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CHEMICAL PROBLEMS 2021 no. 1 (19) ISSN 2221-8688
7
UDC 542.971+550.4+542.97
PHOTO-CATALYSIS PROCESSES IN NON-REGULAR MEDIA
T.A. Marsagishvili, G.D. Tatishvili, M.N. Matchavariani
RAgladze Institute of Inorganic Chemistry and Electrochemistry I. Javakhishvili Tbilisi State University 1 Ilia Tchavtchavadze Avenue, Academic Building I, 0179 Tbilisi, Georgia, e-mail:tati@iice.ge
Received 29.12.2020 Accepted 24.02.2021
Abstract: Technologies that cause the least harm to nature from global warming standpoint are becoming called-for. In this regard, photo-catalysis which uses the solar energy for chemical transformations of substances is of great interest for studying possible mechanisms of such processes. At the same time, the use ofphoto-catalysis significantly expands capabilities of the synthesis of new organic and inorganic materials. Charge phototransfer is always at the heart of photochemical transformations. Such processes can be considered from the point of view of quantum mechanics, and the probability of various mechanisms of the passing processes can be calculated using the apparatus of Green's functions. Analytical expressions for kinetic parameters of the charge transfer process between polyatomic particles in non-regular condensed medium and expressions for the extinction coefficient for appropriate processes of charge photo-transfer are obtained. Based on the analysis of obtained analytical expressions of dark and optical charge transfer processes, correlations between kinetic parameters of these processes are obtained. A methodology for determining kinetic parameters of the charge photo-transfer process on the basis of the shape of light absorption curve by means of the system functioning as the frequency of absorbed photons in various systems is presented. In this respect the general methodology has been applied to various specific processes. A methodology for determining the kinetic parameters of the charge phototransfer process from the shape of the light absorption curve by the system as a function of the frequency of absorbed photons in various systems is presented. The general methodology presented has been applied to various specific processes. Keywords: photo-catalysis, photo-transfer, optical spectrum, extinction coefficient, reorganization energy DOI: 10.32737/2221-8688-2021-1-7-17
The problem of photo-catalysis becomes more and more urgent as negative tendencies of global warming tend to grow. Technologies that cause the least harm to nature from global warming point of view are becoming called-for. First of all, such technologies that use the energy of the sun and the minimum of fossil fuels. In this regard, photocatalysis [1 - 24], which uses the energy of the sun for chemical transformations of substances, is of great interest for studying the possible mechanisms of such processes.
At the same time, the use of photo-catalysis significantly expands capabilities of synthesis of organic and inorganic materials. Capabilities of using photo-catalysis [10] for the synthesis of new materials is often facilitated by performing spectral studies into individual reagents used for photosynthesis [11], as well as
individual stages of the process proper.
The methods of spectral studies of individual substances [18], both in vacuum and condensed medium, are widely used to identify certain substances and control the quality of compounds obtained. At the same time, when using optical methods, a question arises about an unambiguous interpretation of separate bands in the optical spectrum. More questions arise when interpreting the optical spectra of systems with charge photo-transfer [19].
So, the use of optical methods to study complex condensed systems used for photo-catalysis requires an insight into individual components (reagents) and separate stages of the charge transfer process.
Electron transfer always lies at the heart of photochemical transformations. This can be a photo-transfer of an electron of the reagent to an
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CHEMICAL PROBLEMS 2021 no. 1 (19)
excited state followed by the transition of an electron to another particle and, hence, the implementation of a chemical reaction. This can be a direct photo-transfer of an electron to the final state (from one reagent to another reagent) and thus the implementation of a photochemical reaction. But more complex processes can also occur, in which molecules of the medium in which the reaction takes place take an active part.
where I0 is intensity of light on the input of measurement cell with thickness d; I is intensity of light on the output of the cell. Proportionality coefficient k may be substituted for extinction
K = la CaEa
These processes can be considered from quantum mechanics point of view and the probability of various mechanisms of the passing processes can be calculated if we apply the apparatus of Green's functions for finite (nonzero in the Kelvin scale) temperatures [25]. When using optical methods to study complex charge transfer systems, it is convenient to use Lambert Beer's law, according to which the optical density D is determined by the relation:
(1)
coefficients of particles adsorption sa and extinction coefficient of the direct outer-sphere transition of the system with photon adsorption
Sab:
.*+■■■ (2)
where Ca is concentration of a-type particles. coefficient of the process will be proportional to
In case of photo-transfer of an electron the concentration of particles C on a particle to an excited state, the extinction
Let us write the perturbation leading to the photo-transfer in the form:
V = -fdrP(r)E(r) (4)
Where P(r )the polarization operator of the of the photon electric field strength which can
, . • ^y-h^u • * be written in the representation of secondary
particles-reagents is, Efr)there is the operator . , „ ^ J
quantization in the form:
lkt7<
(5)
Where V0 is main volume of periodicity, ekcr is unit vector of photon polarization with wave vector k and polarization a (o= 1, 2), a^ and aba are photon creation and annihilation operators, ojk is frequency of photons, i is complex unit.
