Download An Overview of Computational Chemistry

Chem. 141
Dr. Mack
An Overview of Computational Chemistry
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Theoretical Chemistry:
The mathematical description of chemistry
Computational Chemistry:
A mathematical method that is sufficiently well developed
that it can be automated for implementation on a
computer.
Chemical Problems
Computer Programs
Physical Models
Math formulas
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Note that the words exact and perfect do not appear in
these definitions.
•Computational chemistry is based on a approximations
and assumptions.
•Only a real experimental measurement can approach the
limits of exactness!
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What does Computational
Chemistry Calculate?
Energy, Structure, and Properties
• Molecular Geometries:
• What is the energy for a given geometry?
• How does energy vary when geometry changes?
• Which geometries are stable?
• How does energy change w/r extenal
perturbation?
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2
What else can be computed:
•Enthalpies of formation
•Dipole moment
•Orbital energy levels (HOMO, LUMO, others)
•Ionization energy (HOMO energy)
•Electron affinity (LUMO energy)
•Electron distribution (electron density)
•Electrostatic potential
•Vibrational frequencies and normal modes (IR
spectra)
•Electronic excitation energy (UV-Vis spectra)
•NMR chemical shifts and coupling constants
•Reaction path and barrier height
•Reaction rate
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What can one learn from computational calculations?
calculations?
• Molecular Visualization (Graphic Representation)
•
•
•
•
Molecular Mechanics (Classical Newtonian Physics)
Semi-empirical Molecular Orbital Theory
Quantum
Ab Initio Molecular Orbital Theory
Mechanical
Methods
Density Functional Theory
• Geometry Optimization
• Molecular Dynamics
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3
Types of Calculations:
Molecular Mechanics (MM)
Empirical energy functions parameterized against experimental
dataFast, simple, generally not very accurate (> 104 atoms)
Semi-empirical Molecular Orbital (MO) Methods
Can treat moderate sized molecules (> 102 atoms).
Accuracy depends on parameterization.
Ab-initio Molecular Orbital Methods
Computationally demanding (> 10 atoms)
Accuracy can be systematically improved.
Density Function Theory (DFT)
I will focus on
these topics
More efficient than ab-initio calculations (> 10 atoms)
Accuracy varies, however, there is no systematical way to improve
the accuracy.
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Types of Computational Calculations:
Ab Initio:
Initio: The term "Ab Initio" is latin for "from the beginning".
•Computations of this type are derived directly from
theoretical principles, with no inclusion of experimental data.
•Mathematical approximations are usually a simple functional
form for a an approximate solution to a differential equation.
A wave function! (The Shrö
Shrödinger Equation)
The most common type of ab initio calculation is called a
HartreeHartree-Fock calculation.
The ? of complicated system (molecule
molecule) is generated from a
linear combination of simple functions (basis set).
The parameters for the SE are varied until the solution (energy
energy of
the system)
system is optimized.
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4
Molecular
properties
Transition States
Reaction coords.
Ab initio electronic structure theory
Hartree-Fock (HF)
Electron Correlation (MP2, CI, CC, etc.)
Geometry
prediction
Spectroscopic
observables
Prodding
Experimentalists
Benchmarks for
parameterization
Goal: Insight into chemical phenomena.
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Setting up the problem…
What is a molecule?
A molecule is “composed” of atoms, or, more generally as a collection of
charged particles, positive nuclei and negative electrons.
The interaction between charged particles is described by;
Coulomb Potential
Vij = V (rij ) =
qi q j
4πε0 rij
=
rij
qiq j
rij
qj
qi
Coulomb interaction between these charged particles is the only
important physical force necessary to describe chemical phenomena.
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5
But, electrons and nuclei are in constant
motion…
In Classical Mechanics, the dynamics of a system (i.e. how the
system evolves in time) is described by Newton’s 2nd Law:
F = ma
−
dV
d 2r
=m 2
dr
dt
F = force
a = acceleration
r = position vector
m = particle mass
In Quantum Mechanics, particle behavior is described in terms of a
wavefunction, Y.
