Department of Physics, Chemistry and Biology
Master’s Thesis
Theoretical studies of optical absorption in
low-bandgap polymers
Daniel Karlsson
LiTH-IFM-EX-05/1498-SE
Department of Physics, Chemistry and Biology Link¨opings universitet
Master’s Thesis LiTH-IFM-EX-05/1498-SE
Theoretical studies of optical absorption in
low-bandgap polymers
Daniel Karlsson
Adviser: Sven Stafstr¨om
IFM, Link¨oping
Examiner: Sven Stafstr¨om
IFM, Link¨oping
Avdelning, Institution Division, Department Computational Physics
Department of Physics, Chemistry and Biology Link¨opings universitet
SE-581 83 Link¨oping, Sweden
Datum Date 2005-09-13 Spr˚ak Language Svenska/Swedish Engelska/English Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats ¨Ovrig rapport
URL f¨or elektronisk version
ISBN
ISRN
Serietitel och serienummer Title of series, numbering
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Titel Title
Teoretiska studier av optisk absorption i polymerer med l˚aga bandgap Theoretical studies of optical absorption in low-bandgap polymers
F¨orfattare Author
Daniel Karlsson
Sammanfattning Abstract
The absorption spectra of a recently designed low-bandgap conjugated polymer has been studied using the semi-empirical method ZINDO and TDDFT/B3LYP/6-31G. The vertical excitation energies have been calculated for monomer up to hexamer. Two main absorption peaks can be seen, the one largest in wavelength corresponding to a HOMO to LUMO transition, and one involving higher or-der excitations. TDDFT results are red-shifted compared to the ZINDO results. Comparison with experiment yields that short conjugation lengths are dominat-ing. This is possibly due to steric interactions between polymer chains, breaking the conjugation length. Such effects are also studied.
Nyckelord Keywords
ZINDO, Density functional theory, Hartree-Fock method, Low bandgap polymers, Optical absorption, Absorption spectra, Semi empirical methods, Plastic solar cells
http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-4349
—
LiTH-IFM-EX-05/1498-SE
Abstract
The absorption spectra of a recently designed low-bandgap conjugated polymer has been studied using the semi-empirical method ZINDO and TDDFT/B3LYP/6-31G. The vertical excitation energies have been calculated for monomer up to hexamer. Two main absorption peaks can be seen, the one largest in wavelength corresponding to a HOMO to LUMO transition, and one involving higher or-der excitations. TDDFT results are red-shifted compared to the ZINDO results. Comparison with experiment yields that short conjugation lengths are dominat-ing. This is possibly due to steric interactions between polymer chains, breaking the conjugation length. Such effects are also studied.
Acknowledgements
First and foremost I would like to thank my supervisor Professor Sven Stafstr¨om for giving me the opportunity of working with this project, and for helping and assisting throughout the work. I would also like to thank the rest of the Com-putational physics group for their support. Acknowledgements go to Professor Olle Ingan¨as for taking the initiative of the project. Other people that deserves a thank you include Shimelis Admassie for discussions and experimental data and my opponent Andreas Gerhardsson for reading and commenting.
A big thank you to all my friends and classmates who make my days something more than just work. Finally, I want to thank my family for their support and care, and always being there for me.
Contents
1 Introduction 1 1.1 Project description . . . 1 1.2 Thesis outline . . . 1 2 Background 3 2.1 Conjugated polymers . . . 3 2.2 Applications . . . 43 Quantum chemistry and methodology 5 3.1 The Schr¨odinger equation . . . 5
3.1.1 The Born-Oppenheimer approximation . . . 6
3.2 Orbitals, Spin and Slater determinants . . . 6
3.2.1 The Pauli exclusion principle . . . 6
3.2.2 Hartree products . . . 7
3.2.3 Slater determinants . . . 7
3.2.4 The energy of a Slater determinant . . . 8
3.3 Hartree-Fock theory . . . 10
3.3.1 The Hartree-Fock equations . . . 10
3.3.2 Roothan-Hall equations . . . 11
3.3.3 The self-consistent-field (SCF) procedure . . . 12
3.4 Basis functions . . . 12
3.5 Electron correlation . . . 13
3.5.1 Configuration interaction . . . 13
3.6 Semi-empirical methods . . . 15
3.6.1 Expression for the Fock matrix . . . 15
3.6.2 Complete Neglect of Differential Overlap . . . 16
3.6.3 Intermediate Neglect of Differential Overlap . . . 16
3.6.4 ZINDO . . . 17
3.7 Density functional theory . . . 18
3.7.1 The Kohn-Sham equations . . . 19
3.7.2 Time-dependent DFT . . . 21 ix
x Contents
4 Results and discussion 23
4.1 Computational details . . . 23
4.2 The polymer wm356 . . . 23
4.2.1 Experimental results . . . 23
4.3 Geometry . . . 25
4.4 Excitation energy calculations . . . 26
4.4.1 ZINDO . . . 26
4.4.2 TDDFT . . . 34
4.4.3 Difference densities . . . 35
4.5 Geometry deformations . . . 37
4.5.1 Torsion . . . 37
4.5.2 Phenyl groups removed . . . 38
5 Conclusions and future work 39 5.1 Conclusion . . . 39
5.2 Future work . . . 39
Chapter 1
Introduction
Over the last decade conjugated polymers have gained much interest due to their electronic and optical properties. One of the most important factors of controlling physical properties is the bandgap, which is a current topic of research. In particu-lar, polymers with a low bandgap are desired in applications such as light emitting diodes (LEDs) and solar cells. Other applications, e.g. electronic transistors, also take benefit of the sometimes very high conductive properties possessed by these polymers.
1.1
Project description
In this thesis we study the electronic structure and optical absorption of a recently designed polymer showing very good absorption properties in the low frequency (IR) region. Focus is put on determining molecular geometry and absorption spectra. Calculated results are compared with experimental data, and based on such a comparison we try to obtain a qualitative understanding of the physical properties of the polymer. Both density functional theory and the semi-empirical ZINDO method are used for calculating molecular geometry, electronic structure and excited states.
1.2
Thesis outline
Chapter 2 gives some background of conjugated polymers, their properties and applications. In Chapter 3 a brief introduction to quantum chemistry is given, together with a more specific description of the theory used in the calculations. The results of the calculations are then presented and discussed in Chapter 4, while in Chapter 5 the conclusions are summarized and possible ways to continue and improve are discussed.
Chapter 2
Background
2.1
Conjugated polymers
The word polymer originates from the Greek words ’poly’ meaning many and ’mer’ meaning part. A polymer is built up from a repeated unit called a monomer, or simply unit or unit cell. Two repeated units constitute a dimer, three a trimer and so on.
Conventional polymers, plastics, are traditionally used for their mechanical and insulating properties. The use of organic “π-conjugated” polymers as electrically conducting materials in molecular based electronics is relatively new. The great diversity of different organic polymers issues from the property of the carbon atom capable of forming different types of chemical bonds with atoms of different kind. This property is due to hybridization between the two electrons in the 2s2
level and the six electrons in the 2p2 level, forming four equivalent sp3-orbitals.
The bonds between hybridized sp-orbitals are called σ-bonds. The large band gap in σ-bonded polymers with sp3 hybridized carbon atoms renders them to be
electrically insulating. In conjugated molecules and polymers, the carbon atom hybridize to three sp2-orbitals while the p
z-orbital remains. The sp2-orbitals form
σ-bonds in the plane of the structure, connecting adjacent atoms. The remaining pz-orbitals, perpendicular to the backbone plane, overlap and form the weaker
π-bonds (π-orbitals) with neighbouring pz-orbitals. For extended structures the
π-overlap of pz-orbitals tends to delocalize the electronic wavefunction along the
polymer chain. The delocalized π-orbitals usually constitute the higher occupied and lower unoccupied electronic states, which are responsible for the rather small band gap, typically 1 to 4 eV [1], of the conjugated polymers. The result is that the conjugated polymers exhibit semi-conducting properties.
The extension of delocalization of the π-orbitals defines the conjugation length of the polymer. The range of delocalization may differ, but often orbitals delocalize over parts of the polymer chain, allowing for the conjugation length to be measured in the number of unit cells.
In order to alter the electronic properties, as well as to enhance their pro-cessibility (solubility), chemically active groups are often attached to the main
4 Background
conjugated backbone. One method is to use either electron accepting or electron donating groups, giving rise to charge transfer, which directly affect the delocalized electronic levels.
