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Statistical Mechanics and MultiScale Simulation Methods
ChBE 591-009
Prof. C. Heath Turner
Lecture 03
• 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
Properties of Slater Determinants:
• Every e- appears in every spin orbital (indistinguishability)
• Includes QM exchange
• Accounting for spin introduces a new energy quantity. Assume we want to
calculate the interelectronic repulsion (Erep) for our wavefunction 3YSD:
3
E rep 
E rep 
Y SD 

3

1
2
Y SD 
a
(1 )
 a (1) (1) b ( 2 ) ( 2 )   a ( 2 ) ( 2 ) b (1) (1) 
1
r12
2
E rep  J ab  K ab
1
r12

3
Y SD  d r1 d  1 d r2 d  2
2
 b ( 2 ) d r1 d r2    a (1) b (1)
1
r12
 a ( 2 ) b ( 2 ) d r1 d r2
Introduction to Quantum Chemistry
Properties of Slater Determinants:
• What happens to the Kab integral when describing e- of opposite spin?
3
Y SD 
1
2
 a (1) (1) b ( 2 )  ( 2 )   a ( 2 ) ( 2 ) b (1)  (1) 
Introduction to Quantum Chemistry
Hartree-Fock SCF Method:
• The HF SCF method works similar to the Hartree product SCF method,
except that now we include the effects of spin.
• Typically working with closed-shell systems (all e- are spin-paired, 2 e- per
orbital). Wavefunctions described with Slater determinants. Called “restricted
Hatree-Fock” or RHF.
• There is a 1e- Fock operator for each e-, as defined below:
fi  
Vi
HF
1
nuclei
 
2
 j 
2
i

k
Zk
rik
 Vi
HF
 j
2Ji  Ki
Ji and Ki are operators that return the integrals Jij and Kij
The HF equations are used to solve the standard eigenvalue equation:
fi  i   i  i
Introduction to Quantum Chemistry
Hartree-Fock SCF Method:
• The secular equation that corresponds to the HF SCF method is then:
 F11

F
 21


 FN 1
 ES 11
F12  ES 12
 ES 21


 ES N 1




F1 N  ES 1 N 






F NN  ES NN 
• The iterative SCF method is then used to solve for the energy, the wave
function, and the associated coefficients.
DRAWBACKS of HF SCF:
1. Electron correlation is still neglected
2. LCAO approach requires intensive four-index integrals (not shown here)
3. The # of four-index integrals calculated scale as N4 (poor scaling)
Introduction to Quantum Chemistry
Hartree-Fock SCF Method:
TWO GENERAL SOLUTIONS:
1. “Semiempirical MO Theory” – simplification of HF calculations by estimating
or fitting parameters from experiment.
2. “Ab initio” – use HF as a stepping stone to the exact solution of Schrödinger
Eq. Concentrate on more efficient and more elegant methods for
performing these type of calculations.
The second approach is the more fundamental approach. It has greater
potential for describing new systems, without the inherent need for any “fitting”.
We will begin by concentrating on the 2nd approach: ab initio = “from the
beginning”
Ab initio Molecular Orbital Theory
GOAL: Solve the HF equations with an “infinite” basis set, with no additional
approximations.
Then, any remaining error can be assigned to correlation effects:
E corr  E  E HF
Introduction to Quantum Chemistry
Ab initio Molecular Orbital Theory
Utility:
1. In certain cases Ecorr cancels out.
2. The HF wave functions may accurately describe other properties.
Basis Sets: set of mathematical functions used to construct the wavefunction.
• Each MO in HF theory is expressed as a linear combination of basis functions.
• Full HF wavefunction is expressed as a Slater determinant formed from
individual occupied MOs.
• Desire to have an infinite basis set. While this is impossible, we can get close.
• Useful to choose a basis set functional form that can be evaluated efficiently.
• Basis set form should be chemically useful.
Gaussian Type Orbitals
• Slater type orbitals (STOs) closely resemble hydrogenic AOs, but they are
computationally inefficient – no analytical solution for these functions when
evaluating the SCF integrals.
Introduction to Quantum Chemistry
Ab initio Molecular Orbital Theory
Need an analytical solution to the four-index integral.
Solution: modify the STOs to allow analytic integration – change the radial
decay from e-r to (e-r)2. General form of these GTOs:
 2 
 x, y , z; , i, j, k   

  
3/4
 8 
i ! j ! k !








