Applications of DMRG to

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Applications of DMRG to Conjugated Polymers S.
Ramasesha Solid State and Structural Chemistry
Unit Indian Institute of Science Bangalore 560
012, India
Collaborators H.R. Krishnamurthy
Swapan Pati Anusooya Pati
Kunj Tandon C. Raghu Z.
Shuai J.L. Brédas
Funding DST, India CSIR, India BRNS,
India
ramasesh_at_sscu.iisc.ernet.in
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Plan of the Talk

• Introduction to conjugated polymers
• Models for electronic structure
• Modifications of DMRG method
• Computation of nonlinear optic coefficients
• Exciton binding energies
• Ordering of low-lying states
• Geometry of excited states
• Application to phenyl based polymers
• Future issues

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Introduction to Conjugated Polymers
Contain extended network of unsaturated (sp2
hybridized) Carbon atoms
Eg Poly acetylene (CH)x, poly para phenylene
(PPP) poly acene and poly para phenylene
vinylene (PPV)
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Early Interest

• High chemical reactivity
• Long wavelength uv absorption
• Anisotropic diamagnetism

Current Interest

• Experimental realization of quasi 1-D system
• Organic semiconductors
• Fluorescent polymers
• Large NLO responses

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Theoretical Models for ?-Conjugated Systems

• Hückel Model
• Assumes one orbital at every Carbon site
• involved in conjugation.
• Assumes transfer integral only between
• bonded Carbon sites.

tij is resonance / transfer integral between
bonded sites and ai, the site energy at site i.
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• Drawbacks of Hückel model
• Gives incorrect ordering of energy levels.
• Predicts wrong spin densities and spin-spin
  correlations.
• Fails to reproduce qualitative differences
  between closely related systems.
• Mainly of pedagogical value. Ignores explicit
  electron-electron interactions.

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Interacting p-Electron Models

• Explicit electron electron interactions
  essential for realistic modeling

ijkl ??i(1) ?j(1) (e2/r12) ?k(2) ?l(2)
d3r1d3r2
This model requires further simplification to
enable routine solvability.
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Zero Differential Overlap (ZDO) Approximation
ijkl ??i(1) ?j(1) (e2/r12) ?k(2) ?l(2)
d3r1d3r2
ijkl ijkl?ij ?kl
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Hubbard Model

• Hückel model on-site repulsions
• iijj iijj ?ij Ui

?
?

• Introduced in 1964.
• Good for metals where screening lengths
• are short.

• Half-filled one-band Hubbard model yields
• antiferromagnetic spin ½ Heisenberg model
• as U / t ? ?.

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Pariser-Parr-Pople (PPP) Model

• zi are local chemical potentials.
• V(rij) parametrized either using
• Ohno parametrization
• V(rij) 2 / ( Ui Uj ) 2
  rij2 -1/2
• Or using Mataga-Nishimoto
  parametrization
• V(rij) 2 / ( Ui Uj )
  rij -1
• PPP model is also a one-parameter (U / t) model.

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Model Hamiltonian
PPP Hamiltonian (1953)
?
?
?
?
?
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Status of the PPP Model

• PPP model widely applied to study excited
  electronic states in conjugated molecules and
  polymers.
• U for C, N and t variety of C-C and C-N bonds are
  well established and transferable.
• Techniques for exact solution of PPP models with
  Hilbert spaces of 106 to 107 states well
  developed.
• Exact solutions are used to provide a check on
  approximate techniques.

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Symmetries in the PPP and Hubbard Models
Electron-hole symmetry

• When all sites are equivalent, for a bipartite
• lattice, we have electron-hole or charge
• conjugation or alternancy symmetry, at
• half-filling.
• Hamiltonian is invariant for the transformation

• Polymers also have end-to-end interchange
• symmetry or inversion symmetry.

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E-h symmetry divides the N Ne space into two
spaces, one containing both covalent and
ionic bases, the other containing only ionic
bases. Dipole operator connects the two spaces.

