CHAPTER 1 INTRODUCTION
1.3 Literature Surveys
1.3.1 Theoretical Development
1.3.1.1 Schrödinger equation
The foundation of the theory of electronic structure of matter is the non-relativistic Schrödinger equation for the many-electron wave function Ψ [Schrödinger, 1926]. The much heavier nuclei are considered as fixed in space by the Born-Oppenheimer approximation, so the wave function Ψ depends on the
position and spins of the N electrons [Michael, 2000].
1.3.1.2 Hartree-Fock Theory
Within Hartree-Fock theory one assumes the wave function of the N particle (electron) system to be an anti-symmetrized product of one particle (electron) functions [Markus, 2002]. Moreover, the HF is an approximation, as it does not account for dynamic correlation due to the rigid form of single determinant wave function. To account for dynamic correlation, one has to go to correlated methods, which use multi-determinant wave functions, and these scales as fifth, or even greater powers with the size of a system [Jan, 1996].
The calculation of the many-body wave function of a system of interacting electrons is a formidable task, which can only be carried out – and is only
meaningful – for systems with a few tens of electrons [Kohn, 1999]. Its observables for larger systems are to be determined; the calculation of the many-body wave functions has to be avoided due to the seemingly formidable computational difficulty.
1.3.1.3 Hohenberg & Kohn Theorem
In the year 1964 Hohenberg and Kohn published a paper in Physical Review, where they stated two fundamental theorems, which gave birth to modern density functional theory, an alternative approach to deal with the many body problem in electronic structure theory [Hohenberg and Kohn, 1964].
Up till now, both the exact ground state density as well as the Hohenberg-Kohn functional is still unknown, so one cannot make use of the Hohenberg-Kohn theorems to calculate the molecular properties.
1.3.1.4 Kohn-Sham Equations
Kohn and Sham introduced a fictitious system of N non-interacting electrons to be described by a single determinant wave function in N “orbits” [Kohn and Sham, 1965]. The construction of the density explicitly from a set of orbits ensures that it is legal – it can be constructed from an asymmetric wave function. Due to the second part of the H&K theorem, namely that the total energy is minimized by the true ground-state density, the variational principle can now be utilized. With the standard functional derivatives and the additional definition of the so-called
exchange-correlation potential. The exchange-correlation functional ,εxc, which is
simply the sum of the error made in using a non-interacting kinetic energy and the error made in treating the electron-electron interaction classically [Harrison].
All that remains now is the question what to do with theεxc, without which one
cannot do any practical calculations. As mentioned above, this term has to tread on an approximate manner. Although there are many different functions available, almost all of them are derived from the electron density of a uniform electron gas, which can be calculated by means of statistical thermodynamics.
1.3.1.5 A New Density Functional Theory Formulation
In a recent paper, a generic derivation from cluster expansion results in a new DFT formulation without the exchange-correlation term that makes the computation much traceable in physics without ad hoc assumption as mentioned in the above [Hsu, 2003].
1.3.2 Numerical Methods
In the past, numerical methods for solving DFT formulation can be divided into two categories. One is orbital-type method and the other is real-space method. The former includes computations using slater-type orbitals (STOs), Gaussian-type orbitals (GTOs) and plane waves, while the latter is the real-space method that can be
loosely categorized as finite differences (FD), finite elements (FE), and wavelets.
1.3.2.1 Orbital-type method
Linear combinations of analytical functions =
∑
j i
i(r) cκχκ(r) ϕ
z Plane Waves, χκ(r)=exp(ikκr)
z Slater Type Orbitals, χκ(r)=xkκ ylκzmκ ⋅exp(−ζκr), whose shape close to true orbits (hydrogen atom) but evaluation of integrals is expensive.
z Gaussian Type Orbitals, χκ(r)=xkκylκzmκ ⋅exp(−ζκr2), whose evaluation of integrals cheap but different from true orbital shape.
Linear combination of GTOs to approximate STOs that is 1 STO ≈ 3 GTOs, but
analytical integration is still much faster than with STOs (Fig. 1.4).
1.3.2.2 Real-space method
Real-space methods can loosely be categorized as one of three types: finite differences (FD), finite elements (FE), or wavelets. All three lead to structured, very sparse matrix representations of the underlying differential equations on meshes in real space. Applications of wavelets in electronic structure calculations have been thoroughly reviewed recently [Arias, 1999]. As implied in the title, the primary focus is on calculations in density functional theory (DFT); real-space methods are in no way limited to DFT, but since DFT calculations comprise a dominant theme in modern electrostatics and electronic structure, the discussion here will mainly be
restricted to this particular theoretical approach.
The early development of FD and FE methods for solving partial differential equations stemmed from engineering problems involving complex geometries, where analytical approaches were not possible [Strang and Fix, 1973]. Example applications include structural mechanics and fluid dynamics in complicated geometries. However, even in the early days of quantum mechanics, attention was paid to FD numerical solutions of the Schrödinger equation [Kimball and Shortley, 1934; Pauling and Wilson, 1935].
Real-space calculations are performed on meshes; these meshes can be as simple as Cartesian grids or can be constructed to conform to the more demanding geometries arising in many applications. Finite-difference representations are most commonly constructed on regular Cartesian grids. They result from a Taylor series expansion of the desired function about the grid points. The advantages of FD methods lie in the simplicity of the representation and resulting ease of implementation in efficient solvers. Disadvantages are that the theory is not variational, and it is difficult to construct meshes flexible enough to conform to the physical geometry of many problems. Finite-element methods, on the other hand, have the advantages of significantly greater flexibility in the construction of the mesh and an underlying variational-type formulation. Other advantages include easier
parallel implementation using domain decomposition and possible mesh refinement in regions where solution changes rapidly, as mentioned earlier. However, the cost of the flexibility is an increase in complexity and more difficulty in the implementation of multiscale or related solution methods [Thomas, 2000].