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

• 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?

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)

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ΦSD =

φ1 1( ) φ2(1) L φN (1)

φ1 2( ) φ2(2) L φN (2)

L L L L

φ1 N( ) φ2(N) L φN (N)

, φi |φ j = δij

Slater Determinant

Φ0 is a single determinantal wavefunction.

RHF dissociation problem

Consider H2 in a minimal basis composed of one atomic 1s orbital on each atom. Two AOs () leads to two MOs ()…

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φ1 = N1(χ A + χ B ); bonding MO

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φ2 = N2 (χ A − χ B); antibonding MO

H 1s H 1s

H H

H H

The ground state wavefunction is:

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Φ0 =φ1α (1) φ1β (1)

φ1α (2) φ1β (2)Slater determinant with two electrons in the bonding MO

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Φ0 = φ1α (1)φ1β(2) − φ1α (2)φ1β (1)

Φ 0 = φ1(1)φ1(2) α (1)β(2) − β(1)α (2)[ ]

Φ 0 = φ1(1)φ1(2) = (χ A (1) + χ B (1))(χ A (2) + χ B (2))

Φ 0 = χ A (1)χ A (2) + χ B (1)χ B (2) + χ A (1)χ B (2) + χ B (1)χ A (2)

Expand the Slater Determinant

Factor the spatial and spin parts

H does not depend on spin

Four terms in the AO basis

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A χ A

χ B χ BIonic terms, two electrons in one Atomic Orbital

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A χ B

χ B χ ACovalent terms, two electrons shared between two AOs

H2 Potential Energy Surface

0

E

H H

H. + H.H H

At the dissociation limit, H2 must separate into two neutral atoms.

Bond stretching

At the RHF level, the wavefunction, Φ, is 50% ionic and 50% covalent at all bond lengths.

H2 does not dissociate correctly at the RHF level!! Should be 100% covalent at large internuclear separations.

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A χ A

χ B χ B

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A χ B

χ B χ A

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 Φ0 and Φ1

Ionic terms Covalent terms

First, write a new wavefunction using the anti-bonding MO.

The form is similar to Φ0, but describes an excited state:

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Φ1 =φ2α (1) φ2β(1)

φ2α (2) φ2β(2)= φ2α (1)φ2β(2) − φ2α (2)φ2β (1)

Φ1 = φ2 (1)φ2(2) α (1)β (2) − β(1)α (2)[ ]

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Φ1 = φ2 (1)φ2(2) = (χ A (1) − χ B (1))(χ A (2) − χ B (2))

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Φ1 = χ A (1)χ A (2) + χ B (1)χ B (2) − χ A (1)χ B (2) − χ B (1)χ A (2)

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φ2 = N2 (χ A − χ B); antibonding MO

MO basis

AO basis

Trial function - Linear combination of Φ0 and Φ1; two electron configurations.

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Ψ =a0Φ 0 + a1Φ1 = a0(φ1φ1) + a1(φ2φ2 )

Ψ = (a0 + a1) χ A χ A + χ B χ B[ ] + (a0 − a1) χ A χ B + χ B χ A[ ]

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. Ψ = ΨCI, 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).

Ionic terms Covalent terms

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:

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Ψ =a0Φ HF + aii =1∑ Φ i

Multi-determinant wavefunction

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:

1

2

3

4

5

6

7

8

9

i

a a a

b b

c

i i

j j

k

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ΨHF

€ €

Ψia

€

Ψijab

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Ψijkabc

a,b,c… =virtual MOs

i,j,k… = occupied MOs

a,b

i,j

€

Ψijab

a

b

c,d

i

j

k,l

€

Ψijklabcd

Single Double Triple QuadrupleRef.Excitation level …

Represent the space containing all N-fold excitations by Ψ(N).Then the COMPLETE CI wavefunction has the form

Where

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ΨCI = C0Φ HF + Φ (1) + Φ(2) + Φ (3) + ... + Φ(N )

