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Bloch's Theorem and Plane Wave Basis Sets

 

As yet there has been no mention of how to handle the infinite number of interacting electrons moving in the static field of an infinite number of ions. Essentially, there are two difficulties to overcome: a wavefunction has to be calculated for each of the infinite number of electrons which will extend over the entire space of the solid and the basis set in which the wavefunction will be expressed will be infinite.

The ions in a perfect crystal are arranged in a regular periodic way (at 0K). Therefore the external potential felt by the electrons will also be periodic - the period being the same as the length of the unit cell, tex2html_wrap_inline5530 . That is, the external potential on an electron at tex2html_wrap_inline5508 can be expressed as tex2html_wrap_inline5534 . This is the requirement needed for the use of Bloch's theorem. By the use of this theorem, it is possible to express the wavefunction of the infinite crystal in terms of wavefunctions at reciprocal space vectors of a Bravais lattice.

Bloch's theorem uses the periodicity of a crystal to reduce the infinite number of one-electron wavefunctions to be calculated to simply the number of electrons in the unit cell of the crystal (or half that number if the electronic orbitals are assumed to be doubly occupied - that is, spin degenerate). The wavefunction is written as the product of a cell periodic part and a wavelike part:

  eqnarray432

The first term is the wavelike part which will be discussed below. The second term is the cell periodic part of the wavefunction. This can be expressed by expanding it into a finite number of plane waves whose wave vectors are reciprocal lattice vectors of the crystal

  eqnarray439

where tex2html_wrap_inline5536 are the reciprocal lattice vectors which are defined by tex2html_wrap_inline5538 for all tex2html_wrap_inline5530 where tex2html_wrap_inline5530 is a lattice vector of the crystal and m is an integer. Therefore each electronic wavefunction is written as a sum of plane waves

  eqnarray451

By the use of Bloch's theorem, the problem of the infinite number of electrons has now been mapped onto the problem of expressing the wavefunction in terms of an infinite number of reciprocal space vectors within the first Brillouin zone of the periodic cell, tex2html_wrap_inline5546 . This problem is dealt with by sampling the Brillouin zone at special sets of k-points discussed in Section 3.6.

The electronic wavefunctions at each k-point are now expressed in terms of a discrete plane wave basis set. In principle this Fourier series is infinite. However, the coefficients for the plane waves, tex2html_wrap_inline5552 , each have a kinetic energy tex2html_wrap_inline5554 . The plane waves with a smaller kinetic energy typically have a more important role than those with a very high kinetic energy. The introduction of a plane wave energy cutoff reduces the basis set to a finite size.

This kinetic energy cut-off will lead to an error in the total energy of the system but in principle it is possible to make this error arbitrarily small by increasing the size of the basis set by allowing a larger energy cut-off. The cut-off that is used in practice depends on the system under investigation. For example, to describe the 2p valence electrons in carbon will require a large plane wave basis set to span the high energy states described by the wavefunction close to the carbon nucleus. However in silicon, the 3p valence electrons do not acquire such a high energy since they are repelled from the nucleus by orthogonalisation to the lower lying 2p core states. This will be discussed in more detail in later chapters.

Another advantage of expanding the electronic wavefunctions in terms of a basis set of plane waves is that the Kohn-Sham equations take a particularly simple form. Substitution of Equation (3.24) in to the Kohn-Sham equations, (3.17), gives

  eqnarray469

It can be seen in this form that the reciprocal space representation of the kinetic energy is diagonal and the various potentials can be described in terms of their Fourier components. Usual methods of solving the plane wave expansion of the Kohn-Sham equations is by diagonalisation of the Hamiltonian matrix whose elements tex2html_wrap_inline5562 are given by the term in curly brackets. It follows that the size of the Hamiltonian matrix is determined by the energy cut-off

  eqnarray497

It will be shown in a later section that it is not necessary to solve this by conventional matrix diagonalisation techniques, but a more computationally efficient method exists where the plane wave coefficients are treated as dynamical variables.


next up previous
Next: k-point Sampling Up: First Principles Molecular Dynamics Previous: ExchangeCorrelation and the

Stewart Clark
Thu Oct 31 19:32:00 GMT 1996