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324
CHEMICAL PROBLEMS 2024 no. 3 (22) ISSN 2221-8688
UDC 535.32
THEORETICAL MODELS OF CHARGE TRANSFER PROCESSES IN NONPOLAR
MEDIA
T. Marsagishvili, M. Machavariani
Ivane Javakhishvili Tbilisi State University, R. Agladze Institute of Inorganic Chemistry and Electrochemistry, Mindeli str. 11, 0186, Tbilisi, Georgia, e-mail: tamaz.marsagishvili@gmail.com
Received 06.01.2024 Accepted 17.03.2024
Abstract: The charge transfer processes in nonpolar media are considered in the work. The nonpolar medium is regarded as a field of random forces or impulses, and this field acts on the reacting particles. The interaction of reagents with the medium is described through the function offluctuation of the impulse of a particle of the medium with the speed of the reagent particle. When calculating the kinetic parameters of such a system, the quantum theory of charge transfer processes was used using Green's functions of the impulse density operators of the condensed medium. Analytical expressions for the kinetic parameters of the process are obtained.
Keywords: Green's functions, nonpolar medium, multichannel collision theory, charge transfer, quantum theory.
DOI: 10.32737/2221-8688-2024-3-324-331
Introduction
Kinetics of the elementary act of charge transfer processes in nonpolar media
Charge transfer processes in nonpolar media are of great interest of entire fields of knowledge in chemistry, electrochemistry, and photochemistry [1-11]. At the same time, theoretical models for charge transfer processes in polar condensed media have been developed to a much greater extent, see, for example, works [12 - 18].
The transition from polar to non-polar systems by formal extrapolation of the results to low
values of dielectric constant at zero frequencies, to low energies of reorganization of the medium, etc. can hardly be considered acceptable. Charge transfer processes in nonpolar media require a completely different approach [19, 20]. One of the approaches of taking into account the non-polar medium in the kinetics of the elementary act of charge transfer processes will be presented in this work. We will use the multichannel collision theory. We write the complete Hamiltonian of the system in the form:
H = Hi+Vi = Hf + Vf
(1)
Where Vt and Vf are the interactions between reacting particles in the initial and final states, Hi and Hf are the channel Hamiltonians of the
initial and final channels. We write the channel system in the form:
Hamiltonians of the
Я— Um _L_ UP J- ui'f
i,f = Hi,f + Hl,f + Hint
(2)
CHEMICAL PROBLEMS 2024 no. 3 (22)
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Here Hjj is Hamiltonian of medium, Hf is the
Hamiltonian of reacting particles, H- is the Hamiltonian of the interaction of particles with
the medium, for which we use the following representation:
HLt = -fP(r)v(r) dr
(3)
Where P(r) - is the fluctuation of impulse at the point r, v(r) is the speed of the reacting particles.
Interaction in form (3) means that we consider a non-polar medium in the form of a field of random forces or impulses, and this field dynamically affects the reacting particles. Integration is carried out over the entire space
over the dimensionless variable r. If we move from r to integration over Cartesian variables, then in formula (3) P should be understood as the impulse density.
The probability of an elementary act of charge transfer between particles with the transition of the system from electronic state i to electronic state f has the form:
Wur = 2rl
,ß(Fi-Ein)
i n Erw )
(4)
Where ty£ and tyf are the wave functions of the initial and final states of the system, Ein and Efn, are the corresponding energy levels, n and n' are all quantum numbers of the system,
except electronic ones, Fi is the free energy of the initial state.
Using the integral representation for the S -function, we reduce formula (4) to the form:
Wir =ßeßFi f^deSp[e-ß(1-0)HiVie-ßeHfVi]
(5)
Where the spur (trace) is assumed to be in all coordinates of the system, except electronic ones.
Substituting expressions for channel Hamiltonians (2) into formula (5), we have:
Wir = ß-eßFi filddSp \e-ß(1-e)[Hin+Hi}+Hiint}vie-ße[H^"+H^+H^nt]vi
(6)
We will carry out further calculations in the approximation in which the spur can be divided into two spurs using formula (6) - according to the coordinates of the medium, and according to the coordinates of the reacting particles. This is
possible if
\j4m -L.
[Hi,r + HiJ'
the
movement of the reactants is of a classical nature or, in the case of a quantum nature, the changes in the velocities of the reacting particles during the charge transfer process are small, then the commutator can be neglected and for the probability of an elementary act we obtain:
Wir = eßFn filiSp
-ßa-e)[Hm+Hiint]-ße[Hm+H(nt]
]fP(9)de
(7)
fP(9) = eßF?Sp [e-ß(1-0)Hi3Vie-ßeHf]
(8)
Where F™ is the free energy of the subsystem H + H\nt, Ff is the free energy of the reactants in the initial state. When calculating states for non-polar media, the corresponding gas-phase
(vacuum) values should be taken as the frequencies of intramolecular vibrations and the equilibrium lengths of the corresponding chemical bonds.
