Journal of Quantum Informatio n Science, 2011, 1, 87-95
doi:10.4236/jqis.2011.12012 Published Online September 2011 (http://www.SciRP.org/journal/jqis)
Copyright © 2011 SciRes. JQIS
Modeling of the Chemico-Physical Process of
Protonation of Molecules Entailing Some
Quantum Chemical Descriptors
Sandip K. Rajak, Nazmul Islam, Dulal C. Ghosh*
Department of Chemistry, University of Kalyani, Kalyani, India
E-mail: *dcghosh1@rediffmail.com
Received May 27, 2011; revised July 25, 2011; accepted August 6, 2011
Abstract
Relying upon the basic tenets of scientific modeling, an ansatz for the evaluation of proton affinity of mole-
cules are evolved in terms of a four component model. The components of the model chosen are global de-
scriptors like ionization energies, global softness, electronegativity and electrophilicity index. These akin
quantum mechanical descriptors of atoms and molecules are linked with the charge rearrangement and po-
larization that occur during the physico-chemical process of protonation of molecules. The suggested ansatz
is invoked to compute the protonation energy of as many as 43 compounds of diverse physico-chemical na-
ture viz, hydrocarbons, alcohols, carbonyls, carboxylic acids, esters, aliphatic amines and aromatic amines. A
detailed comparative study of theoretically evaluated protonation energies of the above mentioned molecules
vis-à-vis their corresponding experimental counterparts reveals that there is a close agreement between the
theory and experiment. Thus the results strongly suggest that the proposed modeling and the ansatz for
computing PA, the proton affinity, of molecules for studying the physico-chemical process of protonation
may be valid proposition.
Keywords: Physico-Chemical Process of Protonation, Proton Affinity, Conceptual Density Functional
Descriptors, Commonality between Density Functional Descriptors and Proton Affinity,
Muliti-Linear Regression Model
1. Introduction
The protonation reaction or the physico-chemical process
of protonation is ubiquitous in almost all the areas of
chemistry and biochemistry [1-5]. The majority of che-
mical reaction occurs in acid medium. The chemical
process of protonation is fundamental first step of many
chemical rearrangements, and enzymatic reactions [4].
The resulting protonated molecule is frequently a pivotal
intermediate that guides the succeeding steps of the
overall process. The knowledge of the intrinsic basicity
and the site of protonation of a compound is central for
the understanding of the mechanism of chemical reac-
tions. The legend proton affinity is defined as the nega-
tive of the enthalpy change of a protonation reaction at
the standard conditions. The gas-phase proton affinities
are a quantitative measure of the intrinsic basicity of a
molecule [6]. The study of thermochemistry of the pro-
ton transfer reaction in the gas phase is well-known ex-
periment of acid-base reaction [7]. Dynamics of proton
transfer is also important for ionization processes in mass
spectroscopy [8]. Basicity is defined [9] as the tendency
of a molecule, B, to accept a proton, H+, in the following
Base-Acid (Proton) adduct BH+ formation reaction
BHBH PA

 (1)
where PA is the proton affinity of the base B.
This concept of basicity was generalized further and
freed from reference to a specific acid (H+) by Lewis
[10]. During the physico-chemical process of protonation,
electronic charge is soaked by the proton from the entire
skeleton of the molecule. As a result, all the structural
parameters i.e. bond lengths and bond angles, and other
charge dependent physical properties like the polarizabil-
ity and the dipole etc are affected. A plethora of informa-
tion has appeared on the study of this important chemico-
physical process [6,7].
Although, experimentally the proton affinity can be
S. K. RAJAK ET AL.
88
determined by several techniques like the measurement
of the heats of formation [5] of the species involved in
the adduct formation reaction, by mass spectroscopic
measurement techniques [7-9,11] and by the measure-
ment of the ionization thresholds [7]. The “acid-base
adducts” are not always stable and/or does not exist in all
cases and also it is well known [12] that the experimental
determination of the proton affinities of molecules is not
easy always. For this reason, in recent years, much em-
phasis has been given to the calculation of proton affini-
ties through some quantum mechanical and its young
branch, the density functional theoretical models [13-17].
