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Introduction to Computational Chemistry
NSF Computational Nanotechnology and Molecular Engineering
Pan-American Advanced Studies Institutes (PASI) WorkshopJanuary 5-16, 2004
California Institute of Technology, Pasadena, CA
Andrew S. Ichimura
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For the Beginner
There are three main problems:
1. Deciphering the language.
2. Technical implementation.
3. Quality assessment.
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Focus on
Calculating molecular structures and relative
energies.
1. Hartree-Fock (Self-Consistent Field)
2. Electron Correlation
3. Basis sets and performance
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Ab initio electronic structure theory
Hartree-Fock (HF)Electron Correlation (MP2, CI, CC, etc.)
Molecular
properties
Geometryprediction
Benchmarks for
parameterization
Transition States
Reaction coords.
Spectroscopicobservables
Prodding
Experimentalists
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
Coulomb interaction between these charged particles is the only important
physical force necessary to describe chemical phenomena.
Vij V(rij ) qiqj
40rijqiqj
rijrij
qi
qj
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But, electrons and nuclei are in constant motion
In Classical Mechanics, the dynamics of a system (i.e. how the systemevolves in time) is described by Newtons 2nd Law:
F maF = force
a = acceleration
r= position vector
m = particle mass
dV
dr m
d2r
dt2
In Quantum Mechanics, particle behavior is described in terms of a wavefunction, .
HY i
Y
t
Hamiltonian OperatorH
Time-dependent Schrdinger Equation
i 1; h 2
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Time-Independent Schrdinger Equation
IfH is time-independent, the time-
dependence ofY may be separated out as a
simple phase factor.
H(r,t) H(r)
Y(r,t) Y(r)eiEt/
H(r)Y(r) EY(r) Time-Independent Schrdinger Equation
Describes the particle-wave duality of electrons.
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Hamiltonian for a system with N-particles
Sum of kinetic (T) and potential (V) energy
HT
V
T Ti 2
2mii1
N
i1
N
i2 2
2mii1
N
2
xi2
2
yi2
2
zi2
i2
2
xi2
2
yi2
2
zi2
Laplacian operator
Kinetic energy
V Vij
j1
N
i1
N
qiqj
rijj1
N
i1
N
Potential energy
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 Schrdinger equation can beseparated into two parts:
One part describes the electronic wavefunction for a fixed nucleargeometry.
The second describes the nuclear wavefunction, where the electronicenergy plays the role of a potential energy.
So far, the Hamiltonian contains the following terms:H Tn Te Vne Vee Vnn
Tn Te
Vne Vee Vnn
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Born-Oppenheimer Approx. cont.
In other words, the kinetic energy of the nuclei can be treated separately. This
is theBorn-Oppenheimer approximation. As a result, the electronicwavefunction depends only on thepositions of the nuclei.
Physically, this implies that the nuclei move on a potential energy surface
(PES), which are solutions to the electronic Schrdinger equation. Under the
BO approx., the PES is independent of the nuclear masses; that is, it is the
same for isotopic molecules.
Solution of the nuclear wavefunction leads to physically meaningful
quantities such as molecular vibrations and rotations.
0
E
H H
H. + H.
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Limitations of the Born-Oppenheimer approximation
The total wavefunction 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 wavefunction - a product of nuclear and
electronic wavefunctions - must be used.
In writing the Hamiltonian as a sum of electron kinetic and potential energyterms, relativistic effects have been ignored. These are normally negligible
for lighter elements (Z
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Self-consistent Field (SCF) Theory
GOAL: Solve the electronic Schrdinger equation, HeY=EY.
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 wavefunction is used inHeY=EY, then the energy must be greater than or equal to the exactenergy. The equality holds when Y is the exact wavefunction.
In practice: Generate the best trial function that has a number of
adjustable parameters. The energy is minimized as a function of theseparameters.
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SCF cont.
The energy is calculated as an expectation value of the Hamiltonian operator:
E
Y
HeYd
YYdIntroduce bra-ket notation,
Y HeYd Y | He | Y
Y
Yd Y | Y
bra n complex conjugate , left
ketm
rightCombined bracketdenotes integration over all coordinates.
EY | He | Y
Y | Y
If the wavefunctions are orthogonal and normalized (orthonormal),
Yi |Yj ijij 1
ij 0
Then,E Y | He |Y
(Kroenecker delta)
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SCF cont.
Antisymmetric wavefunctions can be written as
Slater determinants.
Since electrons are fermions, S=1/2,the total electronic wavefunction must beantisymmetric (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.)
