## Periodic Systems

Periodic systems can be described as a fundamental unit cell being repeated to form an infinite system. The periodicity can be in one dimension (e.g. a polymer), two dimensions (e.g. a surface) or three dimensions (e.g. a crystal), with the latter being the most common. The unit cell in three dimensions can be characterized by three vectors a1, a2 and a3 spanning the physical space, with the length and the angles between them defining the shape.37 There are seven possible shapes, the simplest of which is cubic, where all vector lengths are equal and all angles are 90°.

Figure 3.8 A cubic unit cell defined by three vectors

A unit cell can have atoms (or molecules) occupying various positions within the cell (corners, sides, centre), and the combination of a unit cell and its occupancy is called a Bravais lattice, of which there are fourteen possible forms. The periodic (infinite) system can then be generated by translation of the unit cell (Bravais lattice) by lattice vectors t.

The reciprocal cell is defined by three vectors b1, b2 and b3 derived from the a1, a2 and a3 vectors of the direct cell, and obeying the orthonormality condition a^ = 2rc8y.

The reciprocal cell of a cubic cell with side length L is also a cube, with the side length 2n/L. The equivalent of a unit cell in reciprocal space is called the (first) Brillouin zone. Just as a point in real space may be described by a vector r, a "point" in reciprocal space may be described by a vector k. Since k has units of inverse length, it is often called a wave vector. It is also closely related to the momentum and energy, e.g. the momentum and kinetic energy of a (free) particle described by a plane wave of the form eik'r is k and 1/2k2, respectively.

The periodicity of the nuclei in the system means that the square of the wave function must display the same periodicity. This is inherent in the Bloch theorem (eq. (3.75)), which states that the wave function value at equivalent positions in different cells are related by a complex phase factor involving the lattice vector t and a vector in the reciprocal space.

Alternatively stated, the Bloch theorem indicates that a crystalline orbital (f) for the nth band in the unit cell can be written as a wave-like part and a cell-periodic part (j), called a Bloch orbital.

The Bloch orbital can be expanded into a basis set of plane wave functions (cPW).

Alternatively, the basis set can be chosen as a set of nuclear-centred (Gaussian) basis functions, from which a set of Bloch orbitals can be constructed.

«basis «basis jnk(r) = X Cnajka(r) = X XCnaeik tCGTO(r +1)

The problem has now been transformed from treating an infinite number of orbitals (electrons) to only treating those within the unit cell. The price is that the solutions become a function of the reciprocal space vector k within the first Brillouin zone. For a system with «basis functions, the variation problem can be formulated as a matrix equation analogous to eq. (3.51).

The k appears as a parameter in the equation similarly to the nuclear positions in molecular Hartree-Fock theory. The solutions are continuous as a function of k, and provide a range of energies called a band,, with the total energy per unit cell being calculated by integrating over k space. Fortunately, the variation with k is rather slow for non-metallic systems, and the integration can be done numerically by including relatively few points.38 Note that the presence of the phase factors in eq. (3.76) means that the matrices in eq. (3.79) are complex quantities.

For a given value of k, the solution of eq. (3.79) provides «basis orbitals. In molecular systems, the molecular orbitals are filled with electrons according to the aufbau principle, i.e. according to energy. The same principle is used for periodic systems, and the equivalent of the molecular HOMO (highest occupied molecular orbital) is the Fermi energy level. Depending on the system, two situations can occur.

• The number of electrons is such that a certain number of (non-overlapping) bands are completely filled, while the rest are empty.

• The number of electrons is such that one (or more) band(s) are only partly filled.

The first situation is analogous to that for molecular systems having a closed-shell singlet state. The energy difference between the "top" of the highest filled band and the "bottom" of the lowest empty band is called the band gap, and is equivalent to the HOMO-LUMO gap in molecular systems. The second situation is analogous to an open-shell electronic structure for a molecular system, and corresponds to a band gap of zero. Systems with a band gap of zero are metallic, while those with a finite band gap are either insulators or semiconductors, depending on whether the band gap is large or small compared with the thermal energy kT.

As mentioned above, the basis functions within a unit cell can be either localized (Gaussian) or delocalized (plane wave) functions. For a Gaussian basis set, the computational problem of constructing the Fk matrix is closely related to the molecular cases, involving multi-dimensional integrals over kinetic and potential energy operators. The periodic boundary condition means that the terms involving the potential energy operators in eq. (3.79) become infinite sums over t vectors. Since the operators involve both positive and negative quantities and only decay as r1, they require special care to ensure convergence to a definite quantity, as for example Ewald sum39 or fast multipole methods.40 For a plane wave basis, the construction of the energy matrix can be done efficiently by using fast Fourier transform (FFT) methods for switching between the real and reciprocal space. All local potential operators are easily evaluated in real space, while the kinetic energy is just the square operator in reciprocal space. FFT methods have the big advantage that the computational cost only increases as NlnN, with N being the number of grid points in the Fourier transform.

The solution of eq. (3.79) can be done by repeated diagonalization of the Fk matrix, analogously to the situation for non-periodic systems. A plane wave basis, however, often involves several thousand functions, which means that alternative methods are used for solving the equation.

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