CHEM 834: Computational Chemistry

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CHEM 834 Computational Chemistry
Parameterized Methods
March 31, 2009
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Topics
last time

• basis sets

• accuracy versus cost

• solvation

today

• semi-empirical molecular orbital methods

• molecular mechanics/force-fields

• multi-level methods

• summary of energy calculation methods

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Hartree-Fock
constructing and diagonalizing the Fock matrix is
the computationally intensive part of a
Hartree-Fock calculation
Fock equations (what we diagonalize to get
molecular orbitals)
FC SC?
Fock matrix
orbital energies
overlap matrix
molecular orbital coefficients
C is a K x K matrix whose columns define the
coefficients, c?i
? is a diagonal matrix of the orbital energies,
?i
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Hartree-Fock
constructing and diagonalizing the Fock matrix is
the computationally intensive part of a
Hartree-Fock calculation
Secular determinant (what we have to solve for
diagonalization)
K x K determinant

• K number of basis functions

• more basis functions ? bigger determinant to solve

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Hartree-Fock
constructing and diagonalizing the Fock matrix is
the computationally intensive part of a
Hartree-Fock calculation
Fock matrix elements (closed-shell system)
two-electron integrals cause K4 scaling of
Hartree-Fock
overlap matrix elements
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Semi-Empirical Molecular Orbital Methods
semi-empirical molecular orbital methods reduce
the effort needed to construct and diagonalize
the Fock matrix by eliminating some terms and
fitting others to experimental data
General Approximations
1. core electrons are neglected

• assumes changes in chemical environment do not
  affect core orbitals

• reduces K ? smaller secular determinant, fewer
  integrals

2. Slater functions are used as basis functions
for valence orbitals

• Slater functions capture cusp behaviour and
  exponential decay with fewer functions that
  Gaussian basis sets

• reduces K ? smaller secular determinant, fewer
  integrals

• overlap integrals can be calculated easily with
  Slater functions

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Semi-Empirical Molecular Orbital Methods
semi-empirical molecular orbital methods reduce
the effort needed to construct and diagonalize
the Fock matrix by eliminating some terms and
fitting others to experimental data
General Approximations

• most two-electron integrals are neglected, others
  are fit to experimental data

if basis functions ?, ?, ?, ? are on three or
four different atomic centers

• eliminates most of the two-electron integrals

are fit to experimental data if basis functions
?, ?, ?, ? are on only one or two atomic centers

• simplifies the calculation of the two-electron
  integrals that are not eliminated

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Complete Neglect of Differential Overlap (CNDO)
early semi-empirical method
Assumptions

• each valence orbital is represented by 1 Slater
  function

• the basis functions are orthonormal in the
  secular determinant

• simplifies the diagonalization of the Fock matrix

• only two-electron integral involving one or two
  basis functions are kept

• the integrals that remain involve at most two
  basis functions on at most two atoms

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Complete Neglect of Differential Overlap (CNDO)
the remaining terms are fit to experimental data
Two-electron integrals

• define a parameter, ?AA

electron affinity of atom A
ionization potential of atom A

• assign integrals as

basis functions on same atom
basis functions on different atoms
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Complete Neglect of Differential Overlap (CNDO)
the remaining terms are fit to experimental data
One-electron integrals
ionization energy from orbital ? on atom A, e.g.
2s orbital on carbon

• overlap between orbitals ? and ?

• parameters fit to experimental data to give the
  interaction strength between atoms A and B

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Complete Neglect of Differential Overlap (CNDO)
the remaining terms are fit to experimental data
Altogether, we have parameters for all of the
integrals

• these parameters are selected so that the results
  of CNDO reproduce experimental data

• only having integrals with 2 basis functions
  makes the scaling K2 instead of K4

• parameterization reduces the effort needed to
  construct the Fock matrix in the first place

once the Fock matrix is constructed, it is
diagonalized to get the molecular orbital
coefficients
In general

• the results are not very accurate ? bad
  geometries, etc.

