Exact Exchange in Density Functional Calculations

More documents

Recommendations

Info

18 3. Density Functional TheoryModeling the nuclei density by a homogeneous background charge (a jellium), the eigenstatesof the non-interacting (Kohn-Sham) Hamiltonian becomes plane waves, for which it can be shownshow that the kinetic- and exchange energies becomeT s = 3V10 (3π2 ) 2/3 n 5/30 E x = − 3V (3π2 ) 1/3 4/3 n 04πwhere n 0 is the (homogeneous) density of the electrons and V is the volume of the system.The basic idea of Thomas-Fermi theory is that if the charge density is not uniform, but variesslowly, the energy expressions will be the same as above, but evaluated locally and then integratedover space. Thus the approximation for the kinetic energy functional, T s [n], and the exchangefunctional, E x [n], becomes:∫T s [n] = dr 3 10 (3π2 ) 2/3 n 5/3 (r)∫E x [n] = −Neglecting correlation altogether, the total energy functional thus becomesE[n] = T s [n] + U H [n] + V ext [n] + E x [n]∫= dr 3 10 (3π2 ) 2/3 n 5/3 (r) + 1 ∫drdr ′ n(r)n(r′ )2 |r − r ′ |∫+dr 3 (3π2 ) 1/3n 4/3 (r)4π∫drn(r)v ext (r) −dr 3 4( 3π) 1/3n 4/3 (r)Making explicit density-functional approximations of all components of the HK theorem is amajor simplification, as the minimization of the energy functional can then be done directly byapplying the Euler-Lagrange equation δE[n]/δn(r) = µ, where µ is a Lagrange multiplier (thechemical potential), which is adjusted such that ∫ drn(r) = N. This results in∫12 (3π2 ) 2/3 n 5/3 (r) +dr ′ n(r ′ ( ) 1/3)3|r − r ′ | + v ext(r) − n 4/3 (r) = µ (3.29)πFrom which the ground state density can be determined, and thereby also the ground state energy.Unfortunately the Thomas-Fermi model makes very poor predictions of the energetics of realsystems. The problem is that the kinetic energy is the dominant energy term, making it crucialthat this is described correctly, but assuming that the electron structure is a homogeneous electrongas, all information on the formation of electronic shells near the nuclei, is obviously lost. Suchstructure is naturally included, when the kinetic energy is determined by solving the Schrödingerequation of a real system (free electron gas), as in the KS scheme.The original model proposed independently by Thomas in 1927 [10] and Fermi in 1928 [11]included only the kinetic part, and among other failures predicts that formation of moleculesis always energetically unfavorable, i.e. that all molecules are unstable. The inclusion of theexchange term was done by P. A. M. Dirac in 1930 [12], and the resulting model makes even worseresults. Because of the computational advantages of the model, several attempts have been madeto improve the model, but none coming close in accuracy to KS-type schemes. For a nice reviewof different TF models, and the systems they can describe see [13].It should be noted that TF theory predates DFT, which is based on the two articles by Hohenbergand Kohn [4] from 1964 and by Kohn and Sham [5] from 1965, by more than threedecades.3.2.2 Comparison of KS and hybrid HF-KS SchemesTo compare the KS scheme with the hybrid HF-KS schemes, we compare the two energy expressions:∫E KS = T KS [n] + U H [n] + ExKS [n] + Ec KS [n] + drn(r)v ext (r)∫E Hy,a = T Hy,a [n] + U H [n] + aExexact [Φ Hy,a ] + (1 − a)ExHy,a [n] + Ec Hy,a [n] + drn(r)v ext (r)
3.3 Exchange and Correlation 19There is a subtle difference between the functionals that the approximated exchange and correlationfunctionals of the two methods are designed to model. This difference stems from thedifference between the KS and the hybrid HF-KS orbitals. We can express the hybrid functionalsin terms of the KS functionals byE Hy,axE Hy,ac= E KSx + ∆E x , ∆E x = 〈Φ Hy,a |̂V ee |Φ Hy,a 〉 − 〈Φ KS |̂V ee |Φ KS 〉 (3.30)= E KSc + ∆E c , ∆E c = 〈Φ KS | ̂T + ̂V ee |Φ KS 〉 − 〈Φ Hy,a | ̂T + ̂V ee |Φ Hy,a 〉 (3.31)Usually the density functional approximations for the exchange and correlation used in hybridschemes are just those established within KS theory. The error caused by this is∆E Hy,axc= (1 − a)∆E x + ∆E c= 〈Φ KS | ̂T + âV ee |Φ KS 〉 − 〈Φ Hy,a | ̂T + âV ee |Φ Hy,a 〉Using the Hellmann-Feynman theorem, and that ∆E Hy,0xc∆E Hy,axc ==∫ a0∫ a0= 0, this can be written as[]dα 〈Φ KS |̂V ee |Φ KS 〉 − 〈Φ Hy,α |̂V ee |Φ Hy,α 〉dα [ Exexact [Φ KS ] − Ex exact [Φ Hy,α ] ] (3.32)which is believed to be small in general [9]. From the success of the hybrid HF-KS