2.2. Density functional theory
The idea to use the electron density ρ instead of a wave function dates back to Thomas and Fermi. In 1927 they independently assumed that the ground state of a system can be obtained through an energy functional of the density.[79,80] The existence of such an
energy functional was proven by Hohenberg and Kohn in 1964, who showed that the energy of the electronic ground state is completely determined by its density.[81] Thus,
according to the first Hohenberg–Kohn theorem, there has to be a density functional (DF) E[ρ] that directly connects the ground state electron density to the exact energy. The second Hohenberg–Kohn theorem is the analogue to the variation principle in wave function theory (WFT), and shows that for any valid trial density ˜ρ, E[ ˜ρ] yields the upper bound to the true ground state energy.[82,83] However, the Hohenberg–Kohn theorems do not provide a construction formalism for the shape of this functional.
As the exact density functional is not known, finding a good approximation to E[ρ] is the goal of density functional theory (DFT) methods. E[ρ] can conveniently be divided into separate functionals that include different contributions.
E[ρ] = Te[ρ] + Ven[ρ] + Vee[ρ] (2.15)
Vee[ρ] = J [ρ] + K[ρ] (2.16)
Te[ρ] gives the kinetic energy, Ven[ρ] yields the Coulomb attraction between electrons and
nuclei and Vee[ρ] describes the electron-electron interaction, that can be split further into
a Coulomb part J [ρ] and an exchange part K[ρ]. Up to this point, the formulation of DFT is orbital-free. The advantage compared to wave function based theories is the dependence of the density on only three variables, i.e. the three Cartesian coordinates, instead of three variables per electron. The functionals Ven[ρ] and J [ρ] can be described
by their classical expressions.
Ven[ρ] =− M X A Z ZAρ(r) |RA− r| dr (2.17) J [ρ] = 1 2 Z ρ(r)ρ(r0) |r − r0| drdr 0 (2.18)
2. Theoretical background
respectively, based on the uniform electron gas (UEG).
TeU EG[ρ] = 3 10(3π) 2 3 Z ρ(r)53dr (2.19) KD[ρ] =−3 4 3 π 13 Z ρ(r)43dr (2.20)
These early attempts to approximate the true functional do not yield useful results for chemical problems, as chemical bonding cannot be described. The main reason for this is the inaccurate expression of the kinetic energy in the Thomas–Fermi model. Kohn and Sham introduced the calculation of the kinetic energy via a fictitious reference system of non-interacting quasi-particles which is supposed to have the same density as the true system.[84] This Kohn–Sham (KS) approach to DFT (KS-DFT) is nowadays the most common one and therefore, the KS prefix will be dropped. The drawback is that orbitals (KS-orbitals) have to be introduced in order to evaluate the kinetic energy TKS. Thus,
the number of variables grows to 3N as in wave function theory. Usually, this approach yields 98 to 99 % of the true kinetic energy. The missing difference in the kinetic energy for the independent compared to the correlated electrons, as well as the overall correlation and the exchange effects are described by the exchange-correlation functional EXC. EXC
is usually divided into an exchange EX and a correlation part EC. The sum of all theses
contributions is the total DFT energy.
EDF T = TKS[φ] + Ven[ρ] + J [ρ] + EXC[ρ] (2.21) EXC[ρ] = EX[ρ] + EC[ρ] (2.22) TKS[φ] =− 1 2 N X i hφi| ˆ∇2i|φii (2.23)
The electronic energy and the respective set of KS-orbitals are obtained iteratively by solving the KS-equations.
ˆ fiKS[ρ]φi = " ˆ hi[ρ] + X j ˆJij[ρ] + vXC[ρ] # φi = εiφi (2.24)
Analogous to the Fock-operator in HF, the Kohn–Sham operator ˆfKS
i [ρ] is an effective
one-electron operator. But instead of the exchange operator in HF, DFT uses an exchange- correlation potential vXC[ρ] that is the derivative of the EXC[ρ] functional with respect
to the density.
