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Simulated molecular dynamical studies of conjugated polymers
Synthetic Metals, 28 (1989) D457-D461
Da57
SIMULATED MOLECULAR DYNAMICAL STUDIES OF CONJUGATED POLYMERS
D.S.WALLACE Clarendon Laboratory, Parks Road, Oxford (U.K.)
ABSTRACT A method similar to that of Car and Parrinello [i] is used self-consistently to relax atomic positions and calculate LCAO wave functions for quite large CnHn+ 2 molecules, representing chains of trans-polyacetylene ( t-PA ).
Electron-electron
interaction is considered to a higher level than in the SSH model [2] by the use of semi-empirical Hartree-Fock methods such as CNDO/2 and INDO.
Genuinely
minimised ground states can be obtained for charged and uncharged chains, and the method can also be used to obtain good approximate dynamics.
THE CAR-PARRINELLO METHOD IN AN LCAO BASIS Car and Parrinello suggested that the self-consistent ground state of a system could be found by treating the wave functions and atomic co-ordinates in the same way, i.e. by writing down dynamical "equations of motion" for the wave functions as well as the atomic co-ordinates;
the constraint that the wave functions remain
orthonormal can be expressed in terms of a matrix of Lagrange multipliers.
In
density functional theory, this method has two advantages over traditional minimisation techniques:
it avoids both the necessity of diagonalising very large
matrices, and that of calculating an exact wave function at each geometry encountered along the minimisation path.
The first of these is not such an
important consideration in an LCAO basis as in the plane wave basis used in density functional theory ( in fact, the effort involved in applying the orthonormality constraint is possibly greater than in matrix diagonalisation ), as the size of the basis set does not grow as quickly with the system size, but the second remains an important consideration. The method used here involves atomic relaxation by the same annealing process as the Car-Parrinello method, but obtains the wave function solution by performing one matrix diagonalisation at each step, rather than repeated diagonalisation to electronic self-consistency.
By not damping the atomic motion, we can obtain a
good approximation to the dynamical behaviour of the system, provided that the
0379-6779/89/$3.50
© Elsevier Sequoia/Printed in The Netherlands
D458 time step is sufficiently small that the Fock matrix calculated remains a good approximation to the exact solution. Energy derivatives with respect to atomic co-ordinates are performed analytically within the CNDO/2 or INDO approximations
( the extra terms considered
in INDO are one-centred, so the derivatives are the same in each case ); the derivatives needed are those of the Coulomb and overlap integrals.
Derivatives of
the former are obtained by differentiating the analytic formulae given by Roothaan [3] while derivatives of the overlap integrals can be expressed in terms of similar integrals [4], and these evaluated in the normal way.
GROUND STATE GEOMETRY Relaxation of a C24H26 molecule produced bond lengths of 1.409~, 1.329~ and i.i13~ for C-C, C=C and C-H respectively,
in good agreement with previous
theoretical studies
[5] and with experiment
ends of the chain.
The C-C=C bond angle is found to he 125.1 °, again in good
[6].
Dimerisation is enhanced at the
agreement with earlier studies, and this angle increases to about 130 ° near the chain ends.
SOLITON AND POLARON GEOMETRY Unconstrained minimisation was performed on a singly charged C24H26 molecule and an uncharged C25H27 molecule, and results are shown in Fig. 1 and 2.
The
dimerisation amplitude plotted is (-l)n(bn+l-bn)/2, where b n is the n'th C-C bond length.
This has been found to give a fairly smooth plot, while directly
representing the magnitude of the dimerisation.
0.075
O. 050
O. 025
I
o. ooo
E
-0.025
-0. 050
-0. 075 0.0
2.0
Fig. i. chain.
4.0
6.0
8.0
I0.0
12.0
14.0
16.0
18.0
20.0
22.0
24.0
Dimerisation of singly charged C24H26 ( in Angstroms 1, plotted along the *
:
CNDO results
: fit to sum of tanh functions.
D~59 O. 0 7 5
O. 050
0.025
O. OOO E -O. 0~5
-0. 050
-0. 075 0.0
Fig. 2. chain.
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
18.0
20.0
22.0
24.0
Dimerisation of uncharged C25H27 ( in Angstroms ), plotted along the * : CNDO results
: fit to tanh function.
