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Theory
From Quantum Mechanics to Density Functional Theory
[based on Chapter 1, Sholl & Steckel (but at a more advanced level)]
• Quantum mechanics (~1920s)
• Hartree & Hartree-Fock-Slater methods
(~1930s-1950s)
• Density functional theory (~1960s)
Quantum theory
• Wave-particle duality of light (“wave”) and electrons
(“particle”)
• Many quantities are “quantized” (e.g., energy,
momentum, conductivity, magnetic moment, etc.)
• For “matter waves”: Using only three pieces of
information (electronic charge, electronic mass, Planck’s
constant), the properties of atoms, molecules and solids
can be accurately determined (in principle)!
Quantum theory – Light as particles
• Max Planck (~1900): energy of electromagnetic (EM) waves can
take on only discrete values: E = nħw
from density of states
from equipartition theorem
– Why? To fix the “ultraviolet catastrophe”
– Classically, EM energy density, e(w) ~ w2eavg = w2(kT)
– But experimental results could be recovered only if energy of a mode is
an integer multiple of ħw as
 (n w )e

e w
n w / kT
e avg
n
n
/ kT

w
e
w / kT
1
n
e(w)

The ultraviolet
catastrophe
Classical (~w2kT)
experimental
w
Quantum theory – Light as particles
• Einstein (1905): photoelectric effect
– No matter how intense light is, if w < wc  no photoelectrons
– No matter how low the intensity is, if w > wc, photoelectrons result
– Light must come in packets (E = nħw)
• Compton scattering (1923): establishes that photons have
momentum!
– Scattering of x-rays of a single frequency by electrons in a graphite target
resulted in scattered x-rays
– This made sense only if the energy and the momentum were conserved, with the
momentum given by p = h/l = ħk (k = 2p/l, with l being the wavelength)
•
By now, it is accepted that waves may display particle features …
Quantum theory – Electrons as waves
• Rutherford (~1911): Experiments indicate that
atoms are composed of positively charged
nuclei surrounded by a cloud of “orbiting”
electrons. But,
– Orbiting (or accelerating charge radiates energy 
electrons should spiral into nucleus  all of matter
should be unstable!)
– Spectroscopy results of H (Rydberg states) indicated
that energy of an electron in H could only be -13.6/n2
eV (n = 1,2,3,…)
Quantum theory – Electrons as waves
• Bohr (~1913):
– Postulates “stationary states” or “orbits”, allowed only if electron’s
angular momentum L is quantized by ħ, i.e., L = nħ implies that E = 13.6/n2 eV
– Proof:
• centripetal force on electron with mass m and charge e, orbiting with velocity v at
radius r is balanced by electrostatic attraction between electron and nucleus 
mv2/r = e2/(4pe0r2)  v = sqrt(e2/(4pe0mr))
• Total energy at any radius, E = 0.5mv2 - e2/(4pe0r) = -e2/(8pe0r)
• L = nħ  mvr = nħ  sqrt(e2mr/(4pe0)) = nħ  allowed orbit radius, r =
4pe0n2ħ2/(e2m) = a0n2 (this defines the Bohr radius a0 = 0.529 Å)
• Finally, E = -e2/(8pe0r) = -(e4m/(8e02h2)).(1/n2) = -13.6/n2 eV
– The only non-classical concept introduced (without justification): L = nħ
Quantum theory – Electrons as waves
• de Broglie (~1923): Justification: L = nħ is equivalent to
nl = 2pr (i.e., circumference is integer multiple of
wavelength) if l = h/p (i.e., if we can “assign” a
wavelength to a particle as per the Compton analysis for
waves)!
– Proof: nl = 2pr  n(h/(mv)) = 2pr  n(h/2p) = mvr  nħ = L
• It all fits, if we assume that electrons are waves!
Quantum theory – Electrons as waves
The Schrodinger equation: the jewel of the crown
•
•
Schrodinger (~1925-1926): writes down “wave equation” for any single
particle that obeys the new quantum rules (not just an electron)
A “proof”, while remembering: E = ħw & p = h/l = ħk
–
–
–
For a free electron “wave” with a wave function Ψ(x,t) = ei(kx-wt), energy is purely kinetic
Thus, E = p2/(2m)  ħw = ħ2k2/(2m)
A wave equation that will give this result for the choice of ei(kx-wt) as the wave function is
2

2
i
(x,t)  
(x,t)
t
2m x 2
•
Schrodinger then “generalizes” his equation for a bound particle

