Lecture 2 of BIOS 203 mini-course taught by Heather Kulik at Stanford University. Introduction to electronic structure theory. http://bios203.stanford.edu or email bios203.course@gmail.com for more information.

1. Introduction to electronic
structure theory!
Heather J Kulik!
hkulik@stanford.edu!
02/27/13!

2. Why we need quantum!
Potential energy surfaces: Bonding and structure:
explicit or for force field from first principles!
development!
experiment!
!
QM!
bent linear!
Sc! Ti! V! Cr! Mn! Fe! Co! Ni! Cu! Zn!
!"

3. Why we need quantum!
Interesting phenomena depend on what the electrons are doing!!
Optical properties! Catalysis!
Magnetic
properties!

4. Quantum mechanics in brief!
Time-dependent Schrödinger equation!
"
!2 2 " " " $#(r, t)
! " #(r, t) +V (r, t)#(r, t) = i!
2m $t
Stationary ! !
V (r, t) = V (r )
potential!
1) Spatial, time- # !2 2 !& ! !
independent %! " +V (r )(! (r ) = E! (r ) from
Schrödinger equation! $
2m ' wikipedia!
d What we usually are
2) Temporal part! i! f (t) = Ef (t) solving in quantum
dt
chemistry!

5. Infinite square well!
n 2! 2 !2
# !2 2 "& " " En =
! " +V (r )(! (r ) = E! (r ) ∞! 2ma 2 ∞!
TISE:" %
$ 2m '
# 0,
% 0!x!a 16 16
for:! V (x) = $ n=4!
% ",
& x < 0, x > a
! 2 d 2! (x)
Inside well:! ! 2
= E! (x)
2m dx 9 n=3! 9
ODE Solution:! ! (x) = Aeikx + Be!ikx , k =
2mE
!
n=2!
Boundary conditions:! ! (0) = ! (a) = 0
4 4
Simplified solution:"
1 n=1! 1
2 ! n! $
! n (x) = sin # x &, n = 1, 2, 3… 0 a
a " a % Related to probability of
where the electron is!!

6. Hydrogen atom!
3
TISE ! 2 $ (n ' ! '1)! ' ! /2 ! 2!+1
Solution:" ! n!m (r, ! , " ) = # & e ! Ln'!'1 ( ! )(Y!m (! , " )
" na0 % 2n(n + !)!
Laguerre Spherical
polynomials! harmonics!
Quantum
numbers:"
n = 1, 2, 3,…
! = 0,1, 2,…, n !1
m = !!,…, !.

7. A note about notation!
!"#$%&%'()*#+,-./012#(#*%'-3()4
! * ! ! !
! = ! (r ) = ! ! !i('6-7*"(285=9:"(;<1#5=ij
(r )! j (r )dr !i ! j = "
ket" bra" ket"
?! >
@
" ?! > # ?! > $! " # "#
* ! ! !
! ! (r )! j (r )dr = !i ! j = "ij
i
B
@
?! > ' ?! > A
" " ? ! > $! " & "
B(

8. A note about notation!
!"#$%&'()$*+,"#%)-'./0
1
" .# 0 ! .# 0 1
" !
&$ $ $ '7')$#5)6)-",'2+-3#%)-4
$ 98 %
1
" !
' '
&$ 1
" !&'
' $
$ 98 %
(:;'9<'=>><'''''?@?=>'A#)*%4#%3'!)B:,%-6')2'!"#:$%",4''CC D:$;$"-B'E:B:$'"-B'F%3),"'!"$G"$%

9. Atoms: Electrons and Nuclei!
ˆ = T + V + V +V
H ˆ ˆ ˆ
e e!e e!N N!N
1" 2" 3" 4"
1 # ! ! &
1" Te = ! # "i
2
3" ˆ
Ve!N (
= "%"V RI ! ri ( )
2 i i $ I '
quantum kinetic
energy of electrons" electrostatic electron-
nucleus attraction"
2" ˆ = "" ! 1 !
Ve!e 4" VN!N
i j>i r ! rj
i electrostatic nucleus-
electron-electron interaction" nucleus repulsion"

10. Born-Oppenheimer!
ˆ ! ! ! ! ! ! ! !
H ! (r1,…, rn , R1,…, RN ) = Etot! (r1,…, rn , R1,…, RN )
We decouple the electrons and nuclei. Only the
electrons are treated as quantum wavefunctions
in the field of fixed or slow nuclei.!
!
Adiabatic: no coupling between electronic
surfaces.!
"
Born-Oppenheimer: Ionic motion does not
influence electronic surface.!

