INTERNATIONAL IOURNAL OF QUANTUM CHEMISTRY, VOL. XVI, 691-706

Excitons and Bose-Einstein Condensation in Living Systems

R. K. MISHRA Department of Biophysics, All India Institute of Medical Sciences, New Delhi-110016, India

K. BHAUMIK Department of Biosciences, Birla Institute of Technology, Pilani, India

S . C . MATHUR AND S. MITRA Department of Physics, Indian Institute of Technology, New Delhi-110016, India

Abstract

The problems connected with the transformations of energy in the living systems are reviewed. Possibility of Bose-Einstein condensation by input of radiant energy over a critical level is considered as a route for overcoming the barriers of activation energy under the conditions of a pump and a therrnalizing environment. Molecular force fields constitute the pump. Under our scheme excitons constitute the major fraction of bosons to so condense. Instantaneous dipoles of London theory are then examined as excitons. Lastly an energy packet from a quantized dipolar field is suggested rather than the concept of “conformons.” Questions of charge and of other modes of energy transfer are briefly discussed.

The subject of energy and charge in living systems is beset by concepts, theories, and work that often form a confusing medley. This situation is, of course, a result of the nature of data in this field, that have certainly been insufficient or irrelevant and often contradictory.

Do we or our molecules work by heat, or by diverse modalities of electromag- netic irradiation that we receive, or by our position in geomagnetic field or gravitational field or by the molecular force fields? How does the nearly forbidden come to pass, that is, the making and breaking of a covalent bond? This happens regularly and with remarkable precision and specificity in a system that is essentially open. Activity involving large work are triggered off with essentially minute signals. Changes are brought about through change of a fraction of bond length, or a dihedral angle, embedded in an intraorganismic sea. Once the above is done the molecules straighten or fold or bend and twist to give sheets, tubes, balls, helices, and ropes of helices, and further balls, the building bricks of an organism with fluctuation and beat and throb of molecular conformation and association. Much of it happens in such time that makes us suspect if we are dealing with rates of entropy consumption fastest in the Universe, at the molecular level.

Whatever the external field or forms of energy, in the last analysis, it is the relevant chemical reaction that one must understand. Homoiothermy or poik- ilothermy [l-31, Qlo [4], or other derived indices [ 5 ] , point to the chemical basis of heat production and even of physical mechanisms of heat production and

@ 1979 John Wiley & Sons, Inc. 0020-7608/79/00 16-069 1$01 .OO

692 MISHRA ET AL.

dissipation. Similar explanations have to be provided in case of responses to other types of energy input, for example, in the optical region leading to photo- periodism, circadian rhythms, and in gravitation leading possibly to geotropism and orientation. Similarly one must understand the basis of movements in magnetic [6-91 and electrical fields and in response to ultrasound. Classical mechanics, hydrodynamics, and rheology of biofluids, autorhythmic biodynamic fields, intrinsic fluorescence, and corona discharge photography offer some other intriguing features of handling energy in living systems.

Notwithstanding the complexity of the systems and issues the problems to be resolved in the energy domain fall into three groups:

(a) How do reactions that are almost impossible on the consideration of energy become possible? We ask thus for light on the mechanisms of the biochemistry of the covalent bond. We inquire about the mechanism of enzyme action, the “high-energy bond,” the negotiation of the energy of activation, and attendant problems of recognition of the specific substrate and its migration to the binding site given atoms chosen by Nature [lo-161.

(b) The second set of problems involve nonenzymatic recognition, approach, gathering, folding, and twisting in an assembly of atoms and bonds. Formation of the hydrogen bonds would belong to this group. Aspects of action of hormones, antigens, and drugs are also for the most part in this group. We seek a microscopic trigger of macroscopic action.

(c) Lastly, one has to understand how reactions of types (a) and (b) and their rates are affected by the number of molecules involved and their state of aggregation, i.e., properties emergent from the mere fact of collection which give to the living system its apparent strangeness from the point of view of the students of matter. It so happens that all three groups are basically understood by elucidating the facts concerning the ways in which living systems handle energy, often a consequence of charge and molecular (or atomic) force fields and external inputs of energy. No final answers can possibly be given but a direction as to where they may be found can perhaps be indicated.