 2  2


i
(x,t)  
2 V (x) (x,t)
t
 2m x

K.E.
P.E.
Hamiltonian operator

The Schrodinger equation
• In 3-d, the time-dependent Schrodinger equation is
 2   2


 2  2 
i
(x, y,z,t)  
 2  2  2  V (x, y,z) (x, y,z,t)
t
 2m x y z 

• Writing Ψ(x,y,z,t) = ψ(x,y,z)w(t), we get the timeindependent Schrodinger equation

 2   2

 2  2 

 2  2  2  V (x, y,z) (x, y,z)  E (x, y,z)
2m
x y z 


Hamiltonian, H
• Note that E is the total energy that we seek, and
 Ψ(x,y,z,t) = ψ(x,y,z)e-iEt/ħ
The Schrodinger equation
H  E
• An eigenvalue problem
– Has infinite number of solutions, with the solutions being Ei
and i
– The solution corresponding to the lowest Ei is the ground state
– Ei is a scalar while i is a vector
– The is are orthonormal, i.e., Int{i(r)j(r)d3r} = dij
– If H is hermitian, Ei are all real (although i are complex)
– Can be cast as a differential equation (Schrodinger) or a
matrix equation (Heisenberg)
– ||2 is interpreted as a probability density, or charge density
Applications of 1-particle Schrodinger equation
• Initial applications
– Hydrogen atom, Harmonic oscillator, Particle in a box
• The hydrogen atom problem
 2   2

 2  2 

 2  2  2  V (x, y,z) nlm (x, y,z)  E nlm nlm (x, y,z)
 2m x y z 



1   2  
1
 
 
1
2
2
  2 r

sin 

r r  dr  r 2 sin   
d  r 2 sin 2   2
e2
4pe 0 r

Solutions: Enlm = -13.6/n2 eV;
ψnlm(r,θ,ϕ) = Rn(r)Ylm(θ,ϕ)
http://www.falstad.com/qmatom/
http://panda.unm.edu/Courses/Finley/P262/Hydrogen/WaveFcns.html
The many-particle Schrodinger equation
• The N-electron, M-nuclei Schrodinger (eigenvalue) equation:
 (r1 , r2 ,..., rN , R1 , R2 ,..., RM )  E(r1 , r2 ,..., rN , R1 , R2 ,..., RM )
The N-electron, M-nuclei wave function
The total energy that we seek
The N-electron, M-nuclei Hamiltonian
2
2
2
2
N
M M
N N
M N
Z
Z
e
2

1
1
e
Z
e
  
 2I  
i2   I J  
  I
2 I 1 J  I RI  RJ 2 i 1 j  I ri  rj I 1 i 1 RI  ri
I 1 2 M I
i 1 2m
M
Nuclear kinetic
energy
Electronic
kinetic energy
Nuclear-nuclear
repulsion
Electron-electron
repulsion
Electron-nuclear
attraction
• The problem is completely parameter-free, but formidable!
– Cannot be solved analytically when N > 1
– Too many variables – for a 100 atom Pt cluster, the wave function is a
function of 23,000 variables!!!
The Born-Oppenheimer approximation
• Electronic mass (m) is ~1/1800 times that of a nucleon mass (MI)
• Hence, nuclear degrees of freedom may be factored out
• For a fixed configuration of nuclei, nuclear kinetic energy is zero
and nuclear-nuclear repulsion is a constant; thus
H elec (r1 , r2 ,..., rN )  Eelec (r1 , r2 ,..., rN )
M N
 2 2 1 N N e2
Z I e2
 
i  
 
2 i 1 j  I ri  rj I 1 i 1 RI  ri
i 1 2m
N
H elec
1 M M Z I Z J e2
E  Eelec  
2 I 1 J  I RI  RJ
Electronic eigenvalue problem is still difficult to solve!
Can this be done numerically though? That is, what if we chose a known
functional form for  in terms of a set of adjustable parameters, and figure out a
way of determining these parameters?
In comes the variational theorem
The variational theorem
• Casts the electronic eigenvalue problem into a minimization problem
• Lets introduce the Dirac notation
 (r1 , r2 ,..., rN )  
H elec   Eelec 
*
3
3
3
...

(
r
,
r
,...,
r
)

(
r
,
r
,...,
r
)
d
r
d
r
...
d
   1 2 N 1 2 N 1 2 rN   
*
3
3
3
...

