Emergence of the Coherent Structure of Liquid Water

Abstract

We examine in some detail the interaction of water molecules with the radiative electromagnetic field and find the existence of phase transitions from the vapor phase to a condensed phase where all molecules oscillate in unison, in tune with a self-trapped electromagnetic field within extended mesoscopic space regions (Coherence Domains). The properties of such a condensed phase are examined and found to be compatible with the phenomenological properties of liquid water. In particular, the observed value of critical density is calculated with good accuracy.

Full text

Water 2012, 4, 510-532; doi:10.3390/w4030510
water
ISSN 2073-4441
www.mdpi.com/journal/water
Article
Emergence of the Coherent Structure of Liquid Water
Ivan Bono 1, Emilio Del Giudice 2, Luca Gamberale 3 and Marc Henry 4,*
1 Consultant via Malta 24, Torino 10141, Italy; E-Mail: ivanbono.mail@libero.it
2 Consultant via Friuli 21, Milano 20135, Italy; E-Mail: emilio.delgiudice@mi.infn.it
3 Consultant via Mauro Macchi 26, Milano 20124, Italy; E-Mail: luca.gamberale@gmail.com
4 Laboratoire de chimie moléculaire de l’état solide, Université de Strasbourg, UMR 7140
UdS/CNRS, 4 rue Blaise Pascal, CS 90032, Strasbourg Cedex 67081, France
* Author to whom correspondence should be addressed; E-Mail: henry@unistra.fr;
Tel.: +39-02-5031-7734; Fax: +39-02-5031-7208.
Received: 29 May 2012; in revised form: 29 June 2012 / Accepted: 2 July 2012 /
Published: 9 July 2012
Abstract: We examine in some detail the interaction of water molecules with the radiative
electromagnetic field and find the existence of phase transitions from the vapor phase to a
condensed phase where all molecules oscillate in unison, in tune with a self-trapped
electromagnetic field within extended mesoscopic space regions (Coherence Domains).
The properties of such a condensed phase are examined and found to be compatible with
the phenomenological properties of liquid water. In particular, the observed value of
critical density is calculated with good accuracy.
Keywords: liquid phase; quantum electrodynamics; coherence; liquid water; critical density
1. Introduction
In the last two decades a conceptual change has occurred in the physics of condensed matter. For a
long time condensed matter has been conceived as an ensemble of atoms/molecules kept together by
static short-range forces. Liquid water has been no exception to this trend. In references [1,2] much
material has been presented to describe the formation of a network of water molecules tied by the
bindings named H-bonds, which should emerge from highly directional protuberances protruding from
the molecule electron clouds. However, a conceptual difficulty emerges here, since, in order to fit
experimental data [3], H-bonds should be conceived as short-lived; the lifetime varies with
OPEN ACCESS

Water 2012, 4 511
temperature from 20 ps at 250 K to 2 ps at 300 K. How is it possible to treat in a static approximation
an electric charge structure (the H-bond) which lives so short a time? An electromagnetic (EM) field
having at least the same time of oscillation should be necessarily introduced. Therefore the physical
system of water molecules tied together would demand necessarily, in the framework of present
knowledge, the explicit consideration of the interaction between water molecules and a radiative EM
field. However when this problem is addressed in the theoretical framework of Quantum
Electrodynamics (QED), a configuration of the system molecules + field emerges different from the
usual network of molecules kept together by forces. In the new configuration, molecules oscillate in
unison between two single-particle states, in tune with a non-vanishing EM field trapped in the
ensemble of molecules. In this new state, termed coherent, the relevant physical variable is phase
φ
.
QED connects
φ
to the EM potentials through the equations (in SI units)
·
2
·
2
h
Ae
hd
Vedt
φ
π
φ
π
=∇
=−


(1)
In the above equation, Ā is the magnetic vector potential, V is the electric potential, h is Planck’s
constant and e is the electron electric charge. Consequently, the phase correlations within the ensemble
of coherent molecules are kept not by the EM fields but by their potentials, which, by the way,
propagate in space at the phase velocity, which, as is well-known, could be larger than c. In Reference [4] a
detailed mathematical theory, based on QED, is presented, discussing the appearance of coherent
solutions in the system water molecules + electromagnetic field, which give rise to the liquid.
On the other hand, the presence of coherence in nature has been recently revealed within living
organisms. For instance, an article [5] appeared in Nature showing that within photosynthetic systems
molecules are “wired together” in a coherent way. We quote verbatim from the abstract
of Reference [5]:
“Intriguingly, recent work has documented that light-absorbing molecules in some photosynthetic
proteins capture and transfer energy according to quantum-mechanical probability laws instead of
classical laws at temperatures up to 180 K. This contrasts with the long-held view that long-range
quantum coherence between molecules cannot be sustained in complex biological systems, even at low
temperatures. Here, we present two-dimensional photon echo spectroscopy measurements on two
evolutionarily-related light harvesting proteins isolated from marine cryptophyte algae, which reveal
exceptionally long-lasting excitation oscillations with distinct correlations and anti-correlations, even
at ambient temperature. These observations provide compelling evidence for quantum coherent sharing
of electronic excitation across the 5-nm-wide proteins under biologically relevant conditions,
suggesting that distant molecules within the photosynthetic proteins are ‘wired’ together by quantum
coherence for more efficient light-harvesting in cryptophyte marine algae.”
Therefore, coherent correlations span at least 5 nm. The correlated molecules, as all biomolecules,
are of course immersed in water, which accounts for the large majority (70% of mass and 99% of
molar concentration) of the components of living organisms. Therefore, the coherent correlations
among biomolecules should be propagated by water. However, G. Scholes et al. [6] produced a study,
based on Molecular Dynamics, showing that EM correlations within a liquid obeying the assumptions

