04/09/18

COMPUTER AIDED DRUG DESIGN

23. Quantum Mechanics

Mukesh Doble
Professor
DEPARTMENT OF BIOTECHNOLOGY
IIT MADRAS

ØMolecular Mechanics - force fields

üNuclei and electrons are lumped into atom-like particles.

üInteractions are based on springs and classical potentials.
ücalculate structure and dynamics, total energies, entropies, free energies, and diffusion

ØDifferent functional forms of force fields

ØParameters - atom types, based on bonding and environment

ØBoundary conditions

ØGeometry optimisation

ØMinimum energy confirmation – Energy minimisation

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2. Quantum mechanics

ü Nuclei and electrons of the molecules are seperately

considered
ü Two approaches
Ab initio - more rigorous, from first principles (no
stored parameters or data), takes a long time,
restricted to small molecules
Semi-empirical - faster, but less accurate, can be used on
larger molecules
ü Useful for MO energies, partial charges, electrostatic
potentials, dipole moments

Quantum theory is based on Schrodinger's equation:

electrons are considered as wave-like particles whose "waviness" is

mathematically represented by a set of wave functions

addresses the following questions:

Where are the electrons and nuclei in space? configuration,

conformation, size, shape, etc.
•what are their energies? heat of formation, conformational
stability, chemical reactivity, spectral properties, etc

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Quantum mechanics methods are based on the following principles:

•Nuclei and electrons are distinguished from each other.

•Electron-electron (usually averaged) and electron-nuclear interactions are
explicit.
•Interactions are governed by nuclear and electron charges (i.e. potential
energy) and electron motions.
•Interactions determine the spatial distribution of nuclei and electrons and
their energies

•Ab initio

•Limited to tens of atoms and best performed using a supercomputer.

•Can be applied to organics, organo-metallics, and molecular fragments (e.g.
catalytic components of an enzyme).
•Vacuum or implicit solvent environment.
•Can be used to study ground, transition, and excited states (certain methods).
•Specific implementations include: GAMESS and GAUSSIAN.

•Semi empirical

•Limited to hundreds of atoms.

•Can be applied to organics, organo-metallics, and small oligomers (peptide,
nucleotide, saccharide).
•Can be used to study ground, transition, and excited states (certain methods).
•Specific implementations include: AMPAC, MOPAC, and ZINDO.

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Basis Sets

•The set of one-electron wavefunctions used to build

molecular orbital wavefunctions is called the basis
set
Minimal basis set is one in which only occupied orbitals of each
isolated atom are used to compose the molecular orbitals.

Unoccupied molecular orbitals are called virtual orbitals

Born-Oppenheimer Approximation
Decouples the electronic and nuclear degrees of freedom. Assumes the
nuclear centers of mass are fixed for a given calculation. I.e., the wave
function is parameterized with respect to the nuclear coordinates.

Hartree-Fock Approximation
The many electron problem is approximated by a sequential calculation of
the response of the ith electron in the average potential of the rest of the
electrons. This one-electron operator is called the Fock operator.

Anti-symmetric wave function.

Wave functions describing electrons obey Fermi Dirac statistics, that is,
they must be anti-symmetric with respect to an interchange of coordinates.
This is conveniently expressed in terms of a determinant called the Slater
Determinant.

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. Roothan Equations
The Hartree-Fock equations are a set of coupled integro-differential equations.
Using basis vectors of Hilbert space, this can be transformed into a set of algebraic
equations This gives a generalized eigenvalue problem to solve

HF-SCF Hartree-Fock Self Consistent Field

Name of the procedure to calculate the charge distribution. The problem is non-
linear (the Fock matrix is dependent on the variational coefficients that are being
calculated) thus an iterative approach is used (self consistency)

Post Hartree-Fock Calculations

The Hartree-Fock calculation has a built in deficiency; no electron-electron
correlation. Thus, further work needs to be performed. Lumped as post Hartree-
Fock calculations. Most of these schemes involve the diagonalization of large
matrices.

