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The calculation of magnetic effects and RYDMR spectra for intermediate short-lived complexes of paramagnetic species
31March 1995
CMEMICAL PHYSICS LETTERS
ELSEVIER
Chemical Physics Letters 235 (1995) 591-595
The calculation of magnetic effects and RYDMR spectra for intermediate short-lived complexes of paramagnetic species S.l. Kubarev, I.S. Kubareva, E.A. Ermakova Institute of Chemical Physics, RussianAcademy of Sciences, Moscow, Russian Federation
Rcccived 13 January 1995
Abstract A method based on the symbolic solution of the stochastic Liouville equation for the calculation of magnetic effects and RYDMR spectra is suggested. It allows the .umber of interrelated equations to be cut down and it is also attractive for time-resolved problems and intermediate short-lived complexes of paramagnetic species. By way of illustration quantum beats are considered and a high sensitivity to changes in radical pair parameters and variable magnetic field intensity is detected.
1. Introduction
The starting point in the theory of magnetic effects for arbitrary spin systems is the stochastic Liouville equation (SLE), which is given by (h = 1) 8/3 i~t
where f3 is the system density matrix; ~ is the spin-Hamiltonian descriptive of the spin coherent motion; P is the operator descriptive of the random motion of the spin complex components, including multichannel decay processes; K is the linear operator in the electron-nuclear- spin space which is responsible for recombination, annihilation and like changes within the complex. The traditional method for attacking problems devoted to investigations on the role of intermediate short-lived states of paramagnetic species in the kinetics of chemical transformations [1-6] is representElsevier Science B.V. SSDI 0009- 2614(95 )00158-1
ing the matrix equation of (1) as a set of interrelated differential equations. If the system can be characterized by the time-independent Hamiltonian ~ or if or~e can eliminate the time dependence of ~ (for example, by turning to the rotating frame of refere~tce, which is typical for RYDMR spectroscopy), then the problem reduces to the solution of a set of algebraic equation, s. As a rule, the set of interrelated differential or algebraic equations breaks up into isolated diagonal blocks only in some specific cases. At the same time not all the functions are needed for the ultimate result. The diagonal elements of the density matrix ~ are of interest, i.e. the number of functions required for the specific applications is not more than v/N where N is the total number of equations. Although, up to now, only pairs of paramagnetic species (such as radical pairs (RP), positronium ( e . . . e +), electron-hole ( e . . . h), two triplets (T...T), doublet-triplet ( D . . . T ) and so on) haw been considered as spin systems, it is evident that
592
S.L Kubaret' et al. / Chemical Physics Letters 235 (1995) 591-595
equation type (1) can be used for the investigation of intermediate short-lived states of complexes, i.e. three or more paramagnetic species. Such complexes may also result from different chemical reactions, It is reasonable that the number of interrelated equations for intermediate complexes is much more than the number of equations for pairs of paramagnetic species. In the case when ~ is the time-dependent Hamiltonian and there is no way to get rid of the time-dependence by any manipulation, then the difficulties increase because of the rise in the number of equations. There also exists a problem in searching for the time-dependent solution of the SLE. This problem for tasks related to constant magnetic field effects has been discussed previously [6-9]. The importance of solving this problem increases when using pulsed methods upon spin systems. The solution of this problem is of great importance for RYDMR spectroscopy and, especially, for time-dependent RYDMR spectroscopy. The detection of quantum beats ',by hyperfine interaction in Ref. [10] and by the Ag-mechanism in Ref. [11]) has been one of the first steps in this promising field. The increasing experimental possibilities require further improvements in the test models. The possibilities of the cage model, for example, may be extended if the rate constants of dissociation and recombination are considered as functions of time. The spin-Hamiltonian parameters may also be functions of time. Thus optically induced spin conversion effects may be simulated by the discontinuous jump of one of the radical g factors (of RP or complex), as well as by the discontinuous jump of the local magnetic field acting on one of the radicals, The structural rearrangement (such as proton transfer, hydrogen atom transfer and so on) involved during the lifetime of the intermediate complex may be approximated by the discontinuous jump of hyperfine interaction constants (hfi constants). Thus with the help of magnetic effects and especially with the help of RYDMR spectroscopy it is possible to study not only the structure of the intermediate short-lived states, but also the dynamics of their transformations, From what has been said it is evident that a search for other methods to find iS(t) is an important task,
especially such methods which allow cutting down the number of equations required for the final result.