Consider a two electron system, e.g. He or H2
. A suitable antisymmetric
wavefunction to describe the ground state is:
1,2 1(1)2(2) 1(2)2(1)
Each electron resides in a spin-orbital, a product of spatial and spin functions.
(Spin functions are orthonormal: | = | =1; | | 0)
2,1 1(2)2(1) 1(1)2(2) 2,1 1,2
Interchange the coordinates of the two electrons,
(He: 1 =2 = 1s)
(H2: 1 = 2 = bonding MO)
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A more general way to represent antisymmetric electronic wavefunctions is in the
form of a determinant. For the two-electron case,
1,2 1(1) 2(1)
1(2) 2(2) 1(1)2(2) 1(2)2(1)
For an N-electron N-spinorbital wavefunction,
SD
1 1 2(1) N(1)
1 2 2(2) N(2)
1 N 2(N) N(N)
, i |j ij
ASlater Determinant (SD) satisfies the antisymmetry requirement.
Columns are one-electron wavefunctions, molecular orbitals.Rows contain the electron coordinates.
One more approximation: The trial wavefunction will consist of a single SD.
Now the variational principle is used to derive the Hartree-Fock equations...
SCF cont.
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Hartree-Fock Equations
(1) Reformulate the Slater Determinant as,
A 1(1)2(2) N(N) A is the diagonal producA the antisymmetrizer
A 1
N!
(1)p Pp 0
N1
1
N!
1 Pij Pijkijk
ij
Pis the permutation operator. Pij permutes two electron coordinates .
(2) He Te
Vne Vee
Vnn
Te 12
i2
i
N
Vne
ZaRa ria
i
N
Vee 1ri rjji
N
i
N
Vnn ZaZbRa Rbb a
a
One electron
termsDepends on
two electrons
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hi 1
2i
2 Za
Ra ria
gij 1
ri rj
He hi
i1
N
gijji
N
i
N
Vnn
One-electron operator - describes electron
i, moving in the field of the nuclei.
Two-electron operator - interelectron
repulsion.
Hamiltonian
Expectation value over
Slater Determinant
Ee | He |
Ee
A |
He |
A (1)p
p 0
N1
|
He |
P
(3) Calculation of the energy.
Examine specific integrals:
| Vnn | VnnNuclear repulsion does not depend
on electron coordinates.
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The one-electron operator acts only on electron 1 and yields
an energy, h1, that depends only on the kinetic energy and
attraction to all nuclei.
For coordinate 1,
| h1 | 1(1)2(2) N(N) | h1 | 1(1)2(2) N(N) 1(1) |
h1 |1(1) 2(2) |2 (2) N(N) |N(N) h1
| g12 | 1(1)2(2) N(N) | g12 | 1(1)2(2) N(N) 1(1)2 (2)| g12 |1(1)2 (2) 3 (3) |3 (3) N(N) |N(N)
= 1(1)2 (2)| g12 |1(1)2 (2) J12
Coulomb integral,J12: represents the classical repulsion
between two charge distributions 12(1) and 2
2(2).
| g12 |P12 1(1)2(2) N(N) | g12 | 2(1)1(2) N(N)
1(1)2 (2)| g12 |2 (1)1(2) 3 (3) |3 (3) N(N) |N(N)
= 1(1)2 (2)| g12 |2 (1)1(2) K12
Exchange integral,K12: no classical analogue. Responsible for
chemical bonds.
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The expression for the energy can now be written as:
Sum of one-electron, Coulomb,
and exchange integrals, and Vnn.
To apply the variational principle, the Coulomb and Exchange integrals are
written as operators,
Ee i |
hi |ii1
N
1
2 j |
Ji |j j |
Ki |j
jN
iN
Vnn
Ji |j(2) i(1) | g12 |i(1) j (2)
Ki |j (2) i(1) | g12 |j(1) i(2)
Ee
hii1
N
1
2 (JijKij )j
N
iN
Vnn
The objective now is to find the best orbitals (i, MOs) that minimize theenergy (or at least remain stationary with respect to further changes in i),
while maintaining orthonormality ofi.
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Employ the method of Langrange Multipliers:
f(x1,x2, xN)
g(x1,x2, xN) 0L(x1,x2, xN,) f(x1,x2, xN) g(x1,x2, xN)
OptimizeL such thatLxi
0,Li
0
Function to optimize.
Rewrite in terms of another function.
Define Lagrange
function.
Constrained optimization ofL.
L E ij i |j ij ij
N
L E ij i |j i |j ij
N
0
In terms of molecular orbitals, the Langrange function is:
Change inL with respect to small
changes in i should be zero.
E i |hi |i i |
hi |i i1
N
i | Jj Kj |i i | Jj Kj |iij
N
Change in the energy with respect changes in i.
