CHEM 834: Computational Chemistry

1
CHEM 834 Computational Chemistry
Quantum Chemical Methods 3
March 17, 2009
2
Topics
last time

• review of quantum mechanics

• basic principles of quantum chemistry

• Hartree-Fock methods

today

• Post-Hartree-Fock calculations

• electron correlation

• multi-determinant wavefunctions

• perturbation theory

3
Hartree-Fock SCF Procedure

• specify molecule (nuclear coordinates, charges,
  number of electrons) and basis set ?

• calculate X S-1/2

• obtain a guess density matrix, P (more on this
  later)

• build Fock matrix, F Hcore G

• transform Fock matrix, F XFX

• diagonalize F to get C and ?

self-consistent field cycle

• calculate C XC

• form a new density matrix P using C

• if new P differs from old P, go back to 5. If
  new P is the same as old P, stop.

this procedure yields the set of coefficents C
that define the lowest energy single Slater
determinant wavefunction for a given set of basis
functions ?
4
Initial Guess of Molecular Orbitals
Hartree-Fock calculations are based on a
self-consistent optimization of the electronic
structure ? where does the original guess come
from?
1. take a superposition of atomic states

• usually a good guess of the distribution of
  electron density

2. diagonalize the core Hamiltonian
3. use orbitals from a lower-level calculation
4. use orbitals from a previous calculation
5
Initial Guess of Molecular Orbitals
Hartree-Fock calculations are based on a
self-consistent optimization of the electronic
structure ? where does the original guess come
from?
HOMO-LUMO mixing

• for singlet diradicals we need to break the
  symmetry between the ? and ? HOMOs

LUMO
HOMO
6
Convergence of Molecular Orbitals
SCF iterations continue until the wavefunction
stops changing ? how do we know when it stops
changing?
Convergence criteria
1. maximum change in any element of the density
matrix
Gaussian
1.0 x 10-4 for single point calculations
1.0 x 10-5 for geometry optimizations
2. change in total energy of the system
Gaussian
5.0 x 10-5 Hartree for single point calculations
1.0 x 10-6 Hartree for geometry optimizations
7
Hartree-Fock in Gaussian/Gaussview
Gaussian has the capability to run

• restricted Hartree-Fock

• HF or RHF on the route line

• default method in Gaussian

• restricted open-shell Hartree-Fock

• ROHF on the route line

• be sure to assign the proper multiplicity

• unrestricted Hartree-Fock

• UHF on the route line

• be sure to assign the proper multiplicity

• care needed in assigning guess density matrix for
  singlet diradicals

8
Hartree-Fock in Gaussian/Gaussview
Hartree-Fock methods are specified under the
Method tab in the Calculation Setup window

• HF set by default in route line

• can specify multiplicity for open-shell systems

• can selected different types of HF methods

9
Guess Wavefunction in Gaussian/Gaussview
Gaussian has the capability to generate various
guesses

• Harris functional

• guessharris on route line

• default method in Gaussian

• basically superposition of atomic densities

• core Hamiltonian

• guesscore on route line

• diagonalizes Hamiltonian without
  electron-electron interactions

• Huckel method

• guesshuckel on route line

• does a quick semi-empirical calculation

• read from checkpoint file

• guessread on route line

• uses orbitals from a previous calculation

10
Guess Wavefunction in Gaussian/Gaussview
initial guess is specified under the Guess tab
in the Calculation Setup window

• different types of guesses are available from
  drop-down window

11
Guess Wavefunction in Gaussian/Gaussview
initial guess is specified under the Guess tab
in the Calculation Setup window

• HOMO-LUMO mixing can be selected for singlet
  diradicals

12
Examples
1. Restricted Hartree-Fock calculation of
twisted ethene
2. Restricted Open-Shell Hartree-Fock
calculation of twisted ethene
3. Unrestricted Hartree-Fock calculation of
ethene without HOMO-LUMO mixing
4. Unrestricted Hartree-Fock calculation of
ethene with HOMO-LUMO mixing
90
Also look at
all of the Gaussian/Gaussview examples provided
in the notes on search the potential energy
surface used restricted Hartree-Fock calculations
13
RHF Calculation of Twisted Ethene - Input

