig. 1) and that elongation to ca. 2.0 Å leads to a
destabilization of over 10 kJ mol − 1. However, there is no reason to assume that
real metal ions would not capture another ligand present in solution (solvent,
anions) to produce five- or six-coordinate [M(bispidine)Xn ]m + species (n= 1, 2)
with different cavity shapes and sizes. Also, the ratio of ideal bond distances and
force constants involving the bonds to the amine and to the pyridine donors might
differ from metal ion to metal ion. This indicates that, dependent on the problem
to solve, the fictive metal ion independent cavity size and shape may or may not be
a relevant parameter. The simple answer to the second question is: there is no way
to check the correctness of the curve. Experimental proof for the metal ion
independent curve (measurement of stability constants) will not be possible (see
above) and the computation of strain energies of individual metal complexes does
involve (individual) metal dependent terms. That is, data points that are based on
computed structure and strain energy pairs are not expected to coincide with he
computed metal-independent curve. Structures (and thermodynamic properties) of
transition metal coordination compounds are the result of a compromise between
metal ion and ligand preferences. Metal ion independent cavity size (and shape)
computations may be used to quantify the ligand preferences. The problem with the
interpretation of these curves arises because it is not an easy task to compute the
metal ion preferences separately and to predict the metal ion dependent balance
between ligand and metal ion dictation.

3.4. Quantitati6e structure stability relationships

Molecular and materials properties are related to structures [62,76,79,84]. Obvi-

ous examples include the destabilization and enhanced reactivity due to steric
strain, the dependence of the ligand field strength from metal–donor distances and
angular distortions, Karplus relations in NMR spectroscopy, stabilization by
hydrogen bonding, etc. Hence, there must be algorithms that relate (calculated or
observed) structures and strain energies (that is, the results of force field calcula-
tions) with thermodynamic properties, reactivities, electronic and spectroscopic
properties. The mathematical models used are generally based on pattern recogni-
tion methods and the resulting models may be used for interpolations. That is, the
quality and applicability of these models depends on the quality and variability of
the ‘training set’ of observables.
92 P. Comba / Coordination Chemistry Re6iews 185–186 (1999) 81–98

A number of quantitative structure–property relationships (QSPR) have been

used for the design of new compounds, for the interpretation of their properties and
for the determination of structures [76,79,84,90,115 –117]. These include linear and
non-linear correlations of (computed) structural parameters and strain energies with
complex stabilities [6,66,70,71,118], reduction potentials [32,33,39,119], electron
transfer rates [32,120,121] and ligand field properties [41,107,122–124], and the
computation of IR, NMR and EPR spectra, based on structural information
[84,117,125 – 128]. Quantitative structure–activity relationships (QSAR), which are
used in drug design for almost 30 years [129], use a mathematical model to relate
numerical properties of the molecular structure to the activity of the substance. The
electron topological approach (ET) is an extension of simple QSAR methods which,
as a third dimension, includes information on electronic features [130].
Linear regressions have been used for the correlation of complex stabilities with
the strain energies of transition metal coordination compounds [6,66,70,71,118].
For lanthanoid(III) compounds with bis-alkylhydrogenphosphates (Chart 5) the
strain energy of the complexation reaction DUM is (Eqs. (9) and (10)):
(Chart 5)

M(OH2)39 + +6HRM{(HR)(R)}3(OH2)3 + 3H + + 6H2O (9)

DU M 6U HR U Mcom 3U H 6U aq

DUM =UMcom +3UH +6Uaq −UM − 6UHR (10)

With M= La(III) as a reference the relative strain of the complexation reaction is
(Eq. (11)):
 P. Comba / Coordination Chemistry Re6iews 185–186 (1999) 81–98 93

DUM −DULa =(UMcom −ULacom) − (UM − ULa) (11)

The correlation of the relative strain energies of the complexation reaction with the
relative extraction constants leads to a linear plot (Eqs. (12) and (13); Fig. 2; a is
the apparent QSPR constant with an expected value of 1.08) [70].
DGM = − RT ln Kex,M (12)
DUM −DULa =a log (Kex,M/Kex,La) (13)
The small deviation of a from the expected value (1.26 vs. 1.08) is probably due to
the neglect of entropy, ion-pairing and solvation (see above). The good linearity
suggests that the neglected terms are constant or linearly dependent on the strain
energy. A similar observation was made when reduction potentials of hexaamine-
cobalt(III/II) and tetraaminecopper(II/I) couples were correlated with strain energy
differences between the oxidized and the reduced forms [32,33].
What is the value of a correlation such as that presented in Fig. 2? All three
ligands (Chart 5) fit to the same curve with a slope of 1.26, and other derivatives
with similar donor sets that lead to complexes with the same stoichiometry (Eq. (9))
will probably also fit to the curve. Thus, a plot such as that of Fig. 2 is a valuable
test for the type of structure and extraction mechanism of a new ligand system.
Does it also help to design new ligands with increased selectivities? The linear
correlation between the relative complexation strain energy and the relative ex-
tractabilities indicates that, as expected, increasing bulk of the substituents of the
organophosphato ligands results in increasing selectivity. With the force field used

