CHEM 834: Computational Chemistry

1
CHEM 834 Computational Chemistry
Quantum Chemical Methods 1
March 16, 2009
2
Topics
last time

• quantum chemical methods

• review of quantum mechanics

• variational principle

• molecular orbitals

• Hartree product trial wavefunction

today

• Hartree-Fock calculations

• Slater determinant trial wavefunction

• Hartree-Fock energy

• Fock operator

• self-consistent solution of the Fock operator

• availability in Gaussian

3
Variational Principle
energy of trial function, ?
true ground state energy of the system
normalization factor
for any suitable trial wavefunction

• the energy of the trial function is guaranteed to
  be higher than the real ground state energy

• if we happen upon the real wavefunction we get E0

• otherwise, we get an upper bound on E0

• basic principle

trial wavefunction with lower energy is a
better trial wavefunction
4
Hartree Product
lets test ?HP against the criteria for a valid
wavefunction
1. Indistinguishabilty

• distinguishes between electrons

• invalid form of wavefunction

2. Antisymmetry

• not antisymmetric

• invalid form of wavefunction

5
Slater Determinant Wavefunction
deficiencies of ?HP can be overcome by taking
linear combinations of product wavefunctions
1. Indistinguishabilty

• does not distinguish between electrons

• electron 1 can be found in either orbital

• electron 2 can be found in either orbital

• valid form of wavefunction

2. Antisymmetry

• antisymmetric

• valid form of wavefunction

6
Slater Determinant Wavefunction
?12 is a Slater determinant
normalization constant
determinant
Determinants

• square array of elements

• lines on left and right (dont confuse with
  matrices that use brackets)

• functions as a shorthand for writing the sum of
  products

7
Slater Determinant Wavefunction
Properties of Determinants (D)
1. if all elements in a column or row are 0, D 0
2. multiplying a row or column by k multiplies D
by k
3. switching two columns or two rows changes the
sign of D
4. if two columns or two rows are identical, D 0
5. if two columns or two rows are multiples of
each other, D 0
6. multiplying a column (or row) by k and adding
it to another column (or row) leaves D unchanged
8
Slater Determinant Wavefunction
virtually all quantum chemical methods use Slater
determinants
For an N electron system

• columns labeled by molecular orbitals

• rows labeled by electrons

• shorthand notation

Based on properties of determinants

• switching two rows changes sign of ?SD ?
  antisymmetry

• if two columns (molecular orbitals) are identical
  ?SD 0 ? two electrons cant occupy same orbital
  (Pauli exclusion principle)

9
Slater Determinant Wavefunction
Hartree product was an independent electron
wavefunction
what about Slater determinants?
consider two electron system with electrons of
opposite spin
square to get probability
define P(r1,r2)dr1dr2 probability of finding
electron 1 in dr1 and electron 2 in dr2

• probability is just products ? electrons of
  opposite spin are independent with Slater
  determinant

• in fact, P(r1,r1) ? 0 ? with Slater determinant
  two electrons of opposite spin can occupy the
  same point in space

10
Slater Determinant Wavefunction
Hartree product was an independent electron
wavefunction
what about Slater determinants?
consider two electron system with electrons of
same spin
using analysis on preceding slide

• probability is not just products ? electrons of
  same spin are correlated with Slater determinant

• called exchange correlation because last two
  terms exchange electron coordinates between
  different spatial orbitals

• consequence of antisymmetrizing the wavefunction,
  and related to Pauli exclusion principle

• P(r1,r1) 0 ? with Slater determinant two
  electrons of same spin cannot occupy the same
  point in space

11
Slater Determinant Wavefunction
For an N electron system

• columns labeled by molecular orbitals

• rows labeled by electrons

• shorthand notation

Physics captured

• electrons of opposite spin are treated
  independently

? neglects instantaneous Coulomb interactions

• electrons of opposite spin are correlated through
  exchange interactions

? accounts for Pauli exclusion principle
? significant improvement over Hartree product
12
Hartree-Fock Calculations
Hartree-Fock calculations are the basis for
virtually all quantum chemical methods

