Chemistry 6440 / 7440
Introduction to
Molecular Orbitals
Resources
• Grant and Richards, Chapter 2
• Foresman and Frisch, Exploring Chemistry
with Electronic Structure Methods
(Gaussian Inc., 1996)
• Cramer, Chapter 4
• Jensen, Chapter 3
• Leach, Chapter 2
• Ostlund and Szabo, Modern Quantum
Chemistry (McGraw-Hill, 1982)
Potential Energy Surfaces
• molecular mechanics uses empirical functions for
the interaction of atoms in molecules to calculate
potential energy surfaces
• these interactions are due to the behavior of the
electrons and nuclei
• electrons are too small and too light to be
described by classical mechanics
• electrons need to be described by quantum
mechanics
• accurate potential energy surfaces for molecules
can be calculated using modern electronic
structure methods
Schrödinger Equation
Ĥ  E
• H is the quantum mechanical Hamiltonian for the
system (an operator containing derivatives)
• E is the energy of the system
•  is the wavefunction (contains everything we
are allowed to know about the system)
• ||2 is the probability distribution of the particles
(as probability distribution, ||2 needs to be
continuous, single valued and integrate to 1)
Hamiltonian for a Molecule
ˆ 
H
electrons

i
  2 2 nuclei   2 2 electronsnuclei  e 2 Z A electrons e 2 nuclei e 2 Z A Z B
i  
A   
 

2me
riA
rij
rAB
A 2m A
i
A
i j
A B
• kinetic energy of the electrons
• kinetic energy of the nuclei
• electrostatic interaction between the electrons
and the nuclei
• electrostatic interaction between the electrons
• electrostatic interaction between the nuclei
Solving the Schrödinger Equation
• analytic solutions can be obtained only for
very simple systems
• particle in a box, harmonic oscillator,
hydrogen atom can be solved exactly
• need to make approximations so that
molecules can be treated
• approximations are a trade off between
ease of computation and accuracy of the
result
Expectation Values
• for every measurable property, we can
construct an operator
• repeated measurements will give an
average value of the operator
• the average value or expectation value of
an operator can be calculated by:
  Ôd 
  d
*
*
O
Variational Theorem
• the expectation value of the Hamiltonian is the
variational energy
* ˆ

 Hd
  d
*
 Evar  Eexact
• the variational energy is an upper bound to the lowest
energy of the system
• any approximate wavefunction will yield an energy
higher than the ground state energy
• parameters in an approximate wavefunction can be
varied to minimize the Evar
• this yields a better estimate of the ground state energy
and a better approximation to the wavefunction
Born-Oppenheimer Approximation
• the nuclei are much heavier than the electrons
and move more slowly than the electrons
• in the Born-Oppenheimer approximation, we
freeze the nuclear positions, Rnuc, and calculate
the electronic wavefunction, el(rel;Rnuc) and
energy E(Rnuc)
• E(Rnuc) is the potential energy surface of the
molecule (i.e. the energy as a function of the
geometry)
• on this potential energy surface, we can treat the
motion of the nuclei classically or quantum
mechanically
Born-Oppenheimer Approximation
• freeze the nuclear positions (nuclear kinetic energy is
zero in the electronic Hamiltonian)
ˆ 
H
el
electrons

i
  2 2 electrons
i  
2me
i
nuclei

A
 e 2 Z A electrons e 2 nuclei e 2 Z A Z B
 

riA
rij A B rAB
i j
• calculate the electronic wavefunction and energy
ˆ   E , E 
H
el el
el
* ˆ

 el Hel el d
*

 el el d
• E depends on the nuclear positions through the nuclearelectron attraction and nuclear-nuclear repulsion terms
• E = 0 corresponds to all particles at infinite separation
Nuclear motion on the
Born-Oppenheimer surface
• Classical treatment of the nuclei (e,g. classical
trajectories)
F  ma , F  E / R nuc , a   R nuc / t
2
2
• Quantum treatment of the nuclei (e.g. molecular
vibrations)
ˆ   
total  el  nuc , H
nuc nuc
nuc
ˆ 
H
nuc
nuclei

A
 2 2
  E (R nuc )
2m A
Hartree Approximation
• assume that a many electron wavefunction
can be written as a product of one electron
functions
(r1 , r2 , r3 ,)   (r1 ) (r2 ) (r3 )
• if we use the variational energy, solving the
many electron Schrödinger equation is
reduced to solving a series of one electron
Schrödinger equations
• each electron interacts with the average
distribution of the other electrons
Hartree-Fock Approximation
• the Pauli principle requires that a wavefunction for electrons
must change sign when any two electrons are permuted
– since |(1,2)|2=|(2,1)|2, (1,2)=(2,1) (minus sign for fermions)
• the Hartree-product wavefunction must be antisymmetrized
• can be done by writing the wavefunction as a determinant
– determinants change sign when any two columns are switched

