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Statistical Mechanics and MultiScale Simulation Methods
ChBE 591-009
Prof. C. Heath Turner
Lecture 02
• Some materials adapted from Prof. Keith E. Gubbins: http://gubbins.ncsu.edu
• Some materials adapted from Prof. David Kofke: http://www.cbe.buffalo.edu/kofke.htm
Introduction to Quantum Chemistry
Born-Oppenheimer Approximation
The Hamiltonian (H) contains pairwise attraction and repulsion terms.
 No particle moves independently (correlation)
SIMPLIFICATION: BO approximation – decouple motions of nucleus and the electrons.
Valid Assumption? Electron motion is much faster than nucleus (mass ratio ~ 1800)
Electron relaxation is essentially instantaneous.
Result – electronic Schrödinger equation:
H el  VN el qi ; qk   Eel el qi ; qk 
qi – electron coordinates (variables); qk – nuclear coordinates (parameters)
Eigenvalues of this equation = “electronic energy”.
VN is constant for a set of fixed nuclear coordinates.
SOLUTION: Wavefunctions invariant to constant terms in H, thus VN can be removed
and the eigenvalue = “pure electronic energy”
Eel = “pure electronic energy” + VN
•
Overall BO approximation is a very mild one, and is used throughout quantum
chemistry: construction PES, finding equilibrium molecular geometries, and
locating TS species.
Introduction to Quantum Chemistry
LCAO Basis Set Approach
Question: How do we choose the mathematical functions to construct the trial wave
function?
 The convenient functions are called a “basis set”.
 If only 1 e- and 1 nucleus – exact solution of Schrödinger eq. possible (see Fig 2.3):
1s, 2s, 2p, 3s, 3p, 3d, etc.
These mathematical functions (hydrogenic atomic orbitals) are
useful in constructing MO’s.
Imagine a guess wavefunction, f, as a linear combination of atomic wave functions:
N
f   aii
i 1
This set of N functions i is called the “basis set”. This is known as LCAO approach.
The basis functions provide the form of the electron density. We desire basis
functions that allow the e- density to persist in regions that lower the overall energy.
LOCATIONS of the basis functions NOT predetermined.
Introduction to Quantum Chemistry
Example: Basis Functions Applied to a Molecule
Challenge: Describe the bonding between C and H.
• Add a p-function (in addition to the s-function)
• Is H sp-hybridized?
NOT NECESSAIRILY:
• Same result accomplished with multiple s-orbitals
• Calculations simpler, results similar
Conclusion: Relax some of your chemical intuition! Don’t always think in terms of
preconceived chemistry – think in terms of the mathematics.
Introduction to Quantum Chemistry
The Secular Equation


 

  i aii  H  j a j j dr
E




a

a



  i i i  j j j dr
Energy of our guess wavefunction
 a a   H  dr  a a H
E

a
a


d
r
a a S
 
i
j
i
j
ij
i
j
ij
i
j ij
ij
i
ij
j
i
j
Atomic orbital basis set (while efficient) is
no longer orthonormal.
ij
Hij = “resonance integral”, Hii = energy of single e- in basis function i = ionization
potential of the AO in the environment of the surrounding molecule
Sij = “overlap integral”, amount of overlap between 2 basis functions (in a phasematched fashion) in space.
GOAL: Choose ai to minimize E. (variational principle)
Minimum (necessary condition):
E
 0 k
ak
Introduction to Quantum Chemistry
N
Result:
 a H
i 1
i
ki
 ESki   0 k
N equations, N unknowns (ai). Nontrivial solution: if and only if determinant formed
from coefficients of the unknowns (ai) is equal to zero. This is the secular equation:
 H11  ES11
 H  ES
21
 21



