Statistical Mechanics and MultiScale Simulation Methods ChBE 591009

1
Statistical Mechanics and Multi-Scale Simulation
MethodsChBE 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

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Density Functional Theory

• HF
• optimize the e- wavefunction
• the wavefunction is essentially uninterpretable,
  lack of intuition
• e- correlation is only accounted for using
  post-HF methods
• DFT
• 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

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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 EY to Er
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..
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Density Functional Theory

• Local functional
• Non-local or gradient-corrected
• 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
• First term interaction with external potential
  (nuclei)
• Second term KE(e-) e-/e- interactions

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Density Functional Theory

• Solution optimize e- density until E is
  minimized.
• Constraints on e- density?
• How do we include this constraint? Lagrange
  multipliers (m)
• This is the DFT equivalent of the Schrödinger Eq.
  Vext indicates constant external potential
  (nuclear positions).
• Central crux of DFT What is the function,
  Fr(r)?

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Density Functional Theory

• Kohn and Sham split Fr(r) into three terms
• Fr(r) EKEr(r) EHr(r) EcCr(r)
• EKEr(r) e- kinetic energy
• EHr(r) e-/e- Coulombic interaction
• EcCr(r) e- exchange/correlation KE
  correction E(self-interaction)
• One-electron Kohn-Sham equations
• Solution (SCF approach)
• guess density
• derive orbitals
• calculate new density from orbitals
• repeat

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Density Functional Theory

• The solution hinges on VCCr(r)
• We must find the functional ECCr(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
• Local-density approximation (LDA)
• Generalized gradient approximation (GGA)
• LDA
• The value of the exchange energy depends only on
  the local density.
• 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 ECC at r.
• LSDA variation of LDA accounting for spin
  polarization (open-shell systems), similar to UHF
  method, which splits solutions in to a and b
  spins.
• ECC is based on the uniform electron-gas model,
  which is known accurately, and can be cast into
  an analytical form.
• Functionals VWN, VWN5 (Vosko-Wilk-Nusair)

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Density Functional Theory

• 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

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Density Functional Theory

• 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
• 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.

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Density Functional Theory

• 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.
• 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.
• In general, literature provides guidance with
  regards to performance.