Brillouin zone sampling An infinite crystal contains an infinite number of electrons. Exploiting translational symmetry o f crystals and introducing the reciprocal lattice, it is possible to consider only the number of electrons contained within one unit cell. However, the HF or Kohn-Sham equations must be solved for
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each point of the first Brilluoin zone. The wavevector k becomes a new quantum number.
By Bloch theorem, an z-th crystal orbital with the wavevector k can be written as: 'I'ikW = e""- Wk(r) = e"" . (5.3.5.1) i.e. is a product o f a periodic function w^Cr) and a plane wave. In the last equality of (5.3.5.1) the periodic function w,^(r) is expressed as a Fourier series with coefficients Q +k (K is a reciprocal lattice vector). All the z-th orbitals with different k-vectors form a band, in which the orbital energy generally depends on the k-vector, much like phonon frequencies depend on it. In the exact solution one has to solve the one- electron equations at every k-point and perform integration over the Brillouin zone. In practice, only a finite set of k-points can be used, and it is important to achieve convergence vsdth as small a number of k-points as possible. Generally, for metals one needs many more k-points than for insulators or semiconductors. Smaller unit cells also require more k-points to be included. Monkhorst and Pack (1976) suggested a recipe, which has become the most commonly used one. They defined a uniform set of special k-points as:
k = Upb| + u^bj + Ujbj , (5.3.5.2)
where b^, bj, and bj are the reciprocal lattice vectors and Up, u,, u, are numbers from the sequence:
u^ = (2r-q-l)/2q (r=l,2,3,...q) (5.3.5.3)
The total number of ^-points is q \ but due to symmetry the number o f independent k-points can be much smaller.
Basis sets. Here I shall consider only the traditional atomic-orbital and plane wave basis sets. There exist other possibilities, e.g. hybrid basis sets of the LAPW method.
In the LCAO (Linear Combination of Atomic Orbitals) scheme one defines atom- centred orbitals as a product o f the angular (Y,„,) and radial (%(r)) parts:
4>i(r) = Y,^%(|r|) , (5.3.5.4)
where the radial part is a linear combination of either Slater-type functions:
X(|r|) = r”e-«'- (5.3.5.S)
or Gaussian-type functions:
where m, and a are parameters. Slater orbitals are more accurate and require fewer functions in an accurate representation of orbitals. However, they are much more computationally expensive, and Gaussian functions are usually preferred.
From atomic orbitals we form a set of Bloch functions, whose linear combinations give the crystal orbitals. In practice, LCAO basis sets almost always suffer from incompleteness. In molecules and crystals^ parameters (e.g., exponents) o f the orbitals (5.3.5.5) and (5.3.5.6) depend on the atomic positions, but are only optimised for one structure. Dependence of the basis set on atomic positions implies the presence of Pulay forces, which must be evaluated when optimising crystal structures. Although very economical for insulators, LCAO basis needs very many Gaussian functions for studies of metals. LCAO basis also suffers from the basis set superposition error. It is also difficult to systematically increase the basis set expressed in local orbitals.
Plane wave basis set is the most natural and general basis set for crystals, following directly from the Bloch theorem. A single plane wave is:
<t'CfK(r) = e'‘'‘+'‘'' (5.3.5.?)
This basis set is complete and very convenient for many applications. Its main shortcoming is that a huge number of plane waves are needed to describe rapidly changing wavefunctions in solids. E.g., for the valence electrons in Al, an estimated
10^ plane waves are needed to reproduce oscillations of the valence electron wavefunction in the core region. For the core electrons this problem is even much more serious - because the core electron wavefunctions are more rapidly changing and there is a cusp in the density (eq. 5.3.4) at the nucleus. There are several ways to overcome this problem - e.g., the LAPW method, PAW (Projector Augmented- Wave) method, linear muffm-tin method, and the pseudopotential method, which I shall describe below. For details of all these methods see Singh (1994), Blochl (1994), Thijssen (1999).
By construction, plane wave basis set can be used only in conjunction with periodic boundary conditions. Atoms, molecules, and surfaces can be treated in an approximate fashion by using sufficiently large unit cells preventing significant interaction between their periodic images. A very large number of plane waves is required to treat such systems.
