Electronic structure and Hartree

2.2 Density functional theory calculations

2.2.1 Electronic structure and Hartree–Fock theories

In quantum mechanics, one of the fundamental postulate is that the state and conse- quently all physical properties of a system can be fully described by a wavefunction Ψ. If a static system is considered,Ψcan by solved with the time-independent, non- relativistic form of the Schrödinger equation:

HÒΨ=EΨ (2.3)

where HÒandE are the Hamiltonian operator and its corresponding observable, the total energy of the system, respectively. The Hamiltonian operator for theN_e- electron andM-nuclei system have the following form in atomic units:

HÒ=TÒ_e+TÒ_n+VÒ_en+VÒ_ee+VÒ_nn

=−

N_e

∑︂

i=1

∇²_i 2 −

∑︂

A=1

∇²_A 2m_A−

N_e

∑︂

i=1 M

∑︂

A=1

Z_A r_iA+

N_e

∑︂

i=1 N_e

∑︂

j>i

1 r_{i j} +

∑︂

A=1 M

∑︂

B>A

Z_AZ_B

r_AB (2.4)

whereiandjrun over the electrons,AandBover the nuclei,m_Ais the mass of the nucleus,∇²is the Laplacian operator,Z_Ais the atomic number of nucleusA, andr_ab is the distance between particles theaandb. In Equation 2.4, the first two terms,TÒ_e andTÒ_n, are the kinetic energy of the electrons and nuclei, respectively, and the three last terms,VÒ_en,VÒ_ee, andVÒ_nn, represent the potential energy due to the attraction between the electrons and nuclei, the electron-electron repulsions, and the nucleus- nucleus repulsions, respectively.

Analytical solving of Equation 2.4 is impossible for systems larger than two particles (i.e. the hydrogen atom) due to the complexity caused by pairwise attraction and repulsion terms. This problem can be addressed by adopting the Born–Oppenheimer approximation[106], which takes advantage of the fact that the nuclei are much heavier and move much slower than the electrons. Thus, the electrons can be assumed to respond instantaneously to the movement of the nuclei. Due to the different timescales of nuclear and electron motion, they can be decoupled and solve the Schrödinger equation for the molecular system at the fixed nuclear positions. Con- sequently, in Equation 2.4, the kinetic energy of the nuclei (TÒ_n) is zero, correlation inVÒ_enis eliminated, andVÒ_nn becomes a constant that is easy to evaluate. The remaining terms account for the electronic Hamiltonian,HÒ_el, which has the following form in atomic units:

HÒ_el=−

N_e

∑︂

i=1

∇²_i 2 −

N_e

∑︂

i=1 M

∑︂

A=1

Z_A r_iA+

N_e

∑︂

i=1 N_e

∑︂

j>i

r_{i j} (2.5)

WhenHÒ_el is employed in Equation 2.3 with an electronic wavefunction,Ψ_el, the electronic energy, E_el, is obtained. The total energy of the system (with the fixed nuclei) is obtained by addingE_el to a (classical) nucleus-nucleus repulsion energy:

E_tot=E_el+E_nn.

Although Equation 2.5 has a simpler form than Equation 2.4, it is still insolvable for a many-electron system due to the remaining correlation between the individual electrons, and thus additional approximations are required. In independent-particle models, the electrons are assumed to move independently of each other. In the HF theory, this is achieved by treating the electron–electron repulsion in an averaged way, i.e. each electron moves in the electrostatic field of the nuclei and the average field generated by the other electrons. The many-electron problem reduces to the set of one-electron problems, where each electron in a molecule is described by a spin orbital,χ_i(x_j), which is a product of a spatial orbital (also referred to as a MO),ψ_i, and an electron spin eigenfunction (αorβ). The total, many-electron wavefunction is then expressed by a product of one-electron spin orbitals.

