Applications of DMRG to Conjugated Polymers

S. RamaseshaSolid State and Structural Chemistry Unit

Indian Institute of ScienceBangalore 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@sscu.iisc.ernet.in

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

Introduction to Conjugated Polymers

Contain extended network of unsaturated(sp2 hybridized) Carbon atoms

n

Eg: Poly acetylene (CH)x, poly para phenylene (PPP) poly acene and poly para phenylene vinylene (PPV)

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

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 i, the site energy at site ‘i’.

o = tij (ai aj + H.c.) + i ni

†

<ij> i

Hückel model Single bandfor tight – bindingmolecules model in Solids

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.

Interacting Electron Models

Explicit electron – electron interactions essential for realistic modeling

[ij|kl] = i*(1) j(1) (e2/r12) k

*(2) l(2) d3r1d3r2

This model requires further simplification to enable routine solvability.

HFull = Ho + ½ Σ [ij|kl] (EijEkl – jkEil)ijkl

Eij = a†i,aj,

Zero Differential Overlap (ZDO) Approximation

[ij|kl] = [ij|kl]ij kl

[ij|kl] = i*(1) j(1) (e2/r12) k

*(2) l(2) d3r1d3r2

Hückel model + on-site repulsions

[ii|jj] = [ii|jj] 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 .

Hubbard Model

HHub = Ho + Σ Ui ni (ni - 1)/2

i

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.

[ii|jj] parametrized by V( rij )

HPPP = HHub + Σ V(rij) (ni - zi) (nj - zj)i>j

Model Hamiltonian

PPP Hamiltonian (1953)

HPPP = Σ tij (aiσ ajσ + H.c.) + Σ(Ui /2)ni(ni-1)

+ Σ V(rij) (ni - 1) (nj - 1)

†

<ij>σ i

i>j

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.

Symmetries in the PPP and Hubbard Models

When all sites are equivalent, for a bipartitelattice, we have electron-hole or charge conjugation or alternancy symmetry, at half-filling.

Hamiltonian is invariant for the transformation

Electron-hole symmetry:

ai † = bi ; ‘i’ on sublattice A

ai † = - bi ; ‘i’ on sublattice B

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

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.

Ne = NEint. = 0, U, 2U,···

Eint. = 0, U, 2U,···

Eint. = U, 2U,···

Even e-h space

Odd e-h space

Dipole operator

Includes covalent states

Excludes covalent states

Spin symmetries:

Hamiltonian conserves total spin and z – component of total spin.

[H,S2] = 0 ; [H,Sz] = 0

Exploiting invariance of the total Sz is trivial, but of the total S2 is hard.

When MStot. = 0, H is invariant when all the

spins are rotated about the y-axis by This operation corresponds to flipping all the spins in the basis – called parity.

MS = 0

Stot. = 0,1,2, ···

Stot. = 0,2,4, ···

Stot. =1,3,5, ···

Even parity space

Odd parity space

Parity divides the total spin space into spacesof even total spin and odd total spin.

Important states in conjugated polymers:

Ground state (11A+g);

Lowest dipole excited state (11B-u);

Lowest triplet state (13B+u);

Lowest two-photon state (21A+g) 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.

Why do we need symmetrization

Matrix Representation of Site e-h and Site Parity Operators

• Fock space of single site:

|1> = |0>; |2> = |>; |3> = |>; & |4> = |>

• The site e-h operator, Ji, has the property:

Ji |1> = |4> ; Ji |2> = |2> ; Ji |3> = |3> & Ji |4> = - |1>

= +1 for ‘A’ sublattice and –1 for ‘B’ sublattice

• The site parity operator, Pi, has the property:

Pi |1> = |1> ; Pi |2> = |3> ; Pi |3> = |2> ; Pi |4> = - |4>

The C2 operation does not have a site representation

Matrix representation of system J and P

J of the system is given by

J = J1 J2 J3 ····· JN

P of the system is, similarly, given by

P = P1 P2 P3 ····· PN

The overall electron-hole symmetry and paritymatrices can be obtained as direct products

of the individual site matrices.

Symmetrized DMRG Procedure

At every iteration, J and P matrices of sub-blocks are renormalized to obtain JL, JR, PL and PR.

From renormalized JL, JR, PL and PR, the super block matrices, J and P are constructed.