(
r
,
r
,...,
r
)
H

(
r
,
r
,...,
r
)
d
r
d
r
...
d
rN   H 
1 2
N
1
2
  1 2 N
• Note that the above eigenvalue equation has infinite solutions: E0, E1, E2,
… & correspondingly 0, 1, 2, …
• Our goal is to find the ground state (i.e., the lowest energy state)
• Variational theorem
– choose any normalized function F containing adjustable parameters, and
determine the parameters that minimize <F|Helec|F>
– The absolute minimum of <F|Helec|F> will occur when F = 0
– Note that E0 = <0|Helec|0> thus, strategy available to solve our problem!
The Hartree method
•
•
•
The first attempt to solve the Schrodinger equation for atoms other than
hydrogen (i.e., containing more than one electron) was by Hartree &
Hartree (father & son)
Hartree suggested that  H (r1 , r2 ,..., rN )  1 (r1 )2 (r2 )... N (rN )
Thus
H elec H (r1 , r2 ,..., rN )  EH H (r1 , r2 ,..., rN )
Involves only
one electron
 2 2 M N Z I e2
1 N N e2
 
 i  
 
2
m
R

r
2 i 1 j  I ri  rj
i 1
I 1 i 1
I
i
N
H elec
N
H elec
•
1 N N
  h(i )   v(i, j )
2 i 1 j  I
i 1
Involves two
electrons
Apply variational theorem: minimize <H|Helec|H> subject to the
constraint <H|H> = 1 (normalization)
• This converts the many-electron Schrodinger equation to a set of 1electron equations [proved by Slater] which are much easier to
solve
The Hartree method (contd.)
N
EH   H H elec  H    H h(i )  H
i 1
1 N N
   H v(i, j )  H
2 i 1 j  I
 H h(i )  H   ... 1* (r1 ) 2* (r2 )... N* (rN )h(i )1 (r1 ) 2 (r2 )... N (rN )d 3 r1d 3r2 ...d 3rN
  i* (ri )h(i )i (ri )d 3 ri  Eii
Electronic energy when electrons
do not interact with each other
 H v(i, j )  H   ... 1* (r1 ) 2* (r2 )... N* (rN )v(i, j )1 (r1 ) 2 (r2 )... N (rN )d 3r1d 3 r2 ...d 3 rN
   i* (ri ) *j (rj )v(i, j )i (ri ) j (rj )d 3ri d 3 rj  e 2 
ni (ri )n j (rj )
ri  rj
d 3 ri d 3 rj J ij
ni(ri) = i*(ri)i (ri)
Thus,
EH   H H elec  H
N
1 N N
  Eii   J ij
2 i 1 j  I
i 1
Classical electrostatic interaction
between electrons i and j
We are not done yet (!), as we still need to know all the s to determine EH
(We need to use the variational theorem)
The Hartree method (contd.)
•
Minimize EH with respect to each of the s, say *i(ri), subject to the
constraint <i|i> = 1

 
*
3


E

e

(r
)

(r
)d
r


H
j
j
j
j
j
j 0
*i (ri ) 



j
Lagrange multipliers


h(i)i (ri )    *j (rj )v(i, j) j (rj )i (ri )d 3 rj  e ii (ri )  0




j i


h(i)    *j (rj )v(i, j) j (rj )d 3 rj i (ri )  e ii (ri )




j i
•
Resubstituting h(i) and v(i,j), we get the 1-electron Hartree equation

 2
n j (rj ) 3 
Z I e2
2
2

i  
 e 
d rj i (ri )  e ii (ri )
2
m
r

r
R

r


I
j i
i
j
I
i
nj(rj) = j*(rj)j (rj)
Hartree Hamiltonian, hH
Compare!
The “Hartree” potential:
The electrostatic potential seen by an
electron i due to all other electrons (note
the summation over j)
H elec H (r1 , r2 ,..., rN )  EH H (r1 , r2 ,..., rN )
The Hatree method (contd.)
•
•
•
•
The Hartree 1-electron equation needs to be solved “self-consistently” to
obtain the solutions (i.e., ei and i) for all the electrons! Why?
Because the Hartree potential is written in terms of the solutions
Thus, Hartree “guessed” the solutions, used these guesses to compute
the Hartree potential, after which they solved the equation to get new
solutions, used these to calculate the new Hartree potential, and so on,
till the input and output solutions were close to each other  selfconsistency
Finally, the total energy is given by
1
ZI ZJ e2
1
1
ZI ZJ e2
E  E H  
  E ii    J ij   
2 I J I RI  RJ
2 i j i
2 I J I RI  RJ
i
But,
e i  i h H i  E ii   J ij
j i
Thus,
Equation A
1
1
ZI ZJ e2
E  e i    J ij   
2 i j i
2 I J I RI  RJ
i
Equation B
The Hartree prescription for the total energy
Guess i(ri) for all the electrons
Remember that i(ri) is a 1-electron wave function
ni(ri) = i*(ri)i (ri)
 2
n j (rj ) 3 
Z I e2
2
2