2. Method In this Letter a method, which not only cuts down the number of interrelated equations, but is also convenient for models of complexes and for new interactions with the surroundings, is suggested. This method is based on the symbolic solution of the SLE (1), which may be given for many tasks by [12] 0_~= _ i [ , g ~ ( t ) ~ ( t ) - ~(t),,~'* (t)] (2) ~tt Here ,~'(t) is some non-Hermitian operator, it is equal to the sum of the Hermitian spin-Hamiltonian of the system and the non-Hermitian members connected with the decay and transformation process in the complex. We can correlate the operator ,~'(t) to the evolution operator :~./(t), which must satisfy the following equation:
i - - = ~? (t)~')(t), (3) dt with the initial conditions ~'/(0) = 1. With the help of the evolution operator it is easy to construct the formal solution of Eq. (2),
= (t). (4) We will illustrate the new method with the example of the RP with no magnetic nuclei. It is the simplest, system and extension of the method to more complicated systems would be no special problem. A typical form of the radical pair SLE with different rates of singlet and triplet recombination is given by [1-61 St - Hs[/~s, ~] + - HT[ PT, ~1 +,
(5)
where [ ]_ and [ ]+ denote the commutator and anticommutator, respectively; H d is the rate constant of dissociation; H s and H r are the rate constants of singlet and triplet recombination; /~s is the projection operator into the singlet state IS); PT is the projection operator into the triplet state IT). It should be pointed out that in Eq. (5) and in the
S.I. Kubarec et al. / Chemical Physics Letters 235 (1995) 591-595
following equations throughout this Letter all the terms have dimensions of magnetic field [4,5]. The investigation of equations such as (1) and (5) allows almost total information on the spin system. For the singlet conversion one must calculate the value Tr(/~s t3(t)) or the value fo Tr(/~s t3(t)). For example, the RP recombination probability is expressed by
Thus when Eq. (2) is compared with Eq. (5), it is apparent that ^
^
I o
,7g'.(t) =,,~,,(t) + iHs/~s + iHT/a.r + ~lHd,
G,,,,(t) =,
(8) 8,,,,,. Here
with the ob,,ious initial condition G,m(O) = In), Ira) are total-spin basis functions. The diagonal elements of the density matrix ~(t) at any yield and at any predecessor, required for some special tasks, may be expressed by the functions (8), Oss(t) --pl~(t) -- IGit(t) l 2, for singlet predecessor PSS(t)
=Pll(t)
E IG~m(t) l 2 ,
"~
m = 2
for triplet predecessor 4
Pss(t)=pll(t)=¼
~ IG~,,,(t)l", m
= 1
for random predecessor
(9)
4
• (t) = E g,,,(t) = IG21(t)l 2 11 = 2
+ IG3,(t) 12+ IG41(t) l z, 4
4
q~(t)-- E g,,,(t)--½ E [IGz,,,(t) l z n=2
hi=2
+la3m(t) 12+ IG4,,(t) I 2], 4
• (t)=
4
Y'~ p,,,(t) =¼ Y'. n=2
• dGh,
dt
4
= Y'.