 2i ! 2 j ! 2 k ! 
i jk
1/ 2
i
j
k
x y z e

 x  y  z
2
2
2

 = width of the GTO; i, j, k = non-negative integers that dictate the nature of
the orbital.
Possible cases:
Introduction to Quantum Chemistry
Contracted Gaussian Functions
GTOs are computationally efficient, but they are not as accurate as STOs.
Important Concern: radial portion of the orbital.
• should have a ‘cusp’ at r = 0
• should decay exponentially at long time (e-r versus (e-r)2)
• this can be addressed with linear combination of GTOs:
 x, y , z; , i, j, k  
M
 c  x , y , z; , i, j, k 
a
a 1
‘contracted’ basis function: basis function is represented as a linear combination of
Gaussians. ‘c’ chosen to optimize the shape and ensure normalization.
‘Primitive Gaussians’: individual Gaussian functions.
** Researchers (Hehre, Stewart, and Pople) have previously found optimal
contraction coefficients and exponents for mimicking the STOs.
** These are designated STO-MG, where M is the number of primitive Gaussians.
Optimal combination of speed/accuracy: STO-3G, defined for most atoms in
periodic table.
STOs vs GTOs
GTOs are mathematically easy to work with
STOs vs GTOs
GTOs are mathematically easy to work with, but the
shape of a Gaussian is not that similar to that of an
exponential.
STOs vs GTOs
Therefore, linear combinations of Gaussians are used to
imitate the shape of an exponential. Shown is a
representation of the 3-Gaussian model of a STO.
Introduction to Quantum Chemistry
Contracted Gaussian Functions
GTOs fail to exhibit radial nodal behavior (present in AOs).
Solution: contraction coefficients allow the nodal behavior to be reproduced
(by manipulating the sign of the coefficients).
Modified Basis Sets
STO-3G = ‘single-z’ basis set, ‘minimal’ basis set, 1 basis function / orbital. Thus:
H, He: 1s
Li to Ne: 1s, 2s, 2px, 2py, 2pz
Na to Ar: 1s, 2s, 2px, 2py, 2pz, 3s, 3px, 3py, 3pz
Increase flexibility: ‘decontract’ the basis set, this is called a ‘double-z’ basis set.
Instead of 3 primitive Gaussians / basis function, the primitives are split:
original: 1 basis function (3 primitives)  1 basis function (2 primitives) ‘contracted’ +
1 basis function (1 primitive) ‘diffuse’
* secular equation is increased.
‘triple-z’ – treat each primitive as a separate basis function.
Introduction to Quantum Chemistry
Modified Basis Sets
Core Orbitals – weakly affected by chemical bonding, whereas valence
orbitals participate significantly.
Result: ‘split-valence’ or ‘valence-multiple-z’ basis sets, 3-21G, 4-31G, 6-31G.
• double # of functions for valence e-, but keep single function for inner shells.
• nomenclature:
• 3-21G: 3 Gaussian primitives for core orbitals, valence e- described by 3
Gaussians – 2 Gaussians for contracted part and 1 Gaussian for the
diffuse part.
• 6-31G: 6 Gaussian primitives for core orbitals, valence e- described by
4 Gaussians – 3 Gaussians for contracted part and 1 Gaussian for the
diffuse part.
• 6-311G: 6-31G with a third layer of valence functions composed of a
single, uncontracted set
Hartree-Fock is not well suited to
calculation of excited state properties
• Hartree-Fock theory works well for ground state properties
because the energies of occupied orbitals are relatively
accurately determined. However, the energy of
unoccupied orbitals is not well determined and therefore
excited state properties and transition energies are not well
determined within the HF approach.
• To account for excited state properties one can include
excited electron configurations. Such ab initio approaches
that move beyond HF theory are collectively called
configuration interaction (CI).