•Ab initio Methods

Some of the major ab initio programs are

1.GAMESS
2.Gaussian

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Semi-Empirical Methods

•Parameterize the Hamiltonian by fitting it to experimental data or

the results of first principles calculations.

•These calculations are less demanding but less accurate.

•They cannot be applied to systems which are radically different

from those used in the parameterization procedure.

•Can be used to obtain heats of formation, ionization potentials,

optical spectra, electrostatics

Semiempirical Molecular Orbital Methods

•Semiempirical approximations involve the neglect of three- and four-

center integrals arising in the J and K terms of the Hartree-Fock
equation.
•For molecules with large numbers of electrons, evaluation of these
integrals can be computationally impractical.
•Experimentally determined parameters (or in some cases,
parameters determined from ab initio calculations on model systems)
are used to compensate for the missing integrals.
•Such parameters are obtained from measured or calculated
ionization potentials, electron affinities, and spectroscopic quantities.

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Useful Results From Molecular Orbital Calculations

•Molecular orbital energies, and coefficients .
•Total electronic energy, Elec, calculated from the sum of the Coulomb integrals and
the molecular orbital energies for all molecular orbitals in a molecule.
•Total nuclear repulsion energy Vnucleus-nucleus.
•Total energy, Etot, calculated from Eelec + Vnucleus-nucleus.
•Heat of Formation, calculated from Etot - Eisolated atoms. The heat of formation is
used for evaluating conformational energies.
•Partial atomic charges, q, calculated from the molecular orbital coefficients using
methods such as Mulliken population analysis, or using electrostatic potential fitting
methods (Wendy Cornell, UCSF).
•Electrostatic potential.
•Dipole moment.

Semiempirical HF Methods

1.Extended Huckel Theory (EHT) - FORTICON8.

2.Complete Neglect of Differential Overlap (CNDO) and enhancements (CNDO/
1, CNDO/2, CNDO/S, etc.).
3.Intermediate Neglect of Differential Overlap (INDO).
4.Modified INDO (MINDO) and enhancements (MINDO/2, MINDO/2', and
MINDO/3) - AMPAC (MINDO/3 only).
5.Michael Zerner's INDO (ZINDO) - ZINDO.
6.Modified Neglect of Diatomic Overlap (MNDO) - AMPAC, MOPAC, Gaussian.
7.Austin Model 1 (AM1) - AMPAC, MOPAC, Gaussian.
8.Parametric Method 3 (PM3) - AMPAC, MOPAC, Gaussian.
9.SemiChem Austin Model 1 (SAM1) - AMPAC. Explicitly treats d-orbitals.

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Semi-empirical methods

• Time required for HF - ab initio scales as N4

– calculation of one-electron and two-electron integrals
• Semi-empirical methods speed things up by
– Considering only the valence electrons
– Using only a minimal basis set
– Setting some integrals to zero, only N2 to evaluate
– Use simple formulae to estimate the values of the remaining integrals
• The evaluated integrals are parameterised based on calculations or
experimental results (hence “semi-empirical”)
– This compensates (to some extent) for the simplifications employed
– Implicitly includes electron correlation neglected by HF

Semi-empirical methods
• CNDO (Complete Neglect of Differential Overlap)
– CNDO (Pople, 1965)
• INDO (Intermediate Neglect of Differential Overlap)
– INDO (Pople, Beveridge, Dobosh, 1967)
– INDO/S, “ZINDO/S” (Ridley, Zerner, 1973)
– MINDO/3 (Bingham, Dewar, Lo, 1973)
– SINDO1 (Nanda, Jug, 1980)
• NDDO (Neglect of Diatomic Differential Overlap)
– MNDO (Dewar, Thiel, 1977)
– AM1 (Dewar, ..., Stewart, 1985) - MNDO/d (Thiel, Voityuk, 1996)
– PM3 (Stewart, 1989) - AM1/d (Voityuk, Rosch, 2000)
- RM1 (Rocha, ..., Stewart, 2006)
– SAM1 (Dewar, Jie, Yu, 1993) - PM6 (Stewart, 2007)

