Statistical Mechanics and MultiScale Simulation Methods
ChBE 591-009
Prof. C. Heath Turner
Lecture 07
• 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
Semiempirical Molecular Orbital Theory
Background
• HF MO theory limited due to computational complexity
• approximations (fitting to experimental data) to improve speed
• approximations to improve accuracy
Implementations
•
Most demanding step of HF calculations: J and K integrals – numerical solutions
required, N4 scaling
•
SOLUTION: estimate these terms
•
J (coulomb integral): repulsion between 2 e-. Estimate: If basis functions
of e- #1 are far from e- #2, integral is zero.
•
Electron Correlation: dE of He with and without e- correlation = 26 kcal/mol
•
Analytic Derivatives: geometry optimizations need to calculate dE/dr. Early
approximations of HF were used to develop analytic derivatives.
Semiempirical Methods
• Hückel Theory/Extended Hückel Theory
• CNDO, INDO, NNDO
• MINDO/3, MNDO, AM1, PM3
• SAR
• QM/MM
Theory and Simulation Scales
Based on SDSC Blue Horizon (SP3)
512-1024 processors
1.728 Tflops peak performance
CPU time = 1 week / processor
TIME/s
100
(ms)
Continuum
Methods
Atomistic
Simulation
Methods
10-3
Mesoscale methods
Lattice Monte Carlo
Brownian dynamics
Dissipative particle dyn
(ms) 10-6
(ns) 10-9
(ps) 10-12
Semi-empirical
methods
Ab initio
methods
tight-binding
MNDO, INDO/S
(fs) 10-15
10-10
10-9
(nm)
NC State University 2002
Monte Carlo
molecular dynamics
10-8
10-7
10-6
10-5
10-4
(mm)
LENGTH
/meters
Semiempirical Molecular Orbital Theory
Hückel Theory (Erich Hückel)
• Illustration of LCAO approach
• Used to describe unsaturated/aromatic hydrocarbons
• Developed in the 30’s (not used much today)
Assumptions:
1. Basis set = parallel 2p orbitals, one per C atom (designed to treat planar hydrocarbon
p systems)
2. Sij = dij (orthonormal basis set)
3. Hii = a (negative of the ionization potential of the methyl radical)
4. Hij = b (negative stabilization energy). 90º rotation removes all bonding, thus we can
calculate DE: DE = 2Ep - Ep where Ep = a and Ep = 2a + 2b (as shown below)
5. Hij = between carbon 2p orbitals more distant than nearest neighbors is set to zero.
Semiempirical Molecular Orbital Theory
Hückel Theory: Application – Allyl System (C3H5)
• 3 carbon atoms = 3 carbon 2p orbitals
• Construct the secular equation:
H11  ES11
H12  ES12 
H 21  ES21


H N 1  ESN 1


H1N  ES1N


 H NN  ESNN
a E
b
0
a E
b 0
 b
0
b
a E
E  a  2b , a
E  a  2b
Q: What are the possible energy values (eigenvalues)?
Q: What is the lowest energy eigenvalue?
Q: What is the molecular orbital associated with this energy?
Solve:
N
 a H
i 1
i
ki
 ESki   0
Answer:
a2  2a1
a3  a1
Semiempirical Molecular Orbital Theory
Hückel Theory: Application – Allyl System (C3H5)
We need an additional constraint: normalization of the coefficients:
2
c
 i 1
i
a112  2a112  a112  1
a11  1 / 2, a21  2 / 2, a31  1 / 2
** Second subscript has been added to designate the 1st energy level (bonding).
1
2
1
p1 
p2  p3
2
2
2
2
2


p

0
p

p3
Next energy level:
2
1
2
2
2
Lowest energy Molecular Orbital:
Highest energy level:
Allyl cation: 2e- =

