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Electronic Structure Theory
Lecture Schedule
1. Born-Oppenheimer approx.- energy surfaces
2. Mean-field (Hartree-Fock) theory- orbitals
3. Pros and cons of HF- RHF, UHF
4. Beyond HF- why?
5. First, one usually does HF-how?
6. Basis sets and notations
7. MPn, MCSCF, CI, CC, DFT
8. Gradients and Hessians
9. Special topics: accuracy, metastable states
Jack Simons, Henry Eyring Scientist and Professor
Chemistry Department
University of Utah
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To whom are these lectures directed?
Ph. D. level students who have completed first-year graduate classes
in quantum mechanics in a chemistry or physics department and who
wish to learn more about electronic structures.
Research-active experimental chemists who have at least this same
background and who are presently or wish to use electronic structure
calculations to help interpret and guide their scientific studies.
Ph. D. and postdoctoral students specializing in theoretical chemistry
but with emphasis outside electronic structure theory can also
benefit.
Faculty at primarily undergraduate institutions who wish to include
more theory in their classes and who have had sufficient background.
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Sources of additional information- beyond the tip of the iceberg.
Molecular Electronic Structure Theory, Helgaker, Jørgensen, Olsen
Second Quantization Based Methods in Quantum Chemistry,
Jørgensen, Simons
Quantum Mechanics in Chemistry, Simons, Nichols
(http://simons.hec.utah.edu/TheoryPage/quantum_mechanics_in_ch
emi.htm)
My theoretical chemistry web site:
http://simons.hec.utah.edu/TheoryPage
An Introduction to Theoretical Chemistry, Simons
(http://simons.hec.utah.edu/NewUndergradBook/)
Qutantum Chemistry, 5th Ed., I. N. Levine
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Lecture Schedule:
Session 1: Born-Oppenheimer approximation: adiabatic and diabatic surfaces;
non-BO couplings; surface crossings; the electronic and vibration-rotation
Schrödigner equations; minima and transition states.
Session 2: Hartree-Fock: atomic units; electron-nuclear and electron-electron
cusps, antisymmetry; Coulomb holes, mean –field potential, Slater determinants,
spin-orbitals, spin functions, Slater-Condon rules, The HF equations, Coulomb
and exchange, Koopmans’ theorem, orbital energies, problems arising when
using single determinant approximations; certain states require more than one
determinant; restricted and unrestricted wave functions.
Session 3: Pros and cons of HF: limitations of single determinants,
configuration state functions, homolytic bond cleavage and the need for CI,
dynamical and essential electron correlation; restricted and unrestricted HF
(RHF, UHF),
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Session 4: Beyond HF-Why? Bond breaking, configuration mixing, polarized
orbital pairs for essential and dynamical correlation, how important correlation
is; reminder about cusps and introduction to explicitly correlated wave
functions.
Session 5: First, one usually does HF-how? LCAO-MO, scaling with basis size,
SCF process, how to start the SCF, meaning of occupied and virtual orbitals,
spin problems in UHF,
Basis notations; complete-basis extrapolation of the Hartree-Fock and
correlation energies.
Session 6: Basis sets and notations ; STOs and GTOs, nuclear cusps, contracted
GTOs, core, valence, polarization, diffuse, and Rydberg basis functions,
notations, complete basis extrapolation, basis set superposition errors (BSSE).
Session 7: Why beyond HF?MPn, MCSCF, CI, CC, DFT. MP theory, E1, E2, y1,
Brillouin theorem, divergence and why, size-extensivity, multi-configuration
SCF (MCSCF), AO-to-MO integral transformation, configuration interaction
(CI)); coupled-cluster (CC), density functional theory (DFT), Kohn-Sham
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equations, strengths and weaknesses, scaling with basis size.
Session 8: Gradients, Hessians, minima and transition states, reaction paths,
harmonic vibrational frequencies, Hellmann-Feynman theorem,
orbital
responses.
Session 9: Typical error magnitudes for various methods and various basis sets.
Special tricks for studying metastable anions; variational collapse; virtual
orbitals are difficult to identify- examples; long-range potentials and the
centrifugal potential; valence and long-range components of the wave function;
relation to electron scattering; charge stabilization method; the stabilization
method.
Some of the material will be covered in problem/discussion sections.
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The following several slides contain material I ask you to read
and refresh your memory about prior to the School. I am asking
you to do so because the electronic structure lectures are the first
you will hear at the School, and I would like you to prepare before
hand. During the School, we will provide you with material from
the other Lecturers that you can view prior to those lectures to
similarly prepare.
You should have had this material in your quantum, spectroscopy,
or angular momentum classes. If there is something you don’t
know about, please have a friend teach you so you will be versed
in it when the School begins. I. N. Levine’s text Quantum
Chemistry 5th Ed. contains much of this material (look under
ladder operators, angular momentum, spin, Slater determinants,
and Condon-Slater rules.
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A brief refresher on spin
  |  1
SZ 1/2 
SZ   1/2 
S 2 
  |   0
  |  1
1/2(1/2 1)  3/4 2
2
S 2 
1/2(1/2  1)  3/4 2