• ab initio methods are built upon Hartree-Fock
  calculations

• in practice, density functional theory
  calculations look very much like Hartree-Fock
  calculations

• semi-empirical molecular orbital methods involve
  approximations with the framework of Hartree-Fock
  theory

Other common names for Hartree-Fock calculations

• self-consistent field (SCF) calculations

• these names are technically incorrect, but you
  may see them in the literature

• molecular orbital (MO) calculations

13
Hartree-Fock Calculations
The basic idea of Hartree-Fock calculations
1. the full Hamiltonian within the
Born-Oppenheimer approximation
2. a trial wavefunction consisting of one Slater
determinant
3. molecular orbitals expressed as linear
combinations of basis functions
basis function with a fixed form
artificial spin function
coefficient in linear expansion (called molecular
orbital coefficients)
14
Hartree-Fock Calculations
The basic idea of Hartree-Fock calculations
1. the full Hamiltonian within the
Born-Oppenheimer approximation
2. a trial wavefunction consisting of one Slater
determinant
3. molecular orbitals expressed as linear
combinations of basis functions
4. variational optimization of ? using the
molecular orbital coefficients as variational
parameters
15
Linear Combination of Basis Functions
Hartree-Fock molecular orbitals are expressed as
linear combinations of basis functions
Basic motivation

• linear combination of atomic orbitals

• consider H2

• molecular orbitals are expressed as linear
  combinations of atomic orbitals

• K atomic orbitals yields K molecular orbitals

H
H

• N molecular orbitals with lowest energy are
  occupied and all others are called unoccupied

1sL
1sR

• applies to all molecules

16
Linear Combination of Basis Functions
Hartree-Fock molecular orbitals are expressed as
linear combinations of basis functions
basis functions other than atomic orbitals

• linear combination of atomic orbitals

• basis functions dont have to be atomic orbitals

• we can choose any suitable well-defined
  mathematical function

• well discuss basis functions more in a later
  lecture

atomic orbital basis set
alternative basis set

• chemically intuitive

• still mathematically valid

C
H
17
Hartree-Fock Energy
to use the variational method, we need a
relationship between the energy and the MO
coefficients
for any normalized wavefunction
in Hartree-Fock theory
and if you work it out, the energy becomes
18
Hartree-Fock Energy
to use the variational method, we need a
relationship between the energy and the MO
coefficients
recall that molecular orbitals are linear
combinations of basis functions
so, now we have a direct relationship between the
Hartree-Fock energy and molecular orbital
coefficients

• later, well look at how to variationally
  optimize the MO coefficients

• now, lets look at the energy expression itself

19
Hartree-Fock Energy
in terms of the MOs, the Hartree-Fock energy is
or in standard short-hand notation
exchange integral
Coulomb integral
one electron energy
two electron energy
20
One Electron Energy
same molecular orbital
details

• involves only 1 molecular orbital ? one-electron
  energy

21
One Electron Energy
kinetic energy of electron in ?a
Coulombic attraction between M nuclei and
electron in ?a
details

• involves only 1 molecular orbital ? one-electron
  energy

• electronic kinetic energy

• nuclear-electron Coulombic attraction

• must sum over all electrons in the system

22
Coulomb Integral
two different molecular orbitals
details

• involves 2 molecular orbitals

• called a two-electron integral

• have to sum over all pairs of electrons to get
  total energy

23
Coulomb Integral
probability of finding electron a in dx1
details

• involves 2 molecular orbitals

• called a two-electron integral

• have to sum over all pairs of electrons to get
  total energy

• represents Coulomb interaction between two charge
  clouds representing the electrons in molecular
  orbitals a and b

• probability of finding electron a in dx1 is
  independent of finding electron b in dx2