The soliton formed in C25H27 is fitted well by a tanh function of width 8.5 carbon units, fairly close to the SSH value of 7 carbon units and much larger than is found using a different semi-empirical Hartree-Fock method, MNDO [7].
The
polaron in C48H50 can be fitted to a pair of tanh functions, of width 7 units and separation 9 units. As in the SSH model, it is predicted that adding charge to a molecule of t-PA will affect the chain length; calculations on C10Hl2 show that the length is reduced by about 0.01~ on removal of an electron, and increased by about 0.03A on addition of an electron.
Futhermore, approximately half of the charge resides on
the hydrogen atoms, rather than in the carbon Pz orbitals as assumed by SSH; while the charge distribution on the carbon atoms does display the characteristic alternation between low and high density, the hydrogen atoms are charged roughly uniformly around the polaron.
DYNAMICAL STUDIES We can obtain an estimate of the propagation velocity of a soliton along a chain by starting a dynamical run with a long bond, and therefore a localised electronic state, at one end of the chain.
Early results are shown in Fig. 3,
where the dimerisation parameter is plotted along the chain after increasing lengths of time.
The basic time step is J(mp/m e) atomic units, or 1.04 x lO-15s.
The soliton velocity is found to remain fairly constant at about 0.i carbon units per time step, and taking 1.2~ as the repeat distance we obtain a soliton velocity of 1.2 x iO4ms -I.
This is rather lower than theoretical values obtained elsewhere
[8], but is of the same order of magnitude, and is less than the estimated speed of sound in t-PA [9], as seems intuitively reasonable.
D460 O. 0 7 5
O. O S D
o. o2s I
o.ooo E
-0.025 -0.050 -0.075 O.O
2.0
Fig. 3.
4.0
G.O
8.0
iO.O 12.0 l,~.O IG.O 18.0 20.0 22.0 24.0
Dimerisation of C25H27
different times.
* : 5 units
( in Angstroms + : 15 units
) along the chain, at three ~
: 25 units.
CONCLUSIONS This method can be used to verify the results of the SSH mode] in such areas as the basic nature of defects in conjugated polymers, areas.
and to go beyond SSH in other
The most obvious weaknesses of the SSH model are its treatment of
electron-electron
interaction and its assumption that polymer molecules can be
treated as entirely one-dimensional systems, needing to be considered.
with only the carbon Pz orbitals
In both of these areas, the self-consistent
outlined in this paper represents an improvement:
LCAO method
the electron-hole symmetry
responsible for some of the shortcomings of SSH ( such as the failure to account for luminescence quenching in cis-polyacetylene systems are treated as three-dimensional, Further calculations, progress,
[i0] ) has been removed and the
with a full valence basis set.
on larger model systems and on different polymers,
are in
and it is hoped that investigation into the question of non-radiative
transitions in conjugated polymers will lead to results in the near future.
ACKNOWLEDGEMENTS I should like to express my gratitude to W.Hayes and A.M.Stoneham for many useful discussions,
and to A.H.Harker for permission to use some of the
integration routines from his quantum chemical program MOSES
[ii].
This work has
been supported by the Underlying Research Programme of the United Kingdom Atomic Energy Authority.
D461 REFERENCES
i
R.Car and M.Parrinello, Phys: Rev:_B 55 (1985) 2471.
2
W.P.Su, J.R.Schrieffer and A.J.Heeger, Phzs. Rev. B 22 (1980) 2099.
3
C.C.J.Roothaan, J. 9hem: PbY~ ~
4
P.Pulay and F.T~rLk, Nol: Ph~s~_25 (1973) i153.
(1951) 1445.
5
H.Teramae, T.Yamabe and A.Imamura, J. Chem. Ph~s- 81 (1894) 3564.
6
c.s.yannoni and T.C.Clarke, Phys. Rev. Left. 51 (1983) 1191.
7
D.S.Boudreaux, R.R.Chance, J.L.Br4das and R.Silbey, phys~ Rev. 828 (1983) 6927.
8
W.Forner, C.L.Wang, F.Hartino and J.Ladik, Phys. R e W B37 (1988) 4567.
9
F.Guinea, Phi_± Rev. 830 (1984) 1884.
i0 W.Hayes, C ont@m~ Phys,_26 (1985) 421. ii A.H.Harker and S.B.Lyon, AERE-R8598, Harwell Laboratory, 1979.
