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Molecular orbital models of poly(p-phenylene)
ELSEVIER
SyntheticMetals 101(1999) 381-382
Molecular orbital models of poly(p-phenylene) H.Dalya, W. Barforda, R. J. Bursillb. a Department of Physics, The University of Sheffield, Sheffield, S3 7RH, UK
b
School of Physics, University of New South Wales, Sydney, NSW 2052, Australia
Abstract
A two molecularorbital modelof poly(para-phenylene) basedon the under-lyingPariser-Parr-Pople modelis introduced. The modelis solvedusingthe densitymatrix renormalisation groupmethodfor oligomersof up to 15repeatunits. Key~~ords:semi-empirical modelsandcalculations,poly(para-phenylene) andderivatives. 1. Introduction
@3 l
+
Zt-
.
-
The observation of electroluminescensein organic semiconductors suchaspoly(para-phenylene) (PPP)hasresulted in much experimental and theoretical activity due to their potentialasdevice materials[l-7]. Here,we usea 2 bandmodel of PPPto calculatethe low lying excitationenergies. The model is basedon the extendedHubbard(or Pariser-ParrPople(P-P-P))Hamiltonian.The molecularorbitalsof the repeat unit in the chain- phenylene- arefound andusedto constructthe Hamiltonianfor PPPwhich is then solvedby the densitymatrix renormalisation groupmethod(DMRG), 2. The Molecular
Orbital
Model
Firstly, startingfrom the Pariser-Parr-Pople (P-P-P)method,the Hamiltonianfor the rt electronsin benzenecanbeconstructed.
H = - c r,(c;c,,
+ h.c.) + UC nitniJ
ci/>o
i
(1)
+C Yij(H)q - l)(nj,n - 1) it
j'
+
/P\
+ +
+
.
-63
+
a3 ,2
:
14)
o--
+ + -CD
x3 ,I’,
13) +
a-0
Fig. I. The six orthogonalised wave’?unctions in benzene. Exactly diagonalisingthis Hamiltonianresultsin six allowed eigenstates,sk=-2tcosk,k=O, b/3, &2rr/3, rr, with angular quantumnumberj=3k/rr. The correspondingorthogonalised wavefunctionsarerepresented in figure 1. The molecularorbitalsof benzenearenow usedasthe basisfor PPP. One notesthat in states13)and 14)there are nodesof electron amplitude on the carbon atoms which bond with neighbouringrepeat units. These stateswill then be nonbonding,or localised,in the PPPchain. States]l), /2), 15)and]6), however, will be delocalised,or bonding, and there will be a hybridisationof thesefour bondingorbitalsforming bands.
where ciacreatesa 7celectronwith spin o on site i, ni,= ciOc,,, ni=niT+niJand 0 represents nearestneighbours. The Coulomb interaction, Vij, is defined by the Ohno parameterisation [S] as
F’(/=
u (1+0.4881R;)“2
(2)
whereRij isthe interatomicdistanceandthe P-P-Ppotential,U = 10.06eV. t=2.539 is the hop ing integralfor a phenyl bond,and tij=t(l-a6) where ~r=l.l4k Pand S=differencein bond length between phenyl bond and other bond 191.
Fig; 2. The molecularorbitalsof benzeneandthe bandsformed by theseMO in a chain.
