a. Semi-empirical methods

1
Additional Quantum-based Methods
a. Semi-empirical methods b. Density functional
theory c. Molecular Orbital Applications
2
Semi-empirical methods
3
Semiempirical Molecular Orbital Calculation
4
LCAO-MO fi ?r cri yr   ?s ( Heffrs - ei
Srs ) csi 0   Heffrs ? lt ?r Heff ?s
gt Srs ? lt ?r ?s gt

• Parametrization
• Heffrr ? lt ?r Heff ?r gt
• minus the valence-state
  ionization
• potential (VISP)

5
Atomic Orbital Energy
VISP --------------- e5 -e5 --------------- e
4 -e4 --------------- e3 -e3 --------------
- e2 -e2 --------------- e1 -e1   Heffrs
½ K (Heffrr Heffss) Srs K 1?3
6
CNDO, INDO, NDDO (Pople and co-workers) Hamiltoni
an with effective potentials Hval ?i -(h2/2m)
?i2 Veff(i) ?i?jgti e2 / rij
two-electron integral (rstu) lt?r(1) ?t(2)
1/r12 ?s(1) ?u(2)gt   CNDO complete neglect of
differential overlap (rstu) ?rs ?tu (rrtt) ?
?rs ?tu ?rt
7
INDO intermediate neglect of differential
overlap (rstu) 0 when r, s, t and u are not on
the same atom. NDDO neglect of diatomic
differential overlap (rstu) 0 if r and s (or t
and u) are not on the same atom. CNDO, INDO are
parametrized so that the overall results fit well
with the results of minimal basis ab initio
Hartree-Fock calculation. CNDO/S, INDO/S are
parametrized to predict optical spectra.
8
MINDO, MNDO, AM1, PM3 (Dewar and co-workers,
University of Texas, Austin)   MINDO modified
INDO MNDO modified neglect of diatomic overlap
AM1 Austin Model 1 PM3 MNDO parametric method
3   based on INDO NDDO reproduce the binding
energy
9
Density functional theory
10
Quantum Methods
Electron Density
Wavefunctions
DFT
Hartree-Fock
TD-DFT
MP2-CI
The HF equations have to be solved iteratively
because VHF depends upon solutions (the
orbitals). In practice, one adopts the LCAO
scheme, where the orbitals are expressed in terms
of N basis functions, thus obtaining matricial
equations depending upon N4 bielectron integrals.

11
Information provided by ? is redundant
N 42e-
benzene

• Number of terms in the determinantal form ? N!
  1.4 ??1051
• Number of Cartesian dimensions 3N 126
• is a very complex object including more
  information than we need!
• The use of electron density allows to limit the
  redundant information
• The electron density is a function of three
  coordinates no matter of the electron number.

12
What is Density?

• Density provides us information about how
  something(s) is(are) distributed/spread about a
  given space
• For a chemical system the electron density tells
  us where the electrons are likely to exist (e.g.
  allyl)