i  
 e 
d rj i (ri )  e ii (ri )
2
m
r

r
R

r


I
j i
i
j
I
i
Solve!
Is new i(ri) close to old i(ri) ?
Yes
Calculate total energy
Z I Z J e2
1
1
E   e i   J ij  
2 i, j
2 I J  I RI  RJ
i
No
The Hartree-Fock-Slater method
•
The Hartree method has deficiencies: the wave function does not obey
the Pauli exclusion principle
•
The Pauli principle can be stated in many ways
–
–
–
–
No 2 electrons can be in the same state
2 electrons with the same spin cannot be in the same spatial orbital
Exchange of 2 electrons will result in a sign change of the total wave function
(we need to explicitly consider spin, but we are going to get by without it!)
 (r1,r2 ,...,rN )   (r2,r1,...,rN )
But,
 H (r1,r2 ,...,rN )   H (r2 ,r1,...,rN )
as
1(r1)2 (r2 )...N (rN )  1(r2 )2 (r1 )...N (rN )

The Slater determinant
Spin orbital
1(x1)  1(r1)1(1)

x1  r1,1
(x1, x 2 ,..., x N ) 
Position & spin variable

•
•
1 (x1 )
1 (x 2 )
1
N!
.
.
2 (x1) . . . N (x1 )
2 (x 2 ) . . . N (x 2 )
.
.
. . .
. . .
.
.
.
.
. . .
.
1 (x N ) 2 (x N ) . . . N (x N )
Exchanging 2 rows (or columns) of a determinant results in a sign change of the
value of the determinant!
Using 
this wave function instead of the Hartree product wave function, and going
through the same exercise results in the Hartree-Fock-Slater equation
Hartree electrostatic interaction
(classical electron-electron
interaction)
The exchange operator
(quantum mechanical
electron-electron interaction)


*
3
ˆ
h(i )     j (rj )v(i, j ) j (rj )d rj  K j i (ri )  e ii (ri )
j i




*
3
ˆ
K ji (ri )      j (rj )v(i, j )i (rj )d rj  j (ri )
 j i

Beyond Hartree-Fock-Slater
•
The Hartree-Fock-Slater method still has deficiencies as it allows 2
electrons of unlike spin to be at the same spatial location (i.e., it includes the
“exchange” interaction exactly, but completely ignores “correlation”)
•
This may be overcome be using a linear combination of Slater determinants
(“Configuration Interaction”), or by perturbation treatments, which have been
shown to provide extremely accurate results (and hence are considered the
“gold standard” of electronic structure computations even today)
•
A note about scaling: If N is the number of electrons, the Hartree, HartreeFock-Slater, and more advanced methods scale roughly as N3, N4, and N6,
respectively, in terms of computational time
•
Density functional theory (DFT) is an alternative approach, that includes
both “exchange” and “correlation” in an approximate way (in practice)
•
DFT scales as N3 or better, and comprises the best trade-off between
accuracy and practicality at the present time
Density functional theory (DFT)
[Hohenberg-Kohn (1965)]
•
DFT is a reformulation of Schrodinger’s quantum mechanics
•
In Schrodinger’s quantum mechanics, observables are functionals of
ψ(r1,r2,…,rN). For e.g., E = <ψ(r1,r2,…,rN)|H|ψ(r1,r2,…,rN)>
•
Note: A functional is a function of a function; e.g., E[f(r)] is a functional of f(r),
but f(r) is a function of r
– i.e., for different choices of the functional form of f(r), E will take on different
values!
•
In DFT, the total energy of a system (or any property, including ψ(r1,r2,…,rN))
is a unique functional of the total electronic charge density, n(r)
[Theorem 1]
•
The correct n(r) minimizes the total energy  equivalent to the variational
theorem
[Theorem 2]