The similarity of DFT and HF is also reflected in the same computational effort. Reduc- ing the formal scaling ofO(N4) toO(N3) can be achieved by using the efficient resolution
2.2. Density functional theory
of the identity approximation for the Coulomb integrals (RI-J),[74] The major advantage
of DFT over HF is the inclusion of correlation effects when a sufficiently accurate approx- imation for EXC[ρ] is applied. While DFT is formally exact, the commonly used density
functional approximations (DFAs) are not. A drawback of DFT compared to WFT is, that it cannot be systematically improved, e.g. by increasing the number of excitation configurations as in coupled cluster theory. Therefore, the development of DFAs is often based on a trial and error approach, i.e. the accuracy of the functional is evaluated by testing it on typical, exemplary systems. As only a small part of the chemical space can be covered, uncertainties of the accuracy remain.
2.2.1. Hierarchy of density functional approximations
Although the systematic improvement of density functionals is difficult, the accuracy and computational effort of a density functional can be classified. Perdew introduced the picture of ’Jacob’s ladder’ which ascends from the ’Hartree-hell’ to the ’heaven of chemical accuracy’.[85] Density functionals can be categorized into rungs of that ladder. The higher the rung, the more information of the systems is used in the functional and the more expensive it is. The increase in accuracy with each rung cannot be guaranteed, but that the general picture holds has been verified statistically on a large scale.[86]
The local density approximation
The first rung of Jacobs ladder includes functionals that only take into account the local electron density. This local density approximation (LDA) is based on the assumption that the electron density varies slowly, and that it behaves like the uniform electron gas (UEG). The LDA exchange functional is a modification of the Dirac functional (Eq. 2.20) for which a pre-factor was introduced by Slater (also called Xα method).[87] An LDA
description of the correlation energy was derived by Vosko, Wilk and Nusair (VWN)[88]
via analytic interpolation formulae based on accurate Monte-Carlo calculations.
The LDA is widely and successfully used for the description of metallic solids, as their electronic structures are similar to the UEG. LDA functionals yield reasonable molecular structures but unfortunately, tend to overbind most molecular systems.
The general gradient approximation
The major reason for the inaccuracies of LDA methods for molecular systems is the strongly varying electron density of a molecule, which cannot be described by an approx- imation solely based on he UEG. Therefore, in addition to the local density, functionals on the second rung of the ladder take the gradient of the density ∇ρ into account. This
2. Theoretical background
general gradient approximation (GGA) is based on an LDA description modified by an enhancement factor FGGA
XC , which depends on both the electron density and its gradient.
EXCGGA = X
σ=α,β
Z
ρ EXCLDA[ρ] FXCGGA[ρ,∇ρ] dr (2.25) σ is the spin variable for α or β spin, and the LDA exchange and correlation function- als are usually modified separately. Common GGA functionals are the PBE exchange and correlation functional by Perdew, Burke, and Ernzerhof,[89,90] Beckes’s B88 exchange
functional[91] and the LYP correlation functional by Lee, Yang, and Parr.[92]
meta-GGA functionals
The third rung of Jacobs ladder comprises functionals, which additionally include higher order derivatives of the electron density, such as the electron density Laplacian ∇2ρ.
The Laplacian was found to be numerically unstable and thus, meta-GGA functional are based on enhancement factors that include the related orbital kinetic energy density instead. Meta-GGA functionals are often more accurate than GGA functionals, although this cannot be generalized and the improvement is by far not as large as going from LDA to GGA. The probably most popular meta-GGA functional is the TPSS functional developed by Tao, Perdew, Staroverov, and Scuseria,[93] which was also extensively used
in this thesis for the optimization of geometries. GGA and meta-GGA functionals are called semi-local functionals, since they are not only evaluated based on local electron density, but also take information about its close proximity into account.