We write V[f the electronic resonant integral of state "/' in the form: the transition of the system from state "i" to
%= V* + ZsbsQs
(9)
where Qs are the coordinates of intramolecular After calculating the function fp(6) we obtain: vibrations of the reagents.
P(0) = e
6(I-6)Ü>1sÜ>{ -ßhs Ers-T2-—TT
ßFf+ße(]-]f)e (1-8WS +e(<Js)
7 0
Kl2 + ZsWfb; + bsy-*
(i-e)(^s)2+e(Js)
(10)
Here Ji and J are the minimum energies of frequencies of oscillations of the 5-th degree of
impurity in the initial and final states, freedom of reagents in the initial and final
respectively; F? is the free energy of the states, respectively.
reactants in the initial state, Ers is the Calculation of a spur using the coordinates of
reorganization energy of the 5-th degree of the medium can only be carried out
freedom of the regents, œls
«
are
approximately. Let's rewrite it in the form:
fm(0) = {ef
= {eße{Hm+Hlint) -ße(
Hm+Hint+H&t
h,
where,
Hi
fi _
int
J P(r) [v?(r) — vl(r)]dr = J dr Pffià'v' (r)
If we introduce the S-matrix for the interaction , then
where
fm(d) = {S(ß0)) S(ß9)= Trexp[—jHeH^^t(T)dT]
(11) (12)
(13)
(14)
Let us expand the exponential in the S-matrix into a series:
ZW rP8 r rP6 r rP6
MJ dnj dTiJdr^J dT2...J drj?J dTkPai(r\T1)
X Pa2 (J2^2).. . Pak Qr\T1)Ava2 i^). •. Qf^Tk))i
(15)
We will accept the approximation of decoupling this type of decoupling is described in some quantum statistical averages of fluctuation detail in [20]. In addition, for fp(6) we will operators of impulse density into pair averages restrict ourselves to the classical approximation. according to rules similar to those of Wick's As a result we get: theorem [19]. The possibility of carrying out
(S(pe))i = --J drd? fPedT1tfedT2GPaPp(r,?,T-T')Ava(?)Avp(?)
Where GPaPp is the temperature Green function density of the medium: of the fluctuation operators of the impulse
(16)
2
2
GPaPp (fP.T -T')= - (TTPa(r,x)Pp (?,T'))i (17)
A A v(r) = vf (r) - vl(r) (18)
In the Green function GPaP we move on to the Fourier representation by t-t':
Gpapp(fP,T-T') = -IZ^GpaPpify^^e-1^-^ (19)
In order to be able to calculate the sum over n function at finite temperatures GPap . To do
and obtain the value of the Green function , .. , .. p, .
. ... this, we use the correlation relations between
GPaPp not only at discrete points Mn, but also on the Green functions GPP and GRP:
the entire complex frequency m plane, it is
necessary to move on to the time retarded Green
GPP(un) = GPP(iun), Mn>0 (20)
GPP(Mn) = GPP(-Mn)
and dispersion relations for the Green function GPP:
GRP(un) = du'JmGrM (21)
ppv n nJ-z u'-iun v '
As a result we get:
fm(6) =
exp{ß6 J drdr' GRPaPß(r,r'. m = 0)Ava(r)Avß(r) +
UZ dfd?,m GRPcipf(7,?. u)Ava(.mvß(f) (22)
The Green function of fluctuations of the obtain an expression for the mean square force
impulse density of a medium is closely related acting on the molecule; calculate the diffusion
to all kinds of functions of velocities, forces, coefficient for the friction coefficient in a liquid;
and impulses in condensed media. Such study the processes of thermal conductivity and
correlations make it possible to study the electrical conductivity (for details on transfer
influence of the medium on a specific molecule; phenomena in condensed systems, see [20]).
Correlation function
The correlation function K^ifP.u) is relationship between these functions has a
the most convenient function for relation with particularly simple form for the Fourier
the Green function GRPaPB(f,r' ,m). The components:
GRP p(r,?,u)=1Jm dM s ^ 0 (23)
pcpß\ ' ' J 2nJ-z u'-u-iS ' v '
We write the correlation function Kaß(r;:P,a) of impulses and the correlation fonction in the factorization approximation, expressing it over spatial coordinates Gaß(r.r'): through the normalized autocorrelation function
Kaß (r. t) = Gaß (r. P) W(t) (24)
For radially symmetric interaction, the function can use a number of model functions of the Gap(r,r') can be replaced by the function type:
SaßG — r'). As a function G(r — r'), you G(r r ) ~ S (r r )
G (|r — ?"'|) ~ exp [A(r — 7"')] G (r — r' )~ exp [A(r — r' )] • cos [^(r — r' )]
(25)
Where the parameters A and p can be defined as fitting ones.
The time dependence of the normalized autocorrelation functions of impulses for various condensed systems has been studied in some detail in a number of works [20]. The most commonly used model of the medium is
one in which the motion of atoms is assumed to be oscillations near some average position, which in turn experiences Brownian motion. To describe Brownian motion, the Langevin equation is used for the motion of a molecule in a statistically equilibrium medium.