 H N 1  ES N 1
H12  ES12  H1N



 H NN
 ES1N 


0



 ES NN 
Result:
• N energies Ej (some may be degenerate)
• each Ej will have a different set of coefficients (aij)
• aij can be found by solving set of N equations (using Ej)
• Result is the optimal wave function (within the given basis set):
N
f j   aij i
i 1
o In a 1 e- system, the lowest energy MO would define the ground state
of the system. Higher energy orbitals would be “excited states”.
o All of the MO’s determined this way are orthogonal. (For degenerate
MO’s some minor complications arise).
Introduction to Quantum Chemistry
Hückel Theory (Erich Hückel)
• Illustration of LCAO approach
• Used to describe unsaturated/aromatic hydrocarbons
• Developed in the 30’s (not used much today)
Assumptions:
1. Basis set = parallel 2p orbitals, one per C atom (designed to treat planar hydrocarbon
p systems)
2. Sij = dij (orthonormal basis set)
3. Hii = a (negative of the ionization potential of the methyl radical)
4. Hij = b (negative stabilization energy). 90º rotation removes all bonding, thus we can
calculate DE: DE = 2Ep - Ep where Ep = a and Ep = 2a + 2b (as shown below)
5. Hij = between carbon 2p orbitals more distant than nearest neighbors is set to zero.
Introduction to Quantum Chemistry
Hückel Theory: Application – Allyl System (C3H5)
• 3 carbon atoms = 3 carbon 2p orbitals
• Construct the secular equation:
H11  ES11
H 21  ES21

H N 1  ES N 1
H12  ES12  H1N



 H NN
 ES1N
a E
b
0

a E
b 0
 b

0
b
a E
 ES NN
E  a  2b , a
E  a  2b
Q: What are the possible energy values (eigenvalues)?
Q: What is the lowest energy eigenvalue?
Q: What is the molecular orbital associated with this energy?
Solve:
N
 a H
i 1
i
ki
 ESki   0
Answer:
a2  2a1
a3  a1
Introduction to Quantum Chemistry
Hückel Theory: Application – Allyl System (C3H5)
We need an additional constraint: normalization of the coefficients:
2
c
 i 1
i
2
2
2
a11
 2a11
 a11
1
a11  1 / 2, a21  2 / 2, a31  1 / 2
** Second subscript has been added to designate the 1st energy level (bonding).
1
2
1
p1 
p2  p3
2
2
2
2
2


p

0
p

p3
Next energy level:
2
1
2
2
2
Lowest energy Molecular Orbital:
Highest energy level:
Allyl cation: 2e- =

1 
1
2
1
 3  p1 
p2  p3
2
2
2
2 a  2b

Introduction to Quantum Chemistry
Picture taken from: CJ Cramer, Essentials of Computational Chemistry, Wiley (2004).
Introduction to Quantum Chemistry
Many Electron Wave Functions:
• Previously (Hückel) electron-electron repulsion accounted for in an
approximate way, a (ionization potentials) and b (rotational barriers), called
“effective Hamiltonian” method.
• One-electron Hamiltonian is separable (neglect nuclear contribution and
electron-electron correlation):
N
H   hi
(N = total # of electrons, hi = one-electron Hamiltonian)
i 1
1 2 M Zk
hi   i  
2
k 1 rik
hi i   i i
(M = total # of nuclei)
(one-electron Schrödinger equation)
• Many electron eigenfunctions a product of the one-electron eigenfunctions:
HP   1 2  N
(“Hartree Product” wave function)
Introduction to Quantum Chemistry
• It can be shown that the eigenvalue of HP is the sum of the one-electron
eigenvalues:
N