Given DMRG basis states ’, ’> ( |’> are eigenvectors of right & left block density matrices, L & R and |’> are Fock states of the two single sites in the super-block), super- block matrix J is given by

J’’ ’’ = ’, ’|J|’’>

= < JLJ1|> ’J1|’> < ’JR’

similarly, the matrix P.

C2| ’,’> = (-1) | ’,’,>;

= (n’ + n’)(n+ n)

and from this, we can construct the matrix for C2.

Operation by the end-to-end interchange on theDMRG basis yields,

Since J, P and C2 all commute, they form an

Abelian group with irreducible representations,eA+, eA-, oA+, oA-, eB+, eB-, oB+, oB-; where ‘e’ and

‘o’ imply even and odd under parity; ‘+’ and ‘-’

Imply even and odd under e-h symmetry.

Ground state lies in eA+, dipole allowed optical

excitation in eB-, and the lowest triplet in oB+.

Projection operator for a chosen irreducible representation , Pis

P = R) R

R

1/h

The dimensionality of the space is given by,

D = 1/h R) red.R)

R

Eliminating linear dependencies in the matrix P yields the symmetrization matrix S with D rowsand M columns, where M is the dimensionalityof the unsymmetrized DMRG space.

The symmetrized DMRG Hamiltonian matrix, H S , is obtained from the unsymmetrized DMRG Hamiltonian, H ,

H S = S H S†

The symmetry operators JL, JR, PL, and PR for the augmented sub-blocks can be constructed and renormalized just as the other operators.

To compute properties, one could unsymmetrizethe eigenstates and proceed as usual.

To implement finite DMRG scheme, C2 symmetry is used only at the end of each finite iteration.

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

Eg,N = 1.278, U/t =4

Eg,N = 2.895, U/t =4

DMRG

DMRG

PRB, 54, 7598 (1996).

The spin gap in the limit U/t should vanishfor Hubbard model.

PRB, 54, 7598 (1996).

Dynamic Response Functions from DMRG

Commonly used technique in physics is Lanczostechnique

In chemistry, sum-over-states (SOS) techniqueis 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).

†

0

ˆ ˆ1 | | | |( ) Im

R R

G O R R O GI

E E i

Correction Vector Technique

Correction vector ) is defined as

(1)0

† (1)

ˆˆ( ) ( ) |

then,

1 ˆ( ) Im | | ( )

H E i O G

I G O

We can solve for ) in a chosen basis bysolving a set of inhomogeneous linear algebraicEquations, using a small matrix algorithm. J. Comput. Chem., 11, 545 (1990).

Need for Symmetrization

In systems with symmetry, dipole operator maps

{eA+} {eB-}i

Therefore, lies in eB- subspace. The unsymmetrized

matrix (H-E0I) is singular while in the eB- subspace it is

nonsingular; allowing solving for from

Similarly, , lies in the singlet or odd parity subspace.Using parity eliminates singularity of the matrix (H E0 ħ)for ħ= ET.

Computation of NLO Coefficients

To solve for dynamic nonlinear optic coefficients, we solvea hierarchy of correction vectors:

and the linear and NLO response coefficients are given by

(1) (1)

(1) (2)ijkl 1 2 3 1 2 1

1( ) [ ( ) | | ( ) | | ]

2and

ˆ( , , ) ( ) | | ( , )

ij i j i j

i j kl

G G

P

% %

%

Where, P permutes the frequencies and the subscripts in pairs and = .

To test the technique, we compare the rotationally averaged linear polarizability and THG coefficient

3 3

1 , 1

1 1; (2 )

3 15ii iijj ijjii i j

Computed at = 0.1t exactly for a Hubbard chain of 12 sites at U/t=4 with DMRG computation with m=200

5.343 5.317 598.3 591.1exact DMRG exact DMRG

in 10-24 esu and in 10-36 esu in all cases

The dominant xx) is 14.83 (exact) and 14.81 (DMRG)and xxxx) 2873 (exact) and 2872 (DMRG).

THG coefficient in Hubbard models as a functionof chain length, L and dimerization :

Superlinear behavior diminishes both with increase in U/t and increase in .

av. vs Chain Length and in U-V Model

For U > 2V, (SDW regime) av. shows similar dependence on L as the Hubbard model, independent of .