It is now established [18] that the ab initio quantum
mechanical approaches and its numerous variants are
very successful in providing reliable values of proton
affinity and gas phase basicity for small molecules.
However, due to the reason of heavy computational cost ,
application of ab initio methods for the estimation of
proton affinities is still impractical for larger molecules
[19]. It is also recognized [20] that the popular semi em-
pirical methods such as AM1, MNDO and PM3 etc are
not that reliable in calculating proton affinities. Although
there are some attempts of modeling to compute protona-
tion energy for specific groups of compounds [6,21-26],
but fact remains that no universal model has been, so far,
put forward subsuming the energetic effects necessarily
appearing in the physico-chemical process of protonation
as a substitute for experimental or theoretical measure-
ment of the energy of protonation.
Currently the conceptual density functional theory
[27-44] of chemical reactivity have introduced many de-
scriptors, global and local, like electronegativity, hard-
ness, softness, fukui functions, electrophilicity index etc
in theoretical chemistry. Such descriptors have made
serious inroad in science and opened a new paradigm of
chemical thinking, modeling and computation [27-44].
In this work, we have developed a model for the
evaluation of proton affinity in terms of some akin con-
ceptual reactivity descriptors which can be conceptually
linked and associated with the physico-chemical process
of protonation. The akin descriptors are the ionization
energy (I), the global softness(S), the electronegativity
(χ), and the global electrophilicity index (ω).
1.1. The Physico-Chemical Process of
Protonation
In the terminology classification of chemical reaction
according to the reagent and the substrate, proton is an
electrophile. In the physico-chemical process of protona-
tion, when a proton dynamically approaches towards a
nucleophile from a long distance it is attracted by the
electron cloud of the molecule. The proton acting as an
electrophile soaks the electron density from the entire
skeleton of the nucleophile [45]. As a result, the electron
cloud of the nucleophile is redistributed and remains
under the influence of the electrophile, the proton. In
some circumstances, the proton fixes at a site of lone pair,
if available, in the molecule. However, if there is no lone
pair in the structure of the molecule, the proton remain-
ing attached to the sphere of the charge cloud of the
molecule. The polarizing power of the proton induces a
physical process of structural and energetic changes in
the molecule and the effect is expected to be at its maxi-
mum at the gas phase of the molecule. Thus, the gas-
phase basicity is certainly the ideal revelator of the
structural and energetic characteristics of the molecular
protonation process.
1.2. The Physico-Chemical Process of
Protonation Entailing the Ionization
Energy, the Electronegativity, the
Chemical Hardness, the Softness,
and the Electrophilicity Index
In order to suggest a mathematical modeling of comput-
ing the protonation energy of molecules involving the
above akin theoretical descriptors that may be associated
and directly linked with the physico-chemical process of
protonation, we depict the glimpses of the role of each
descriptor in the process separately viz.
1) The ionization energy (I)
This is a fundamental descriptor of the chemical reac-
tivity of atoms and molecules. High ionization energy
indicates high stability and chemical inertness and small
ionization energy indicates high reactivity of the atoms
and molecules [46]. Mills et al. [47] discovered a linear
relationship between the proton affinity and the additive
inverse of the ionization energies of molecules.
2) Electronegativity (χ)
Electronegativity though defined in many different
ways, the most logical and rational definition of it is the
electron holding power of the atoms or molecules. Elec-
tronegativity is defined and measured as the power (force)
with which the valence electron of an atom is held by its
screened nuclear charge. The more electronegative ele-
ments hold electrons more tightly and the less electro-
negative elements hold less tightly. Lohr [48] has dis-
cussed the physico-chemical process of protonation from
a deeper insight and discovered the important relation-
ship between the protonation and electronegativity. He
[48] further went to conclude that there is a protonic
counter part of electronegativity as a organizing principle
of acidity and basicity. However, the inverse relationship
between the electronegativity and protonation process
and associated energetic effect is straight forward.
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S. K. RAJAK ET AL.