HHP     i HP
 i 1 
• If all i are normalized, then HP is also normalized:
HP   1  2  N
2
2
2
2
• Hartree Hamiltonian (includes e-,e- interaction in an approximate way)
1 2 M Zk
hi   i    Vi  j
2
k 1 rik
Vi  j   
j i
Recall:
j   j
j
rij
dr
Vi{j} = interaction potential with all of
the other electrons occupying
orbitals {j}
j = charge density (probability)
associated with electron j
2
** PROBLEM: j are NOT known yet. Unfortunately, j appear in both the
j term and in the 1-electron Schrödinger Eq.
View of the Hartree method
The kth electron is now treated as a point charge in the
field of all of the other electrons. This procedure takes
the many-electron problem and simplifies it to many
one electron problems.
-
+
-
+
-
Many-electron system
All electron-electron repulsion
is included explicitly.
One-electron system
with remaining electrons
represented by an average
charge density.
Introduction to Quantum Chemistry
•
Q: How do we solve this problem?
•
A: Hartree (1928) proposed an iterative “self-consistent field
(SCF)” method:
1. Guess the wavefunctions i for all of the occupied MOs
2. Construct the operators hi
3. Solve the 1e- Schrödinger eq. for the new set of i
4. Repeat procedure with the new i as the initial guess, until
converged.
Convergence: usually dE falls below a threshold
Important Note: The energy calculated using this approach needs a
correction because of double counting of the electron-electron repulsion:
hi includes repulsion between electron i and electron j, but so does hj.
Correction:
2
i  j
1
E  i    
dri dr j
2 i j
rij
i
2
The double-integral in this equation in called the “coulomb integral”, abbreviated Jij
Introduction to Quantum Chemistry
•
The Hartree approximation works well for atoms. However, the
form of the wavefunction is not correct, and the method fails for
molecules. In the 1930’s, Fock and Slater fixed this problem:
Electron Spin and Antisymmetry
•
All e- are characterized by a spin quantum number
•
Electron spin function is an eigenfunction of the operator Sz and
has only 2 possible values: +/- ħ/2
•
Spin eigenfunctions are orthonormal and typically denoted as a
and b
•
Pauli exclusion principle: no 2 electrons may be characterized by
the same set of quantum #’s. Thus in a given MO, only 2 e- may
be accommodated, with quantum #’s a and b.
Say we have 2 e- of the same spin (a). We could construct a ground state
Hartree product wave function: 3    (1)a (1) ( 2)a ( 2)
HP
a
b
“3” in the wave function indicates triplet state (2 parallel electron spins)
Introduction to Quantum Chemistry
•
a and b are distinct and orthonormal, thus we seem to have satisfied the
Pauli exclusion principle. However, relativistic quantum theory imposes an
additional constraint (within the Pauli exclusion principle):
** The wavefunctions must change sign whenever the coordinates of the two
electrons are interchanged. (“Antisymmetry” requirement)
TEST: check our HP wavefunction (with permutation operator P12)
P12  a (1)a (1) b (2)a (2)   b (1)a (1) a (2)a (2)
  a (1)a (1) b (2)a (2)
THUS: this does not rigorously satisfy the Pauli principle. However, we
can make a correction:
3
SD   a (1)a (1) b (2)a (2)  a (2)a (2) b (1)a (1)
Does it satisfy the criterion? YES
Is the wavefunction normalized? NO
Introduction to Quantum Chemistry
•
Q: How do we normalize the wavefunction?

3
SD dr1d1dr2 d2  1  0  1  2
2
1
2
•
We need to include the normalization factor:
•
We can mathematically express our wavefunction as a determinant:
3
1  a (1)a (1)  b (1)a (1)
SD 
2  a (2)a (2)  b (2)a (2)
•
Commonality: P operator switches two rows – determinant changes sign
when any two rows are switched.
•
Useful for constructing antisymmetric wavefunctions. These are called
“Slater Determinants”, and their general form is shown on the next slide…
Introduction to Quantum Chemistry
SD
1 (1)
1 (2)
 2 (1)
  N (1)



1




N! 
1 ( N )  2 ( N )   N ( N )
N = total # of electrons
 = spin orbital: (spatial orbital) x (electron spin eigenfunction)
Abbreviated Notation:
SD  1 2   N
For doubly-filled orbitals, it is often written:
1   1 (1)a (1) and 2   1 (2)b (2)
SD   12  3   N