U=2V (SDW/CDW crossover point) Hubbard chains havelarger av. than the U-V chains

PRB, 59, 14827 (1999).

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,

lim ( );

( ) ( 1) ( 1) 2 ( )

g gN

g N e N e N e

E E N

E N E N N E N N E N N

1lim [1 ( )]b g uN

E E E B N

PRB, 55, 15368 (1997)

• Nonzero V is required for nonzero Eb

• V < U/2, Eb is nearly zero

• V > U/2, Eb strongly depends upon

• Charge gap Eg not independent of V in the SDW limit.

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) < E (21Ag) …. Polymer is fluorescent

E (21Ag) < E (11Bu) …. Polymer nonfluorescent

Level ordering controlled by polymer topology, correlation strength and conjugation length

PRL, 71, 1609 (1993).

For small U/t, (11Bu) is below (21Ag). As U/t increases,weight of covalent states in 21Ag increases. 11Bu hasno covalent contribution and hence its energy increaseswith U/t.

PRB 56, 9298 (1997)

Crossover of the 11Bu and 21Ag states can also be seen to occur as a function of . As U/t increases, crossover occurs at a higher value of .

The 21Ag state can be described as two triplet excitons only at large U/t values and small dimerization.

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)

Lattice Relaxations of Excited StatesPoly acetylene (CH)x can support different topologicalexcitations made up of solitons:

Equilibrium geometry of even carbon polyene is

x x

Equilibrium geometry of odd carbon polyene is solitonic

x

Adv. Q. Chem., 38,123 (2000).

Noninteracting theories - soliton mid-gap state.

Soliton treated as an elementary electronic excitation.

Eg: Triplet – a soliton and anti-soliton pair

2A – two soliton and anti-soliton pairs

Electron correlations remove the association betweensoliton 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:

Use the PPP model Assume each bond has a distortion i.

Include a strain energy term (1/)i i2 ; = 22/kt0,

k is force constant, is e-p coupling strength defined by i=xi/t0, xi is equilibrium bond length.

Constrain total chain length.

Obtain self-consistent is for each state.

= 0.1 long coherence length. We need to compute excited state geometries of long chains.

Bond order profile of aneutral and charged oddpolyene chain of 61 sites.

Bond order profile for,11Ag+,

11Bu- , 21Ag

+ ,13Bu+states in a

40 site polyene chain.

13Bu+ is a pair of soliton and

anti-solitons.

21Ag+ is two pairs of solitons

And anti solitons.

Polymers with Nonlinear Topologies

Many interesting phenyl, thiophene and other ringbased 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.

Some Interesting Questions

Is there a Peierls’ instability in polyacene?

Is the ground state geometry

Uniform

Cis

Trans

Band structure of polyacenescorresponding to the three cases.

Matrix element of symmetric perturbation between A and S band edges is zero.

Conditional Peierls Instability.

Role of Long-range Electron Correlations

Used Pariser-Parr-Pople model within DMRG scheme

Polyacene is built by adding two sites at a time.

cis

cis

trans

trans

E) = E(N,0) – E(N,) ; A = cis / trans

E) = Lim. N {E) / N}

For both cis and trans distortions, E) 2

Peierls’ instability is conditional in polyacenes

Bond order – bond order correlations

bi,i+1 = (a†i,ai,+ H.c.)

= 0

= 0

Bond order – bond order correlations and the bondstructure factors show that polyacene is not distortedin the ground state.

Spectral gaps in polyacenes.

Interesting to study one and two photon gaps as well asspin gaps in polyacenes

Comparison of DMRG and exact optical gap in Hückel model for polyacenes with up to 9 rings.

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.

Bond Orders in Different States

Ground state - legs

Ground state - rung

Future Issues An important issue in conjugated polymers – explanation for singlet to triplet branching ratio, 0.25 in e-h recombination. free spin statistics = 0.25, experiments 0.25 0.6.

Exact time dependent quantum many- body studies on short chains emphasize role of electron correlations, yield > 0.25

Nature (London), 409, 494 (2001), PRB 67, 045109 (2003).

,

, ,

, 2, ,

1(0) | | | | ,

2ˆ ˆ

(1 ) ( ) (1 ) ( )2 2

( ) | ( ) | |

S T

S T S T

S Tm n S T m n

P P P P

iH t iH tt t t

I t t

h h

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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