Nanomaterials and Energy
Volume 3 Issue NME4
Equilibrium configuration of
(H2O)n, for n=1–3
Lamsal, Mishra and Ravindra
ice | science
Pages 129–138 http://dx.doi.org/10.1680/nme.14.00005
Research Article
Received 02/05/2014 Accepted 23/06/2014
Published online 27/06/2014
Keywords: ab initio/binding energy/density functional theory/
dimer/ground-state energy/hydrogen bond/trimer/water molecule
ICE Publishing: All rights reserved
Equilibrium configuration of
(H2O)n, for n = 1–3
Chiranjivi Lamsal
Nuggehalli M. Ravindra*
PhD Candidate, Department of Physics, New Jersey Institute of Technology,
Newark, NJ, USA
Professor, New Jersey Institute of Technology, Newark, NJ, USA
Central Department of Physics, Tribhuvan University, Kirtipur,
Kathmandu, Nepal
Devendra Raj Mishra
Professor, Central Department of Physics, Tribhuvan University, Kirtipur,
Kathmandu, Nepal
The equilibrium configurations of water molecule (H2O)n=1, its dimer (H2O)n=2 and trimer (H2O)n=3 have been studied in this
paper. The ionic character of the OHO hydrogen bond, formed between the electronegative oxygen atoms, in (H2O)n=2,
appears as a change in bond lengths in one of the water molecules while the bond lengths in the other molecule remain
same as in (H2O)n=1. The increased hydrogen bond strength, caused by the cooperative effect of three-body forces (H2O)n=3, as
compared to (H2O)n=2, results in the equilibrium structure of the water trimer to be much more rigid than that of the dimer.
An attempt has been made, from the energy standpoint, to understand the controversy on chemical formula of water as
H2O or H1·5O by studying the equilibrium configuration of (H2O) OH, which corresponds to 2H1·5O. The ground-state energy of
(H2O) OH is found to be lower than the sum of the ground-state energies of OH and H2O at infinite separation. Calculations
also show the ground-state energy of (H2O) OH to be lower than the sum of the energies of the constituent atoms at infinite
separation. This shows that the binding energy of (H2O) OH is positive, which indicates that (H2O) OH is relatively stable.
1.Introduction
The study of hydrogen bonding is of great significance as the
hydrogen bond plays a vital role in physiological phenomena
such as the contraction of muscle and the transmission of
impulses along nerves and in the brain.1 For a long molecule,
a conjugated system provides the only way of transmitting an
effect from one end to the other; and the hydrogen bond is the
only well-known strong and directed inter-molecular interaction
that can come into operation quickly. For a better understanding
of the hydrogen bonding interactions, the study of equilibrium
configuration of small water clusters2–4 has attracted a great
deal of attention. Along with the well-known density anomaly
of water at ~ 4°C (Figure 1), it has several other peculiar
properties ranging from physical to thermodynamics anomalies
as listed by Chaplin.5 Most of them are explained on the basis of
hydrogen bonding of water.5,6 However, it has not been possible
to obtain a satisfactory explanation of the causes of intriguing
anomalies and singularities that have been observed in different
phases of water.3,7 Researchers have always been interested in
implementing novel experimental and theoretical techniques to
understand the various kinds of interactions that are responsible
for the complex behaviour observed in apparently simple
molecule of water.8
Although water is a well-studied nanomaterial, its chemical formula
and structure continue to challenge scientists – we still do not have
the complete and simple explanation of the peculiar behaviour of
water. Various controversies, even on the structure of water, have
also emerged time and again. For instance, Wernet et al.11 argued
that the structure of water molecules was found to be bound into
linear chains and rings – an assertion against well-established
notion of tetrahedral coordination of water molecules. According to
Chatzidimitriou-Dreismann,12 neutrons and electrons colliding with
water molecule for just attoseconds, in a neutron Compton scattering
(NCS) experiment, see the hydrogen to oxygen ratio to be 1·5:1
instead of 2:1. Such claims, to date, are disputed to some extent.13–15
However, there are still some questions under invetigations that carry
‘both arguments’ side by side.16 Structure of water and formation of
hydrogen bonds in bulk water are one of the 100 outstanding unsolved
problems in science.17 In this context, the following statement of
Lawrence18 seems still relavant: ‘Water is H2O, hydrogen two parts,
and oxygen one. But there is also a third thing, that makes it water.
And no one knows what that is (I believe God knows)’.
