Semiempirical and Density Functional Theory Methods

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Semi-empirical and Density Functional Theory
Methods
Computational Chemistry 5510 Spring 2006 Hai Lin
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Practice with HF Theory

• Computational cost N4 (N is the number of basis
  sets)
• STO-3G 6-31G 6-31G 6-31G
• H 1 2 2 5
• C 5 9 15 15
• C6H6 36 66 102 120

• Accuracy is limited (not account for electron
  correlation)
• Two electrons having the same spins avoid
• each other. (Accounted for by the
• Slater determinant)
• Two electrons having the opposite spins
• avoid each other. (Not considered)

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Correlate Electrons
HF Theory
Implicit Electron Correlations
Explicit Electron Correlations
Semi-empirical Wave Function Methods
Electron-Correlated Wave Function Methods

• Møller-Plesset Perturbation Theory
• MPn (n 2, 3, 4)
• Configuration Interaction Methods
• Full CI, CISD, MRCI, etc.
• Coupled Cluster Methods
• CCSD, CCSD(T), etc.

AM1, PM3, MNDO, etc.
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Semi-empirical Methods

• Employing simplifying assumptions when solving
  Hartree-Fock (actually Roothaan-Hall) equations.
• The simplifications neglect certain integral
  terms that are expensive to calculate, and rely
  on empirically parameters to make calculations in
  agreement with experimental data.
• (That is why these methods are called
  semi-empirical methods.)

HF Theory

• The parameterization procedure accounts for
  electron correlations implicitly.
• Scaling behavior N3

Additional Approximations
Semi-empirical Methods
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Common Semi-empirical Methods

• 1965 Pople CNDO
• 1967 Pople INDO
• 1975 Dewar MINDO/3
• 1977 Dewar MNDO
• 1985 Dewar AM1
• 1989 Stewart PM3
• 1970s Zerner ZINDO
• 1996 Thiel MNDO/d

• Accuracy depends on parameterizations and can not
  be systematically improved (like molecular
  mechanics).
• Validate the method before using it.
• Applied to molecules of relatively large size (
  300 atoms).
• Obtain preliminary results that are refined by
  high-level calculations.

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Extended Hückel Theory

• Most semi-empirical methods above make
  approximations at the integral level, i.e.,
  neglecting certain integral terms when
  calculating the Fock matrix.
• Extended Hückel Theory (EHT) makes approximation
  directly on the Fock matrix elements, and again
  needs parameterization.
• EHT is often used to provide an initial guess for
  the orbitals that are used in higher-level (e.g.,
  HF and DFT) calculations.
• Originally developed by Hoffmann et al. in 1960s.
  Recently, it becomes hot again for treating large
  systems, especially in studies of proteins
  containing metals, and is called tight-binding
  theory.

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What is a Functional?

• A function depends on a set of variables.
• y f (x)
• E.g., wave function depend on electron
  coordinates.

• A functional depends on a functions, which in
  turn depends on a set of variables.
• E F f (x)
• E.g., energy depends on the wave function, which
  depends on electron coordinates.

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Density Functional Theory
y

• The electron density is the square of wave
  function and integrated over electron
  coordinates.
• The complexity of a wave function increases as
  the number of electrons grows up, but the
  electron density still depends only on 3
  coordinates.

x
r

• 1964 Hohenberg Kohn
• The ground-state electronic energy is determined
  completely by the electron density r.
• E F r (x)
• If we knows electron density r, we can bypass the
  wave function and calculate energy directly!

x
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Density Functional Theory (2)

• There is a one-to-one connection between energy E
  and density r E F r (x) , but we do not
  know the F! (A trade-off!)
• The central task in DFT is to find the F.
• Thomas-Fermi-Dirac Model Non-reacting uniform
  electron gas
• Kohn-Sham Theory (1965) Atoms Molecules
• EDFTr Tr Ener Jr Excr

electrons-electrons Exchange energy
Electronic Kinetic energy
electrons-electrons Coulombic energy
Nuclei-electrons Coulombic energy
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Density Functional Models

• The central task in DFT is to find the functional
  that is as accurate as possible. (Maybe we will
  never know the exact functional)
• Various functions proposed whose coefficients are
  adjusted to obtained agreements with experiments
  or high-level ab intio calculations