k=J
Ill)
As seen from Eq. (11), the functions G2,,,(t), G3,,(t) and G4o,(t) also appear in the motion equation for G~,,(t). Setting up the motion equation for these functions as well, will give the complete system of four differential equations of first order which can be represented in matrix form, i
dG
(12)
dt
where
c],,,'
4 I
The functions G,,,(t) must obey the motion equation which directly follows from Eq. (3). Now we will demonstrate that the group of functions appearing in expressions (9) and (10) satisfy the same set of four differential equations of first order. It should be noted that particular groups of these functions represent different solutions corresponding to different initial conditions. By way of example let us consider the function Gin(t)= ( l l ~ ( t ) l m ) and then set up the motion equation for it,
(7)
where ~:]~ is the spin-Hamiltonian of the RP. Furthermore, some auxiliary functions which are important for the following are identified,
593
[IG2m(t) l z
m=2
+lG3m(t) lZ + [G4,,(t) I2]. (10)
As the matrix elements of ~ are independent of the index 'm' (see Eq. (1 1)) for m = 1, 2, 3, 4 we will have four different solution groups of the same set of equations. It is easily seen that these groups of solutions correlate with different initial conditions. Eventually we will have a complete set of functions solving the given problem of searching for the density matrix elements according to (9) and (10). It stands to reason that such a complete set is not always required• For example, in the case of a singlet predecessor for the calculation of singlet yields only the function Gll(t) is needed; if the triplet yield is also required, we will also need the functions G,j(t), G31(t) and G41(t) (see (9)). However, all these four functions appear in the same solution set and so one needs to refer to the system only once (see Eq. (11) for m = 1). So far the general case has been under consideration. In specific cases some further facilities are
594
S.L Kubarec et aL /Chemical Physics Letters 235 (1995) 591-595
possible. For instance, in the case when the operators ,,~)'(t) and ~ ( t ) are commutative (such a situation always takes place if ~ is the time-independent Hamiltonian), the equation for the function G~=(t) may be returned in the form • dGlm
dt
06
Pss 0,4
-- (1 t ~ ; ~ ( t ) ~ ' ( t ) I m ) 0.2
=
(ll~'(t)~(t)Im) 4
= E
~"*.(t)G,k(t),
(13)
k=l
0,0
V 0,0
Putting m = !, 2, 3, 4 we have a complete set of four equations for the functions Gll(t), Gl,(t), Gi3(t) and Gi4(t), which govern the matrix element Pss(t) = pll(t) for all kinds of predecessors, This system in matrix form is dG =~ i ' ~ = G~',
(14)
where G = ( G t l , G=2, Gt3, Gi4); the initial condition is G(0) = (1, O, 0, 0). We emphasize that for the general case, as well, the problem is reduced to consideration of only the total system which consists of four equations, because we treat the system separately every time. The traditional way gives a set of 16 interrelated equations even when only one function is needed. ~0
Pss O8
016
, 0,5
1.0
15
2,0
2,5
3,0
T
Fig, 2, The quantum beats curves for RP with no Magnetic nuclei for tile triplet predecessor. Parameters are the same as for Fig, I, (-----) The curve in the absence of a variable magnetic field; (.... ) the curve in the presence of a variable magnetic field of 4.20e.
Next let us take a look at the RP with magnetic nuclei. In a secular approximation one needs only to make a substitution of G with (~(J~..... J.) and with ,,~"~J*..... ~-~,where Jl . . . . . j, determine different subensembles of the RP, answering the particular sets of quantum numbers: Jl = 1, 2 . . . . . 211 4. 1; j,, = 1, 2 . . . . . 2/,, + 1; here ! is nuclear spin. As in a secular approximation each of the ensembles is treated separately, so once again, the problem is reduced to examination of the complete set of four equations. It can be shown that when rigorously solving the problem on the 'RP with arbitrary number of magnetic nuclei and with arbitrary nuclear spin' system one must deal with a set of equations, the number of which coincides with the basis dimension.
04
3. Some applications
0,0 00
05
1.0
15
20
25
30
1" Fig, I, The quantum heats curves in the case of a A g mechanism of spin conversion for RP w'tb,no magnetic nuclei fin the singlct prede~.'~,~or, H,~ = 4(XI(| Oc; to = 4000 Oe; J = 0; g= = 2.0063; g~ = 2,0013; H s = 1,50e; H d = 0.567 Oe, ( ) The curve in the absence of a variable magnetic field H=: ( . . . . ) the curve in the presence of a variable magnelic field of 4 . 4 0 e ,
Since this Letter is mainly devoted to the presentation of the method, we shall restrict our consideration of its applications only to quantum beats. We do not demonstrate RYDMR spectra for RP, because the results gained by using (14) coincide with those obtained in Refs. [1,4,5]. The investigation of the quantum beats phenomena is of interest because of its high sensitivity to changes in RP parameters and variable magnetic field intensity.
S.I. Kubarer et al. / Chemical Physics Letters 235 (1995) 591-595 1.0
,Oss 0.8
0.6
0.4
0.2
0.0
0,0
0.5
t .0
1,5
20
2,5
3.0
"T
Fig. 3. The dynamics of the changes in the quantum beats spectra under discontinuous jumping of the g-factor, H i = 4 . 4 0 ~ ; H o = 400(I Oe; ¢o = 4011(I Oe; It d = 11.567 Oe; H s = 1 . 5 0 e . ( ~ - ) The curve in the absence of the discontinuous jump of gl (Ag = 0.002); ( . . . . ) the curve in the presence of the discontinuous jump of gl (Ag =0.009) at t = 5 . 6 7 9 × 10 -H s.