Water clusters are formed due to grouping of water molecules
that have fewer chemical reactions with other water molecules as
compared to the molecules of the bulk. Small water clusters are
*Corresponding author e-mail address: [email protected]
129
Nanomaterials and Energy
Volume 3 Issue NME4
Equilibrium configuration of
(H2O)n, for n=1–3
Lamsal, Mishra and Ravindra
which is the first-order perturbation energy correction to the
unperturbed energy E (0) .
Density: Kg/m3
1000
990
980
970
960
0
–30
30
60
100
Temperature: °C
Figure 1. Variation of density of water with temperature9,10
particularly important because they offer a model for characterising
structural changes and bonding mechanism in transitioning from
isolated molecules to bulk phase environments.19 Furthermore, water
clusters help in explaining many anomalous properties of water.20
In this paper, we have presented the equilibrium configurations for
water molecule (H2O)n=1 and its dimer (H2O)n=2 and trimer (H2O)n=3
using Gaussian 98 set of programs, ‘Gaussian 98’.21
2.
Simulation approach
Ab initio calculations are becoming widely popular in studying the
electronic structures and various physical properties (e.g. groundstate energy, dipole moment, polarisability, vibrational frequencies
and nuclear quadrupole moment) of many-electron systems.22–24 The
first principles approaches can be classified into three main categories:
the Hartree–Fock approach (HF), the density functional theory
(DFT) approach and the quantum Monte–Carlo (QMC) approach,23
of which the first two methods are used in our calculations. The HF
self-consistent method is based on the one-electron approximation in
which the motion of each electron in the effective field of all the other
electrons is governed by a one-particle Schrödinger equation. In this
approximation, the ‘Hartree–Fock’ energy of many-electron system
having N-electrons can be written as follows:
1.
E ( 0 ) + E (1) = E ( HF )
where
2.
∧
E (0) = ψ (0) H0 ψ (0) = ∑ εi
i
is the expectation value of sum of the one-electron Fock operators
and is known as the lowest-energy eigenvalue of the unperturbed
system.
Similarly, the expectation value of first-order perturbed
Hamiltonian Hˆ ′ over the unperturbed state Ψ(0) of the system is
written as follows:
∧
3.
130
E (1) = ψ ( 0 ) H ′ ψ ( 0 )
The HF wave function satisfies the antisymmetry requirement and
it includes the correlation effects arising from the pairs of electrons
of the same spin. However, the motions of the electrons of opposite
spin remain uncorrelated in this approximation. The methods
beyond the HF approximation, which deal with the phenomenon
associated with many-electron system, are known as electron
correlation methods. One of the approaches to electron correlation
is the Møller–Plesset (MP) perturbation method that adds higher
excitations to the HF approximation as a non-iterative correction
utilising techniques from many-body perturbation theory.21,24 The
HF procedure, used in our work, utilises variational procedure called
Hartree–Fock–Roothaan approach.25 However, MP calculations
are not variational and can produce an energy value below the true
energy.22 MP second-order perturbation theory (MP2) considers the
correction up to second order as follows:
4.
E = E ( 0 ) + λ E (1) + λ 2 E ( 2 )
where λ is the expansion parameter and
∧
5.
E (2) = ∑
t
2
ψ (0 ) H′ ψ t
Et − E ( 0 )
is the first-order perturbation to the HF energy.
Another first principles approach to calculate the electronic
structure of many-electron systems is DFT. In this theory, exchangecorrelation energy is expressed, at least formally, as a functional of
the resulting electron density distribution, and the electronic states
are solved self-consistently as in the HF approximation.23 In the HF
approximation, the exchange interaction is treated exactly but the
dynamic correlation, arising due to Coulomb repulsion, between
the electrons is neglected. The DFT, in principle, is exact but, in
practice, both exchange and dynamic correlation effects are treated
approximately.26
Perturbation theory may also be used to calculate the expectation
value of static electric dipole moment, -er , in a stationary state of
the one-electron atom.27 In the lowest approximation, the dipole
moment is expressed as follows:
2
6.
p 0 = − e rnn = − e∫ r Ψn (0) (r ) d 3r
This is called the permanent electric dipole moment of the system
because it represents a vector defined by the unperturbed state of
the system. It vanishes for all states that possess definite parity.
Considering an electron bound in an atom and placed in a weak,
uniform, constant electric field E, the dipole moment of the oneelectron atom can be expressed as follows:
Nanomaterials and Energy
Volume 3 Issue NME4
7.