Another approach used for creating a low bandgap polymer is to use alternat-ing electron donatalternat-ing and electron acceptalternat-ing units in the actual backbone of the polymer. Electron donating units have a high highest occupied molecular orbital (HOMO) and electron accepting units have a low lowest unoccupied molecular orbital (LUMO). Alternating these units in a polymer chain causes intra-chain charge transfer and tends to destabilize HOMO and stabilize LUMO of the poly-mer. This is an effective way of reducing the bandgap. For more information on this type of polymer see Ref. [2]. For further reading about conjugated polymers the reader is referred to Ref. [3, 1].
2.2
Applications
The semi-conducting properties of conjugated polymers have made it possible to use them in various electronic devices. In particular they may be used in applications such as transistors and integrated circuits [4, 5, 6, 7], photo-voltaic devices (solar cells) [8, 9] and their counterparts light emitting diodes (LEDs) [10]. Even solid-state lasers are under development [11, 12].
Regarding the extremely low-bandgap polymers [13], they are gaining most interest in the areas of LEDs and solar cell devices. While being mouldable, lightweight and manufactured at low costs, the issue with solar cells has been the low energy conversion efficiency, only a few percent. In order to increase the efficiency a low bandgap is desired, allowing electronic transitions at low energies and thus taking advantage of a wider spectra of the solar emission. The basic concept of a solar cell using polymers is to first excite an electron of the polymer, and then to let the polymer act as electron donor. The excited electron has to be transfered to an electron acceptor, like the C60. These two molecular complexes
are then embedded between metal layers, one anode and one cathode, in order to get a voltage.
Chapter 3
Quantum chemistry and
methodology
The initial sections in this chapter closely follow the first 4 chapters in Ref. [14], and for further insight the reader is referred to Ref. [14].
3.1
The Schr¨
odinger equation
The central problem in quantum chemistry is solving the time-independent Schr¨odinger equation
HΨ = EΨ (3.1)
where H is the Hamiltonian, Ψ is the wavefunction and E is the energy of the system. The solution to this many body problem is analytically possible only for systems with up to two particles, for example the hydrogen atom. However, one is most often faced with larger systems, and various approximations and compu-tational methods have to be used. This chapter gives a short introduction to the theory and methods used in this diploma work.
In atomic units (~ = e = me= 1) the total non-relativistic, time-independent
Hamiltonian for a system with N electrons and M nuclei is
H = − N X i=1 1 2∇ 2 i − M X iA=1 1 2MA∇ 2 A− N X i=1 M X A=1 ZA riA + + N X i=1 N X j>i 1 rij + M X A=1 M X B>A ZAZB RAB (3.2) where A, B designate nuclei, i, j electrons, and Z are the atomic numbers. The first term in Eq. (3.2) is the operator for the kinetic energy of the electrons; the
6 Quantum chemistry and methodology
second term is the operator for the kinetic energy of the nuclei; the third term represents the Coulomb attraction between electrons and the nuclei; the fourth and fifth terms represents the repulsion between electrons and between nuclei, respectively.
3.1.1
The Born-Oppenheimer approximation
Since the nuclei are much heavier than the electrons their velocity is much lower in comparison. This allows for the approximation that the electrons are moving in the field of fixed nuclei, the Born-Oppenheimer approximation. The second term in Eq. (3.2), the kinetic energy of the nuclei, can now be neglected, and the last term, the repulsion between the nuclei, can be considered constant, Vnn. A
constant added to the Hamiltonian only adds to the energy-eigenvalue and doesn’t affect the wavefunction.
H = − N X i=1 1 2∇ 2 i − N X i=1 M X A=1 ZA riA + N X i=1 N X j>i 1 rij + Vnn (3.3)
The new Hamiltonian is called the electronic Hamiltonian, and the solution to the resulting Schr¨odinger equation is called the electronic wavefunction and electronic energy. The area of quantum chemistry deals with solutions to the electronic problem, and when writing Ψ and H one usually means the electronic wavefunction and the electronic Hamiltonian.
3.2
Orbitals, Spin and Slater determinants
The electronic Hamiltonian depends only on spatial coordinates. To completely describe an electron it is necessary to specify the spin degree of freedom. This is done by introducing two orthonormal spin functions, α(ω) and β(ω), corresponding to spin up and spin down. Thus each electron is described by four variables, ¯
x = (¯r, ω).
An orbital is a wave function for a single electron. A spatial orbital ψi(¯r)
describes the spatial distribution of the electron, and the spatial molecular orbitals are assumed to form an ON basis, hψi|ψji = δij. A spin orbital also depends on
electron spin, and from one spatial orbital one can form two different spin orbitals
χ(¯x) = ψ(¯r)α(ω) or ψ(¯r)β(ω) (3.4) Spin orbitals are also known as Molecular Orbitals (MO).
3.2.1
The Pauli exclusion principle
The many electron wavefunction Ψ for an N-electron system is a function of ¯
x1, ¯x2, . . . , ¯xN. The electrons are indistinguishable so apart from a phase
3.2 Orbitals, Spin and Slater determinants 7
of any two electrons. According to the spin-statistic theorem the total wavefunc-tion for fermions (for example electrons) must be antisymmetric with respect to interchange of the coordinates ¯x of any two electrons,
Ψ(¯x1, . . . , ¯xi, . . . , ¯xj, . . . , ¯xN) = −Ψ(¯x1, . . . , ¯xj, . . . , ¯xi, . . . , ¯xN) (3.5)
It follows that if i = j the antisymmetric wavefunction is identically zero, i.e. two electrons cannot occupy the same spin orbital. This is what is called the Pauli exclusion principle.
3.2.2
Hartree products
If we consider a system of non-interacting electrons and thus neglect electron-electron repulsion the electron-electronic Hamiltonian is a sum of one-electron-electron Hamiltonians
H = N X i=1 h(i), h(i) = −1 2∇ 2 i − X A ZA riA (3.6) The eigenfunctions to the operators h(i) can be taken to be a set of spin orbitals, {χa},
h(i)χa(¯xi) = aχa(¯xi) (3.7)
Now, since the total Hamiltonian is a sum over one-electron Hamiltonians, the many electron wavefunction, which is an eigenfunction of H, will be a product of spin orbitals, one for each electron
ΨHP(¯x1, ¯x2, . . . , ¯xN) = χ1(¯x1)χ2(¯x2) . . . χN(¯xN) = Π (3.8)
and is called a Hartree product. A useful short-hand notation is Π. The eigenvalue, E, will be a sum of the spin orbital energies
HΨHP = EΨHP, E = 1+ 2+ . . . + N (3.9)
Unfortunately, the Hartree product does not satisfy the required antisymmetry of many electron wavefunctions.
3.2.3
Slater determinants
There is a certain form of the many electron wavefunction that fulfils the require-ment of antisymmetry, namely a Slater determinant (SD)
Ψ(¯x1, ¯x2, . . . , ¯xN) = 1 √ N ! χ1(¯x1) χ2(¯x1) · · · χN(¯x1) χ1(¯x2) χ2(¯x2) · · · χN(¯x2) .. . ... . .. ... χ1(¯xN) χ2(¯xN) · · · χN(¯xN) (3.10)
An interchange of two electrons corresponds to interchange of two rows in the determinant, and thus a change of sign. Two electrons occupying the same spin
8 Quantum chemistry and methodology
orbital corresponds to having two columns in the determinant equal, which makes the determinant equal to zero. A useful short-hand notation of the Slater deter-minant is
Ψ(¯x1, ¯x2, . . . , ¯xN) = A[χ1χ2· · · χN] = AΠ (3.11)
where the antisymmetrizing operator A acts on the Hartree product (the product of the diagonal elements of the determinant) to get a correctly antisymmetrized wavefunction.
