- Email: [email protected]
Calculations of the optical line shapes for disordered molecular aggregates using the resolvent of the Hamiltonian
JOURNAL OF
LUMINESCENC Journal of Luminescence 72-74 (1997)439-441
ELSEVIER
Calculations of the optical line shapes for disordered molecular aggregates using the resolvent of the Hamiltonian D.V. Makhov*,
V.V. Egorov,
A.A. Bagatur’ants,
M.V. Alfimov
Institute qf Chemical Physics, 4 Kosygina street. Moscow I1 7977, Russian Federation
Abstract
An effective method is proposed for numerical calculation of the absorption line shapes of molecular aggregates with diagonal disorder. The absorption intensity is expressed as a resolvent of aggregate Hamiltonian and then fluctuations of average transition energy in aggregate are separated from the relative energy fluctuations. Absorption
Keywords:
spectrum; Molecular aggregate; Diagonal disorder -
The theoretical investigations of the optical properties of molecular aggregates are often based on the model of the system of two-level molecules with Gaussian diagonal disorder [l-S]. In the framework of this model, aggregate Hamiltonian has the form fi = i
In)@%
+ &l)
n=l
5
In)~~/,,(ml,
tl=l.m=l nfm (1)
where E, is the fluctuation of the transition energy of nth molecule, and V,, is the intermolecular interaction. The expression for absorption line shape is [l-4] I(w)K
...
de1 ... de,exp
-t
::
1F
+ii2+’
ss
N XC II=1
’&~QJ-
where E, are eigenenergies, and a,,,,, are the components of nth eigenvector of Hamiltonian (1). In numerical calculations of absorption line shapes, b-functions in Eq. (2) are usually replaced by some functions with small but finite width [2,3]. Then, the integral is calculated by Monte-Carlo method with E, and an,,, found by numerical diagonalization of Hamiltonian (1). However, this simplest method of calculations is not a very efficient one, and spectra calculated by this method usually exhibit high level of noise (see Ref. [Z], for example). The efficiency of line-shape calculations can be sufficiently increased by separating the fluctuations of the average transition energy in the aggregate from the relative energy fluctuations [4]. To do so, it is convenient to perform the following linear orthogonal transformation [4]:
~9,
*Corresponding author. Fax: + + 7-095-9361255;e-mail: [email protected]. 0022-2313/97/$17.00 0 1997 Elsevier Science B.V. PII SOO22-23 13(96)00306-7
All
rights reserved
+ cos
(;(n -
l)(m - 1)
)I
E,.
(3)
440
D. V Makhov et al. i Journal of Luminescence
One can see that s;/fi is the arithmetic mean of all E,. So, transformation (3) separates out the fluctuation of average transition energy as one of the new independent variables. This allows to represent Hamiltonian as
where 2 is reduced Hamiltonian that does not depend on e;. Then, substituting Eqs. (3) and (4) in Eq. (2) and integrating over E;, we obtain c41 s;’ + I(o) K
...
. . + a;J2 202
ss
where B, and a,,,,, are eigenenergies and eigenvectors of reduced Hamiltonian 2. One can see that integrand in Eq. (5) is smooth and no longer contains energy &functions. This means that now each of Hamiltonian contributes diagonalization to the spectrum not only at the eigenenergies but at all frequencies at once. As a result, the efficiency of numerical calculations with Eq. (5) is much higher than in the case when Eq. (2) is used directly. Besides foregoing method [4], there is another way to increase the efficiency of line-shape calculations. The fact is that diagonalization of Hamiltonian gives more information than necessary (in particular, only the sum of eigenvector components but not components themselves appears in the expression for absorption intensity). A more efficient way is to express the absorption intensity as a resolvent of Hamiltonian [6]: I(o) ac lim Im y-0
x
... s
exp
E: +
where (7) One can see that Eq. (6) is the same as Eq. (2) with S-functions replaced by Lorentz functions with half-width il. Integral (6) is calculated by MonteCarlo method with (Pl(fi - w - iy))‘IP) found by solving the corresponding set of linear equations (parameter y should be small enough to approximate &functions well). In the case of molecular chain with only nearest-neighbour interaction, the numerical procedure can be reduced to the recurrence relation which is precisely the same as the one derived in Refs. [7, 81. This method of calculations is especially efficient for large aggregates because it allows to avoid the diagonalization of high-dimension matrixes. The efficiency of this method of calculations can be further increased by separating out the fluctuations of average transition energy in the same way as in the case when diagonalization of Hamiltonian is used. Let us perform the same linear transformation (3). Then, Eq. (6) takes the form I(w) cc lim Im 7-0
x (Pj(A - 0 - iy)-‘IP),
d& ... drh s
El2 +
. . . + E;3
2a2 x (Pl(,% - (o - c’,/fi)
- iy)-‘1P).