• accounts for electrostatic energy resulting from
  electron a moving in average field of electron b

• Jab is non-zero even when x1 x2

• electrons can exist at same point in space

• violates Pauli exclusion principle for electrons
  of same spin

24
Exchange Integral
two different molecular orbitals
details

• involves 2 molecular orbitals

• called a two-electron integral

• have to sum over all pairs of electrons to get
  total energy

• exchanges coordinates of electrons in molecular
  orbitals a and b

• does not have a simple classical analogue like
  Coulomb integral

• accounts for exchange energy (Pauli repulsion
  between electrons)

• consequence of using antisymmetric wavefunction

• is only non-zero for electrons of the same spin

• Pauli repulsion doesnt affect electrons of
  opposite spins

• cancels out Coulomb energy arising from two
  electrons with the same spin being at the same
  point in space

25
Two Electron Energy
sum over all pairs of electrons

• prefactor of ½ prevents double-counting of
  interactions

Jab has a positive sign

• Coulomb interaction between electrons is repulsive

Kab has a negative sign

• exchange interactions make electrons of the same
  spin avoid each other

• Jab does not account for this correlation,
  overestimates repulsive energy

• -Kab removes this overestimation by reducing the
  energy

Jaa Kaa

• Jaa is the interaction of an electron with itself

• Kaa cancels out exactly this spurious interaction

26
Hartree-Fock Energy
Hartree-Fock energy captures

• kinetic energy of electrons

• nuclear-electron attraction

• electron-electron Coulomb repulsion

• this interaction is treated in an average sense

• instantaneous electron-electron Coulomb
  interactions are not considered

• exchange interactions

• accounts for Pauli repulsion/exclusion principle

• electrons of the same spin avoid each other ?
  decrease in Coulomb repulsion

All of these quantities are approximate because
the wavefunction has an approximate form!!!!
27
Variational Minimization of the Hartree-Fock
Energy
we want to minimize E by altering the molecular
orbital coefficients subject to the constraint
that the MOs are orthogonal
if you do this (we wont), you find out that the
best set of molecular orbitals are given by
we have to solve this eigenvalue problem to get
the molecular orbital, ?a, and its energy, ?a
28
Conversion to Eigenvalue Problem
recall that eigenvalue problems are of the form
operator expression for a particular property
eigenvalue value of property when system is in
state i
eigenfunction
and the expectation value of B is
29
Conversion to Eigenvalue Problem
we are going to convert
into
energy operator
orbital energy
molecular orbital
fa will be decomposed into a sum of operators for
different kinds of interactions
30
Core Operator
Expectation value
kinetic energy operator
Coulomb potential due to M nuclei

• kinetic energy of electron in ?a

• Coulomb attraction between M nuclei and the
  electron in ?a

31
Coulomb Operator

• charge cloud due to electron in ?b

expectation value of Jb(x1)

• accounts for average Coulomb repulsion between
  electrons in molecular orbitals a and b

32
Coulomb Operator
since the electron in ?a interacts with N-1 other
electrons
charge cloud of electron in molecular orbital a
Coulomb potential arising from other N-1 electrons
33
Exchange Operator

• exchanges the coordinates of the electrons in
  molecular orbitals a and b

• can only be defined in terms its effect on a
  molecular orbital

• accounts for Pauli repulsion between electrons in
  molecular orbitals a and b

• does not have a classical analogue

• non-local operator ? result of operating on ?a at
  x1 depends on value of ?a at x2

34
Exchange Operator
expectation value of Kb(x1)
exchange energy of electron in ?a
35
Fock Operator
to get molecular orbital ?a, we must solve
Fock operator
36
Fock Operator