• this shouldnt be surprising considering the
  significant approximations

• CNDO calculations are rarely performed today, but
  the concepts form the basis for other
  semi-empirical methods

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Intermediate Neglect of Differential Overlap
(INDO)
instead of treating two-electron integrals
generically, lets treat s-s, s-p, and p-p
differently
In CNDO

• property of atom A

• does not distinguish between s, p, d, etc.
  orbitals on the same atom

• CNDO only considers up to 2 different basis
  functions on atom A

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Intermediate Neglect of Differential Overlap
(INDO)
instead of treating two-electron integrals
generically, lets treat s-s, s-p, and p-p
differently
In INDO, consider up to four basis functions on
atom A
including only s and p functions (and applying
symmetry)

• these parameters can be fit to spectroscopic data

• most common INDO method is INDO/S (also called
  ZINDO), which is use to simulate excited states

• INDO/S can be very accurate if the molecules
  studied are similar to those used to fit the
  parameters

• more combinations if d orbitals are included

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Neglect of Diatomic Differential Overlap (NNDO)
INDO and CNDO do not distinguish between orbital
types when treating two-center two-electron
orbitals
the NNDO method considers all combinations of
orbitals on atoms A and B

• these integrals are approximated with point
  charges, dipoles, quadrupoles, etc.

• the values are selected to reproduce experimental
  data

• three- and four-center integrals are still
  neglected ? still have K2 scaling

. . .
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Neglect of Diatomic Differential Overlap (NNDO)
all modern semi-empirical calculations are
performed within the NNDO approximation
the most common NNDO models are
MNDO (modified neglect of differential overlap)
AM1 (Austin model 1)
PM3 (parameterized model 3)
comments on semi-empirical models

• can treat systems with thousands of atoms

• accuracy can be good if applied to molecules
  similar to those used for parameterization

• transferability is questionable (i.e. you cant
  trust semi-empirical models when applied to
  systems that are quite different from those used
  in parameterization)

• thermal effects and zero-point energies are
  included by fitting to experimental data ? do not
  add them in calculated quantities

• semi-empirical methods can be useful for
  preliminary qualitative work

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Accuracy of Semi-Empirical Methods
semi-empirical methods can be used to locate
minima and transition states on the potential
energy surface, but you should be skeptical of
the results
Mean unsigned errors (kcal/mol) in predicted
heats of formation
Lighter includes C, H, N, O, F and Heavier
includes Al, Si, P, S, Cl, Br, I, Hg
errors on reaction barriers can be much larger
because the models are not parameterized using
data pertaining to transition states
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Summary of Semi-Empirical Methods
semi-empirical methods make various
approximations to the Hartree-Fock method

• neglect core electrons

• minimal basis set of Slater functions

• neglect three- and four-center integrals

• treat all remaining integrals as parameters that
  are fit to experimental data

semi-empirical methods are fast, but of limited
accuracy for systems that werent used in the
parameterization

• can treat systems with thousands of atoms

• accuracy is qualitative at best

• not suited for quantitative work

• common methods include AM1, PM3, MNDO and ZINDO
  (excited state calculations)

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Current Directions in Semi-Empirical Methods
the development of semi-empirical techniques is
still a very active area of research
treatment of transition metals

• d orbitals were neglected in early semi-empirical
  models

• recently, models for treating transition metals
  such as PM3(tm) and MNDO/d have been reported

application in structure-activity relationship
studies

• semi-empirical models are used extensively to
  calculate the properties of large sets of
  molecules for the development of
  structure-activity relationship, e.g. in drug
  design

tight-binding density functional theory

• there has been significant recent work in the
  development of semi-empirical methods based on
  DFT instead of Hartree-Fock

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Semi-Empirical Methods in Gaussian/Gaussview
Gaussian is capable of running semi-empirical
calculations at the AM1, PM3, MNDO and ZINDO
levels (as well as some others)