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Define the Fock Operator, Fi
Fi hi Jj Kj j
N
Effective one-electron operator, associatedwith the variation in the energy.
E i |Fi |i i |
Fi |ii1
N
Change in energy in termsof the Fock operator.
L i | Fi |i i | Fi |i i1N
ij i |j i |j ijN
0
According to the variational principle, the best orbitals, i, will make L=0.
Fii ijjj
N
After some algebra, the final expression becomes:
Hartree-Fock Equations
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After a unitary transformation, ij0 and iii.
Fii ' ii ' HF equations in terms ofCanonical MOs anddiagonal Lagrange multipliers.
i i '| Fi |i ' Lagrange multipliers can be interpreted asMO energies.
Note:
1. The HF equations cast in this way, form a set of pseudo-eigenvalue
equations.
2. A specific Fock orbital can only be determined once all the other
occupied orbitals are known.
3. The HF equations are solved iteratively. Guess, calculate the
energy, improve the guess, recalculate, etc.4. A set of orbitals that is a solution to the HF equations are called
Self-consistent Field (SCF) orbitals.
5. The Canonical MOs are a convenient set of functions to use in the
variational procedure, but they are not unique from the standpoint
of calculating the energy.
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Koopmans Theorem
The ionization energy is well approximated by the orbital energy, i.
* Calculated according to Koopmans theorem.
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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 aknown 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 MObasis because they are easy to handle computationally.
Each MO is expanded in a set of basis functions centered at the nuclei and arecommonly called Atomic Orbitals.
(Molecular orbital = Linear Combination of Atomic Orbitals - LCAO).
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MO Expansion
i ci
M
Fi ci
M
i ci
M
FC SC
F |F |
S
|
LCAO - MO representation
Coefficients are variational parameters
HF equations in the AO basis
Matrix representation of HF eqns.
Roothaan-Hall equations (closed shell)
F - element of the Fock matrix
S - overlap of two AOs
Roothaan-Hall equations generate M molecular orbitals from M basis functions.
N-occupied MOs
M-N virtual or unoccupied MOs (no physical interpretation)
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Total Energy in MO basis
One-electron integrals, M2 Two-electron integrals, M4
Computed at the start; do not change
Products of AO coeff form Density Matrix, D
E i |hi |i
i1
N
12
ij | g|ij ij | g|ji j
N
i
N
Vnn
E cici
M
| hi | i1
N
12
cicjcicj | g| | g| Vnn
M
ij
N
Total Energy in AO basis
D cjcjj
occ.MO
; D cicii
occ.MO
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General SCF Procedure
Obtain initial guess
for coeff., c i,form
the initial D
Form the Fock matrix
Diagonalize the Fock Matrix
Form new Density Matrix
Two-electronintegrals
Iterate
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Computational Effort
Accuracy
As the number of functions increases, the accuracy of the Molecular Orbitals
improves.
As M, the complete basis set limit is reached Hartree-Fock limit.
Result: The best single determinant wavefunction that can be obtained.
(This is not the exact solution to the Schrodinger equation.)
Practical Limitation In practice, a finite basis set is used; the HF limit is never reached.
The term Hartree-Fock is often used to describe SCF calculations with
incomplete basis sets.
Formally, the SCF procedure scales as M4 (the number of basis
functions to the 4th power).
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Restricted and Unrestricted Hartree-Fock
1
2
3
4
5
RHF
Singlet
ROHF
Doublet
UHF
Doublet
Energy
Restricted Hartree-Fock (RHF)
For even electron, closed-shell singlet states, electrons in a given MO
with and spin are constrained to have the same spatial dependence.
Restricted Open-shell Hartree-Fock (ROHF)
The spatial part of the doubly occupied orbitals are restricted to be the same.
Unrestricted Hartree-fock (UHF)
and spinorbitals have different spatial parts.
Spinorbitalsis(n)
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Comparison of RHF and UHF
R(O)HF
and spins have same spatialpart
Wavefunction, , is an
eigenfunction of S2 operator.
For open-shell systems, theunpaired electron () interacts
differently with and spins.
The optimum spatial orbitals are
different. Restricted
formalism is not suitable for spin
dependent properties.
Starting point for more advanced
calculations that include electron
correlation.
UHF
and spins have differentspatial parts
Wavefunction is notan
eigenfunction of S2. may be
contaminated with states of
higher multiplicity (2S+1).
EUHF ER(O)HF
Yields qualitatively correct
spin densities.
Starting point for more
advanced calculations that
include electron correlation.
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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.
RHF, ROHF, UHF
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 theMean Field Approximation.
Electron correlation has been neglected!!!