Thus Figs. 1 and 2 give the relationship between the quantum beats signal (for RP with no magnetic nuclei for singlet and triplet predecessors, respee-
595
tively) and the variable magnetic field intensity. For the singlet predecessor (see Fig. 1) later in the process of quantum beats a detectable increase in the signal at certain variable magnetic field intensity is evident. For the triplet predecessor (see Fig. 2) an increase in the signal integrated intensity is observed. Analogous effects are observed for RP with magnetic nuclei. Figs. 3 and 4 show the dynamics of the changes in the quantum beats spectra on discontinuous jumping of the g-factor and hfi constant of one of the radicals, respectively. As noted above, the discontinuous jump of the g-factor simulates optically induced spin conversion effects, and the discontinuous jump of the hfi constant simulates the structural rearrangement process occurring during the lifetime of the intermediate complex. Notice that the abscissa in all the figures, time (1/Oe), is plotted in the following scale: 1
( O e ) - l = 5 . 6 7 9 × 10 - 8 s.
References 10
Pss 08
0,6
04
0,2
0,0 0.0
0.5
1.0
1,5
20
2,5
3,0
T
Fig. 4. The quantum beats under discontinuousjumping of the hfi constant A. H a = 4000 Oe; H d = 0.567 Oe; H s = 1 . 5 0 e ; d = 0; a~ = 4000 Oe. H I = 3 0 e . ( - ) The curve in the absence of the discontinuous jump of A (A = 16); ( . . . . ) the curve in the presence of the discontinuous jump of A (A = 64) at t = 5.679 × 10 -~ s.
[1] U.E. Steiner and T. Ulrich, Chem. Rev. 89 (1989) 51. [2] M.E. MicheI-Beyerle, R. Haberkorn, W. Bude, E. Steffens, H. Schroder, H.J. Neusser, E.W. Schlag and H. Heidlitz, Chem. Phys. 17 (1976) 139. [3] K. Schulten, H. Staerk, A. Weller, H.J. Werner and B. Nickel, Z. Physik. Chem. II)1 (1976) 371. [4] E.A. Ermakova and S.I. Kubarev, J. Chem. Phys. I1 (1992) 73. [5] E.A. Ermakova and S.I. Kubarcv, J. Chem. Phys. I1 (1992) 857. [6] J. Tang and J.R. Norris, Chem. Phys. Letters 92 (1982) 136. [7] K. Lendi, Chem. Phys. 20 (1977) 135. [8] H.J. Wcrner, Z. Schulten and K. Schulten, J. Chem. Phys. 67 (1977) 646. [9] R. Haberkorn, Chem. Phys. 26 (1977) 35. [10] O.A. Anisimov, V.L. Bizyaev, N.N. Lukzen V.M. Grigoryants and Yu.N. Molin, Chem. Phys. Letters 101 (1983) 131. [11] A.V. Veselov, V.I. Melekhov, O.A. Anisimov and Yu.N. Molin, Chem. Phys. Letters 136 (1987) 263. [12] R. Haberkorn and W. Dietz, Solid State Commun. 35 (1980) 505.
CMEMICAL PHYSICS LETTERS
ELSEVIER
Chemical Physics Letters 235 (1995) 591-595
The calculation of magnetic effects and RYDMR spectra for intermediate short-lived complexes of paramagnetic species S.l. Kubarev, I.S. Kubareva, E.A. Ermakova Institute of Chemical Physics, RussianAcademy of Sciences, Moscow, Russian Federation
Rcccived 13 January 1995
Abstract A method based on the symbolic solution of the stochastic Liouville equation for the calculation of magnetic effects and RYDMR spectra is suggested. It allows the .umber of interrelated equations to be cut down and it is also attractive for time-resolved problems and intermediate short-lived complexes of paramagnetic species. By way of illustration quantum beats are considered and a high sensitivity to changes in radical pair parameters and variable magnetic field intensity is detected.