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INDO
• Compared to CNDO, INDO allows different values for the one-centre
two-electron integrals depending on the orbital types involved (s, p, d,
etc.)
• INDO more accurate than CNDO at predicting valence bond angles but
poor overall at predicting molecular geometry
• ZINDO/S is a parameterisation of INDO using spectroscopic data
– First described by Ridley, Zerner (1973)
– Since then, Zerner and co-workers extended to include most of the
elements in the periodic table
– ZINDO/S still widely used for prediction of electronic transition
energies and oscillator strengths, particularly in transition metal
complexes (UV-vis spectrum)
• ZINDO/S results can be of comparable accuracy to those obtained with
the more rigorous (slower) TD-DFT method

NNDO
• Compared to INDO, NNDO allows different values for the two-centre
two-electron integrals depending on the orbital types involved (s, p, d,
etc.)
• AM1 “Austin Model 1”
– Dewar, Zoebisch, Healy, Stewart (1985)
• PM3 – “Parameterised Model 3”
– Stewart (1989)
– Essentially AM1 with an improved parameterisation (automatic
rather than manual, large training set)
• Both AM1 and PM3 are still widely used
• PM6 – Stewart (2007)
– Main objective to improve handling of hydrogen bonds
– PM6 was included in Gaussian09
http://openmopac.net/Manual/accuracy.html

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Density Functional Theory (DFT)

• Walter Kohn - Nobel Prize in Chemistry 1998
– "for his development of the density-functional theory“
• Based on the electron density rather than the wavefunction
– DFT optimises the electron density while MO theory (HF, etc.) optimises the
wavefunction
• Components of the Hamilitonian are expressed as a function of the electron
density (ρ), itself a function of r
– Function of a function = “functional”
– Composed of an exchange functional and a correlation functional
• Variational principle can also be shown to hold
– Hohenberg-Kohn Variational Theorem
– The correct electron density will have the lowest energy
• The approach used to solve for the density is the Kohn-Sham (KS) Self-
consistent Field (SCF) methodology

Density Functional Theory (DFT)

• Local spin density approximation (LSDA)
– Functional values can be calculated from the value of ρ at a point
– Exchange functionals: LDA, Slater (S), Xα
– Correlation functionals: VWN
• Generalised gradient approximation (GGA)
– Functional values use gradient of ρ
– Exchange functionals: B (Becke), B86, PBE
– Correlation functionals: P86, PW91, LYP (Lee, Yang, Parr)
• Name of DFT method is made by joining the exchange and correlation functionals
into one word
– E.g. B + PW91 = BPW91

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Density Functional Theory (DFT)

• Hybrid DFT methods

– Includes some HF exchange – not “pure” DFT
– B3 - Becke’s 3-parameter functional for exchange (includes some HF, LDA
and B)
– Also some one-parameter models: B1, PBE1, mPW1
• B3LYP is the most popular DFT functional to date

DFT compared to MO theory

• DFT is an exact theory (unlike HF which neglects electron correlation) but the
functionals used are approximate
• DFT calculations scale as N3
• More rapid basis-set convergence
• Much better for transition metal complexes
• Known problems with DFT
– Not good at dispersion (van der Waals)
– H-bonds are somewhat too short
– overdelocalises structures
• Hard to know how to systematically improve DFT results
– In MO, can keep increasing the basis set and level of theory

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Semi-Empirical Methods

• Some of the electrons are considered explicitly

• Reduces computational demand of the problem

Pilar, F.L. Elementary Quantum Mechanics. Second Edition. Dover

Publications, Inc. Mineola, New York, 1990, 454.

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