• hf specifies restricted Hartree-Fock calculation

• 3-21g specifies the basis set

14
RHF Calculation of Twisted Ethene - Output
we are interested in

• whether the SCF cycle converged

• the SCF energy

• level of convergence and number of SCF cycles
  needed to achieve convergence

• E(RHF) Hartree-Fock energy in Hartree

15
ROHF Calculation of Twisted Ethene - Input

• rohf specifies restricted open-shell Hartree-Fock
  calculation

• 3-21g specifies the basis set

16
RHF Calculation of Twisted Ethene - Output
we are interested in

• whether the SCF cycle converged

• the SCF energy

• level of convergence and number of SCF cycles
  needed to achieve convergence

• E(ROHF) Hartree-Fock energy in Hartree

• information regarding spin contamination

• no spin contamination for ROHF

17
UHF Calculation of Twisted Ethene, No Mixing -
Input

• uhf specifies unrestricted Hartree-Fock
  calculation

• 3-21g specifies the basis set

18
UHF Calculation of Twisted Ethene, No Mixing -
Output
we are interested in

• whether the SCF cycle converged

• the SCF energy

• level of convergence and number of SCF cycles
  needed to achieve convergence

• E(UHF) Hartree-Fock energy in Hartree

• information regarding spin contamination

• no spin contamination for because electrons
  remained paired due to symmetry in initial guess
  wavefunction

19
UHF Calculation of Twisted Ethene, Mixing - Input

• uhf specifies unrestricted Hartree-Fock
  calculation

• guessmix specifies HOMO-LUMO mixing

• 3-21g specifies the basis set

20
UHF Calculation of Twisted Ethene, Mixing - Output
we are interested in

• whether the SCF cycle converged

• the SCF energy

• E(UHF) Hartree-Fock energy in Hartree

• number of SCF cycles needed to achieve convergence

• information regarding spin contamination

• spin contamination because UHF wavefunction is
  not an eigenfunction of S2

• Gaussian projects out (annihilation) the largest
  component of the spin contaminant

21
Comparison of Hartree-Fock Results for Twisted
Ethene
Method
Energy
RHF
-77.372645
ROHF
these are all the same
-77.372645
UHF, no mixing

• RHF, ROHF and UHF without mixing all yield the
  same electronic structure with paired electrons
• mixing allows spins to localize independently

-77.372645
UHF, with mixing
-77.412018
which result is best?

• UHF with mixing gives the lowest energy solution
  ? best approximation of the energy

22
Electron Correlation
based on the variation principle
Hartree-Fock energy
real ground state energy
why is the Hartree-Fock energy too high?
Hartree-Fock method uses a single Slater
determinant wavefunction

• electron-electron interactions are treated in an
  average sense

• each electron feels the average Coulomb
  repulsion arising from the static distrubution of
  the other N-1 electrons

• instantaneous electron-electron interactions are
  neglected

• causes Coulomb energy to be too high

• remedied partially by exchange interactions,
  which correlate positions of electrons with the
  same spin (this is kind of correlation)

23
Electron Correlation
the correlation energy is

• the correlation energy is everything that is not
  included in a Hartree-Fock calculation

• Ecorr is similar to bond energies ? important if
  calculating reaction energies

there are two types of electron correlation
1. Dynamic correlation

• accounts for fact that electrons move such that
  they avoid other electrons

• this is the instantaneous electron-electron
  interactions missing in Hartree-Fock

2. Static correlation

• in systems with multiple resonance states,
  electrons can avoid each other by occupying
  different resonance states

• a single Slater determinant only represents 1
  resonant state, so Hartree-Fock cannot describe
  this type of correlation