Water 2012, 4 512
usually made for water, fall off within a distance smaller than 2 nm. Consequently, we should
conclude that the usual picture of liquid water is unable to account for the observations of the coherent
structure of living organisms.
Additional evidence of the fact that biomolecules are phase-correlated is provided by
thermodynamics. The Second Law of Thermodynamics prescribes that the yield
η
= W/Q of a thermal
engine, where Q is the heat absorbed at the temperature T and W is the performed work, should not
exceed the quantity ∆T/T where ∆T is the temperature decrease of the energy flow within the engine.
In a typical living organism at room temperature
η
cannot exceed a few percent, whereas
bio-electrochemical evidence of the membrane processes gives a value of about 70% [7]. To avoid a
thermodynamic inconsistency, we are forced to conclude that a living organism is not a thermal
engine, so that its component molecules are not independent but are phase-correlated; actually, the
ensemble of coherent molecules is able to receive energy as free energy, since its entropy is vanishing.
The above statement is corroborated by the result quoted in the literature [8] that a muscle is not a
thermal engine but a quantum machine. In conclusion, we should accept that liquid water, at least
within living organisms, should exist in a coherent state, supporting the suggestion coming from
QED analysis.
In the following we will present a description of the QED results in more physical terms than in
Reference [4], giving a physical intuition of the process of the emergence of coherence. The role of
water coherence in biological organisms has been discussed in Reference [9].
2. Emergence of Coherence
In this section we will try to provide a physical intuition of the emergence of coherence within a
composite quantum system, such as an ensemble of atoms/molecules. According to first principles, a
quantum system (either particle or field) cannot but fluctuate; in particular vacuum also fluctuates, as
shown by the phenomenon of the Lamb shift of the energy levels of hydrogen atoms. These levels
exhibit energy values slightly below (about 1 ppm) the ones calculated by quantum mechanics, which
is based on the assumption that the Coulomb field that connects the proton and the electron is classical
(non-fluctuating). The above difference between measured and calculated values of the hydrogen
energy levels, first detected by Lamb in 1947 [10], is the evidence that actually a fluctuating field
should be present, just the one produced by the quantum fluctuations of the vacuum oscillators.
Therefore, according to quantum physics, we should always assume that molecules are coupled to a
quantized EM field.
In order to give to the reader a qualitative understanding of the origin of the collective interaction
among molecules induced by their interaction with the EM field, we give a simple scale argument. The
size of a molecule is about a few Angstroms (Å); in the case of water slightly more than 1 Å. The
typical energy difference Eexc between two levels of an atom/molecule is in the order of some eV
(say 10 eV); the photon supplying this energy could be extracted from the environment, at the very
least from the quantum fluctuations of the vacuum, and its size (we mean by size the extent of the
region where the photon can be located) is of course its wavelength
λ
= hc/Eexc
which for Eexc = 10 eV
gives
λ
= 1200 Å. We find, therefore, the surprising result that the tool able to change the internal
structure of the molecule is about one thousand times more extended than the molecule itself. At the

Water 2012, 4 513
usual densities of gases on Earth (in the case of water vapor at the boiling point 2 × 1019 molecules⋅cm3)
one single photon able to affect the molecular structure would include within its own volume some
20,000 molecules! Hence the collectivizing feature of the interaction between molecules and
EM fields.
Let us describe now how this process of interaction could occur. A fluctuation of the EM mode
having the frequency corresponding to the energy jump among molecule levels emerges either from
the quantum vacuum or from the thermal bath. This fluctuation involves simultaneously all
the molecules present in the volume
λ
3 spanned by its wavelength
λ
, in the case of water some
20,000 molecules, as candidates to be excited. It excites one of them with a probability P which can be
calculated from the Lamb shift. The molecule remains excited for a time tdecay (the decay time of the
excited level) and then decays giving back the EM fluctuation, which could fly away or excite a
second molecule. The probability PN that this fluctuation would give rise to an excitation of one out of
the N molecules present within the ensemble would be, of course:
3(/)
N
PPNP NV
λ
=× = ×
When this number is smaller than 1, the EM fluctuation would disappear eventually in the original
background, but when the density reaches the critical value (N/V)crit such that:
3(/) 1
crit
PNV
λ
×= (2)
The EM field loses the chance of leaving the ensemble of molecules and gets permanently trapped
within the region, bouncing from one molecule to another. In a short time more photons undergo
the same fate until a sizeable EM field grows in this region that will be called from now a
Coherence Domain (CD) attracting from the environment other molecules of the same species that are,
by definition, able to resonate with the growing EM field. In this way the system self-produces a
negative pressure which counteracts the positive pressure of the vapor. An attractive dynamic arises
which binds molecules together, giving rise to a large increase of the molecular density and of the EM
field until a value of the molecular density is reached, where, as we will see in detail in the following,
the trapped EM field grows exponentially (runaway) inducing a further increase of density. This
runaway brings the density of the system to a limiting value given by the intermolecular distance at
which the repulsive forces produced by the molecule hard cores come into play. The final result is a
closely packed ensemble of molecules oscillating in phase with an EM field. This field is trapped
within the CD since its oscillation time is renormalized by the interaction with molecules; the time of
oscillation of the free photon should be actually supplemented by the time spent within the molecules
in the form of excitation energy. Consequently, the frequency of the fields (matter and EM) in the
CD,
ν
CD, becomes smaller than the frequency
ν
0 of the free field and the squared mass of the photon,
m2, becomes, according to the Einstein equation:
22
24 2 2 2 2
0
22
0
CD
cc
mc h h
νν
λλ

=−<−=


(3)
since we know that the mass of a free photon is zero. On the contrary, the above equation tells us that
the mass of the photon oscillating in the CD becomes imaginary; therefore this photon is no longer a
particle, is unable to propagate and gets trapped in the CD in form of a cohesion energy, not static but

Water 2012, 4 514
at the origin of the coherent oscillation of molecules. The outcome of the above complex multi-step
dynamics is the actual spontaneous formation of a cavity for the EM field.
Let us address now the problem of the selection of one specific level in the above lasing dynamics
in the case of multi-level molecules, as is the case of water molecules. Phenomenological evidence
gives us the indication that the transition vapor-liquid occurs basically in two steps. In a first step,
molecule condensation occurs at several points simultaneously giving rise to a slow growth of density;
this step takes almost the whole time of transition, until a threshold is reached where very fast
condensation dynamics takes place, bringing the system to the final liquid phase.
In the framework of the present quantum field theory approach, the process can be described as
follows. Each molecular level is able a priori to give rise to a condensed phase based on the coherent
oscillation of all molecules between the molecular ground state and the given excited level. For each
molecular level this process implies a specific time of growth of the matter density and of the
associated cavity. The level able to produce the highest negative pressure through the above dynamics
would give rise to the dominant condensation, ruling out all the other molecular levels. A cavity
having the size of the wavelength of the winner EM mode, which resonates with the excitation energy
of the level, is therefore self-produced. When, as we will see in next section, the molecular density
reaches the critical density for runaway, the second part of the transition takes place until the system
settles in the final coherent configuration.
3. Mathematical Derivation of Coherence
Let us now rephrase the above argument in mathematical terms. In Reference [11], the general
Lagrangian description of the interaction between an ensemble of N microscopic units and the EM
field has been written down and the corresponding Euler-Lagrange equations of motion have been
derived. In the present article we will start just from these equations, where the variables are the
matter-field
Ψ
that describes the space-time distribution of molecules among the different individual
molecular states and the EM field A

.
The mathematical description of the dynamics sketched in the above section is extremely complex
when addressed in the case of multi-level molecules. Since this process is intrinsically a dispersive
one, it is conceivable, as discussed in [12,13], that one should drop the usual rotating wave
approximation (RWA) in the equations in order to get, as suggested in the abstract in [13], the
irreversible time evolution of the system into a generalized coherent state, exhibiting entanglement of
the modes in which the counter-rotating terms are expressed. We will do this in a future publication. In
this paper we will omit the description of the process of selection of the winner mode and adopt the
procedure of choosing the level according to the best-fit of the phenomenological consequences.
Let us address now the problem of the mathematical description of runaway. Since we are interested
in the investigation of the onset of the coherent oscillation between two specific molecule levels, we
limit ourselves to the consideration of the two-component matter-field describing the space-time
distribution of molecules between the two levels. The contribution to the dynamics of the remaining
levels will come into play via second-order radiative corrections and will be taken into account in due
course. Moreover, since we expect that the outcome of this mathematical treatment will be the
emergence of a coherent state where the molecule radiative dipoles i
J

