Background. ⢠1920s: Introduction of the Thomas-Fermi model. ..... Perdew, J.P.; Burke, K.; Ernzerhof, M. "Generalized Gradient Approximation Made. Simple" ...
Density Functional Theory
Density Functional Theory (DFT) DFT is an alternative approach to the theory of electronic structure; electron density plays a central role in DFT. Why a new theory? HF method scales as CI methods scale as MPn methods scale as CC methods scale as
N4 N6-N10 >N5 >N6
(N - # of basis functions)
Correlated methods are not feasible for medium and large sized molecules!
1 2 − 2 ∇i + v i Φi (r) + N
N
j =1
1 Φ j (r' ) Φi (r)dτ j − r - r' 2
1 Φ (r' )Φi (r' ) Φ j (r)dτ j = εi Φi (r) r - r' j =1 * j
“Density functional theory” Search in Web of Science 1986 50 items 1996 1,100 items 2006 5,600 items 2015 12,900 items 2016~
8,200 items
2006~2016 135,600 items Using Google Scholar > 1,300,000 items The electron density
- it is the central quantity in DFT - is defined as:
Background •
1920s: Introduction of the Thomas-Fermi model.
•
1964: Hohenberg-Kohn paper proving existence of exact DF.
•
1965: Kohn-Sham scheme introduced.
•
1970s and early 80s: LDA. DFT becomes useful.
•
1985: Incorporation of DFT into molecular dynamics (Car-Parrinello) (Now one of PRL’s top 10 cited papers).
•
1988: Becke and LYP functionals. DFT useful for some chemistry.
•
1998: Nobel prize awarded to Walter Kohn in chemistry for development of DFT.
1 2 − 2 ∇i + v i Φi (r) + N
N
j =1
1 Φ j (r' ) Φi (r)dτ j − r - r' 2
1 Φ (r' )Φi (r' ) Φ j (r)dτ j = εi Φi (r) r - r' j =1 * j
Basic Theory:
The electron density is the essential
Properties of the electron density
Function: y=f(x)
ρ= ρ(x,y,z)
Functional: y=F[f(x)]
E=F[ρ(x,y,z)]
Hohenberg–Kohn Theorems
ρ(r)
ρ(r)dr = N First HK Theorem (HK1)
N
ν(r)
The external potential Vext(r) is (to within a constant) a unique functional of ρ(r). Since, in turn Vext(r) fixes H, the full many particle ground state is a uniquefunctional of ρ(r). Thus, the electron density uniquely determines the Hamiltonian operator and thus all the properties of the system.
H
Hˆ Ψ = EΨ
E
Proof:
Ψ’ as a test function for H:
Ψ as a testfunction for H’:
Summing up the last two inequalities: Contradiction!
Variational Principle in DFT Second HK Theorem The functional that delivers the ground state energy of the system, delivers the lowest energy if and only if the input density is the true ground state density. Total energy function:
- variational principle For any trial density ρ(r), which satisfies the necessary boundary conditions such as: ρ(r)≥0 and and which is associated with some external potential Vext, the energy obtained from the functional of FHK represents an upper bound to the true ground state energy E0.
First attempt: Thomas-Fermi model (1927) L.H. Thomas, Proc. Camb. Phil. Soc., 23, 542-548 (1927) 3 2 2/3 5/3 E. Fermi, Rend. Acad., Lincei, 6, 602-607 (1927) TTF [ρ(r )] = (3π ) ρ (r )dr 10 ρ ( r )ρ ( r 3 ρ(r ) 1 1 2) 2 2/3 5/3 (3π ) ETF [ρ(r )] = ρ (r )dr − Z r dr + 2 r 12 dr1dr2 10
Problem in TF approximation: ignore correlation, and use local density approximation
Kohn and Sham (1965) T[ρ]
– kinetic energy of the system
Kohn and Sham proposed to calculate the exact kinetic energy of a non-interacting system with the same density as for the real interacting system. – kinetic energy of a fictitious non-interacting system of the same density ρ(r) Ψi - are the orbitals for the non-interacting system (KS orbitals) Tks is not equal to the true kinetic energy of the system, but contains the major fraction of it T = Tks + (T-Tks)
1 N TKS = − Ψi ∇2 Ψi 2 i=1
TKS
Hohenberg-Kohn (1964) and Kohn-Sham (1965) ---Modern DFT
FHK [ρ] = TKS[ρ] + J[ρ] + Enon-cl[ρ] E[ρ] = ENe [ρ] + TKS[ρ] + J[ρ] + Exc [ρ] = N
2 ZA ϕi (r1 ) dr1 A=1 r1A M
- i=1
1 N − ϕi ∇2 ϕi 2 i=1 2 2 1 1 N N ϕ j (r2 ) dr1dr2 + ϕi (r1 ) 2 i=1 j=1 r12
+ Exc [ρ]
Exc[ρ] includes everything which is unknown: - exchange energy - correlation energy - correction of kinetic energy (T-TKS)
Question: How can we uniquely determine the orbitals in our non-interacting reference system? How can we define a potential such that it provides us with a slater determinant which is characterized by the same density as our real system?