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Parity divides the total spin space into
spaces of even total spin and odd total spin.
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Why do we need symmetrization

• Important states in conjugated polymers
• Ground state (11Ag)
• Lowest dipole excited state (11B-u)
• Lowest triplet state (13Bu)
• Lowest two-photon state (21Ag) etc.
• In unsymmetrized methods, the serial index of
• desired eigenstate depends upon system
  size.
• In large correlated systems, where only a
• few low-lying states can be targeted, we
• could miss important states altogether.

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Matrix Representation of Site e-h and Site
Parity Operators

• Fock space of single site
• 1gt 0gt 2gt ?gt 3gt ?gt 4gt
  ??gt

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?
The overall electron-hole symmetry and
parity matrices can be obtained as direct
products of the individual site matrices.
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Symmetrized DMRG Procedure
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Operation by the end-to-end interchange on
the DMRG basis yields,
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Projection operator for a chosen irreducible
representation G, PG , is
The dimensionality of the space G is given by,
Eliminating linear dependencies in the matrix P
G yields the symmetrization matrix S with D G
rows and M columns, where M is the
dimensionality of the unsymmetrized DMRG space.
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The symmetrized DMRG Hamiltonian matrix, H S ,
is obtained from the unsymmetrized DMRG
Hamiltonian, H ,
H S S H S
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Checks on SDMRG

• Optical gap (Eg) in Hubbard model known
  analytically.
• In the limit of infinite chain length, for
• U/t 4.0, Egexact 1.2867 t U/t 6.0
  Egexact 2. 8926 t

PRB, 54, 7598 (1996).
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The spin gap in the limit U/t ?? should
vanish for Hubbard model.
PRB, 54, 7598 (1996).
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Dynamic Response Functions from DMRG
Commonly used technique in physics is
Lanczos technique
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In chemistry, sum-over-states (SOS) technique is
widely used

• The Lanczos technique has inherent truncation
• in the size of the small matrix chosen.
• SOS technique limits number of excited states.
• Correction vector technique avoids truncation
• over and above the Hilbert space truncation
• introduced in setting up the Hamiltonian
  matrix.

J. Chem. Phys., 90, 1067 (1989).
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Correction Vector Technique
Correction vector f(1)(w) is defined as
We can solve for f(1)(w) in a chosen basis
by solving a set of inhomogeneous linear
algebraic Equations, using a small matrix
algorithm.
J. Comput. Chem., 11, 545
(1990).
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Need for Symmetrization
In systems with symmetry, dipole operator maps
Therefore, fi(1)(w) lies in eB- subspace. The
unsymmetrized matrix (H-E0I) is singular while in
the eB- subspace it is nonsingular allowing
solving for fi(1)(0) from
Similarly, fi(1)(w) , lies in the singlet or odd
parity subspace. Using parity eliminates
singularity of the matrix (H - E0 - hw) for hw
ET.
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Computation of NLO Coefficients
To solve for dynamic nonlinear optic
coefficients, we solve a hierarchy of correction
vectors
and the linear and NLO response coefficients are
given by
Where, P permutes the frequencies and the
subscripts in pairs and ws -w1-w2-w3 .
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Computed at w 0.1t exactly for a Hubbard chain
of 12 sites at U/t4 with DMRG computation with
m200

5.343 5.317 598.3 591.1
The dominant a (axx) is 14.83 (exact) and 14.81
(DMRG) and g (gxxxx) 2873 (exact) and 2872 (DMRG).
a in 10-24 esu and g in 10-36 esu in all cases
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THG coefficient in Hubbard models as a
function of chain length, L and dimerization d
Superlinear behavior diminishes both with
increase in U/t and increase in d.
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gav. vs Chain Length and d in U-V Model
For U gt 2V, (SDW regime) gav. shows similar
dependence on L as the Hubbard model,
independent of d. U2V (SDW/CDW crossover point)
Hubbard chains have larger gav. than the U-V
chains
PRB, 59, 14827 (1999).
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Exciton Binding Energy in Hubbard and U-V Models

• We focus on lowest 11Bu exciton.
• The conduction band edge Eg is assumed to be

corresponding to two long neutral chains
giving well separated, freely moving
positive and negative polarons

• Exciton binding energy Eb is given by,

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• Nonzero V is required for nonzero Eb
• V lt U/2, Eb is nearly zero
• V gt U/2, Eb strongly depends upon d
• Charge gap Eg not independent of V in the SDW
  limit.