connecting the two levels are

Water 2012, 4 515
all aligned, we limit ourselves to consider only the component of the EM field A parallel to this
common radiative dipole. Finally, for normalization reasons we introduce the reduced fields obtained
by dividing the original fields by N, so that we will reason on a per-molecule basis. Notice that in
the following we will use the standard unit system of field theorists, namely the Natural Unit System
where the Planck constant h, the speed of light c and the elementary charge e are without dimension
and assume the values 2π, 1 and 0.302814 (Lorentz-Heaviside units) respectively. Consequently the
energy difference ∆E = Eq − E0 = h
ω
q/2π can be written simply as
ω
q; in the natural unit system energy
and frequency have the same dimension as well as space and time. Energy therefore assumes the
dimension of the inverse of a length. In this unit system 1 eV is equal to 50,677 cm−1.
Moreover we introduce the adimensional time
τ
=
ω
q·t and factor out the term exp(−i
τ
) from all the
fields. We define therefore the following reduced field:
()
()
()
00
1exp
1exp
1exp
qq
i
N
i
N
AAi
N
χ
τ
χ
τ
τ
=Ψ
=Ψ
=
(4)
where
χ
0 is the field describing the molecules in the ground state |0> and
χ
q describes the molecules in
the excited state |q>.
Three equations of motion, named coherence equation (CE), arise from the general formalism of
QED [11] using the rotating wave approximation (RWA). The first equation corresponds to a process
where matter in the excited state
χ
q de-excites and reaches the ground state
χ
0, releasing an EM
field A. The second equation describes a process where matter in the ground state
χ
0 is shifted to the
excited state
χ
q by the absorption of the EM mode described by A. The amplitude of the above
processes is governed by the coherent coupling constant:
0
2
p
q
q
gf
ωπ
ω
=
between the matter field and the EM field, where:
p
e
N
eVm
ω
=
is the electron plasma frequency and f0q is the oscillator strength [14] of the optical transition between
the states |0> and |q>. By neglecting the spatial dependence, the first two coherence equations are:
*
0
0
() () ()
() () ()
q
q
igA
igA
χ
τ
χ
ττ
χ
τ
χ
ττ
=
=

 (5)
where the dot over the symbols indicates the derivative with respect to the adimensional time
τ
; the
photons being described by the field A if absorbed and by the field A* if emitted. Now, by multiplying
the first equation by
χ
0* and the second one by
χ
q* and by adding them together with their conjugates,
it is possible to derive the conservation relationship:

Water 2012, 4 516
()
**** ** **
00 00 00 00
01
qq qq qq qq
dcste
dt
χχ χχ χχ χχ χχ χχ χχ χχ
+++= + =+==
 (6)
meaning that the number of molecules is conserved.
The third equation of the CE describes the evolution of the EM field which receives three
contributions. The first contribution comes from the direct transitions governed by the constant g
between the two levels. The second contribution comes from the two-step transitions (named
dispersive transitions) governed by the constant µ, called photon mass term, where matter makes a
first jump from the ground state to an intermediate virtual state |n> (n ≠ q) and a second jump from the
latter to the final state. The third contribution is given by the vacuum through the so-called gauge
properties of the EM field. The name for the coupling constant µ is connected to the fact, mentioned
above, that the EM field acquires an imaginary mass because of the increased oscillation time (and
consequently of the reduced frequency) of the field when tightly coupled with molecules that
continuously emit and reabsorb the photons that compose the coherent EM field. In mathematical
terms, the third coherence equation may be written:
*
0
1() () () () ()
2q
AiA A g
ττ
μ
τ
χ
τ
χ
τ
−+−=
  (7)
Moreover, we are able to show the existence of two conserved quantities, Q and H. It is very easy to
check that the quantity
()
****
2qq
i
QAA AAAA
χχ
=+ − +
 (8)
is a conserved quantity. Actually, if we differentiate Equation (8) with respect to
τ
and replace the
derivatives with the values given by the CE we get zero. Similarly the quantity
()
** ** *
00
1
2qq
H Q AA AA g A A
μχχχχ
=+ + + +
 (9)
can also be shown to be conserved. In Reference [11], Q has been identified with the momentum per
particle and H with total energy per particle. Both Q and H are adimensional quantities in units of
ω
q.
The ensemble of the three CEs and the two conservation equations gives a system of five equations.
The adimensional photon mass term in Equation (7) is proportional to the square of the photon
mass. It may be decomposed into two contributions: µ = µ0 + µcont. The contribution µ0 is related to
the transitions from the ground state to the excited states, described by the oscillator strengths f0n, that
have an energy smaller than the first ionization threshold h
ω
q/2π (the discrete spectrum not including
the state |q>):
20
022
3
2
n
p
nq nq
f
μω
ωω
≠
=− −
 (10)
The second contribution µcont is related to the dispersive transitions occurring via the excited states
belonging to the continuum spectrum:
20
22
()
3
2cont
cont p
q
f
d
ω
ωω
μω
ωω
+∞
=− −
 (11)

Water 2012, 4 517
where f0(
ω
) is the oscillator strength per unit energy of the transition from the ground state level to the
continuum states [14]; The parameters g and µ depend on the values of the physical variables defining
the state of the system molecules + field, so that they become renormalized when the transition of the
system from the non-coherent to the coherent state has taken place, as will be shown below. When we
consider this dependence, the equations of motion that describe the new ground-state of the system
become highly non-linear. The three CE admit always as a (trivial) solution the extreme case of the
absence of coherence; in this case the EM field is vanishing, apart from quantum fluctuations of the
order
()
1/ON
, the excited field
χ
q is vanishing too and all molecules are in the ground state (
χ
0 = 1).
However, under suitable conditions, the above equations admit a non-trivial solution too, whose
energy is lower than the trivial one. In order to show the existence of such a non-trivial state,
physically obtainable in a physical process which starts from the trivial state, we need to prove that the
trivial state could become dynamically unstable. Under suitable conditions, the three CE equations
may develop a runaway solution for the field A, namely a solution exponentially increasing with time
until a saturation value is reached.
3.1. Runaway
The condition for runaway can be found by studying the short-time evolution of the coherence
equations leading to runaway solutions starting from the zero-field ground state. To this purpose we set
the initial conditions as:
0
11
(0) , (0) , (0) 1
q
ANN
χχ
 (12)
From the CE we easily obtain by differentiation:
**
00
1() () () () () () () 0
2qq
AiA Ag
τ τ μ τ χτχτ χτχτ

−+−− + =

   
(13)
and introducing the initial conditions and making use of Equation (5) we find the equation which in the
early stage of the process is linear:
2
1() () () () 0
2AiA AigA
ττμτ τ
−+−+ =
   (14)
By searching for solutions of the form exp(ip
τ
) we arrive at the third-order algebraic equation for
the exponent coefficient:
32 2
10
2pp pg
μ
−− += (15)
As clearly represented in Figure 1, there will be three real solutions or one real plus two complex
conjugate solutions, depending on the values of g and µ. In the first case the dynamical evolution of
the system remains stable and no runaway appears; in the second case a runaway will always be
present, driving the system to a configuration totally different from the initial one. The minimal
condition for runaway is found by requiring:

Water 2012, 4 518
32 2
32 2
10
2
10
2
c
c
dpp pg
dp
pp pg
μ
μ

−− + =



−− +=


(16)
whose solution is
3/2
282 42
27 3 9 3
c
g
μμ

=+ ++


(17)
Figure 1. Graphical representation of the behavior of Equation (15) for µ = −0.47 and for
three different values of g. Note that when g = gc, the curve is tangential to the straight line
y = 0. When g < gc, we get three real solutions and when g > gc, only one real solution
exists, implying that there exist complex conjugate solutions, giving rise to runaway.
The condition for runaway is given by: 22
c
gg≥ implying a threshold for density, since g2 depends
on N/V. We are now in the position of finding the non-trivial solution (limit-cycle) towards which the
system evolves and we will prove eventually that this energy is lower than zero.
3.2. Coherent Stationary Solutions
We now afford the solution of the coherence Equations (5,7) for liquid water. In this way we are
able to derive the properties of the coherent state (liquid) as a function of the excited level involved in
the molecule’s coherent oscillation. The selection of the specific excited level will be addressed in the
next section. It should be stressed that once the conditions for runaway are met, an avalanche process
drives the system from the zero-field configuration to a radically different quantum configuration,
releasing an energy corresponding to the latent heat of vaporization to the environment [4]. Once this
process is completed the dynamics of the system have dramatically changed. We are therefore

Water 2012, 4 519
interested to the determination of the coherent solutions after condensation has reached its stationary
final configuration. The values of µ and g become renormalized by the process of condensation and by
frequency renormalization. The stationary solution of the CE is found by writing the fields in a
phase representation:
[
]
[]
00
0
() cos exp ()
() sin exp ()
() exp ()
qq
i
i
AAi
χ
ταθτ
χ
ταθτ
τφτ
=⋅


=⋅


=
(18)
In Appendix A we derive explicitly such solutions by introducing the Ansatz Equation (18) into
Equations (5,7–9). This solution will be expressed through an elegant geometrical method [15] by
transforming the above equations into parametric equations. The solution is given by the intersection
of the following parametric curves in the two-dimensional strip:
0tan2 ,02x
αφ
≤= ≤+∞ <<

()
2
12
2
12
2
22
22
(): ():
11
11
4
1
21
12
1
x
g
x
x
g
x
ε
φ
φφ
φ
φφμ
Γ= Ψ=


−Ψ= −

+

Γ=

Ψ= − −



Γ=−
+





(19)
The complete solution for the amplitude of the fields and the energy gap Ecoh is given by
the formula (see appendix A)
()
0
00
0
22
00
1arctan
2
2
tan
tan
3
12 sin sin2
2
1
q
coh q r
rq
x
x
Ag
gA
gA
EA gA
α
φ
θα
θα
ω
μ
αα
ωω φ

=


=


=

=


=++−



=−






(20)
It is very easy to check, by combining the first four equations in and using some algebra, that
0q
φ
θθ
=−
 
This equation tells us that the EM field oscillates in tune with the matter field. But our task is not
complete yet. The photon mass term after condensation becomes substantially renormalized due to two
important facts: frequency renormalization and migration of the initial quantum state from |0> to
cos
α
|0> + sin
α
|q>. As said above, a new expression for µ must be considered, which depends on the
parameters of the coherent solution. The new expression depends both on Ecoh and
ω
r, so that we end

Water 2012, 4 520
up with a highly non-linear system of equations that must be solved consistently. The renormalized
expression, whose explicit calculation is described in Appendix B, contains not only the amplitudes f0n
and f0(
ω
) connecting the molecule ground-state |0> to the excited states of the discrete and continuum
spectrum respectively, but also the amplitudes fqn and fq(
ω
) connecting the excited partner in the
coherent oscillation with the discrete and continuum spectrum. This necessity could give rise to
possible troubles since these quantities are not known experimentally. Luckily, we will learn from the
following analysis that the contribution of these unknown variables to the renormalized expression of
µ in the case of water is negligible.
4. The Case of Liquid Water
We now apply the above general mathematical results to the particular case of liquid water.
4.1. Calculation of Critical Density for Water
The critical density dc of water is the minimal density at which water can exist as a liquid at
thermodynamic equilibrium. It is reasonable to assume that such a condition corresponds in our
theoretical framework to the minimal density at which a coherent state can exist at equilibrium. Since
the coupling constant g depends on density, dc is the solution of the equation:
gc = g(dc) (21)
The condensation of the liquid occurs starting from the vapor. The first step, as discussed in the
previous section, is the selection of the lasing mode, namely the mode coupled to the ground state
by the coherent oscillation; we are still unable to identify it by a formal derivation, so that we
will provisionally select the mode corresponding to the coherent solution which best describes
the phenomenology.
First of all, we will evaluate the critical density for each excited level by using Equation (21). We
will do that by introducing into Equation (21) the available experimental data on the oscillator
strengths of water. The evaluation of µ in Equation (17) requires an experimental input (the UV
spectrum of water) provided by Reference [16], to be inserted in Equations (10,11). We should note
that the uppermost region of the discrete spectrum, ranging from 11.122 eV up to 12.62 eV (see Figure 7
of Reference [16]) has not been assigned in [16]. This is the consequence of the presence in this energy
interval of a very large number of discrete levels gathering near the ionization threshold at 12.62 eV,
each of them with a very low f-value. Being unable to distinguish among these closely packed lines,
we have decided to treat such levels as if they were part of the continuum spectrum and we have called
this contribution the continuum tail. Moreover, a few levels are simultaneously present in the same
energy interval whose oscillator strength is substantially larger than average value and must therefore
be taken into account as potential levels for coherence and assigned as discrete energy levels. We have
been able to derive the oscillator strengths of the discrete peaks included in this region by an
elaboration of the graph exhibited in Figure 7 of Reference [16]. To separate the contributions of the
discrete and tail part of the spectrum in the region between 11.122 eV and 12.62 eV, we have made a
few reasonable assumptions, accepting an error of the estimates in the range of 20%. We have
estimated the continuum tail contribution in the region from 11.122 eV to 12.62 eV by assuming that

Water 2012, 4 521
the continuum tail starts just at the end of the first discrete peak at 11.3 eV (see Figure 2). Furthermore,
by assuming a linear approximation and by requiring the continuity of f0(
ω
) and its derivative
df0(
ω
)/d
ω
at 12.62 eV, the continuum tail may be expressed by: f0(
ω
) = 0 (
ω
< 11.3 eV);
f0(
ω
) = 0.0724
ω
− 0.81639 (
ω
< 11.85 eV) and f0(
ω
) = 0.108625
ω
− 1.2459 (
ω
< 12.62 eV).
Figure 2. Oscillator strength in the overlap (11–13 eV) between discrete and continuous
UV-spectrum of the water molecule as reported in Figure 7 of Reference [16]. The red line
indicates the extrapolation of the tail of the continuum spectrum.
This estimated tail should therefore be added to the experimental continuum spectrum above 12.62 eV
quoted in Reference [16]. It turns out that the lower limit of the integral (Equation (11)) lies within the
region of the discrete spectrum, giving rise for some levels to the appearance of a singularity of the
type 1/(
ω
−
ω
q) . For these levels the integral should be understood as the well-known principal value
integral. The discrete spectrum in the region from 11.122 eV to 12.62 eV is obtained by evaluating the
peak heights p0n from the experimental curve after subtraction of the estimated continuum tail (Figure 3).
The last step is the calculation of the discrete f-values using the following argument. By using the
Thomas-Reiche-Kuhn sum rule [14] and the value 0.3362 quoted in [16] as the sum of all the oscillator
strengths of the levels below 12.62 eV we can write:
12.62
00 0
11.12
11.12 12.62 11.12
( ) 0.3362
n n
eV
nn
eV
eV E eV E eV
f
fd f
ωω
<< <
+=−