(8)
Let us make now the following variable substitution: o’ = w - E; /JN.
(9)
Then
... +E,:
2a2
... s
x exp( (yiz’)
s
(
de1 ... dsN
72-74 (1997) 439-441
E;2+ 1
. + s;j’ 2a2
V-5)
x
(P/(2@-
w’ -
iy))‘IP).
(10)
D.V. Makhov et al. /Journal of Luminescence 72-74 (1997) 439-441
One can see that parameter w no longer appears under the sign of resolvent but has been moved to the argument of the exponential function. This means that now each calculation of resolvent contributes at all frequencies of spectrum at once, contrary to Eq. (6) where each calculation of resolvent contributed only at one given frequency o. Calculations show that this increases the efficiency of numerical procedure by several tens. This work was supported by Russian Foundation for Fundamental Research (grant 96-0333906).
441
References [l] M. Schreiber and Y. Toyozawa, J. Phys. Sot. Japan 51 (1982) 1528. [Z] H. Fidder, J. Terpstra and D.A. Wiersma, J. Chem. Phys. 94 (1991) 6895. [3] H. Fidder, J. Knoester and D.A. Wiersma, J. Chem. Phys. 95 (1991) 7880. [4] D.V. Makhov et al., Chem. Phys. Lett. 246 (1995) 371. [5] J. Klafter and J. Jortner, J. Chem. Phys. 68 (1978) 1513. [6] E.W. Knapp, Chem. Phys. 85 (1984) 73. [7] J. KGhler, A.M. Jayannavar and P. Reineker, Z. Phys. B 75 (1989) 451. [8] Th. Neidlinger and P. Reineker, J. Lumin. 60&61 (1994) 413.
LUMINESCENC Journal of Luminescence 72-74 (1997)439-441
ELSEVIER
Calculations of the optical line shapes for disordered molecular aggregates using the resolvent of the Hamiltonian D.V. Makhov*,
V.V. Egorov,
A.A. Bagatur’ants,
M.V. Alfimov
Institute qf Chemical Physics, 4 Kosygina street. Moscow I1 7977, Russian Federation
Abstract
An effective method is proposed for numerical calculation of the absorption line shapes of molecular aggregates with diagonal disorder. The absorption intensity is expressed as a resolvent of aggregate Hamiltonian and then fluctuations of average transition energy in aggregate are separated from the relative energy fluctuations. Absorption
Keywords:
spectrum; Molecular aggregate; Diagonal disorder -
The theoretical investigations of the optical properties of molecular aggregates are often based on the model of the system of two-level molecules with Gaussian diagonal disorder [l-S]. In the framework of this model, aggregate Hamiltonian has the form fi = i
In)@%
+ &l)
n=l
5
In)~~/,,(ml,
tl=l.m=l nfm (1)
where E, is the fluctuation of the transition energy of nth molecule, and V,, is the intermolecular interaction. The expression for absorption line shape is [l-4] I(w)K
...
de1 ... de,exp
-t
::
1F
+ii2+’
ss
N XC II=1
’&~QJ-
where E, are eigenenergies, and a,,,,, are the components of nth eigenvector of Hamiltonian (1). In numerical calculations of absorption line shapes, b-functions in Eq. (2) are usually replaced by some functions with small but finite width [2,3]. Then, the integral is calculated by Monte-Carlo method with E, and an,,, found by numerical diagonalization of Hamiltonian (1). However, this simplest method of calculations is not a very efficient one, and spectra calculated by this method usually exhibit high level of noise (see Ref. [Z], for example). The efficiency of line-shape calculations can be sufficiently increased by separating the fluctuations of the average transition energy in the aggregate from the relative energy fluctuations [4]. To do so, it is convenient to perform the following linear orthogonal transformation [4]:
~9,
*Corresponding author. Fax: + + 7-095-9361255;e-mail: [email protected]. 0022-2313/97/$17.00 0 1997 Elsevier Science B.V. PII SOO22-23 13(96)00306-7
All
rights reserved
+ cos
(;(n -
l)(m - 1)
)I
E,.