∂Ψ
Hˆ Ψ = ih
∂t
Hˆ
Time-dependent Schrödinger Equation
(i =
−1;h = h 2π
)
Hamiltonian Operator
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Time-Independent Schrödinger Equation
Hˆ (r,t) = Hˆ (r)
Ψ(r,t) = Ψ(r)e−iEt / h
Hˆ (r)Ψ(r) = EΨ(r)
If H is time-independent, the timedependence of Y may be separated out
as a simple phase factor.
Time-Independent Schrödinger Equation
Describes the particle-wave duality of electrons.
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Hamiltonian for a system with N-particles
Hˆ = Tˆ + Vˆ
Sum of kinetic (T) and potential (V)
energy
N
N
N
h2  ∂ 2
h2 2
∂2
∂2 
ˆ
ˆ
Kinetic energy
T = ∑ Ti = −∑
∇ i = −∑
 2 + 2 + 2
2m i  ∂x i ∂y i ∂zi 
2m i
i=1
i=1
i=1
 ∂2
∂2
∂2 
∇ 2i =  2 + 2 + 2 
 ∂x i ∂y i ∂zi 
N
N
N
Laplacian operator
N
qq
Vˆ = ∑ ∑Vij = ∑ ∑ i j
rij
i=1 j>1
i=1 j>1
Potential energy
(Coulombic)
When these expressions are used in the time-independent Schrodinger
Equation, the dynamics of all electrons and nuclei in a molecule or atom
are taken into account.
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Born-Oppenheimer Approximation
• Since nuclei are much heavier than electrons, their velocities are much
smaller. To a good approximation, the Schrödinger equation can be
separated into two parts:
– One part describes the electronic wave function for a fixed nuclear
geometry.
– The second describes the nuclear wave function, where the
electronic energy plays the role of a potential energy.
+
Hˆ = Tˆn + Tˆe + Vˆne + Vˆee + Vˆnn
n = nuclear
e = electronic
ne = nucleus-electron
ee = electron-electron
nn = nucleus-nucleus
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•
Since the motions of the electrons and nuclei are on different time
scales, the kinetic energy of the nuclei can be treated separately. This
is the Born-Oppenheimer approximation. As a result, the electronic
wave function depends only on the positions of the nuclei.
•
Physically, this implies that the nuclei move on a potential energy
surface (PES), which are solutions to the electronic Schrödinger
equation. Under the BO approx., the PES is independent of the
nuclear masses; that is, it is the same for isotopic molecules.
.
E
.
H + H
0
H H
•
Solution of the nuclear wave function leads to physically meaningful
quantities such as molecular vibrations and rotations.
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Limitations of the Born-Oppenheimer approximation
•The total wave function is limited to one electronic surface, i.e. a
particular electronic state.
•The BO approx. is usually very good, but breaks down when
two (or more) electronic states are close in energy at particular
nuclear geometries.
•In such situations, a “ non-adiabatic” wave function - a product
of nuclear and electronic wave functions - must be used.
The electronic Hamiltonian becomes,
Hˆ = Tˆe + Vˆne + + Vˆee + Vˆnn
B.O. approx.; fixed nuclear coordinates
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Hartree-Fock Self-consistent Field (SCF) Theory
GOAL: Solve the electronic Schrödinger equation, HeΨ=E Ψ.
PROBLEM:
– Exact solutions can only be found for one-electron systems, e.g., H2+.
SOLUTION:
– Use the variational principle to generate approximate solutions.
Variational principle
– If an approximate wave function is used in He Ψ =E Ψ, then the
energy must be greater than or equal to the exact energy.
– The equality holds when Ψ is the exact wave function.
In practice:
– Generate the “best” trial function that has a number of adjustable
parameters.
– The energy is minimized as a function of these parameters.