= -JC6 M Sdrd?(Er(?,e)Ea<T,
Where integration contour C6 over 6 runs parallel to the imaginary axis, angle brackets denote quantum statistical averaging over the initial state of the system Hamiltonian Ht The
Separating electronic and nuclear states for reagents and molecules in the adiabatic approximation, the transfer probability in the first order of the perturbation theory can be written in the form:
^ : (6)
Heisenberg operators are determined using the Hamiltonian of the free electromagnetic field with the renormalized photon propagation velocity:
E (?,&) = exp(p9Hph)E(r)exp(-p8Hph) (7) It is not difficult to calculate a correlator of operators of the intensity E in formula (6):
(8)
The first term in the equation in curly brackets describes processes with light emission, and the second term - with light absorption. Reactions with light radiation are important for strongly exothermic processes, in particular, for processes involving electron-excited particles.
For definiteness, we will consider only
processes with light absorption to comply with
photo-catalysis processes.
Let's assume that the source produces
photons with polarization o and wave vector —> —t —t
within interval from k to k+ Ak. In this case:
'JS^tl
expf/ÏÉ^ + ifep-r)] =
2it<tiklea eYexp[p9cok +ik(r- r')]
Where the frequency interval Arok and space angle of the direction of the wave vector AQk is
determined by the vector Afe, the flux density / coincides with a number of photons, which passing per unit time through a unit area
(9)
perpendicular to the vector k. For simplicity, we will omit the indices k and o in the future. The expression for the extinction coefficient of the catalytic reaction can be found from formulas (6) and (9):
dV^ — Sdx
(10)
It is obvious that matrix elements of the where the transfer occurs is determined by
polarization vector in the obtained expression atomic scales, k(r - r7) = 0. As a result, for
are nonzero only for the points r and r' which the extinction coefficient we have: near to reacting particles. Since the region
(11)
Where dfi = [dfie J, dfl is the dipole moment of the transfer.
A further simplification of formula (11) can be carried out for the case when no exchange (transfer) of heavy fragments between reagents occurs during the reaction. For such reactions the set of coordinates of the centers of gravity"pj" and angles (pi, which determine the
spatial orientation of the reacting particles, is identical in the beginning and the end of the process. Using the classical approximation for the translational-rotational motion of particles, from formula (11) we obtain:
Where pfl are radius-vectors of the gravity centers, respectively, for example, for the first particle; analogously, angles <picc. Depending on the symmetry of reagents, Q0 is equal to either
(12)
8n , 4n, or 1, Hamiltonians Hit and Hft differ from Hamiltonians Hi and Hf only by kinetic energy of translational-rotational motion, a correlation function has the form:
dPi f no dyia
-PFi(.p,<p)
(13)
Where Fi is the free energy of the system with We accept the Condon approximation and Hamiltonian Hit,, assume that the photons are unpolarized, then:
(14)
Here, the factor 1/3 arises when integrating over We also accept the one-electron approximation the angles that determine the orientation of the assuming that the state of one electron only first particle. changes during photo-transfer:
Where Sf, denotes the overlap integral of the first term, in the formula (15) cf(ic) can be taken electronic wave functions of the initial and final out of the integral sign at the point x*, where Sfl states. Since d(x) is a slow function of x, in the has maximum:
dfi =[d(.x*')-dii]Sfi(pr<p')=d,Sfi(pr<p) Since d * weakly depends on p, 9, from formula (14) we obtain:
(16)
(17)
Under similar assumptions, the rate constant of as follows: the thermal reaction can be reduced to the form
Where V* is connected with perturbation which leads to the reaction:
Comparing formulas (17) and (18), we obtain a the charge photo-transfer process and the correlation between the kinetic parameters of corresponding thermal process:
£a |y * I
Zttcj 3(7
(20)
Although the (20) correlation was obtained in one-electron approximation, it can be expected that it will be valid in many-electron approximation for matrixes dfi andly,. This requires that the function dfi be a slow function of p, cp coordinates.