• also called non-dynamic correlation

24
Dynamic Correlation

• electrons are charged particles moving in space

• their motions are correlated so that they avoid
  each other

• Hartree-Fock calculations do not capture this

• consequence of considering static electron
  distributions

• direct consequence of using a single Slater
  determinant wavefunction

• only correlation is from exchange interactions

Solution
change the form of the wavefunction
1. Multi-determinant Wavefunction

• called post-Hartree-Fock methods

2. Perturbation Theory
25
Multideterminant Wavefunctions
since a single Slater determinant is not
sufficient, lets try a linear combination of
Slater determinants
motivated from 4th postulate of quantum mechanics
any wavefunction can be expressed as a linear
combination of the eigenfunctions of any quantum
mechanical operator
details

• to get the best multideterminant wavefunction we
  must variationally optimize the coefficients, ai

• if i ?, we will get the exact wavefunction

• in practice, we can only include a finite number
  of determinants

• which determinants do we include?

• where do they come from?

26
Multideterminant Wavefunctions
Slater determinants in a multideterminant
wavefunction are derived from the Hartree-Fock
Slater determinant wavefunction
in Hartree-Fock calculations

• molecular orbitals are expanded as a linear
  combination of K basis functions

• taking linear combinations of K basis functions
  yields 2K molecular orbitals (considering spin)

• only the lowest energy N orbitals are used to
  build the Hartree-Fock Slater determinant

• the remaining 2K N orbitals are called
  unoccupied or virtual orbitals

• we can switch occupied and unoccupied orbitals
  to make new Slater determinants

27
Multideterminant Wavefunctions
Slater determinants in a multideterminant
wavefunction are derived from the Hartree-Fock
Slater determinant wavefunction

• switching orbitals corresponds to exciting
  electrons into higher energy states

virtual
energy
occupied
28
Multideterminant Wavefunctions
Slater determinants in a multideterminant
wavefunction are derived from the Hartree-Fock
Slater determinant wavefunction

• switching orbitals corresponds to exciting
  electrons into higher energy states

• we can use single excitations

virtual
labeled r, s,
energy
occupied
labeled i, j,
29
Multideterminant Wavefunctions
Slater determinants in a multideterminant
wavefunction are derived from the Hartree-Fock
Slater determinant wavefunction

• switching orbitals corresponds to exciting
  electrons into higher energy states

• we can use single excitations

virtual
labeled r, s,

• we can use double excitations

energy

• we can excitations from up to all N occupied
  orbitals into any of the 2K N virtual orbitals

occupied

• this gives a whole series of new Slater
  determinant wavefunctions

labeled i, j,

• note that the Hartree-Fock molecular orbitals, ?,
  are not re-optimized in this procedure

30
Multideterminant Wavefunctions
Slater determinants in a multideterminant
wavefunction are derived from the Hartree-Fock
Slater determinant wavefunction
how does including excited Slater determinants
improve the wavefunction?
1. we get more variational parameters

• more parameters to variationally optimize in
  guess wavefunction ? lower energy solution

2. including extra orbitals gives electrons more
space to avoid each other
?3
?2
consider allyl anion

• excitation allows one of the original ?2
  electrons to spread out into ?3

?1
31
Multideterminant Wavefunctions
how are multideterminant wavefunctions used in
practice?
Full configuration interaction

• also called Full CI

• includes entire set of excitations from N
  occupied orbitals into all possible combinations
  of 2K N virtual orbitals

• this will give the exact wavefunction for a
  specific basis set

• the number of Slater determinants is tremendous

• computational effort is prohibitive, and full CI
  calculations are only used on very small molecules

32
Multideterminant Wavefunctions
how are multideterminant wavefunctions used in
practice?
Truncated configuration interaction

• only consider certain levels of excitations

• most common include all single and double
  excitations (CISD)

• reduces significantly the number of determinants
  in the wavefunction, making the calculations more
  computationally tractable

• double excitations are most important for
  capturing correlation energy

• still recovers 95 of the correlation energy

• has problems related to size-consistency

33
Size Consistency
the energy of two well-separated molecules should
be the sum of their energies

• truncated CI methods are not size-consistent

• consider the double excitations for an A B system

• all excitations are from 2 occupied orbitals on A
  into 2 virtual orbitals on A

A

• all excitations are from 2 occupied orbitals on B
  into 2 virtual orbitals on B