(3)
440
D. V Makhov et al. i Journal of Luminescence
One can see that s;/fi is the arithmetic mean of all E,. So, transformation (3) separates out the fluctuation of average transition energy as one of the new independent variables. This allows to represent Hamiltonian as
where 2 is reduced Hamiltonian that does not depend on e;. Then, substituting Eqs. (3) and (4) in Eq. (2) and integrating over E;, we obtain c41 s;’ + I(o) K
...
. . + a;J2 202
ss
where B, and a,,,,, are eigenenergies and eigenvectors of reduced Hamiltonian 2. One can see that integrand in Eq. (5) is smooth and no longer contains energy &functions. This means that now each of Hamiltonian contributes diagonalization to the spectrum not only at the eigenenergies but at all frequencies at once. As a result, the efficiency of numerical calculations with Eq. (5) is much higher than in the case when Eq. (2) is used directly. Besides foregoing method [4], there is another way to increase the efficiency of line-shape calculations. The fact is that diagonalization of Hamiltonian gives more information than necessary (in particular, only the sum of eigenvector components but not components themselves appears in the expression for absorption intensity). A more efficient way is to express the absorption intensity as a resolvent of Hamiltonian [6]: I(o) ac lim Im y-0
x
... s
exp
E: +
where (7) One can see that Eq. (6) is the same as Eq. (2) with S-functions replaced by Lorentz functions with half-width il. Integral (6) is calculated by MonteCarlo method with (Pl(fi - w - iy))‘IP) found by solving the corresponding set of linear equations (parameter y should be small enough to approximate &functions well). In the case of molecular chain with only nearest-neighbour interaction, the numerical procedure can be reduced to the recurrence relation which is precisely the same as the one derived in Refs. [7, 81. This method of calculations is especially efficient for large aggregates because it allows to avoid the diagonalization of high-dimension matrixes. The efficiency of this method of calculations can be further increased by separating out the fluctuations of average transition energy in the same way as in the case when diagonalization of Hamiltonian is used. Let us perform the same linear transformation (3). Then, Eq. (6) takes the form I(w) cc lim Im 7-0
x (Pj(A - 0 - iy)-‘IP),
d& ... drh s
El2 +
. . . + E;3
2a2 x (Pl(,% - (o - c’,/fi)
- iy)-‘1P).
(8)
Let us make now the following variable substitution: o’ = w - E; /JN.
(9)
Then
... +E,:
2a2
... s
x exp( (yiz’)
s
(
de1 ... dsN
72-74 (1997) 439-441
E;2+ 1
. + s;j’ 2a2
V-5)
x
(P/(2@-
w’ -
iy))‘IP).
(10)
D.V. Makhov et al. /Journal of Luminescence 72-74 (1997) 439-441
One can see that parameter w no longer appears under the sign of resolvent but has been moved to the argument of the exponential function. This means that now each calculation of resolvent contributes at all frequencies of spectrum at once, contrary to Eq. (6) where each calculation of resolvent contributed only at one given frequency o. Calculations show that this increases the efficiency of numerical procedure by several tens. This work was supported by Russian Foundation for Fundamental Research (grant 96-0333906).
441
References [l] M. Schreiber and Y. Toyozawa, J. Phys. Sot. Japan 51 (1982) 1528. [Z] H. Fidder, J. Terpstra and D.A. Wiersma, J. Chem. Phys. 94 (1991) 6895. [3] H. Fidder, J. Knoester and D.A. Wiersma, J. Chem. Phys. 95 (1991) 7880. [4] D.V. Makhov et al., Chem. Phys. Lett. 246 (1995) 371. [5] J. Klafter and J. Jortner, J. Chem. Phys. 68 (1978) 1513. [6] E.W. Knapp, Chem. Phys. 85 (1984) 73. [7] J. KGhler, A.M. Jayannavar and P. Reineker, Z. Phys. B 75 (1989) 451. [8] Th. Neidlinger and P. Reineker, J. Lumin. 60&61 (1994) 413.