Equilibrium configuration of
(H2O)n, for n=1–3
Lamsal, Mishra and Ravindra
p = -e∫ ρ r d 3r = p0 − e2 ∑
k≠n
rnk rkn + rkn rnk
E n (0 ) − E k (0 )
⋅E
where ∫ Ψn (0)*r Ψk (0)d 3r = rnk and ∫ Ψk (0)*r Ψn (0)d 3r = rkn . The last term
of Equation 7 represents the induced dipole moment in the state n
and can be written as follows:
8.
p1 = − e2 ∑
rnk rkn + rknrnk
E n (0 ) − E k (0 )
k≠n
⋅E = α ⋅E
where α is the polarisability for the state n given by:
9.
α = e2 ∑
k≠n
rnk rkn + rknrnk
E k (0 ) − E n (0 )
and is symmetric tensor of second rank. For many instances,
the polarisability is scalar. That is, α xy = α yz = α zx = 0, and
α xx = α yy = α zz . The polarisability of a molecule is responsible
for the London–Van der Waals interaction energy or dispersion
energy that results from the instantaneous correlation between the
fluctuating dipole moments, due to the motion of electrons.28
The first principles methods (i.e. HF, HF plus MP2 and DFT)
discussed above can be implemented with the aid of the Gaussian
98 set of programs.21 The ‘Gaussian’ input file for optimising
the geometry and calculating the energy of molecules needs the
specification of basis sets to be used. The choice of a set of basis
functions χ α used to express the molecular orbitals Φ i can be
written as follows:
10.
Φ i ( x ) = ∑ ciα χ α ( x )
The basis functions, in both atomic and molecular calculations, are
of the form given by:
11.
χ n l m m s ( x ) = r n −1e − ζnl rYl m (θ, φ) α or r n −1e − ζnl rYl m (θ, φ) β
The space part of Equation 11 is known as a Slater Type Orbital
(STO). In order to speed up molecular integral evaluation, STOs
for the atomic orbitals, in a linear combination of atomic orbitals
(LCAO) wave function, can be substituted by Gaussian-Type
Functions (GTFs)
2
12.
gijk = Nxai yaj zak e − α ra
where, i, j and k and are the nonnegative integers, α is a positive
orbital exponent determining the size (radial extent) of the function
and xa, ya, za are Cartesian coordinates with origin at the nucleus
‘a’. The Cartesian–Gaussian normalisation constant, N, is given by:
3
13.
1
i+ j+k
i!j !k !  2
 2α  4  (8α )
N=  

 π   (2i )!(2 j )!(2 k )! 


Because of the reduction in the computational time, given by
Gaussians in multicentre-integral evaluation, contracted Gaussian
basis sets can be used:
14.
χα = ∑ diα gi
i
where the contraction coefficients di α are the fixed constants
within a given basis set and the gi’s are the normalised Cartesian
Gaussians (Equation 12) centred on the same atom and having the
same i, j, k values as one another, but different α’s. In Equation 14,
χα is known as a contracted Gaussian-type function (CGTF) and
the gi’s are called primitive Gaussians. Several methods exist to
form contracted Gaussian sets. One of the ways to form a minimal
(minimum) CGTF set is to fit a linear combination of N Gaussian
functions to a STO per atomic orbital, where the coefficients in the
linear combination and the Gaussian orbital exponents are chosen
to yield the best least-squares fit. Most commonly, N=3 is chosen,
which gives a set of CGTFs called STO-3G.
A basis set can be made larger by increasing the number of basis
functions per atom. Split-valence basis sets of CGTFs, such as
3-21G, 6-31G and 6-311G, have two or more sizes of contracted
basis functions for each valence orbital. In the 3-21G set, each
inner-shell atomic orbital (1s for Li-Ne; 1s, 2s, 2px, 2py, 2pz for
Na-Ar and so on) is represented by a single CGTF that is a linear
combination of three primitive Gaussians; for each valence-shell
atomic orbital (1s for H; 2s and the 2p’s for Li-Ne;…;4s and the
4p’s for K, Ca, Ga-Kr; 4s, the 4p’s, and the five 3d’s for Sc-Zn),
there are two basis functions, one of which is a CGTF that is a
linear combination of two Gaussian primitives and one which is
a single diffuse function. The 6-31G set uses six primitives in
each inner-shell CGTF and represents each valence-shell atomic
orbital by one CGTF with three primitives and one Gaussian
with one primitive while 6-311G set uses another additional
Gaussian with single primitive in each valence-shell atomic
orbital.