1. Introduction
The starting point in the theory of magnetic effects for arbitrary spin systems is the stochastic Liouville equation (SLE), which is given by (h = 1) 8/3 i~t
where f3 is the system density matrix; ~ is the spin-Hamiltonian descriptive of the spin coherent motion; P is the operator descriptive of the random motion of the spin complex components, including multichannel decay processes; K is the linear operator in the electron-nuclear- spin space which is responsible for recombination, annihilation and like changes within the complex. The traditional method for attacking problems devoted to investigations on the role of intermediate short-lived states of paramagnetic species in the kinetics of chemical transformations [1-6] is representElsevier Science B.V. SSDI 0009- 2614(95 )00158-1
ing the matrix equation of (1) as a set of interrelated differential equations. If the system can be characterized by the time-independent Hamiltonian ~ or if or~e can eliminate the time dependence of ~ (for example, by turning to the rotating frame of refere~tce, which is typical for RYDMR spectroscopy), then the problem reduces to the solution of a set of algebraic equation, s. As a rule, the set of interrelated differential or algebraic equations breaks up into isolated diagonal blocks only in some specific cases. At the same time not all the functions are needed for the ultimate result. The diagonal elements of the density matrix ~ are of interest, i.e. the number of functions required for the specific applications is not more than v/N where N is the total number of equations. Although, up to now, only pairs of paramagnetic species (such as radical pairs (RP), positronium ( e . . . e +), electron-hole ( e . . . h), two triplets (T...T), doublet-triplet ( D . . . T ) and so on) haw been considered as spin systems, it is evident that
592
S.L Kubaret' et al. / Chemical Physics Letters 235 (1995) 591-595
equation type (1) can be used for the investigation of intermediate short-lived states of complexes, i.e. three or more paramagnetic species. Such complexes may also result from different chemical reactions, It is reasonable that the number of interrelated equations for intermediate complexes is much more than the number of equations for pairs of paramagnetic species. In the case when ~ is the time-dependent Hamiltonian and there is no way to get rid of the time-dependence by any manipulation, then the difficulties increase because of the rise in the number of equations. There also exists a problem in searching for the time-dependent solution of the SLE. This problem for tasks related to constant magnetic field effects has been discussed previously [6-9]. The importance of solving this problem increases when using pulsed methods upon spin systems. The solution of this problem is of great importance for RYDMR spectroscopy and, especially, for time-dependent RYDMR spectroscopy. The detection of quantum beats ',by hyperfine interaction in Ref. [10] and by the Ag-mechanism in Ref. [11]) has been one of the first steps in this promising field. The increasing experimental possibilities require further improvements in the test models. The possibilities of the cage model, for example, may be extended if the rate constants of dissociation and recombination are considered as functions of time. The spin-Hamiltonian parameters may also be functions of time. Thus optically induced spin conversion effects may be simulated by the discontinuous jump of one of the radical g factors (of RP or complex), as well as by the discontinuous jump of the local magnetic field acting on one of the radicals, The structural rearrangement (such as proton transfer, hydrogen atom transfer and so on) involved during the lifetime of the intermediate complex may be approximated by the discontinuous jump of hyperfine interaction constants (hfi constants). Thus with the help of magnetic effects and especially with the help of RYDMR spectroscopy it is possible to study not only the structure of the intermediate short-lived states, but also the dynamics of their transformations, From what has been said it is evident that a search for other methods to find iS(t) is an important task,
especially such methods which allow cutting down the number of equations required for the final result.
2. Method In this Letter a method, which not only cuts down the number of interrelated equations, but is also convenient for models of complexes and for new interactions with the surroundings, is suggested. This method is based on the symbolic solution of the SLE (1), which may be given for many tasks by [12] 0_~= _ i [ , g ~ ( t ) ~ ( t ) - ~(t),,~'* (t)] (2) ~tt Here ,~'(t) is some non-Hermitian operator, it is equal to the sum of the Hermitian spin-Hamiltonian of the system and the non-Hermitian members connected with the decay and transformation process in the complex. We can correlate the operator ,~'(t) to the evolution operator :~./(t), which must satisfy the following equation:
i - - = ~? (t)~')(t), (3) dt with the initial conditions ~'/(0) = 1. With the help of the evolution operator it is easy to construct the formal solution of Eq. (2),
