The information for this section were taken from Ref. [62, 110, 111].
To obtain the energy E0 of the ground state and the ground state wavefunction ψ0(r;R), one needs to solve the electronic Schrödinger equation Eq. (2.5). Unfortunately, for many body systems, this is not feasible. The standard quantum chemistry approach to such a problem is to apply the variational principle which states that the energy of any trial wavefunction will be an upper bound of the true ground state energy of the system
D
ψk|Hˆe|ψkE
hψk|ψki ≥E0. (2.10)
and equality will only be reached whenψk =ψ0. The denominator on the l.h.s of Eq. (2.10) will be 1 when the electronic wave functions are normalized. This means that, the lower the energy value obtained with an electronic guess wave function, the closer this wave function will be to the ground state electronic wave function, therefore oering a tool to judge how good a guess wave function is.
Methods to evaluate the wave function depend on which approximations are made to the Hamiltonian to make it calculable. The only information that is needed for the construction of the electronic Hamiltonian for a given system is the number of electronsNe of this system and the potential VN,e which in turn is determined by the nuclear chargeZ and the positions of the nuclei. With these information, the ground state wave function can, in principle, be calculated and the energy of the system can be determined. The number of electrons is in direct relation to the electron density n(r) of the system where rdenes the position from where the electron density is taken:
Ne = Z
drn(r) (2.11)
Furthermore, the density has also cusps at the nuclei position, therefore, the positions of the nuclei are derivable from the electron density. Additionally, the density at the position of the nuclei contain information about the nuclear charge. This means that the electron density in fact contains all information one needs to construct the electronic Hamiltonian for a system.
Instead of calculating the energy of a system from the wave function, which depends on the positions of all the electrons in the system (and their spin), the electron density could be used which depends only on three coordinates.
All density functional theory starts with the Hohenberg-Kohn theorem which states that the ground state of a given system has only one specic electron density associated with it. This ground state electron density n0(r) uniquely denes the system's Hamiltonian and with that makes it possible to calculate any observable of the system. Secondly, the energy of the ground state also has variational property with respect to the electron density. This means that the
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electron density can be used in lieu of the wave function to calculate a system's energy. The ground state energy of a system can therefore be expressed as a functional
E0[n0] =Te[n0] +Vee[n0] +VN,e[n0] =F[n0] +V[n0]. (2.12) Te andVee are universal functionals, they can be combined to the Hohnberg-Kohn functionalF that, if it were known, would be a universal key to all imaginable systems. VN,e where I have dropped the indices on the right hand side of Eq. (2.12) is an external potential that varies for dierent systems.
The electrostatic interaction energyVeebetween the electrons can be split up into the Hartree energyVH whose functional form is known and a nonclassical contributionEncl to the electron-electron interaction that contains a self-interaction correction (the interaction of the electron-electron density with itself in the Hartree energy), exchange and Coulomb correlation:
Vee[n] = 1 2
Z Z
drdr0n(r)n(r0)
|r−r0| +Encl[n] =VH[n] +Encl[n] (2.13) In 1965, Kohn and Sham [112] suggested an approach to treat the universal functional F that avoids the shortcomings in treatment of the kinetic energy functional of direct methods such as the Thomas-Fermi-method. In the Kohn-Sham approach, a part of the kinetic energy functional is treated in terms of single-particle orbitals (i.e., one-electron functions)φ(r)of a noninteracting system. This allows to treat a large part of the kinetic energy functional exactly. For it, the kinetic energy functional is decomposed into a partTs[n](s for single-particle) that represents the kinetic energy of individual, noninteracting particles and the remainderTc[n](c for correlation).