0379-6779/99/$ - seefrontmatter0 1999ElsevierScience S.A. All rightsreserved. HI: SO379-6779(98)00806-6
H. Daly et al. I Synthetic
382
As the molecular orbitals 15) and 16) are far in energy from the Fermi energy, it can be assumed that they contribute very little to low lying excitations and they can be disregarded. Also, the nonbonding states, (3) and /4), do not contribute to the band structure. One can then assume that the low lying excitations can be modelled with only the 2 bands, which are due mainly to molecular orbitals 11) and 12). The P-P-P Hamiltonian for the chain, based on these bands, is
Metals
IO1 (1999) 381-382
Tables 3. Hamiltonian -3.338 3.33s 6.699
E2
E3 u22, u33
4.188
v,:
0.691 -0.668 0.668
x23 b2
43
H =-ciC#~,,(ai:~a~+~~~ +h.c.)+ciuw, +$,
(3)
where aioa creates an electron in state CLwith spin o on repeat unit i. X,s is the exchange energy and v& is the Coulomb repulsion of a pair of molecular orbitals j units apart. All 3 and 4 centre integrals, as well as 1 and 2 centre integrals beyond nearest neighbour, are neglected. Parameters
The contribution of each molecular orbital to each band can be found by obtaining the Wannier functions from the molecular orbitals. Tables 1 and 2 show the Wannier functions for bands 2 and 3 (the two bands retained in the Hamiltonian). Table 3 shows the Hamiltonian parameters calculated from the usual 1 and 2 electron integrals. Table 1. Molecular orbital contribution to band 2 centred on repeat unit m and neighbouring repeat unit mltl 15) m til
0 -0.226
11) 0.961 0.023m
1%
16)
0 -0.010 0.146
m m&l
11)
-0.010 -0.061
4. Transition
0 0.146
2 0 L 0
l
++.*..
+ Charge
5
,2’Ag+
gap
-/-I IO
.l’B,
15
L
Figure 4. The calculated l’B,’ and 2’A,+ transition energies, and the charge gap for oligomers of 3 to 15 repeat units, L. energy. The 1 ‘B,‘ is the lowest lying singlet excitation. It can be considered bound as it lies below the charge gap. The transition energy of the I ‘B,‘ state is ca. 1eV higher than the experimental result. This is a result of the fact that the sing& particle basis of the conduction and valence bands does not provide an adequate representation of the many body interactions. Instead, all six molecular orbitals need to be retained. However, a re-parameterised two state model does provide a good description of PPP [ 10,l 11 5. References.
-0.062
Table 2. Molecular orbital contribution to band 3 centred on repeat unit m and neighbouring repeat unit m&l 15)
I2 T IO -
i
JL'D(% - l)(%+,p - 1)
3. Hamiltonian
0
t23, t32
+C Uuun,,dk+1’1 C=piJwpiuniu
parameters in eV.
12)
16)
0.961 0.023
0 -0.226
Energies
The Hamiltonian was then solved using the DMRG method to find the transition energies of the low lying PPP excitations, from the even parity singlet ground state, l’A,‘, to the odd parity singlet state, l’B,,-, and the even parity singlet state, “A,+. These are shown in figure 4. Also shown is the charge gap. In long chains, there is a set of continuum states above the ground state. The charge gap is the energy difference between the ground state and the lowest continuum state. It is given by the difference between the energy of a hole in the valence band plus the energy of an electron in the conduction band, and the Fermi
[I] J. M. Leng, S. Jeglinski, X. Wei, R. E. Benner, 2. VI Vardeny, F. Guo, S. Mazumbdar, Phys. Rev. Lett. 72 (1994) 15% [2] S. Heun, R. F. Mahrt, A. Greiner, U. Lemmer, H. Bessle, D. A. Halliday, D. D. C. Bradley, P. L. Burn, A. B. Holmes, J. Phys, C 5(1993)247 [3] J. Cornil,
D. Beljonne, R. I-I. Friend, J. L. Bredas, Chem. Phys. Lett 223 (1994) 82-88. [4] M. Chandros, S. Mazumbdar, S. Jeglinski, X. Wei, 2. V. Vardeny, E. W. Kwock, T. M. Miller, Phys. Rev. B. 50 (1994) 14702. [5] M. J. Rice, Yu. N. Garstein, Phys. Rev. Lett. 73 (1994) 2504 [6] Yu. N. Garstein, M. J. Rice, E. M. Conwell, Phys. Rev. B. 72 (1995)1683. [7] W. Barford,
R. J. Bursill, Chem. Phys. Lett. 268 (1997) 535. [S] K. Ohno, Theor. Chim. Acta (1964) 219. [9] R. J. Bursill, C. Castleton, W. Barford, Chem. Phys. Lett. (in press) [lo] W. Barford, et al., J. Phys.: Condens. Matt., 10, 6429 (1998) [ 1 l] M. Yu Lavrentiev, et al., these proceedings.