To obey the Pauli exclusion principle, the total wavefunction should be antisymmet- ric, which is ensured by writing it in the form of a single Slater determinant of spin orbitals. For a molecular system withN_eelectrons, the wavefunction can be thus written in the form of

Ψ(x₁,x₂, ...,x_N_e) = 1 pN_e!

|︁

χ₁(x₁) χ₂(x₁) . . . χ_N_e(x₁) χ₁(x₂) χ₂(x₂) . . . χ_N_e(x₂)

... ... ... ... χ₁(x_N_e) χ₂(x_N_e) . . . χ_N_e(x_N_e)

|︁

(2.6)

where each row corresponds to an individual electron and each column to a spin orbital.

The MOs can be obtained as linear combinations of nuclei-centered basis functions, i.e. atomic orbitals,ϕ:

ψ_i=

basis functions

∑︂

a_iµϕ_µ (2.7)

where the set of basis functionsϕ_µis also referred to as the basis set associated with a set of MO coefficientsa_i_µ. In a basis set, a finite set of mathematical functions are used to represent the MOs and build the wavefunction. The quality of the quantum mechanical calculations depend on the size of the basis set, i.e. number of the basis functions it includes. Using an infinite, complete set of basis functions would be required to represent MOs exactly. However, this is impossible in practice and finite basis sets are used instead. In larger basis sets, fewer constraints have been set on electrons leading to more accurate description of the wavefunction, but also computationally more demanding calculations. Thus, suitable balance between the size of the basis set and accuracy should be found.

Current basis functions typically make use of the Gaussian-type orbitals (GTOs) as basis functions. Valence orbitals take part in the chemical bonding, while core orbitals are usually only weakly affected by it. In so-called split-valence basis sets,

this is utilized by representing core orbitals with a single (contracted) basis function, whereas valence orbitals are represented by two or more basis functions of different size. The Pople’s split-valence basis sets have been generally popular and are also a common choice in the quantum chemical studies of PSC systems. While other type of basis sets, such as Dunning’s, have also been employed for their modeling, only those of Pople’s have been considered here, as the main focus of this work has been on comparison of different functionals.

In this thesis, the Pople’s 6-31G** split-valence basis set, which consists of one core orbital basis function comprised from six primitive GTOs and two valence orbitals comprised from three and one primitive GTOs, has been mostly used. The 6-31G**

basis set includes polarization functions for each atom (denoted by the asterisks), which allow more mathematical flexibility for description of MOs. The first "star"

denotes to addition of sixd-type polarization functions for each atom but hydrogen, whereas the second star refers to the addition of a set of threep-type polarization functions for each hydrogen atom. Furthermore, some calculations in this work have been carried out with the 6-31+G* basis set, which includes diffuse functions, i.e. large-size versions of thes- andp-type functions, which improve the description of molecules of lone pairs, e.g. anions.

As the MO coefficients, i.e. a_i_µ in Equation 2.7 are not known, a trial wavefunction is used to solve the energy. The best set of MOs, which yield the lowest energy, will be obtained with a iterative self-consistent-field (SCF) approach. The minimized energy is defined by using the variational principle, according to which any approx- imate wavefunction will have an energy above or equal to the exact energy.

The HF equations that are solved self-consistently have the form

Fbχ_i= (−1 2∇²_i−

∑︂

A=1

Z_A

r_iA+V_i^HF)χ_i=ε_iχ_i (2.8) whereFbis a one-electron Fock operator andV_i^HFis the HF potential,bJ−K, withb the components of the Coulomb (Jb) and exchange (Kb) operators, which represent

The hindrance of the HF method is that it neglects the correlation between electrons, which leads to overestimation of electron localization[26]. Post-HF methods have been developed to account for the missing electron correlation. For example, in Configuration Interaction[107, 108]and Coupled Cluster methods[109], electron correlation is taken into account by using a multi-determinant wave function. In Møller–Plesset (MP) perturbation theory[110], the perturbation theory is applied to the Hamiltonian. These methods are computationally more demanding than HF, but have a high accuracy and offer results that systematically approach the exact solution of the Schrödinger equation. There are some approximations of these methods, e.g. the resolution of the identity MP2 (RI-MP2), which have lower computational cost, but produce the results nearly identical to those of full MP2 (i.e. the second or- der MP). In the theoretical studies of the PSC compounds, the RI-MP2 method has been used for example in calculating reference torsional potentials for monomers of D–A copolymers[111].

