# Emergence of the Coherent Structure of Liquid Water

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Emergence of Coherence

_{exc}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/E

_{exc}which for E

_{exc}= 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 usual densities of gases on Earth (in the case of water vapor at the boiling point 2 × 10

^{19}molecules∙cm

^{3}) 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.

^{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 t

_{decay}(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 P

_{N}that this fluctuation would give rise to an excitation of one out of the N molecules present within the ensemble would be, of course:

_{crit}such that:

_{CD}, becomes smaller than the frequency ν

_{0}of the free field and the squared mass of the photon, m

^{2}, becomes, according to the Einstein equation:

## 3. Mathematical Derivation of Coherence

_{q}− E

_{0}= hω

_{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}.

_{q}·t and factor out the term exp(−iτ) from all the fields. We define therefore the following reduced field:

_{0}is the field describing the molecules in the ground state |0> and χ

_{q}describes the molecules in the excited state |q>.

_{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:

_{0q}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:

^{*}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:

_{q}. The ensemble of the three CEs and the two conservation equations gives a system of five equations.

_{0}+ µ

_{cont}. The contribution µ

_{0}is related to the transitions from the ground state to the excited states, described by the oscillator strengths f

_{0n}, that have an energy smaller than the first ionization threshold hω

_{q}/2π (the discrete spectrum not including the state |q>):

_{cont}is related to the dispersive transitions occurring via the excited states belonging to the continuum spectrum:

_{0}(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 , 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

**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 = g

_{c}, the curve is tangential to the straight line y = 0. When g < g

_{c}, we get three real solutions and when g > g

_{c}, only one real solution exists, implying that there exist complex conjugate solutions, giving rise to runaway.

^{2}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

_{coh}is given by the formula (see appendix A)

_{coh}and ω

_{r}, so that we end 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 f

_{0n}and f

_{0}(ω) connecting the molecule ground-state |0> to the excited states of the discrete and continuum spectrum respectively, but also the amplitudes f

_{qn}and f

_{q}(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

#### 4.1. Calculation of Critical Density for Water

_{c}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, d

_{c}is the solution of the equation:

_{c}= g(d

_{c}) (21)

_{0}(ω) and its derivative df

_{0}(ω)/dω at 12.62 eV, the continuum tail may be expressed by: f

_{0}(ω) = 0 (ω < 11.3 eV); f

_{0}(ω) = 0.0724ω − 0.81639 (ω < 11.85 eV) and f

_{0}(ω) = 0.108625ω − 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.

_{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 p

_{0n}from the experimental curve after subtraction of the estimated continuum tail (Figure 3).

_{0}(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 f

_{0n}= α∙p

_{0n}. It is easy to get α as:

**Figure 3.**Oscillator strength with continuum tail subtracted for the UV spectrum of the water molecule between 11 eV and 13 eV.

^{−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}.

^{3}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.

**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 g

^{2}are also given for completeness.

ω_{q} | f | g^{2} | µ_{0} | µ_{c} | µ_{tail} | 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) | g^{2} | g_{c}^{2} | µ | V_{mol}^{crit} (cm^{3}) | d_{crit} (g∙cm^{3}) |
---|---|---|---|---|---|

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

_{r,cont}is negligible after renormalization. The only 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

_{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}. The parameter

^{2}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 f

_{0q}, whose denominator could be made to vanish by the physical condition:

_{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 obtained may now be reported in Equation (20) in order to retrieve the values of sin

^{2}α, E

_{coh}and ω

_{r}. Inserting these three new values into Equation (27) allows the computation of µ

_{r}(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) → → (sin

^{2}α, E

_{coh}, ω

_{r}) → µ

_{r}(ω

_{r}) is performed until a positive value of µ

_{r}(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.

_{coh}, completing the demonstration that the non-coherent gas-like state 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 5.**Self-consistent space-independent solution of Equation (19) for the level E = 12.07 eV with a coherent coupling constant g

^{2}= 0.02, an electron plasma frequency ω

_{p}= 6.7783 eV and a renormalized photon mass µ

_{r}= −0.48425.

**Table 3.**Solutions of the coherence equations for each discrete excited level ω

_{q}of the water molecules and associated coherent energies E

_{coh}and critical densities d

_{crit}.

ω_{q} | g^{2} | μ_{r} | x | φ | A_{0} | sin^{2}α | ω_{r} | E_{coh} (eV) | d_{crit} (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

#### Appendix A

_{0}and ϑ

_{q}are arbitrary integration constants. Equation (A5) may then be rewritten in the following form:

_{q}≡ 0, meaning that the field is zero and no molecules are excited. By replacing Equation (A4) into Equation (A2), we get:

#### Appendix B

_{r}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 = 2πν. As explained in [11], the photon mass term is defined by:

_{r,cont}is given by:

#### Appendix C

_{coh}/Ω and η = ω

_{r}/Ω. We obtain finally:

#### Appendix D

_{q}(ω).

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**MDPI and ACS Style**

Bono, I.; Del Giudice, E.; Gamberale, L.; Henry, M.
Emergence of the Coherent Structure of Liquid Water. *Water* **2012**, *4*, 510-532.
https://doi.org/10.3390/w4030510

**AMA Style**

Bono I, Del Giudice E, Gamberale L, Henry M.
Emergence of the Coherent Structure of Liquid Water. *Water*. 2012; 4(3):510-532.
https://doi.org/10.3390/w4030510

**Chicago/Turabian Style**

Bono, Ivan, Emilio Del Giudice, Luca Gamberale, and Marc Henry.
2012. "Emergence of the Coherent Structure of Liquid Water" *Water* 4, no. 3: 510-532.
https://doi.org/10.3390/w4030510