All further analysis for extinction
for a number of cases.
If there is no entanglement of normal coordinates of intramolecular vibrations of the reacting particles in the initial and final states and if the interaction of intramolecular vibrations of reacting particles with the medium polarization can be neglected, and the harmonic
approximation can be used for intramolecular coefficient and calculations for specific models
.... ^ „ vibrations, then we obtain the following
onM 00 /-» /1 iM 1+ 1 M fin tv» /-» t 1 r n n T f ^
can be carried out in the same way as for thermal processes. So, we merely present results
expression for the extinction coefficient:
(21)
Where the function <p,6*) and Vv{6*) looks like:
tFm(p,<p,8i)= - - J dp dp'dip dip' AEt(p ,<p)AEf{p' ,<p ) dio Imgff(ptq>,p',v')*
, Pui(l-9) J Puj9 . , 2 1
.s/7—-- sh-— /(at-sk—)
2 2 v 2 '
(22)
where AEt{p r<p) is the change in the electric medium [10].
field strength of the reactants during the charge If, moreover, the frequencies of
transfer process, lmgff0 ,<p,p- ,f) is retarded intramolecular vibrations of reacting particles
Green's function of polarization operators of do not chan§e durin§the reaction'then
x„shx„
vvm=xBiJEr
(23)
Here Ern is the reorganization energy of the n-th degree of freedom [10].
To obtain final results for the adduced processes from the shape of the light absorption
models, it is necessary to move in two curve by the system.
directions: when calculatingfm{p,<p), one First of all, consider the sum rules that
should use statistical-mechanical models and for allow us to determine the transition dipole
calculating dj, - quantum-mechanical methods. moment. For this, both parts of correlation (10)
Below is the calculation of kinetic should be divided into and integrated over
parameters of the charge phototransfer when adJusted for integral representation of the
g function. As a result we get:
In the long-wave approximation, correspondingly:
(25)
If the translational and rotational coordinates do them can be considered as classical, then: not change during the reaction and motion along
Zir1 f rip^flg riiPLi
^CMOfetf,«or
For bimolecular reactions, if the transfer occurs form: upon contact of reagents, the sum rules take the
(26)
(27)
If the characteristic distance at which the d -% and the initial distribution function
transition dipole moment changes is A, 1 . , , , .
tP[ {Rc) can be considered equal to unity, then
(28)
Finally, if the radii of particles are the same and equal to r0, then
J0 cj % "
32 jz-
d%r¿ A
(29)
Proceeding from these sum rules, it is possible dependence of the extinction coefficient on to determine the transition dipole moment only frequency is known. For example, from (29): if the entire experimental curve of the
dh =
J° "ft
(30)
To determine the mechanism of the charge of the bond between the reagent and the photo-transfer process in non-regular condensed medium. The parameter X can be used for system and calculate reaction parameters, first estimation: of all, it is necessary to determine the strength
(31)
À = 2^-cthp^ u™ 2
If X<<1, bond force is weak, if X>>1, bond force is strong.
DÀ = DÀ ■ hv—
1. With a weak bond with the medium, the absorption curve will have a Lorentzian shape:
rz
(32)
the absorption
Where hv is the photon energy, iiv^ is the peak (atkv = hvmax position of the maximum of the absorption decreases in two times).
peak, D;'^. is the absorption at the Calculations are more convenient to be made if
max
point hv = hvmni, /' is half-width of absorption formula (32) is rewritten as follows:
hv _ _1
r^D.