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Pseudopotentials. T he m ain ideas o f the pseudopotential approach are: 1) to exclude chem ically inactive core electrons from explicit consideration and 2) to replace (w ithin the sphere o f the radius r j the true C oulom bic potential due to the core by a sm oother effective potential acting on the valence electrons. In this approach, w e use the frozen core approxim ation, w hich usually w orks w ell. W here this is not the case, it is alw ays possible to use ‘sm all c o re ’ pseu d o p o ten tials w ith som e o f the core orbitals treated as valence orbitals.
O utside the ‘c o re ’ radius the potential and the w avefunction are correct. W ithin the core region the w avefunction is different from the exact one in that it is n o deless and is sm oother (Fig. 5-6 and Fig. 5-7). N odes and oscillations o f the exact valence functions in the core region are due to the orthonorm ality c o n stra in ts'”.
The definition o f a pseudopotential is inherently non-unique: it depends on and other technical details. L arger result in sm oother potentials, w hich allow a sm aller num ber o f plane w aves to be used, but degrade the accuracy. In m olecular and solid- state calculations particular care m ust be taken to avoid large overlaps o f the core
Fig. 5-6. Construction of a pseudopotential. Beyond /*, the wavefunction and potential match the true all-electron ones (with modifications after Payne et al., 1992).
Valence orbitals must be orthogonal to the core electrons. If they have a different angular momentum, orthogonality is achieved automatically due to their angular parts; orbitals with the same angular momentum must be orthogonal due to the sign-changing radial part o f the wavefunction. This implies that if orbitals o f a given angular momentum appear only in the valence shell, there w ill be no radial orthogonality with the core, no radial nodes, and these valence electrons will better penetrate the core and experience very strong potentials. As a consequence, for elements where this occurs (F* row elements, 3d- and 4f-elem ents) pseudopotentials must be very hard. The way out was found in the formulation o f ultrasoft pseudopotentials (Vanderbilt, 1990).
-i
Fig. 5-7. All-electron (a) and pseudowavefunction (b) of a 5d-orbitaI in An. Solid contours - positive, dashed contours - negative wavefinction (taken from A. Rappe’s w eb-page at: http://www.sas. upenn.edu/chem/faculty/rappe/rappe.html).
regions. A t not only the w avefunction, b ut also its first derivative are exact, therefore sm all overlaps are not critical. P seudopotentials are constructed so as to m atch the all-electron eigenvalues in as m any different atom ic config u ratio n s as possible, to ensure transferability o f the pseudopotential to chem ically different system s.
U sually, pseudopotentials are generated under a constraint o f norm -conservation:
E jl P d r = X ! | | <t>i“ ( r ) P d r , (5.3,5.8)
' 0 ' 0
i.e. the pseudow avefunction gives the correct (equal to the all-electron result) n um ber o f valence electrons w ithin the core region. T hese pseudopotentials are called hard, or norm -conserving. It is possible to use m uch softer p seudopotentials (called ultrasoft, or non-norm conserving) by relaxing the n o rm -conservation co n d itio n (V anderbilt, 1990). This allow s one to use relatively large and reduce the nu m b er o f plane w aves by a factor o f ~2 w ithout any loss in accuracy. In V an derbilt p seudopotentials, one has to add the so called ‘augm entation c h a rg e ’ (w hich is the strongly atom -localised part o f the valence electron density) to (|)T'(r)|^ in order
/•
to obtain the charge density w ithin the core spheres. U ltrasoft pseu d o p o ten tials are alm ost indispensable in sim ulations o f com pounds o f ‘h a rd ’ elem ents - such as O and other U* row elem ents, Fe and other 3d-elem ents, and rare earths.