 (22)
where f0(
ω
) is the above estimated tail. The oscillator strengths of the individual levels are derived by
assuming that all the peaks in the unassigned region have a gaussian shape and an equal width so that
in the region under investigation f0n =
α
⋅p0n. It is easy to get
α
as:
12.62
00
11.12
11.12
0
11.12 12.62
0.3362 ( )
n
n
eV
neV
EeV
n
eV E eV
f
fd
p
ωω
α
<
<<
−−
=

 (23)

Water 2012, 4 522
Figure 3. Oscillator strength with continuum tail subtracted for the UV spectrum of the
water molecule between 11 eV and 13 eV.
Since the above estimate could appear somehow arbitrary, we have checked a posteriori for the
stability of the results by varying the straight line describing the tail. We have found that the
differences always lie within the quoted uncertainty. The estimated values are shown in Table 1, where
we also list the oscillator strengths of the levels below 11.12 eV given in Reference [16]. Coming back
to the solution of Equation (21), the last problem is the evaluation of the contribution coming from the
continuum spectrum. The integral Equation (11) between 12.62 eV and 200 eV is easily calculated by
using the experimental data of Reference [16]. The contribution of the far spectrum above
Ω
= 200 eV
has been shown to be negligible (of order 10−4) in Appendix C. In Table 2 we show the contribution to
the mass photon term µ coming from the discrete spectrum, the continuum spectrum and the
continuum tail, computed at the nominal density of 1 g⋅cm−3.
The exact solution of Equation (21) for the various energy levels is given in Table 2, together with
the estimate of the experimental uncertainty of the critical densities. Please note that all the physical
quantities shown in Table 2 are calculated at the critical density. From Table 2 we see that all the
levels have critical densities in the range of the experimental value 0.322 g⋅cm3 of liquid water. The
level at 12.07 eV, which we are going to show as the likeliest on phenomenological grounds, has a
critical density quite close to the experimental value. Actually a coherent state produced by the
oscillation between the ground state and this level would include a plasma of very loosely bound
electrons whose energy per particle reaches a value slightly lower than the ionization threshold (12.62 eV).
This property would account for the appearance of the relevant phenomenon of electron transfer
occurring in water and aqueous systems, which is particularly important in living organisms [17]. For
these reasons we should presume that the level at 12.07 eV is the winner of the competition among
levels for the self-production of the cavity where the coherent field becomes self-trapped. In a future
publication we will address the derivation from first principles of the selection of such a level.

Water 2012, 4 523
Table 1. Contribution to the mass photon term µ coming from the discrete spectrum, the
continuum spectrum and the continuum tail, computed at the nominal density of 1 g⋅cm−3.
The oscillator strengths and the coupling constants g2 are also given for completeness.
ω
q
f g2 µ0µcµtail µ
7.400 0.0500 0.264 −0.279 −0.853 −0.053 −1.185
9.700 0.0732 0.225 −0.384 −1.028 −0.092 −1.504
10.000 0.0052 0.015 0.225 −1.064 −0.104 −0.943
10.170 0.0140 0.039 0.130 −1.087 −0.113 −1.070
10.350 0.0107 0.029 0.243 −1.114 −0.123 −0.994
10.560 0.0092 0.024 0.133 −1.149 −0.139 −1.155
10.770 0.0069 0.017 −0.068 −1.188 −0.161 −1.418
11.000 0.0218 0.052 −0.297 −1.238 −0.198 −1.733
11.120 0.0223 0.052 0.698 −1.267 −0.227 −0.796
11.385 0.0098 0.022 0.381 −1.343 −0.359 −1.321
11.523 0.0086 0.019 0.521 −1.389 −0.381 −1.250
11.772 0.0178 0.037 0.486 −1.492 −0.362 −1.368
12.074 0.0101 0.020 0.501 −1.558 −0.384 −1.441
12.243 0.0053 0.010 0.626 −1.437 −0.569 −1.381
12.453 0.0025 0.005 0.551 −1.089 −0.923 −1.461
Table 2. Critical molar volumes and densities for the various discrete transitions of water.
Values in brackets represent estimated standard errors.
Energy (eV) g2 gc2µ Vmolcrit (cm3) dcrit (g⋅cm3)
7.40 0.092 0.092 −0.41(1) 52(1) 0.35(1)
9.70 0.065 0.065 −0.44(1) 62(2) 0.29(1)
10.00 0.008 0.008 −0.49(10) 34(7) 0.52(10)
10.17 0.018 0.017 −0.48(5) 40(5) 0.45(5)
10.35 0.014 0.014 −0.49(5) 37(4) 0.49(5)
10.56 0.010 0.010 −0.49(4) 42(3) 0.42(3)
10.77 0.006 0.006 −0.49(3) 52(3) 0.35(2)
11.00 0.015 0.014 −0.49(4) 64(5) 0.28(2)
11.12 0.031 0.031 −0.47(7) 31(5) 0.59(9)
11.38 0.008 0.008 −0.49(3) 48(3) 0.37(2)
11.52 0.007 0.007 −0.49(3) 46(3) 0.39(3)
11.77 0.013 0.013 −0.49(2) 51(2) 0.36(1)
12.07 0.007 0.007 −0.49(2) 53(2) 0.34(1)
12.24 0.004 0.004 −0.50(2) 50(2) 0.36(1)
12.45 0.002 0.001 −0.50(1) 53(1) 0.34(1)
4.2. Coherent Ground-State of Liquid Water
In this subsection, and in the particular case of water, we discuss the solutions Equation (20) of
Equations (5,7) that define the coherent stationary-state that the water system approaches after
runaway. As noted at the end of Section 3, we have to re-determine the parameter µ. First of all, we
observe (see Appendix D) that the contribution of µr,cont is negligible after renormalization. The only

Water 2012, 4 524
possible contribution to the photon mass term comes from singularities appearing in the discrete sum
of Equation (B5). In order to evaluate this sum we show at first by a general argument that
1
rq
ωω
φ
=−

must be substantially smaller than
ω
q. The numerical value of the 1-component of
Γ
in Equation (19)
cannot be lower than
Γ
min = −1/4. Therefore we get from the graphical solution of Equation (19)
(see Figure 5) that:
()
2
2
1
10
44
g
φφ
−
−< <