Hybrid functionals
The forth rung of the ladder contains functionals that use additional non-local information based on the occupied KS-orbitals. This is achieved by substituting a part of the DFT exchange with non-local Fock-exchange evaluated with the KS-orbitals (EHF
X ), also called
’exact’ exchange. This approach can be motivated based on the adiabatic connection.[94]
EXChybrid = EC(meta−)GGA+ (1− aX)E
(meta−)GGA
X + aXEXHF (2.26)
Functionals of this class are named hybrid functionals and its most prominent example is the B3LYP functional. It contains 20 % of Fock-exchange (aX = 0.2), 0.08 % of Slater’s
LDA and 0.72 % B88 exchange, and 0.19 % VWN-LDA and 0.81 % LYP correlation.[95,96]
Further examples of typical hybrid functionals are BHLYP[97] (a
X = 0.5) and PBE0[98]
(aX = 0.25). Popular hybrid functional based on meta-GGA functionals are Zhao and
Truhlar’s M05[99] and M06[100] classes of functionals, also frequently called Minnesota
2.2. Density functional theory
functionals, which are highly parametrized with up to 40 parameters. The PW6B95[101]
hybrid functional (aX = 0.28) is also based on a meta-GGA and used in this thesis to
a large extent. All the so far mentioned hybrids have in common, that they employ the same amount of Fock-exchange over the whole space. Therefore, these hybrid functionals are also called global hybrids.
The principal problem with global hybrid functionals is that the underlying (meta-)GGA potential decays exponentially. Thus, in the asymptotic limit the hybrid exchange poten- tial decays with ax/r instead of the correct 1/r behavior. This behavior can be corrected
by introducing a range separation that retains the short-range error cancellation between exchange and correlation of a global hybrid and yields the correct potential of 100 % Fock exchange in the asymptotic limit. In these so called range-separated hybrid functionals, the two-electron operator r1
12 is partitioned into a short-range and a long-range component
using the error function (erf).[102,103]
1 r12 = 1− erf(µr12) r12 +erf(µr12) r12 (2.27)
The short-range part is then treated by an exchange functional, and the long-range part by HF exchange. Examples of this type of functionals are ωB97 and ωB97X developed by Chai and Head-Gordon,[104] and their advancements ωB97X-D[105] and ωB97X-D3.[106] Further examples are LC-BLYP and CAM-B3LYP by Handy,[107] though the latter one
is not asymptotically correct as it contains a finite amount of GGA exchange in the long- range limit. Nevertheless, CAM-B3LYP has been used successfully for the computation of electronic circular dichroism spectra within this thesis. In general, range-separated functionals perform good for the calculations of excited states by means of time-dependent DFT, due to the much more appropriate orbital energies.
Virtual-orbital dependent functionals
Functionals on the fifth and last rung of the ladder take the virtual KS-orbitals into account when calculating the correlation energy. Several approaches to accomplish this have been published, including perturbation methods as done by G¨orling and Levy,[108,109]
and random phase approximation (RPA) methods.[110] The probably most widely used
approach is that of a double-hybrid density functional (DHDF) proposed by Grimme.[111]
A part of the correlation energy is computed by second order Møller–Plesset perturbation theory (MP2) from the KS-orbitals of a preceding hybrid functional SCF calculation.
EXCDHDF = (1− aX)E
(meta−)GGA
X + aXEXHF + (1− aC)E
(meta−)GGA
2. Theoretical background
The most accurate DHDFs are those containing rather high amounts of Fock exchange and much smaller amounts of non-local correlation. E.g. the B2PYLP[111] functional employs
53 % exact exchange (aX = 0.53) and 27 % MP2 correlation (aC = 0.27). Double-hybrid
functionals are very accurate for reactions energies and basic molecular properties.[86] However, as the MP2 part with its formal scaling of O(N5) is rather expensive for large molecules and MP2 in general cannot accurately treat π − π stacked systems, DHDFs were not employed in the work of this thesis.