2.3 Electronic coupling matrix elements
2.3.1 Density Functional Theory
This equation approximates the two redox partners exchanging a charge
q as spheres of radii r1 and r2, respectively, at a center separation of R and
immersed in a dielectric medium of static dielectric constant εsand optical di-
electric constant εop. While the latter experimentally falls in the range of 1.5-
2.5 for most media [72], it is 1 with a nonpolarizable force field. Hence, the term in the central bracket (the “Pekar factor”) is too large and thus is λouter.
A polarizable force field provides an approximative way of explicitly incor-
porating electronic polarizability, thus increasing the simulated medium’s εop
and hence decreasing the Pekar factor in Eq. 2.39.
A comparison of nonpolarizable and polarizable force fields for ET in a cytochrome yielded a decrease by about a third when both dynamics and per-snapshot calculations of ∆E used a polarizable force field; however, most of this difference could already be attained by only calculating ∆E for each snapshot with a polarizable force field while taking the snapshots themselves from dynamics with a nonpolarizable force field. [5] In the light of these previous results, the slight increase in accuracy by running dynamics with a polarizable force field did not seem worth the large additional computational cost this would mean for all pairs in MtrF. Hence, for the calculation of λ for heme pairs in MtrF we decided to only use a polarizable force field (based on induced atomic dipoles, see Section 2.2.1) to calculate ∆E for each snap- shot while taking the snapshots from nonpolarizable force field dynamics. As the nonpolarizable dynamics used in calculating the redox potentials for each heme include a window with the heme in question fully reduced (and all oth- ers oxidized), we can in fact use these fully-reduced windows for hemes i and j as the initial and final states A and B for ET from heme i to heme j - reusing their snapshots but calculating a different quantity (∆E for ET rather than∂Vε
∂ ε for oxidation). Thus, after the calculation of redox potentials, no new
dynamics are required for calculating λ .
2.3
Electronic coupling matrix elements
2.3.1
Density Functional Theory
Electronic couplings need to be computed for many snapshots, restricting the acceptable computational costs of the quantum chemical method chosen. This makes the use of Density Functional Theory (DFT) appealing as DFT is gen- erally computationally cheaper than wave function methods based on solving the Schroedinger equation [73]. DFT is based on the fact that the ground state
2.3. ELECTRONIC COUPLING MATRIX ELEMENTS
electron density uniquely determines the electronic Hamiltonian and thus the energy and all other properties of the system, i. e. the ground state energy is a functional of the corresponding density; furthermore, for a given exter- nal potential V (~r) (from e. g. a specific nuclear geometry) and fixed num- ber of electrons the correct ground state electron density is the density that minimizes this functional. These two fundamental findings are known as the Hohenberg-Kohn theorems [74]. The energy functional E [ρ] was defined as:
E[ρ] =
V(~r) ρ (~r) d~r + F [ρ] (2.40)
where ρ (~r) denotes the electron density at point ~r and F [ρ] denotes that fraction of the total energy not due to the interaction of the electron density with the external potential; finding an actual expression for this functional is nontrivial due to the difficulty of expressing the electronic kinetic energy as a functional of the density.
A practical way of putting the energy functional E [ρ] to use was provided by the Kohn-Sham approach [75] which defines an electronic wave function
ψs for a hypothetical reference system of noninteracting electrons producing
the same density as the actual system:
ψs=
1 √
N!det [φ1φ2· · · φN] (2.41)
where the φi are the individual spin orbitals occupied by the reference
system’s N electrons. The kinetic energy Ts[ρ] for this refence system is then
given by: Ts[ρ] = N
∑
i ψs −1 2∇ 2 i ψs (2.42) This allows for F [ρ] in Eq. 2.40 to be expressed as the sum of Ts[ρ] and aneffective one-electron potential. To this end, F [ρ] is split into several terms:
F[ρ] = Ts[ρ] + J [ρ] + Exc[ρ] (2.43)
where J [ρ], the so-called Hartree term, denotes the classic Coulomb en- ergy of a charge density ρ:
J[ρ] = 1 2 d~rd~r0ρ (~r) ρ (~r 0) |~r −~r0| (2.44)
and the so-called exchange-correlation energy Exc[ρ] is given by:
Exc[ρ] = T [ρ] − Ts[ρ] +Vee[ρ] − J [ρ] (2.45)
2.3. ELECTRONIC COUPLING MATRIX ELEMENTS
its total electronic interaction energy, including exchange and correlation en- ergy.