(24)
The third-order inequality has the approximated solution:
2
11g
φ
<<+
 (25)
where use has been made of the smallness of g2. The parameter
1
rq
ωω
φ
=−

cannot exceed the limiting value g2
ω
q and therefore is so small with respect to all the excitations
ω
n to
make all the denominators appearing in the sum Equation (B5) large enough to allow us to neglect the
corresponding terms, except for the term proportional to f0q, whose denominator could be made to
vanish by the physical condition:
coh r
E
ω
 (26)
In conclusion we get:
()
()
22 2
0
22 2
3sin
2
pr q
rr
qcoh r
f
E
ωω α
μω ωω
=− − (27)
We are equipped now with the theoretical tools necessary for the calculation of the self-consistent
condition. Starting from the lowest possible value of the photon mass µ = −0.5, we look for which
couple of values ( , )
x
φ
 Equation (19) are simultaneously satisfied (
Γ
1 −
Ψ
1 = 0 =
Γ
2 −
Ψ
2). This may
be done in MATLAB with the built-in numerical solver FSOLVE or by using multidimensional secant
methods such as Broyden’s algorithm. The values of x and
φ
 obtained may now be reported in
Equation (20) in order to retrieve the values of sin2
α
, Ecoh and
ω
r. Inserting these three new values into
Equation (27) allows the computation of µr(
ω
r). If this value is negative, a small positive increment
ε
is added to the starting µ-value and a new cycle (µ +
ε
) → (,)
x
φ
 → (sin2
α
, Ecoh,
ω
r) → µr(
ω
r) is
performed until a positive value of µr(
ω
r) is obtained. As shown in Figure 4, this positive value means
that Equation (26) has been satisfied with an accuracy given by the chosen increment
ε
.
Figure 5 shows the graphical solution corresponding to the choice of the level at 12.07 eV as the
excited partner of the ground state in the coherent oscillation, while the calculated values are shown in
Table 3. The values corresponding to this particular level are different from the ones given in
Reference [4] due to the different values of oscillator strengths we have used. Nevertheless, the
differences do not change the global landscape of the predicted features. It is worth noting that a
negative value is obtained for Ecoh, completing the demonstration that the non-coherent gas-like state

Water 2012, 4 525
becomes unstable above a critical density. In Table 3 we have also listed the values of the critical
densities and of the energies of the coherent states that would emerge from a hypothetical coherent
oscillation between the ground state of the water molecule and each excited level of the UV spectrum.
We can see that the energy of the state corresponding to the level at 12.07 eV, which is at present our
preferred candidate for playing the role, is reasonable according to the thermodynamic evaluations
given in Reference [4].
Figure 4. Variation of the computed solution as a function of the running µ.
Figure 5. Self-consistent space-independent solution of Equation (19)
11 2 2
( 0.76687, 1.013661 0.17554, 0.7 3 )953,x
φ
=− = Γ =Ψ −Γ=Ψ=−=
 for the level
E = 12.07 eV with a coherent coupling constant g2 = 0.02, an electron plasma frequency
ω
p = 6.7783 eV and a renormalized photon mass µr = −0.48425.

Water 2012, 4 526
Table 3. Solutions of the coherence equations for each discrete excited level
ω
q of the
water molecules and associated coherent energies Ecoh and critical densities dcrit.
ω
q
g2
μ
rx
φ
A0sin2
α
ω
rEcoh (eV) dcri
t
(g.cm-3)
7.400 0.264 −0.3589 1.457 1.091 1.55 0.22 0.672 −0.67(17) 0.348(8)
9.700 0.225 −0.3759 1.403 1.082 1.60 0.21 0.795 −0.79(10) 0.290(7)
10.000 0.015 −0.4879 0.713 1.011 2.94 0.09 0.107 −0.11(4) 0.522(102)
10.170 0.039 −0.4715 −0.909 1.023 −2.36 0.13 0.238 −0.24(3) 0.451(51)
10.350 0.029 −0.4781 −0.843 1.019 −2.52 0.12 0.192 −0.19(6) 0.489(51)
10.560 0.024 −0.4815 −0.804 1.016 −2.64 0.11 0.168 −0.17(5) 0.424(32)
10.770 0.017 −0.4864 0.736 1.012 2.86 0.10 0.129 −0.13(4) 0.348(23)
11.000 0.052 −0.4634 −0.978 1.029 −2.21 0.14 0.322 −0.32(10) 0.280(21)
11.120 0.052 −0.4634 −0.978 1.029 −2.21 0.14 0.325 −0.33(10) 0.590(92)
11.385 0.022 −0.4828 −0.787 1.015 −2.69 0.11 0.168 −0.17(6) 0.372(24)
11.523 0.019 −0.4850 −0.757 1.013 −2.78 0.10 0.151 −0.15(5) 0.394(25)
11.772 0.037 −0.4728 0.897 1.023 2.38 0.13 0.265 −0.27(8) 0.356(13)
12.074 0.020 −0.4842 −0.767 1.014 −2.75 0.10 0.165 −0.16(5) 0.342(12)
12.243 0.010 −0.4916 0.642 1.008 3.23 0.08 0.093 −0.09(2) 0.359(13)
12.453 0.005 −0.4956 −0.535 1.004 −3.80 0.06 0.051 −0.05(1) 0.341(8)
5. Discussion and Outlook
Let us now compare the results of our theoretical approach with the phenomenology of water.
In this paper we have not addressed the problem of temperature, which is extensively discussed in [4];
the present discussion holds rigorously at T = 0. In previous articles [4,18–21] we have calculated the
thermodynamic quantities for a liquid corresponding to the choice of the level at 12.07 eV and we have
found a good agreement with the observations, including density and specific heat. However the main
element to be taken into account is the property of water in biological organisms to release electrons.
This property cannot be understood in the conceptual framework where water molecules are assumed
to be uncorrelated and subjected only to mutual static interactions. In this case the excitation of each
molecule cannot be larger than a couple of eVs, much smaller than the ionization threshold which is at
12.62 eV. On the contrary, in our approach, when we choose as excited partner in the coherent
oscillation the level at 12.07 eV, each coherent molecule would produce during its oscillation a
quasi-free electron, very weakly bound. The ensemble of these electrons would be very easily
excitable, giving rise to collective coherent excitations (vortices) which are able to explain a large
number of biological phenomena arising as a consequence of the possibility of having an electron
transfer in aqueous systems. This circumstance strongly suggests that the level to be chosen for the
coherent oscillation is the one at 12.07 eV. Other candidate levels would give rise to a liquid having a
higher density due to the smaller molecular volume in the excited state and to a much smaller ability to
release electrons. The determination of the thermodynamic and electromagnetic variables (specific
heat, viscosity, dielectric permittivity and so on) of liquid water would allow us to test the predictive
power of this approach. As a first example, properties of water glasses at low temperature have been
analyzed [19] and found in good agreement with the observed facts. Finally, we want to stress that
since the parameters g and µ controlling the onset of coherence for different choices of the excited
level depend on the thermodynamic parameters, our approach allows the appearance of different

Water 2012, 4 527
phases of liquid water corresponding to the choice of different excited levels. This opens the way to discuss
the thorny problem of the existence of more than one phase of liquid water at low temperature.
Appendix A
In this Appendix we evaluate the non-trivial limit cycle of CE (minimum energy state) by using a
method introduced in [4]. We re-write here Equations (5,7)
*
0
0
*
0
() () ()
() () ()
1() () () () ()
2
q
q
q
igA
igA
AiA A g
χτ χτ τ
χτ χτ τ
ττ
μ
τ
χ
τ
χ
τ
=
=
−+−=


