Post Hartree-Fock Methods
(Lecture 2)
NSF Computational Nanotechnology and Molecular Engineering
Pan-American Advanced Studies Institutes (PASI) Workshop
January 5-16, 2004
California Institute of Technology, Pasadena, CA
Andrew S. Ichimura
Outline
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•
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Shortcomings of the SCF-RHF procedure
Configuration Interaction
MCSCF
Size-consistency and size-extensivity
Perturbation theory
Coupled Cluster Methods
What is electron correlation and why do we need it?
0 is a single determinantal wavefunction.
 SD 
1 1
1 2
 2 (1)
 2 (2)
 N (1)
 N (2)
1 N   2 (N)
 N (N)
,
 i |  j  ij
Slater Determinant
Recall that the SCF procedure accounts for electron-electron repulsion by
optimizing the one-electron MOs in the presence of an average field of the
other electrons. The result is that electrons in the same spatial MO are too
close together; their motion is actually correlated (as one moves, the other
responds).
Eel.cor. = Eexact - EHF
(B.O. approx; non-relativistic H)
RHF dissociation problem
Consider H2 in a minimal basis composed of one atomic 1s orbital on
each atom. Two AOs (c) leads to two MOs ()…
H
H
2  N2 ( cA  c B);
antibonding MO
H 1s
H 1s
1  N1( cA  cB );
H
H
bonding MO
The ground state wavefunction is:
1(1) 1 (1)
0 
1(2) 1 (2)
 0  1 (1)1(2)  1(2)1 (1)
Slater determinant with two electrons
in the bonding MO
Expand the Slater Determinant
 0  1 (1)1 (2) (1)(2)  (1)(2)
Factor the spatial and spin parts
 0  1 (1)1 (2)  ( cA (1)  c B (1))( c A (2)  c B (2)) H does not depend on spin
Four terms in
 0  cA (1) cA (2)  c B (1) cB (2)  c A (1) cB (2)  c B (1) cA (2)
the AO basis
cA c A
cB c B
cA c B
cB c A
Ionic terms, two electrons in one Atomic Orbital
Covalent terms, two electrons shared between two AOs
H2 Potential Energy Surface
.
E
. At the dissociation
H + H
0
H
H
Bond stretching
limit, H2 must separate
into two neutral atoms.
H H
At the RHF level, the wavefunction, , is 50% ionic and 50% covalent at all
cA c B
cA c A
bond lengths.
cB c B
cB c A
H2 does not dissociate correctly at the RHF level!!
Should be 100% covalent at large internuclear separations.
RHF dissociation problem has several consequences:
•
Energies for stretched bonds are too large. Affects transition state structures Ea are overestimated.
•
Equilibrium bond lengths are too short at the RHF level. (Potential well is too
steep.) HF method ‘overbinds’ the molecule.
•
Curvature of the PES near equilibrium is too great, vibrational frequencies are
too high.
•
The wavefunction contains too much ‘ionic’ character; dipole moments (and
also atomic charges) at the RHF level are too large.
On the bright side, SCF procedures recover ~99% of the total electronic energy.
But, even for small molecules such as H2, the remaining fraction of the energy - the
correlation energy - is ~110 kJ/mol, on the order of a chemical bond.
To overcome the RHF dissociation problem,
Use a trial function that is a combination of 0 and 1
First, write a new wavefunction using the anti-bonding MO.
2  N2 ( cA  c B);
antibonding MO
The form is similar to 0, but describes an excited state:
 2 (1) 2  (1)
1 
  2 (1)2  (2)  2 (2) 2 (1)
 2 (2) 2  (2)
1  2 (1) 2 (2) (1) (2)   (1) (2)
MO basis
1  2 (1)2(2)  ( cA (1)  c B (1))( c A (2)  cB (2))
1  cA (1) c A (2)  cB (1) c B (2)  c A (1) cB (2)  cB (1) cA (2)
Ionic terms
Covalent terms
AO basis
Trial function - Linear combination of 0 and 1;
two electron configurations.
  a0  0  a11  a0 (11)  a1 (2 2 )
  (a0  a1 )cA c A  c B cB  (a0  a1 )cA c B  c B cA 
Ionic terms
Covalent terms
Three points:
1. As the bond is displaced from equilibrium, the coefficients (a0, a1) vary
until at large separations, a1 = -a0: Ionic terms disappear and the molecule
dissociates correctly into two neutral atoms.  = CI, an example of
configuration interaction.
2.
The inclusion of anti-bonding character in the wavefunction allows the
electrons to be farther apart on average. Electronic motion is correlated.
3.
The electronic energy will be lower (two variational parameters).
Configuration Interaction - Excited Slater Determinants
Since the HF method yields the best single determinant wavefunction
and provides about 99% of the total electronic energy, it is commonly
used as the reference on which subsequent improvements are based.
As a starting point, consider as a trial function a linear combination of Slater
determinants:
  a0 HF   ai i
Multi-determinant wavefunction
i 1
a0 is usually close to 1 (~0.9).
• M basis functions yield M molecular orbitals.
• For N electrons, N/2 orbitals are occupied in the RHF wavefunction.
• M-N/2 are unoccupied or virtual (anti-bonding) orbitals.
Generate excited Slater determinants by promoting up to N
electrons from the N/2 occupied to M-N/2 virtuals:
b
a
9
a,b,c… =
virtual MOs
a
8
a,b
b
a
b
a
c
c,d
k
i
k,l
i
j
j
abc
ijk
abcd
ijkl
7
6
5
i
i
4
i,j
3
i,j,k… =
occupied MOs 2
j
1
HF
Excitation level
Ref.
ia
Single
ijab
Double
ijab
Triple
Quadruple
…
Represent the space containing all N-fold excitations by (N).
Then the COMPLETE CI wavefunction has the form
CI  C0 HF   (1)  (2)   (3)  ...  (N )
Where
 HF  Hartree  Fock
occ virt
    Cia ia
i a
(1)

