Applications of DMRG to Conjugated Polymers S. Ramasesha Solid State and Structural Chemistry Unit Indian Institute of Science Bangalore , 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

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 (sp 2 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. t ij is resonance / transfer integral between bonded sites and  i, the site energy at site ‘i’.  o =  t ij (a i  a j  + H.c.) +  i n i † i    

Hückel model Single band for tight – binding molecules 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) (e 2 /r 12 )  k * (2)  l (2) d 3 r 1 d 3 r 2 This model requires further simplification to enable routine solvability. H Full = H o + ½ Σ [ij|kl] (E ij E kl –  jk E il ) ijkl E ij =  a † i,  a j,           

Zero Differential Overlap (ZDO) Approximation [ij|kl] = [ij|kl]  ij  kl [ij|kl] =  i * (1)  j (1) (e 2 /r 12 )  k * (2)  l (2) d 3 r 1 d 3 r 2

 Hückel model + on-site repulsions [ii|jj] = [ii|jj]  ij = U i  Introduced in  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  H Hub = H o + Σ U i n i (n i - 1)/2 i  

Pariser-Parr-Pople (PPP) Model  z i are local chemical potentials.  V(r ij ) parametrized either using Ohno parametrization: V(r ij ) = { [ 2 / ( U i + U j ) ] 2 + r ij 2 } -1/2 Or using Mataga-Nishimoto parametrization: V(r ij ) = { [ 2 / ( U i + U j ) ] + r ij } -1  PPP model is also a one-parameter (U / t) model. [ii|jj] parametrized by V( r ij ) H PPP = H Hub + Σ V(r ij ) (n i - z i ) (n j - z j ) i>j   

Model Hamiltonian PPP Hamiltonian (1953)     H PPP = Σ t ij (a iσ a jσ + H.c.) + Σ(U i /2)n i (n i -1) + Σ V(r ij ) (n i - 1) (n j - 1) † 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 ~10 6 to 10 7 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 bipartite lattice, we have electron-hole or charge conjugation or alternancy symmetry, at half-filling.  Hamiltonian is invariant for the transformation Electron-hole symmetry:   a i † = b i ; ‘i’ on sublattice A a i † = - b i ; ‘i’ on sublattice B    Polymers also have end-to-end interchange symmetry or inversion symmetry.

E-h symmetry divides the N = N e space into two spaces, one containing both ‘covalent’ and ‘ionic’ bases, the other containing only ionic bases. Dipole operator connects the two spaces. N e = N E int. = 0, U, 2U,··· E int. = 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,S 2 ] = 0 ; [H,S z ] = 0  Exploiting invariance of the total S z is trivial, but of the total S 2 is hard.  When M S tot. = 0, H is invariant when all the spins are rotated about the y-axis by p. This operation corresponds to flipping all the spins in the basis – called parity.       

M S = 0 S tot. = 0,1,2, ··· S tot. = 0,2,4, ··· S tot. =1,3,5, ··· Even parity space Odd parity space Parity divides the total spin space into spaces of even total spin and odd total spin.

 Important states in conjugated polymers: Ground state (1 1 A + g ); Lowest dipole excited state (1 1 B - u ); Lowest triplet state (1 3 B + u ); Lowest two-photon state (2 1 A + 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, J i, has the property: J i |1> = |4> ; J i |2> = h|2> ; J i |3> =  |3> & J i |4> = - |1> h = +1 for ‘A’ sublattice and –1 for ‘B’ sublattice     The site parity operator, P i, has the property: P i |1> = |1> ; P i |2> = |3> ; P i |3> = |2> ; P i |4> = - |4>    

 The C 2 operation does not have a site representation  Matrix representation of system J and P    J of the system is given by J = J 1  J 2  J 3  ·····  J N         P of the system is, similarly, given by P = P 1  P 2  P 3  ·····  P N     The overall electron-hole symmetry and parity matrices 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 J L, J R, P L and P R.  From renormalized J L, J R, P L and P R, 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|  ’  ’> =  ’  J 1 |  ’> <  ’  J R  ’  similarly, the matrix P.   

 C 2 |  ’,  ’> = (-1)  |  ’,  ’,  >;  = (n  ’ + n  ’ )(n  + n  ) and from this, we can construct the matrix for C 2.  Operation by the end-to-end interchange on the DMRG basis yields, Since J, P and C 2 all commute, they form an Abelian group with irreducible representations, e A +, e A -, o A +, o A -, e B +, e B -, o B +, o B - ; where ‘e’ and ‘o’ imply even and odd under parity; ‘+’ and ‘-’ Imply even and odd under e-h symmetry. Ground state lies in e A +, dipole allowed optical excitation in e B -, and the lowest triplet in o B +.   