• one-electron operator ? only operates on electron
  in molecular orbital a

• acts as an effective Hamiltonian for the electron
  in molecular orbital a

Hartree-Fock molecular orbitals are
eigenfunctions of the Fock operator
Fock operator
energy of molecular orbital a
molecular orbital a
we have to solve this eigenvalue problem for each
molecular orbital to generate the best Slater
determinant wavefunction
37
Hartree-Fock Energy vs. Orbital Energies
the total Hartree-Fock energy is not the sum of
the orbital energies!!!
based on the expectation values of the core,
Coulomb and exchange operators
so, the sum of the orbital energies is
noting that
yields
38
Hartree-Fock Energy vs. Orbital Energies
the total Hartree-Fock energy is not the sum of
the orbital energies!!!
factor of ½ prevents double counting of
electron-electron interactions
fa?a includes interaction of electron a with b
fb?b includes interaction of electron b with a
Take home message
You have to solve for the individual molecular
orbitals with the Fock operator, then use the
orbitals to construct the Slater determinant
wavefunction, and then evaluate properties like
the total energy.
39
Solving the Hartree-Fock Equations
Specific case restricted closed-shell
Hartree-Fock

• assume all electrons are paired

• two electrons per spatial orbital

• restricted Slater determinant

• only have to solve for N/2 spatial orbitals

Closed-shell Fock operator
40
Closed-Shell Hartree Fock
General Fock operator
Closed-shell Fock operator

• one electron terms same because they are
  independent of N

• prefactor of 2 before J term

• Coulomb terms only depend on spatial electron
  distribution

• sum over N/2 orbitals ? we only capture half of
  the relevant interactions

• other N/2 interactions are identical ? we can
  multiply by 2

• prefactor of 1 before K term

• exchange is only non-zero for electrons of same
  spin

• only N/2 interactions are relevant for each
  orbital

41
Introduction of a Basis
we want to solve
using a linear combination of basis functions
we can solve this more easily by converting it
into a matrix problem
multiply on the left by ?? and integrate
F?? is an element of the Fock matrix
S?? is an element of the overlap matrix
Note there will be K of these equations because
there are K basis functions
42
Conversion to a Matrix Problem
Overlap matrix, S

• K x K Hermitian matrix

• S?? represent spatial overlap between basis
  functions ? and ?

• diagonal elements 1 (a basis function overlaps
  perfectly with itself)

• off-diagonal elements depend on sign of basis
  functions and relative positions in space

43
Conversion to a Matrix Problem
Fock matrix, F

• K x K Hermitian matrix

• matrix representation of Fock operator for basis
  set ?i

44
Constructing the Fock Matrix
using closed-shell Fock operator
one-electron contributions

• effort scales with K2 ? double K, increase effort
  4x

45
Constructing the Fock Matrix
two electron contributions

• effort scales with K4 ? double K, increase effort
  16x

• (????) two electron integral in terms of basis
  functions

• analogous expression for the exchange term (?aa?)

46
Constructing the Fock Matrix
sum only over occupied orbitals
Density matrix

• for a known set of ?, P?? specifies the
  distribution of electron density

• also useful for getting atomic charges, bond
  orders, etc.

47
Conversion to a Matrix Problem
with F and S defined, we can transform
into a matrix equation
FC SC?
C is a K x K matrix whose columns define the
coefficients, c?i
? is a diagonal matrix of the orbital energies,
?i
48
Solving the Fock Equations
FC SC?

• we know how to build matrices F and S

• lets solve to get C and ?

• standard way to solve eigenvalue problems is
  through matrix diagonalization (analogous to
  solving secular determinant/equations)

diagonalization

• square matrix, A, can be broken down into the
  product of three square matrices

• P square matrix, columns are eigenvectors

• D diagonal matrix, diagonal elements are
  eigenvalues

we saw diagonalization when calculating normal
modes
49
Solving the Fock Equations
FC SC?

• we know how to build matrices F and S

• lets solve to get C and ?

• standard way to solve eigenvalue problems is
  through matrix diagonalization (analogous to
  solving secular determinant/equations)

Two problems
1. FC SC? is not in an diagonalizable form

• we need something like AP PD

2. we are trying to solve for C, but F is a
function of C
F(C)C SC?