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The energy is calculated as an expectation value of the Hamiltonian operator:
E=
Ψ | Hˆ e | Ψ
Ψ |Ψ
If the wave functions are orthogonal and normalized (orthonormal),
Ψi | Ψ j = δ ij
Then,
δij = 1
δij = 0
(Kroenecker delta)
E = Ψ | Hˆ e | Ψ
Hˆ = Tˆe + Vˆne + + Vˆee + Vˆnn
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Hartree Approximation
• assume that a many electron wave function can be
written as a product of one electron functions
Ψ (r1 , r2 , r3 , L) = φ ( r1 )φ (r2 )φ ( r3 ) L
• Use the variational method energy, solving the many
electron Schrödinger equation is reduced to solving a
series of one electron Schrödinger equations
• In this approximation, each electron interacts with the
average distribution of the other electrons
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Variational Theorem
• the expectation value of the Hamiltonian is the variational
energy
*
∫ Ψ Hˆ Ψdτ = E ≥ E
var
exact
*
∫ Ψ Ψ dτ
• the variational energy is an upper bound to the lowest energy of
the system
• any approximate wave function will yield an energy higher
than the ground state energy
• parameters in an approximate wave function can be varied to
minimize the Evar
• this yields a better estimate of the ground state energy and a
better approximation to the wave function
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Since electrons are fermions, S = 1/2, the total electronic wave
function must be antisymmetric (change sign) with respect to the
interchange of any two electron coordinates. (Pauli principle - no
two electrons can have the same set of quantum numbers.)
Each electron resides in a spin-orbital, a product of spatial
and spin functions.
Φ(1,2) = φ1α(1)φ 2β(2) − φ1α (2)φ 2β (1)
ϕI – spin-orbit wave function
α = spin up state and β = spin down state (for electrons 1 & 2)
(Spin functions are orthonormal: α | α = β | β = 1; α | β = β | α = 0)
Interchange the coordinates of the two electrons,
Φ(2,1) = φ1α(2)φ 2β(1) − φ1α (1)φ 2β (2)
Φ(2,1) = −Φ (1,2)
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A more general way to represent antisymmetric electronic wave functions is in the
form of a determinant. For the two-electron case,
Φ(1,2) =
φ1α (1) φ 2 β(1)
φ1α (2) φ 2 β(2)
= φ1α(1)φ 2β(2) − φ1α (2)φ 2β (1)
For an N-electron N-spinorbital wave function,
φ1 (1)
Φ SD =
φ1 (2)
φ 2 (1)
L
φ N (1)
φ 2 (2)
L φ N (2)
L
L
L
L
φ1 (N ) φ 2 (N) L φ N (N)
,
φ i |φ j = δij
A Slater Determinant (SD) satisfies the antisymmetry requirement.
Columns are one-electron wave functions, molecular orbitals.
Rows contain the electron coordinates.
One more approximation: The trial wave function will consist of a single SD.
Now the variational principle is used to derive the Hartree-Fock equations...
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Fock Equation
• take the Hartree-Fock wavefunction
Ψ = φ1 φ2 L φn
• put it into the variational energy expression
∫ Ψ ĤΨdτ
=
∫ Ψ Ψd τ
*
Evar
*
• minimize the energy with respect to changes in the orbitals
∂Evar / ∂φi = 0
• yields the Fock equation
F̂φi = ε iφi
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F̂φi = ε iφi
• the Fock operator is an effective one electron
Hamiltonian for an orbital φ
• ε is the orbital energy
• each orbital φ sees the average distribution of all the
other electrons
• finding a many electron wave function is reduced to
finding a series of one electron orbitals
ˆ + Jˆ − K
ˆ
Fˆ = Tˆ + V
NE
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ˆ + Jˆ − K
ˆ
Fˆ = Tˆ + V
NE
2
ˆ = − h ∇2
T
2me
kinetic energy operator
nuclear-electron attraction operator
V̂ne =
nuclei
∑
A
− e2Z A
riA
Coulomb operator (electron-electron repulsion)
electrons
∑
Jˆ φi = {
j
exchange operator
(antisymmetry requirement)
∫φ j
electrons
ˆ φ ={
K
i
∑
j
e2
φ j dτ }φi
rij
∫φ j
e2
φi dτ }φ j
rij
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Solving the Fock Equations
F̂φi = ε iφi
1.
2.
3.
4.
5.
obtain an initial guess for all the orbitals φi
use the current φI to construct a new Fock operator
solve the Fock equations for a new set of φI
if the new φI are different from the old φI, go back to step 2.
When the values converge (do not change to a prescribed
limit), the calculation is complete.