Atomic orbitals are distorted in shape and have their centres of
charge shifted upon molecule formation. Split valence basis sets
allow orbitals to change size, but not to change shape and, for this
polarisation to allow, we add basis-function STOs whose l quantum
numbers are greater than the maximum l of the valence shell of
ground-state atom. Any such basis set is a polarised basis set. For
example, polarised basis sets add d functions to oxygen atoms and
f functions to transition metals, and some of them add p functions
to hydrogen atoms. The 6-31G* basis set (defined for the atoms H
through Zn) is a valence double-zeta polarised basis set that adds to
the 6-31G set six d-type Cartesian–Gaussian polarisation functions
on each of the atoms Li through Ca and ten f-type Cartesian
Gaussian polarisation functions on each of the atoms Sc through
Zn. This basis set is also known as 6-31G (d). Another popular
polarised basis set is 6-31G**, also known as 6-31G (d, p), which
adds p functions to hydrogen atoms in addition to the d functions on
131
Nanomaterials and Energy
Volume 3 Issue NME4
Equilibrium configuration of
(H2O)n, for n=1–3
Lamsal, Mishra and Ravindra
heavy atoms. Other polarised basis sets are STO-3G*, STO-3G**,
3-21G*, 3-21G**, 6-311G*, 6-311G** and so on.21
molecule was found to be -76·235 Hartree (Ha). The calculated
and experimentally determined configuration of water molecule
resembles very close to each other.22
3.
Results and discussion
The main results of the present work can be summarised as follows.
We have calculated (a) the ground-state energy and the equilibrium
geometry of water molecule, and its dimer and trimer; (b) the
dipole moment of water molecule, in the HF and HF including MP
perturbation (HF plus MP2) levels of approximation. Calculations
have also been performed to obtain (c) the ground-state energy of
water-like molecules such as (H2O) OH, H2O2 and (H2O) OH- in
different levels (i.e. HF, HF plus MP2 and DFT) of approximation.
These calculations have been carried out using the Gaussian 98
set of programs. The effect of MP perturbation is to lower the
ground-state energy, due to the correlation between the motions of
the electrons within a molecular system. The difference between
the HF plus MP2 energy and the HF energy provides an estimate
for the contribution of many-body interactions to the ground-state
energy of the molecular system.
Figure 3 shows the equilibrium configuration of (H2O)n=2 obtained
in the HF plus MP2 level of approximation using the basis set
6-311G* and variation of the equilibrium energy of water dimer
in its ground state with respect to the O1-O2 distance, the bond
angles (H3-O1-H4 and H6-O2-H5) and the dihedral angle H3-O1O2-H5. On varying the distance (d) between two oxygen atoms
O1 and O2 from 2·4 to 5 Å, the change in the energy of (H2O)n=2
is of the order of 1·3 × 10–2 a.u. with energy minimum occurring
at around d = 2·85 Å. It is seen from Figure 3(c) that the variation
of bond angles (H3-O1-H4 and H5-O2-H6) from 90° to 125° will
be accompanied by energy change of the order of 10–2 a.u., with
energy minimum observed at around the angle of 107°. Similarly,
the variation of dihedral angle H5-O2-O1-H3 from 70° to 175°
causes the energy change within even smaller range, which is of
O
Figure 2 shows the equilibrium configuration of (H2O)n=1 obtained
in HF plus MP2 level of approximation using the basis set 6-311G*
and variation of equilibrium energy of water molecule in its ground
state with respect to bond length and bond angle. With this analysis,
we have estimated the values of the bond length, the bond angle
and the ground-state energy for (H2O)n=1 as shown in Table 1. As
can be seen from Table 1, a molecule of water (i.e. (H2O)n=1) has
the structure of the form of an isosceles triangle with side 0·957 Å
and vertex angle of 106·7°. The ground-state energy of water
132
H
H
A water molecule
(a)
–76·322
–76·16
–76·233
Energy: Ha
Energy: Ha
In order to optimise the geometry or minimise the energy of a
molecule, we find a local minimum in the neighbourhood of the
initially assumed geometry with the use of different basis sets in the
HF- and HF plus MP2-level calculations and then repeat the local
minimum search procedure so as to locate the global minimum on
the potential energy surface. To be sure that a minimum has been
found and not a saddle point, a frequency calculation is done at the
geometry found; one way to avoid getting a saddle point instead of
a minimum is to eliminate all symmetry in the starting geometry.
For a true minimum, all 3N-6 calculated vibrational frequencies
would be real; for instance, 3, 12, 21 real frequencies for (H2O)n
with n = 1, 2 and 3, respectively. For an nth-order saddle-point, the
structure would have n imaginary frequencies. The basis sets used
in the variational procedure are based on the Gaussian functions.