Projection operator for a chosen irreducible representation , P   is 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   rows and M columns, where M is the dimensionality of 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 J L, J R, P L, and P R for the augmented sub-blocks can be constructed and renormalized just as the other operators. To compute properties, one could unsymmetrize the eigenstates and proceed as usual. To implement finite DMRG scheme, C 2 symmetry is used only at the end of each finite iteration. 

Checks on SDMRG  Optical gap (E g ) in Hubbard model known analytically. In the limit of infinite chain length, for U/t = 4.0, E g exact = t ; U/t = 6.0 E g exact = t E g,N   = 1.278, U/t =4 E g,N   = 2.895, U/t =4 DMRG PRB, 54, 7598 (1996).

The spin gap in the limit U/t   should vanish for Hubbard model. PRB, 54, 7598 (1996).

Dynamic Response Functions from DMRG Commonly used technique in physics is Lanczos technique

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).

Correction Vector Technique Correction vector    ) is defined as We can solve for    ) in a chosen basis by solving a set of inhomogeneous linear algebraic Equations, using a small matrix algorithm. J. Comput. Chem., 11, 545 (1990).

Need for Symmetrization In systems with symmetry, dipole operator maps {eA+}{eA+} {eB-}{eB-} ii Therefore,     lies in e B - subspace. The unsymmetrized matrix (H-E 0 I) is singular while in the e B - 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  E 0  ħ  ) for ħ  = E T.

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   =      .

To test the technique, we compare the rotationally averaged linear polarizability and THG coefficient Computed at  = 0.1t exactly for a Hubbard chain of 12 sites at U/t=4 with DMRG computation with m=  in esu and  in esu in all cases The dominant  xx ) is (exact) and (DMRG) and  xxxx ) 2873 (exact) and 2872 (DMRG).

THG coefficient in Hubbard models as a function of 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 have larger  av. than the U-V chains PRB, 59, (1999).

Exciton Binding Energy in Hubbard and U-V Models  We focus on lowest 1 1 B u exciton.  The conduction band edge E g is assumed to be corresponding to two long neutral chains giving well separated, freely moving positive and negative polarons  Exciton binding energy E b is given by,

PRB, 55, (1997) Nonzero V is required for nonzero E b V < U/2, E b is nearly zero V > U/2, E b strongly depends upon  Charge gap E g 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 (1 1 B u ) and the lowest two-photon state (2 1 A g ).  Kasha rule in organic photochemistry – fluorescent light emission always occurs from lowest excited state.  Implications for level ordering E (1 1 B u ) < E (2 1 A g ) …. Polymer is fluorescent E (2 1 A g ) < E (1 1 B u ) …. Polymer nonfluorescent  Level ordering controlled by polymer topology, correlation strength and conjugation length PRL, 71, 1609 (1993).

For small U/t, (1 1 B u ) is below (2 1 A g ). As U/t increases, weight of covalent states in 2 1 A g increases. 1 1 B u has no covalent contribution and hence its energy increases with U/t. PRB 56, 9298 (1997)

 Crossover of the 1 1 B u and 2 1 A g states can also be seen to occur as a function of . As U/t increases, crossover occurs at a higher value of .  The 2 1 A g 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 States Poly acetylene (CH) x can support different topological excitations made up of solitons: Equilibrium geometry of even carbon polyene is xx 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 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:  Use the PPP model  Assume each bond has a distortion  i.  Include a strain energy term (1/  )  i  i 2 ; = 2  2 /  kt 0, k is force constant,  is e-p coupling strength defined by  i =  x i /t 0, x i is equilibrium bond length.  Constrain total chain length.  Obtain self-consistent  i s for each state.  = 0.1 long  coherence length. We need to compute excited state geometries of long chains.

Bond order profile of a neutral and charged odd polyene chain of 61 sites. Bond order profile for,1 1 A g +, 1 1 B u -, 2 1 A g +,1 3 B u + states in a 40 site polyene chain. 1 3 B u + is a pair of soliton and anti-solitons. 2 1 A g + is two pairs of solitons And anti solitons.

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.

Some Interesting Questions Is there a Peierls’ instability in polyacene? Is the ground state geometry Uniform Cis Trans 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.

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 b i,i+1 =   (a † i,   a i,   + H.c.)  = 0

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.

 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, (2003).

 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