3.2.4
The energy of a Slater determinant
The antisymmetrizing operator can be written as a sum of permutation operators Pij which generates all possible permutations of two coordinates, Pijk generates
all possible permutations of three coordinates and so on. A = √1 N ![1 − X ij Pij+ X ijk Pijk− . . .] (3.12)
The electronic Hamiltonian Eq. (3.3) can be written in terms of one and two electron operators H =X i h(i) +X i X j>i gij+ Vnn (3.13)
where h(i) is the one electron operator from Eq. (3.6), and gij =
1
rij (3.14)
is a two electron operator. It can be shown that A commutes with H and that AA =√N !A. Now the energy of a Slater determinant can be written as
E = hΨ|H|Ψi = hAΠ|H|AΠi =√N !hΠ|H|AΠi = hΠ|X i h(i) +X i X j>i gij+ Vnn|Πi− (3.15) −X ij hΠ| · · · |PijΠi + X ijk hΠ| · · · |PijkΠi + · · ·
The one electron operator only gives a non-zero value for the first term in Eq. (3.15) coming from the identity operator in A, since permutations yield vanishing overlap integrals of orthonormalized spin orbitals χi.
hΠ|h(i)|Πi = hχi(i)|h(i)|χi(i)i = hi (3.16)
For the two electron operator only the terms arising from the identity and Pij
3.2 Orbitals, Spin and Slater determinants 9
integrals between orthogonal spin orbitals. The term from the identity operator is hΠ|gij|Πi = hχi(i)χj(j)|gij|χi(i)χj(j)i (3.17)
= Z
d¯rid¯rj|χi(¯ri)|2rij−1|χj(¯rj)|2
= Jij
and is called a Coulomb integral, which represents a classical repulsion between electron densities |χi(¯xi)|2. The Coulomb integral can be written
Jij = hχi(i)χj(j)|gij|χi(i)χj(j)i (3.18)
= hχi(i)|hχj(j)|gij|χj(j)i|χi(i)i
= hχi(i)|Jj(i)|χi(i)i
where Jj(i) is called the Coulomb operator. The effect of the Coulomb operator
when acting on a spin orbital is Jj(i)χa(i) =
Z
d¯xjχ∗j(j)r−1ij χj(j)
χa(i) (3.19)
The second non-zero term, involving the two electron operator and two coor-dinates exchanged, is
hΠ|gij|PijΠi = hχi(i)χj(j)|gij|χj(i)χi(j)i (3.20)
= Z
d¯rid¯rjχ∗i(¯ri)χ∗j(¯rj)r−1ij χj(¯ri)χi(¯rj)
= Kij
and is called an exchange integral. It has no classical interpretation like the Coulomb integral. Writing it in a different manner we get
Kij= hχi(i)χj(j)|gij|χj(i)χi(j)i (3.21)
= hχi(i)|hχj(j)|gij|χi(j)i|χj(i)i
= hχi(i)|Kj(i)|χi(i)i
where Kj(i) is called the exchange operator and is defined by its effect when acting
on a spin orbital χa(i),
Kj(i)χa(i) =
Z
d¯xjχ∗j(j)rij−1χa(j)
χj(i) (3.22)
It involves exchange of two electrons which is due to the antisymmetric nature of the single determinant. The energy of the SD can now be written
E = N X i hi+ N X i N X j>i
10 Quantum chemistry and methodology
3.3
Hartree-Fock theory
We now make another approximation, to let the wavefunction consist of a single Slater determinant. This implies that electron - electron repulsion only is included as an average effect. Each electron is considered to be moving in the field of the nuclei and the average field of the other N-1 electrons. The variational principle can now be used to derive the Hartree-Fock (HF) equations.
3.3.1
The Hartree-Fock equations
The optimal spin orbitals that give the best SD are according to the variational principle those that minimize the electronic energy
E0= hΨ0|H|Ψ0i (3.24)
where H is the full electronic Hamiltonian as in Eq. (3.3) and |Ψ0i is a single SD.
The optimal spin orbitals are found as solutions to the Hartree-Fock equations f (i)χa(¯xi) = aχa(¯xi) (3.25)
where f (i) is the Fock operator and ais the orbital energy. (For a full derivation of
the Hartree-Fock equation the reader is referenced to Ref.[14]) The Fock operator is a sum of a one-electron Hamiltonian, h(i) see Eq. (3.6), and the Hartree-Fock potential, vHF(i), which is the average potential felt by electron i due to
electron-electron repulsion
f (i) = h(i) + vHF(i) (3.26)
The Hartree-Fock potential can be expressed in terms of the Coulomb and ex-change operators from Eq. (3.19) and (3.22)
vHF(i) =
N
X
a
(Ja(i) − Ka(i)) (3.27)
The Fock operator is dependent on its eigenfunctions χa through the Coulomb
and exchange operators, which makes the Hartree-Fock equation non-linear. Thus it will need to be solved by an iterative procedure. (See Section 3.3.3)
It should be noted that the Hartree-Fock energy, the energy of the SD formed by the HF-optimized spin orbitals (see Eq. (3.23)), is not a sum of orbital energies a. This is due to the form of the Coulomb and exchange interactions which get
counted twice if we just sum up the orbital energies. Since the eigenvalues a are
interpreted as orbital energies it follows from Koopmans’ theorem [15] that the ionization energies and electron affinities are given by a.
Another note is that the electronic Schr¨odinger equation is just solved ap-proximately by using the variational principle to find an approximation to the
3.3 Hartree-Fock theory 11
exact electronic wavefunction. But there is another Hamiltonian and correspond-ing eigenvalue equation to which the Hartree-Fock SD is an exact solution, the Hartree-Fock Hamiltonian HHF = N X i f (i) (3.28)
Its eigenvalue is not the HF energy, but the sum of orbital energies.
3.3.2
Roothan-Hall equations
Restricted closed-shell Hartree-Fock
Assume that the spatial part of the spin orbital ψa(¯r) is required to be the same
irrespective of spin. The spin orbitals are then called restricted, which is known as restricted Hartree-Fock (RHF) formalism. If we also require each spatial orbital to be doubly occupied, a determinant formed from them is called a closed-shell determinant. These restrictions means that the spin functions can be integrated out, so that the calculation of spin orbitals now become equivalent of solving the Hartree-Fock equation for just the spatial orbitals. This results in the closed-shell spatial Hartree-Fock equation
f (i)ψj(i) = jψj(i) (3.29)
where the closed-shell Fock operator has the form
f (i) = h(i) + N/2 X a 2Ja(i) − Ka(i) (3.30) Introduction of a basis
The unknown spatial orbitals can be expanded in terms of known basis functions, called atomic orbitals (AO), and we now introduce a set of K AOs, φν. (Basis
functions are further described in Section 3.4.) ψa =
K
X
ν
Cνaφν, a = 1, 2, . . . , K (3.31)
This is known as a Linear Combination of Atomic Orbitals, LCAO. Using a set of K basis functions φν, will then generate a set of K spatial orbitals and 2K spin
orbitals. The N spin orbitals with the lowest energy will be occupied and form the Slater determinant of the Hartree-Fock ground state. The remaining 2K − N spin orbitals are unoccupied and sometimes called virtual.
The Hartree-Fock equations (3.25) may now be written as f (i)X ν Cνaφν(i) = a X ν Cνaφν(i) (3.32)
12 Quantum chemistry and methodology
Multiplying by φ∗
µ(i) on the left and integrating we get
X
ν
Cνa
Z
d¯riφ∗µ(i)f (i)φν(i) = a
X ν Cνa Z d¯riφ∗µ(i)φν(i) (3.33) where we define Sµν = Z d¯riφ∗µ(i)φν(i) (3.34)
as the matrix elements of the overlap matrix S, and Fµν =
Z
d¯riφ∗µ(i)f (i)φν(i) (3.35)
as the matrix elements of the Fock matrix F. Both are Hermitian K ×K matrices, where K is the number of AOs. All equations may now be collected in a matrix notation called Roothaan-Hall equations
FC= SCε (3.36)
where C is a K × K matrix of the expansion coefficients Cνa and ε is a diagonal
matrix of the orbital energies a. The Hartree-Fock differential equation has now
been converted to a matrix equation which can be solved by standard matrix calculations. The problem of calculating MOs reduces to the problem of calculating the set of expansion coefficients. However, because the basis functions are not generally orthogonal, the overlap matrix S will make the Roothaan equations non-linear.
3.3.3
The self-consistent-field (SCF) procedure
Because the Hartree-Fock equation is non-linear it needs to be solved by an itera-tive procedure. This is done by the self-consistent-field procedure and is a method for solving the Roothaan-Hall equations. The procedure starts off with an initial guess of the expansion coefficients, forms the F matrix, and diagonalizes it. The new set of coefficients obtained is then used for calculating a new Fock matrix, and so on until convergence is reached.