On increasing the size of basis sets (i.e. STO-3G, 3-21G, 6-31G
and 6-311G), the HF value of the ground-state energy of the
equilibrium geometry gets lowered. Lowering of the HF energy
value of the equilibrium geometry is also seen by adding the
polarisation functions in the basis sets. The effect of increasing the
size and polarisation functions to the basis sets is also noticed for
the equilibrium geometry. However, the consistency of the results
obtained has been tested by studying their convergence with respect
to the use of basis sets of different size and complexity.29
–76·234
–76·235
95
–76·18
–76·20
–76·22
–76·24
100 105 110 115
Bond angle: °
(b)
0·7
1·1
1·3
0·9
Bond Length: Å
(c)
Figure 2. (a) Equilibrium configuration of (H2O)n=1; and variation of
ground-state energy of water molecule with (b) bond angle (H-O-H)
at the bond length (O-H) of 0·9567Å and (c) bond length (O-H) at the
bond angle (H-O-H) of 106·70°
Estimated
value
Experimental
value22
Bond length: Å
0·957
0·958
Bond angle:°
106·7
104·5
−76·235
−76·480
Parameter
Energy: Ha
Table 1. Optimised parameters for (H2O)n=1
Nanomaterials and Energy
Volume 3 Issue NME4
Equilibrium configuration of
(H2O)n, for n=1–3
Lamsal, Mishra and Ravindra
B
O2
H6
H4
01-02
H5
–152·472
Energy: Ha
H3
O1
–152·478
–152·484
A
1
2
3
A water dimer
Energy: Ha
Energy: Ha
–152·4816
H3-O1-H4
H6-O2-H5
–152·470
–152·475
–152·480
–152·485
5
(b)
(a)
–152·465
4
Distance: Å
H3-O1-O2-H5
–152·4820
–1524824
–152·4828
90
105
120
Bond angle: °
135
70
(c)
105
140
175
Dihedral angle: °
(d)
Figure 3. (a) Equilibrium configuration of (H2O)n=2; and variation of
ground-state energy of water dimer with (b) distance between two
oxygen atoms O1-O2, (c) bond angles (H3-O1-H4 and H6-O2-H5) and
(d) dihedral angle H3-O1-O2-H5
the order of 10–3 a.u. and the minimum in energy occurs at the
angle of 122·03°. These results lead us to a conclusion that the
variation in energy with the dihedral angle is not as sensitive as
the variation in energy with the distance between two oxygen
atoms and the bond angle. With this analysis, we have estimated
the values of the inter-atomic distances, the bond angles, the
dihedral angles and the ground-state energy for the equilibrium
configuration of (H2O)n=2 as shown in Table 2. As can be seen from
Table 2, the intra-molecular geometry (i.e. bond distances and
bond angles in the water molecules A or B) of (H2O)n=2 is similar to
that of (H2O)n=1 and the corresponding values of the bond distances
and the bond angles do not differ by more than 1%. However, it
should be remarked that the bond length O2-H6 in (H2O)n=2 (where
the hydrogen atom H6 of molecule B is close to the oxygen atom
O1 of molecule A as compared to the other hydrogen atoms in
molecule B) is slightly stretched, with a slight decrease in the bond
length O2-H5 as compared to the corresponding bond lengths in
(H2O)n=1. This stretching of O-H bond can be attributed to the
small ionic character of the OHO hydrogen bond formed between
the electronegative oxygen atoms O1 and O2 of (H2O)n=1; with
the O-H-O distance of 2·85 Å, the equilibrium position of the
hydrogen atom H6 of 0·96 Å from the oxygen atom labelled O2
and 1·89 Å from the oxygen atom O1. These bond distances are
typical of the OHO hydrogen bonds. Our calculated value of the
distance (d) between two oxygen atoms, O1 and O2, is around
2·85 Å, which agrees well with the previously reported values of d
equal to 2·95 Å3 and 2·94 Å4, respectively, within 5%.
Intra-molecular geometry:
Bond length: Å
O1-H3
0·9578
O1-H4
0·9578
O2-H6
0·9626
O2-H5
0·9555
Bond angle: °
H3-O1-H4
107·34
H5-O2-H6
106·81
Inter-molecular geometry:
Inter-atomic distance: Å
O1-O2
2·8498
O1-H6
1·8909
H3-H6
2·3846
Dihedral angle: °
H3-O1-O2-H5
122·03
H4-O1-O2-H5
−122·03
Energy:
Ground-state energy: Ha
HF value
−152·0751
HF plus MP2 value
−152·4828
Table 2. Optimised parameters for (H2O)n=2
133
Nanomaterials and Energy
Volume 3 Issue NME4
Equilibrium configuration of
(H2O)n, for n=1–3
Lamsal, Mishra and Ravindra
Figure 4 shows the equilibrium configuration of (H2O)n=3 obtained
in the HF plus MP2 level of approximation using the basis set
6-311G* and variation of the equilibrium energy of water trimer in
its ground state with respect to the O-O distances, the bond angles
and the dihedral angles. It can be seen from Figure 4(b) that O1-O2
and O1-O3 variations are very close to each other throughout, with a
slight deviation from O2-O3 variation in the long-distance range. It
should be noted here that the lowest-energy structure of water trimer
consists of strong non-linear hydrogen bonds30 between three pairs
of oxygen and hydrogen atoms; the remaining three hydrogen atoms
lie above and below the plane containing the three oxygen atoms.