• this is a non-linear problem

• we cant solve this exactly

50
Solutions to our Problems
Problem 1 FC SC? is not in an diagonalizable
form
Solution

• define a transformation matrix X S-1/2

• called orthogonalizing the basis (details not
  really important here)

• gives a standard eigenvalue problem that we can
  solve through matrix diagonalization

• the eigenvalues are unaffected by the
  transformation

• we know how to transform from C to C to get the
  molecular orbital coefficients

51
Solutions to our Problems
Problem 2 F is dependent on C (same for F and
C)
Solution

• solve with iterative techniques

• make an initial guess of C1

• build F1 and solve for C2

• use C2 to build F2 and solve for C3

. . .

• use CN to build FN and solve for CN1

• stop when CN and CN1 are the same (or very
  similar)

• called a self-consistent field (SCF) approach

• final matrix C defines the wavefunction

52
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 ?
53
Open-Shell Systems
what about systems with unpaired electrons?
Restricted Open-Shell Hartree-Fock

• some electrons are unpaired

• number of unpaired electrons is determined by the
  multiplicity

• remaining electrons are paired in molecular
  orbitals as in restricted Hartree-Fock

Unrestricted Hartree-Fock

• ? and ? electrons are allowed to occupy different
  molecular orbitals

• no pairing of electrons in molecular orbitals

• number of ? and ? molecular orbitals determined
  by multiplicity

54
Restricted Open-Shell Hartree-Fock
force some electrons to be paired, some left
unpaired
Details

• orbitals optimized in same manner as restricted
  HF method

unoccupied orbitals

• density matrix is modified

energy
unpaired electrons

• wavefunction is eigenfunction of S2 operator

electrons of opposite spin paired in the same
spatial orbital

• does not account for spin polarization

• alpha electrons in paired orbitals should be
  repelled by unpaired alpha electrons more than
  beta electrons in paired orbitals due to Pauli
  repulsion

ROHF calculations are not often used because lack
of spin polarization leads to erroneous results
55
Unrestricted Hartree-Fock
? and ? electrons are treated separately
Details

• double the number of molecular orbitals as in
  restricted Hartree-Fock (including unoccupied MOs)

• wavefunction is still treated with one Slater
  determinant

• have to construct separate density matrices

energy

• have to build separate Fock matrices (analogous
  for ? Fock matrix)

• accounts for Coulomb repulsion between electrons
  of same and opposite spin

• exchange interactions only affect electrons of
  the same spin

56
Unrestricted Hartree-Fock
? and ? electrons are treated separately
Details

• unrestricted Hartree-Fock wavefunctions are not
  eigenfunctions of S2

• suffer from spin contamination

• mixing in of higher multiplicity states into the
  wavefunction

• unrestricted Hartree-Fock calculations do account
  for spin polarization

energy

• allowing ? and ? electrons to occupy different
  spatial orbitals lets electrons of like spin
  avoid each other

UHF calculations are most commonly used to treat
open-shell systems at the Hartree-Fock level
57
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
58
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
59
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
60
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

61
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

62
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

63
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

64
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

65
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
66
RHF Calculation of Twisted Ethene - Input

• hf specifies restricted Hartree-Fock calculation

• 3-21g specifies the basis set

67
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

68
ROHF Calculation of Twisted Ethene - Input

• rohf specifies restricted open-shell Hartree-Fock
  calculation

• 3-21g specifies the basis set

69
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

70
UHF Calculation of Twisted Ethene, No Mixing -
Input

• uhf specifies unrestricted Hartree-Fock
  calculation

• 3-21g specifies the basis set

71
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

72
UHF Calculation of Twisted Ethene, Mixing - Input

• uhf specifies unrestricted Hartree-Fock
  calculation

• guessmix specifies HOMO-LUMO mixing

• 3-21g specifies the basis set

73
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

74
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