Transcript Slide 1

Statistical Mechanics and MultiScale Simulation Methods
ChBE 591-009
Prof. C. Heath Turner
Lecture 06
• Some materials adapted from Prof. Keith E. Gubbins: http://gubbins.ncsu.edu
• Some materials adapted from Prof. David Kofke: http://www.cbe.buffalo.edu/kofke.htm
Density Functional Theory
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HF
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optimize the e- wavefunction
the wavefunction is essentially uninterpretable, lack of intuition
e- correlation is only accounted for using post-HF methods
DFT
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optimize the e- density
Increased in popularity within last 2 decades.
Hamiltonian depends ONLY on the positions and atomic
number of the nuclei and the number of e-.
Given a known e- density  form the H operator  solve the
Schrödinger Eq.  determine the wavefunctions and energy
eigenvalues.
Hohenberg and Kohn – proved ground-state E is uniquely
defined by the e- density. E is a unique functional of r(r).
Functional example:
Q f (r )   f (r )dr
Significance
(1). The wave function Y of an N-electron system includes 3N
variables, while the density, r no matter how large the system is,
has only three variables x, y, and z. Moving from E[Y] to E[r] in
computational chemistry significantly reduces the computational
effort needed to understand electronic properties of atoms,
molecules, and solids.
(2). Formulation along this line provides the possibility of the linear
scaling algorithm currently in fashion, whose computational
complexity goes like O(NlogN), essentially linear in N when N is
very large.
(3). The other advantage of DFT is that it provides some chemically
important concepts, such as electronegativity (chemical potential),
hardness (softness), Fukui function, response function, etc..
Density Functional Theory
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‘Local’ functional: f (r )  r (r )
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‘Non-local’ or ‘gradient-corrected’:
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As with MO theory, the density (in exact DFT) obeys a
variational principle – the lower E is more accurate.
In DFT, the E functional is written as:
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f (r )  r (r )  r ' (r )
E r (r )   Vext (r ) r (r )dr  F r (r )
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First term: interaction with external potential (nuclei)
Second term: KE(e-) + e-/e- interactions
Density Functional Theory
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Solution: optimize e- density until E is minimized.
Constraints on e- density? N   r (r )dr
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How do we include this constraint? Lagrange
multipliers (m):
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
E r (r )  m  r (r )dr   0
r (r )
 E r (r ) 

  m
 r (r ) Vext
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This is the DFT equivalent of the Schrödinger Eq. Vext
indicates constant external potential (nuclear positions).
** Central crux of DFT: What is the function, F[r(r)]?
Density Functional Theory
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Kohn and Sham split F[r(r)] into three terms:
F[r(r)] = EKE[r(r)] + EH[r(r)] + EcC[r(r)]
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EKE[r(r)] = e- kinetic energy
EH[r(r)] = e-/e- Coulombic interaction
EcC[r(r)] = e- exchange/correlation + KE correction + E(selfinteraction)
One-electron Kohn-Sham equations:
 12  M Z A 

r (r2 )
     
dr2 VC r1  i (r1 )   i i (r1 )

r12
 2  A1 r1 A 

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Solution (SCF approach):
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guess density
derive orbitals
calculate new density from orbitals
repeat
Density Functional Theory
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The solution hinges on VC[r(r)]:
VC r (r ) 
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EC r (r )
r (r )
We must find the functional: EC[r(r)]. Unfortunately, there is no way to solve
for this functional, but we can attempt to find expressions that work well.
Since we must invoke approximations for this term, the implementation of DFT
is no longer variational (unlike HF).
DFT remains size consistent (despite losing variational behavior).
There are two basic implementations (approximations) of DFT:
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Local-density approximation (LDA)
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Generalized gradient approximation (GGA)
LDA
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The value of the exchange energy depends only on the local density.
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The e- density may vary as a function of r, but r is single-valued, and the
fluctuations in r away from r do not affect the value of EC at r.
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LSDA: variation of LDA accounting for spin polarization (open-shell
systems), similar to UHF method, which splits solutions in to  and b
spins.
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EC is based on the uniform electron-gas model, which is known
accurately, and can be cast into an analytical form.
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Functionals: VWN, VWN5 (Vosko-Wilk-Nusair)
Density Functional Theory
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GGA
– The value of the exchange energy depends on the local density AND
on the gradient of the density. Overcomes LDA tendency to overbind.
Adds ~20% to compute time.
– Exchange and Correlation contributions usually calculated separately.
– BLYP – popular functional including exchange contribution from
(B)ecke and correlation contribution from (L)ee, (Y)ang, and (P)arr.
– Functionals: BLYP, BP86, BPW91
Hybrid Functionals
– Incorporate HF exchange contribution into the DFT functional
– Exact exchange for a non-interacting system can be calculated using
HF (using KS orbitals).
– Very popular
– Functionals: B3LYP, B3PW91, B1PW91, PBE1PBE
Periodic Systems
– DFT often used
– Periodic plane waves
Car-Parrinello MD
– Ab initio MD
– Chemical reactions
– On-the-fly potentials
Density Functional Theory
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Similarities with HF:
– A basis set is still needed, but can be more flexible (numerical
basis functions)
– Solution of secular equation
– SCF procedure is still used
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Differences with HF:
– e- correlation is implicitly included
– The solution of the secular equation is computationally more
efficient – formally scales as N3 as opposed to N4.
– Sometimes empirical parameters are included
– Some properties are easier to extract from HF than from DFT
– DFT has challenge of systematic improvability – difficult to
predict performance of 2 different functionals. HF is more
predictable, with full CI (with an infinite basis set) as the
ultimate goal.
Density Functional Theory
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PERFORMANCE:
– Formally scales as N3, but improvements are possible
– Convergence w.r.t. basis set size is more rapid
– DFT SCF is sometimes more problematic, thus HF orbitals can be
used as an initial guess for KS orbitals
– Not capable of describing London dispersion forces – dispersion not
included in functionals. This can artificially arise from BSSE.
– H-bonded systems: heavy-atom/heavy-atom distances typically too
short by 0.1 angstrom, but energetics o.k. (need diffuse functions in
basis set).
– Complexes with charge-transfer interactions, DFT overpredicts the
interactions.
– DFT sometime overstabilizes systems, increasing symmetry.
– Increasing basis set size does not always improve the accuracy.
– Hybrid functionals typically outperform pure functionals.
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In view of exceptions, DFT usually performs at level of MP2 theory or
better, but not as consistent. DFT does a much better job with transition
metals than MO theory.
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In general, literature provides guidance with regards to performance.