11. Variational Principle!
ˆ | !#
"! | H
E[!] =
"! | !#
Energy of a trial wavefunction is greater Unless the trial wavefunction is
than the energy of the exact solution:! the exact solution:!
E[!] " E0 E[!] = E0
If we have the ground state energy, we have the ground
state wavefunction.!

12. Hartree Equations!
If we assume many-body wavefunction can be represented as single-particle
orbitals:!
! ! ! ! ! ! !
! ! (r , r ,…, r ) = ! (r )! (r )"! (r )
1 2 n 1 1 2 2 n n
!
We obtain Hartree equations directly from variational principle + Time-
independent Schrodinger equation:!
!
& 1 ! ! ! 2 1 !) ! !
(! "i + %V ( RI ! ri ) + % # ! j (rj ) ! ! drj +! i (ri ) = !" i (ri )
2
( 2
' I j$i rj ! ri +
*
Pros: Self-consistent equations – each equation for single particle depends
on others, can solve together iteratively to reach a solution.!
!
Cons: Wave function does not obey Pauli exclusion principle, no explicit
correlation.!

13. Slater determinant!
! ! !
!" (r1 ) ! " (r1 ) " !" (r1 )
! ! !
! ! ! 1 !" (r2 ) ! " (r2 ) " !" (r2 )
! (r1, r2 ,…, rn ) =
n! " " # "
! ! !
!" (rn ) ! " (rn ) " !" (rn )
Satisfies anti-symmetry for Pauli exclusion principle:
Slater determinant changes signs when we exchange
two electrons (columns in the matrix).!

14. Hartree-Fock Equations!
With a Slater determinant wavefunction to represent our many-body
wavefunction:! ! ! !
! (r ) ! (r ) " ! (r )
! ! ! !
! !
"
!
1 ! (r ) ! (r ) " ! (r )
1 " 1 " 1
" 2 " 2 " 2
! ! (r , r ,…, r ) =
n!
1
"
2 n
" # "
! ! !
! (r ) ! (r ) " ! (r )
! " n " n " n
We may now obtain analogous Hartree-Fock equations:!
Standard notation! Bra-ket notation!
$ 1 2 ! !' ! Coulomb integral!
&! "i + #V ( RI ! ri ))! " (ri ) + 1-electron integrals! ˆ
Jij = ! i! j " ee ! i! j
% 2 I (
"i2 Exchange integral!
$ hi = ! i ! +Ve!N ! i
* ! 1 ! !' ! 2 ˆ
K ij = ! i! j " ee ! j! i
&# * ! µ (rj ) ! ! ! µ (rj )drj )! " (ri ) !
&µ
% rj ! ri )
(
$ N N!1 N
! 1 ! !' ! ! ˆ
ESlater [! ] = ! Slater H ! Slater = " hi + " " ( Jij ! K ij )
#& * ! µ* (rj ) ! !
rj ! ri
! " (rj )drj )! µ (ri ) = !" # (ri )
µ &
% )
( i=1 i=1 j=i+1
Pros: Anti-symmetrized, exchange is exact.!
Cons: Only correlation comes from anti-symmetrized Slater.!

15. Self-consistency!
Initial guess Form
Diagonalize
MOs from operators/
Fock matrix!
XYZ! Fock matrix!
Are we no!
Done! yes!
converged?!