• specify the method on the route line

• do not specify a basis set

• AM1, PM3, and MNDO are used for ground states

• ZINDO is available for excited state calculations

• only certain elements may be available

• input and output are analogous to Hartree-Fock
  calculations

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Semi-Empirical Methods in Gaussian/Gaussview
semi-empirical methods can be selected from the
Calculation Setup Menu in Gaussview
set method to Semi-empirical
select specific model
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Semi-Empirical Methods in Gaussian/Gaussview
semi-empirical methods can be selected from the
Calculation Setup Menu in Gaussview
set method to Zindo
set up details of excited state calculations
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Force-Fields/Molecular Mechanics
instead of treating the electronic structure of
molecules with quantum mechanics, lets consider
the change in energy due to deformations from the
molecules minimum energy geometry
Types of deformations
butane
1. bond stretching
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Force-Fields/Molecular Mechanics
instead of treating the electronic structure of
molecules with quantum mechanics, lets consider
the change in energy due to deformations from the
molecules minimum energy geometry
Types of deformations
butane
1. bond stretching
2. angle bending
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Force-Fields/Molecular Mechanics
instead of treating the electronic structure of
molecules with quantum mechanics, lets consider
the change in energy due to deformations from the
molecules minimum energy geometry
Types of deformations
butane
1. bond stretching
2. angle bending
3. torsional rotation
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Force-Fields/Molecular Mechanics
instead of treating the electronic structure of
molecules with quantum mechanics, lets consider
the change in energy due to deformations from the
molecules minimum energy geometry
Types of deformations
butane
1. bond stretching
2. angle bending
3. torsional rotation
4. non-bonded interactions

• van der Waals interactions

• Coulomb interactions

force-fields describe the changes in energy due
to these deformations using experimental data
instead of first-principles methods
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Force-Fields/Molecular Mechanics
instead of treating the electronic structure of
molecules with quantum mechanics, lets consider
the change in energy due to deformations from the
molecules minimum energy geometry
with a force-field, we write the energy as a sum
of terms that depend only on the geometry
distance between atoms A and B
torsional angle formed by atoms A, B, C, and C
angle formed by atoms A, B, and C
distance between atoms A and B
charges on atoms A and B
non-bonded interactions, depend on atoms that are
not connected to each other
bonded interactions, depend on atoms that are
connected to each other
each term is then described with a parameterized
potential energy expression
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Force-Fields/Molecular Mechanics
instead of treating the electronic structure of
molecules with quantum mechanics, lets consider
the change in energy due to deformations from the
molecules minimum energy geometry
with a force-field, we write the energy as a sum
of terms that depend only on the geometry
each term is then described with a parameterized
potential energy expression
Obvious questions

• what potential energy expressions do we use?

• where do we get the parameters?

• is it valid to break down the energy this way?

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Bond Stretching Terms
consider the change in energy due to stretching a
bond between atoms A and B
we can expand the energy as a Taylor series about
R0
DAB
R0
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Bond Stretching Terms
consider the change in energy due to stretching a
bond between atoms A and B
looking at the first three terms
DAB
we have
R0
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Bond Stretching Terms
consider the change in energy due to stretching a
bond between atoms A and B
this gives us a harmonic or quadratic bonding
potential
experimental
energy
DAB
spring constant for the bond between atoms A
and B
RAB
R0
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Bond Stretching Terms
consider the change in energy due to stretching a
bond between atoms A and B
the harmonic potential

• provides a good description of the potential
  around R0

energy

• energy goes to ? for large RAB

? bonds cannot break
? no reactions

• requires 2 parameters

? R0 equilibrium bond distance
RAB
? kAB(2) vibrational force constant

• parameters are obtained through spectroscopic
  measurements or quantum chemical data (recall
  normal mode calculations)

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Bond Stretching Terms
consider the change in energy due to stretching a
bond between atoms A and B
the cubic potential

• provides a good description of the potential
  around R0

energy

• energy goes to -? for large RAB

? all molecules are unstable

• requires 3 parameters

diverges to -?
? R0 equilibrium bond distance
? kAB(2) vibrational force constant
RAB
? kAB(3) anharmonic force constant
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Bond Stretching Terms
consider the change in energy due to stretching a
bond between atoms A and B
the quartic potential
energy

• provides a better description of the potential
  around R0 than the harmonic and cubic potentials

• energy goes to ? for large RAB

? no bond dissociation

• requires 4 parameters

RAB
? R0 equilibrium bond distance
? kAB(2) vibrational force constant
? kAB(3) anharmonic force constant
? kAB(4) 4th order force constant
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Bond Stretching Terms
consider the change in energy due to stretching a
bond between atoms A and B
the Morse potential

• provides a good description of the potential
  across a large range of RAB

energy

• can account for bond dissociation

• requires 3 parameters

? R0 equilibrium bond distance
? DAB bond dissociation energy
? ?AB generic fitting parameter
RAB