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Milman and co-workers (2000) have presented an analysis of the performance of plane wave pseudopotential calculations for compounds of almost all elements. Comparison of the performance of pseudopotential and all-electron methods was made in many works, e.g. Holzwarth et al. (1997). The general conclusion is that pseudopotential calculations are very accurate, except in cases where core polarisation effects are significant (e.g., Ca atom in CaFj). Another source of errors of pseudopotential calculations is significant overlap of the valence and core orbitals for some elements (e.g., Na). In such cases, non-linear core corrections (Louie et al., 1982) significantly improve pseudopotentials. It is important that the same functional is used in generating the pseudopotential and in performing solid-state calculations (Fuchs et al., 1998); this point, not taken into account by many researchers, was always kept in my works, all of which used the PW91 functional for both solid-state calculations and pseudopotential generation.
In conjunction with pseudopotentials, plane wave basis sets become extremely useful. Using ultrasoft pseudopotentials, it is possible to give a satisfactory description o f solids by using typically a basis set of -1 0 0 plane waves per atom. Only the plane waves with the lowest kinetic energy need to be included. The number of plane waves is controlled by the kinetic energy cut-off parameter only plane waves with the kinetic energy below are included;
^ |k+Kp < (in atomic units) (5.3.5.9)
The number of plane waves is roughly proportional to the volume of the unit cell and depends on the cut-off as ; an approximate estimate of the number of plane waves:
N = - ^ { 2 E J ’^ (5.3.5.10)
bn
differs only slightly from the actual number It must be always checked that E^„, is high enough for good convergence of results.
A plane wave basis set does not depend on atomic positions; therefore, there are no Pulay forces. However, this basis does depend on the volume when a finite number of plane waves are included". This is the origin of the so-called Pulay stress.
" There are even discontinuities in the basis set related to volum e change during structural optim isation. Such effects are significantly reduced when working with large sets o f plane w aves.
This stress is always negative, i.e. tends to compress the structure. It is nearly isotropic, has zero shear components and is roughly independent of volume:
2 dE
d\aEcm ’ (5.3.5.11)
Pulay stress affects mainly the pressure, which can be easily corrected by shifting the calculated values by a constant:
p { V )= P \iy )-^ ^ p , (5.3.5.12) where is the pressure calculated at the constant number of plane waves. The origin o f the Pulay stress is in the basis set incompleteness due to the presence of only a finite number of plane waves. A related correction to the total energy (Francis & Payne, 1990) is:
E,^ = E J N ) - \ ^ ^ \ n ^ (5.3.5.13)
3 omEcvt Nc
Increasing the basis set, it is possible to reduce the Pulay stress and errors in the total energy to arbitrarily small values; all my calculations have negligible Pulay stresses and very small total energy errors avoiding the use of the approximate equations (5.3.5.11-13). For more details of the plane wave pseudopotential method see Singh (1994) and Payne et al. (1992).
Fig. 5-8 presents valence electron distributions calculated using this method. Analysis of the charge and spin densities is a powerful tool for investigation of chemical bonding and interatomic interactions in crystals (Coppens, 1992, 1997; Tsirelson, 1986, 1993). Figure 5-8 already gives some indication of the ionicity of bonds and degree of charge transfer. More quantitative information can be obtained by integration of the charge density within a sphere and comparison o f results obtained for different systems. This is shown in Fig. 5-9.
