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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: http://gubbins.ncsu.edu
- Some materials adapted from Prof. David Kofke: http://www.cbe.buffalo.edu/kofke.htm

- 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

Continuum

Methods

Based on SDSC Blue Horizon (SP3)

512-1024 processors

1.728 Tflops peak performance

CPU time = 1 week / processor

TIME/s

100

Atomistic SimulationMethods

Mesoscale methods

(ms)

10-3

Lattice Monte Carlo

Brownian dynamics

Dissipative particle dyn

(ms)

10-6

Semi-empirical

(ns)

10-9

methods

Monte Carlo

molecular dynamics

(ps)

10-12

Ab initio

methods

tight-binding

MNDO, INDO/S

(fs)

10-15

10-10

10-9

10-8

10-7

10-6

10-5

10-4

(mm)

(nm)

LENGTH/meters

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?

Solve:

Answer:

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.
- 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.