3) Global Softness (S)
The softness is in fact the inverse concept of hardness,
a fundamental descriptor of the stability and reactivity of
atoms and molecules. It is apparent that the chemical
hardness fundamentally signifies the resistance towards
the deformation or polarization of the electron cloud of
the atoms, ions or molecules under small perturbation of
chemical reaction. The softness is simply the reciprocal
of the hardness. Thus the general operational signifi-
cance of the hard-soft chemical species may be under-
stood in the following statement. If the electron cloud is
strongly held by the nucleus, the chemical species is
“hard” but if the electron cloud is loosely held by the
nucleus the system is “soft” [40,49]. Hence the polariza-
tion process associated with protonation should be di-
rectly controlled by the softness of the molecule. Or in
other words, the energetic effect associated with the pro-
tonation should be directly proportional to the softness of
the molecule.
4) Electrophilicity Index (ω)
In reference to nucleophilic-electrophilic, acid-base or
donor-acceptor reaction, the electrophilicity index [50,51]
of atoms and molecules seems to be an absolute and
fundamental property of such chemical species because it
signifies the energy lowering process on soaking elec-
trons from the donors. This tendency of charge soaking
and energy lowering must emanate and develop from the
attraction between the soaked electron density and screened
nuclear charge of the atoms and molecules.
As the process of electron soaking by proton continues,
the accumulated electron density will, however, shield
the proton. However, the electrophilicity of the substrate
will oppose the charge soaking by the proton during the
physico-chemical process of protonation and hence the
protonation is hindered by the electrophilicity and hence
protonation energy should bear an inverse relationship
with electrophilicity index.
2. The Modeling of the Physico-Chemical
Process of Protonation and Algorithm
for Computing the Proton Affinity of
the Molecules
The descriptors like the ionization process of atoms and
molecules, the physical property like hardness, softness,
the electronegativity and the electrophilicity have close
relation i.e. akin with each other in their operational sig-
nificance and origin.
We have tried to posit above that the physico-chemical
process of protonation can be linked to the above akin
descriptors –the ionization process, the hardness, soft-
ness, electronegativity and electrophilicity. Recently, we
[52-61] have published good number of papers where we
have discussed that the three descriptors, the electro-
negativity, the hardness and the electrophilicity index of
atoms and molecules are fundamentally qualitative per se
and operationally the same. All these three descriptors
represent the attraction of screened nuclei towards the
electron pair/bond. Thus, we can safely and reasonably
conclude that the proton affinity and the three descriptors
have inverse relationship.
Thus, since the above four parameters have dimension
of energy and can be linked to the process of charge re-
arrangement and polarization during the physico-che-
mical process of protonation, they can be components of
a probabilistic scientific modeling of proton affinity. The
physico-chemical process of protonation has direct link
to the charge polarization and alteration of electron dis-
tribution in the molecule.
The proton affinity or the ability of donating the lone
pair of a Lewis base and the ability for the deformation
of electron cloud of a species, the softness, and /or the
tendency of the molecule to lose electron, the ionization
potential, are fundamentally similar in physical appear-
ance stemming from the attraction power of the nuclei of
the atoms forming the molecule. The softness, the ioni-
zation energy, the electronegativity (chemical potential)
and the electrophilicity index have direct link to the
process of polarization and transfer of charge from a
substrate and hence control the energetic effect—the
protonation energy. Considering all the above mentioned
fundamental nature of the physico-chemical process of
protonation and its probable relationship with the quan-
tum mechanical descriptors, we suggest an ansatz for the
computation of the proton affinity in terms of these
theoretical descriptors. The physico-chemical process
and the energetic effect must entail the above stated four
parameters. To derive an explicit relation to compute the
proton affinity in terms of the above stated descriptors,
we suggest explicit inter relationships between the pro-
tonation energy and the descriptors relying upon their
response towards the protonation.
PA I
(2)
PA S
(3)
PA 1
(4)
PA 1
(5)
Combining the above four relations we get,
 
123 4
PACCICC 1C1S
  (6)
where PA is proton affinity, C, C1, C2, C3 , and C4 are the
constants I is ionization energy, S is global softness, χ is
the electronegativity and ω is the global electrophilicity
index of the molecule.
Copyright © 2011 SciRes. JQIS
S. K. RAJAK ET AL.
90
3. Method of Computation
Ab initio Hartree-Fock SCF method and the Koopmans’
theorem are invoked to compute the ionization potential
(I) and electron affinity(A), which in turn, are used in
computing the descriptors invoked in this studies.