This small deviation can be understood as an effect of triggering
the non-linear bond, within the inherent ‘asymmetry’ of water
trimer. With this analysis, we have estimated the intra-molecular
geometry, inter-molecular geometry and the ground-state energy
for (H2O)n=3 as shown in Table 3. It is seen from this Table 3 that
our calculated value of the average O-O distance (d) in (H2O)n=3
is around 2·74 Å, which agrees well with the previously reported
value of 2·80 Å3 within 2·2%. It is also noticed that the value of d in
(H2O)n=3 is significantly shorter than in (H2O)n=2 (where the value of
d is equal to 2·85 Å). This shortening of the distance between two
oxygen atoms in (H2O)n=3 can largely be attributed to the increased
hydrogen bond strength caused by the cooperative effect of threebody forces.3 It is seen from Figure 4(a) that the water trimer has a
cyclic equilibrium structure with each water monomer acting as a
single hydrogen bond donor and acceptor. The equilibrium structure
of the water trimer is much more rigid than that of the dimer.
In order to estimate the magnitude of the dipole moment for
(H2O)n=1, we used the 6-311G* basis set in HF plus MP2 level
of calculation and found it to be 2·3 debye, which is greater, by
around 20%, than the previously reported value of 1·9 debye.31 As
the use of basis set of higher flexibility is usually expected to be a
better approximation, we have used double star in the basis sets,
which indicates the inclusion of p functions to hydrogen atoms in
addition to the d functions on oxygen atom. Values of the dipole
moment for (H2O)n=1, obtained with the HF and HF plus MP2
level of calculations using 6-311G**, are 2·14 and 2·19 debye,
respectively, as shown in Table 4. However, these values differ
from the value reported in the literature31 by more than 10%. With
these observations, calculations are performed based on the DFT;
since the functional of Becke32 and Lee–Yang–Parr33 is used, it is
also called BLYP calculations.
It is seen from Table 4 that the values of the bond length and the
bond angle for (H2O)n=1, obtained with the density functional
(BLYP) calculations, performed using the basis set 6-311G
and its single- and double-starred counterparts, agree with the
corresponding HF plus MP2 values within 3%. Furthermore,
the values of the ground-state energy, obtained with the density
functional (BLYP) calculations, get lowered by around 0·5% as
compared to the corresponding HF plus MP2 values. The value
of the ground-state energy, obtained with the density functional
(BLYP) calculations performed using the basis set 6-311G**,
is −76·43 a.u., which is close to the experimentally observed
H5
–228·64
Energy: Ha
C
B
H9
O2
–228·76
H8
A water trimer
(a)
–228·742
–228·735
H7-O2-H4-O3
–228·743
–228·744
4
6
Distance: Å
90
8
–228·738
–228·741
96 108 120 132
Dihedral angle: °
(d)
Figure 4. (a) Equilibrium configuration of (H2O)n=3; and variation
of ground-state energy of water trimer with (b) distance between
two oxygen atoms (O1-O2, O2-O3 and O3-O1), (c) bond angles
105 120 135
Bond angle: °
(c)
H8-O3-O2-O1
–228·744
–228·745
134
2
(b)
Energy: Ha
Energy: Ha
–228·72
–228·740
–228·745
H7
O3
–228·68
–228·730
Energy: Ha
H6
H4
H5-O1-H4
H6-O2-H7
H9-O3-H8
–228·735
O1-O2
O2-O3
O3-O1
Energy: Ha
A
O1
H9-H8-O2-O1
–228·735
–228·740
–228·745
255 270 180 190
Dihedral angle: °
(e)
255
270 285 300
Dihedral angle: °
(f )
(H5-O1-H4, H6-O2-H7 and H9-O3-H8), (d) dihedral angle H7-O2H4-O3, (e) dihedral angle H8-O3-O2-O1 and (f) dihedral angle
H9-H8-O2-O1
Nanomaterials and Energy
Volume 3 Issue NME4
Equilibrium configuration of
(H2O)n, for n=1–3
Lamsal, Mishra and Ravindra
value of −76·48 a.u..22 The correlation energy,34 the difference
between the MP2 or BLYP values of the ground-state energy
and the corresponding HF value, of water molecule obtained
with the density functional (BLYP) calculations performed
using the basis set 6-311G** is −0·38 a.u., which agrees well
with the previously reported value of -0·37a.u.24 within around
3%. The BLYP calculations give the value of the dipole moment
for (H2O)n=1 to be 2·0 debye which is close to the previously
reported value of 1·9 debye within around 5%.
