threshold o f the starting energ\% then the new wavefunction is accepted. It is then used
to calculate the forces and update the ion positions. Otherwise the new charge density
and wavefunction are used as the initial input and the procedure is repeated self-
consistentiy until the convergence criteria are finally met.
2.3.5 Force M odel and Geometry Optimization
The forces on the ions are calculated via the Hellmann-Feynman theorem'’^. Classically,
force is the derivative of energv" with respect to position. The simplest way to estimate
the force is to calculate it num erically by m oving an ion in all directions. F o r a system co m p o sed o f N ions this will require 6N calculations o f the energy. H ow ever, using the H ellm ann-F eynm an theorem , all forces (acting o n aU the ions) can be calculated by calling the energ\^ subroutine only once, w hich results in a considerable speed-up. T h e th eo rem states that the force on any fixed nucleus in a system o f nuclei and electrons is just the classical electrostatic attraction exerted o n the nucleus. T h e force can th ere fo re be estim ated direcdy from the w avefunction w hich has previously b een calculated.
O n e problem th at arises in force calculations is that there should be an additional term to represent the derivative o f the basis set w ith respect to the p o sition o f the ion. T his con trib u tio n to the force on the ion is called the Pulay force*^^ (also referred to as Pulay stress). I f the value o f the Pulay force is n o t calculated, there is a further e rro r in the value o f the H ellm ann-Feynm an force. Pulay forces m u st always be considered w hen localized basis sets are used. A lthough for a plane w ave basis set, the use o f pseudopotentials has been show n to substantially reduce these com plications, the Pulay forces m ust be properly treated for the case o f volum e relaxations.
T he m inim um energ}^ ionic configuration corresponds to the situation w here there are zero n et forces on each ion. T h e calculated forces are used to p e rfo rm Q u a si-N ew to n relaxations, w hich instead o f obtaining an estim ate o f the H essian m atrix (the square m atrix o f second partial derivatives) at a single point, gradually builds up an approxim ate H essian m atrix by using gradient inform ation from som e o f the previous iterates, until the forces on the atom s have converged to a set criteria.
2.4 Experimental Electronic Structure Techniques
W ithin a rigorous theoretical approach, aU the necessary inform ation relating to the underlying chem ical structure is in principle accessible. H ow ever, experim ental ap p roaches to m easure the electronic structure require a series o f com plim entary techniques to provide a substantial body o f inform ation on the electronic strucm re o f solids. T h ree such techniques, photoem ission spectroscopy, soft X-ray em ission spectroscopy and X -ray adsorption spectroscopy are detailed below.
2.4.1 Photoem ission Spectroscopy
X -ray pho to em issio n spectroscopy (XPS) was developed in the 1960’s and is based o n the p hotoelectric effect outlined by E instein in 1905, w here the con cep t o f the p h o to n was used to describe the ejection o f electrons from a surface w hen p h o to n s im pinge u p o n it. In a p h o toem ission experim ent, a solid sam ple is exposed, in vacuum , to ultraviolet (UPS) o r X-ray radiation (XPS) T he incident p h o to n s excite electrons from states below the Ferm i level to states above the vacuum level, from w here they can escape the soUd, Figure 2.6. By m easuring the kinetic energy^ and intensit)" o f the em itted electrons over a large n u m b er o f em ission directions, the binding energy' and total density o f states o f the occupied electronic states can be determ ined. H igh energy p h o to n s are used to m easure core level spectra in X-ray photoem ission spectroscopy. XPS m easurem ents o f core level binding energies aUow the chem ical state o f atom s to be detertnined due to the characteristic energies o f the core electrons. In principle, it is possible to determ ine b o th the elem ent t\^ e as weU as oxidation state. T he m o st versatile p h o to n source fo r photoem ission experim ents is synchrotron radiation, since it is continuous in energy from the UV to the hard X -ray region. T he energj- resolution available from state o f the art electron energy analyzers is less than 5 m eV for electrons w ith 15 eV kinetic energy'^'*.
0> c la