3.4
Basis functions
If the set of basis functions used in Eq. (3.31) was complete the expansion would be exact. Larger basis sets would lower the Hartree-Fock energy until the basis set approaches completeness, when the Hartree-Fock limit is reached. A complete basis set would consist of an infinite number of functions. For practical computa-tional reasons one is restricted to a finite set of basis functions.
There are essentially two types of basis functions used in electronic structure calculations: Slater Type Orbitals (STO) and Gaussian Type Orbitals (GTO). The STOs (φ ∼ e−ζr) describe the physics of the atomic orbitals more correctly,
3.5 Electron correlation 13 but two-electron integral evaluation becomes computationally demanding. GTOs (φ ∼ e−ζr2) are more computationally efficient, since analytical expressions are obtained for the integrals (3.34) and (3.35). By using linear combinations of GTOs as basis functions they might be chosen to approximate STOs, atomic orbitals or any set of functions desired. This is known as a contraction of primitive Gaussian type orbitals (PGTO) to a contracted Gaussian type orbital (CGTO).
A basis where there are as many basis functions as there are atomic orbitals in the shells that are occupied is called a minimal basis. In this thesis the basis set 6-31G* has been used. This is a so called split valence basis, where the number of basis functions for valence shells are doubled compared to the minimal basis. The core orbitals are a contraction of six PGTOs, the inner part of the valence orbitals is a contraction of three PGTOs and the outer part is represented by one PGTO. In this case p- and d-type polarization functions have been added to hydrogen and the second row, respectively, which is indicated by the star.
3.5
Electron correlation
Because of the antisymmetry of the Slater determinant, this kind of single determi-nantal wave function incorporates so called exchange correlation. This means that the motion of two electrons with parallel spins is correlated, i.e. the probability of finding these electrons at the same point in space is exactly zero. However, in the opposite spin case, there is always a probability of finding two electrons with opposite spins at the same point in space. Their motion is then said to be uncorre-lated. Therefore, a single determinantal wave function, such as the Hartree-Fock wavefunction, is often referred to as an uncorrelated wave function. This is an approximation within the HF model, which is why electron - electron interaction only is included as an average effect. On average, the electrons are further apart than described by the HF model.
As a consequence from the fact that electrons with opposite spins are not correlated within the HF model, the HF energy will be too high. The difference between the exact non-relativistic energy (E0) and the HF-limit energy (E0) is
called the correlation energy
Ecorr= E0− E0 (3.37)
In order to obtain the correlation energy the starting point must be a wavefunction consisting of more than one Slater determinant. This implies that one can no longer picture the electrons as residing in orbitals, but has to consider the electron density instead.
3.5.1
Configuration interaction
One way of incorporating electron correlation is the method of configuration in-teraction (CI). The basic idea is to diagonalize the N-electron Hamiltonian in a basis of N-electron Slater determinants. The exact wave function of the N-electron problem can be written as a linear combination of all possible N-electron Slater
14 Quantum chemistry and methodology
determinants formed from a set of spin orbitals. These determinants can be con-structed with reference to the HF ground state
|Ψ0i = |χ1χ2· · · χaχb· · · χNi (3.38)
and can be viewed as approximate excited states of the system. As we have seen, a set of K basis functions will lead to a set of N occupied and 2K − N unoccupied MOs. By replacing MOs which are occupied in the HF determinant with unoccu-pied MOs, a set of 2KN determinants can be constructed. A determinant where one occupied MO χa has been replaced by an unoccupied MO χris called a singly
excited determinant and is denoted
|Ψrai = |χ1χ2· · · χrχb· · · χNi (3.39)
A doubly excited determinant is one where two occupied MOs have been replaced |Ψrsabi = |χ1χ2· · · χrχs· · · χNi (3.40)
and so on.
We may now use these N-electron Slater determinants as a basis in which to expand the exact many electron wave function |Φ0i
|Φ0i = c0|Ψ0i + X ar cra|Ψrai + X a<b r<s crsab|Ψrsabi + X a<b<c r<s<t crstabc|Ψrstabci + · · · (3.41)
where the expansion coefficients are what is left to be optimized. If all possible determinants are included in the given basis, all electron correlation is accounted for. With a complete basis set, and all determinants included, the solution would be exact within the Born-Oppenheimer approximation (Section 3.1.1).
The energy of the CI wavefunction can be found by using the variational prin-ciple and minimize the energy with respect to the expansion coefficients under the constraint that the CI wavefunction is normalized. Since the expansion is linear the problem reduces to diagonalizing and finding the eigenvalues of the matrix rep-resentation of the Hamiltonian in the basis of the N-electron determinants, with elements Hij = hΨi|H|Ψji, where Ψi,j are determinants in the expansion. The
resulting CI secular equations are
H00− E H01 · · · H0j · · · H10 H11− E · · · H1j · · · · · · · Hj0 · · · Hjj− E · · · · · · · c0 c1 .. . cj .. . = 0 0 .. . 0 .. . (3.42)
In order to reduce the computational effort, the CI expansion must be trun-cated. An expansion containing only singly excited determinants besides the ground state is called CI with singles (CIS). Including only doubly excited deter-minants we get CI with doubles (CID), including both singly and doubly excited
3.6 Semi-empirical methods 15
determinants is called CISD and further on. As matrix elements between the HF ground state determinant and a singly excited determinant is zero (see [14]), CIS does not give any direct improvement over the HF energy. However, singly excited determinants are important for calculating excited states, which can be approxi-mately computed as higher roots to the CI secular equation (Eq. (3.42)). Including doubly excited determinants a great deal (80 - 90 % [15]) of the correlation energy can be recovered.
For further reading about configuration interaction, see Ref. [14].
3.6
Semi-empirical methods
There are quite a few semi-empirical methods available for solving the Hartree-Fock equations today. The semi-empirical quantum chemical methods gain com-putational speed by neglecting many of the integrals which are tedious to evalu-ate. The remaining integrals are often taken to be parameters with values that are determined through comparison with experiment. The semi-empirical method Zerners Intermediate Neglect of Differential Overlap (ZINDO) has been used in this thesis and will be studied. First we will look at the expression for the Fock matrix elements, needed for proceeding with the semi-empirical approximations.
This section follow the theory put forward by Zerner in [16], which should be consulted for further reading about semi-empirical methods.
3.6.1
Expression for the Fock matrix
An element of the Fock-matrix from Eq. (3.35) with the closed shell Fock operator from Eq. (3.30) is
Fµν = hφµ(i)|f(i)|φν(i)i = hφµ(i)|h(i)|φν(i)i + N/2
X
a
hφµ(i)|2Ja(i) − Ka(i)|φν(i)i
= Hµν+ N/2
X
a
2hφµ(i)ψa(j)|gij|φν(i)ψa(j)i − hφµ(i)ψa(j)|gij|ψa(i)φν(j)i
= Hµν+ N/2 X a X λσ
CλaCσa∗ [2hφµ(i)φλ(j)|gij|φν(i)φσ(j)i − hφµ(i)φλ(j)|gij|φσ(i)φν(j)i]
= Hµν+
X
λσ
Pλσ[hµλ|νσi − 1
2hµλ|σνi] (3.43)
where we have defined the core-Hamiltonian matrix Hµν =
Z
16 Quantum chemistry and methodology
the density matrix
Pλσ= 2 N/2
X
a
CµaCνa∗ (3.45)
and two-electron integrals without the g operator present hµλ|νσi =
Z
d¯rid¯rjφ∗µ(i)φ∗λ(j)gijφν(i)φσ(j) (3.46)
3.6.2
Complete Neglect of Differential Overlap
Differential overlap is defined as the product of atomic orbitals depending on the same electron coordinates φA
µ(i)φBν(i)d¯riand is the integrand of an overlap integral.
The superscripts A and B refers to atomic centers, and the subscripts µ and ν to individual atomic orbitals. In the complete neglect of differential overlap (CNDO) model we make the replacement
φAµ(i)φBν(i)d¯ri→ δµνφAµ¯(i)φAν¯(i)d¯ri (3.47)
wherever it occurs in a two-electron integral. One of the requirements of the method is to maintain rotational invariance [17, 18]. This is indicated by the bar over the subscript µ which means that the actual orbital is replaced by an “s” symmetry orbital of the same spatial extent.
This approximation implies that all differential overlap between orbitals centred on different atoms is set to zero, as is the case for different orbitals centred on the same atom. Thus all three- and four-center two-electron integrals are neglected, and the overlap matrix S becomes the unit matrix. Also neglected are many of the one- and two-center two-electron integrals.