Kohn-Sham Equations: Minimize E[ρ] with the conditions:
ρ(r)dr = N ϕi ϕ j = δij
M 1 2 ZA ρ(r2 ) − ∇ + + − dr v (r ) 2 xc 1 ϕi = εiϕi r12 A=1 r1A 2
with:
Kohn-Sham Formalism
Kohn-Sham equations
1 2 − ∇ + v(r) + ρ(r') dr' − Ki (r) ϕ j = εjϕ j r − r' 2 i Hartree-Fock equations
Kohn-Sham Orbitals The orbitals satisfying the Kohn-Sham orbitals have no physical significance. Their only connection to the real world is that the sum of their squares add up to the exact density. However, many authors recommend the KS orbitals as legitimate tools in qualitative MO considerations and this is due to the fact that the KS orbitals are not only associated with a one electron potential which includes all non-classical effects, but they are also consistent with the exact ground state density. Thus, in a sense, these orbitals are in a sense much closer to the real systems than the HF orbitals that neither reflect correlation effects nor do they yield the exact density. On the other hand, the Slater determinant generated from the KS orbitals will not be confused with the true many-electron wave function! The exact wave function of the target system is simply not available in density functional theory! Accordingly, the eigenvalues εi connected to the KS orbitals do not have a strict physical meaning. In Kohn-Sham theory there is no equivalent of Koopmans’ theorem, which could relate orbital energies to ionization energies. There is one exception though: as a direct consequence of the long range behavior of the charge density (its asymptotic exponential decay for large distances from all nuclei) the eigenvalue of the highest occupied orbital, εmax, of the KS orbitals equals the negative of the exact ionization energy. This holds strictly only for εmax resulting from the exact VXC, not for solutions obtained with approximations to the exchange-correlation potential.
Exc[ρ] = ?? Local Density Approximation (LDA) – uniform electron gas
Exc [ρ] = ρ(r)εxc (ρ(r))dr
The exchange energy is about ten times larger than correlation in “standard” systems
εxc – the exchange-correlation energy per particle of a uniform electron gas of density ρ(r) - depends on the density at r split into exchange and correlation contributions represents the exchange energy of an electron in a uniform electron gas of a particular density
Slater exchange functional (S) For the correlation part: Monte-Carlo simulations of the homogenous electron gas – Ceperly and Alder --- interpolation of these results analytical expressions of εc
Vosko, Wilk & Nusair (1980) most widely used LDA → SVWN Perdew &Wang (1992)
Local Spin Density Approximation (LSDA) - variant of LDA for unrestricted formalism (open-shell systems) Two spin-densities:
Performance of LDA (LSDA) • • •
for atoms and molecules the exchange energy is usually underestimated by 10%, but this is compensated by an overestimation of correlation by 2 or 3 times. underbind core electrons and overbind atoms in molecules not able to reproduce the effects of bond breaking and forming
Molecules do not resemble a uniform electron gas!
LDA • Used by physicists for 40 years. Exc [ρ] = ρ(r)εxc (ρ(r))dr
• εxc(n) for homogenous electron gas. – exchange-correlation energy per electron
• Assumption: grad n is small in some sense. • Accurate for nearly homogeneous system and for limit of large density.
