Quantum mechanics fully describe the phenomena of the nano world. Most ob-served properties of solid materials can be explained by solving the full (non-relativistic) many-body Schrödinger equation
i ∂
∂t|Ψfulli=Hˆfull|Ψfulli (2.1) with the full many-body Hamiltonian for electrons and nuclei
Hˆfull =
Here, the Hamiltonian ˆHfullcontains (ordered as in the equation) the kinetic energy of the electrons, the kinetic energy of the atomic nuclei and the interactions of the electrons with the atomic cores as well as the interaction among electrons and the interaction among cores, respectively.
Already in 1929 Dirac [3] stated that the Equations (2.1) and (2.2) describe “a large part of physics and the whole of chemistry”but “are too complicated to be soluble”analytically for more than two particles. However, the very different scale of masses,ma ≈ 3676Zame, justifies the separation of motions according to the different time scales, the so calledBorn-Oppenheimerapproximation [4], has shown to hold in most cases. It fails only in very special scenarios wherevibronic (com-bined vibrational and electronic) states play a central role. It is possible to treat
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only the electronic part of the Hamiltonian quantum mechanically assuming the atomic positionsRato be parameters. The atomic motion can be treated classically neglecting the quantum nature of their vibrational motion.
The many-body (MB) Schrödinger equation in the Born-Oppenheimer approxi-mation forms the basis for a (non-relativistic) quantum mechanical description of the interacting electron system at zero temperature, i.e. no atomic movements are considered. It is given by
i ∂
The challenge arising from this equation is the dimensionality of the underly-ing Hilbert space. Considerunderly-ing for example a system where each particle has two accessible eigenstates, e.g. a spin-12 system, the many-body Hilbert space is 2N -dimensional. The exponential growth makes it problematic to find eigensolutions of the Schrödinger equation, since conventional numerical solvers for eigenvalue problems scale with the third power of the dimension. This leads to a total work-load proportional to∝23Nfor finding the exact solutions of a spin system withN particles.
The problem becomes even more challenging giving more degrees of freedom to the electrons. Quantum mechanics tells us to express position and momentum of the electrons by a continuous distribution function. So we have to consider an entire function space to represent their state. Most approximations are based on truncating these function spaces to subspaces of a finite number of dimensions.
However, there is a constant tradeoff: On the one hand the number of basis func-tions has to be kept large to preserve the accuracy of results. On the other hand the number of basis functions needs to be kept small at the same time since using d basis requires a storage ofdNnumbers and the number of computation operations scales proportional to∝ d3N which makes this practically impossible already for systems with a few electrons.
In the following we consider the electronic Hamiltonian of the many body
sys-2.1. The Many-Body System 15 whereVext(rˆ)stands for the one-particle external potential of the atom cores like the electron-core interaction in Equation (2.2). Optionally, additional electric or magnetic fields can be included into the external potential.
The MB wave function for the electrons needs to be antisymmetric under ex-change of any two particles since electrons are fermions. Consider a two-particle wave functionΨMB(r1,s1;r2,s2)with the spatial coordinate riand the spin statesi
of thei-th particle. The Pauli exclusion principle demands that no two fermions occupy the same state. The consequence is a wave function that is antisymmetric under the exchange of the particles, i.e. simultaneously interchangingr1↔r2and s1↔s2 produces a factor−1 such that
ΨMB(r2,s2;r1,s1) = −ΨMB(r1,s1;r2,s2). (2.6) Now, ifr1 =r2 =rands1 =s2 =sholds this results in
ΨMB(r,s;r,s) = −ΨMB(r,s;r,s), (2.7) i.e. ΨMB(r,s;r,s) = 0. Therefore, no two electrons can simultaneously have the same position and spin coordinates. This holds for more than two particles, too. In practice this means that two electrons with the same spin will avoid being near to each other, giving a minimum in the spatial electron-electron correlation function known as the exchange hole. Because the electrons avoid coming close to each other, where the energy contributions from the Coulomb repulsion are high, the exchange hole leads to a lowering of the total energy.
2.1.1. Hartree- and Hartree-Fock approach
A first attempt to model the many-body wave function is a product ansatz (some-times called Hartree ansatz) of one-particle wave functions
ΨH(r1,r2, . . . ,rN) =φq1(r1)φq2(r2)· · ·φqN(rN), (2.8) where theqiare different sets of quantum numbers. This wave function can easily be stored due to the separation of variables. A basis set of sizedwould then require the storage ofNdnumbers. However, the Hartree ansatz leads to a description of the electrons without the explicit consideration of the Pauli principle. This was
fixed by the ansatz of a determinant of single particle wave functions proposed by
The mathematical construct of determinants intrinsically satisfies the antisymme-try constraint imposed by the Pauli principle. Variation of the total energy
Etot =hΨMB|HˆMB|ΨMBi (2.10) with respect to the single particle states φi(r) leads to the Hartree-Fock [62, 63]
equations
with the Hartree-Fock energy parametersǫHFi . This leads to a computational very expensive scheme that does not contain the full electron-electron interaction. Even though the exchange interaction is treated in an exact manner the effects of electron-electron correlations are not included.
Correlations in the statistical meaning are cross dependencies between the parti-cles coordinates and spins. Assuming the many-body wave function as a product of single particle states as Equation (2.8) gives a totally uncorrelated description for the electrons, i.e. their probability distributions are independent of each other.
A rather different approach towards a solution of the MB problem was found by considering the degrees of freedom of the electron density rather than the full MB wave function.