According to Koopmans’ theorem the ionization po-
tential (I) and the electron affinity (A) are computed as
follows:
HOMO
I
 (7)
LUMO
A
 (8)
where εHOMO and εLUMO are the orbital energies of the
highest occupied and the lowest unoccupied orbitals.
Parr et al. [40,41,44] defined the chemical potential, μ ,
electronegativity, χ, and hardness, η, in the framework of
density functional theory, DFT as



Vr
EN IA2

  (9)
 
22
Vr Vr
12N12EN12 IA


 
 (10)
where E, N ,v(r), I and A are the energy, the number of
electrons, the external potential, the ionization energy
and the electron affinity of a chemical system respec-
tively.
Softness is a reactivity index and is defined as the re-
ciprocal of hardness
S1
(11)
Parr et al. [36] defined electrophilicity index (ω) as
 
22
 
(12)
In this study we have taken some hydrocarbons in
Set-1, alcohols, carbonyls, carboxylic acids and esters in
Set-2, aliphatic amines in Set-3 and aromatic amines in
Set-4. The molecules are so chosen whose experimental
protonation [7,62-65] energy are known. The PQS Mol
1.2 - 20-win software [66] has been used to calculate the
global descriptors using the ab initio Hartree-Fock SCF
method with 6 - 31 g basis set. The geometry optimiza-
tion technique is adopted. The ionization energy, the
electronegativity, the global softness, and the global elec-
trophilicity index of the molecules are computed by in-
voking the Koopmans’ theorem and Equation (7), Equa-
tion (9), Equation (11) and Equation (12) respectively.
A multi linear regression analysis [67] is performed
using Minitab 15 [68] to compute the correlation coeffi-
cients C, C1, C2, C
3 and C4 by plotting experimental PA
along the abscissa and the values of the quantum me-
chanical descriptors along the ordinate. The computed
correlation coefficients C, C1, C2, C3 and C4, for the Set 1,
Set 2, Set-3 and Set-4 are tabulated in Table 1.
Thereafter, we have computed the P.A’s of four sets of
molecules invoking the suggested ansatz, Equation (6),
and putting the quantum mechanical descriptors and the
respective correlation coefficients of each set of mole-
cules under our study. The comparative study of theo-
retically evaluated and experimentally determined PA’s
of the Set 1 - Set 4 is performed in the Tables 2-5 re-
spectively.
Table 1. Correlation coefficients and R2 value for the Set 1,
Set 2, Set 3 and Set 4.
Sets C C1 C
2 C
3 C
4 R
2
1 450 18.0 24100 –40539 8019 0.992
2 –113 –4.66–1810 3561 –705 0.818
3 17.1 0.666–0.1 –0.1 –0.150.995
4 –129 +7.94147 167 –11.10.916
Table 2. Experimental P.A (eV), calculated P.A (eV) and R2
for the Set 1.
Molecule Experimental P.A Calculated P.A R2
Methane 5.63294 6.01418 0.99
Ethane 6.17932 6.56332
Propane 6.48286 6.83883
Butane* 6.83237 7.07331
Isobutane 7.02491 7.34303
Pentane* 6.86533 7.13276
Hexane* 7.01407 7.37095
*P.A calculated by Wróblewski et al.45
Table 3. Experimental P.A (eV), calculated P.A (eV) and R2
for the Set 2.
Molecule Experimental P.A Caculated P.A R2
Formaldehyde 7.38916 7.90889 0.817
Formic acid 7.68837 8.13938
Methanol 7.81846 8.57263
Ketene 8.55564 8.96532
Acetaldehyde 7.9659 8.34277
Ethanol 8.04829 8.57857
Acetic acid 8.12201 8.55023
Acetone 8.41254 8.90774
Propanol 8.15236 8.47112
Propionic acid8.26077 8.50291
Methyl acetate8.28679 8.61904
Butanol 8.17838 8.46674
Copyright © 2011 SciRes. JQIS
91
S. K. RAJAK ET AL.
Table 4. Experimental P.A (eV), calculated P.A (eV) and R2
for the Set 3.