We have also studied the equilibrium configurations of waterrelated materials (H2O) OH, H2O2 and (H2O) OH-. Motivation
behind this extension was ‘physics news update:’12 ‘A WATER
MOLECULE’S CHEMICAL FORMULA IS REALLY NOT
H2O…. … neutrons and electrons colliding with water for just
attoseconds will see a ratio of hydrogen to oxygen of roughly 1·5
to 1… The story begins in 1995. At the ISIS neutron spallation
facility in the UK, a German-British collaboration collided
epithermal neutrons (those with energies of up to a few hundred
electron volts) with a target that included water molecules.35
Detecting the number and energy loss of the scattered neutrons
in the resulting attosecond-scale collisions, the researchers
noticed that the neutrons were scattering from 25% fewer
protons than expected. Apparently, the protons in hydrogen were
sometimes “invisible” to the neutron probes. While the exact
details are still being debated by theorists, the researchers’ own
theoretical considerations suggest the presence of short-lived
(sub-femtosecond) entanglement, in which protons in adjacent
hydrogen atoms (and possibly the surrounding electrons) are all
interlinked in such a way as to change the nature of the scattering
results…’.
The equilibrium configuration of (H2O) OH (which corresponds
to 2H1·5O), H2O2 and (H2O) OH- have been estimated using the
basis set 6-311G** in different levels (i.e. HF, HF plus MP2
and BLYP) of approximation. Figure 5 shows the equilibrium
configuration determined by the density functional (BLYP)
calculations with the basis set 6-311G**. Table 5 shows the
ground-state energy of (H2O) OH, H2O2, and (H2O) OH- along
with that of H, O, OH and OH- in the three different levels of
approximation.
The above energy calculations for (H2O) OH, H2O2 and (H2O) OH-,
with the corresponding values of the ground-state energy, can be
expressed in the following form:
15.
H 2O2
H 2O2
16.
+
H
→
(H2 O) OH
( − 151 ⋅ 5740 a.u.) ( − 0 ⋅ 4976 a.u.) ( − 152 ⋅ 1822 a.u.)
+
OH
→ (H 2 O ) OH
( − 76 ⋅ 4285 a.u.) ( − 75 ⋅ 7449 a.u.) ( − 152 ⋅ 2499 a.u.)
H 2O2
17.
+
OH
−
→ (H 2 O ) OH −
( − 76 ⋅ 4285 a.u.) ( − 75 ⋅ 7449 a.u.) ( − 152 ⋅ 2499 a.u.)
Equations 15, 16 and 17 can also be written as follows:
18.
E [H 2 O2 ] + E [H ] – E [(H 2 O) OH ] = 3·0096 eV
19.
E [H 2 O ] + E [OH ] – E [(H 2 O) OH ] = 0·3755 eV
Intra-molecular geometry:
Bond length: Å
O1-H4
0·9703
O1-H5
0·9560
O2-H6
0·9707
O2-H7
0·9564
O3-H8
0·9709
O3-H9
0·9562
Bond angle: °
H4-O1-H5
108·17
H6-O2-H7
107·73
H8-O3-H9
107·79
Inter-molecular geometry:
Inter-atomic distance: Å
O1-O2
2·7294
O2-O3
2·7372
O3-O1
2·7466
O1-H6
1·8441
O2-H8
1·8501
O3-H4
1·8693
O1-H7
3·2524
O3-H5
3·3186
O2-H9
3·2755
Dihedral angle: °
H7-O2-H4-O3
106·88
H8-O3-O2-O1
173·19
H9-H8-O2-O1
280·09
Energy:
Ground-state energy: Ha
HF value
−228·1281
HF plus MP2 value
−228·7454
Table 3. Optimised parameters for (H2O)n = 3
135
Nanomaterials and Energy
Volume 3 Issue NME4
Bond
length: Å
Bond
angle: °
Energy: a.u.