εi = φ i '| Fˆi |φ i '
The results are interpreted as MO energies.
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Hartree-Fock Orbitals
•
•
Hartree-Fock orbitals (φ ‘s ) can be computed numerically
for atoms
These resemble the shapes of the hydrogenic orbitals (s, p, d
and f)
•
for homonuclear diatomic molecules, the Hartree-Fock
orbitals resemble the MO’s for species like H2+
•
σ, π, bonding and anti-bonding orbitals
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LCAO Approximation
•
•
•
•
numerical solutions for the Hartree-Fock orbitals only
practical for atoms and diatomics
diatomic orbitals resemble linear combinations of atomic
orbitals
e.g. sigma bond in H2
σ ≈ 1sA + 1sB
for polyatomics, approximate the molecular orbital by a
linear combination of atomic orbitals (LCAO)
φ = ∑ cµ χ µ
µ
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Basis Functions
φ = ∑ cµ χ µ
µ
•
•
•
•
χ’s are called basis functions
usually centered on atoms
can be more general and more flexible than atomic
orbitals
larger number of well chosen basis functions yields
more accurate approximations to the molecular
orbitals
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Basis Set Approximation
• For atoms and diatomic molecules, numerical HF methods
are available.
• In most molecular calculations, the unknown MOs are
expressed in terms of a known set of functions - a basis set.
Two criteria for selecting basis functions.
I) They should be physically meaningful.
ii) computation of the integrals should be tractable.
• It is common practice to use a linear expansion of Gaussian
functions in the MO basis because they are easy to handle
computationally.
• Each MO is expanded in a set of basis functions centered at
the nuclei and are commonly called Atomic Orbitals.
(Molecular orbital = Linear Combination of Atomic Orbitals LCAO).
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Basis sets
• Basis functions approximate orbitals of atoms in molecule
• Linear combination of basis functions approximates total
electronic wave function
• Basis functions are linear combinations of Gaussian functions
– Contracted Gaussians
– Primitive Gaussians
STOs vs. GTOs
•Slater-type orbitals (J.C. Slater)
•Represent electron density well in valence region and beyond
(not so well near nucleus)
•Evaluating these integrals is difficult
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Gaussian-type orbitals (F. Boys)
•Easier to evaluate integrals, but don’t represent electron density
well
•This can be overcome this by using linear combination of GTOs
Minimal basis set
• One basis function for every atomic orbital required to
describe the free atom
• Most-common: STO-3G
• Linear combination of 3 Gaussian-type orbitals fitted to
one Slater-type orbital
• CH4: H(1s); C(1s,2s,2px,2py,2pz)
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More basis functions per atom
• Split valence basis sets
• Double-zeta:
• Triple-zeta:
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Ways to increase a basis set
• Add more basis functions per atom
•allow orbitals to “change size”
• Add polarization functions
•allow orbitals to “change shape”
• Add diffuse functions for electrons with large radial extent
• Add high angular momentum functions
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Ab Initio (latin, “from the beginning”) Quantum
Chemistry Summary of approximations
•
•
•
•
•
Born-Oppenheimer Approx.
Non-relativistic Hamiltonian
Use of trial functions, MOs, in the variational procedure
Single Slater determinant
Basis set, LCAO-MO approx.
Consequence of using a single Slater determinant and
the Self-consistent Field equations:
Electron-electron repulsion is included as an average effect. The electron
repulsion felt by one electron is an average potential field of all the others,
assuming that their spatial distribution is represented by orbitals. This is
sometimes referred to as the Mean Field Approximation.
Electron correlation has been neglected!!!
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How does one know what to use when?
• Considerations:
–
–
–
–
–
Computational resources
Molecule size
Kind of chemical question
Desired accuracy
Recommendations from literature
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•Books
Literature
•F. Jensen, Introduction to Computational
Chemistry, Wiley, 1999 (our textbook)
•A. R. Leach, Molecular Modelling, 2nd Ed.,
Prentice Hall, 2001
•C. J. Cramer, Essentials of Computational
Chemistry, 2nd Ed., Wiley, 2004
There are more, but these three should be good at the beginning.
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