Understanding Density Functional Theory in Chemistry
Density Functional Theory (DFT) plays a crucial role in chemistry by uniquely determining molecular properties based on electron density. The Hohenberg-Kohn Theorem establishes the foundation, with the goal of finding an exact energy functional expressed in terms of density. Various concepts like the Thomas-Fermi theory, Kohn-Sham DFT, and the use of orbitals are discussed in relation to determining kinetic and exchange-correlation potentials. Ultimately, DFT provides insight into electronic structures without relying on wave functions.
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Chem Chem 580: DFT 580: DFT
Density Functional Theory Density Functional Theory (Chapter 6, Jensen) (Chapter 6, Jensen) Hohenberg-Kohn Theorem (Phys. Rev., 136,B864 (1964)): For molecules with a non degenerate ground state, the ground state molecular energy and all other electronic properties are uniquely determined by the ground state density 0. Appealing since density depends on 3 coordinates whereas wave function depends on 3N coordinates. Problem: Don t know the density. So, HK theorem is an existence theorem. Also, need to know the density at every point in 3N-dimensional space. Functional: Function of a function. E[ 0(x,y,z)]
Density functional theory Density functional theory Goal of DFT development: Find exact energy functional that expresses the ground state energy in terms of the density: E=E[ (x,y,z)] = + + = + + + E T V V T V J K ne ee ne T=kinetic energy, Vne=el-nuc attraction, J=Coulomb, K=exchange In the limit of the exact density, J[ ] and K[ ] implicitly include electron correlation. Vne [ ]and J[ ] are treated normally: ( ) r Z = V dV A ne r r A ( ) ( ') r r r = J dV '
Density functional theory Density functional theory ne ee T V V + + = = + + + E T V J K ne Question: How do we determine T[ ] and K[ ]? In QM, T is differential operator; K depends on orbitals Thomas-Fermi theory (Physics): Provides T[ ] of a uniform, non- interacting electron gas. Can solve exactly Spherical cow Problem: Does not work for chemistry since chemical bonds don t exist in this model. = 5/3 ( ) r dr T C 3 ( ) 3/2 = 2 3 C TF F F 10 3 3 4 = 4/3 ( ) r dr K C 1/3 = C Dirac TF X X
Density functional theory Density functional theory Kohn & Sham (KS) proposed using orbitals to construct functional. (Phys. Rev., 140, A1133 (1965)). Similar to HF independent particle model Assume non-interacting electrons. The Hamiltonian of the reference system is: 1 2 n = h = + 2 i KS i ( ) r v KS i H h s i s = 1 i This gives us starting orbitals like 1st step in HF Now add in repulsions: = + ( ) + H T V V ext ee = no repulsion: independent particle model = repulsion term Vee depends on orbitals; Vext=Vne SCF similar to HF
Kohn Kohn- -Sham (KS) DFT Sham (KS) DFT What is the wave function for the reference system ( =0)? Independent particle model: Must be a Slater determinant! = 1... ,0 1 s N N The molecular orbitals iare called Kohn-Sham orbitals with orbital energies i Note: orbitals, NOT density DFT is not supposed to have a wave function!
KS Density functional theory KS Density functional theory The exact kinetic energy functional for the reference system is: 1 2 T = 2 i s i i i For interacting electrons, this is only an approximation (similar to HF). Since the density is still not known, Kohn-Sham calculates the density as: = 2 = ( ) r ( ) r ( ) r Same as for HF s i i = + + + Then, we have: E T V J E DFT S Ne xc Exc contains exchange and correlation corrections. Exchange is the largest part Usually highly parametrized
KS Density functional theory KS Density functional theory Major problem and effort is to find a reasonable expression for Exc[ ]. The KS procedure: (1) (1) = KS (1) h i i i ( ) r 1 2 Z r + + = 2 1 r (1) (1) (1) 2 d v A 2 xc i i i 12 r A 1 A Solved iteratively as in HF The exchange-correlation potential vxc is given by: ( ) r r E = xc ( ) r v xc ( )
KS Density functional theory KS Density functional theory There are MANY choices of exchange-correlation functionals. One approach, so far not broadly successful, is to use very high level of ab initio theory to get density, then use it to determine Exc[ ]. This approach is difficult to generalize. Most common to separate into two independent parts: xc x E E = + E c Jacob s ladder: Unoccupied orbitals Generalized RPA (Random Phase Approximation) Hybrid Add HF Exact exchange , 2 , And/or Meta-GGA , GGA LDA All levels build upon LDA as the base: House of cards All levels build upon LDA as the base: House of cards Lowest rung is unreliable Lowest rung is unreliable
Local density approximation (LDA): The local density can be treated as a uniform electron gas. Basically Thomas-Fermi Theory (spherical cow) xc xc E ( ) ( ) xc x = ( ) ( ) + = LDA r ( ) r d c Cx 1/3 3 3 4 ( ) ( ) ( ) r 1/3 x = Early variant was X method devised by Slater where correlation is ignored. The exchange part: 3 2 ( ) ( ) ( ) r 1/3 = C is a parameter (usually 0.75) x x
LDA LDA Correlation functional calculated ad hoc, often determined using Monte Carlo methods and then fitted to analytic formula. The two most common correlation functionals: Vosko-Wilk-Nusair (VWN): Can. J. Phys,58,1200 (1980) Perdew-Wang: Phys. Chem Rev. B, 45,13244(1994) Both formulas given in Jensen. Lots of fitted parameters. 1) 2) LDA tends to overestimate binding energies, does not describe hydrogen bonds properly.