Special case of
2
J 2 | j,m 
2
j( j  1) | j,m 

S 
1 1
1 1
( 1)  ( 1)  
2 2
2 2
S  0
Special case of
J | j,m 
j( j 1)  m(m 1) | j,m 1 

For acting on a product of spin-orbitals, one uses
SZ   SZ ( j)
j
Examples:

S   S ( j)
S 2  S S  SZ2  SZ
j
SZ(1)(2) 1/2 (1)(2) 1/2 (1)(2)  (1)(2)
S(1)(2)  (1)(2)  (1)(2)

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Let’s practice forming triplet and singlet spin functions for 2 e’s.
We always begin with the highest MS function because it is “pure”.
 (1) (2)
So, MS =1; has to
be triplet
SZ(1)(2) 1/2 (1)(2) 1/2 (1)(2)  (1)(2)
SZ (1)(2)  1/2 (1)(2) 1/2 (1)(2)   (1)(2) So, MS =-1; has to
be triplet
S (1) (2)   (1) (2)   (1) (2)

1(1 1) 1(11) | S  1, M S  0 
So, |1,0 
1
[ (1) (2)   (1) (2)]
2
This is the MS =0 triplet
 to have MS = 0 and be orthogonal
How do we get the singlet? It has
1
to the MS = 0 triplet. So, the singlet is
| 0,0 
[ (1) (2)   (1) (2)]
2
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Slater determinants (Pi,j) in several notations. First, for two electrons.
1  (r1 )   (r1 )
y (r1 ,r2 ) 
2  (r2 )   (r2 )
Shorthand
y (r1 ,r2 )   
1  (r1 ) (1)  (r1 ) (1)

2  (r2 ) (2)  (r2 ) (2)
y (r1 ,r2 ) 
1
2
 (r ) (1) (r ) (2)   (r ) (1) (r ) (2)
1
  (r1 ) 2 (r2 )
| 12 |
2
1
2
1
2
 (1) (2)   (1) (2)
Symmetric space;
antisymmetric spin
(singlet)
1
1
[1 (1)2 (2)  1 (2)2 (1)] 
[1 (1) 2 (2)  2 (1)1(2)] (1) (2)
2
2
y (r1 ,r2 )  y (r2 ,r1 )
Antisymmetric space; symmetric spin (triplet)
1  (r1 )   (r1 )
1  (r2 )   (r2 )

2  (r2 )   (r2 )
2  (r1 )   (r1 )
Notice the Pi,j
antisymmetry
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More practice with Slater determinants
y (r1 ,r2 ,...,rn )  1 , 2 ....., n
1 (1)
y (r1 ,r2 ,...,rn ) 
Shorthand notation for general case
 2 (1)
 n (1)
1 (2)  2 (2)
 n (2)
1 (n)  2 (n)
 n (n)
Odd under interchange of
any two rows or columns
The dfn. of the Slater determinant
contains a N-1/2 normalization.
y (r1 , r2 ,..., rn ) 
1
p(P)