Much work has centered on the mechanisms of enzyme action [17-231, which is virtually the biochemistry of the formation of the covalent bond of which the major problem in the energy domain is the overcoming of the barrier of activation energy. This has been an area of fruitful experimentation and enormous specula- tion and has enriched language with terms like high-energy bond, polarons, and conformons [24-271. Quite unwittingly we have come face to face with the hydrogen bonds and weaker interactions, number-dependent reactions, phase transitions, cooperativity, molecular recognition, association, self-organization, phase fluctuations which may lie at the very roots of the living state [28-321 giving rise to its complexity, nonlinearity, and its emergent and apparent mystery.

As if this was not enough there is also evidence of generation, migration, and interaction of charge in the living systems [33-361.

It is not possible to do justice to any one of the above questions in the confines of a paper such as this. We would rather discuss approaches which we believe go the the font et origo of these problems and address ourselves to three problems

LIVING SYSTEMS 693

stated before. No final answers can be given but a direction as to where they may be found can perhaps be indicated.

Many of the reaction mechanisms well known to the organic chemists have been invoked in biochemistry [37-381. The role of the enzyme is to provide suitable energetics and a variety of mechanisms have been postulated in this connection to account for the observed enzyme kinetics. A mechanism that could probably be in operation would be a Bose-like condensation speculated by Frohlich 139-431. The biochemical systems may be regarded as open systems in a bath whose main role is thermalization. The system may possess collective longitudinal modes (phonons) which may store the energy in the lowest-frequency mode if the energy is pumped into the vibrational modes exceeding a certain critical rate. Experiments with 6-7 mm coherent electromagnetic radiation support the idea [44-471. Frohlich’s ideas are based on rate equations, parti- cularly the rate of occupancy (n,) of the ith mode, which is zero at stationarity as described by

s, + 4(T) [ (n , + 1) - n, epfiwi]

+x(T) 1 [(n, + l)n, epfiwl - n,(n, + 1) ePfiwc] = 0 (1) I

where 6 = l /KBT, T is the temperature of the bath, Ks is the Boltzmann constant, and si is the rate of inflow of quanta from the external pump. The second term refers to the first-order generation and destruction of oscillator quanta by the reservoir which would normally bring the system to thermal equilibrium. 4( T ) is the rate of this process. In the second order, a process involving the absorption of a quantum hwi and the emission of h q or vice versa occurs, transferring quanta from one mode to another, where x( T ) measures this.

The above formulation leads to the Bose distribution.

where the chemical potential p is extractable from

and

p vanishes if si = 0 or the X-type process is zero, when thermal distribution of the Bose gas of noninteracting phonons results. However, pumping of energy and the two-quantum process leads to the possibility of Bose condensation. As per Frohlich [48] one could consider a system consisting of a polarization field p(r, t ) , elastic field A(r, t ) , coupling constant c, and a space-time point (r, t ) . We can consider a lamellar aggregate of hydrocarbon chains or a macromolecule along

694 MISHRA ET AL

the length of chains in the molecule and the Hamiltonian density is then written

A(r, t ) being slower than the p(r, t ) field, it acts more like a parameter countering deformation and arranges to minimize potential energy and u, being the elastic wave velocity, leading to

-+ dA - =--( C

ax (t;) (+2 p 2 ) Angular brackets denote an average over a time longer than the time scale of the polarization field. If 2 oscillating units, say hydrogen bonds, are considered with average spacing of a, length L of the molecule will equal Zu and the normal modes would form a narrow band of frequencies W k (from W, to W,) = [(wz + A2)]1’2 = w i + A2/200, where A is the bandwidth. The density of levels in the region of giant dipole or small wave-number oscillation is expressed by

2 1 /2 D ( w k ) = (Zwk/rA){[(w2k -wO)] 1

In a narrow-band approximation the average total energy in these modes is given by

Fr = (p2)Lw:, Introducing creation and annihilation operator c: ( c k ) for the polarization modes, the effective Hamiltonian becomes

h @ k represents the amount by which the elastic deformation depresses the energy of each polarization mode, apart from the overall strain-energy contribution. The depression is proportional to the total energy of the modes and inversely to its frequency. Polar oscillations couple with strength (g) with the environs which in effect thermalizes the system. The bath has a set of independent oscillator frequencies with the annihilation operations 6,. We expect a shift in frequency of polar modes, which are ignored, and a broadening of width y occurs. Oscillators absorb energy from the pump because they can relax with the bath. The Hamil- tonian