Limitations • • • • •
Band gap problem Overbinding (cohesive energies 10-20% error). High spin states. Hydrogen bonds/weak interactions Graphite
GEA Gradient Expansion Approximation • Early method attempt to go beyond the LDA. • Based on the idea that for slowly varying density, we could develop an expansion:
ε xc [n, ∇n ] = ε LDA [n] + f1 (n ) ∇n + f 2 (n ) ∇n
In fact the first order term is zero. Made things much worse. Why?
2
Exchange-Correlation Hole • Due to phenomena of exchange there is a depletion of density (of the same spin) around each electron. • Mathematically described as ρ xc (r , r′) • The exchange correlation energy written as
Exc =
ρ (r )ρ xc (r, r′) r − r′
drdr′
Properties of the hole • Subject of much research.
ρ xc (r, r ') = ρ x (r, r ') + ρc (r, r ') ρ x (r, r ') ≤ 0
ρ x (r, r') dr' = −1 ρc (r, r') dr' = 0
The LDA must obey these. The GEA does not need to.
Why is this important? • Huge error made to the integral would occur if the hole is not normalised correctly. • The LDA has this correct – it is the correct expression for a proper physical system. • In fact, only need the spherical average of the hole is needed.
Gunnarsson and Lundqvist [1976].
GGA idea • • • •
A brute force fix. If rx(r,r’)>0, set it to zero. If sum rule violated, truncate the hole. Resulting expressions look like:
E x [GGA] = Ex [LSDA] − F (sσ ) nσ4 / 3dr
•
Note that ss is large when
sσ =
∇nσ (r ) nσ4 / 3 (r )
– Gradient is big – n is low (exponential tails; surfaces)
•
ss is small when
– Gradient is small – n is large (including core regions)
•
Sometimes written as enhancement factor.
σ
Generalized Gradient Approximation (GGA) Exc [ρ] = ρ(r)εxc (ρ(r), ∇ρ(r),...)dr
to account for the non-homogeneity of the true electron density → gradient
εxc depends on the density and its gradient at r GGA EXC is usually split into its exchange and correlation contributions:
GGA is not always better than LDA Sometimes, worest - the reduced gradient density - interpreted as a local inhomogeneity parameter - it has large values for large gradients and in regions of small densities - it is zero for the homogenous electron gas (1)
Adjust εxc such that it satisfies all (or most) known properties of the exchangecorrelation hole and energy. PW91, PBE… (2) Fit εxc to a large data-set own exactly known binding energies of atoms and molecules. BLYP, OLYP, HCTH…
Forms of F for exchange functionals First class (A.D. Becke, Phys. Rev. A, 38, 3098, 1988)
β= 0.0042 – empirical Derived functionals: FT97, PW91, CAM(A) and CAM(B) Second class (A. D. Becke, J. Chem. Phys 84, 4524, 1986; J. P. Perdew, Phys. Rev. B 33, 8822, 1986;J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett 77, 3865, 1996)
Derived functionals: B86, P, PBE Correlation functionals P86, PW91, LYP
PW91
Perdew, J. P.; Chevary, J. A.; Vosko, S. H.; Jackson, K. A.; Pederson, M. R.; Singh, D. J.; Fiolhais, C. "Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation", Phys. Rev. B, 46, 6671-6687 (1992).
PBE
Perdew-Burke-Ernzerhof functional Give essentially the same results as the PW91 functional but it is more robust in systems with rapidly varying electron density
•
The physics stable: – Principled, parameter free – Numerous analytic properties – Slow varying limit should give LDA response. This requires Fx →ms2 , m=0.21951 – Density scaling, n(r)→l3n(lr), Ex→l Ex
κ , κ = 0.804 F (s ) = κ − 2 1 + μs / κ Perdew, J.P.; Burke, K.; Ernzerhof, M. "Generalized Gradient Approximation Made Simple", Phys. Rev. Lett., 77, 3865-3868 (1996).
Revised Perdew-Burke-Ernzerhof functional improve the DFT description of the adsorption energies of molecules on metallic surfaces Marlo, M.; Milman, V. "Density-functional study of bulk and surface properties of titanium nitride using different exchange-correlation functionals", Phys. Rev. B, 62, 2899-2907 (2000).
PBEsol improve the first-principles gradient expansion for exchange over a wide range of density gradients, and hence to improve the description of equilibrium properties of densely packed solids and their surfaces.