16. Basis sets: a technical note!
Molecular wavefunction from the combination of some basis
of atomic-like wavefunctions.!
Minimal: one basis function per A.O.!
Double/triple/4/5/6 zeta: two/three/four/five/six
basis functions per A. O.!
Split valence: one A.O. for core, more for
valence.!
Polarization: From mixing l orbital with l+1!
Diffuse functions: letting electrons move away
from the nucleus (key for anions)!
Notation examples:
6-31G: Split-valence, double-zeta: Core is 6,
valence is 3 in one, 1 in the other.!
6-31G(d)/6-31G*: adds d polarization.!
Easier to
6-311G: split-valence triple-zeta.! work with:!
6-31+G: adds diffuse functions.!

17. Basis sets: other
considerations!
Condensed matter simulations often ∞! ∞!
use plane wave basis sets.!
!
We construct our wave function from a summation 16 16
n=4!
of standing waves, like the solutions to the particle
in a box.!
!
!
! 9 n=3! 9
Pros: Can extrapolate to complete basis set limit,
unlike challenges with localized basis set. Can n=2!
describe delocalized electrons straightforwardly.!
! 4 4
Cons: Need many functions to describe strong
oscillations, e.g. near core of nucleus. Often have 1 n=1! 1
to use pseudopotentials (effective core potentials).! 0 a

18. What about correlation?!
What kind of correlation are we talking about?"
A common definition:" Ecorr = Eexact ! EHF
In Hartree-Fock, each electron experiences repulsion ! ! !
!" (r1 ) ! " (r1 ) " !" (r1 )
from an average electron cloud, motion of individual ! ! !
! ! ! 1 !" (r2 ) ! " (r2 ) " !" (r2 )
electrons is not correlated.! ! (r , r ,…, r ) =
1 2 n
n! " " # "
! ! ! !
!" (rn ) ! " (rn ) " !" (rn )
But Slater determiant enforces: two electrons of same
spin are in the same space, a kind of correlation.!
Dynamic correlation: how one electron repels another when they are nearby.!
!
Static correlation: multi-determinantal character: i.e. one Slater is not enough!!
!
Why we need correlation: accurate relative energetics, van der Waals, etc.!

19. Configuration Interaction!
! 0 = !1! 2 !! I !! K
HF
Ehf 0 dense 0
!1 = !1! 2 !! K+1 !! K
HF
0 dense sparse very sparse
d
e extremely
n sparse sparse
Pros: Introduces dynamic correlation.! s sparse
e
!
Cons: Expensive calculation – many
possible excitations, poor scaling. ! very extremely extremely
0
! sparse sparse sparse
In practice: Truncated excitations to
maintain accuracy and minimize cost. Diagonalize CI, roots are solutions:
(e.g. CCSDT, CISD…)" ground state, first excited state…!

20. Multi-Configuration SCF!
• CSF - Configuration state function.!
! = c0 ! 0 + c1!1 + c2 ! 2 +…
• WFN gets multiple CSFs, multiple determinants
per CSF.!
• Obtain variational optimum for shape of MO s
and for weight of CSF.!
• Orbital energies are undefined, only occupation
matters:!
Pros: Introduces both Cons: Difficult to find true minimum in coefficient
static and dynamic space, combinatorial explosion of CSFs. Still
correlation.! lacking dynamic correlation (MRCI)!
!
n!(n +1)!
N= CH3OH!
!m$ !m $ ! m$ ! m $ m=14,n=12!
# &!# +1&!# n ' &!# n ' +1&! N=169,884!!
"2%"2 %" 2%" 2 %

21. Density functional theory!
Is there a way to work with the probability
# !2 2 "& " " density instead of the wave function?!
TISE:" %! " +V (r )(! (r ) = E! (r )
$ 2m '
Ψ
?!
ρ
Fundamental principles of density functional theory:!
Vext!
1st Hohenberg-Kohn
theorem (1964):"
E,
Ψ
Vext and N,ρ define the
quantum problem!
Charge density uniquely
determines the potential."
N,ρ

22. 2nd Hohenberg-Kohn theorem!
A variational principle for DFT:"
! ! ! ! !
Ev [ ! '(r )] = F[ ! '(r )]+ ! vext (r )! '(r )dr " E0
Trial wavefunction
from trial density:" Our universal Energy of the
functional with an exact solution."
Ψ
ρ
initial guess."
BUT we still donʼt know what our functional form is!!