B

• some excitations are from occupied orbitals on A
  into virtual orbitals on A

50 Å
B
A

• some excitations are from occupied orbitals on B
  into virtual orbitals on B

• some excitations will be from 1 occupied orbital
  on A into 1 virtual orbital on A and 1 occupied
  orbital on B into 1 virtual orbital on B

more excitations are included in the supersystem
than in the sum of the individual molecules with
truncated CI
34
Coupled-Cluster Calculations
if we include the right excitations, we can use a
truncated CI expansion and retain size-consistency
coupled-cluster operator
generates all Nth excitations
generates all double excitations
generates all single excitations
coupled-cluster wavefunction
35
Coupled-Cluster Calculations
truncated coupled-cluster wavefunction

• includes all possible single and double
  excitations

• product of operating with the singles operator
  twice

• gives excitations the prevent CISD from being
  size consistent

currently, the CCSD(T) method is the gold
standard quantum chemical method

• includes all possible single, double and some
  triple excitations

• wavefunction involves as huge number of Slater
  determinants

• computational effort prevents application of
  CCSD(T) to systems with more than 20 atoms

36
Perturbation Theory
basic idea
start with a problem we can solve and add in
parts we cannot solve as small perturbations
part we cannot solve (perturbation)
total Hamiltonian
part we can solve
dimensionless parameter

• varies from 0 to 1

• turns perturbation on or off

part we can solve gives
zeroth order ground state energy
zeroth order ground state wavefunction
37
Perturbation Theory
we are interested in ground state eigenfunctions
and eigenvalues for the whole Hamiltonian
expand ? and E as Taylor series about ?(0) and
E(0)
first order correction to the wavefunction
first order correction to the energy
38
Perturbation Theory
inserting the Taylor expansions into the
eigenvalue expression
collecting terms with the same power of ?
set of equations to get the corrections to the
energy and wavefunction
39
Perturbation Theory
lets enforce orthonormality
then multiply each equation on the left by ?0(0)
and integrate
this gives use the zeroth order energy
40
Perturbation Theory
lets continue
1
0
0
zeroth order wavefunctions
first order correction to the energy
in general, the Nth order correction to the
energy depends on the N-1, N-2, corrections to
the wavefunction
41
Perturbation Theory
how do we get the wavefunction corrections?
using 4th postulate of quantum mechanics
eigenfunctions of H(0)
linear expansion coefficients
first order correction to the wavefunction
42
Perturbation Theory
setting
yields (after some algebra)
zeroth order ground state wavefunction
zeroth order wavefunction for the ith excited
state
zeroth order ground state energy
zeroth order energy of the ith excited state
43
Perturbation Theory
first order correction to the wavefunction
we also have
so we can solve for E0(2) from
giving (we wont work it out)
44
Perturbation Theory
in summary
we split the Hamiltonian into parts we can solve
and parts we cannot solve
starting from a wavefunction that we can solve,
we get corrections to the energy and wavefunction
that account of the effect of V
45
Perturbation Theory
how do we apply perturbation theory to chemical
problems?
we split H into a part we can solve, and a part
we cannot solve
we can solve the Fock operator, so
the difference between H and H(0) gives
46
Perturbation Theory
we also take the zeroth order wavefunction to be
a single Slater determinant
the zeroth order energy is then
energy of molecular orbital i

• the zeroth order energy is just the sum of the
  orbital energies

• recall, this does not equal the Hartree-Fock
  energy

• it double counts the electron-electron
  interactions

47
Perturbation Theory
now we can solve for the energy corrected to
first order
Hartree-Fock wavefunction
full electronic Hamiltonian
in perturbation theory, the energy corrected to
first order is the Hartree-Fock energy!!!
1st order corrections get a single Slater
determinant result
48
Perturbation Theory
consider the second order correction to the
energy and first order correction to the
wavefunction
ground state Slater determinant
excited Slater determinant
these corrections mix in extra Slater
determinants!!!
49
Perturbation Theory
lets look at the second order correction to the
energy
if you work this out
50
Møller-Plesset Perturbation Theory
if we put this all together, we get whats called
second order Møller-Plesset Perturbation Theory
basically, if we do a Hartree-Fock calculation we
have all of the basic quantities needed to
calculate the corrections to the wavefunction and
energy