methods usingstandard xc functionals, it might seem that these are actually better approximations of ExcHy,a thanthe functional ExcKS which they where designed to approximate.3.3 Exchange and CorrelationThe Hartree and Exchange terms of the total energy is defined by the expression (3.19):U H [n] + E x [n] = 〈Φ minn(r) |̂V ee |Φ minn(r) 〉 (3.33)If one considers the electron-electron interaction as a perturbation to the non-interacting Hamiltonian,we can interpret the above quantity as the first order correction to the total energy due tothis perturbation. The explicit form of U H isU H [n] = 1 2∫drdr ′ n(r)n(r′ )|r − r ′ |= 1 2∑nn ′ f n f n ′∫drdr ′ φ∗ n(r)φ n (r)φ ∗ n ′(r′ )φ n ′(r ′ )|r − r ′ |(3.34)which is just the classical electrostatic energy of the charge density n(r). This interaction clearlyincludes a spurious self-interaction of each state with itself.The explicit form of the exchange energy is∑∫E x [n] = − 1 2f n f n ′δ σnσ n ′nn ′drdr ′ φ∗ n(r)φ n ′(r)φ ∗ n ′(r′ )φ n (r ′ )|r − r ′ |(3.35)Here spin has been included, according to the prescription in section 2.4, as it is crucial for theinterpretation of exchange.The object ‘a Slater determinant’ is constructed to satisfy the antisymmetry principle, whichis the principle leading to Pauli repulsion, so this effect is expected to show in the expression〈Φ|̂V ee |Φ〉. In the exchange energy, (3.35), we see that the diagonal terms, n = n ′ , exactly cancelsthe self-interaction of the Hartree term, and is thus just a classical correction of this. The remainingn ≠ n ′ parts of E x is a direct consequence of Pauli repulsion. It can be seen that E x correlatespairs of states with parallel spin, and gives a large negative value for any such pair, which havenon-zero overlap. This is because, in the true ground state, the Pauli exclusion principle would
Page 1: Exact Exchange in DensityFunctional
Page 5: AbstractThis thesis discuss the the
Page 8 and 9: viii
Page 11 and 12: Chapter 1IntroductionElectronic Str
Page 13: 3magnetic and electric field streng
Page 16 and 17: 6 2. Basic Wave Function TheoryIt d
Page 18 and 19: 8 2. Basic Wave Function Theory2.3
Page 21 and 22: Chapter 3Density Functional TheoryD
Page 23 and 24: 3.2 The Generalized Kohn-Sham Schem
Page 25 and 26: 3.2 The Generalized Kohn-Sham Schem
Page 27: 3.2 The Generalized Kohn-Sham Schem
Page 31 and 32: 3.3 Exchange and Correlation 21Figu
Page 33 and 34: 3.3 Exchange and Correlation 23is t
Page 35 and 36: 3.3 Exchange and Correlation 25The
Page 37 and 38: Chapter 4Extended SystemsAlthough w
Page 39: 4.2 Basis Sets and Boundary Conditi
Page 42 and 43: 32 5. Orbital Dependent Functionals
Page 44 and 45: 34 5. Orbital Dependent Functionals
Page 47 and 48: Chapter 6Projector Augmented WaveMe
Page 49 and 50: 6.1 The Transformation Operator 39i
Page 51 and 52: 6.4 Densities 41For local operators
Page 53 and 54: 6.5 Total Energies 43where the nota
Page 55 and 56: 6.5 Total Energies 45i.e. that the
Page 57 and 58: 6.6 The Transformed Kohn-Sham Equat
Page 59 and 60: 6.7 Exact Exchange in PAW 49The Val
Page 61 and 62: 6.7 Exact Exchange in PAW 51where t
Page 63 and 64: 6.8 Summary 53where∫ṽ nn ′(r)
Page 65 and 66: Chapter 7Implementing PAWFor an imp
Page 67 and 68: 7.2 The Atomic Data Set of PAW 57Th
Page 69 and 70: 7.2 The Atomic Data Set of PAW 59Th
Page 71 and 72: Chapter 8Numerical ResultsThe main
Page 73 and 74: 8.2 Molecules 63Figure 8.1: Compari
Page 75 and 76: 8.2 Molecules 65Mol Expt LDA PBE
Page 77 and 78: 8.3 Miscellaneous 67Poisson equatio
Page 79 and 80:
8.3 Miscellaneous 69And the express
Page 81:
HG¡"!$#&%')(+*,*-/.$0'#&%1"(+*,*>>
Page 85 and 86:
References[1] J. P. Perdew, S. Kurt
Page 87 and 88:
REFERENCES 77[26] P. E. Blöchl, C.
Page 89 and 90:
REFERENCES 79[52] A. GörlingOrbita
Page 91 and 92:
Appendix AExact ExchangeThe non-int
Page 93 and 94:
Appendix BWannier Functions and Exa
Page 95 and 96:
Appendix CFourier TransformTo repre
Page 97 and 98:
Appendix DMultipoles and SphericalH
Page 99 and 100:
D.3 Spherical Harmonics 89Generaliz
Page 101 and 102:
D.3 Spherical Harmonics 91The RSH s
Page 103:
D.3 Spherical Harmonics 93L l m ȳl
Page 106 and 107:
96 E. Approximations of the Exchang
Page 109 and 110:
Appendix FForces in PAWIn the groun
Page 111 and 112:
Appendix GThe External Potential in
show all

Exact Exchange in Density Functional Calculations

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?