W(œ) = 2j^V(t)cosœtdt,
(26)
The following expression was obtained in work [20]:
D ( y2y2 . yy Ate2
+
(te2+y2)te2+y2y2 te2 M2+yr
+
2 A2
(27)
Here D is diffusion coefficient in medium, Y is friction coefficient, y' is characteristic time of decrease of the autocorrelation function of the stochastic force, is the cutoff frequency of the spectrum of stochastic forces in the medium, A is the ratio of the root-mean-square value of the fluctuation of the force acting on the oscillator (an atom performing oscillatory motion) to the root-mean-square value of the fluctuation of the force acting on the center of
oscillations and determining the Brownian motion of atoms.
Finally, the velocity for the reactant particles must be determined. The movement of reactants in a field of random forces can be considered in the same way as a condensed nonpolar medium itself is considered.
For reactants, equations for stochastic motion with damping can be written in the form [20]:
f+ J0>r(t — t')V(t')dt' = F(t)
(28)
This equation is written in matrix form, and here
V(t) =
dr/dt dR/dt
; r(t) =
KV + v?
—M2
—M?
y(t) + v2
; F(t) =
A(t) lB(t)
(29)
Where r - describes the instantaneous position of the particle's center of gravity, R is the average position of the particle's center of oscillation, A(t) is the fluctuation force in the medium leading to particle oscillations; B(t) is the fluctuation force in the medium, leading to Brownian motion; ^(t)- takes into account the effect of friction force retardation, y(t) is
dependence of the friction coefficient on time, can be expressed through the corresponding
frequency value m0: where m is
the mass of the particle, m* is its effective mass (takes into account the mass of the particle and several atoms of the medium surrounding the particle).
te
Conclusion
The processes of charge transfer in nonpolar media were considered. The nonpolar medium is represented as a field of random forces or impulses, and this field acts on the reacting particles. The interaction of reagents with the environment is represented through the function of fluctuation of the impulses of the particles of the medium with the velocities of the reagent particles.
When calculating the kinetic parameters of such a system, the quantum theory of charge transfer processes was used using Green's
functions of the impulse density operators of the condensed medium. Analytical expressions for the kinetic parameters of the process are obtained.
Further calculations require detailing the potentials of intermolecular interactions; the nature of fluctuations of random forces and impulses in the medium; the selection of parameter values from experiments and boundary conditions for solving the given equations.
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QEYRÍ-POLYAR MÜHÍTDO YÜKLORÍN DAÇINMASI PROSESÍNÍN NOZORi
MODELLORÍ
T.Marsaqiçvili, M.Maçavariani
ivane Cavaxiqvili adina Tbilisi Dövldt Universiteti, R. Aqladze adina Qeyri-üzvi Kimya vd Elektrokimya
institutu,
Mindeli kûç., 11, 0186, Tbilisi, Gürcüstan, e-poçt: tamaz.marsagishvili@gmail.com
Xülasa: í§da qeyri-polyar mühitlarda yüklarin daçinmasi proseslari ara§dirilmi§dir. Qeyri-polyar mühit kimi tasadüfi qüvvalar va ya impulslar sahasi hesab olunur va bu saha qar§iliqli tasirda olan hissaciklara tasir göstarir. Reagentlarin mühitla qar§iliqli tasiri mühitda olan hissaciklarin impulsunun reagentin hissaciklarinin süratindan asililiq funksiyasi vasitasila tasvir edilir. Bela sistemlarin kinetik parametrlarinin hesablanmasi zamani kondensa olunmu§ mühitin impuls sixligi operatorlarinin Qrin funksiyalarindan va yük daçinmasi proseslarinin kvant nazariyyasindan istifada edilmiçdir. Prosesin kinetik parametrlari ûçûn analitik ifadalar alinmiçdir.
Açar söztari: Qrin funksiyalari, qeyri-polyar mühit, çoxkanalli toqquçma nazariyyasi, yük daçinmasi, kvant nazariyyasi.
ТЕОРЕТИЧЕСКИЕ МОДЕЛИ ПРОЦЕССОВ ПЕРЕНОСА ЗАРЯДА В НЕПОЛЯРНЫХ
СРЕДАХ
Т.Марсагишвили, М.Мачавариани
Тбилисский государственный университет имени Иване Джавахишвили, Институт неорганической
химии и электрохимии им. Р. Агладзе, ул. 11, 0186, Тбилиси, Грузия, e-почта: tamaz.marsagishvili@gmail.com
Резюме: В работе рассмотрены процессы переноса заряда в неполярных средах. Неполярная среда рассматривается как поле случайных сил или импульсов, и это поле действует на реагирующие частицы. Взаимодействие реагентов со средой описывается через функцию изменения импульса частицы среды со скоростью частицы реагента. При расчете кинетических параметров такой системы использовалась квантовая теория процессов переноса заряда с использованием функций Грина операторов плотности импульса конденсированной среды. Получены аналитические выражения для кинетических параметров процесса.
Ключевые слова: функции Грина, неполярная среда, многоканальная теория столкновений, перенос заряда, квантовая теория.
