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Statistical Mechanics and Multi-Scale Simulation Methods ChBE 591-009. Prof. C. Heath Turner Lecture 07. Some materials adapted from Prof. Keith E. Gubbins: Some materials adapted from Prof. David Kofke:

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Statistical Mechanics and Multi-Scale Simulation Methods ChBE 591-009

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Statistical Mechanics and Multi-Scale Simulation MethodsChBE 591-009

Prof. C. Heath Turner

Lecture 07

  • Some materials adapted from Prof. Keith E. Gubbins:

  • Some materials adapted from Prof. David Kofke:

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


  • 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



Atomistic SimulationMethods

Mesoscale methods



Lattice Monte Carlo

Brownian dynamics

Dissipative particle dyn







Monte Carlo

molecular dynamics



Ab initio
















NC State University 2002

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:

  • Basis set = parallel 2p orbitals, one per C atom (designed to treat planar hydrocarbon p systems)

  • Sij = dij(orthonormal basis set)

  • Hii = a (negative of the ionization potential of the methyl radical)

  • 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:

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?



Semiempirical Molecular Orbital Theory

  • Hückel Theory: Application – Allyl System (C3H5)

  • We need an additional constraint: normalization of the coefficients:

** Second subscript has been added to designate the 1st energy level (bonding).

Lowest energy Molecular Orbital:

Next energy level:

Highest energy level:

Allyl cation: 2e- =

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):

  • 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:

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):

  • 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):

  • The one-electron integrals for the diagonal matrix elements can be approximated as (m is centered on atom A):

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):

  • 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.


  • 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 (

    • Quantitative Structure-Property Relationship (QSPR) – correlation developed between structure and physical properties. Typically used to design materials and predict properties.

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