occ virt
(2)
   Cijab ijab
i, j

a,b
occ virt
(3)
Linear combination of Slater
determinants with single excitations
Doubly excitations

C
ijabc
k
abc
ij k
Triples
i, j ,k a,b,c

(N )

occ
virt
i, j ,k...
a,b,c...
 
abc...
N-fold excitation
Cijabc...

k...
ij k...
The complete CI expanded in an infinite basis yields the exact solution
to the Schrödinger eqn. (Non-relativistic, Born-Oppenheimer approx.)
abc...
The various coefficients, Cijk... , may be obtained in a variety of ways.
A straightforward method is to use the Variation Principle.
ECI 
ECI



CI | H | CI
CI | CI
C
abc...
ijk...
0
HCK  E K CK
Expectation value of He.
Energy is minimized
wrt coeff
In a fashion analogous to the HF eqns,
the CI Schrodinger equation can be
formulated as a matrix eigenvalue
problem.
abc...
The elements of the vector, CK , are the coefficients, Cijk...

And the eigenvalue, EK, approximates the energy of the Kth state.
E1 = ECI for the lowest state of a given symmetry and spin.
 and spin, and so on.
E2 = 1st excited 
state of the same symmetry
Some nomenclature…
One-electron basis (one-particle basis) refers to the basis set. This limits
the description of the one-electron functions, the Molecular Orbitals.
The size of the many-electron basis (N-particle basis) refers to the number
of Slater determinants. This limits the description of electron correlation.
In practice,
• Complete CI (Full CI) is rarely done even for finite basis sets - too expensive.
Computation scales factorialy with the number of basis functions (M!).
• Full CI within a given one-particle basis is the ‘benchmark’ for that basis
since 100% of the correlation energy is recovered. Used to calibrate
approximate correlation methods.
• CI expansion is truncated at a some excitation level, usually Singles and
Doubles (CISD).
(1)
(2)
CI  C0 HF    
Configuration State Functions
Consider a single excitation from the RHF
reference.
Both RHF and (1) have Sz=0,
but (1) is not an eigenfunction
of S2.
RHF
(1)
Linear combination of singly excited
determinants is an eigenfunction of S2.
Configuration State Function, CSF
(Spin Adapted Configuration, SAC)
Singlet CSF
Only CSFs that have the same multiplicity
as the HF reference contribute to the
correlation energy.
1,2  1(1) 2(2)  1 (2) 2 (1)
Example H2O:
(19 basis functions)
CISD
(~80-90%)
Full CI
Example: Neon Atom
Relative
importance
Ref.
Singles
Doubles
Triples
Quadruples
2
1
4
3
abc...
Weight =
abc... 2
(C
 ij k... )
ij k...
for a given excitation level.
(Frozen core approx., 5s4p3d basis - 32 functions)
1.
2.
CISD (singles and doubles) is the only generally applicable method. For modest
sized molecules and basis sets, ~80-90% of the correlation energy is recovered.
CISD recovers less and less correlation energy as the size of the molecule increases.
Size Consistent and Size Extensive
Size consistent method - the energy of two molecules (or fragments)
computed at large separation (100 Å) is equal to the twice energy of the
individual molecule (fragment). Only defined if the molecules are noninteracting.
Ex. (ECISD of two H2 separated by 100Å) < 2(ECISD of one H2)
Size extensive method - the energy scales properly with the number of
particles. (Same fraction of correlation energy is recovered for CH4, C2H6,
C3H8, etc.)
1.
Full CI is size consistent and extensive.
2.
All forms of truncated CI are not. (Some forms of CI,
esp. MR-CI are approximately size consistent and size
extensive with a large enough reference space.)
Multi-configuration Self-consistent Field (MCSCF)
9
Carry out Full CI and orbital optimization within a
small active space. Six-electron in six-orbital MCSCF
is shown. Written as [6,6]CASSCF.
8
7