T[n] =− ¯h2 2me
Ne
X
i
φi(r)|∇2i|φi(r)
+Tc[n] =Ts[n] +Tc[n] (2.14) Tc and Encl are combined into the exchange correlation functional Exc so that the total energy expression assumes the following form:
E[n] =Ts[n] +VH[n] +Exc[n] +V[n] (2.15) The single particle orbitals are chosen such that they reproduce the density of the original system:
n(r) =
Ne
X
i
hφi|φii (2.16)
To obtain the ground state energy of the system, the variation principle can be applied:
0 = δE[n]
δn(r) = δTs[n]
δn(r) +δVH[n]
δn(r) +δVN,e[n]
δn(r) +δExc[n]
δn(r) = δTs[n]
δn(r) +vH(r) +vN,e(r) +vxc(r) (2.17)
If Eq. (2.17) is compared with a system of noninteracting particles moving in an external potential vs(r)it becomes clear that one can treat the entire problem as a pretend-noninteracting single-particle problem where the potentials contributing to Eq. (2.17) can be seen as making up the external noninteracting-particle potential:
vs(r) =vH(r) +vN,e(r) +vxc(r) (2.18) Eq. (2.16) and Eq. (2.18) are the Kohn-Sham equations. The Kohn-Sham orbitals φi can be obtained by solving the one-electron Schrödinger equation
−¯h2∇2 2me
+vs(r)
φi=εiφi(r). (2.19)
These orbitals dene the noninteracting system which, according to Eq. (2.16) has the same density as the real system. Up until this point, the scheme is exact in so far that, if all func-tionals that make up vs(r) were known, one could calculate the exact energy of the system.
Unfortunately,vxc is unknown and needs to be approximated.
An early approximation to the exchange-correlation functional was made with the local density approximation (LDA) that treats the exchange-correlation energy as that of a locally homoge-neous electron gas
ExcLDA[n] = Z
drn(r)εLDAxc (n(r)). (2.20) The per volume exchange of a homogeneous electron gas is known exactly and the correlation energy of a homogeneous liquid can be calculated with Quantum Monte Carlo [113] and interpo-lated. The LDA approximation has proven itself to be quite accurate, due mostly to systematic error cancellations: the exchange is overestimated while the correlation is underestimated. It provides reasonable geometries and vibrational frequencies but greatly overestimates atomiza-tion energies [114] and fails to predict chemical bond energies within chemical accuracy (energy errors of the order of 0.0434 eV). An improvement to it is the generalized gradient approximation that treats not only the local density n(r)but also its gradient in general functions:
ExcGGA[n] = Z
drn(r)εGGAxc (n(r),∇n(r)) (2.21) The general functions can either be parametrized to test sets of selected molecules or using exact constrains (e.g. the Perdew-Burke-Ernzerhof- (PBE) [115], Perdew-Wang-`91- (PW91) [116, 117]
or revised PBE (RPBE) functionals [118]). The GGA functionals do not provide chemical accu-racy but provide reliable results for covalent, ionic, metallic and to some extend hydrogen bond interactions.
LDA and GGA are the rst two rungs of the so-called Jacob's ladder of DFT [114] which reaches from Hartree calculation that do not treat exchange correlation eects to chemical accuracy with accurate treatment of exchange correlation eects. Further rungs (that is, further
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improvement) on this ladder include the treatment of Kohn-Sham kinetic energy densities or second derivatives of the density in the meta-GGAs. Hybrid functionals follow that mix Hartree-Fock exchange into the DFT exchange functional. Spin can also be dealt with in DFT by including individual functionals for the α andβ spin densities.
While the electron density should, in theory, contain information about all states of the system, calculating any but the ground state in DFT is not easy, for the variation principle does not apply to excited states. On the level of GGA, the description of non-local interactions such as van-der-Waals fails (although on higher levels of DFT progress has been made in that direction [119]), same as an accurate description of dative bonds cannot be achieved. Hydrogen bonds are often predicted to be too short.
SRP-Functional
Commonly used DFT GGA-functionals for gas phase particles in interaction with metal surfaces are the PBE [115], PW91 [116, 117] or RPBE functionals [118]). However, PW91 overestimates binding energies while RPBE underestimates them [39]. If functionals both over- and underes-timate experimental properties, to obtain chemical accuracy for a system, the specic reaction parameter (SRP) DFT approach introduced by Chuang et al. [120] can be taken. This has been done by Diaz et al. [121] for reactive scattering of H2 from Cu(111). The version used in this work consists of a mixture of the PBE- and RPBE-functionals [12]:
ESRPxc =xExcRPBE+ (1−x)ExcPBE (2.22) with a weighting factor of x = 0.48. The resulting functional will be referred to as `SRP48'.
Although the SRP48 has been optimized for H2 on Cu(111), I expect it to perform similarly well for H with Au(111), since PW91 (whose energetic behavior PBE [115] was designed to mimic) performs already quite well for H2 with Au clusters but could be improved by a mixing with RPBE [122].