1 (1) 1 (2)  1 (n)
1 2 (1) 2 (2)  2 (n)
n




n (1) n (1)  n (n)
 1 2  n
Spin Orbitals
• each spin orbital I describes the distribution of one
electron
• in a Hartree-Fock wavefunction, each electron must be
in a different spin orbital (or else the determinant is zero)
• an electron has both space and spin coordinates
• an electron can be alpha spin (, , spin up) or beta spin
(, , spin down)
• each spatial orbital can be combined with an alpha or
beta spin component to form a spin orbital
• thus, at most two electrons can be in each spatial orbital
Fock Equation
• take the Hartree-Fock wavefunction
  1 2  n
• put it into the variational energy expression
Evar 
*

 Ĥd
*

  d
• minimize the energy with respect to changes in the orbitals while
keeping the orbitals orthonormal
Evar / i  0,
*

 i  j d   ij
• yields the Fock equation
F̂i   ii
Fock Equation
F̂i   ii
• the Fock operator is an effective one electron
Hamiltonian for an orbital 
•  is the orbital energy
• each orbital  sees the average distribution of
all the other electrons
• finding a many electron wavefunction is reduced
to finding a series of one electron orbitals
Fock Operator
ˆ V
ˆ  Jˆ  K
ˆ
Fˆ  T
NE
• kinetic energy operator
2


ˆ
T
2
2me
• nuclear-electron attraction operator
V̂ne 
nuclei

A
 e2 Z A
riA
Fock Operator
ˆ V
ˆ  Jˆ  K
ˆ
Fˆ  T
NE
• Coulomb operator (electron-electron repulsion)
2
e
Jˆ i  {    j  j d }i
rij
j
• exchange operator (purely quantum mechanical
-arises from the fact that the wavefunction must
switch sign when you exchange to electrons)
electrons
e2
  j rij i d } j
electrons
ˆ  {
K
i

j
Solving the Fock Equations
F̂i   ii
1. obtain an initial guess for all the orbitals i
2. use the current I to construct a new Fock
operator
3. solve the Fock equations for a new set of I
4. if the new I are different from the old I, go
back to step 2.
Hartree-Fock Orbitals
•
•
•
•
for atoms, the Hartree-Fock orbitals can be computed
numerically
the  ‘s resemble the shapes of the hydrogen orbitals
s, p, d orbitals
radial part somewhat different, because of interaction
with the other electrons (e.g. electrostatic repulsion
and exchange interaction with other electrons)
Hartree-Fock Orbitals
•
•
•
for homonuclear diatomic molecules, the
Hartree-Fock orbitals can also be computed
numerically (but with much more difficulty)
the  ‘s resemble the shapes of the H2+
orbitals
, , bonding and anti-bonding orbitals
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
  1sA + 1sB
for polyatomics, approximate the molecular
orbital by a linear combination of atomic
orbitals (LCAO)
   c  

Basis Functions
   c  
•
•
•
•

’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
Roothaan-Hall Equations
•
choose a suitable set of basis functions
   c  
•

plug into the variational expression for the
energy
*
Evar
•
 Ĥd


   d
*
find the coefficients for each orbital that
minimizes the variational energy
Roothaan-Hall Equations
•
•
•
•
•
basis set expansion leads to a matrix form of
the Fock equations
F Ci = i S Ci
F – Fock matrix
Ci – column vector of the molecular orbital
coefficients
I – orbital energy
S – overlap matrix
Fock matrix and Overlap matrix
•
Fock matrix
F     F̂ d
•
overlap matrix
S       d
Intergrals for the Fock matrix
•
Fock matrix involves one electron integrals of kinetic
and nuclear-electron attraction operators and two
electron integrals of 1/r
ˆ V
ˆ )  d
h     (T
ne

•
•
one electron integrals are fairly easy and few in
number (only N2)
1
(  |  )     (1)  (1)   (2)  (2)d 1d 2
r12
two electron integrals are much harder and much
more numerous (N4)
Solving the Roothaan-Hall Equations
1. choose a basis set
2. calculate all the one and two electron integrals
3. obtain an initial guess for all the molecular
orbital coefficients Ci
4. use the current Ci to construct a new Fock
matrix
5. solve F Ci = i S Ci for a new set of Ci
6. if the new Ci are different from the old Ci, go
back to step 4.
Solving the Roothaan-Hall Equations
•
•
•
•
also known as the self consistent field (SCF) equations,
since each orbital depends on all the other orbitals, and
they are adjusted until they are all converged
calculating all two electron integrals is a major
bottleneck, because they are difficult (6 dimensional
integrals) and very numerous (formally N4)
iterative solution may be difficult to converge
formation of the Fock matrix in each cycle is costly,
since it involves all N4 two electron integrals
Summary
•
•
•
•
•
start with the Schrödinger equation
use the variational energy
Born-Oppenheimer approximation
Hartree-Fock approximation
LCAO approximation