%= x0+Cha:6:bm+ c hg(Cib;bl+Cjb:bm)+P ( 5 ) m m. I , i

(where P denotes pump) leads to the following rate of the change of occupancy of the kth polar mode:

sk +4(T)[(nk + 1 ) - n k ePh(wk-ek) 1 l = O (6) 6h(wl-e1) - nk(nr + 1) e @ h ( w k - e k ) + x ( T ) C [(a + 1)nl e

I

LIVING SYSTEMS 695

p = l/KBT, where KB is the Boltzmann constant, T is the temperature of the bath, s k is the rate of inflow of quanta from the pump, and the second and third terms represent the creation and annihilation of quanta in first and second order of perturbation. We then essentially get a Bose-type distribution

n k = [I + (4/,y)(sk/S)(1- e - p l r ) l / ( e p r * ( o k - e k ) - l r ’ - 1) (7)

p, the chemical potential, can be inferred in a manner after Eq. ( 2 ) and thus

and

s -c sk = 4 c [nk e P h ( o k - e k ) - ( n k + 111 k k

Since quanta are pumped in at a critical pumping rate S,, population and “condensation” in the lowest mode will occur. S, can be found by calculating the upper limit in the number of quanta in excited state N*,

( k # 0) and CL = h(wo- O0)

in the high-temperature approximation

where w: is the next to lowest frequency and is given by

[ ( w r + A2.rr2/Z2)]”2 = wo + A27r2/2w0Z2‘

S , is now associated with the critical energy in the polarization mode, which must be exceeded and

The rate for N*

is given by

(9)

differs from the usual pumping rate by a factor due to elastic restoring force, equal to

(1 +c2&,/(+2w40L)-’

It is interesting to note that the elastic restoring force lowers the S,, and that 80 is proportional to E~ and as O0 increases, the mode becomes softer and the frequency becomes imaginary at certain levels, and the material may be torn apart. This system is stabilized, however, by a nonlinear term which produces a restoring force which is cubic in the polarization field. The effective potential energy that includes this stabilization is given by

696 MISHRA ET AL.

Po is the contribution of the lowest mode to the polarization field, and d is the strength of stabilizing interaction. At E~ 2 u2Lw:/c2, the mode goes soft. Below this, Po is zero and therefore Po at equilibrium is

This indicates the possibility of metastable state displacement appearing like a ferroelectric phase transformation. Thus either Bose condensation or a “ferro- electric” state occurs, the former at a lower pumping rate. The critical flux without restoring force from adenosine ~ 0 . 4 mW/cm2. This is already in the order of magnitude observed in experiments with Escherichia coli. In addition to the experiments with E. coli there has also been the enhancement of specific activity of chymotrypsinogen when in low concentration, by laser irradiation [49]. Phyto- chrome induced root growth can be caused by laser irradiation as far away as a mile [50]. Bhaumik et al. [5 1-54] have tried to explain this “switching on” of an enzyme into a giant dipolar state, and have formulated two separate enzyme kinetic schemes [54] including both excitation and collisional de-excitation in the two schemes: one parallel case, in which the enzyme is activated by a conventional channel as well as by laser irradiation and the other, in series, in which only the excited-state enzyme participates in the reaction. The two channels suffer de- excitation by various processes. In the series formation it is excited by collisions and de-excited. Excitation by irradiation leads to excited enzyme substrate complex formation and the usual resultant products. Relevance of this analysis to photoexcitation of chlorophyll and to such occurrences as thymine dimerization, photochemical reaction [35, 36, 55-57], photoviscosity effects [57], and photo- pyroelectricity [58] is evident. The above viewpoint, however, begs some ques- tions. It may be possible that the “condensation” or “alignments” occur but is it the way of Nature? It is quite evident that the interior of bigger organisms is not available to a variety of radiations relevant to the concept. Other points to consider are: high body temperature (300°K) in homo-iotherms, the fluxes and flow of ions, presence of a material of variable dielectric, heterogeneity of molecular species, as well as the identification of bosons. Some of these are unnecessary and others demand extension of the concept provided below.