• Morse potential is computationally intensive to
  compute

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Bond Stretching Terms
consider the change in energy due to stretching a
bond between atoms A and B
Overall

• most force-fields describe bond stretching with
  harmonic, cubic, or quartic potentials

• computationally easy

• harmonic parameters are readily accessible from
  experiments

• more parameters required for cubic and quartic
  forms

• provide a good description of bonding around
  equilibrium distance

• not suitable for describing bond
  dissociation/formation

• Morse potential is not used very often

• computationally intensive

• not all parameters are readily accessible from
  experiments

• description of bonding around equilibrium
  distance is similar to polynomial potentials

• suitable for describing bond dissociation/formatio
  n (sometimes used in reactive force-fields)

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Angle Bending
in molecules, angles between bonds take on
preferred values

• ?ABC is the angle formed between bonds A-B and B-C

• in force-fields bonds and angles between atoms
  are specified explicitly

B
C
A

• so, for water, you would have bonds A-B and B-C
  and angle A-B-C, but not bond A-C and angle A-C-B

?ABC
angle bending is usually described with a
polynomial

• most common force-fields use N 2 - 6

• force constants and equilibrium angles are taken
  from experimental data and/or quantum chemical
  calculations

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Torsional Rotation
we also have to account for rotation about bonds
Rotation is periodic

• one whole rotation is 360º

• for some rotations, the initial structure is
  recovered every 360/n degrees

• e.g. rotation about the C-C bond in ethane is
  3-fold degenerate

Standard torsional potential
looking down the C-B bond
rotational barrier
equilibrium torsional angle - 180º
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Torsional Rotation
we also have to account for rotation about bonds
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Non-Bonded Interactions
atoms that arent connected through bonds still
interact with each other
van der Waals interactions

• instantaneous dipole moments cause atoms to
  attract each other at long distances

• at short distances, repulsion between electron
  clouds (Pauli repulsion) overcomes this attraction

repulsive region
E0

• in between there is an equilibrium distance for
  non-bonded interactions

energy
attractive region
RAB
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Non-Bonded Interactions
atoms that arent connected through bonds still
interact with each other
van der Waals interactions

• these non-bonded interactions are generally
  modeled with a Lennard-Jones potential

E0

• also called a 12-6 potential

?AB

• other models are in common use, too

energy
?AB

• parameters are obtained from experiments and
  high-level calculations (vdw interactions are due
  to electron correlation)

RAB
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Non-Bonded Interactions
atoms that arent connected through bonds still
interact with each other
Coulomb interactions

• we like to think of atoms as having partial
  charges (this is nonsense, but we think that way
  anyhow)

• in force-fields charged atoms interact through a
  Coulomb potential

• charges are usually obtained through quantum
  chemical calculations

• models exist to let the charges on the atom
  change if there is a change in the chemical
  environment

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Force-Fields
putting together all of the bonded and non-bonded
interactions gives a force-field (also called
molecular mechanics)
this gives a simple representation of the energy
based only on the molecular geometry and a set of
parameters
evaluating these types of energy expressions is
much easier than solving for a wavefunction
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Atom Types
the parameters for a force-field depend on the
nature of the bonds angles, etc.
Force-fields define various atom types

• its a bit more complicated to set up force-field
  calculations than quantum chemical calculations

• e.g. in the AMBER force-field

CT any sp3 carbon
C an sp2 carbon in a carbonyl
CA an aromatic sp2 carbon

• you also have to explicitly define all bonds,
  angles, torsions, etc.

CM an sp2 carbon, double bonded
CC an sp2 aromatic carbon in a 5-membered ring
with one substituent and next to a nitrogen
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Connectivity
the atom types, bonds, angles, and torsions that
define structural isomers are different
carbonyl oxygen
double bond
sp3 carbon
alcohol oxygen
single bond
sp2 carbon
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Connectivity
the atom types, bonds, angles, and torsions that
define structural isomers are different
carbon bonded to alcohol oxygen
C-O bonds in ethers

• the force-field parameters for structural isomers
  will be different

you cannot compare the energies of structural
isomers (or other systems with the same numbers
of atoms) with force-fields!!!