3.6.3
Intermediate Neglect of Differential Overlap
The intermediate neglect of differential overlap (INDO) model contains all integrals that CNDO does, plus all one-center two-electron integrals. These integrals have proven to be of great importance in the prediction of molecular spectral properties. With the approximations above the INDO Fock-matrix for the closed shell becomes
FAA µµ = HµµAA+ A X λσ Pλσ[hµλ|µσi − 1 2hµλ|σµi] + X B6=A B X λ Pλλhµλ|µλi (3.48) − X B6=A ZBh¯µ|1/RB|¯µi (3.49) FAA µν = HµνAA+ A X λσ Pλσ[hµλ|νσi − 1 2hµλ|σνi] (3.50) FAB µν = Hµν− 1 2Pµνhµν|µνi (3.51)
3.6 Semi-empirical methods 17 HµµAA= Uµµ− X B6=A ZBh¯µ|1/RB|¯µi (3.52) where Uµµ= hµ| −1 2∇ 2 − ZA/RA|µi (3.53)
is the one-center core integral.
3.6.4
ZINDO
There are several possible parameterizations available for INDO. INDO/S is an INDO model parameterized by Ridley and Zerner [19] and is modified to repro-duce electronic spectra. It is also known as ZINDO/S and is a part of the ZINDO program package developed by the group of Zerner. This model is parameterized at the CIS level of theory and calibrated directly on electronic spectroscopy. The parameters are optimized for giving accurate energy differences at fixed geome-tries, and not for calculating geometry or energy. The procedure of an INDO/S calculation of UV visible spectra is to first perform an SCF calculation and con-tinue with a CI calculation. Although parameterized at the CIS level this model is capable of calculating configuration interaction at other levels.
The scheme of parameterization is a generalization of the CNDO method intro-duced by Pople [17, 20]. To begin with, the one-electron matrix H is parameterized as HµµAA= UµµAA− X B6=A ZBγAB (3.54) HµνAA= 0 (3.55) HAB µν = (βA+ βB) ¯Sµν/2 (3.56)
Here βA and βB are atomic specific parameters. ¯Sµν is a proportionality constant
related to the overlap [16]. γAB is the two-center two-electron integral set to the
integral over s symmetry orbitals.
γAB= h¯µ¯ν|¯µ¯νi (3.57)
All two-center two-electron integrals in the INDO Fock-matrix are evaluated using the Mataga-Nishimoto formula [21]
γAB =
fy
2fy/(γAA+ γBB) + RAB
(3.58) Here γAAis a one-center two-electron integral evaluated using Pariser’s observation
[22]
18 Quantum chemistry and methodology
as the difference between the ionization potential, taken from atomic spectra, and the electron affinity of an s, p or d electron. Here F0(AA) is a Slater-Condon factor [23]. Other one-center two-electron integrals that involve Slater-Condon F and G factors are retained, for example
hss|ssi = F0(ss) hspx|spxi = 1/3G1(sp)
hpxpx|pxpxi = F0(pp) + 2F2(pp)/25
The one-center Coulomb integrals hµν|µνi = F0(µν) are calculated analytically
while G integrals are taken as parameters. Mixed one-center integrals of Slater-Condon R type are set to zero.
The one-center core integral UAA
µµ is estimated empirically from atomic
ion-ization energies Iµ. As an example, the ionization energy when removing an s
electron of an atom with a {slpmdn} configuration can be expressed in terms of Uss, F and G Is= E(sl−1pmdn) − E(slpmdn) = −Uss− (l − 1)F0(ss) − m F0(sp) −16G1(sp) − n F0(sd) − 101G2(sd) (3.60) where Uss is obtained after rearrangement.
The oscillator strength, which is a measure of the absorption intensity for a transition O → I, is calculated in the INDO/S model using the dipole length operator including all one-center terms
fO,I = 4.7092 ∗ 10−7µ2O,I(EI− EO) (3.61) where µO,I = hΨ(O)|µ|Ψ(I)i = X A QARO,I+ X µ,ν Gµνhµ|r|νi (3.62)
is the dipole length operator, and µ is the expectation value of the same. EI is
the energy of Ψ(I), Gµν is the transition density, and QAis the transition charge
between the two states.
Lately a few improvements have been introduced to the method, see for exam-ple Ref. [24].
3.7
Density functional theory
The theory presented in this section is obtained from Ref. [15, 25], and a more in depth study can be found in Ref. [25].
3.7 Density functional theory 19
In density functional theory (DFT) the focus is not on the complicated N-electron wavefunction Ψ(¯x1, ¯x2, . . . , ¯xN) and the associated Schr¨odinger equation,
but instead on the much simpler electron density ρ(¯r). The electron density is the number of electrons per unit volume for a given state. It is dependent only on 3 coordinates independently of the number of electrons of the system. Thus we have that
Z
ρ(¯r)d¯r = N (3.63)
The fundamental concept of DFT rely on the proof by Hohenberg and Kohn that the ground state energy and all other ground state electronic properties are uniquely determined by the electron density. Furthermore, the energy variational principle holds, i.e., the exact ground state of the system corresponds to the elec-tronic density for minimal total energy. The problem is then trying to determine the functional dependence E[ρ] which is not known, and using the variational principle in order to minimize the energy.
The total energy functional can be written
E[ρ] = T [ρ] + Vne[ρ] + Vee[ρ] = Z ρ(¯r)V (¯r)d¯r + F [ρ] (3.64) where F [ρ] = T [ρ] + Vee[ρ] (3.65) Vee[ρ] = J[ρ] + non-classical term (3.66) V (¯r) =X A ZA |RA− ¯r| (3.67) J[ρ] =1 2 Z Z ρ(¯r)ρ(¯r0) |¯r − ¯r0| d¯rd¯r 0 (3.68)
Here T [ρ] is the total kinetic energy, Vne[ρ] represents the electron - nuclei
attrac-tion, and Vee[ρ] represents the electron-electron interaction. V (¯r) is the external
potential felt by an electron due to the nuclei. The term J[ρ] is the classical coulombic electron repulsion term (analogue to the Coulomb term in HF theory), and the non-classical term includes the exchange-correlation energy. The prob-lem is the term F [ρ] which we don’t have an exact expression for, except for the Coulomb interaction.
3.7.1
The Kohn-Sham equations
The contribution by Kohn and Sham was to introduce orbitals in such a way that the kinetic energy can be simply computed exactly for non-interacting electrons,
20 Quantum chemistry and methodology
leaving only a small correction term handled separately. Introducing the Kohn-Sham orbitals φi we get the expression
Ts[ρ] = N X i hφi| − 1 2∇ 2 |φii (3.69)
for the kinetic energy and
ρ(¯r) =
N
X
i
|φi(¯r)|2 (3.70)
for the electron density. The remaining kinetic energy is included in the exchange-correlation term, by rewriting Eq. (3.65)
F [ρ] = Ts[ρ] + J[ρ] + Exc[ρ] (3.71)
where
Exc[ρ] = T [ρ] − Ts[ρ] + Vee[ρ] − J[ρ] (3.72)
is defined as the exchange correlation energy and contains the correction to the kinetic energy and the non-classical part of Vee[ρ]. The main problem now lies
in finding a functional which describes the exchange-correlation. Although many attempts have been made, no exact functional has been found, and we have to rely on the numerous approximate functionals available. If the exact Exc[ρ] would be
used, DFT provides the exact total energy.
By a procedure much like the derivation of the Hartree-Fock equations, the variational principle may be used to derive the Kohn-Sham equations
hKSφi= iφi (3.73) hKS = − 1 2∇ 2+ V ef f(¯r) (3.74) Vef f(¯r) = V (¯r) + Z ρ(¯r0) |¯r − ¯r0|d¯r 0+ V xc(¯r) (3.75)
with the exchange-correlation potential Vxc(¯r) =
δExc[ρ]
δρ(¯r) (3.76)
Analogously to the HF method, the unknown Kohn-Sham orbitals may be ex-panded in a set of basis functions, and an iterative procedure for solving the Kohn-Sham equations is employed. Starting with a guess of ρ(¯r), one is then able to construct Vef f(¯r), compute the Kohn-Sham orbitals from Eq. (3.73), and then
find a new ρ(¯r) from Eq. (3.70).