• second order Moller-Plesset theory is called MP2

• Nth order Moller-Plesset theory is called MPn

• calculations up to MP4 are not unusual

• for larger values of n, the corrections to the
  wavefunction and energy become very complex

51
Møller-Plesset Perturbation Theory
MP2 calculations were the standard way of
capturing electron correlation prior to the
advent of density functional theory methods
Advantages of MP methods

• MP2 captures 90 of electron correlation

• necessary integrals in corrections can be
  evaluated relatively rapidly

Disadvantages of MP methods

• MP methods are not variational

• we cant be sure the real energy is lower than
  the MPn energy

• electron correlation is not a minor perturbation

• Taylor series only converges if perturbation is
  small

• going from MP2 ? MP3 ? MP4 ? does not guarantee
  an improvement in the wavefunction and energy

52
Static Correlation
the electronic structure of some systems are best
described as a combination of several states
(think resonance structures)

• allowing electrons to spread out among more
  that one state decreases electron-electron
  repulsion

• called static correlation

• has nothing to do with the correlated motions of
  electrons

Example
? system of singlet trimethylenemethane
?4

• 4 ? electrons

• 2 degenerate frontier orbitals

energy
?2
?3

• 4 ? orbitals in total

?1
how do we distribute the ? electrons?
53
Static Correlation
energy
energy
possible electron configurations
energy
energy
54
Static Correlation
?4
?4
?2
?3
?2
?3
energy
energy
?1
?1

• the molecular orbitals with these two electron
  configurations are different

• putting electrons in ?2 will affect ?1 in a
  different way than putting electrons in ?3

• each configuration has its own Slater determinant

55
Multi-Reference Methods
a single Slater determinant represents 1
electronic state

• to describe systems with many accessible states
  we need to consider multiple determinants

• each Slater determinant is fully optimized

• molecular orbitals coefficients are variationally
  optimized to minimize the total energy

• linear expansion coefficients ai are optimized
  variationally to minimize the energy

Questions
1. how do we get the extra Slater determinants?
2. which extra Slater determinants do we include
in the expansion?
56
Multi-Reference Methods
generating additional Slater determinants

• additional Slater determinants are generate by
  excitations out of Hartree-Fock Slater
  determinant as in CI, coupled-cluster, etc.

• orbitals ?3 and ?4 are still present as virtual
  states

?4

• so, we can still excite electrons into these
  virtual orbitals to generate new electronic
  configurations

energy
?2
?3
?1
57
Multi-Reference Methods
which Slater determinants do you use

• we want to include all Slater determinants
  corresponding to relevant resonance states in
  the multi-determinant wavefunction

Complete Active Space Self-Consistent Field

• pick a number of orbitals that may be occupied in
  various resonance states

?4

• pick how many electrons can be distributed
  through those states

• take all possible combinations

energy
?2
?3
e.g., trimethylenemethane
?1
4 ? electrons
20 possible configurations
4 ? orbitals
58
Multi-Reference Methods
Complete Active Space Self-Consistent Field
(CASSCF) is the most common method for treating
static correlation

• involves generating Slater determinants by
  considering all possible excitations that can be
  generated by distributing m electrons in n
  molecular orbitals

• set of orbitals is called the active space

• standard notation is CASSCF calculation with
  (m,n) active space

number of active electrons
number of active orbitals

• large active spaces yield a large number of
  determinants

• selection of active space is black art, and
  probably not accessible to novices (and many
  experienced computational chemists)

CASSCF calculations are necessary when static
correlation is significant

• singlet diradicals

• reactions at some transition metal centers

• excited states/electron transfer

• describing bond dissociation

59
Multi-Reference Methods
Complete Active Space Self-Consistent Field
(CASSCF) is a generalization of Hatree-Fock
calculations

• one Slater determinant will be the Hartree-Fock
  wavefunction

• if static correlation is negligible, aHF ? 1, and
  it will be a Hartree-Fock calculation

CASSCF can be the basis for CI, MPn and
coupled-cluster calculations

• generate excited states using CASSCF
  wavefunction instead of Hartree-Fock wavefunction

• called MRCI, MRMP2, CASMP2, CASPT2, etc.