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Φ HF = Hartree − Fock

Φ (1) = CiaΨi

a

a

virt

∑i

occ

∑

Φ (2) = CijabΨij

ab

a,b

virt

∑i, j

occ

∑

Φ (3) = CijkabcΨijk

abc

a,b,c

virt

∑i, j ,k

occ

∑

Φ (N ) = Cijk...abc...Ψijk ...

abc...

a,b,c...

virt

∑i, j ,k...

occ

∑

Linear combination of Slater determinants with single excitationsDoubly excitations

Triples

N-fold excitation

The complete ΨCI expanded in an infinite basis yields the exact solution to the Schrödinger eqn. (Non-relativistic, Born-Oppenheimer approx.)

The various coefficients, , may be obtained in a variety of ways. A straightforward method is to use the Variation Principle.

The elements of the vector, , are the coefficients, And the eigenvalue, EK, approximates the energy of the Kth state.

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∂ECI

∂Cijk...abc... = 0

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Cijk...abc...

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ECI =ΨCI | H | ΨCI

ΨCI | ΨCI

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Hr C K = EK

r C K

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Cijk...abc...

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rC K

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.

E1 = ECI for the lowest state of a given symmetry and spin.E2 = 1st excited state of the same symmetry and spin, and so on.

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).

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ΨCI = C0Φ HF + Φ (1) + Φ(2)

Configuration State Functions

Consider a single excitation from the RHF reference.

ΦRHF Φ(1)

Both ΦRHF and Φ(1) have Sz=0, but Φ(1) is not an eigenfunction of S2.

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.

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Φ 1,2( ) = φ1α (1)φ2β(2) − φ1α (2)φ2β (1)

Example H2O:

Full CI

(19 basis functions)

CISD(~80-90%)

Example: Neon Atom

Ref.Singles 2Doubles 1Triples 4Quadruples 3

Weight =

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(Cijk...abc...)2

ijk...

abc...

∑for a given excitation level.

Relativeimportance

(Frozen core approx., 5s4p3d basis - 32 functions)

1. 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.2. 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 non-interacting.

Size extensive method - the energy scales properly with the number of particles. (Same fraction of correlation energy is recovered for CH4, C2H6, C3H8, etc.)

Ex. (ECISD of two H2 separated by 100Å) < 2(ECISD of one H2)

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)

1

2

3

4

5

6

7

8

9

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ΨHF

H2O MOs

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.

Complete Active Space Self-consistent Field (CASSCF)

Why? 1. To have a better description of the ground or

excited state. Some molecules are not well-described 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 (MR-CISD). 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.

C N HC N

H

C NH

Transition State

H

C

H

C

H

H

Singlet state of twisted ethene, biradical.

O -

O+

O O

O

O

zwitterionic biradical

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.

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ˆ H = ˆ H 0 + λ ˆ H '

ˆ H 0Φ i = E iΦ i, i = 0,1,2,...,∞

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ˆ H Ψ = WΨ

W = λ 0W0 + λ1W1 + λ 2W2 + ...

Ψ = λ 0Ψ0 + λ1Ψ1 + λ 2Ψ2 + ...

Hamiltonian with pert.,

Unperturbed Hamiltonian

As the perturbation is turned on, W (the energy) and Ψ change. Use a Taylor series expansion in .

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ˆ H 0 = ˆ F ii =1

N

∑ = ˆ h i + ˆ J ij − ˆ K ij( )j =1

N

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟

i=1

N

∑

ˆ H '= gijj >1

N

∑i =1

N

∑ − gijj =1

N

∑i =1

N

∑

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W0 = sum over MO energies

W1 = Φ 0| | ˆ H ' | Φ 0 = E(HF)

W2 =Φ 0| | ˆ H ' | Φ ij

ab Φ ijab | ˆ H ' | Φ 0

E0 − E ijab

a< b

vir

∑i < j

occ

∑

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E(MP2) =φiφ j | φaφb − φiφ j | φbφa[ ]

2

ε i + ε j −ε a − ε ba <b

vir

∑i< j

occ

∑

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Define ˆ H 0 and ˆ H '

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.