As stated above the restoring force reduces the critical pumping rate of 0.4 mW/cm2. Effects of electrical fields are sometimes observable in chemical systems like field effect in liquid crystals [59]. Even so the molecular force field with a mosaic of effective residual electronic charges, in a bed of gegenions located on atoms and bonds provide a satisfactory level of radiant energy. From the data of Boyd on ATP and spectral data of adenosine studied by us, it appears that the molecules can provide such a force field, using the narrow-band approximation. Mutual distorsion mechanisms [55-571, whether photoinduced or otherwise, are cases in point. Using the procedure by Bykov [60], Jain [61] calculated atomic and bond charges in molecules. The procedure suggests the possibility of damped oscillation of charge on the bond during the period of bond formation under the

LIVING SYSTEMS 697

influence or electronegativity which could be detected by appropriate resonance spectroscopy. Even luminescence can result as is already well known in the case of the lucifarin-luciferase system. The net result in the case of enzyme substrate, drug receptor, or antigen-antibody union will be to reduce E of the whole system.

Judging from the orientational and rotational freezing of a pair of molecules like pentane [62] or others, by the full molecular force field of pentane [28, 63-66], the entire subject of the conformational hyperspace, intramolecular coupling by mutual interactions [67-701, variation in charge densities on molecu- larization [69,70], lengthening of CN bond in haem-haem interaction [71], cascading of conformations in solutions [72-741, coenzyme conformation [75], lowering of barriers of internal rotation [76], and intramolecular acyl migration in lipids point to the effectiveness of molecular and atomic force fields at distances relevant to living state. Many such interactions are strengthened in a hydrophobic environment. Lowering of the internal energy of the system by Bose pumping strengthens the notion of its feasibility. If it is true as a general mechanism, then the consequences are recognition of a specific substrate or an antigen (in case of an antibody), and holding the molecule for a sufficiently long period for necessary transformation. Rhythmic variation in crystallization [77] or oscillation in enzyme activity [78] are explainable as being due to elastic field coupling.

Protons and Hydrogen Bonds as Bosons

Frohlich and earlier workers suggested the hydrogen bonds as bosons. Rates of proton transport [79-8 11, proton pumping and hydrogen bond formation, formation of “icebergs” would tend to support the idea superficially, awaiting actual calculations and proof. Unfortunately the molecular force field has to contend with a high dielectric constant in water, with gegenions in solutions and with a substantial dielectric constant in lipids. Evidence for the existence of vacuum in the living systems is not at hand. Lastly the density of protons required, lOZ3/cm3, will lead to an explosive situation. Thus we may also have to consider other candidates for the status of bosons.

Fluctuating Excitons

The framework of the original London description [82,83] of uncertainty generated instant dipole over an atom giving rise to charge fluctuating forces, after two-photon exchange, and cascading through virtual transitions, in low-lying levels, would indicate virtual Frenkel excitons, localized bound electron-hole pairs, transient, and fluctuating around an atom in an S-type orbitals or, in a more complicated way, in other types of orbitals. The traditional low-frequency spec- troscopy refer to the limit nu; + 0, where n is the excitonic concentration and a. is the excitonic Bohr radius. In this limit, excitons obeys the Bose commutation rules and can be considered as a perfect boson. While the relative singlet-state excitons are always much more attractive as compared to their Van der Waals attraction, the relative triplet excitons are repulsive. Here, however, the instantaneous dipoles always have an attractive interaction, according to the London theory.

698 MISHRA ET AL.

They offer a special case since repulsive elements increase sharply when distances are closer than equilibrium position. In dense systems thermal quasiequilibrium is reached in a relatively short time and one can expect that with increasing excitonic densities, phase transitions or condensations should be evident in both momen- tum space and in localized real space [84] . This will be facilitated further by the London charge fluctuation interactions, which are additive [29] and which can reach fairly high values in similar molecules [29] . Due to experimental difficulties it is not readily possible to get information on the possibility of a contribution of Bose-Einstein condensation to the attractive London interaction in large mole- cules.

Keldysh and Kozlov [130] described the pair theory of a Bose-Einstein condensed spinless electron-hole system. Their theory leads to the Bogoliubov spectrum [85 ] for weakly interacting Bose particles. Neglecting exchange poten- tial and adding grand canonical terms they wrote the Hamiltonian

k k

where Vq = 4 ~ r e ~ / ~ ~ q ~ is the Coulomb potential and E, (k ) = E, + h2k2/2m, and E h ( k ) = h2k2/2rnh, where me and mh are the effective electron and hole masses and E, is the energy band gap a’s refer to electron and B to hole. Agranovich [86] analyzed the possibility of Bose condensation of Frenkel excitons. We describe a view on the London excitations as Frenkel excitons in the Appendix.