1
P 1 (1)2 (2)...n (n)  O 1 2 ... n 



n! P
P permutation operator
O
1
p( P)

1
P



n! P
 1
p( P)
antisymmetrizer
parity ( p(P) least number of
transpositions that brings the indices
back to original order)
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Example : Determinant for 3-electron system


O 1  2  3 

1 
1

P

P
  ij  ijk  1  2  3

6
i, j
i, j,k


1 (1) 2 (2) 3 (3)   2 (1)1 (2) 3 (3) 

1 



(1)

(2)

(3)


(1)

(2)

(3)
 3

2
1
1
3
2
6



(1)

(2)

(3)


(1)

(2)

(3)
3
1
3
1
2
 2

permutations
1, P12 , P13 , P23 , P231 , P312
transpositions
0 1
 
parity
3

1

2

2

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The good news is that one does not have to deal with most of these
complications. Consider two Slater determinants (SD).
1
yA 
(1) p P1(1) 2 (2) 3 (3)... N (N)

N!
P
P
yB 
1
qQ
(1)
Q '1 (1) '2 (2) '3 (3)... 'N (N)

N! Q

Assume that you have taken t permutations1 to bring the two SDs into
maximalcoincidence. Now, consider evaluating the integral
*
d1d2d3...dN
y

A [  f ( j) 
j1,N
 g(i, j)]y
B
jk1,N
where f(i) is any one-electron operator (e.g., -ZA/|rj-RA|) and g(i.j)
is any two-electron operator (e.g., 1/|rj-rk|). This looks like a
2) x N! terms).
horrible
task
(N!
x
(N
+
N

1. A factor of
(-1)t
will then multiply the final integral I
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I
 d1d2d3...dNy [ 
*
A
f ( j) 
j1,N
yA 

 g(i, j)]y
B
jk1,N
1
pP
(1)
P1(1) 2 (2) 3 (3)... N (N)

N! P
yB 
Q
1
(1) q Q '1 (1) '2 (2) '3 (3)... 'N (N)

N! Q
1.The permutation P commutes with the f + g sums, so

1
I
N!
2.
I

 d  (1)
pP
 *1 (1) *2 (2) *3 (3)... *N (N)[  f ( j) 
P
j1,N
Py B  (1) y B and
pP
N!
N!
 d   * (1) *
1
2
 (1) p (1) p  N!
P
P
(2) *3 (3)... *N (N)[  f ( j) 
j1,N
 *N (N)[  f ( j) 
 d *1 (1) *2 (2) *3 (3)...
j1,N

jk1,N
jk1,N
so
P
P
 g(i, j)]Py
 g(i, j)]y
B
jk1,N
g(i, j)] (1) q Q '1 (1) '2 (2) '3 (3)... 'N (N)
Q
Q
Now what?
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B



I
2 (2) *3 (3)... *N (N)[  f ( j) 
 d * (1) *
1
j1,N

g(i, j)](1) q Q '1 (1) '2 (2) '3 (3)... 'N (N)
Q
jk1,N
Q
Four cases: the Slater-Condon rules (memorize them)
yA and yB differ by three or more spin-orbitals: I = 0
yA and yB differ by two spin-orbitals-AkAl;BkBl
I
 dkdl *
Ak
(k) *Al (l)g(k,l)[Bk (k)Bl (l)  Bl (k)Bk (l)]
yA and yB differ by one spin-orbital-Ak;Bk
I
  dkdj * (k) * ( j)g(k, j)[ (k) ( j)   (k)
  dk * (k) f (k) (k)
Ak
j
Bk
j
j
Bk
( j)]
j A,B
Ak
Bk
yA and yB are identical
I    dkdj *k (k) * j ( j)g(k, j)[ k (k) j ( j)   j (k) k ( j)]

k j A

k A
 dk *
k
(k) f (k) k (k)
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