= (t). (4) We will illustrate the new method with the example of the RP with no magnetic nuclei. It is the simplest, system and extension of the method to more complicated systems would be no special problem. A typical form of the radical pair SLE with different rates of singlet and triplet recombination is given by [1-61 St - Hs[/~s, ~] + - HT[ PT, ~1 +,
(5)
where [ ]_ and [ ]+ denote the commutator and anticommutator, respectively; H d is the rate constant of dissociation; H s and H r are the rate constants of singlet and triplet recombination; /~s is the projection operator into the singlet state IS); PT is the projection operator into the triplet state IT). It should be pointed out that in Eq. (5) and in the
S.I. Kubarec et al. / Chemical Physics Letters 235 (1995) 591-595
following equations throughout this Letter all the terms have dimensions of magnetic field [4,5]. The investigation of equations such as (1) and (5) allows almost total information on the spin system. For the singlet conversion one must calculate the value Tr(/~s t3(t)) or the value fo Tr(/~s t3(t)). For example, the RP recombination probability is expressed by
Thus when Eq. (2) is compared with Eq. (5), it is apparent that ^
^
I o
,7g'.(t) =,,~,,(t) + iHs/~s + iHT/a.r + ~lHd,
G,,,,(t) =
(8) 8,,,,,. Here
with the ob,,ious initial condition G,m(O) = In), Ira) are total-spin basis functions. The diagonal elements of the density matrix ~(t) at any yield and at any predecessor, required for some special tasks, may be expressed by the functions (8), Oss(t) --pl~(t) -- IGit(t) l 2, for singlet predecessor PSS(t)
=Pll(t)
E IG~m(t) l 2 ,
"~
m = 2
for triplet predecessor 4
Pss(t)=pll(t)=¼
~ IG~,,,(t)l", m
= 1
for random predecessor
(9)
4
• (t) = E g,,,(t) = IG21(t)l 2 11 = 2
+ IG3,(t) 12+ IG41(t) l z, 4
4
q~(t)-- E g,,,(t)--½ E [IGz,,,(t) l z n=2
hi=2
+la3m(t) 12+ IG4,,(t) I 2], 4
• (t)=
4
Y'~ p,,,(t) =¼ Y'. n=2
• dGh,
dt
4
= Y'.
k=J
Ill)
As seen from Eq. (11), the functions G2,,,(t), G3,,(t) and G4o,(t) also appear in the motion equation for G~,,(t). Setting up the motion equation for these functions as well, will give the complete system of four differential equations of first order which can be represented in matrix form, i
dG
(12)
dt
where
c],,,'
4 I
The functions G,,,(t) must obey the motion equation which directly follows from Eq. (3). Now we will demonstrate that the group of functions appearing in expressions (9) and (10) satisfy the same set of four differential equations of first order. It should be noted that particular groups of these functions represent different solutions corresponding to different initial conditions. By way of example let us consider the function Gin(t)= ( l l ~ ( t ) l m ) and then set up the motion equation for it,
(7)
where ~:]~ is the spin-Hamiltonian of the RP. Furthermore, some auxiliary functions which are important for the following are identified,
593
[IG2m(t) l z
m=2
+lG3m(t) lZ + [G4,,(t) I2]. (10)
As the matrix elements of ~ are independent of the index 'm' (see Eq. (1 1)) for m = 1, 2, 3, 4 we will have four different solution groups of the same set of equations. It is easily seen that these groups of solutions correlate with different initial conditions. Eventually we will have a complete set of functions solving the given problem of searching for the density matrix elements according to (9) and (10). It stands to reason that such a complete set is not always required• For example, in the case of a singlet predecessor for the calculation of singlet yields only the function Gll(t) is needed; if the triplet yield is also required, we will also need the functions G,j(t), G31(t) and G41(t) (see (9)). However, all these four functions appear in the same solution set and so one needs to refer to the system only once (see Eq. (11) for m = 1). So far the general case has been under consideration. In specific cases some further facilities are
594
S.L Kubarec et aL /Chemical Physics Letters 235 (1995) 591-595
possible. For instance, in the case when the operators ,,~)'(t) and ~ ( t ) are commutative (such a situation always takes place if ~ is the time-independent Hamiltonian), the equation for the function G~=(t) may be returned in the form • dGlm
dt
06
Pss 0,4
-- (1 t ~ ; ~ ( t ) ~ ' ( t ) I m ) 0.2
=
(ll~'(t)~(t)Im) 4
= E
~"*.(t)G,k(t),
(13)
k=l
0,0
V 0,0
Putting m = !, 2, 3, 4 we have a complete set of four equations for the functions Gll(t), Gl,(t), Gi3(t) and Gi4(t), which govern the matrix element Pss(t) = pll(t) for all kinds of predecessors, This system in matrix form is dG =~ i ' ~ = G~',
(14)
where G = ( G t l , G=2, Gt3, Gi4); the initial condition is G(0) = (1, O, 0, 0). We emphasize that for the general case, as well, the problem is reduced to consideration of only the total system which consists of four equations, because we treat the system separately every time. The traditional way gives a set of 16 interrelated equations even when only one function is needed. ~0
Pss O8
016
, 0,5
1.0
15
2,0
2,5
3,0
T
Fig, 2, The quantum beats curves for RP with no Magnetic nuclei for tile triplet predecessor. Parameters are the same as for Fig, I, (-----) The curve in the absence of a variable magnetic field; (.... ) the curve in the presence of a variable magnetic field of 4.20e.