PRB, 55, 15368 (1997)
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Ordering of Low-lying Excitations

• Two important low-lying excitations in
  conjugated
• Polymers are the lowest one-photon state
  (11Bu) and
• the lowest two-photon state (21Ag).
• Kasha rule in organic photochemistry
  fluorescent
• light emission always occurs from lowest
  excited state.
• Implications for level ordering
• E (11Bu) lt E (21Ag) . Polymer is
  fluorescent
• E (21Ag) lt E (11Bu) . Polymer
  nonfluorescent
• Level ordering controlled by polymer topology,
• correlation strength and conjugation length

PRL, 71, 1609 (1993).
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For small U/t, (11Bu) is below (21Ag). As U/t
increases, weight of covalent states in 21Ag
increases. 11Bu has no covalent contribution and
hence its energy increases with U/t.
PRB 56, 9298 (1997)
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• Crossover of the 11Bu and 21Ag states can also
  be seen
• to occur as a function of d. As U/t
  increases, crossover
• occurs at a higher value of d.
• The 21Ag state can be described as two triplet
  excitons
• only at large U/t values and small
  dimerization.

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• Crossover of 2A and 1B also occurs for
  intermediate
• correlation strengths.
• For small U/t, 2A is always above 1B. For large
  U/t,
• 1B is always above 2A.

2A state is more localized than 1B state. As
system size increases 1B descends below 2A.
PRB 56, 9298 (1997)
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Lattice Relaxations of Excited States
Poly acetylene (CH)x can support different
topological excitations made up of
solitons Equilibrium geometry of even carbon
polyene is
Equilibrium geometry of odd carbon polyene is
solitonic
Adv. Q. Chem., 38,123 (2000).
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Electron correlations remove the association
between soliton topology and energy of the
state. Do electron correlations also remove the
association of excited state molecular geometry
with solitons?
Obtaining equilibrium geometries of excited
states
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Polymers with Nonlinear Topologies
Many interesting phenyl, thiophene and other
ring based polymers

• Poly para phenylene, (PPP)
• Poly para phenylene vinylene, (PPV)
• Poly acenes, (PAc)
• Poly thiophenes (PT)
• Poly pyrroles
• Poly furan

All these are one-dimensional polymers but
contain ring systems. Incorporating long range
Coulomb interactions important.
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Some Interesting Questions
Is there a Peierls instability in polyacene?
Is the ground state geometry
Band structure of polyacenes corresponding to the
three cases. Matrix element of symmetric
perturbation between A and S band edges is
zero. Conditional Peierls Instability.
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Role of Long-range Electron Correlations
Used Pariser-Parr-Pople model within DMRG scheme
Polyacene is built by adding two sites at a time.
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DEA(N,d) E(N,0) E(N,d) A cis /
trans DEA(?,d) Lim. N ? ? DEA(N,d) / N For
both cis and trans distortions, DEA(?,d) ?
d2 Peierls instability is conditional in
polyacenes
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Bond order bond order correlations
bi,i1 Ss (ai,s ai,s H.c.)
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Bond order bond order correlations and the
bond structure factors show that polyacene is not
distorted in the ground state.
Spectral gaps in polyacenes. Interesting to
study one and two photon gaps as well as spin
gaps in polyacenes
Comparison of DMRG and exact optical gap in
Hückel model for polyacenes with up to 9 rings.
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• Crossover in the two-photn and optical gap at
• pentacene, experimnetally seen.
• One photon state more localized than two photon
  state.
• Unusually small triplet or spin gap.

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Bond Orders in Different States
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Future Issues

• An important issue in conjugated polymers
• explanation for singlet to triplet branching
  
• ratio, h gt 0.25, in e-h recombination. free
  spin
• statistics h 0.25, experiments 0.25 ? h ?
  0.6.
• Exact time dependent quantum many-
• body studies on short chains emphasize
• role of electron correlations, yield h gt 0.25

Nature (London), 409, 494 (2001), PRB 67, 045109
(2003).
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• Studies required for long chains and real
• polymers to explain some experimental
• observations.
• Other questions
• Triplet-triplet scattering in polymers
• exciton migration in polymers
• exciton dissociation in polymers

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Thank You