Like in the Hartree-Fock case, we can interpret the eigenvalues i as
molec-ular orbital energies, and the highest occupied orbital energy is the ionization potential (Koopmans’ theorem [15]), under the constraint that the exact exchange-correlation functional is used. Since this is not possible, the orbital energies will be somewhat different from the ionization potential.
3.7 Density functional theory 21
3.7.2
Time-dependent DFT
The density functional theory, which is a theory for the ground state, has been extended to the treatment of excitations. Through a time-dependent formalism the excitation energies are obtained from frequency-dependent response functions. Here follow a very brief description of the most important concepts in time-dependent DFT (TDDFT). The theory is described in detail in Ref. [26, 27, 28].
A system, initially in its ground state, subject to a time-dependent perturbation will lead to a modified effective potential
Vef f(¯r, t) = V (¯r, t) +
Z ρ(¯r0, t)
|¯r − ¯r0|d¯r 0+ V
xc(¯r, t) (3.77)
which can be used to derive the time-dependent Kohn-Sham equation −12∇2+ Vef f(¯r, t) ψi(¯r, t) = i ∂ ∂tψi(¯r, t) (3.78) The exchange-correlation potential here is the functional derivative of the time-dependent exchange-correlation functional Axc[ρ], which is approximated by Exc[ρt]
(the exchange-correlation functional of time-independent Kohn-Sham theory) for the density at fixed t
Vxc(¯r, t) =
δAxc[ρ]
δρ(¯r, t) ≈
δExc[ρt]
δρt(¯r) (3.79)
This is known as the adiabatic (low frequency) approximation. The time-dependent Kohn-Sham orbitals give the true time-dependent charge density
ρ(¯r, t) =X
iσ
fiσ|ψiσ(¯r, t)|2 (3.80)
The linear response to a perturbation w(t) is given by δρσ(¯r, ω) = X ij ψiσ(¯r)δPijσ(ω)ψ∗jσ(¯r) (3.81) where δPijσ(ω) = fjσ− fiσ ω − (iω− jω)× wijσ(ω) + X klτ Kijσ,klτδPklτ(ω) (3.82) is the response of the Kohn-Sham density matrix in the basis of the unperturbed molecular orbitals. The coupling matrix K describes the linear response of the self-consistent field vSCF (the last two terms in Eq. (3.77)) to changes in the charge density.
Since the dynamic polarizability, ¯α(ω), describes the response of the dipole moment to a time-dependent electric field, it may be calculated using the response of the charge density. For the time-dependent, electric perturbation
22 Quantum chemistry and methodology
where Ez(t) is the applied field strength, the xz-component of the dynamic
polar-izability can be expressed as
αxz(ω) = 2¯x†P−1/2[Ω − ω21]−1P−1/2z¯ (3.84)
by using Eq. (3.82). Here we have defined Pijσ,klτ = δσ,τδi,kδj,l (fkτ − flτ)(lτ− kτ) > 0 (3.85) and Ωijσ,klτ = δσ,τδi,kδj,l(lτ − kτ)2 (3.86) + 2q(fiσ− fjσ)(jσ− iσ) × Kijσ,klτp(fkτ− flτ)(lτ − kτ)
According to the sum-over-states relation, ¯ α(ω) =X I fI ω2 I − ω2 (3.87) the poles of the dynamic polarizability determine the excitation energies, ωI, and
the residues, fI, determine the corresponding oscillator strengths. A careful
com-parison of Eqs. (3.84) and (3.87) shows that excitation energies and oscillator strengths can be obtained by solving the matrix eigenvalue problem
Ω ¯FI = ω2IF¯I (3.88)
Chapter 4
Results and discussion
4.1
Computational details
All calculations were performed with the Gaussian 03 [29] software , running on the computer cluster Green at NSC, Link¨oping. Geometry optimizations have been carried out at the DFT (see Section 3.7) level of theory, using the Becke + Lee-Yang-Parr (B3LYP) functional [15] and the basis set 6-31G* (Section 3.4). The semi-empirical ZINDO model (described in Section 3.6.4) and time-dependent DFT (TDDFT) (Section 3.7.2) have been used in calculations of electronic struc-ture and spectra. The TDDFT calculations also made use of the B3LYP functional but with the 6-31G basis set without polarization functions in order to reduce the computational effort. The results have been interpreted and visualized with the Molden, Molekel, Matlab and Povray softwares.
4.2
The polymer wm356
The polymer called wm356 is a recently designed polymer that shows very good absorbance in the low frequency (IR) region. The bandgap is experimentally esti-mated to ∼1 eV. [30] A schematic picture of one unit of the polymer, a monomer, can be seen in Fig. 4.1. It is built up by a kind of acceptor unit in the middle (two hexagons with one pentagon at the bottom), and a donor unit consisting of a thiophene ring on each side. As discussed in Section 2.1 this mixing of segments with high HOMO (Highest Occupied Molecular Orbital) and low LUMO (Lowest Unoccupied Molecular Orbital) gives rise to intra-chain charge transfer and favours a low bandgap. For information about the synthesis of this kind of polymer the reader is referenced to Ref. [2].
4.2.1
Experimental results
Fig. 4.2 shows the absorption spectra of the polymer wm356 as obtained from ex-perimental measurements. [30] The longest wavelength absorption peak is located
24 Results and discussion
Figure 4.1. Schematic picture of one unit of wm356 with bonds numbered.
400 600 800 1000 1200 1400 1600 1800 2000 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 A b s o rb a n c e Wavelength/nm WM356n
4.3 Geometry 25
at 940 nm and corresponds to ∼1.3 eV in energy. The absorption is estimated to be zero after 1550 nm, ∼0.8 eV. It should be noted that there are additional effects that affect these measurements, compared to what is obtained from the ide-alized picture in the calculations. For example steric interactions between polymer chains will make the chains twist and turn. Since this polymer has quite long side chains, this will most certainly be of importance. The conjugation length of the polymers in the sample is not known, but it is likely to consist of several different conjugation lengths.
4.3
Geometry
Geometries for the monomer, dimer and trimer have been optimized using B3LYP/6-31G*. As termination of the sequence, phenyl rings were attached to the ends of the sequence. The carbon chains attached to the oxygen atoms were replaced by only hydrogens, as these chains are expected to be of minimal importance for the bandgap and excitation energies.
Geometries were optimized for the so called trans-state, were the S atoms of the thiophene rings are faced downwards in the same direction as the S atom in the middle. This state is slightly larger in energy than the cis-state, where thiophene rings are flipped. It is difficult to estimate the amount of cis- and trans-states in the sample used for experiment, but both states probably exist since they are close in energy. The optimized geometries for the dimer and trimer can be seen in
(a) Dimer (b) Trimer from side
Figure 4.3. Optimized structures.
Fig. 4.3. Notable is that the two phenyl rings with oxygen on top of the monomer are in different planes than the backbone of the polymer. Also worth mentioning is that the chain is curling along the backbone, most obvious for the trimer.
26 Results and discussion The geometry of the tetramer was created “by hand” from the optimized ge-ometry of the trimer, and then turning it into C2 rotational symmetry using Gaussview (for reducing computational effort). The pentamer and hexamer were created in the same manner, but without forcing the geometry into a certain sym-metry.
Bond lengths for the monomer and the central unit of the trimer are presented in Table 4.1, where bonds are numbered according to Fig. 4.1. The bond lengths are for the monomer without phenyl rings on the sides. There is an evident
Bond number Monomer Trimer
1 1.4576 1.4505 2 1.3118 1.3150 3 1.3654 1.3613 4 1.4563 1.4546 5 1.4198 1.4291 6 1.4279 1.4313 7 1.4599 1.4593 8 1.3428 1.3398 9 1.6324 1.6355 10 1.4596 1.4484 11 1.7710 1.7736 12 1.3885 1.3983 13 1.4168 1.4008 14 1.3704 1.3891 15 1.7255 1.7518 16 - 1.4348
Table 4.1. Bond lengths in ˚Angstr¨om for the dimer and trimer with bond numbers according to Fig. 4.1
relaxation of the geometry between the two conjugation lengths. In particular bond numbers 10, 12, 13 and 14 on the thiophene rings are changing, and the maximum difference is ∼ 0.015 ˚A. Other changes like angular and dihedral are not included here. However, it can be seen by just visualizing the geometries, that the plane of the central acceptor unit and the plane of the thiophene ring are more aligned when connecting to other units, as for the middle unit of the trimer.