SyntheticMetals 101(1999) 381-382
Molecular orbital models of poly(p-phenylene) H.Dalya, W. Barforda, R. J. Bursillb. a Department of Physics, The University of Sheffield, Sheffield, S3 7RH, UK
b
School of Physics, University of New South Wales, Sydney, NSW 2052, Australia
Abstract
A two molecularorbital modelof poly(para-phenylene) basedon the under-lyingPariser-Parr-Pople modelis introduced. The modelis solvedusingthe densitymatrix renormalisation groupmethodfor oligomersof up to 15repeatunits. Key~~ords:semi-empirical modelsandcalculations,poly(para-phenylene) andderivatives. 1. Introduction
@3 l
+
Zt-
.
-
The observation of electroluminescensein organic semiconductors suchaspoly(para-phenylene) (PPP)hasresulted in much experimental and theoretical activity due to their potentialasdevice materials[l-7]. Here,we usea 2 bandmodel of PPPto calculatethe low lying excitationenergies. The model is basedon the extendedHubbard(or Pariser-ParrPople(P-P-P))Hamiltonian.The molecularorbitalsof the repeat unit in the chain- phenylene- arefound andusedto constructthe Hamiltonianfor PPPwhich is then solvedby the densitymatrix renormalisation groupmethod(DMRG), 2. The Molecular
Orbital
Model
Firstly, startingfrom the Pariser-Parr-Pople (P-P-P)method,the Hamiltonianfor the rt electronsin benzenecanbeconstructed.
H = - c r,(c;c,,
+ h.c.) + UC nitniJ
ci/>o
i
(1)
+C Yij(H)q - l)(nj,n - 1) it
j'
+
/P\
+ +
+
.
-63
+
a3 ,2
:
14)
o--
+ + -CD
x3 ,I’,
13) +
a-0
Fig. I. The six orthogonalised wave’?unctions in benzene. Exactly diagonalisingthis Hamiltonianresultsin six allowed eigenstates,sk=-2tcosk,k=O, b/3, &2rr/3, rr, with angular quantumnumberj=3k/rr. The correspondingorthogonalised wavefunctionsarerepresented in figure 1. The molecularorbitalsof benzenearenow usedasthe basisfor PPP. One notesthat in states13)and 14)there are nodesof electron amplitude on the carbon atoms which bond with neighbouringrepeat units. These stateswill then be nonbonding,or localised,in the PPPchain. States]l), /2), 15)and]6), however, will be delocalised,or bonding, and there will be a hybridisationof thesefour bondingorbitalsforming bands.
where ciacreatesa 7celectronwith spin o on site i, ni,= ciOc,,, ni=niT+niJand 0 represents nearestneighbours. The Coulomb interaction, Vij, is defined by the Ohno parameterisation [S] as
F’(/=
u (1+0.4881R;)“2
(2)
whereRij isthe interatomicdistanceandthe P-P-Ppotential,U = 10.06eV. t=2.539 is the hop ing integralfor a phenyl bond,and tij=t(l-a6) where ~r=l.l4k Pand S=differencein bond length between phenyl bond and other bond 191.
Fig; 2. The molecularorbitalsof benzeneandthe bandsformed by theseMO in a chain.