Correlation Functionals • Perdew - Zunger 1986 • Perdew Wang (1991) – Part of parameter free PW91
• Perdew, Burke, Ernzerhof (1996) – GGA made simple! – Parameter free – Simplified construction – Smoother, better behaved.
Exchange-Correlation functionals
BLYP
• • • •
Different approach – based on accurate wave functions for the Helium atom. No relation to the homogeneous electron gas at all. One empirical parameter Often combined with Becke exchange to give BLYP.
Atomisation energies (kcal/mol) H2 CH4 C2H2 C2H4 N2 O2 F2
HF 84 328 294 428 115 33 -37
LSD 113 462 460 633 267 175 78
PBE 105 420 415 571 243 144 53
EX 109 419 405 563 229 121 39
Hybrid functionals • Why not just add correlation to HF calculations? We could write EXC=EX[exact]+EC[LSD]
• Correct XC hole is localised. • Exchange and correlation separately are delocalised. • DFT in LDA and GGA give localised expressions for both parts. • Sometimes simpler is better!
Hybrid Functionals Since EX>>EC, an accurate expression for the exchange functional is a prerequisite for obtaining meaningful results from density functional theory. KS GGA Ehyb [ρ ] = αE + (1 − α)E x xc xc
EXKS-the exact exchange calculated with the exact KS wave function α- fitting parameter
B3LYP, B1LYP, mPW0, PBE0, HSE03/06
Meta-GGAs • • • • •
Perdew 1999 Better total energies. 2 Ingredients: ∇ n , KE density Very hard to find potential, so cannot do SCF with this. Therefore structural optimisation not possible.
HSE03/06 Recent development. Several motivations: • B3LYP more accurate than BLYP. Some admixture of exchange needed. • Exact exchange is slow to calculate. • Linear scaling K-builds don’t scale linearly in general. • Plane wave based (physics) codes can’t easily find exact exchange.
Screened Exchange • Key idea (Heyd, Scuseria 2003):
1 erfc(ωr ) erf (ωr ) = + r r r
• First term is short-ranged; second long ranged. • w=0 gives full 1/r potential. • How to incorporate into a functional?
HSE03 E
PBE 0 x
+ (1 − a )E
= aE
HF x
= aE
HF , SR x
+ aE
PBE x
HF , LR x
+ (1 − a )E
(1 − a )E ≈ aE =E
HF , SR x
PBE x
+ (1 − a )E
[
+a E
HF , SR x
PBE , SR x
−E
+E
PBE , SR x
]
PBE , SR x
PBE , LR x
PBE , LR x
+
Where does this leave us? • Need to find short-ranged HF contribution. – – – –
Linear scaling Parallelism is perfect Will not be time consuming for large systems. Can also do with different splittings with only minor modification:
1 exp(− ω 2 r 2 ) 1 − exp(− ω 2 r 2 ) = + r r r
Where does this leave us? • Need short ranged part of PBE exchange energy. Approach this from the standard expression:
E xc =
ρ (r )ρ xc (r, r′) r − r′
drdr′
Modify the interaction to short ranged term Need explicit expression for the hole. Provided by work of EP (1998).
How about the accuracy? • Enthalpies of formation (kcal/mol):
B3LYP PBE PBE0 HSE03
MAE(G2) 3.04 17.19 5.15 4.64
Conclusion: competitive with hybrids.
MAE(G3) 4.31 22.88 7.29 6.57
How about the accuracy? • Vibrational freqs (cm-1); 82 diatomics
B3LYP PBE PBE0 HSE03
MAE(G2) 33.5 42.0 43.6 43.9
Conclusion: competitive with hybrids.
How about the accuracy? • Band Gaps (eV)
C Si Ge GaAs GaN MgO
LDA 4.23 0.59 0.00 0.43 2.09 4.92
PBE 4.17 0.75 0.00 0.19 1.70 4.34
HSE 5.49 1.28 0.56 1.21 3.21 6.50
EXP 5.48 1.17 0.74 1.52 3.50 7.22
Has HSE got legs? • • • • •
Different separations? Improved formalism for GGA then possible. Standard applications: ZnO, Ge etc. Effect on spectral calculations: EELS Possibility of multiplet calculations for defect centres.