Da57
SIMULATED MOLECULAR DYNAMICAL STUDIES OF CONJUGATED POLYMERS
D.S.WALLACE Clarendon Laboratory, Parks Road, Oxford (U.K.)
ABSTRACT A method similar to that of Car and Parrinello [i] is used self-consistently to relax atomic positions and calculate LCAO wave functions for quite large CnHn+ 2 molecules, representing chains of trans-polyacetylene ( t-PA ).
Electron-electron
interaction is considered to a higher level than in the SSH model [2] by the use of semi-empirical Hartree-Fock methods such as CNDO/2 and INDO.
Genuinely
minimised ground states can be obtained for charged and uncharged chains, and the method can also be used to obtain good approximate dynamics.
THE CAR-PARRINELLO METHOD IN AN LCAO BASIS Car and Parrinello suggested that the self-consistent ground state of a system could be found by treating the wave functions and atomic co-ordinates in the same way, i.e. by writing down dynamical "equations of motion" for the wave functions as well as the atomic co-ordinates;
the constraint that the wave functions remain
orthonormal can be expressed in terms of a matrix of Lagrange multipliers.
In
density functional theory, this method has two advantages over traditional minimisation techniques:
it avoids both the necessity of diagonalising very large
matrices, and that of calculating an exact wave function at each geometry encountered along the minimisation path.
The first of these is not such an
important consideration in an LCAO basis as in the plane wave basis used in density functional theory ( in fact, the effort involved in applying the orthonormality constraint is possibly greater than in matrix diagonalisation ), as the size of the basis set does not grow as quickly with the system size, but the second remains an important consideration. The method used here involves atomic relaxation by the same annealing process as the Car-Parrinello method, but obtains the wave function solution by performing one matrix diagonalisation at each step, rather than repeated diagonalisation to electronic self-consistency.
By not damping the atomic motion, we can obtain a
good approximation to the dynamical behaviour of the system, provided that the
0379-6779/89/$3.50
© Elsevier Sequoia/Printed in The Netherlands
D458 time step is sufficiently small that the Fock matrix calculated remains a good approximation to the exact solution. Energy derivatives with respect to atomic co-ordinates are performed analytically within the CNDO/2 or INDO approximations
( the extra terms considered
in INDO are one-centred, so the derivatives are the same in each case ); the derivatives needed are those of the Coulomb and overlap integrals.
Derivatives of
the former are obtained by differentiating the analytic formulae given by Roothaan [3] while derivatives of the overlap integrals can be expressed in terms of similar integrals [4], and these evaluated in the normal way.
GROUND STATE GEOMETRY Relaxation of a C24H26 molecule produced bond lengths of 1.409~, 1.329~ and i.i13~ for C-C, C=C and C-H respectively,
in good agreement with previous
theoretical studies
[5] and with experiment
ends of the chain.
The C-C=C bond angle is found to he 125.1 °, again in good
[6].
Dimerisation is enhanced at the
agreement with earlier studies, and this angle increases to about 130 ° near the chain ends.
SOLITON AND POLARON GEOMETRY Unconstrained minimisation was performed on a singly charged C24H26 molecule and an uncharged C25H27 molecule, and results are shown in Fig. 1 and 2.
The
dimerisation amplitude plotted is (-l)n(bn+l-bn)/2, where b n is the n'th C-C bond length.
This has been found to give a fairly smooth plot, while directly
representing the magnitude of the dimerisation.
0.075
O. 050
O. 025
I
o. ooo
E
-0.025
-0. 050
-0. 075 0.0
2.0
Fig. i. chain.
4.0
6.0
8.0
I0.0
12.0
14.0
16.0
18.0
20.0
22.0
24.0
Dimerisation of singly charged C24H26 ( in Angstroms 1, plotted along the *
:
CNDO results
: fit to sum of tanh functions.
D~59 O. 0 7 5
O. 050
0.025
O. OOO E -O. 0~5
-0. 050
-0. 075 0.0
Fig. 2. chain.
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
18.0
20.0
22.0
24.0
Dimerisation of uncharged C25H27 ( in Angstroms ), plotted along the * : CNDO results
: fit to tanh function.
