To specify the concept of Density Functional Theory (DFT) and Complete Active Space Self Consistent Field (CASSCF) the Hartree-Fock approach will be briefly introduced.
Hartree-Fock
Hartree-Fock is a quantum mechanical method for calculating wave function and energy of the ground state of a many-body system by solving the time-independent Schrödinger equation. Several approximations are used to get a computationally efficient method that gives adequate results at the same time.
• The wave function of each electron is described as an orbital, given by a linear combination of basis functions out of a preassigned set. The product of those single electron orbitals forms the many particle basis for the total electron wave function. The Pauli principle excludes that two electrons share the same orbital. Thus the wave function has to be totally anti-symmetric which means that it changes the sign when any two electrons are exchanged. This totally anti-symmetric product is called slater determinant.
• By using the variational principle, the orbitals are optimised to give the lowest possible energy for the chosen basis set, restricting the total wave function to be described by a single Slater Determinant.
• The mean field approximation used in Hartree-Fock averages the inter-actions between the electrons. A single electron does not interact with each other electron separately, but with an averaged electron cloud.
In contrast to the wave function used in the Hartree-Fock- and post-Hartree-Fock-methods, as CASSCF, which will be described in the following section, the electron densityρ(r)is a physical observable, which only depends on~r. Density Functional Theory (DFT) derives all physical quantities from the electron density ρ(r). This is given for a N-electron wave function Ψby
ρ(r1) =N Z
|Ψ(x1, x2, ...xN)|2dx2...dxN. (2.19)
Hohenberg and Kohn (HK)78 demonstrated that
• the ground state electron density is uniquely defined by the core po-tential and the ground state electron density uniquely defines the core potential (HK theorem I).
• the variational principle holds for calculating the energy from the elec-tron density. The ground state elecelec-tron density minimises the energy (HK theorem II).
From the HK theorem I, it follows that the energy is given by a functional of the electron density E[ρ] split into parts as
E[ρ] =T[ρ] +Ene[ρ] +Eee[ρ]. (2.20) The term Ene[ρ] for the attraction between electrons and nuclei is given by the expression In equation 2.20, the electron-electron repulsion Eee[ρ] is given by the Coulomb J[ρ] and exchange parts K[ρ]. The Coulomb energy term is
J[ρ] = 1 2
Z Z ρ(r)ρ(r´)
|r−r´| drdr´. (2.22) Terms for the kinetic and exchange energy are approximated, e.g., from the electron density of a free electron gas. In these approximations the ex-chage and kinetic terms are given by
K[ρ]' −Cx where Cx and CT are constants.
A solution to this problem was presented by Kohn and Sham (KS)79. They proposed to reintroduce orbitals and for this split the kinetic energy
into one that can be calculated exactly for a non-interacting system and a small correction to the exact term that is absorbed into the exchange-correlation term. By this approach the kinetic energy becomes equivalent to the Hartree-Fock kinetic energy
The total energy for DFT can thus be formulated as
EDF T[ρ] =TS[ρ] +Ene[ρ] +J[ρ] +Exc[ρ]. (2.25) By comparing the DFT energy (Equation 2.25) to the exact energy cal-culated from the electron density (Equation 2.20), the exchange-correlation term is defined as
Exc[ρ] = (T[ρ]−TS[ρ]) + (Eee[ρ]−J[ρ]). (2.26) Exchange-Correlation Functionals
Different DFT methods vary by the exchange-correlation functional used. A correct exchange-correlation functional is only possible for a uniform electron gas which is used in the first method mentioned below. For all other systems, the functional is an approximation.
The popular types of Exchange-Correlation Functionals are the following:
1. The local density approach (LDA) expresses the energy as the integral over a function of the density
EXC = Z
F(ρ)dr. (2.27)
2. The generalised gradient approximation (GGA) expresses the energy as the integral of a function of the density and its gradient
EXC = Z
F(ρ,∇ρ)dr. (2.28)
3. The meta GGAs use higher order derivatives of the density EXC =
Z
F(ρ,∇ρ,∇2ρ)dr. (2.29)
4. The hybrid functionals combine a GGA functional with a portion of exchange energy calculated from orbitals
EXC = Z
F(ρ,∇ρ)dr+ξEX0. (2.30) The most established exchange-correlation functional is the hybrid func-tional B3LYP, which is also used in this work. It is a funcfunc-tional of type 4 with ξ = 20%. This functional’s popularity is because of its good overall performance.
Time-dependent DFT (TD-DFT) is, in contrast to DFT, able to describe excited states that are important for the investigation of the deactivation events in PYP, the topic of this work, and has been used for excited state MD simulations80–82.
DFT as well as Hartree-Fock are single-reference methods. These methods are not able to describe bond breaking and formation sufficiently, as will be described in the following section on the example of breaking the bond in H2
using Hartree-Fock.
Further problems encountered by TD-DFT are the description of valence states in molecules with extendedπ-systems, charge-transfer excited states83 and conical intersections between ground and excited state84, 85. Both an extended π-system as well as conical intersections are present in the deacti-vation events in PYP studied in this thesis. Therefore TD-DFT will not be used. For the excited state simulations CASSCF, described in the following section, is applied.
As DFT, in particular the B3LYP functional what will be described in chap-ter 3, is a reliable and computationally efficient QM method, it is used in this work for ground state calculations.