Complete Active Space Self-consistent Field (CASSCF)
6
H2O MOs
5
4
3
2
1
HF
Why?
1. To have a better description of the ground or
excited state. Some molecules are not welldescribed by a single Slater determinant, e.g. O3.
2. To describe bond breaking/formation; Transition
States.
3. Open-shell system, especially low-spin.
4. Low lying energy level(s); mixing with the
ground state produces a better description of the
electronic state.
5. …
MCSCF Features:
1.
In general, the goal is to provide a better description of the main features of
the electronic structure before attempting to recover most of the correlation
energy.
2.
Some correlation energy (static correlation energy) is recovered. (So called
dynamic correlation energy is obtained through CI and other methods
through a large N-particle basis.)
3.
The choice of active space - occupied and virtual orbitals - is not always
obvious. (Chemical intuition and experience help.) Convergence may be
poor.
4.
CASSCF wavefunctions serve as excellent reference state(s) to recover a
larger fraction of the dynamical correlation energy. A CISD calculation
from a [n,m]-CASSCF reference is termed Multi-Reference CISD (MRCISD). With a suitable active space, MRCISD approaches Full CI in
accuracy for a given basis even though it is not size-extensive or consistent.
Examples of compounds that require MCSCF
for a qualitatively correct description.
H
H
C
C
O+
H
H
O
O
O-
O
zwitterionic
Singlet state of twisted
ethene, biradical.
biradical
H
C
N
H
C
N
Transition State
O
H
C
N
Mœller-Plesset Perturbation Theory
In perturbation theory, the solution to one problem is expressed in terms of
another one solved previously. The perturbation should be small in some
sense relative to the known problem.
Hˆ  Hˆ 0  Hˆ '
Hˆ 0  i  E i i, i = 0,1,2,...,
Hamiltonian with pert., 
Unperturbed Hamiltonian
Hˆ   W
W  0W0  1W1  2W2  ...
  00  11  22  ...
As the perturbation is turned on, W (the
energy) and  change. Use a Taylor series
expansion in .
ˆ and Hˆ '
Define H
0
N


ˆ
ˆ
ˆ
ˆ
ˆ
H0   Fi   hi   J ij  K ij 

i 1
i1 
j 1
N
N

N
N
N

N
Hˆ '   gij    gij
i 1 j 1
i 1 j 1
W0  sum over MO energies
W1 =  0| | Hˆ '|  0  E(HF )
occ vir
W2   
ab
ab
 0| | Hˆ '|  ij  ij | Hˆ '|  0
E 0  E ijab
i  j a b


E(MP2)   
occ vir
i j a b
i
j
Unperturbed H is the sum over Fock
operators  Moller-Plesset (MP) pert th.
Perturbation is a two-electron
operator when H0 is the Fock
operator.
With the choice of H0, the first
contribution to the correlation
energy comes from double
excitations.
| a  b   i j |  b a
i   j  a  b

2
Explicit formula for 2nd
order Moller-Plesset
perturbation theory, MP2.
Advantages of MP’n’ Pert. Th.
• MP2 computations on moderate sized systems (~150 basis functions)
require the same effort as HF. Scales as M5, but in practice much less.
• Size-extensive (but not variational). Size-extensivity is important; there is
no error bound for energy differences. In other words, the error remains
relatively constant for different systems.
• Recovers ~80-90% of the correlation energy.
• Can be extended to 4th order: MP4(SDQ) and MP4(SDTQ). MP4(SDTQ)
recovers ~95-98% of the correlation energy, but scales as M7.
• Because the computational effort is significanly less than CISD and the
size-extensivity, MP2 is a good method for including electron correlation.
Coupled Cluster Theory
Perturbation methods add all types of corrections, e.g., S,D,T,Q,..to a given
order (2nd, 3rd, 4th,…).
Coupled cluster (CC) methods include all corrections of a given type to infinite
order.
The CC wavefunction takes on a different form:
Tˆ
CC  e  0
Coupled Cluster Wavefunction
0 is the HF solution
1 ˆ 2 1 ˆ3
ˆ
ˆ
e 1T + T  T 
2
6
Tˆ
ˆ T
ˆ  Tˆ  Tˆ 
T
1
2
3