The soliton formed in C25H27 is fitted well by a tanh function of width 8.5 carbon units, fairly close to the SSH value of 7 carbon units and much larger than is found using a different semi-empirical Hartree-Fock method, MNDO [7].
The
polaron in C48H50 can be fitted to a pair of tanh functions, of width 7 units and separation 9 units. As in the SSH model, it is predicted that adding charge to a molecule of t-PA will affect the chain length; calculations on C10Hl2 show that the length is reduced by about 0.01~ on removal of an electron, and increased by about 0.03A on addition of an electron.
Futhermore, approximately half of the charge resides on
the hydrogen atoms, rather than in the carbon Pz orbitals as assumed by SSH; while the charge distribution on the carbon atoms does display the characteristic alternation between low and high density, the hydrogen atoms are charged roughly uniformly around the polaron.
DYNAMICAL STUDIES We can obtain an estimate of the propagation velocity of a soliton along a chain by starting a dynamical run with a long bond, and therefore a localised electronic state, at one end of the chain.
Early results are shown in Fig. 3,
where the dimerisation parameter is plotted along the chain after increasing lengths of time.
The basic time step is J(mp/m e) atomic units, or 1.04 x lO-15s.
The soliton velocity is found to remain fairly constant at about 0.i carbon units per time step, and taking 1.2~ as the repeat distance we obtain a soliton velocity of 1.2 x iO4ms -I.
This is rather lower than theoretical values obtained elsewhere
[8], but is of the same order of magnitude, and is less than the estimated speed of sound in t-PA [9], as seems intuitively reasonable.
D460 O. 0 7 5
O. O S D
o. o2s I
o.ooo E
-0.025 -0.050 -0.075 O.O
2.0
Fig. 3.
4.0
G.O
8.0
iO.O 12.0 l,~.O IG.O 18.0 20.0 22.0 24.0
Dimerisation of C25H27
different times.
* : 5 units
( in Angstroms + : 15 units
) along the chain, at three ~
: 25 units.
CONCLUSIONS This method can be used to verify the results of the SSH mode] in such areas as the basic nature of defects in conjugated polymers, areas.
and to go beyond SSH in other
The most obvious weaknesses of the SSH model are its treatment of
electron-electron
interaction and its assumption that polymer molecules can be
treated as entirely one-dimensional systems, needing to be considered.
with only the carbon Pz orbitals
In both of these areas, the self-consistent
outlined in this paper represents an improvement:
LCAO method
the electron-hole symmetry
responsible for some of the shortcomings of SSH ( such as the failure to account for luminescence quenching in cis-polyacetylene systems are treated as three-dimensional, Further calculations, progress,
[i0] ) has been removed and the
with a full valence basis set.
on larger model systems and on different polymers,
are in
and it is hoped that investigation into the question of non-radiative
transitions in conjugated polymers will lead to results in the near future.
ACKNOWLEDGEMENTS I should like to express my gratitude to W.Hayes and A.M.Stoneham for many useful discussions,
and to A.H.Harker for permission to use some of the
integration routines from his quantum chemical program MOSES
[ii].
This work has
been supported by the Underlying Research Programme of the United Kingdom Atomic Energy Authority.
D461 REFERENCES
i
R.Car and M.Parrinello, Phys: Rev:_B 55 (1985) 2471.
2
W.P.Su, J.R.Schrieffer and A.J.Heeger, Phzs. Rev. B 22 (1980) 2099.
3
C.C.J.Roothaan, J. 9hem: PbY~ ~
4
P.Pulay and F.T~rLk, Nol: Ph~s~_25 (1973) i153.
(1951) 1445.
5
H.Teramae, T.Yamabe and A.Imamura, J. Chem. Ph~s- 81 (1894) 3564.
6
c.s.yannoni and T.C.Clarke, Phys. Rev. Left. 51 (1983) 1191.
7
D.S.Boudreaux, R.R.Chance, J.L.Br4das and R.Silbey, phys~ Rev. 828 (1983) 6927.
8
W.Forner, C.L.Wang, F.Hartino and J.Ladik, Phys. R e W B37 (1988) 4567.
9
F.Guinea, Phi_± Rev. 830 (1984) 1884.
i0 W.Hayes, C ont@m~ Phys,_26 (1985) 421. ii A.H.Harker and S.B.Lyon, AERE-R8598, Harwell Laboratory, 1979.