[hv2 - 2hvmaxhv + (hv,nax)2 + f2]
maje
(33)
In considering this relation as a quadratic difficult to calculate coefficients on computer: polynomial (function hv/DÀ of hv), it is not
a„
; a±
2hv,
r-D,
nm. „ _
7- 5 a2 -
(34)
Thus, for the reaction parameters we get:
/ \ 2
I _ ni r2 — °i J "i- 1
---(¿"J '
D
À
: -,
(35)
By the found value of D*nax and relation (21) can be estimated if for calculation of
2. At strong bond with the medium, a number of models should be considered that will describe the most probable processes. 1) Let us consider a model in which the system has only classical degrees of freedom, we'll describe them in a harmonic approximation,
will be used the relation (27) or
(28).
assuming that there is no frequency change during the reaction. If two particles are involved in the reaction and the transfer occurs upon contact of reactants, then the extinction coefficient has the form:
(36)
Where is reorganization energy of the system [8], and 0*is equal to:
The optical density for the considered model is described by the function as follows:
(ii'-fiv+^r)2
D = Dmaxhv ■ exp [-
±KTEr
Let us rewrite formula (38) in the form (AF = 0): Y= A + ^-X + BX*, A = lnDmax-^)B =
2KT
4KT
(37)
(38)
(39)
Where Y = in(%v), X = hv.
For numerical calculations, the last correlation necessary to determine such values of should be rewritten in a form convenient for coefficients A and B so that using the least-squares method. In so doing, it is
should have minimum (here Xi and - are experimental points). After uncomplicated transformations we obtain:
(41)
Dmax = exp [A - Er = -
Where S is the number of experimental points.
It must be noticed, that at practical use of correlations (38) should be used computer programs.
Having determined the reorganization energy Er and the Dmaxvalue by this method, the distribution function of the reactants and the transition dipole moment can be calculated using formula (36) and the sum rules (27-29).
2) As the next model we will consider a system in which, in addition to the classical subsystem shown in the previous model, there is also a quantum degree of freedom with a frequency toq that does not change during the reaction. The extinction coefficient will have the form:
Where E* is the reorganization energy of the similarly toff,,), and 8* is determined from the quantum degree of freedom (calculated equation:
-Cjk + AF(RC) + (1 - 28)ET + -- = 0
The shape of the absorption curve for this case is as follows:
D = Dma*hv-eXP
*KTEr+2&„R
ir
Let us rewrite formula (44) as follows: Where the coefficients of polynomial are:
a0 = lnDmax
(43)
(44)
(45)
an = —-:
,
After computer calculations a± coefficients anda2, kinetic parameters of the
ofii0, reaction can be determined:
a = I—; fev^^T
D,nax = expM ■+
(46)
(47)
The transition dipole moment and the distribution function of the reagents can be calculated similarly to the previous models. 3) Finally let's dwell on a model in which during, the reaction the reorganization of medium occurs, which is described in the classical harmonic approximation without
changing frequencies and two classical degrees of freedom are reorganized, the frequencies of which change during the reaction. Moreover, we will assume, that the frequency of the first degree of freedom in the initial state oj[ will be equal to the second degree of freedom at the end col, and vice versa, wS = oj{ .
For this model the extinction coefficient will have the form:
Where
CO [f
Lr
EE
is
the
reorganization energy of the vibrational
(48)
degree of freedom with changeable frequency (evidently, E^ — E^ — ¿7^), the value of 8* is determined from equation:
+ &F(RC) + (1 - 2B)E? + y
f(i-&)z-gzyz ^ yz(i-&)z-8z
[l-8 + yz8]z [(1 -6)y2+ez]zj The absorption curve has the form (AF = 0):
EE = o
(49)
(50)
To determine reaction parameters E™ and Ej! tn(^) = f(hv) we find the experimental
we will use the following method. From the . hv ,, „„ , c ^ . A , values or 9* at different hv.
slope of the experimental curve exp
Equation (50) may be represented as:
hv
_ — f'n _L
1-20
(1 -9)z-bzy2 y2( 1-8y-6
[i -e+y26]z [(i-0)r2 + 0]2
(51)
This relation is the equation of the straight-line as a function of the expression from curly brackets. Using the experimental values of 9* at different ft v, it is easy to determine and
from equation (51) using the linear regression program.
Pursuant to the extinction coefficient, it is easy to find the product:
(52)
Where the functions f(&) and <p(0)have the form:
Tp exp6 (jr- l)' exp-[y2 6 (l- £)]/<p(.8)
¥(0) = p8(l - Q)E™ + pE\
(53)
Calculating numerically the combination i^ld^l'and using the sum rules (27) it is easy to determine and dfi.
Analytical expressions for kinetic parameters of the charge transfer process between polyatomic particles in an irregular condensed medium and expressions for the extinction coefficient for the corresponding processes of charge photo-transfer are obtained Based on the analysis of the obtained analytical
expressions of dark and optical charge transfer processes, correlations between the kinetic parameters of these processes are obtained. A methodology for determining the kinetic parameters of the charge photo-transfer process from the shape of the light absorption curve by the system as a function of the frequency of absorbed photons in various systems is presented. The general methodology presented has been applied to various specific processes.