“Conformons”

Volkenstein [25] and later Green and Ji [26] and still later Kemeny and Goklany [87] proposed the concept of “conformon.” Volkenstein’s conformon is an electron moving with a polarized medium and is associated with contrans- formation. According to the author lack of periodicity and homogeneity dis- tinguishes this pseudoparticle from “polaron.” His conformon is localized. Kemeny also has considered a somewhat similar system: second quantized electrostatic field-phonon interaction. These lack, however, proper parameters as usually applied for describing a pseudoparticle. They could have validity in case of unshielded charge say in a hydrophobic environment, which does indeed exist in mitochondria. In an aqueous ionized situation, these will be swamped out by gegenions. The electronic promotional energies are too high for biologic systems. As suggested earlier nonpolar charge fluctuation interactions, with the probability of their additivity are liable to be of greater significance, relevance, and effective- ness in a biosystem considered in bulk [28,30,31,64-661. Even then it is logical to quantize the total molecular force field. As pointed out earlier the London interaction is treated as an interaction between an instantaneous dipole, reaching up to low-lying excited levels. If the electrostatic interactions due to the bulk of gegenions are unimportant, most of the other interactions follow a dipole-dipole (induced or otherwise) interaction pattern and we could consider a procrustean

LIVING SYSTEMS 699

field of a dipolar stratum, and assembly of such “particles” and their states and occupancies. A series of phenomena wait to be explained in energy units, if possible, beginning from the crossing of the barriers of internal rotation, leading to generation of sheets, crimping of fibers, and formation of helices, hollow tubes, and induction of changes in hybridization schemes, generating double bonds, cyclization, decyclizations, and annealed structures with the cooperation of local mechanisms and catalytic surfaces. Elsewhere we have discussed the application of the principles of least abrupt change for force fields [31]. We have pointed out that the balance of molecular forces leads not only to the generation of various shapes of molecular aggregates but also to their fluctuations the fluctuating liquid crystallinity as a state of matter essential for the living process. Evidence exists in physical systems of various modalities of energy affecting crystalline versus noncrystalline aggregation [88] in chalcogenide glasses and the presence of semiconduction in amorphous and liquid crystal has been at hand [89,90].

High-Energy Bonds

A concept that has been held and investigated for a long time is that of the high-energy bond.

A. High-Energy Bonds Involving Phosphorus

Adenosine-Ribose

OH; OH I OH

0-d-t. 0 - h - O e d - O H I l l II I l l 0 : 0 ; o

I I I I

Fatty acid and amino acid activating enzymes I I

R - c - O ~ H ? I H+OH ATPase I

I H I H+OR Kinase

R-c-c-O~H I H ~ N H R Creatinine I l and arginine NH2 I I phosphokinase

B. High-Energy Bonds Involving Sulfur

Acetyl COA (acyl esters of thiols) mixed anhydrides of phosphoric acid and sulfuric acid, adenosine-phosphoryl-sulfate for sulfonation; sulfonium compounds.

C. High-Energy Bonds Involving Nitrogen

activation; amino carboxyl is esterified with ribose of t-RNA. Acid imidazoles with acyl group, attached to nitrogen atom in amino acid