1ˆk
 T
k 0 k!
 Tˆ N
Exponential operator generates
excited Slater determinants
Cluster Operator
N is the number of electrons
CC Theory cont.
occ vir
a
a
ˆ 
T
t



1 0
i
i
i
a
occ vir
ˆ  
T
  tijab abij
2
0
The T-operator acting on the HF reference
generates all ith excited Slater Determinants, e.g.
doubles ijab.
i j a b
tia
tijab

Expansion coefficients are called amplitudes; equivalent to
the ai’s in the general multi-determinant wavefunction.
HF ref.
ˆ
1 ˆ 2  ˆ
ˆ
ˆ T
ˆ  1 Tˆ 3   T
ˆ  Tˆ T
ˆ  1 Tˆ 2  1 Tˆ T
ˆ2 1 T
ˆ 4  
e T  1ˆ  Tˆ1  T

T

T

T
2
2 1
3 1
2 1   3
6 1   4
2 2 2 2 1 24 1 
singles
doubles
triples
Quadruple excitations
The way that Slater determinants are generated is rather different…
CC Theory cont.
1ˆ
ˆ
T
HF reference
Singly excited states
1
1 ˆ 2 
ˆ
T

 2 2 T1 
ˆ
ˆ Tˆ  1 T
ˆ 3 
T

T
2 1
 3
6 1 
ˆ
T
2
ˆ2
T
1
ˆ
T
3
ˆ Tˆ
T
2 1
ˆ
T
1
ˆ
ˆ  1 Tˆ 2 
T4  Tˆ 3 T
1

2 2 
 1

1
2
4
ˆ 
ˆ 
 Tˆ 2 T
T
1
 2
24 1 
3
Connected doubles
Dis-connected doubles
Connected triples, ‘true’ triples
‘Product’ Triples, disconnected triples
ˆ
T
4
True quadruples - four electrons interacting
ˆ 2
T
2
Product quadruples - two noninteracting pairs
ˆ Tˆ , T
ˆ Tˆ 2, Tˆ 4
T
3 1 2 1
1
Product quadruples, and so on.
CC Theory cont.
If all cluster operators up to TN are included, the method yields energies that are
essentially equivalent to Full CI.
In practice, only the singles and doubles excitation operators are used forming
the Coupled Cluster Singles and Doubles model (CCSD).
ˆ
ˆ
ˆ  T
ˆ 1T
ˆ 2  Tˆ T
ˆ  1 Tˆ 3   1 Tˆ 2  1 Tˆ T
ˆ2 1 T
ˆ 4 
e T1  T2  1ˆ  T
1
 2 2 1   2 1 6 1  2 2 2 2 1 24 1 
The result is that triple and quadruple excitations also enter into the energy
expression (not shown) via products of single and double amplitudes.
It has been shown that the connected triples term, T3, is important. It can be
included perturbatively at a modest cost to yield the CCSD(T) model. With the
inclusion of connected triples, the CCSD(T) model yields energies close to the
Full CI in the given basis, a very accurate wavefunction.
Comparison of Models
Scale with M
Size-extens ive/cons istent
Variationa l
Gene rall y applicable
Requires ΤgoodΥze ro-order 
Extension to Multi- referenc e
CI-SD
M6
No
Y
Y
Y
Yes
CI-SDTQ
M10
~Yes
Y
No
~No
MP2
M5
Y
No
Y
Y
Yes
MP4(SDTQ )
M7
Y
No
Y
Y
CCSD
M6
Y
No
Y
~No
Not yet
common
CCSD(T)
M7
Y
No
Y
No
Accuracy with a medium sized basis set (single determinant reference):
HF << MP2 < CISD < MP4(SDQ) ~CCSD < MP4(SDTQ) < CCSD(T)
In cases where there is (a) strong multi-reference character and (b) for excited
states, MR-CI methods may be the best option.