Density Functional Theory

The premise of density functional theory (DFT) is based on the proof by Hohenberg and Kohn[62] that the ground state energy can be determined by the electron density, ρ. Once ρ is known, then the wave function can be determined and all the properties of a given system. The ground state energy is therefore a functional of the electron density

i.e., E0 = E[ρ]. Unlike wave function methods, where for an N-electron system there are

4N variables (3 spatial coordinates and 1 spin coordinate), there is only one electron density described by one set of spatial coordinates (e.g., x, y, z), independent of the total number of electrons. DFT methods therefore scale as Nbasis3 or Nbasis4 depending on the

DFT method used. This represents a computational cost savings of about a factor of 1,000 relative to the ab initio methods commonly used. The Hohenberg and Kohn theorem shows that ρ can be used to determine E0, however, it does not show how to calculate E0

from ρ.

The first attempt to determine E0 from ρ is Kohn-Sham[63] method, which is the

basis of the DFT methods used throughout this dissertation. In Kohn-Sham DFT, the energy of a system, where the electrons are correlated, is calculated with respect to a non- interacting system with no electron correlation. The difference in the kinetic and potential energies between the interacting and non-interacting is defined as

(2.45)

where the subscript s denotes the non-interacting system, T[ ]

ρ

is the kinetic energy,

[ ]

ee

V

_ρ

is the interacting electron energy, V_c[ ]

_ρ

is the classic coulomb expression

between two electrons, and

E

_XC

[ ]ρ

is called the exchange-correlation energy. The problem with solving for

E

_XC

[ ]_ρ

is that T[ ]

ρ

and

V

ee

[ ]ρ

are also unknown functionals.

This is the main issue with DFT methods. There is no universal approach for solving

[ ]

XC

E

ρ

and, as a result, there is no universal DFT method that works for every system. Appropriate selection of a DFT method is based on previous experience and on a case by case basis. In developing DFT functionals, there are three general approaches for achieving chemical accuracy. Functionals can either be parameterized to fit a robust set of experimental data, constrained to fulfill well-known universal physical constraints, or a combination of the two. Parameterized functionals offer high accuracy when applied to systems represented by the fitted data, however caution is needed for molecules with properties outside the scope of the parametrization databases. Constrained functionals should in practice offer accuracy over all systems at the cost of less accuracy compared to functionals parameterized for specific systems.

The DFT methods used throughout this dissertation are M06-2X and PBE1PBE, the latter is also known in the literature as PBE0.

M06-2X is a parametrized global hybrid meta-generalized gradient- approximations (hybrid meta-GGA) functional part of the M06 family of functionals developed by the Truhlar group[93]. GGA functionals make the exchange and correlation energies depend on the first derivate electron density as well as the local electron density as given by the local spin density approximation (LSDA). Global hybrid GGA functionals replace a constant percentage of the local exchange by Hartree-Fock (HF) exchange. HF exchange is the energy of a Slater determinant built from Kohn-Sham orbitals solved using the HF self-consistent field (SCF) method. This is not the same as the HF exchange energy from the SCF method since Kohn-Sham orbitals are used.

Adding HF exchange is an improvement over GGA and LSDA functionals because it removes some of the self-interaction energy. That is, the Coulomb and Exchange self- interaction energies cancel each other out in HF theory. Hybrid meta-GGAs include additions to the functional form of GGA functional such as second derivatives of the spin densities or second derivatives of the spin-labeled non-interacting kinetic energy densities[94]. For M06-2X the hybrid exchange-correlation energy is given as

E_XCM 06−2 X = X 100EXC HF + 1− X 100 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ EX DFT + EC DFT (2.46)

where E_XCHF _{is the nonlocal HF exchange energy, X is the percentage of HF exchange,}

E_XDFT _{is the local DFT exchange energy, and E}

C

DFT _{is the local DFT correlation energy.}

For M06-2X the percentage of HF exchange is 54% and the expressions for E_XDFT _and

E_CDFT _{can be found in Zhao and Truhlar[93].}

PBE0[95-97] is a hybrid GGA functional that is constructed to satisfy physical constraints whereas M06-2X is parameterized only for non-metals. In PBE0 the exchange-correlation energy is given as

(

)

0

1

PBE GGA HF GGA

XC XC X X

where GGA XC

E is the Pardew, Ernzerhof, and Burke (PBE) GGA exchange-

correlation functional and a1 is a mixing coefficient set to 25%. From our previous

studies[44] and that of the group of Dibble[64], M06-2X and PBE0 have shown to give reliable structures and accurate thermochemistry for Hg molecules .

An ultrafine pruned integration grid consisting of 99 radials shells and 990 points/shell was used for all DFT/AVTZ calculations. Geometry optimizations were done at the DFT level with a very tight convergence criterion, setting the root meat squared value of the force to 1x10-6.

In document Speciation of gaseous oxidized mercury molecules relevant to atmospheric and combustion environments (Page 71-75)