 
(A1)
Together with the constants of motion, which are in our case the ‘total momentum’ Q and the
Hamiltonian H (see [4], [11]):
()
****
2qq
i
QAA AAAA
χχ
=+ − +
 (A2)
()
** ** *
00
1
2qq
H Q AA AA g A A
μχχχχ
=+ + + +
 (A3)
We now introduce the Ansatz, which accounts for the fact that the fields are complex functions:
[
]
[]
00
0
() cos exp ()
() sin exp ()
() exp ()
qq
i
i
AAi
χ
ταθτ
χ
ταθτ
τφτ
=⋅


=⋅


=
(A4)
into the first two Equation (A1) and we get:
{
}
{}
00
00 0
()sin Aexp () () () cos
()cos Aexp () () () sin
qq
q
gi
gi
θτ α θτ
φ
τθτ α
θτ α θτ
φ
τθτ α

−= +−


−= −−


 (A5)
By equating separately the modulus and the phase factors in Equation (A5), we realize that
0(), ()
q
θτθτ

and ( )
φ
τ
do not depend on
τ
so that 00
q
θθφ
===
   . Moreover, the exponential factor on
the RHS of Equation (A5) should be real and equal to ±1 only. We have therefore:
()
0
11
2
q
π
εθθφε
=± =++− (A6)
These two conditions imply that:
()
000
0
0
() 12
()
()
1
qqq
q
q
π
θτ θτ ϑ ε
θτ θτ ϑ
φτ φτ ϑ ϑ
θθφ
ε

=++−


=+
=+−

=+

=±






(A7)

Water 2012, 4 528
where
ϑ
0 and
ϑ
q are arbitrary integration constants. Equation (A5) may then be rewritten in the
following form:
0
0
0
12
tan
tan tan 2
tan
tan
2
tan 2
1
q
q
gA
gA
g
A
α
αα
θεα
θε α
ε
α
φ
ε
−
−= 
=−


=−

=−


=±





(A8)
By making use of the Ansatz Equation (A4) for the EM field we get:
2
() () () ()AiA A A
τ
φ
ττ
φ
τ
==−
  
(A9)
We replace Equation (A9) into the third Equation (A1) and get:
[] []
2
0
1exp ( ) exp ( ) sin 2
22
g
Ai i
φφμ φ
τε
φ
τα

−− =


 (A10)
2
2
2
2
0
1
022
tan 2 1
sin 2 1tan2
2
tan 2
g
x
gA
x
π
αφφμφε
α
α
α
αφ

≤<−− =


+
=
+

==


 

(A11)
We replace now the Ansatz Equation (A4) into the Equation (A2) getting:
()
2
02
1
1221
QA
x
ε
φ
=−++
+
 (A12)
Equations (A11,A12) can be rewritten in the following form:
()
22
22
2
22
1
221
1
4
1
21
Q
x
gx
g
x
ε
φφ
φε
φφμ

−−
+
−=


−− =


+




(A13)
Now, as 1
rqq
ω
φ
ωω
=− <
, 11
φ
−<
, so that 0 2
φ
<<
, and with 0 ≤ x ≤ +∞, the solution of this
system is therefore given by the intersection of two parametric curves defined by:
()
2
12
2
12
2
22
22
(): ():
11
:1
24
1
:1
:2
:1
x
Qg
x
x
g
x
εφ
εφφ
φ
φφμ
ε
Γ= Ψ=


−− Ψ= −

+

Γ=

Ψ= −−



Γ=
+





(A14)

Water 2012, 4 529
The constants of motion Equations (A2,A3) are fixed by the boundary conditions and are constant
in time provided that the ground state is stable. During the transition from the non-coherent to the
coherent state dissipative processes occur which temporarily spoil the Hamiltonian dynamics, implying
that the transition should occur through the release of heat. Therefore, we expect that Q should not
change during the transition and H should vary by a negative amount. We set Q = 0, considering that
the conditions before runaway are A =
χ
q ≡ 0, meaning that the field is zero and no molecules are
excited. By replacing Equation (A4) into Equation (A2), we get:
22
0
sin (1 ) 0QA
αφ
=+−=
 (A15)
Similarly one gets from Equation (A3):
2
2
00
sin 2
2
HAgA
φ
μ
α

=+ −



(A16)
By making use of the third Equation (A1) we obtain:
2
0
sin 2
22
g
A
φ
α
φμ

−− =−


 (A17)
Inserting Equation (A17) into Equation (A16) we finally get:
()
22
00
3
12 sin sin2
2
coh
q
E
HA gA
μ
αα
ω
==++ − (A18)
that gives the energy of the coherent state in the scale where the energy of the non-coherent state is
zero. Please note that Equation (A18) is the sum of three terms, two of which are positive and one
negative. It turns out by numerical calculation that the negative term corresponding to the matter-field
interaction term is large enough to compensate for the contribution of the positive terms that corresponds
to the energy of the EM field and matter excitation energy, so that the total energy is negative.
Appendix B
In this appendix we derive the renormalized expression µr for the parameter µ in the lowest-energy
coherent state. The calculation is performed by using the Natural Unit System (NU) where h = 2π
and c = 1, so that the energy E becomes equal to the pulsation
ω
= 2π
ν
. As explained in [11], the
photon mass term is defined by:
()
()
2
2
2
22
,,
() rn coh
rr ij
kr kr
rn coh i j
qncoh r
EE
eN coh x n n x coh
VEE
ω
μω ε ε
ωω
≠
−
=− ⋅ ⋅
−−



(B1)
where the state |coh> is the coherent state and can be written in terms of single-particle states as:
cos 0 sin 0coh
αα
=+ (B2)
Please note that, as has been done in the pre-runaway case, the sum in Equation (B1) must be
extended to both the discrete and continuum spectrum in order to fulfill completeness of intermediate
states. Actually the sum in Equation (B1) will contain a sum of discrete terms and an integral over
unbounded intermediate states. By substitution of Equation (B2) into Equation (B1) we find:

Water 2012, 4 530
()
()
()
()
222
,
2
22,
2222
22
,
2
22,
()
sin cos 0
ep rn coh
rr i r cont
kr
ncoh ir
qncoh r
ep rn coh
iircont
kr kr
ncoh ir
qncoh r
mEE coh x n
EE
mEE qxn xn
EE
ωω
μω ε μ
ωω
ωωαε αε μ
ωω
≠
≠
−
=− ⋅ +
−−
−
=− ⋅ + ⋅ +


−−






(B3)
By making use of the definition of oscillator strength [14]:
2
,
3
2( )
mn i
kr
ri
en m
f
mxn
mE E
ε
=⋅
−

(B4)
we can express Equation (B3) as:
() ()
22 2 2
0
,
22
222
3sin cos
() 2
pr qn n
rr rcont
nq nq
qncoh r ncoh r
ff
EE EE
ωω α α
μω μ
ωωω
≠≠