Dipole
moment: debye
HF
0·9455
111·94
−76·0110
2·487
HF+MP2
0·.9687
109·99
−76·1506
2·527
DFT (BLYP)
0·9833
107·67
−76·3986
2·382
HF
0·9395
107·50
−76·0324
2·318
HF+MP2
0·9567
106·71
−76·2350
2·341
DFT (BLYP)
0·9734
105·07
−76·4154
2·219
HF
0·9412
105·34
−76·0470
2·140
HF+MP2
0·9576
102·47
−76·2640
2·196
DFT (BLYP)
0·9725
102·98
−76·4285
2·027
Basis sets used
Level of calculation
6-311G
6-311G*
6-311G**
Equilibrium configuration of
(H2O)n, for n=1–3
Lamsal, Mishra and Ravindra
Table 4. Equilibrium geometry, ground-state energy and dipole moment of water molecule
H5
20. E H 2O + E OH −  − E (H 2 O) OH −  = 2 ⋅ 0817 eV
H3
where E’s are the ground-state energies.
H5
O1
H4
H4
H3
O2
O3
(H2O) OH–
(c)
H2O2
(b)
(H2O) OH
(a)
E [(H 2 O) OH ] < E [H 2 O ] + E [OH ] < E [H 2 O2 ] + E [H ]
Figure 5. Equilibrium configuration of (a) (H2O) OH, (b) H2O2 and
(c) (H2O) OH−
and
E (H 2 O) OH −  < E H 2O + E OH − 
From the above analysis, it is seen that the binding energy of
(H2O) OH is smaller than that of (H2O) OH-. However, it is
clearly seen from Equation 19 that the ground-state energy of
(H2O) OH is lower than the sum of the ground-state energies of
OH and H2O at infinite separation. Calculations also show the
ground-state energy of (H2O) OH to be −152·1822 a.u., which is
below the sum of the energies of the constituent atoms at infinite
separation (i.e. −151·6000 a.u.). This shows that the binding
energy of (H2O) OH is positive, which indicates that (H2O) OH
is relatively stable.
E (Ha) in different levels of approximation
with 6-311G**
HF
HF plus MP2
−0·4998
−0·4998
−0·4976
O STriplet
−74·8052
−74·9181
−75·0734
OH group
−75·4105
−75·5729
−75·7399
OH– group
−75·3612
−75·5736
−75·7449
H2O
−76·0470
−76·2640
−76·4285
(H2O) OH
−151·4669
−151·8492
−152·1822
H2O2
−150·8170
−151·2314
−151·5740
−151·4602
−151·9081
−152·2499
H
(H2O) OH
−
4.Conclusions
The equilibrium configurations of water molecule (H2O)n=1,
its dimer (H2O)n=2 and trimer (H2O)n=3 have been studied
in this paper. The calculations have been performed using
‘Guassian 98’, a simulation package which utilises basis sets of
different size and complexity in different levels (i.e. HF, HF plus
MP2 and DFT) of approximation. The basis sets, used in the
procedure, are based on the Gaussian functions. On increasing
the size of basis sets (i.e. STO-3G, 3-21G, 6-31G and 6-311G),
the HF value of the ground-state energy of the equilibrium
136
O1
H4
It is seen from Equations 18, 19 and 20 that
H2
O1
O2
DFT: BLYP
Table 5. The ground-state energy (E) of H2O and the water-like
molecules
geometry gets lowered. Lowering of the HF energy value of the
equilibrium geometry is also seen by adding the polarisation
functions in the basis sets. The effect of increasing the size and
polarisation functions on the basis sets is also noticed for the
equilibrium geometry. The influence of MP perturbation is to
lower the ground-state energy.
Nanomaterials and Energy
Volume 3 Issue NME4
Equilibrium configuration of
(H2O)n, for n=1–3
Lamsal, Mishra and Ravindra
Global minimum search procedures as well as variation in
bond lengths, bond angles and dihedral angles were used to
determine the ground-state energies and equilibrium geometries.
Correlation energy, defined as the difference between the BLYP
values of the ground-state energy and the corresponding HF
value, of water molecule is −0·38 a.u. and the dipole moment
of water molecule is found to be 2·0 debye. Stretching of
O-H bond of first water molecule near to the other molecule
in water dimer as well as shortening of O-O distance in cyclic
‘equilateral’ structure of (H2O)n=3, as compared to (H2O)n=2, is
due to the presence of OHO hydrogen bonds. From the study
of equilibrium configuration of (H2O) OH, which corresponds
to 2H1·5O,12 the ground-state energy is found to be lower than
the sum of the ground-state energies of OH and H2O at infinite
separation. Calculations also show the ground-state energy
of (H2O) OH to be lower than the sum of the energies of the
constituent atoms at infinite separation. This shows that the
binding energy of (H2O) OH is positive, which indicates that
(H2O) OH is relatively stable.
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