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 TheoryPerturbation 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.

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ΨCC = eˆ T Φ 0

The CC wavefunction takes on a different form:

Coupled Cluster WavefunctionΦ0 is the HF solution

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eˆ T = ˆ 1 + ˆ T +

12

ˆ T 2 +16

ˆ T 3 +L =1k!k =0

∞

∑ ˆ T k Exponential operator generates excited Slater determinants

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ˆ T = ˆ T 1 + ˆ T 2 + ˆ T 3 +L + ˆ T N Cluster Operator

N is the number of electrons

CC Theory cont.

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ˆ T 1Φ 0 = tia

a

vir

∑i

occ

∑ Φ ia

ˆ T 2Φ 0 = tijab

a <b

vir

∑i< j

occ

∑ Φ ijab

The T-operator acting on the HF reference generates all ith excited Slater Determinants, e.g. doubles Φij

ab.

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tia

€

tijab

€

} Expansion coefficients are called amplitudes; equivalent to the ai’s in the general multi-determinant wavefunction.

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eˆ T = ˆ 1 + ˆ T 1 + ˆ T 2 +

12

ˆ T 12 ⎛

⎝ ⎞ ⎠+ ˆ T 3 + ˆ T 2

ˆ T 1 +16

ˆ T 13 ⎛

⎝ ⎞ ⎠ + ˆ T 4 + ˆ T 3

ˆ T 1 +12

ˆ T 22 +

12

ˆ T 2ˆ T 1

2 +1

24ˆ T 1

4 ⎛ ⎝

⎞ ⎠ +L

doubles triples Quadruple excitationssingles

HF ref.

The way that Slater determinants are generated is rather different…

CC Theory cont.

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ˆ 1

€

ˆ T 1

€

ˆ T 2 +12

ˆ T 12 ⎛

⎝ ⎞ ⎠

€

ˆ T 2

€

ˆ T 12

€

ˆ T 3 + ˆ T 2ˆ T 1 +

16

ˆ T 13 ⎛

⎝ ⎞ ⎠

€

ˆ T 3

€

ˆ T 13

€

ˆ T 4

€

ˆ T 4 + ˆ T 3ˆ T 1 +

1

2ˆ T 2

2

+1

2ˆ T 2

ˆ T 12 +

1

24ˆ T 1

4

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

€

ˆ T 22

€

ˆ T 2ˆ T 1

HF reference

Singly excited states

Connected doubles

Dis-connected doubles

Connected triples, ‘true’ triples

‘Product’ Triples, disconnected triples

True quadruples - four electrons interacting

Product quadruples - two noninteracting pairs

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ˆ T 3ˆ T 1,

ˆ T 2ˆ T 1

2, ˆ T 14 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).

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eˆ T 1 + ˆ T 2 = ˆ 1 + ˆ T 1 + ˆ T 2 +

12

ˆ T 12 ⎛

⎝ ⎞ ⎠+ ˆ T 2

ˆ T 1 +16

ˆ T 13 ⎛

⎝ ⎞ ⎠ +

12

ˆ T 22 +

12

ˆ T 2ˆ T 1

2 +124

ˆ T 14 ⎛

⎝ ⎞ ⎠

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

Accuracy with a medium sized basis set (single determinant reference):

HF << MP2 < CISD < MP4(SDQ) ~CCSD < MP4(SDTQ) < CCSD(T)

CI-SD CI-SDTQ MP2 MP4(SDTQ) CCSD CCSD(T)Scale with M M6 M10 M5 M7 M6 M7

Size-extensive/consistent No ~Yes Y Y Y YVariational Y Y No No No No

Generally applicable Y No Y Y Y YRequires ‘good’ zero-order Φ Y ~No Y Y ~No NoExtensi onto Mu -lti reference Yes Yes No t yet

common

In cases where there is (a) strong multi-reference character and (b) for excited states, MR-CI methods may be the best option.