700 MISHRA ET AL.

Although the terminal and preterminal phosphoryl groups of ATP are supposed to be bound to the rest of the molecule by a “high-energy bond,” a majority of postulates [24,61,91-1071 have centered on the relatively large free energy of hydrolysis 7-10 kcal/mol, depending upon circumstances. Meyerhoff and Lohmann [lo51 ascribed this to the relative instability of N-P bond as compared to 0-P bond. This does not explain the actual situation as can be inferred by looking at the sites of attack in the list above. Electrostatic repulsion directed to the same result has been persistently invoked from early days to those of quantum biochemistry. Kirkwood and Westheimer [106,107], Hill and Morales [92], the Pullmans [38, 108-1121, Alving and Laki [93], Boyd and Lipscomb [ 1 131, and Fukui et a1 [ 1 141 reviewed by Fernandez-Alonso [ 1 151 have suggested this mechanism; Wald [17] has suggested involvement of 3d orbitals of phosphorus in attack by lone-pair bearing species. The properties of 3d orbitals also turn up in the work of Boyd and Lipscomb [113] who attribute stabilizing influence to them by .rr-type interaction with oxygen orbitals. These workers by an all-electron calculation find that charges along the backbone of condensed phosphate alternate in sign and bridge oxygen carries a negative charge in contradistinction to the electron calculations of Grabe [ 1161, Pullman and Pullman [l 1 11, and Fukui et al. [ 1141. Fischer-Hjalmars [ 1171 pointed out that the electronegativity of bridge oxygen polarized the rest of the bond. This was considered as a basis of attack by carboxyl phosphate ion on negative ADP giving rise to ATP at substrate level phosphorylation. Grabe [116] postulated transfer of an electron from NADH, via the lowest-unoccupied .rr orbital of FAD to an orbital extending out on oxygen [9] of the flavine and over the P atom of PO4*- ion leading to an energy change necessary for splitting off a phosphoryl and formation of phosphorylated FADH. In the next step when electrons are transferred to the cytochrome the phosphoryl group is activated and can be transferred to ADP. In this mechanism “activation” of phosphoryl along with proof of other details remains to be provided. Cooperative conformational changes for energized states of mitochondria [131] and coupling of energy transduction is postulated by Green and Ji [26] and by Bennun [95]. A critical examination of the role of ATP in vivo is made by Banks and Vernon [94], who object to the term on the grounds that (i) the hydrolysis of ATP is a forbidden reaction in intermediate metabolism, (ii) in v i m situation is not a closed one, and (iii) the free-energy change can be predicted from the property of the whole system and not merely by a free energy of hydrolysis. They also do not consider the magnitude of the free-energy change to be as conceded as usual for theoretical discussions. Russian workers [118] found the free-energy change in glucose-6-phosphate formation to be -3.2 kcal/mol at pH 7.25 and excess magnesium, which according to the authors, is the best estimate.

These objections cannot be ignored and it is attractive to take the view that activation of the species to be phosphorylated, by means of donation of phos- phorus with the 3d orbital, after co-conformational variation causes favorable variation in the internal energy of the system. In other words the phosphorylated product is “active” and efficient in bringing about the concerned physiological or biochemical end result due to conformational and energetic reasons. This may be

LIVING SYSTEMS 701

a more satisfactory way of understanding the phenomenon of the high-energy bond. Electrons, molecular force fields, or proton [Sl] may all add to the system. In numerous cases Bose condensation induced by the substrate may be a sizable factor. It is interesting to note that the magnesium requirement for this reaction may be concerned with the gathering of reactants at one point [119-1211. Perhaps in such a system the capability of magnesium to exist in an unipositive state [81], and due to its electronic structure [121] may offer a bridge for energy transfer for required conformational change, thus coupling electron transport to phos- phorylation of ADP through any macromolecular system [ 122, 1231. Spectral data on the magnesium in chlorophyll suggest its probable utility in energy capture [124].

Other Mechanisms

Charge transfer [ 125-1271, electronic semiconductivity, reversible photo- chemical processes, irreversible photochemical processes, and electrolumines- cence of molecular vapors have been considered [133-1461. There is no doubt that evidence of exciton migrations and their structural basis [128, 1291 or electron [33] luminescence due to electron-transferred organic reactions. They do not, however, seem to offer a universal mechanism of even the predominant modes of energy transport in biology.

Excitons

London [147, 1481 and previous workers [149] had considered excitations of atoms which may even be electrically neutral, to be uncertainty-generated dipoles, the interaction of which, in the second order, was always attractive and followed inverse R6 dependence. In a simple formulation he included polarizabil- ity as an indicator of the ease with which a dipole could be induced in the neighboring atom by the field of an uncertainty caused dipole in an atom. The ionization potentials were introduced in his formula with the notion that the lower their value the stronger the transition, thus accounting for the anomalously high cohesive energy of colored solids.