• very accurate, but very intensive computationally

60
Electron Correlation Summary

• electron correlation is the electron-electron
  energy missing in an Hartree-Fock calculation

there are two types of electron correlation
1. Dynamic correlation

• accounts for fact that electrons move such that
  they avoid other electrons

• captured by using a multi-determinant wavefunction

• each Slater determinant is built by exciting
  electrons within the Hartree-Fock determinant

• the molecular orbitals in each Slater determinant
  are not re-optimized

2. Static correlation

• in systems with multiple resonance states,
  electrons can avoid each other by occupying
  different resonance states

• captured by using multiple Slater determinants in
  the wavefunction

• each Slater determinant is made of a unique set
  of molecular orbitals and represents a different
  electronic configuration

61
Electron Correlation Summary
Configuration Interaction (CI)

• expands the wavefunction as a linear combination
  of Slater determinants

• each determinant is generated by exciting
  electrons from occupied orbitals into virtual
  molecular orbitals

• full CI includes all possible excitations and
  yields the exact wavefunction with a given basis
  set

• usually CI calculation only consider single and
  double excitations (CISD) due to computational
  effort

• truncated CI calculations are not size-consistent

Coupled-Cluster (CC)

• expands the wavefunction as a linear combination
  of Slater determinants

• each determinant is generated by exciting
  electrons from occupied orbitals into virtual
  molecular orbitals

• excitations are selected to preserve
  size-consistency

• currently CCSD(T) is gold standard for quantum
  chemical methods

• CCSD(T) includes all single, double and some
  triple excitations

62
Electron Correlation Summary
Møller-Plesset Perturbation Theory (MPn)

• treats electron correlation as a perturbation

• incorporates additional Slater determinants into
  the wavefunction through corrections to the
  ground state wavefunction

• MP2 calculations are the most common ab initio
  method for capturing electron correlation

• method is not variational ? no guarantee there is
  an upper bound on the energy

Complete Active Space Self-Consistent Field
(CASSCF)

• expands the wavefunction as a linear combination
  of Slater determinants

• each determinant is optimized to represent a
  different resonance structure

• states are generated by considering all possible
  combinations of m electrons in n orbitals

• standard way to capture static correlation

• can act as starting point for CI, CC and MPn
  calculations instead of Hartree-Fock

63
Electron Correlation Summary
in general

• post-Hartree-Fock methods are all based on an
  expansion of the Hartree-Fock Slater determinant
  wavefunction

• the starting point can be an unrestricted
  Hartree-Fock wavefunction, so all of these
  methods can treat open-shell systems

• post-Hartree-Fock methods are very compute
  intensive

• typically they can only be applied to systems
  ranging between 2 and 50 atoms

• some methods require expertise, and should not be
  treated as black-box approaches

well explore the relative accuracies and
computational costs of various quantum chemical
methods in a later lecture
64
Post-Hartree-Fock Methods in Gaussian
various post-Hartree-Fock methods are available
through the Calculation Setup Window in Gaussview
select the method from the drop-down menu
Many methods available

• a full list of keywords for post-Hartree-Fock
  methods is available on the Gaussian website

65
Post-Hartree-Fock Methods in Gaussian
input

• ccsd(t) indicates this is a coupled-cluster
  calculation with all singles and double
  excitations, and perturbative triple excitations

• other relevant methods include

• MP2, MP3, MP4, MP5

• CID, CISD

• CCD, CCSD

66
Post-Hartree-Fock Methods in Gaussian
Toward end of output

• to get the energy find the line with a label for
  your method in the output

• our look for a statement with the method in the
  summary at the end of the Gaussian output file