4.4
Excitation energy calculations
4.4.1
ZINDO
Excitation energies have been calculated for monomer up to hexamer. The results presented here are with phenyl rings terminating the sequence, if nothing else is mentioned. The results of the ZINDO calculations can be seen in Fig. 4.4 and 4.5. Oscillator strengths (see Eq. 3.61) are a relative measure of the strength of
4.4 Excitation energy calculations 27
the absorption and increase with the conjugation length. Therefore, in Fig. 4.5, showing the absorption for all conjugation lengths together, the oscillator strength has been weighted with the inverse of the conjugation length. Gaussian curves have been added to get a more realistic picture of the absorption; for example twists and turns of the polymer chain might change the absorption, see Section 4.5.1. It should be noted that Fig. 4.5 is not directly comparable to experiment since we here assume an equal amount of different conjugation lengths, which is of course not true. The accuracy of both the calculated and experimental spectras can be discussed. The experimental one because of external effects and inaccuracies of the measurement, and the calculated one because of model defects (see further discussion about the accuracy of the ZINDO model in Section 4.4.2). However, keeping this in mind, we can at least employ a qualitative reasoning and roughly compare the experimental spectra with the calculations. Within the accuracy of the model, we get a lower bound of the absorption energies from the calculations, where the conditions are ideal with the polymers in vacuum and no interactions with other chains or solvents.
500 1000 1500 0 0.5 1 1.5 Wavelength (nm) Oscillator strength Monomer 500 1000 1500 0 0.5 1 1.5 2 Dimer 500 1000 1500 0 0.5 1 1.5 2 2.5 Trimer 500 1000 1500 0 0.5 1 1.5 2 2.5 3 Tetramer 500 1000 1500 0 1 2 3 4 Pentamer 500 1000 1500 0 1 2 3 4 Hexamer
Figure 4.4. Absorption for different conjugation lengths, as calculated by ZINDO.
The peaks largest in wavelength mainly corresponds to HOMO - LUMO tran-sitions, and in the peaks further down there are also other orbitals involved in the transitions. We typically see a red-shift (peaks move towards larger wavelength) as the conjugation length increases, Fig. 4.4. This is an effect of atomic orbitals interacting and creating molecular orbitals that are split in energy. Connecting two units of the polymer lead to a larger set of molecular orbitals that get split in
28 Results and discussion energy when interacting, resulting in orbitals closer in energy and thus a lowering of the LUMO and an increase of the HOMO energy. The result is a lowering of the bandgap and lower excitation energies. We observe that the peak largest in wavelength, corresponding to the lowest excitation, is more red-shifted than the peak for the higher excitations, as the conjugation length increases. The red-shift, however, is smaller and smaller going to longer conjugation lengths. Also, looking at the oscillator strengths, we see that the relative strength of the peak largest in wavelength is increased for longer conjugation lengths. It should be noted that the higher excitations of the hexamer were out of range of the calculation, so the corresponding peak is not included here. For the pentamer and hexamer we can see a third peak arising between the two main peaks. This is a result of the wider range of molecular orbitals for these long conjugation lengths.
400 600 800 1000 1200 1400 1600 0 0.5 1 1.5 2 2.5 3 Wavelength (nm) Oscillator strength ZINDO Experiment
Figure 4.5. Total absorption for monomer to hexamer, as calculated by ZINDO.
When comparing the calculated spectra in Fig. 4.5 to the experimental one in Fig. 4.2, we see that they match each other well, though not perfectly. In Fig. 4.2 we see a peak at about 940 nm and then a slope down to about 1550 nm. Com-paring this to the calculated spectra in Fig. 4.5 shows that this peak corresponds to excitation to the first excited state, and indicates that the short conjugation lengths are dominating in the sample. The monomer peak in the calculated spec-tra at 844 nm match the rise in absorbance above 800 nm for the experimental data, and the dimer peak at 1057 nm is slightly above the experimental peak at 940 nm. This suggests that the dominating conjugation length in the sample is somewhere between the monomer and the dimer, as discussed in Section 4.5.1 and
4.4 Excitation energy calculations 29
in connection with Fig.4.6 below.
There can be seen a slight red-shift of the peak for the higher excitations when comparing the calculated spectra to the experimental data. In this wide peak for the higher excitations, the excitations largest in wavelength originates from the longer conjugation lengths. Considering that short conjugation lengths are found to be dominating, the overestimated absorption at 600 nm in the calculations is not strange. However, the red-shift is not large, and taking into account that short conjugation lengths are dominating we get a very good agreement with the experimental data.
Fig. 4.6 shows the excitation energy to the first excited state as a function of the conjugation length in units of [1/n], where n is the conjugation length. There are two sets of values, one with and one without phenyl rings terminating the sequence. The geometries of the polymers without phenyl rings were obtained from the optimized structures by just cutting off and replacing the phenyl rings by hydrogens, and thus their geometries are unrelaxed. The exception is the structure of the monomer without the phenyl rings which is optimized. The reason for this is that the monomer is the one most sensitive to a relaxation of the geometry due to its short conjugation length. As a check, the geometry of the trimer without phenyl rings was relaxed, and the difference in excitation energy to the first excited state between the relaxed and unrelaxed structures was calculated to ∼0.01 eV. Excitation energies for the polymers with phenyl rings are generally lower than
0 0.2 0.4 0.6 0.8 1 0.6 0.8 1 1.2 1.4 1.6 1.8 Conjugation length, [1/n]
Exc. energy [eV]
← E=0.73991+0.86977(1/n)
with phenyl without phenyl
Figure 4.6. Excitation energies to first excited state with and without phenyl rings terminating the sequence, as calculated by ZINDO.
30 Results and discussion for the polymer without them. This is not surprising since the phenyl group involves conjugated π-orbitals, which interact with the conjugated π-orbitals of the polymer responsible for the low bandgap, and extends the conjugation length of the polymer as a whole. For example the HOMO for the monomer is a delocalized π-orbital, see Fig. 4.7(a), which stretches out over the phenyl rings on the sides. Since procentual increase of the conjugation length by adding the phenyl rings is decreasing with larger conjugation lengths, the two sets of excitation energies are converging.
The excitation energy to the first excited state is expected to be proportional to [1/n] [31], which we can see agrees with the linear interpolation of the values obtained in Fig. 4.6. The values for the monomer seems to differ from the other ones and are not included in the linear interpolation. This is probably because of a relaxation of the geometry going from the monomer to the dimer, which affects the bandgap, leading to higher excitation energies. The excitation energy for the dimer is, however, still lower due to the longer conjugation length. The change in geometry between the monomer and the trimer can be seen in Table 4.1 above. It is obvious that the monomer is the most extreme for bond lengths between carbon atoms. The long carbon - carbon single bonds (bond numbers 1, 4, 7 and 10) are longer than for the trimer and the short carbon - carbon double bonds (bond numbers 5, 6, 12 and 14) are shorter than for the trimer.
As the experimental peak at 940 nm, ∼1.3 eV corresponds to excitations to the first excited state (and the peaks for higher energies are corresponding to higher order excitations), we can see in Fig. 4.6, that this energy corresponds to a conjugation length slightly shorter than the dimer, as we have already noted. We will therefore pay more attention to the dimer in the following. The lowest energy for which absorption is recognized in the experimental spectra is ∼0.8 eV at 1550 nm. In Fig. 4.6 this energy corresponds to a conjugation length of about 14 to 15 units, so the longest conjugation length present in the sample used for measurement is 14 to 15 units according to the calculations.
In Table 4.2 are shown the excitation energies to the first excited state and the corresponding orbital transitions. The orbitals are taken from the ZINDO calcula-tions, utilizing CI with singles (CIS), and are the result of a restricted Hartree-Fock SCF calculation. Each excited state corresponds to a CI wavefunction with expan-sion coefficients determining which singly excited determinants that are included and by how much (see Section 3.5.1 for a brief description of configuration inter-action). Due to the normalization condition the square of the coefficients sum up to one. The largest expansion coefficients (we place the cut-off at 0.2 and take the absolute value), and the orbitals added and replaced in that determinant, are pre-sented for the strongly allowed transitions in Table 4.2. By “strongly allowed” we mean transitions with a relatively high oscillator strength. Here we include tran-sitions with oscillator strengths higher than 0.1 for the monomer and higher than 0.2 for the dimer (since the oscillator strength is a relative measure, dependent on the size of the molecule), for excitation energies up to 3.5 eV.