Next let us take a look at the RP with magnetic nuclei. In a secular approximation one needs only to make a substitution of G with (~(J~..... J.) and with ,,~"~J*..... ~-~,where Jl . . . . . j, determine different subensembles of the RP, answering the particular sets of quantum numbers: Jl = 1, 2 . . . . . 211 4. 1; j,, = 1, 2 . . . . . 2/,, + 1; here ! is nuclear spin. As in a secular approximation each of the ensembles is treated separately, so once again, the problem is reduced to examination of the complete set of four equations. It can be shown that when rigorously solving the problem on the 'RP with arbitrary number of magnetic nuclei and with arbitrary nuclear spin' system one must deal with a set of equations, the number of which coincides with the basis dimension.
04
3. Some applications
0,0 00
05
1.0
15
20
25
30
1" Fig, I, The quantum heats curves in the case of a A g mechanism of spin conversion for RP w'tb,no magnetic nuclei fin the singlct prede~.'~,~or, H,~ = 4(XI(| Oc; to = 4000 Oe; J = 0; g= = 2.0063; g~ = 2,0013; H s = 1,50e; H d = 0.567 Oe, ( ) The curve in the absence of a variable magnetic field H=: ( . . . . ) the curve in the presence of a variable magnelic field of 4 . 4 0 e ,
Since this Letter is mainly devoted to the presentation of the method, we shall restrict our consideration of its applications only to quantum beats. We do not demonstrate RYDMR spectra for RP, because the results gained by using (14) coincide with those obtained in Refs. [1,4,5]. The investigation of the quantum beats phenomena is of interest because of its high sensitivity to changes in RP parameters and variable magnetic field intensity.
S.I. Kubarer et al. / Chemical Physics Letters 235 (1995) 591-595 1.0
,Oss 0.8
0.6
0.4
0.2
0.0
0,0
0.5
t .0
1,5
20
2,5
3.0
"T
Fig. 3. The dynamics of the changes in the quantum beats spectra under discontinuous jumping of the g-factor, H i = 4 . 4 0 ~ ; H o = 400(I Oe; ¢o = 4011(I Oe; It d = 11.567 Oe; H s = 1 . 5 0 e . ( ~ - ) The curve in the absence of the discontinuous jump of gl (Ag = 0.002); ( . . . . ) the curve in the presence of the discontinuous jump of gl (Ag =0.009) at t = 5 . 6 7 9 × 10 -H s.
Thus Figs. 1 and 2 give the relationship between the quantum beats signal (for RP with no magnetic nuclei for singlet and triplet predecessors, respee-
595
tively) and the variable magnetic field intensity. For the singlet predecessor (see Fig. 1) later in the process of quantum beats a detectable increase in the signal at certain variable magnetic field intensity is evident. For the triplet predecessor (see Fig. 2) an increase in the signal integrated intensity is observed. Analogous effects are observed for RP with magnetic nuclei. Figs. 3 and 4 show the dynamics of the changes in the quantum beats spectra on discontinuous jumping of the g-factor and hfi constant of one of the radicals, respectively. As noted above, the discontinuous jump of the g-factor simulates optically induced spin conversion effects, and the discontinuous jump of the hfi constant simulates the structural rearrangement process occurring during the lifetime of the intermediate complex. Notice that the abscissa in all the figures, time (1/Oe), is plotted in the following scale: 1
( O e ) - l = 5 . 6 7 9 × 10 - 8 s.
References 10
Pss 08
0,6
04
0,2
0,0 0.0
0.5
1.0
1,5
20
2,5
3,0
T
Fig. 4. The quantum beats under discontinuousjumping of the hfi constant A. H a = 4000 Oe; H d = 0.567 Oe; H s = 1 . 5 0 e ; d = 0; a~ = 4000 Oe. H I = 3 0 e . ( - ) The curve in the absence of the discontinuous jump of A (A = 16); ( . . . . ) the curve in the presence of the discontinuous jump of A (A = 64) at t = 5.679 × 10 -~ s.
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