13
Representations of the electron density of the
water molecule (a) Relief map showing values of
?(r) projected onto the plane, which contains the
nuclei (large values near the oxygen atom are cut
out) (b) Three dimensional molecular shape
represented by an envelope of constant electron
density (0.001 a.u.).
14
Definitions
Function a prescription which maps one or more
numbers to another number
Functional A functional takes a function as
input and gives a number as output. An example
is
Here f(x) is a function and y is a number. An
example is the functional to integrate x from -?
to ?.
15
ab-initio methods can be interpreted as a
functional of the wavefunction, with the
functional form completely known!
Can we write an explicit functional form of
energy E? for DFT? In the general case the
answer is not known. It is the main chalenge in
DFT.
16
Timetable
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
PRLs 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.
17
Thomas-Fermi Energy (1920)
-TF kinetic electron energy is an approximation
of true kinetic electron energy -Potential
nuclear-electron energy -Potential
electron-electron Coulomb interaction
energy exchange and correlation effects are
completely neglected
18
Correlation energy
Exchange correlation Electrons with the same
spin (ms) do not move independently as a
consequence of the Pauli exclusion principle. ?
0 if two electrons with the same spin occupy the
same point in space, independently of their
charge. HF theory treats exactly the exchange
correlation generating a non local exchange
correlation potential. Coulomb
correlation Electrons cannot move independently
as a consequence of their Coulomb repulsion even
though they are characterized by different spin
(ms). HF theory completely neglects the Coulomb
correlation thus generating, in principle,
significant mistakes. Post HF treatments are
often necessary.
19
The Slater exchange functional
The predecessor to modern DFT is Slaters Xa
method. This method was formulated in 1951 as an
approximate solution to the Hartree-Fock
equations. In this method the HF exchange was
approximated by The exchange energy EXa is a
fairly simple function of the electron density
r. The adjustable parameter a was empirically
determined for each atom in the periodic table.
Typically a is between 0.7 and 0.8. For a free
electron gas a 2/3.
20
The VWN Correlation Functional
In ab initio calculations of the Hartree-Fock
type electron correlation is also not included.
However, it can be included by inclusion of
configuration interaction (CI). In DFT
calculations the correlation functional plays
this role. The Vosko-Wilk-Nusair (VWN)
correlation function is often added to the Slater
exchange function to make a combination
exchange-correlation functional. Exc Ex
Ec The nomenclature here is not standardized and
the correlation functionals themselves are very
complicated functions.
21
Correlation Energy Is it important?
N2 molecule Corelation energy 0.5 of the
electronic energy 50 of the binding
energy!
22
DFT energy
EDFT  ENN  ET  Ev  Ecoul  Eexch  Ecorr 
ENN - nuclear-nuclear repulsion energy, Ev -
nuclear-electron attraction energy , Ecoul -
electron-electron Coulomb repulsion energy are
the same as those used in Hartree-Fock theory.
ET - kinetic energy of the electrons  energy
Eexch - electron-electron exchange energy  Are
different from those used in Hartree-Fock theory.
Ecorr -correlation energy of electrons of
different spin is not accounted for in
Hartree-Fock theory. Due to these differences,
the exchange energies calculated exactly in
Hartree-Fock theory cannot be used in density
functional theory.
23
The important advances for practical
calculations are in electronic
density-functional theory.

• From Hohenberg and Kohn (1964)
• Energy is a functional of electron density Er
• Ground-state only, but the exact r minimizes Er
• Then Kohn and Sham (1965)
• Variational equations for a local functional

where Exc contains electron correlation
24
Nonlocal effects, introduced in early 1990s,
have made DFT powerful.

• Kohn and Sham had the local exchange functional

• Need nonlocal effects of gradient,
• Current approach Hybrid functionals
• Combine Hartree-Fock Ex and DFT contributions Ex
  Ec
• Numerous proposed functionals and combinations
• Axel Beckes BLYP, B3LYP, BHHLYP Truhlars M06
• Give excellent structures and frequencies, poorer
  energies.

25
Hybrid functionalsThe basic idea behind the
hybrid functionals is to mix exchange energies
calculated in an exact (Hartree-Fock-like) manner
with those obtained from DFT methods in order to
improve performance. Frequently used methods are
B3LYP method. Becke-3-LYP (B3LYP) uses a
different mixing scheme involving three mixing
parametersEXC  0.2EX(HF) 0.8EX(LSDA)
0.72DEX(B88) 0.81EC(LYP) 0.19EC(VWN)In
this latter case, the B88 (Becke) gradient
correction to the local LSDA exchange energies
carries its own scaling factor of 0.72 and the
LYP (Lee, Yang, and Parr) gradient correction to
the local VWN correlation energies carries its
own scaling factor of 0.81. The three scaling
factors have been derived through fitting the
parameters to a set of thermochemical data 
26
(No Transcript)