• exert caution here, since Gaussian also prints
  out data for methods that are part of the
  calculation, e.g. ccsd energies in a ccsd(t)
  calculation

At end of output
67
Density Functional Theory
density functional theory treats the electron
density quantum mechanically

• currently density functional theory (DFT)
  calculations are the most common type of quantum
  chemical calculation

• generally provides the best balance of accuracy
  and computational efficiency

Canadians have made significant contributions to
DFT
Walter Kohn (B.Sc., University of Toronto)

• 1998 Nobel prize for development of DFT

• considered father of DFT

Axel Becke (Dalhousie, formerly Queens)

• developed density functionals and other methods
  that dramatically improved the accuracy of DFT
  for chemical systems

Tom Ziegler (University of Calgary)

• developed various areas of DFT to make it a
  practical research tool

• one of the first researchers to apply DFT to
  real-world problems

68
Density vs. Wavefunction
Wavefunction

• contains all of the information about the system

• not an observable quantity

• function of 4N variables (3 spatial, 1 spin per
  electron)

Density

• function of only 3 spatial variables

• physically measurable quantity

• connected to the wavefunction

• for a one electron wavefunction

does the electron density contain the same
information that is contained in the wavefunction?
69
First Hohenberg-Kohn Theorem
the ground state energy of a system is uniquely
determined by the ground state electron density
consider two different Hamiltonians
the two Hamiltonians differ in the
electron-nuclear attraction term

• electron-nuclear attraction is called external
  potential in DFT lingo

• each Hamiltonian has a unique ground state
  wavefunction and ground state energy

70
First Hohenberg-Kohn Theorem
can two different wavefunctions give us the same
electron density??
based on the variational principle
we know
71
First Hohenberg-Kohn Theorem
can two different wavefunctions give us the same
electron density??
if we do the same for E0,b
in both cases we assumed that
72
First Hohenberg-Kohn Theorem
can two different wavefunctions give us the same
electron density??

0
clearly, the sum of two energies cannot be less
than the sum of the same two energies
therefore, our assumption that two different
wavefunctions with different energies can give
the same density is wrong
the ground state energy of a system is uniquely
determined by the ground state electron density
73
First Hohenberg-Kohn Theorem
the ground state energy of a system is uniquely
determined by the ground state electron density
the first Hohenberg-Kohn theorem is an existence
theorem

• it tells us that a unique relationship exists
  between the ground state energy and the ground
  state electron density

• it does not tell us what that relationship is

we do know that the energy is a functional of the
density

• a functional is a function whose argument is
  another function

• we wont worry too much about the mathematics of
  functionals (you can take the advanced density
  functional theory course in another year if you
  want the details)

• it can be shown that all other ground state
  properties of the system can be expressed as
  functionals of the electron density, too

74
Second Hohenberg-Kohn Theorem
the ground state density is a variational quantity
Proof

• we know there is unique correspondence between
  the density and the wavefunction

• so, if we pick an arbitrary trial density, we
  will get an arbitrary trial wavefunction

• the variational theory tells us that for any
  trial wavefunction

This means

• if we pick any arbitrary density it will give us
  an upper bound on the true ground state density

• if we compare two densities, the one with the
  lower energy is the better one

• we can employ linear variation techniques to
  optimize the density, like we did with
  wavefunction methods

75
Energy Functionals
First HK Theorem

• there exists an exact functional relationship
  between the ground state density and the ground
  state energy

Second HK Theorem

• the electron density can be optimized
  variationally

to use variational optimization procedures, we
must know how E and ? are related!!!