2.2.2 Density functional theory

Density functional theory provides an alternative method to account for electron correlation with significantly less computational cost compared to post-HF methods. The foundations of DFT were laid by Hohenberg and Kohn[112]. According to their first (existence) theorem, the GS electronic energy of a system is a unique functional of the electron densityρ(r). Thus, the energy and associated GS properties of the system can be derived fromρ(r). The second (variational) theorem proves that the GS energy can be obtained variationally,ρ(r)that minimized the total energy is the exact GS density,ρ₀.

The modern DFT methods are based on the work of Kohn and Sham[113], who in- troduced a fictitious system of non-interacting electrons, which have the sameρ₀as the real system of interest, to avoid the difficulties in deriving the electron–electron interactions. Using a non-interacting system makes the solving of the Schrödinger equation a trivial task, as the Hamiltonian is a sum of one-electron operators with eigenfunctions being the Slater determinants of the individual one-electron eigen-

functions and eigenvalues being the sum of the one-electron eigenvalues. In the Kohn–Sham formalism, the form of the general DFT energy functional is

E_DFT[ρ(r)] =T_s[ρ(r)] +V_ne[ρ(r)] +J[ρ(r)] +E_XC[ρ(r)] (2.9) whereT_s[ρ(r)]is the kinetic energy,V_ne[ρ(r)]is the nuclei-electron potential energy, andJ[ρ(r)]is the Coulomb integral, i.e. Hartree electron-electron repulsion.

The last term,E_XC[ρ(r)], contains the difference between the classical and quantum mechanical electron–electron repulsion and the difference in kinetic energy between the fictitious non-interacting and real systems. The term missing from Equation 2.9 is the HF exchange (K). The pure DFT functionals omit it completely, but in the global hybrid functionals, a small part of exact HF exchange is included in their energy expression. The exact form ofE_XCis not known, and thus several approximations have been developed for it that will be reviewed below.

In the Kohn–Sham formalism[113], the exact GS electron density can be expressed in terms of a set of one-electron orbitals

ρ(r) =

N_e

∑︂

i=1

|ψ_i(r)|² (2.10)

where the sum runs over all the occupied Kohn–Sham orbitals,ψ_i(r). The Kohn–

Sham equations have the form

−1 2∇²+

∑︂

A=1

Z_A r_1A+

∫︂ ρ(r^′)

|r−r^′|dr^′+ν_XC(r)

ψi=εiψi (2.11)

whereε_iare the Kohn–Sham orbital energies andν_XCis the XC potential that is the functional derivative ofE_XC:

ν (ρ) =δE_XC(ρ)

The Kohn–Sham equations (Equation 2.11) are solved self-consistently starting by an initial guess of the electron densityρ(r). Thenν_XCis computed as a function of rwith Equation 2.12 by using some approximation forE_XC, i.e. a XC functional (see below). The Kohn–Sham equations are then solved to obtain an initial set ofε_i, which are used to compute an improvedρfrom Equation 2.10. This is repeated until ρ(r)andE_XChave converged to within a certain tolerance and the total electronic energy is finally obtained with Equation 2.9.

In modern molecular quantum chemistry, DFT methods have become a main tool due to their accuracy and smaller computational cost compared to the post-HF methods. In the accurate MP and coupled cluster theory methods, the computational scaling is fromN⁵toN¹⁰, whereNis the molecular size, which is typically the number of basis functions in the calculation. In DFT methods, the scaling is betweenN² andN³, and developments of the linear-scaling algorithms have resulted in the scaling ofNin some cases enabling the modeling of the systems consisting of thousands of atoms[42, 114]. The main disadvantage of DFT is that there is no systematic approach to improve the results towards the exact solution.