Exchange and Correlation Functionals
In practice: BLYP, B3LYP, BPW91, …
Different functionals for different properties - Structure: bond lengths, bond angles, dihedrals - Vibrational frequencies: wavenumbers, IR intensity, Raman activity - Kinetics: barrier heights - Thermochemistry: atomization energies, binding energies, ionization potentials, electron affinities, heats of formation - Non-bonded interactions: stacking, hydrogen bonding, charge transfer, weak interactions, dipole interactions, π -π interactions What functional should I use?! Depends on: - your problem (system, the property investigated) - availability and the computational costs Atomization energies: Ionization energy: - B3LYP – the best! Electron afinities: Vibrational frequencies: - (BLYP), B3LYP, … MO5-2X - bond dissociation energies, stacking and hydrogen-bonding interactions in nucleobase pairs
Exchange-Correlation Functional
交换相关泛函的“好”与“不好”并不在于 泛函本身，而在于如何在描述合适的体系使 用正确的泛函
Some known limitations of DFT Unknown the exact functional an intrinsic uncertainty in energy between DFT and true ground state energy , and there is no direct way to estimate the magnitude of this uncertainty Limited accuracy in the calculation of excited states. Underestimated the band gaps of semiconductor Inaccurate results in week van der Waals attractions Failure to describe some strong-correlated system
Some recent developments Linear scaling techniques (O(n)) LDA+U for strong correlated system LDA+vdw for van der waals interaction Non-collinear calculation
Dispersion Correction The van der Waals (dispersive) interactions (also known as London dispersion interactions) between atoms and molecules play an important role in many chemical systems. They are in detailed balance with electrostatic (ES) and exchange repulsion (ER) interactions, and together, they control for example, the structures of DNA and proteins, the packing of crystals, the formation of aggregates, host– guest systems, or the orientation of molecules on surfaces or in molecular films. Even though vdW interactions are electrostatic in nature, their distance dependence is 1/r6. London forces are universal attractive forces that arise from the transient dipoles all molecules possess as a result of the fluctuations in the instantaneous positions of electrons. Two instantaneous dipoles are correlated in direction, and because of this correlation, the attraction between the two instantaneous dipoles does not average to zero.
Dispersion correction method vdw-DF
Introduce correction functional: ∫∫ρ(r1)φ(r1,r2)ρ(r2)dr1dr2
Parameterized functional M06-2X
Including dispersion when parameterize exc functionals
DFT-D
Dispersion-Correction Potential (DCP) (1ePOT) 修改单电子有效势算符，在其中引入体系中 周围一个个原子产生的等效VDW势（一个 长程弱吸引势+一个近程弱互斥势）
Non-collinear spin density Spin alignment in the whole structure (collinear magnetic structure) spin-polarization parallel or anti-parallel: FM, AFM…
Some systems：magnetization direction continuous variable of position strong spin-orbital coupling in some f-electron system Requiring: non-collinear magnetic (NCM) with SO calculations
Magnetic interaction
Spin-orbital coupling for NCM
Some progress in NCM calculation • •
Earlier work： atomic sphere approximation 1998, Oda et al. fully unconstrained approach
Spin direction in Fe3 and Fe5 Oda et al. Phys. Rev. Lett. 80, 3622 (1998)
• •
2000, Hobbs et al. fully unconstrained approach in PAW method Currently, NCM implemented in various DFT-packages.
Strong-Correlation Systems • Some materials containing ions with partly filled valence d or f shells • Some molecular system Kohn-Sham approach: no way to improve the correlation/exchange function Problems in : metal-insulator transition, heavy fermion behavior, hightemperature superconductor Mott Insulator: LSDA failure to predict the band gap and magnetism UO2 ： experiment insulator, LSDA metal Reason: magnetism in Mott insulator driven by Hunder’s-rule interaction in homogenous electron gas approx. driven by on-site Column interaction in reality
Orbital-Dependent Functional
Solution : SIC Add “self-interaction correction”, cancelled in Hartree Method Finite system: subtract the interaction directly Infinite system: functional with subtracted interaction developed SIC-LSDA : induce atomic like electronic configruation E XC SIC [n] = E xc [n] – Sigma
i
N
( E H [n I ] + E
xc
[n I ])
Solution: LSDA+U
U and J ：On-site Interaction parameter
J. Phys. Condensed Matter, 9, 767, 1997
