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Transcript
Molecular Modeling:
Beyond Empirical Equations
Quantum Mechanics Realm
C372
Introduction to Cheminformatics II
Kelsey Forsythe
Atomistic Model History

Atomic Spectra


Plum-Pudding Model


Neils Bohr (circa 1913)
Wave-Particle Duality



Planck (circa 1905)
Planetary Model


J. J. Thomson (circa 1900)
UV Catastrophe-Quantization


Balmer (1885)
DeBroglie (circa 1924)
Uncertainty Principle (Heisenberg)
Schrodinger Wave Equation

Erwin Schrodinger and Werner Heisenberg(1926)
Classical vs. Quantum
Trajectory
Real numbers


Deterministic (“The value
is ___”)


Variables
Continuous energy
spectrum





Wavefunction
Complex (Real and
Imaginary components)
Probabilistic (“The average
value is __ ”
Operators
 Discrete/Quantized energy
 Tunneling
 Zero-point energy
Schrodinger’s Equation
Hˆ   E

Hˆ - Hamiltonian operator
Hˆ  Tˆ  Vˆ

N


Gravity?
 
i
2
2mi
N

2
C
i j

e ie j
ri  r j
Hydrogen Molecule
Hamiltonian
Hˆ  Tˆ  Vˆ
2

Hˆ  
2

  2p1  2p 2  e21  e22 





m
m
m
m
 p
p
e
e 

 1
1
1
1
1
1 
C






 re1e 2 rp1 p 2 rp1e1 rp1e 2 rp 2e1 rp 2e 2 
Born-Oppenheimer Approximation (Fix nuclei)
Hˆ el  Tˆel  Vˆel  nuclei  Vnuclei
2
2
2
 1





1
1
1
1 
1
e
1
e
2
ˆ
H el   


C





C



2  me me 
rp1 p 2
 re1e 2 rp1e1 rp1e 2 rp 2e1 rp 2e 2 

Now Solve Electronic Problem
Electronic Schrodinger
Equation

Solutions:
F
(r )   c m  m (r )
m
m (r ), the basis set, are of a known form
 Need to determine coefficients (cm)





Wavefunctions
gives probability of finding electrons in space
(e. g. s,p,d and f orbitals)
Molecular orbitals are formed by linear combinations of
atomic orbitals (LCAO)
Hydrogen Molecule
VBT

HOMO
HOMO 
1
( A   B )
2

LUMO
LUMO 
1
( A   B )
2

Hydrogen Molecule

Bond Density
Ab Initio/DFT



Complete Description!
Generic!
Major Drawbacks:



Mathematics can be cumbersome
Exact solution only for hydrogen
Informatics

Approximate solution time and storage intensive
– Acquisition, manipulation and dissemination problems
Approximate Methods

SCF (Self Consistent Field) Method (a.ka. Mean
Field or Hartree Fock)





Pick single electron and average influence of remaining
electrons as a single force field (V0 external)
Then solve Schrodinger equation for single electron in
presence of field (e.g. H-atom problem with extra force
field)
Perform for all electrons in system
Combine to give system wavefunction and energy (E)
Repeat to error tolerance (Ei+1-Ei)
Recall
Schrodinger Equation
 Quantum vs. Classical
 Born Oppenheimer
 Hartree-Fock (aka SCF/central field) method

Basis Sets

Each atomic orbital/basis function is itself
comprised of a set of standard
functions
Atomic Orbital
F
LCAO
(r )   c m  m (r )
m
N
m   Cmje
 mj r 2
j
Expansion Coefficient
Contraction coefficient
(Static for calculation)
STO(Slater Type Orbital):
~Hydrogen
Atom Solutions

GTO(Gaussian Type Orbital): m 
More Amenable to computation
mj r
2
STO vs. GTO

GTO


Improper behavior for
small r (slope equals
zero at nucleus)
Decays too quickly
Basis Sets
Basis Sets
Molecular Orbital
F


 (r )   cm   m (r ) What “we” do!!
m
Atomic Orbital
N
 m   Cmj  j
STO
GTO/CGTO
j
Optimized using atomic ab initio calculations
PGTO
 j e
 j r 2
Gaussian Type Orbitals

Primitives
 ,n,l,m (r,, )  NYl,m (, )r
(2n2l ) r 2
e
Shapes typical of H-atom orbitals (s,p,d etc)
 Contracted



Vary only coefficients of valence (chemically
interesting parts) in calculation
Minimum Basis Set (STO-3G)

The number of basis functions is equal to the
minimum required to accommodate the # of
electrons in the system
H(# of basis functions=1)-1s
 Li-Ne(# of basis functions=5) 1s,2s,2px, 2y, 2pz

Basis Sets
Types:

STO-nG(n=integer)-Minimal Basis Set




Approximates shape of STO using single contraction of
n- PGTOs (typically, n=3)
Intuitive
The universe is NOT spherical!!
3-21G (Split Valence Basis Sets)