2.2.3 Time-dependent density functional theory

In the studies of excited-state properties of the system, a time-dependent Schrödinger equation is applied instead of Equation 2.3

H(Ò t)Ψ(t) =iħhΨ(t)

d t (2.13)

whereHÒ(t)is the time-dependent Hamiltonian andΨ(t)is the electronic wavefunction as a function of time. Time-dependent extension of DFT, i.e. TDDFT is based on the Runge–Gross theorem[115], which, analogously to Hohenberg–Kohn the- orems[112], states that the time-dependent external potential is a functional of the time-dependent electron density. Similarly to DFT, TDDFT simplifies the solving of Equation 2.13 for the many body system by replacing it with a set of time-dependent

single-electron orbitals. The TD Kohn–Sham equations in TDDFT have the form similar to Equation 2.11:

−1

2∇²+ν(r,t) +

∫︂ ρ(r^′,t)

|r−r^′|dr^′+V_ext(t) +ν_XC(r,t)

ψ_i(r,t)

=iħh ∂

∂tψ_i(r,t) (2.14) where the external potential,V_ext, ν_XC, the Kohn–Sham orbitals, and the density are now time-dependent. In TDDFT, the response of the molecular system to the varyingV_ext(e.g. electric and magnetic fields) is studied, which enables the study of polarizabilities, hyper-polarizabilities, excitation energies, and absorption spectra.

In most quantum chemical codes, the linear response of TDDFT equation, i.e. the linear response of the time-independent GS density to a time-dependentV_ext, is used for calculating excitation energies instead of the full TDDFT. Moreover, the adi- abatic approximation for the XC potential is usually applied, i.e. time-dependent ν_XC(r,t)is replaced with time-independentν_XC(r). This can be justified by an as- sumption thatν_XCis local in time and only the present electron density is taken into account. Thus, the time-dependent density is used in the GS XC potential functionals, i.e. the same (semi)local XC functionals are employed as in DFT. In the Tamm–Dancoff approximation to time-dependent density functional theory (TDA) to TDDFT[116], the Hermitian eigenvalue equation of the full TDDFT is simpli- fied to reduce the computational cost and address the issue of triplet instability, while leaving the excitation energies practically unchanged. In this work, both the linear- responsed TDDFT and TDA have been employed.

2.2.4 Density functional theory with periodic boundary conditions

Several different PBC codes have been developed to model extended, i.e. periodic systems, such as polymers, surfaces, and zeolites. Here, the focus will be on the PBC code developed by Kudin and Scuseria[41, 42], as it is implemented in the Gaussian

09 (and 16) suite of programs. It make use of localized GTOs that are transformed into crystalline orbitals, i.e. Bloch sums, with the following form

Ψ_k=∑︂

⎷Ne^ik·g

ψ_g (2.15)

wherek= (k_x,k_y,k_z)is the reciprocal-lattice vector,ψ_gis a GTOψcentered in cell g, andiis the imaginary unit. Periodic orbitals with different irreducible repre- sentations of the infinite translation group are classified byk. Orbitals belonging to differentkpoints do not interact with each other, and thus the SCF equations, which are similar as in the non-periodic case, can be solved separately for eachkpoint.

Any orbitalµcentered in the central cell (0) is coupled to any orbitalνcentered in theg^thneighbor cell by the Kohn–Sham Hamiltonian matrix elements.

F_µν^0g=T_µν^0g+U_µν^0g+J_µν^0g+V_µν^0g (2.16) where T_µν^0g is the electronic kinetic energy,U_µν^0g is the electron-nuclear attraction term,J_µν^0gis the electron-electron repulsion term, andV_µν^0gis the contribution from the DFT XC potential. The first two terms in Equation 2.16 do not depend on the density matrix, whereas the latter two do. In general, the terms in Equation 2.16 are similar to those of the molecular case. However, one difference from the molecular case is that the electrostatic termsU_µν^0gandJ_µν^0ginclude an infinite number of interactions between a given pair of basis functions and all the other charges of the system.

The energy per unit cell are calculated as

E=∑︂

µ∈0

∑︂

ν∈g(T_µν^0g+U_µν^0g+1

2J_µν^0g) +E_XC+E_NR (2.17) whereE_NRis the nuclear repulsion energy. The implementation of Kudin and Scuse- ria uses the fast multipole method for evaluation of the long-range electrostatic interactions, which enables high accuracy with the O(N) computational cost.

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