=− + +

−− −−


 (B5)
The explicit form of µr,cont is given by:
()
22 2 2
0
,2
22
12.6
3()cos()sin
2
pr q
rcont
qeV coh r
ff
d
E
ω
ωω ω α ω α
μ
ω
ωωω
+∞
=
+
=− −−
 (B6)
Appendix C
The contribution coming from the far spectrum is given by (
Ω
= 200 eV):
()
22
22
()
2
3
pr
qcoh r
fd
E
ω
ωωωω
δμ ωωω
+∞
>Ω
Ω

=− 
 −−

 (C1)
By making the asymptotic ansatz:
2
() ()ff
ω
ω
ω
−
→+∞ 
⎯⎯⎯→Ω

Ω

(C2)
Stemming from high-energy scaling of cross-sections in QED, we have:
()
22
22
22
2()
3
p
r
qcoh r
d
f
E
ω
ωωω
δμ ω
ωωω
+∞ −
>Ω
Ω

=− Ω Ω

 −−
  (C3)
We compute the integral
() ()
2223 3
33
22
32
2
1
11 1
log
21 35 3
r
coh r
dxdx
Ex
ωωω η
η
ηη
ωω εη
+∞ +∞
−−− −
−−
Ω


Ω− Ω
=Ω − + Ω +


+Ω
−− −− 



(C4)
where we have neglected the small contribution of ε = Ecoh/Ω and η = ωr/Ω. We obtain finally:
222
26
2 2 0.00333
() 3.710
9 9 200
pprpr
r
qqq
f
ω
ωωωωω
δμ ω
ωωω
−
>Ω
  
=− Ω =− × =− ×
  
  
Ω  
(C5)
totally negligible in our approximation.

Water 2012, 4 531
Appendix D
In this appendix we evaluate the contribution of the continuum spectrum of water to the
renormalized value of the parameter µ in the coherent state. This contribution turns out to be
negligible. We have:
()
() ()
22
0
22
12.6
22
0
2
12.6 12.6
22
22
()cos ()sin
cos ( ) sin ( ) 10 1
10
12.6 12.6
q
eV coh r
q
eV eV
cohr cohr
ff
Id
E
fd fd
eV
EE
ω
ωω
ωαωα
ω
ωω
αωωαωω
ωω
+∞
=
+∞ +∞
−
==
+
=<
−−
+
<<
−− −−

 (D1)
where the Thomas-Reiche-Kuhn sum rule has been used. The coefficient in Equation (B6) is given by:
22 2
222
22
336.54 0.44 0.01 0.0044
2 2 12.6
pr
r
q
eV eV
ωω ω
ω
×
=×=
× (D2)
An upper bound for the absolute value of Equation (B6) is finally given by:
4
,
0.0044 4.4 10
10
r cont
μ
−
<=× (D3)
a negligible quantity. This solves the problem arising from the lack of experimental knowledge of fq(
ω
).
References
1. Franks, F. Water: A Comprehensive Treatise; Plenum Press: New York, NY, USA, 1982.
2. Teixeira, J.; Luzar, A. Physics of liquid water: Structure and dynamics. In Hydration Processes in
Biology: Theoretical and Experimental Approaches (NATO ASI series A); Bellissent-Funel, M.C.,
Ed.; IOS Press: Amsterdam, the Netherlands, 1999; pp. 35–65.
3. Bertolini, D.; Cassettari, M.; Ferrario, M.; Grigolini, P.; Salvetti, G.; Tani, A. Diffusion effects of
hydrogen bond fluctuations. I. The long time regime of the translational and rotational diffusion
of water. J. Chem. Phys. 1989, 91, 1179–1190.
4. Arani, R.; Bono, I.; del Giudice, E.; Preparata, G. QED coherence and the thermodynamics of
water. Int. J. Mod. Phys. B 1995, 9, 1813–1841.
5. Collini, E.; Wong, C.Y.; Wilk, K.E.; Curmi, P.M.G.; Brumer, P.; Scholes, G.D. Coherently wired
light-harvesting in photosynthetic marine algae at ambient temperature. Nature 2010, 463, 644–647.
6. Scholes, G.D.; Curutchet, C.; Mennucci, B.; Cammi, R.; Tomasi, J. How solvent controls
electronic energy transfer and light harvesting. J. Phys. Chem. B 2007, 111, 6978–6982.
7. Bockris, J.M.; Khan, S.U.M. Bioelectrochemistry; Plenum Press : New York, NY, USA, 1993;
pp. 696–697.
8. Jaynes, E.T. Maximum Entropy and Bayesian Methods; Skilling, J., Ed.; Klüwer Academic
Publishers: Dordrecht, the Netherland, 1989; pp. 1–27.
9. Del Giudice, E.; Spinetti, P.R.; Tedeschi, A. Water dynamics at the root of metamorphosis in
living organisms. Water 2010, 2, 566–586.

Water 2012, 4 532
10. Lamb, W.E.; Retherford, R.C. Fine structure of the hydrogen atom by a microwave method. Phys.
Rev. 1947, 72, 241–243.
11. Preparata, G. QED Coherence in Matter; World Scientific: Toh Tuck, Singapore, 1995.
12. Kurcz, A.; Capolupo, A.; Beige, A.; del Giudice, E.; Vitiello, G. Energy concentration in
composite quantum systems. Phys. Rev. A 2010, 81, 063821:1–063821:5.
13. Kurcz, A.; Capolupo, A.; Beige, A.; del Giudice, E.; Vitiello, G. Rotating wave approximation
and entropy. Phys. Lett. A 2010, 374, 3726–3732.
14. Hilborn, R.C. Einstein coefficients, cross sections, f values, dipole moments, and all that. Am. J.
Phys. 1982, 50, 982–986.
15. Gamberale, L. Quantum Theory of the Electromagnetic Field in Interaction with Matter.
Unpublished work, 2001.
16. Chan, W.F.; Cooper, G.; Brion, C.E. The electronic spectrum of water in the discrete and
continuum regions. Absolute optical oscillator strengths for photoabsorption (6–200 eV). Chem.
Phys. 1993, 178, 387–400.
17. Marcus, R.A. Electron transfer reactions in chemistry: theory and experiment (Nobel lecture).
Angew. Chem. 1993, 32, 1111–1222.
18. Del Giudice, E.; Galimberti, A.; Gamberale, L.; Preparata, G. Electrodynamical coherence in
water: A possible origin of the tetrahedral coordination. Mod. Phys. Lett. B 1995, 9, 953–958.
19. Buzzacchi, M.; del Giudice, E.; Preparata, G. Coherence of the glassy state. Int. J. Mod. Phys. B
2002, 16, 3761–3786.
20. Del Giudice, E.; Tedeschi, A. Water and the autocatalysis in living matter. Electromagn. Biol.
Med. 2009, 28, 46–54.
21. Marchettini, N.; del Giudice, E.; Voeikov, V.L.; Tiezzi, E. Water: A medium where dissipative
structures are produced by a coherent dynamics. J. Theor. Biol. 2010, 265, 511–516.
© 2012 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article
distributed under the terms and conditions of the Creative Commons Attribution license
(http://creativecommons.org/licenses/by/3.0/).