0379-6779/99/$ - seefrontmatter0 1999ElsevierScience S.A. All rightsreserved. HI: SO379-6779(98)00806-6
H. Daly et al. I Synthetic
382
As the molecular orbitals 15) and 16) are far in energy from the Fermi energy, it can be assumed that they contribute very little to low lying excitations and they can be disregarded. Also, the nonbonding states, (3) and /4), do not contribute to the band structure. One can then assume that the low lying excitations can be modelled with only the 2 bands, which are due mainly to molecular orbitals 11) and 12). The P-P-P Hamiltonian for the chain, based on these bands, is
Metals
IO1 (1999) 381-382
Tables 3. Hamiltonian -3.338 3.33s 6.699
E2
E3 u22, u33
4.188
v,:
0.691 -0.668 0.668
x23 b2
43
H =-ciC#~,,(ai:~a~+~~~ +h.c.)+ciuw, +$,
(3)
where aioa creates an electron in state CLwith spin o on repeat unit i. X,s is the exchange energy and v& is the Coulomb repulsion of a pair of molecular orbitals j units apart. All 3 and 4 centre integrals, as well as 1 and 2 centre integrals beyond nearest neighbour, are neglected. Parameters
The contribution of each molecular orbital to each band can be found by obtaining the Wannier functions from the molecular orbitals. Tables 1 and 2 show the Wannier functions for bands 2 and 3 (the two bands retained in the Hamiltonian). Table 3 shows the Hamiltonian parameters calculated from the usual 1 and 2 electron integrals. Table 1. Molecular orbital contribution to band 2 centred on repeat unit m and neighbouring repeat unit mltl 15) m til
0 -0.226
11) 0.961 0.023m
1%
16)
0 -0.010 0.146
m m&l
11)
-0.010 -0.061
4. Transition
0 0.146
2 0 L 0
l
++.*..
+ Charge
5
,2’Ag+
gap
-/-I IO
.l’B,
15
L
Figure 4. The calculated l’B,’ and 2’A,+ transition energies, and the charge gap for oligomers of 3 to 15 repeat units, L. energy. The 1 ‘B,‘ is the lowest lying singlet excitation. It can be considered bound as it lies below the charge gap. The transition energy of the I ‘B,‘ state is ca. 1eV higher than the experimental result. This is a result of the fact that the sing& particle basis of the conduction and valence bands does not provide an adequate representation of the many body interactions. Instead, all six molecular orbitals need to be retained. However, a re-parameterised two state model does provide a good description of PPP [ 10,l 11 5. References.
-0.062
Table 2. Molecular orbital contribution to band 3 centred on repeat unit m and neighbouring repeat unit m&l 15)
I2 T IO -
i
JL'D(% - l)(%+,p - 1)
3. Hamiltonian
0
t23, t32
+C Uuun,,dk+1’1 C=piJwpiuniu
parameters in eV.
12)
16)
0.961 0.023
0 -0.226
Energies
The Hamiltonian was then solved using the DMRG method to find the transition energies of the low lying PPP excitations, from the even parity singlet ground state, l’A,‘, to the odd parity singlet state, l’B,,-, and the even parity singlet state, “A,+. These are shown in figure 4. Also shown is the charge gap. In long chains, there is a set of continuum states above the ground state. The charge gap is the energy difference between the ground state and the lowest continuum state. It is given by the difference between the energy of a hole in the valence band plus the energy of an electron in the conduction band, and the Fermi
[I] J. M. Leng, S. Jeglinski, X. Wei, R. E. Benner, 2. VI Vardeny, F. Guo, S. Mazumbdar, Phys. Rev. Lett. 72 (1994) 15% [2] S. Heun, R. F. Mahrt, A. Greiner, U. Lemmer, H. Bessle, D. A. Halliday, D. D. C. Bradley, P. L. Burn, A. B. Holmes, J. Phys, C 5(1993)247 [3] J. Cornil,
D. Beljonne, R. I-I. Friend, J. L. Bredas, Chem. Phys. Lett 223 (1994) 82-88. [4] M. Chandros, S. Mazumbdar, S. Jeglinski, X. Wei, 2. V. Vardeny, E. W. Kwock, T. M. Miller, Phys. Rev. B. 50 (1994) 14702. [5] M. J. Rice, Yu. N. Garstein, Phys. Rev. Lett. 73 (1994) 2504 [6] Yu. N. Garstein, M. J. Rice, E. M. Conwell, Phys. Rev. B. 72 (1995)1683. [7] W. Barford,
R. J. Bursill, Chem. Phys. Lett. 268 (1997) 535. [S] K. Ohno, Theor. Chim. Acta (1964) 219. [9] R. J. Bursill, C. Castleton, W. Barford, Chem. Phys. Lett. (in press) [lo] W. Barford, et al., J. Phys.: Condens. Matt., 10, 6429 (1998) [ 1 l] M. Yu Lavrentiev, et al., these proceedings.