Density functional theory
total electron
density
[Hohenberg-Kohn (1965)]
Theorem 1: n(r)  v(r)  H  all properties; Thus:
External potential
(e.g., due to nuclei =
ΣIe2ZI/|RI-r|)
The total electronic
kinetic energy
E elec
e2
 T[n(r)] 
2

The exchangecorrelation energy (the
only non-classical
term, or the sum total
of our ignorance!)
n(r)n(r') 3 3
d rd r'  v(r)n(r)d 3 r  E xc [n(r)]
r  r'
Theorem 2: The correct n(r) minimizes Eelec; Thus:
δEelec/δn(r) = 0  correct ground state Eelec
So what? The wave function is a function of 3N variables, but the charge density is a
function of only 3 variable! However, the functional form of Exc[n(r)] is not specified
Density functional theory
[Kohn-Sham (1965)]
• A unique one-to-one mapping is established between a
system containing N interacting electrons with charge
density n(r) moving in an external potential, and a
fictitious system of N non-interacting electrons also with
the same change density n(r)
• What is so great about this?
– The problem of N non-interacting electrons is solvable!
– Each of the N non-interacting electrons exists in an “external
effective potential” that contain the information about the
interactions that have been removed!
Density functional theory
[Kohn-Sham (1965)]
Solvable problem
Difficult problem
N interacting
electrons,
with charge
density n(r)
N noninteracting
electrons, with
charge density
n(r)
Unique mapping
veff = v
+ vH + vxc
v
Theorem 1  v[n(r)], vxc[n(r)]
Potential due to nuclei,
external fields, etc.
δExc/δn(r)
Classical electron-electron
interaction (“Hartree”)
The Kohn-Sham 1-electron equation
Since our electrons do not interact with each other,
we may write a Schrodinger equation for each one!
Non-interacting
electron wave function
Non-interacting
electron energy
 2 2

  veff (r ) i (r )  e i i (r )

 2m

Kohn-Sham Hamiltonian, hKS
veff (r )  v(r )  e
2

n(r ' ) 3 E xc [n(r )]
d r '
r  r'
n(r )
vH(r)
vxc(r)
Nuclear potential, electromagnetic potential, etc.
The above 1-electron equation is EXACT, if we know vxc(r)
Since this is not the case, vxc is approximated (herein lies the
division between DFT and quantum chemistry methods …)
The total energy
• Similar to the Hartree treatment, the Kohn-Sham equations are
solved “self-consistently”
– Why? Because veff(r) depends on n(r) which depends on the ψis (which
are the solutions)!
• The resulting self-consistent eis and ψis may be used to compute the
total energy as follows
Z I Z J e2
1
E  Eelec  
2 I J  I RI  RJ
N
Eelec
1 2 n( r ) n( r ' ) 3 3
  e i  e 
d rd r '  n(r )e xc [n(r )]  vxc [n(r )]d 3r
2
r  r'
i 1
Equation C
<ψi|hKS|ψi>
Note : Exc [n(r )]   n(r)e xc [n(r )]d 3r
The Hohenberg-Kohn-Sham prescription for
the total energy
Guess ψi(r) for all the electrons
Remember that ψi(r) is a 1-electron wave function
occ
n ( r )  2  i ( r )
2
i
 2 2




v
(
r
)
eff

 i (r )  e i i (r )
 2m

No
Solve!
Is new ψi(r) close to old ψi(r) ?
Yes
Calculate total energy
N
1
n( r ) n( r ' ) 3 3
Eelec   e i  e 2 
d rd r '  n(r )e xc [n(r )]  vxc [n(r )]d 3r
2
r  r'
i 1
E  Eelec  nuclear  repulsion  energy
Density functional theory
 (r1 , r2 ,..., rN , R1 , R2 ,..., RM )  E(r1 , r2 ,..., rN , R1 , R2 ,..., RM )
Density Functional Theory (DFT)
[W. Kohn, Chemistry Nobel Prize, 1999]
1-electron wave function
(function of 3 variables!)
 2 2




v
(
r
)
eff

 i (r )  e i i (r )
2
m


occ
The “average” potential
seen by electron i
n ( r )  2  i ( r )
i
2
1-electron energy
(band structure energy)
Energy can be obtained from n(r), or from i and ei (i labels electrons)
•
•
Still parameter-free, but has a few acceptable approximations (next lecture)
DFT is versatile: in principle, it can be used to study any atom, molecule,
liquid, or solid (metals, semiconductors, insulators, polymers, etc.), at any
level of dimensionality (0-d, 1-d, 2-d and 3-d)