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ФОТОКАТАЛИТИЧЕСКИЕ ПРОЦЕССЫ В НЕРЕГУЛЯРНЫХ СРЕДАХ
Т.А. Марсагишвили, Г.Д. Татишвили, М.Н. Мачавариани
Институт неорганической химии и электрохимии им. Р. Агладзе Тбилисского государственного университета им. И. Джавахишвили 0179, Тбилиси, пр. И. Чавчавадзе, 1, e-mail:tati@iice.ge
Все более востребованными становятся технологии, наносящие наименьший вред природе с точки зрения глобального потепления. В связи с этим фотокатализ, использующий энергию солнца для химических превращений веществ, представляет большой интерес для изучения возможных механизмов таких процессов. В то же время использование фотокатализа значительно расширяет возможности синтеза новых органических и неорганических материалов. Фотоперенос заряда всегда лежит в основе фотохимических превращений. Такие процессы можно рассматривать с точки зрения квантовой механики, а вероятность различных механизмов протекающих процессов можно рассчитывать с помощью аппарата функций Грина. Получены аналитические выражения для кинетических параметров процесса переноса заряда между многоатомными частицами в нерегулярной конденсированной среде и выражения для коэффициента экстинкции для соответствующих процессов фотопереноса заряда. На основе анализа полученных аналитических выражений процессов темнового и оптического переноса заряда получены корреляции между кинетическими параметрами этих процессов. Представлена методика определения кинетических параметров процесса фотопереноса заряда по форме кривой поглощения света системой в зависимости от частоты поглощенных фотонов в различных системах. Представленная общая методология применялась к различным конкретным процессам.
Ключевые слова: фотокатализ, фотоперенос, оптический спектр, коэффициент экстинкции, энергия реорганизации.
QEYRI-MÜNTdZdMMÜHiTLORDd FOTOKATALiTiKPROSESLdR T.A. Marsaqa§vili, Q.T. Tati§viU, M.N. Magavariani
i. Cavaxigvili adina Tbilisi Dövldt Universitetinin R. Aqladze adina Qeyri-üzvi vd Elektrokimya institutu 0179, Tbilisi, i. Qavgavadzepr.,1, e-mail:tati@iice.ge
Qlobal istildgmd baximindan tdbidtd daha az zdrdr verdn texnologiyalara tdldbat getdikcd daha da artir. Bu baximdan, gündg enerjisinin kimydvi enerjiyd gevrilmdsi ügün istifadd edildn fotokataliz bu cür prosesldrin mümkün mexanizmldrini öyrdnmdk ügün böyük maraq dogurur. Eyni zamanda, fotokatalizddn istifadd yeni üzvi vd qeyri-üzvi madddldrin sintez imkanlarini genigldndirir. Fotokimydvi gevrilmdldrin dsasini hdmigd yükldrin fotodaginmasi tdgkil edir. Bu cür prosesldrd kvant mexanikasi nöqteyi-ndzdrinddn baxmaq vd bag verdn prosesldrin müxtdlif mexanizmldrinin ehtimalini Qrin funksiyalari aparatindan istifadd etmdkld hesablamaq olar. Qeyri-müntdzdm kondensd olunmug mühitddki gox atomlu hissdcikldr arasinda yük dagima prosesinin kinetik parametrldri vd yükün fotodaginma prosesldrindd müvafiq ekstinsiya dmsali ügün analitik ifaddldr alinmigdir. Qaranliqda vd optik yükdaginma prosesldri ügün alinan analitik ifaddldrin tdhlili dsasinda bu prosesldrin kinetik parametrldri arasinda korrelyasiya dldd edilmigdir. Müxtdlif sistemldrdd udulmug fotonlarin tezliyinddn asili olaraq bir sistem tdrdfinddn igiq udma dyrisinin formasina dsasdn yük fotodaginmasi prosesinin kinetik parametrldrini tdyin etmdk ügün bir metod tdqdim olunmugdur. Tdqdim olunan ümumi metodologiya müxtdlif konkret prosesldrd tdtbiq edilmigdir. Agar sözfor: fotokataliz, fotodaginma, optik spektr, ekstinsiya dmsali, yeniddn tdgkil enerjisi.