• Local Density Approximation (LDA)
• Assuming the functional depends only on the
  electron density
• Example LSDA (e), VWN (c)
• Gradient Corrected or Generalized Gradient
  Approximation (GGA)
• Assuming the functional depends on both the
  electron density and its gradient
• Examples PW86 (e), B88 (e), PW91 (c), LYP (c),
  BLYP
• (e exchange, c correlation)

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Density Functional Models (2)

• Hybrid Models
• Take the exchange as a linear combination of
  terms from LSDA, Hartree-Fock, and GGA.
• Example HH (e), B3 (e), B3LYP
• Doubly Hybrid Models
• In addition to mix the exchange with Hartree-Fock
  exchange, also mix the correlation with MP2
  correlation.
• Examples MC3MPW, MC3BB
• Meta-GGA Models
• Assuming the functional depends on both the
  electron density and its gradient (GGA) as well
  as kinetic energy density.
• Examples VSXC, BB95

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Which DFT Models to Use?

• DFT scaling behavior N4
• Slightly more expensive than HF but more accurate
  
• No systematic way to improve the accuracy (at
  least up till now).
• Accuracy varies from model to model, and from one
  application area to another.
• BLYP good for metals, poor for organic compounds
• B3LYP poor for metals, good for organic
  compounds
• MPW1K specially parameterized for kinetics (good
  for transition states and reaction barrier
  heights, poor for stable molecules)
• Commonly regarded as poor for H-bonding and weak
  (dispersion) interactions, but some people
  disagree.
• Validate a model before use it.
• Test the model on small molecules where reliable
  experimental data are available or high-level ab
  intio calculations can be done.

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Koopmans Theorem

• Frozen MO assumption The molecule orbitals are
  constant for a given neutral molecule and its
  ions (anion and cation)
• For a neutral molecule
• Ionization energy occupied orbital energy
• Electron affinity unoccupied orbital energy
• (It is better to calculate electron affinity as
  occupied orbital energy in the anion.)

E 0
Unoccupied Occupied
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Restricted Unrestricted HF

• Electrons have spins (a and b, or up and down).
  To identify an electron, both spatial orbitals
  and spin orbitals are required.
• Restricted HF accommodate electrons of opposite
  spins in pairs (i.e., in the same spatial
  orbitals).

• Unrestricted HF allow two electrons of opposite
  spins staying in different spatial orbitals.

E 0
Unoccupied Occupied
Closed shell RHF (singlet)
Open shell ROHF (doublet)
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Unrestricted HF

• For a closed-shell system, UHF and RHF gives the
  same results.
• For an open-shell system, UHF gives lower energy
  than RHF (ROHF) does, because the system is now
  relaxed from the restriction of forcing a and b
  electrons in the same spatial orbitals.

• A hydrogen molecule dissociates into ...
• UHF two atoms
• RHF two ions (becuase the a and b electrons have
  to be in the same spatial orbitals even if two
  nuclei are far away from each other).

E
H H-
R
H H
H2
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Problems with UHF

• For an open-shell system, UHF has the spin
  contamination problem The UHF wave function is
  not an eigenfunction of the S2 operator (S is the
  electron spin operator).
• For example, a singlet UHF wavefunction will have
  contributions from triplet and other high-lying
  states. (Not physical!)
• If such a contribution is small, i.e., ?S2? is
  close to 0.75, the wavefunction is approximately
  right.
• If such a contribution is large, e.g., ?S2? is
  much larger than 0.75, the wavefunction is
  qualitatively wrong, and the calculation is
  meaningless.

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Valence Bond Theory

• Close to simple chemical bonding concept two
  electrons pair to form a valence bond.

Y fA(a) fB(b)
Seperated far away
R ? ?
A
B
Y ? fA(a) fB(b) fA(b) fB(a)
Bonding
A
B
A
B
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Summary

• Problems with HF theory
• Semi-empirical theory
• Density functional theory
• Koopmans theorem
• Restricted HF vs. Unrestricted HF
• Valence Bond Theory

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Your Homework

• Read the slides. If you have difficulty in
  understanding the math in the slides, find a
  reference.
• Read textbook (Take notes when you read.)
• 3.4, 3.7, 3.8.4, 3.9, 3.10, 3.11, 3.12,
  and 3.13. You can skip the math in 3.9 and
  3.10.
• 6.1 - 6.5. You can skip the math in those
  chapters.
• Questions
• Why is it better to calculate the electron
  affinity from an anion than from a neutral
  molecule?
• What is a RHF? What is a UHF? What is the
  potential problem of using UHF?
• What are the advantages of using semi-empirical
  and DFT methods?