1 
1
2
1
3  p1 
p2  p3
2
2
2
2 a  2b

Semiempirical Molecular Orbital Theory
Picture taken from: CJ Cramer, Essentials of Computational Chemistry, Wiley (2004).
Semiempirical Molecular Orbital Theory
Extended Hückel Theory (EHT)
• All core e- are ignored (all modern semiempirical methods use this approximation)
• Only valence orbitals are considered, represented as STOs (hydrogenic orbitals).
Correct radial dependence. Overlap integrals between two STOs as a functions of r are
readily computed. (HT assumed that the overlap elements Sij = dij). The overlap between
STOs on the same atom is zero.
• Resonance Integrals:
• For diagonal elements: Hmm = negative of ionization potential (in the
appropriate orbital). Valence shell ionization potentials (VSIPs) have been
tabulated, but can be treated as an adjustable parameter.
• For off-diagonal elements: Hmn is approximated as (C is an empirical
constant, usually 1.75):
H m 
1
Cmn H mm  Hnn S mn
2
RESULT: the secular equation can now be solved to determine MO energies and wave
functions.
** Matrix elements do NOT depend on the final MOs (unlike HF)  the process is NOT
iterative. Therefore, the solution is very fast, even for large molecules.
Semiempirical Molecular Orbital Theory
Extended Hückel Theory (EHT)
• Performance Issues:
• PESs are poorly reproduced
• Restricted to systems for which experimental geometries are available.
• Primarily now used on large systems, such as extended solids (band structure
calculations)
• EHT fails to account for e- spin – cannot energetically distinguish between
singlets/triplets/etc.
Complete Neglect of Differential Overlap (CNDO)
• Strategy – replace matrix elements in the HF secular equation with approximations.
• Basis set is formed from valence STOs
• Overlap matrix, Smn = dmn
• All 2-e- integrals are simplified. Only the integrals that have m and n as identical
orbitals on the same atom and orbitals l and s on the same atom are non-zero
(atoms may be different atoms). Mathematically:
mn | ls   d mn dls mm | ll 
Semiempirical Molecular Orbital Theory
Complete Neglect of Differential Overlap (CNDO)
• The following two-electron integrals remain, and are abbreviated as gAB (A and B
correspond to atoms A and B):
mm | ll   g AB
• These integrals can be calculated explicitly from the STOs or they can be treated as a
parameter. Popular parametric form comes from Pariser-Parr approximation
(IP=ionization potential, EA=electron affinity):
g AA  IPA  EAA
g AA  g BB
g AB 
2  rAB g AA  g BB 
• The one-electron integrals for the diagonal matrix elements can be approximated as (m
is centered on atom A):
1
2
m  2  
k


Zk
m   IPm   Z k  d Z AZk g Ak
rk
k
Semiempirical Molecular Orbital Theory
Complete Neglect of Differential Overlap (CNDO)
• The one-electron integrals for the off-diagonal matrix elements (m is centered on atom
A and n is centered on atom B):
1
2
m  2  
k
b  b B Smn
Zk
n  A
rk
2
• Smn shown above is the overlap matrix element computed using the STO basis set.
This is different than defined previously for the secular equation. b is an adjustable
parameter, seen before in Hückel theory.
PERFORMANCE
•
Reduced number of 2e- integrals from N4 to N2
• The 2e- integrals can be solved algebraically
• CNDO not good for predicting molecular structures
• The Pariser-Parr-Pople (PPP) is a CNDO model that is used some today for
conjugated p systems.
Semiempirical Molecular Orbital Theory
Intermediate Neglect of Differential Overlap (INDO)
• Modification of CNDO method to permit more flexibility in modeling e-/e- interactions on
the same center.
• Modification introduces different values for the unique one-center two-electron
integrals. These values are empirical, adjustable parameters.
• INDO method predicts valence bond angles with greater accuracy than CNDO.
• INDO geometry still rather poor, but does a good job of predicting spectroscopic
properties.
Modified Intermediate Neglect of Differential Overlap (MINDO)
• Previously, semiempirical methods were problem specific
• MINDO developed to become more robust
• Every parameter treated as a free variable, within the limits of physical rules (by
incorporating penalty functions)
Semiempirical Molecular Orbital Theory
Neglect of Diatomic Differential Overlap (NDDO)
• Relaxes constraints on the two-center two-electron integrals. Thus all integrals of
(mn|ls) are retained provided that m and n are on the same center and that l and s are
on the same center.
• Most modern semiempirical models are NDDO models: MNDO, AM1, PM3
Structure-Activity Relationships (SAR)
• Used in the pharmaceutical industry to understand the structure-activity relationship of
biological molecules (use NDDO models)
• Drug companies use SARs to screen through candidate molecules in order to identify
potentially active species – design drugs and predict activity.
• ALSO:
• Quantitative Structure-Activity Relationship (QSAR) – for more information on
the methodology see the guide developed by Network Science Corporation
(http://www.netsci.org/Science/Compchem/feature19.html)
• Quantitative Structure-Property Relationship (QSPR) – correlation developed
between structure and physical properties. Typically used to design materials and
predict properties.