Molecule Experimental P.A Caculated P.AR2
NH3 8.846181 8.860042 0.995
CH3NH2 9.284153 9.341572
CH3CH2NH2 9.409908 9.399806
(CH3)2CHNH2 9.47929 9.499455
(CH3)2NH 9.566017 9.583402
(CH3)3CNH2 9.57469 9.596479
(CH3)3N 9.761153 9.794852
Table 5. Experimental P.A (eV), calculated P.A (eV) and R2
for the Set 4.
Molecules Experimental P.A
(eV))
Calculated P.A
(eV) R2
3-H3C6H4N(C2H5)2 9.925935 9.722904 0.91
4-H3C6H4N(C2H5)2 9.912926 9.706435
C6H5N(C3H7)2 9.912926 9.673925
C6H5N(CH3)(C2H5) 9.84788 9.522402
C6H5NH(C2H5) 9.618053 9.592654
C6H5NHCH3 9.457608 9.44481
C6H5CH2NH2 9.401235 8.976198
2-(OH)C6H4NH2 9.28849 9.197386
3-(OH)C6H4NH2 9.28849 9.197251
4-CH3C6H4NH2 9.266808 9.06326
3-CH3C6H4NH2 9.253799 9.04584
3-CH3C6H4N(CH3)2 9.253799 9.044886
1,2-C6H4(NH2)2 9.22778 9.031081
4-ClC6H4NH2 9.045653 8.720894
3-BrC6H4NH2 9.023971 8.683775
4-FC6H4NH2 9.023971 8.763088
3-CF3C6H4NH2 8.854853 8.674228
For better visualization of the comparative study, the
results of the theoretically computed and experimentally
determined proton affinities of the Set 1 - Set 4 are de-
picted in the Figures 1-4 respectively.
4. Results and Discussion
A deeper look on the Table 2 and Figure 1 (for Set 1),
and the Table 4 and Figure 3 (for Set 3) reveals that
there are excellent correlation between the theoretically
computed proton affinities of the seven hydrocarbons
(Set 1) and seven aliphatic amines (Set 2) respectively.
The R2 value for the correlation of Set1 and Set 3 are
0.99 and 0.995 respectively. A close look at the Figure 1
and Figure 3 reveals that the two sets of PA’s—experi-
mental and theoretical of the two groups of molecules are
so close to each other that one curve just superimposes
upon the other.
A look at the Table 3 and Figure 2 (for Set 2), and
Table 5 and Figure 4 (for Set 4) reveals that there is
fairly a good correlation between the theoretically com-
puted and experimentally determined proton affinities of
as many as twelve compounds containing alcohols, car-
bonyls, carboxylic acids and esters(Set 2), and seventeen
aromatic amines (Set 4) respectively. The R2 value for
correlation of Set 2 and Set 4 are 0.817 and 0.91 respec-
tively.
Figure 1. Plot of calculated P.A Vs Experimental P.A and
P.A calculated by Wróblewski et al. for Set 1.
Figure 2. Plot of calculated P.A Vs experimental P.A for Set
2.
Copyright © 2011 SciRes. JQIS
S. K. RAJAK ET AL.
92
Figure 3. Plot of calculated P.A Vs experimental P.A for Set
3.
Figure 4. Plot of calculated PA Vs experimental P.A for Set
4.
5. Conclusions
In this work, we have presented a scientific model for the
evaluation of protonation energy of molecules in terms
of four quantum theoretical descriptors—the ionization
energy, the global softness, the electronegativity, and the
global electrophilicity index as components. As a basis
of scientific modeling, we have posited that these akin
theoretical descriptors describe the charge rearrangement
and polarization that occur during the physico-chemical
process of protonation. The test molecules chosen are of
diverse physico-chemical nature. A validity test of the
model is performed by comparing the protonation ener-
gies of as many as 43 molecules computed using the
ansatz proposed in this work vis-à-vis their correspond-
ing experimental counterparts. The close agreement be-
tween the theoretically evaluated and experimentally
determined PA’s , the proton affinities, strongly suggests
that the four component modeling in terms of quantum
chemical descriptors having link with the physico-
chemical process of protonation is efficacious and the
suggested ansatz for computing P.A of molecules and
hypothesis relied upon are scientifically acceptable.
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