• unfortunately, the form of the exact energy
  functional remains unknown

• regardless, people have developed approximate
  energy functionals

1. Thomas-Fermi model
2. Kohn-Sham model
76
Thomas-Fermi Energy Functional
in 1927 Thomas and Fermi tried DFT (this was 37
years before Hohenberg and Kohns theorems)
they assumed the total energy is a sum of kinetic
and potential energies
they wrote the potential energy as
nuclear-electron Coulomb attraction
electron-electron Coulomb repulsion
77
Thomas-Fermi Energy Functional
in 1927 Thomas and Fermi tried DFT (this was 37
years before Hohenberg and Kohns theorems)
they assumed the total energy is a sum of kinetic
and potential energies
they wrote the kinetic energy as

• this is the kinetic energy of an ideal electron
  gas, a system in which electrons move against a
  background of uniformly distributed positive
  charge

• the ideal electron gas model doesnt sound much
  like a molecule

78
Thomas-Fermi Energy Functional
in 1927 Thomas and Fermi tried DFT (this was 37
years before Hohenberg and Kohns theorems)
they assumed the total energy is
we can use this to variationally optimize the
density

• and we will find out that the model fails
  miserably

• for example, the TF energy functional predicts
  that molecules are unstable with respect to the
  atoms that form them

79
Thomas-Fermi Energy Functional
whats wrong with the Thomas-Fermi functional?
1. kinetic energy

• based on an ideal electron gas model that doesnt
  even resemble a molecule

• but how do you express the kinetic energy in
  terms of ??

2. electron-electron repulsion

• electron repulsion is treated as the average
  Coulomb interaction

• fails to account for quantum mechanical effects
  like exchange

• fails to account for instantaneous
  electron-electron correlation

80
Kohn-Sham Energy Functional
Kohn and Sham suggested decomposing the total
energy into kinetic and potential energy
contributions

• accounts for the kinetic energy of the electrons

• accounts for nuclear-electron Coulombic attraction

• accounts for average electron-electron Coulombic
  repulsion

• accounts for all electron-electron interactions
  not in

• includes exchange and instantaneous
  electron-electron correlation

81
Kohn-Sham Energy Functional
Kohn and Sham recognized that its easier to
calculate the kinetic energy if we have a
wavefunction
but we dont know the ground state wavefunction
(if we did, we wouldnt bother with DFT)
So, they suggested

• instead of using the real wavefunction and
  density, lets use a Slater determinant
  wavefunction built up from a set of one-electron
  orbitals (like molecular orbitals) to build up an
  artificial electron system that represents the
  ground state density

• artificial system is often called the
  non-interacting reference system

• the one-electron orbitals are called Kohn-Sham
  orbitals

• the Kohn-Sham orbitals are orthonormal

82
Kohn-Sham Energy Functional
the Kohn-Sham orbitals give us an easy way to
construct the density and get an approximate
value of the kinetic energy
kinetic energy of N one-electron orbitals
density from N one-electron orbitals
the density integrates to give the total number
of electrons
83
Kohn-Sham Energy Functional
a Slater determinant wavefunction built up from
Kohn-Sham orbitals will never give the exact
ground state kinetic energy
(weve seen that the closest approximation to the
real wavefunction you can get with a single
Slater determinant is the Hartree-Fock
wavefunction)
so, Kohn and Sham suggested splitting up the
kinetic energy
correction accounting for kinetic energy not
contained in Trs
exact ground state kinetic energy
kinetic energy of the reference system
84
Kohn-Sham Energy Functional
incorporating the orbital-based expressions into
the total energy functional
gives
this is the Kohn-Sham total energy functional
85
Kohn-Sham Energy Functional
with Kohn-Sham orbitals, we can calculate
we dont have an expression for Exc

• Exc is called the exchange correlation functional

• it acts as a repository for all contributions to
  the energy that we do not know how to calculate
  exactly

• if we had an exact expression for Exc we could
  calculate the energy and density exactly

86
The Exchange-Correlation Functional
the exchange-correlation functional includes all
contributions to the energy that are not included
in the other terms in the Kohn-Sham energy
functional
1. exchange energy

• the classical electron-electron energy term does
  not account for Pauli repulsion between electrons
  of the same spin

2. correlation energy

• the classical electron-electron energy term does
  not account for instantaneous Coulomb
  interactions between electrons

3. self-interaction energy

• the classical electron-electron energy term
  includes a spurious contribution from each
  electron interacting with itself

4. kinetic energy correction

• the kinetic energy of the reference system is not
  equal to the ground state kinetic energy of the
  real system

87
(No Transcript)