The functional derivatives δ J[ρ ]
δ ρ (~r) =
d~r0 ρ(~r|~r−~r0)0| and δ Eδ ρ (~r)xc[ρ] =: Vxc(~r) can
now be combined with the external potential V (~r) to yield an effective one- electron potential Vs(~r): Vs(~r) = V (~r) + d~r0 ρ (~r 0) |~r −~r0|+Vxc(~r) (2.46)
This in turn allows the definition of a noninteracting Hamiltonian ˆHs for
the reference system of electrons:
ˆ Hs= N
∑
i −1 2∇ 2 i + N∑
i Vs(~ri) (2.47)where ~riis the position vector of the ith electron. The orbitals constitut-
ing the reference system of noninteracting electrons are then obtained as the eigenfunctions of this Hamiltonian; they are referred to as the Kohn-Sham orbitals: −1 2∇ 2 i +Vs(~r) ψi= εiψi (2.48)
with εi as the ith orbital energy. As for the Hartree-Fock method, the
orbitals have to be determined self-consistently. The density of this reference system and hence of the real system (since their densities were defined to be identical) is then obtained as
ρ (~r) =
N
∑
i
∑
σ|ψi(~r, σ )|2 (2.49)
where σ denotes electron spin (α or β ). Up to here, the theory is ex- act; however, in order to actually use it, an approximation for the exchange-
correlation functional Exc[ρ] needs to be made. The conceptually simplest
approach is to assume this functional to be an integral over a function εxcthat
depends on the local density [75], i. e.:
ExcLDA[ρ] =
ρ (~r)εxc(ρ)d~r (2.50)
Such functionals are consequently referred to as Local Density Approx- imation (LDA) functionals. In contrast, another class of functionals defines
an εxc depending on both the local density ρ(~r) as well as its gradient ∇ρ(~r);
these are known as Generalized Gradient Approximation (GGA) functionals [76]:
ExcGGA[ρ] =
2.3. ELECTRONIC COUPLING MATRIX ELEMENTS
The GGA functionals PBE [76] and BLYP [77] have been employed in this work.
Finally, a third popular class of functionals are so-called hybrid function- als that try to improve on the exchange energy calculation in DFT by calcu- lating the Hartree-Fock exchange energy and mixing it in (i. e., calculate the total exchange energy as a fraction from the exchange functional and another fraction from Hartree-Fock). A popular functional of this kind that is also employed in this work is B3LYP. [78, 79] Thus incorporating a fraction of “exact exchange” is computationally expensive but it does have the advantage of mitigating the self-interaction error in DFT whereby even a single electron interacts with its own charge density, thereby experiencing artificial delocal- ization.
As for methods based on the Schroedinger equation, it is useful to expand the Kohn-Sham orbitals in a basis set. There are two main approaches: Atom- centered basis sets that try to describe the electronic structure around atoms by a relatively small number of well-chosen localized functions; and plane wave basis sets [80] that use a vast number of plane waves of the form fG(~r) = Ω1 ·
ei~G~r where ~Gis the wave vector and Ω a cell volume; this approach assumes
a periodic system with certain cell dimensions, and for cluster calculations (i. e. nonperiodic systems) the system needs to be decoupled from its periodic images (i. e., the interaction energy needs to be subtracted). In this work, plane wave calculations under cluster conditions were used. Unlike atom- centered basis functions, they describe all regions of space equally well and are hence advantageous in describing the region of space between donor and acceptor, which is crucial for the coupling matrix element calculations.
While the procedure described so far treats the system in the gas phase, it is also possible to treat the environment on a classical level, i. e. surround the quantum system with classical point charges and thus take polarization through the environment into account. In the case of MtrF, this allows us to calculate the electron densities of individual cofactors under the influence of the protein and solvent environment. This approach is known as QM/MM [81] where the “QM” stands for the quantum-mechanically treated subsystem and “MM” for the environment treated with Molecular Mechanics.