Gradient corrected methods: Generalized gradient approximation (GGA): Non-uniform electron gas. Both exchange and correlation energies dependent on the gradient of the density. These functionals are sometimes called non-local: Wrong since they depend on the density at a point in space (unlike HF exchange). Exchange part: Best seems to be PW91 (Phys. Rev. B, 46,6671 (1992)) Formula in Jensen, lots of parameters. Built on LDA Correlation part: Most popular by Lee, Yang, Parr (LYP). Contains 4 parameters determined by fitting to data on He atom. (Phys. Rev. B, 37, 785 (1988). Note: LYP provides no correlation for parallel spins.
Meta-GGA: Include either (or both) the second derivative of the density or the orbital kinetic energy density. 1 2 2 = KS i i Most popular functionals: TPSS, B95, M06-L Most popular new functionals are meta-GGAs. Most promising are set of M06 functionals from Truhlar group, but seem to need a different one for different problems. One size does not fit all.
Hybrid functionals: Mix LDA, GGA (or meta-GGA) and exact HF exchange: Called exact exchange Becke proposed (JCP, 98, 5648(1993)): = + b E + + c E + 3 88 B xc LDA x exact x B x LDA c GGA c (1 ) E a E aE E a, b, c are empirically fitted parameters: Semi-empirical! Built on LDA If we use the LYP correlation functional, we get B3LYP. Parameters a, b and c fitted to reproduce the G2 test set. Method Mean absolute deviation (kcal/mol) Maximum Absolute deviation (kcal/mol) G2 1.6 (Composite) 8.2 G2(MP2) 2.0 (Composite 10.1 G2(MP2, SVP) 1.9 (Composite) 12.5 B3LYP 3.1 (Hybrid) 20.1 B3PW91 3.5 (GGA) 21.8 SVWN 90.9 (LDA) 228.7
Hybrid functionals: Different hybrid functionals use different amounts of exact exchange . Works reasonably well for organic molecules Different hybrids work often better for different properties. Often better barrier heights than HF or GGA. Not as good as MP2/RI-MP2 Built on a house of cards Starting point is LDA Better approach is to build DFT on 2-electron functions P.M.W. Gill W. Kutzelnigg
Advantages and disadvantages of DFT Advantages and disadvantages of DFT DFT in GAMESS: $CONTRL DFTTYP= $END and $DFT Can be run with SCFTYP= RHF, UHF, ROHF Numerical grids, cannot integrate analytically: Can make grid denser (better) or sparser using: NRAD= (Number of radial points, default=96) NTHE= (Number of angle theta grids, default=12) NPHI= (Number of angle phi grids, default=24, must be twice NTHE) Larger values= finer grid Can actually avoid grid by using resolution of the identity to simplify integrals. RI based on: . i x y = x y j i m m j m Exact if { m}= complete basis. Can develop auxiliary basis set for this purpose. First suggested by Almlof, implemented by Glaesemann & Gordon. J. Chem. Phys., 112, 10738 (2000).
Advantages and disadvantages of DFT Advantages and disadvantages of DFT Summary of scaling: N7 CCSD(T) N5 MP2 eN Full CI eN MRCI N4 DFT N4 HF DFT and HF cost can be reduced by manipulating long-range: Make linear scaling at very long distances using multipolar expansion or localized orbitals Very demanding methods can be reduced in scaling using localized orbitals: reduces range of interaction. Semi-empirical methods ~ N2-N3: N grows less quickly. Bottleneck is matrix diagonalization. Formal linear scaling method: MM
Advantages and disadvantages of Advantages and disadvantages of DFT DFT Some problems: Weak interactions due to dispersion. For instance, no bonding between two He atoms. Solution: Add dispersion ad hoc (Grimme) Self-interaction error: Density of a single electron interacts with itself. Leads to underestimation of band gaps. Some functionals try to correct for self-interaction. Poor charge transfer predictions. Very bad for metal oxides Ground state theory (need TDDFT for excited states) Poor for conical iontersections Not systematically improvable HF MP2-CCSD(T)-CCSDT-CCSDTQ CIS-CISD-CISDT Nothing comparable for DFT
Further Information http://www.phy.mtu.edu/pandey/talks/trickey_dft-course_2008-07.pdf