• no reaction energies or barriers with force-fields

• you can only compare energies of systems where
  the connectivity and atom types remain constant,
  e.g. conformers

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Reaction Energies and Barriers
force-fields cannot be used to evaluate reaction
energies and barriers

• this energy expression and parameters are defined
  for a specific set of atom types, bond types,
  angle types, etc.

• reactions cause the atom types, bond types, etc.
  to change

• therefore, different sets of force-field
  parameters are used to study the reactant and
  product structures of a reaction

• since, different force-field parameters are used
  in each case, it is not valid to compare the
  energies to evaluate reaction energies and
  barriers

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Steps in a Force-Field Calculation
In a force-field calculation, you must
1. get a molecular geometry
2. specify the atom types and connectivity
3. assign the parameters
4. evaluate energy and forces on nuclei
5. then what?
Force-fields are used for

• geometry optimizations (no reaction energies or
  barriers)

• e.g. determining structures of large molecules

• molecular dynamics

• simulating behaviour of system as it evolves over
  time (well cover this in the next lecture)

• e.g. protein folding

• Monte Carlo calculations

• averaging over the properties of large numbers of
  molecules in different configurations

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Advantages/Disadvantages of Force-Fields
Advantages

• very inexpensive computationally

• force-fields can be used to treat systems with
  millions of atoms

• the energy is broken-down into terms that are
  chemically-intuitive

• can be very accurate if studying systems similar
  to those used to get the parameters

Disdvantages

• cannot be used to study reactions

• limited transferability

• parameterization is specific to a set of
  experimental data

• so, a given set of parameters only apply to a
  narrow set of systems

• parameterization is difficult to do, and there is
  no clear direction for improvement

• limited availability for some systems (e.g.
  transition metals)

• no theoretical justification

• parameters are not orthogonal (changing one
  parameter can significantly affect performance of
  others)

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What Force-Field Do I Use?
a huge number of force-fields exist
Available force-fields
AMBER biomolecules
CHARMM biomolecules and organics
Dreiding main-group organics and inorganic
molecules
MM3 organics and biomolecules
UOPLS biomolecules, some organics, water
UFF general across periodic table
VALBOND transition metal compounds
Which force-field to use

• has the atom types you need

• parameterized using systems similar to yours

• test it against experimental data for your system

• consider what level of accuracy you need

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Force-Field Methods in Gaussian/Gaussview
Gaussian has the Amber, Dreiding, and UFF
force-fields
In Gaussian

• type Amber, Dreiding or UFF on the route line

• do not specify a basis set

• Gaussian will determine the atom types, etc.
  automatically

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Force-Field Methods in Gaussian/Gaussview
Gaussian has the Amber, Dreiding, and UFF
force-fields
In Gaussview

• force-fields are selected from the Calculation
  Setup Menu

select Mechanics from Method Tab
select specific force-field model
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Multi-Layer Methods
you can combine force-fields with quantum
chemical methods to study large systems
my first computational study
What we did

• replaced Mes with H

• what if Mes did matter?

Mes
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Multi-Layer Methods
to reduce computational effort, you can treat
bulky substituents with force-fields
quantum chemical region
force-field regions
Mixing QC and FF methods

• often called QM/MM

• substituents treated with FF methods do not
  contribute directly to the electronic structure

• substituents treated with FF methods do
  contribute to steric effects and long-range
  electrostatics

• can be tricky (need to balance forces across
  boundaries)

• provides a way to study larger systems where
  reactions occur in known localized regions

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Multi-Layer Methods
many different multi-layer methods exist
QM/MM
ONIOM
level 1
level 2
level 3

• mix quantum chemical and force-field methods

• mix several levels of methods (can be quantum
  chemical and/or force-fields)

Applications of Multi-Layer methods

• modeling active sites of catalysts

• modeling active sites of enzymes

• modeling explicit solvation (e.g. solvent with
  force-field, solute with quantum chemistry)

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Summary of Energy Calculation Methods
two main categories of methods
1. Energy Calculation Methods