Frenkel [150], who investigated the quantum basis of optical absorption in pure solids, conceived of packets of their excitations, excitons, based on the excitation of an atom. He conceived of virtual holes and the distance between the electron and hole, in his case zero (0 = 0) excluding the need to diagonalize in the index 0. The energies of his excitons in the first order are eigenvalues of

(6i(Rh) = 4I(Rh, 0) denotes the ground state of all atoms except Rh which is in excited state and which with the antisymmetrizer .FZ, can be written

4 ( R d d a l h - R d * 1 . Q h - R h ) - * . a n b n - R n ) (12)

702 MISHRA ET AL

If 1 = If, then

%(K) = (dJdO)XodJdO))+ c exp (iK - Rh)(dJi(O)Xo+i(Rh)> (13) Rh f O

The second term Xi, is the first-order energy of state when the origin has an atomic site excited and the rest is in ground state. This energy of excitation is E~ - e0. This includes whatever perturbations arise from the neighbors. The second-order excitations, as in the London charge fluctuation case, contribute. This quantity then gives the order of magnitude required to excite the solid. In the case of real transitions this energy corresponds to the excitation energy for the free atom. In excitonic solids this energy must correspond to the exciton energy in the solid. This term therefore, lays the basis of existence of the Frenkel exciton and includes the London excitations as simultaneous excitations of multiple pairs of two neighboring atoms. The second term above in (13) is also responsible for the migration of the excitation, since it connects the excited state at its origin and another at a different site Rh by a process analogous to resonance transfer: motion of the excitons would be specified by a wave vector K, a momentum hK, velocity

V = hK/M* = h-’VKE(K)

and a mass tensor

(&f*)-l= h-’v~ * V&(K)

giving it a sense of reality. It has been stressed previously [151] that in the London case the similarity of

atoms or molecules leads to strongest interactions. In some situations like donor-activator energy transfer we get the formulation

where K is the dielectric constant. The multipole expansion gives the leading term as an electric dipole-dipole term with R-6 dependence. It is interesting to state this aspect because the dipole-dipole term arises from the same part of the interaction Hamiltonian as van der Waals interactions and is proportional to the oscillator strength of sensitisor and activator, the overlap of the curve of Is and Us on the energy scale. Transfer probability then can be as large as 10I2/sec. Along with a number of atomic aggregates of the order of 1023/mol a basis of exciton density is created, which is necessary for condensation purposes. In xenon [152, 1541, an element well known to be subject of London interactions, experimental evidence demonstrates both Frenkel and Wannier excitons. In Foerster’s results [153] the speed of migration of excitation from one molecule to another in a solution of dye varied as R-6 of mean separation of molecules.

So far as London “excitons” are considered, with several assumptions involved, the following values can be estimated in case of one electron excitation: effective mass is mJ3.2, velocity equals 8 x lo7 cm/sec,

E = 1.8 x lo-’* erg or 1.125 eV

LIVING SYSTEMS 703

and

A = 1 . 1 x 1 0 - ~ c m

In the living systems, the energies of excitations and interactions are rather low, fractions of an eV, and may approach phonon energies ( h o , - 0.03 eV). The characteristic temperature to excite these vibrational modes by the relation K B @ = h w , corresponds to room temperature, roughly 300”K, or the body temperature. This leads to a spectrum of excitonic energy and interaction energies of excitons. The oscillatory behavior of excited atoms leads to varying interatomic or intermolecular distances providing thus predominant component of attraction at one time and repulsion at another.

“Jellium ’’ Model and Pseudoparticle

A two-component degenerate Maxwellian plasma can occur in alkali metals. The ionic cores are far apart and only electrostatic forces act amongst the various species. Crystal cohesion is relatively mild and the jellium model can be applied. Unlike a true crystal, transverse acoustical wave is not supported but a good qualitative picture of longitudinal phonons can be verified by measuring longi- tudinal sound velocity in alkali metals [152]. Again it seems to be reasonable to view water and various electrolytes in the body as jellium. Doing this would lead to an understanding of living systems on the basis of Coulombic interactions between permanent and other dipoles of various kinds in which it may be possible to discover fundamental units or packets of energy.

Acknowledgments

The authors wish to express their gratitude to Professor D. S . Kothari, Dr. J. Subba Rao, Dr. M. Kohli, and Dr. K. K. Singh for valuable discussion; to Dr. G. Subba Rao for assistance in preparing this text; and to Dr. R. S . Tyagi for help in estimating power output in the IR range of some molecules. Prior to the con- ference of the International Society of Quantum Biology, the senior author had the privilege of discussing the paper with Professor V. M. Agranovich and Professor M. V. Volkenstein, for which he is grateful to them.

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Received November 1, 1978 Revised March 2, 1979 Accepted for publication May 2, 1979