Density Functional Theory Calculations; The CASTEP Code

Code

The advancement of computational calculations and NMR experiments has led to a symbiotic relationship between experimental spectra and calculated parameters in

recent years. There is a clear need for quantitative theoretical support to all major experiments. First principle quantum mechanical calculations methods have the po- tential to provide information regarding the expected parameters you will see during the NMR experiment. This gives an insight into what experiments maybe needed to characterise a sample or whether NMR is the correct tool for characterisation.

Density functional theory was developed by Kohn10 in the 1960s. [134, 135] The main discovery was that the total electronic energy of a system in an external potential is a unique function proportional to the ground state density. Hence, if the density is known then one should be able to calculate the energy of the system. Unfortunately this ground state density is unknown, but with relatively educated guesses you can obtain excellent results. The particular external potential we are interested in is that generated by nuclei, but this is not necessary for density functional theory itself. The ground state density of the system can be estimated by generating a suitable many-body wavefunction, in principle this is a function of time which takes into consideration the coordinates of the nucleus and electrons. [49]

As the number of nuclear and electronic variables can be large a Born- Oppenheimer approximation is used to decouple the electrons from the nucleus. Hence it is presumed that the electron wavefunction is independent of the nuclear wavefunction. As the electronic wavefunction only depends on the instantaneous nuclear orientation and not its time then the wavefunction of the electronic compo- nent can be described using the time-independent Schrodinger equation. The added advantage of this is the nucleus is massive enough to be treated semi-classically and can be presumed to respond to the electrons in a Newtonian approach. 6

It is important to take into consideration the crystal like arrangement of the sample when applying a wavefunction to it. In a periodic system any wavefunction must be the product of a cell part and a phase factor, this is known as Bloch’s theorem. This is required in order to maintain the translational symmetry of the ground state density. The phase factor takes the form of a plane wave, whose wavevector is a linear combination of reciprocal lattice vectors. From this a plane- wave basis is used to express the wavefunction; both the phase and the cell part. This means that the basis functions are orthogonal and it is possible to Fourier transform between real and reciprocal space.

The Bloch states are labeled by their crystal momentum given the symbol

κ. The problem of solving for an infinite number of electrons has become one of

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This approximation can fall short with small nuclei such as proton and lithium, with the zero point of motion can have substantial effects on the ground state

calculating for a finite number of bands at an infinite number ofκ-points. This is surpassed as the physical properties of the studied system are expected to smoothly vary withκ and hence many integrals can be well approximated by finite sampling ofκ.

If the system studied is not periodic (i.e. contains Frenkel type defects, impurities or disordered) then a supercell is required to be constructed. This is a larger unit cell with a space region to separate the region of interest from the periodic images. The space is usually just supplied as a vacuum. By using these supercells to represent the disorder in the system it still gives the advantages of Bloch’s theorem and limit the size of the calculation. However, they still tend to be enormous calculations which require large national supercomputers.

As the electrons in the system are orthogonal to every other state (due to their Fermi-ionic nature) then as the higher energy states become filled the orthog- onal nature forces the wavefunctions of these states to have increasing numbers of nodes. From this the width of the Fourier spectrum increases and the number of plane-waves needed to represent our wavefunction increase. To limit this, many of the nodes are presumed to be in the core region(near the nucleus). The core region is presumed to contain relatively constant (even with changing chemical environment). Hence the lower energy states in these regions are removed and replaced with an effective potential. This is combined with the nuclear Coulomb potential to create a pseudopotential. The remaining higher states are described by the wavefunctions of the remaining orthogonal electrons, this reduces the number of nodes and the size of the calculation. The use of pseudopotentials dramatically reduces the number of plane waves required to represent the wavefunctions, whilst still giving accurate results.

The major advantages of DFT is that is offers very good scaling with com- putational cost making it an ideal choice for large periodic systems. Also, given the large number of calculations the likely accuracy of property prediction for many properties of differing systems is already known. The obvious disadvantages is that it only applies to ground-state, with many-approximations made and from this it is difficult to improve the accuracy and predict the errors of the calculations.

CASTEP (CAmbridge Serial Total Energy Package) is a code developed to complete first principle calculations to predict the properties (for the purpose of this thesis only the magnetic response is desired) in various materials with periodic order. [136] This utilizes a gauge including projector augmented waves (GIPAW) basis, which maintains the core properties with all-electron accuracy.

orbital current around a nuclear site and to obtain a chemical shielding by Biot- Savart Law. [137] For quadrupolar nuclei only the electric field gradient (EFG) calculation is required and hence a far easier calculation. A minor concern is that the chemical shielding is given with respect to the bare nucleus. Hence, to obtain an isotropic shift an external reference is required to be run for a known isotropic shift, this can prove difficult as most quadrupolar nuclei are shifted by δQIS. If multiple peaks are present then internal referencing can be used or an external reference with a known δiso and is not shifted by 2nd order CQ can also be used. [138, 139] CASTEP calculates the EFG by summing the three discrete terms, first there is a contribution arising from the ionic charge (which is the sum of the nuclear and core electron charge, this appears as an infinite lattice of point charges whose EFG can be obtained by an Ewald summation. [140] 7 Secondly there is a contribution of the valence charge density and finally PAW contribution to account for differences between then pseudo and all electron density charges.

In metallic systems the shielding arises from the previously explained shield- ing and Knight shift. It is possible to use a planewave-pseudopotential approach to compute these shieldings in metals. This is done by GIPAW to calculate the orbital response and PAW to calculate the Knight shift. These calculations are far more demanding than for insulators as you require fineκ-point sampling. For heavy nu- clei relativistic effects dominate and the picture become slightly more difficult. [141] The Schrodinger equation ( ˆHΨ =EΨ) is nonrelativistic meaning it is validated for particles whose velocity is much slower than the speed of light. This approxima- tion fails as you move to heavier elements which contain relativistic effects; these give significant influences on the bonding and hence the structural properties of the materials. This in turn has a significant influence on their magnetic response. To attempt to treat these heavier materials it is more insightful to base a description on the Dirac equation ([iγµ·∂µ−m]Ψ = 0). This is able to add the special relativity onto the quantum states, the main method for employing this is to use the pseu- dopotential to simply add these effects. In the construction of a pseudopotential the free atom is treated relativistically to give the correct eigenvalues, in insulators pseudopotentials the spin-orbit split states are usually averaged however maintain these splits in the potential hence creating a relativistic pseudopotential. The heavy metal Knight shifted samples present in this thesis have so far proven too difficult to

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An Ewald summation is a special case of the Poisson summation formula (Fourier series coeffi- cients of the periodic summation of a function to values of the function’s continuous Fourier trans- form), replacing the summation of interaction energies in real space with an equivalent summation in Fourier space. The advantage of this approach is the rapid convergence of the Fourier-space summation compared to its real-space equivalent when the real-space interactions are long-range.

produce a working relativistic pseudopotential for, hence no CASTEP calculations are present for that work.