• relate the energy of a molecule to its geometry

• fundamental to all calculations

• are used to construct the potential energy surface

• include

1. force-fields/molecular mechanics
- ball-and-spring approach
- neglects electronic structure
2. ab initio methods
- fully quantum mechanical treatment of the
electronic wavefunction
3. semi-emiprical molecular orbital methods
- ab initio methods with an approximate/parameteri
zed Hamiltonian
4. density functional theory methods
- fully quantum mechanical treatment of the
electron density
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Summary of Energy Calculation Methods
ab initio methods
apply quantum mechanics to the electrons in a
molecule using only mathematical approximations
time-independent Shrödinger equation
molecular Hamiltonian within the Born-Oppenheimer
approximation
electronic terms
nuclear-nuclear repulsion from nuclei at fixed
positions
57
Summary of Energy Calculation Methods
ab initio methods
apply quantum mechanics to the electrons in a
molecule using only mathematical approximations
Hartree-Fock

• assume a trial wavefunction comprising 1 Slater
  determinant

• Slater determinant composed of molecular orbitals
  that are linear combinations of basis functions

58
Summary of Energy Calculation Methods
ab initio methods
apply quantum mechanics to the electrons in a
molecule using only mathematical approximations
Hartree-Fock

• variational optimization of molecular orbitals
  shows that the best set of molecular orbitals
  are eigenfunctions of the Fock operator

kinetic energy and nuclear-electron attraction
exchange interactions (Pauli repulsion) between
electrons
average Coulomb repulsion between electrons
59
Summary of Energy Calculation Methods
ab initio methods
apply quantum mechanics to the electrons in a
molecule using only mathematical approximations
Hartree-Fock

• captures exchange interactions, but only treats
  electron-electron repulsion in an average way

• instantaneous interactions between electrons that
  are not captured by Hartree-Fock result in a
  correlation energy

• post-Hartree-Fock methods that capture some
  correlation energy are built on the Hartree-Fock
  wavefunction

• semi-empirical molecular orbitals methods are
  based on approximation to Hartree-Fock

60
Summary of Energy Calculation Methods
ab initio methods
apply quantum mechanics to the electrons in a
molecule using only mathematical approximations
Post-Hartree-Fock methods

• use Hartree-Fock wavefunction to build extra
  determinants

• construct a multi-determinant wavefunction to
  capture electron correlation

• configuration interaction, coupled-cluster, and
  multireference methods use a geometric series of
  determinants in conjunction with the variational
  principle

• perturbation theory methods treat electron
  correlation as a minor perturbation to the
  non-interaction Hamiltonian

• methods such as CCSD(T) can be very accurate, but
  are limited to small system sizes

61
Summary of Energy Calculation Methods
semi-empirical methods
make approximations in the construction of the
Fock matrix
AM1, PM3, MNDO

• neglect all three- and four-center integrals in
  the construction of the Fock matrix

• fit all remaining integrals to experimental data

• use Slater basis functions to minimize the size
  of the Fock matrix

• can treat systems containing thousands of atoms

• of limited accuracy for systems that differ
  significantly from those used to fit the integrals

62
Summary of Energy Calculation Methods
density functional theory
quantum mechanical treatment of the electron
density
Kohn-Sham DFT

• uses fictitious system of one-electron orbitals
  to represent the ground state density

• facilitates evaluation of kinetic energy

• variational treatment of the density shows that
  the best orbitals are eigenfunctions of the
  Kohn-Sham operator

exchange-correlation energy
electron kinetic energy
nuclear-electron attraction
average electron-electron repulsion
63
Summary of Energy Calculation Methods
density functional theory
quantum mechanical treatment of the electron
density
Kohn-Sham DFT

• in principal, Kohn-Sham DFT is exact

• in practice, approximate exchange-correlation
  functionals must be used

• computational cost is similar to Hartree-Fock

• accuracy is similar to post-Hartree-Fock methods

• currently, DFT is the standard quantum chemical
  approach

64
Summary of Energy Calculation Methods
force-fields
neglect electrons, and simply consider changes in
molecular geometry
Force-Fields

• break down energy into contributions from
  bonding, angles, torsions, van der Waals and
  Coulomb interactions

• energy is evaluated using potential energy
  functions that contain parameters

• parameters are fit to experimental data or
  high-level calculations

• can be applied to systems containing millions of
  atoms

• not suitable for studying bond formation/dissociat
  ion, or reaction energies and barriers