Core AOs 3-PGTOs
Valence AOs with 2 contractions, one with 2 primitives
and other with 1 primitive
Basis Sets
Types:
3-21G(*)-Use of d orbital functions (2nd row atoms
only)-ad hoc
 6-31G*-Use of d orbital functions for non-H atoms
 6-31G**-Use of d orbital functions for H as well

Examples

C

STO-3G-Minimal Basis Set
3 primitive gaussians used to model each STO
 # basis functions = 5 (1s,2s,3-2p’s)


3-21G basis-Valence Double Zeta
1s (core) electrons modeled with 3 primitive gaussians
 2s/2p electrons modeled with 2 contraction sets (2primitives and 1 primitive)
 # basis functions = 8 (1s,2s,6-2p’s)

Polarization

Addition of higher angular momentum
functions

HCN

Addition of p-function to H (1s) basis better
represents electron density (ie sp character) of HC
bond
Diffuse functions

Addition of basis functions with small exponents
(I.e. spatial spread is greater)





Anions
Radicals
Excited States
Van der Waals complexes (Gilbert)
Ex. Benzene-Dimers (Gilbert)


w/o Diffuse functions T-shaped optimum
w/Diffuse functions parallel-displaced optimum
Computational Limits

Hartree-Fock limit
NOT exact solution
 Does not include correlation
 Does not include exchange

Exact Energy*
Correlation/Exchange
Basis set size
BO not withstanding
Correcting
Approximations

Accounting for Electron Correlations
DFT(Density Functional Theory)
 Moller-Plesset (Perturbation Theory)
 Configuration Interaction (Coupling single
electron problems)

Computational Reminders



HF typically scales N4
As increase basis set size accuracy/calculation time increases
ALL of these ideas apply to any program utilizing ab initio
techniques NOT just Spartan (Gilbert)
Quick Guide

Basis

Meaning

STO-3G(minimal basis)


3-21G-6-311G(split-valence
basis)



*/**
3 PGTO used for each
STO/atomic orbital
Additional basis functions
for valence electrons
Addition of d-type orbitals
to calculation (polarization)


+/++

** (for H as well)
Diffuse functions (s and p
type) added

++ (for H as well)
Modeling Nuclear Motion
IR - Vibrations
 NMR – Magnetic Spin
 Microwave – Rotations

Modeling Nuclear Motion (Vibrations)
Harmonic Oscillator Hamiltonian
2


1
ˆ
H (r )  
  (r ) 2
2 r 2
8.35E-28
8.35E-28
8.35E-28
8.35E-28
1.4E-18
8.35E-28
8.35E-28
8.35E-28
1.2E-18
8.35E-28
8.35E-28
1E-18
8.35E-28
8.35E-28
8E-19
8.35E-28
8.35E-28
8.35E-28
6E-19
8.35E-28
8.35E-28
4E-19
8.35E-28
8.35E-28
2E-19
8.35E-28
8.35E-28
8.35E-28
0
8.35E-28
0
8.35E-28
8.77567E+14
20568787140
2.03098E-18
1.05374E-18
8.77567E+14
8.77567E+14
8.77567E+14
8.77567E+14
8.77567E+14
8.77567E+14
8.77567E+14
8.77567E+14
8.77567E+14
8.77567E+14
8.77567E+14
8.77567E+14
8.77567E+14
8.77567E+14
8.77567E+14
8.77567E+14
8.77567E+14
8.77567E+14
8.77567E+14
8.77567E+14
0.5
1
8.77567E+14
20568787140
20568787140
20568787140
20568787140
20568787140
20568787140
20568787140
20568787140
20568787140
20568787140
20568787140
20568787140
20568787140
20568787140
20568787140
20568787140
20568787140
20568787140
20568787140
20568787140
1.5
20568787140
1.54682E-18
1.34201E-18
1.15913E-18
9.96207E-19
8.51451E-19
7.23209E-19
6.09973E-19
5.10362E-19
4.2311E-19
3.47061E-19
2.81155E-19
2.24426E-19
1.75987E-19
1.35031E-19
1.0082E-19
7.26787E-20
4.99924E-20
3.22001E-20
1.87901E-20
2 9.29638E-21
2.5
3.29443E-21
8.82365E-19
8.02375E-19
7.26185E-19
6.53795E-19
5.85205E-19
5.20415E-19
4.59425E-19
4.02235E-19
3.48845E-19
2.99255E-19
2.53465E-19
2.11475E-19
1.73285E-19
1.38895E-19
1.08305E-19
8.15147E-20
5.85247E-20
3.93347E-20
2.39447E-20
31.23547E-20
3.5
4.56475E-21
Empirical
for Hydrogen
Molecule9.66155E-19
8.77567E+14Potential
20568787140
1.77569E-18
4