23. The Kohn-Sham approach !
A unique mapping!"
Non-interacting K-S
electrons live in an external
Interacting 1-to-1 Non-interacting potential such that their
mapping! density is the same as the
particles! particles!
fully-interacting density.!
3N variables! 3 spatial variables!
Kohn-Sham equations:"
ˆ ! (r) = #! 1 " 2 + " (r) + " (r) + " (r)&! (r) = # ! (r)
H KS i %
$ 2 H xc ext ( i
' i i
N
n(r') ! E xc 2
! H (r) = " r ! r'
dr' ! xc (r) = n(r) = ! !i (r)
! n(r) i=1

24. Exc: Exchange & Correlation!
In principle, DFT is exact…"
" ! E xc
…but we need the dependence ! xc (r) =
of Exc with our density." ! n(r)
What is Exc?!
E xc [ ! (r)] = !T [ ! (r)] + !Vee [ ! (r)]
Correction to Non-classical
kinetic energy corrections to
from interacting electron-electron
nature of electrons" repulsion"

25. Understanding our Exc!
Typically model exchange and correlation separately. !
!
Exchange-correlation hole is often fit in terms of model system. This is
the space whereby other electrons are excluded around a given electron.!
"
Local density approximation:"
!
Exchange modeled on HEG ! ! !Correlation on HEG!
!
1/3
! LDA 3" 3 % high limit! ! c = A ln(rs ) + B + rs (C ln(rs ) + D)
E x [ ! ] = ! $ ' ( ! (r)4/3 dr
! 4#" &
!
1 ! g0 g1 $
! low limit! ! c = # + 3/2 +... &
2 " rs rs %
Correlated quantum chemical techniques are used
to interpolate between these limits!

26. Many other flavors of Exc!
Generalized gradient approximations: Exc variants are often
including gradient of the density! equated with rungs
! on a ladder…"
Meta-GGAs: Adding in Laplacian of the
density.!
!
Hybrids: Add in H-F exchange (% is
empirical).!
!
DFT+vDW: Add in semi-empirical treatment of
van der Waals!
!
General approaches: fit against analytical
forms in HEG vs. heavily parameterized for test
set against experiment. !

27. DFT in practice!
Errors in geometries can be quite small, considerably
smaller than Hartree-Fock.!
!
Errors in energetics vary widely. Can get errors as little
as a few kcal/mol (needed for mechanistic predictions).!
!
Key challenges: open shell character, multi-reference
character, lack of London dispersion forces, non-
covalently bonded complexes, self-interaction errors:
charge-transfer bound complexes, over-delocalization,
etc…!
!
!

28. Follow-up reading!
• Quantum mechanics!
– C. Cohen-Tannoudji, B. Diu, and F. Laloe Quantum Mechanics: Vol 1
and 2. Wiley-VCH (1992).!
– D. A. Mcquarrie and J. D. Simon Physical Chemistry: A molecular
approach. University Science books (1997).!
– D. J. Griffiths Introduction to quantum mechanics. Addison-Wesley
(2004). !
• Hartree-Fock theory and extensions:!
– R. J. Bartlett “Many-body perturbation theory and coupled cluster theory
for electron correlation in molecules” Ann. Rev. Phys. Chem. (1981).!
– K. B. Lipkowitz, D. B. Boyd, R. J. Bartlett, J. F. Stanton “Applications of
Post-Hartree-Fock Methods: A tutorial” Rev. Comput. Chem. (2007).!
• Density functional theory:!
– K. Burke “Perspective on density functional theory” J. Chem. Phys.
(2012).!
– K. Burke. The ABC of DFT
http://chem.ps.uci.edu/~kieron/dft/book/!
– P. Mori-Sanchez, A. J. Cohen, and W. Yang “Localization and
delocalization errors in density functional theory and implications for
band-gap prediction” Phys. Rev. Lett. (2008).!