Excitations to the first excited state are dominated by HOMO to LUMO tran-sitions, and the higher excitations involve also other orbitals close to HOMO and LUMO. For the higher excitations there are many orbitals involved, and we would
4.4 Excitation energy calculations 31
Exc. energy [eV] Osc. strength Orbital transitions
1.47 0.65 0.69(homo → lumo) + . . . 2.52 0.36 0.66(homo-1 → lumo) + . . . 2.84 0.16 0.36(homo-8 → lumo) +0.49(homo → lumo+1) + . . . 3.01 0.71 0.60(homo → lumo+3) + . . . 3.24 0.65 0.38(homo-2 → lumo) +0.40(homo → lumo+2) +0.26(homo → lumo+4) + . . . (a) Monomer
Exc. energy [eV] Osc. strength Orbital transitions
1.17 1.48 0.62(homo → lumo) +0.30(homo-1 → lumo+1) + . . . 2.28 0.53 0.46(homo-2 → lumo) +0.40(homo-1 → lumo+1) + . . . 2.63 0.51 0.49(homo → lumo+2) +0.24(homo → lumo+4) +0.21(homo-1 → lumo+3) + . . . 2.95 0.37 0.33(homo-1 → lumo+1) +0.31(homo-3 → lumo+1) +0.23(homo-4 → lumo) +0.23(homo → lumo) + . . . 3.03 0.46 0.36(homo → lumo+6) +0.32(homo-1 → lumo+5) 3.19 0.83 0.33(homo-4 → lumo) +0.29(homo-5 → lumo+1) +0.21(homo → lumo+6) + . . . (b) Dimer
Table 4.2. The orbital changes in strongly allowed transitions for the monomer and dimer weighted by the corresponding coefficients in the CI expansion.
32 Results and discussion
(a) HOMO (b) LUMO
(c) HOMO-1 (d) LUMO+2
(e) HOMO-2 (f) LUMO+3
4.4 Excitation energy calculations 33
(a) HOMO (b) LUMO
(c) HOMO-1 (d) LUMO+1
(e) HOMO-2 (f) LUMO+2
34 Results and discussion need to look at the electron density to get a more clear picture of what is happening to the electron structure in the excitation. Anyway, we get an idea of the nature of the transition by looking at what orbitals that are replaced in the determinant. In Fig. 4.7 and 4.8 some of the orbitals that are involved in the lowest strong transitions for the monomer and the dimer are visualized. The orbitals here are delocalised π-orbitals, although there is a tendency of localisation to the acceptor unit for the LUMO and to the thiophene donor units for the HOMO as expected. The HOMO of the monomer is splitting up in two similar orbitals for the dimer, HOMO and HOMO-1. The same can be seen for the LUMO of the monomer, where LUMO and LUMO+1 for the dimer look very much the same. This is not surprising since the lowest excitation for the dimer involves also a HOMO-1 to LUMO+1 transition. We notice that the long - short bond length alternation seems to be important for the structure of the delocalized π-orbitals. The light and dark areas of the orbitals shown in Fig. 4.7 and 4.8 represents a shift of sign in the wavefunction. For example, in Fig. 4.7 we can see that bond numbers 12 and 14 for the monomer are bonding in the HOMO while bond numbers 10 and 13 are antibonding.
It can also be seen from Fig. 4.7 and 4.8 that the phenyl rings, with oxygen on top, are only involved in higher order excitations. A reason for this is that they are twisted compared to the plane of the acceptor - donor structure, and thus there is an angle between pz-orbitals on the phenyl ring and the rest of the structure. This
angle raises the energy and does not favour the occurrence of extended π-orbitals. Excitation energy calculations have also been performed for the cis-state of the monomer (with thiophene rings flipped upside-down). The results are similar to the trans-state used here. The excitation energy to the first excited state is 1.44 eV for the cis-state compared to 1.47 eV for the trans-state.
4.4.2
TDDFT
Conj. length ZINDO TDDFT δ
monomer 1.4682 1.4907 -0.0225
dimer 1.1732 1.1364 0.0368
trimer 1.0287 0.9508 0.0779 tetramer 0.9658 0.8466 0.1192
Table 4.3. Excitation energies [eV] to the first excited state comparing values from ZINDO and TDDFT (B3LYP/6-31G). The right column shows the difference between the ZINDO and TDDFT energies.
Excitation energies have also been calculated using TDDFT (B3LYP/6-31G), and the energies to the first excited state are presented in Table 4.3. The values for TDDFT tend to diverge from the ZINDO values as the conjugation length in-creases. The reason for this is not exactly known. However, TDDFT is reported to underestimate the excitation energies for excitations involving long-range charge transfer [32]. Since excitations in this polymer involves charge transfer between
4.4 Excitation energy calculations 35
donor and acceptor units (see Section 4.4.3), this suggests that the TDDFT calcu-lations here give excitation energies that are too low. The basis set might also play some role here, as the ZINDO model is parameterized to give correct energy differ-ences for a specific basis set, but TDDFT is not. It is not sure that a larger basis set immediately improves the values obtained with TDDFT, but using larger and larger basis sets would definitely improve the results for TDDFT. With a complete basis set and a true exchange-correlation functional the TDDFT results would be exact. ZINDO is reported to reproduce excitation energies within 1000 - 2000 cm−1 (∼ 0.12-0.25 eV) for transitions below 40000 cm−1(∼ 5 eV) [16]. It is also
reported that results tend to improve as the size of the molecule increases. The absorption spectra for the dimer for both of ZINDO and TDDFT can be seen in Fig. 4.9. We see that also excitations to higher excited states are red-shifted for TDDFT compared to ZINDO.
300 400 500 600 700 800 900 1000 1100 1200 1300 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Wavelength (nm) Oscillator strength ZINDO TDDFT
Figure 4.9. Absorption spectra for the dimer.
4.4.3
Difference densities
Because the electronic structure obtained from configuration interaction calcula-tions is described by a linear combination of determinants built up by molecular orbitals, it can be quite hard to describe a transition in terms of molecular or-bitals, when many of them are involved. Instead, we can look at the difference in electron density between an excited state and the ground state. Here we have visualized the difference in electron density between the first excited state and the ground state for the dimer, Fig. 4.10, and trimer, Fig. 4.11, using the electron density from TDDFT/B3LYP/6-31G calculations. The bright surface is where the electrons are going and the dark where they are coming from. It is thus evident that the excitation to the first excited state involves a charge transfer from the donor to acceptor unit. By comparing Fig. 4.10 to the HOMO and LUMO in Fig. 4.8 we can see that the difference density match the molecular orbitals well. This agrees with the fact that the first excited state is mainly a HOMO to LUMO
36 Results and discussion
Figure 4.10. The difference between the electron density of the first excited state and the ground state for the dimer.
Figure 4.11. The difference between the electron density of the first excited state and the ground state for the trimer.
4.5 Geometry deformations 37
transition.
For the trimer in Fig. 4.11 a small tendency of a localized excitation to the central unit can be seen. This tendency is expected to be even stronger for longer conjugation lengths and indicates that a localized exciton is formed in these sys-tems.
4.5
Geometry deformations
From the optimized structures, different kinds of geometry deformations have been introduced and their effect on the absorption has been studied.
4.5.1
Torsion
The optimized structure of the dimer has been manually deformed by introducing an angle between the two unit cells, that is between the two thiophene rings in the middle. This has been done in steps of 10◦ up to 90◦. The result can be seen in
Fig. 4.12, where the excitation energy to the first excited state, as calculated by ZINDO, is plotted versus the angle of torsion θ. A curve of the type a ∗ sin2(θ) + b
has been adjusted to the values to imply such a dependence. The curve seems to fit the values very well.
0 10 20 30 40 50 60 70 80 90 100 1.15 1.2 1.25 1.3 1.35 1.4 Torsion (degrees)
Exc. energy (eV)
← E=0.19*sin2(θ)+1.173
Figure 4.12. Excitation energy for torsion between the units in the dimer.
As the angle between the units approach 90◦, the excitation energy to the
first excited state approaches ∼ 1.36 eV. By this angle the formation of extended π-orbitals is interrupted between the units, so it might be expected that the cor-responding excitation energy would match the values for the monomer ∼ 1.47 eV. Evidently this is not the case, either because there is still some interaction between the orbitals of the units, or because of a relaxation of the geometry, or both.
Anyway, it is interesting to see that the excitation energy for 90◦is about 1.36