Fig. 2. QSPR plot of DUM − DULa vs. log(Kex,M/Kex,La) (for ligand abbreviation see Chart 5).
94 P. Comba / Coordination Chemistry Re6iews 185–186 (1999) 81–98

[70,95], this may be predicted quantitatively. However, there is a limit, where

excessive strain might lead to a change in coordination number, and this cannot be
predicted with harmonic bonding potentials as they were used in the investigation
discussed here.

4. Conclusions and possible extensions

The neglect of entropic terms, of environmental effects (solvation, ion pairing), of

specific metal – donor bonding terms (electronic effects) and of anharmonicity in the
metal–donor stretching potentials leads to a situation that allows accurate predic-
tions of metal ion discrimination by organic ligands only in a very limited range.
While there are similar approximations for molecular mechanics calculations in
other areas, in the field of designing metal ion selective ligands these are too severe
to have molecular mechanics become a generally valuable tool. The neglect of the
metal ion specific electronic effects in the metal–donor bonding interactions
prevents a generally meaningful comparison of the stabilities of a series of com-
pounds with a constant ligand set and variable metal ions. This problem is basically
related to the fact that meaningful thermodynamic predictions are only possible on
the basis of isomers.
There is a very similar difficulty with the computation of reduction potentials.
The free energy of the reduction process may, according to a Born–Haber cycle, be
arbitrarily divided into the ionization potential of the gaseous metal ion (I), the
difference of the complexation free energies of the reduced and the oxidized forms
in the gas phase [D(DG°c )], the corresponding aquation free energies [D(DG°aq)] and
a constant correction parameter C for solvent and electrolyte effects and the
electrode setup (Eqs. (14) and (15)) [31–33]:
DG°red =I + D(DG°c +D(DG°aq) + C (14)
DG°red = − nFE° (15)
It emerges that reduction potentials of transition metal coordination compounds
depend on steric and electronic factors of the metal–donor bonding, specific
solvation effects and entropic terms. These are basically the same factors which are
of interest for metal ion separation, and this is not unexpected (Eq. (16), b is the
Brutto stability constant):
RT b ox
E°complex =E°aquaion − ln red (16)
nF b
Force field calculations have been used to compute reduction potentials of hex-
aaminecobalt(III/II), hexaaminenickel(III/II), hexaaminenickel(II/I) and te-
traaminecopper(II/I) couples, where electronic factors in each set of compounds
were constant and entropic and solvation terms were shown to be linearly related
to strain energy differences [32,33,41,119]. As for the computation of relative
stability constants this approach may only be applied in very restrictive limits.
 P. Comba / Coordination Chemistry Re6iews 185–186 (1999) 81–98 95

Approaches based on ligand field spectroscopy transition [39] and on general

electrochemical parameters [34 – 38] do not specifically account for steric contribu-
tions (Eq. (17), SM is a metal dependent parameter that varies with the relative
metal–donor bond strength of the reduced and the oxidized forms, IM is a metal
dependent parameter that includes factors based on the ionization potential, the
spherical part of the ligand field, the electrode setup and the solvation, EL is a
ligand dependent parameter:
E°= SMSEL +IM (17)
A more generally applicable method for the computation of reduction potentials
and that of relative complex stabilities (see Eq. (16)) must include both, steric and
electronic contributions. This might be based on a combination of the methods
based on steric energies with a variant of Eq. (17). For transition metal coordina-
tion compounds, transferable ligand field or AOM parameters [90,107,122–124]
might be a first approximation for transferable electronic parameters for the
metal–donor bonds.

Acknowledgements

Our studies are supported by the German Science Foundation (DFG), the Fonds
of the Chemical Industry (FCI) and the VW-Stiftung. I am grateful for that, for the
excellent work done by my coworkers, whose names appear in the references, and
to Brigitte Saul, Marlies von Schoenebeck-Schilli and Karin Stelzer for their help in
preparing the manuscript.

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