Existim programs for crystals. A number of codes based on DFT exist for crystals. Plane-wave pseudopotential codes include VASP, CPMD, CASTEP, CETEP, PWSCF, DoD-Planewave, ABINIT. WIEN is an all-electron code based on the linearised augmented plane wave method (see Singh, 1994). SIESTA uses localised basis sets in conjunction with pseudopotentials. All the mentioned programs are based on DFT; CRYSTAL is a unique program employing the Hartree- Fock method, but also having DFT options and hybrid functionals. All these codes develop quickly; frequent updates and documentation can be found on their
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■ijûe -4JOO -9AI -iM (UK 1J 0 2.00 3Ü0 iOO D is ta n c e , Â Distance, t ) Û ’AW iM Distance. Â Distance, Â
D istance, Â -lÜO oooD istance, À
Fig. 5-8. Theoretical valence electron distributions in minerals, (a) MgO ([100] plane), (b) cubic MgSiO^ perovskite ([110] plane), (c) ferromagnetic FeO ([100] plane), (d) spin density in ferromagnetic FeO ([100] plane), (e) ferromagnetic bcc-Fe ([110] plane), (f) spin density in ferromagnetic bcc-Fe ([110] plane). The density units shown in figures are 0.01 e/A^. Contours with density >0.5 e/ (yellow regions) are not shown. M gO : most valence electrons are localised on O atoms, leaving Mg almost completely ionised. M gSiO i: M g atoms are almost fully ionised, but there is a significant electron density on Si atoms, indicating partial covalency o f the S i-0 bonds. FeO: Note ‘bridges’ o f charge density between Fe and O atoms, indicative o f som e covalency. Also note a significant spin density on the O atoms. Fe: Electron density is high everywhere, explaining the metallic conductivity o f iron. In the ferromagnetic phase, the spin density is localised on the atoms and is positive everywhere, perhaps except for small interstitial regions where it is sliglitly negative. Calculations were performed using GGA-PW91 and VASP. LevOO code (Kantorovich, 1996-2001) was used in constructing these images. Experimental structures were used for MgO (ûq = 4.211 A) and Fe {ao = 2.866 A) and theoretical structure ((aro=3.527 A) was used for the cubic MgSiO^ perovskite. For FeO calculations, ao = 4 . 2 \ \ A was used for comparison with MgO.
10.0 p--- 2 0 I in MgO In M gSlOo 2
i
I
c £ Î & O) I Ü 0.0 0 0 0.0 0 3 0.6 0 3 Radius, Â 8 Radius, Â b a 10.0 10.0 in m etallic Fe in FeO 8.0 ÿ 0)I
I
.Ç c z i 4.0 & Ü 2.0 0.6 0.9 Radius, Â 13 0 3 03 Radius, A 1 2 dc
F ig . 5 -9 . A m o u n t o f c h a r g e in s id e a t o m i c s p h e r e s in m i n e r a l s t r u c t u r e s , (a) in MgSiO.i perovskite, (b) Mg in MgO and MgSiO) perovskite, (c) O in MgO, MgSiO^, and FeO, (d) Fe in FeO and metallic Fe. Arrows show the ionic radii (0.72 A for Mg^" and 1.40 A for 0“', 0.40 A for Si”^ , and 0.78 A for the high-spin Fe“ ). Integrating charge within the spheres with these radii gives the atomic charges o f Mg (+1.94 in MgO, +1.96 in M gSiOj), Si (+3.98), Fe (+3.27 in FeO and Fe), and O (-1 .4 in MgO, -1.88 in MgSiOg, -2.00 in FeO). These figures, based on the same calculations as Fig. 5- 8, show that: 1 ) atoms have nearly formal charges within the ionic radius spheres, 2) the sum o f thus determined charges is not zero, because o f the interstitial regions, 3) there is a striking difference between charge distributions around Mg in MgO and M gSiO] and a striking similarity between charge distributions around the Fe atom in both Fe metal and FeO. The latter lends some support to the model o f a metal as a system o f ionic cores immersed in the sea o f the free electrons; FeO is an ionic compound. The horizontal line in (c) indicates the full ionic charge. These images were created using the LevOO code (Kantorovich, 1996-2001).web-pages, which can be accessed from my home page
http://slamdunk. geol. ucl.ac. uk/~artem/7. html).
VASP (Vienna Ab initio Simulation Package: Kresse & Fuithmüller, 1996a,b) code is the main tool used in my work. It is based on DFT within the plane wave pseudopotentials method, and has both static and molecular dynamics regimes. There are numerous options of static structural relaxation in VASP, the most robust of which is the conjugate gradients method (see Payne et al., 1992). For studies of metals, a number of ‘smearing’ methods are available, in which an electronic temperature is specified and the electronic entropy is computed from the density of
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states. Molecular dynamics can be performed with VASP only in the NVE and NVT
ensembles. Equations of motion are solved using the Verlet algorithm; the constant- temperature molecular dynamics is based on the method of Nose (1984). The program uses pseudopotentials of several types (including the ultrasoft ones), very efficient matrix diagonalisation and charge density mixing algorithms; it is very effectively parallelised, and can be used to study relatively large systems. Pseudopotentials are generated by a scalar relativistic code; a library of pseudopotentials for all elements is available.