Lecture 7: Coupled-cluster theory - Helsinki

Lecture 7: Coupled-clustertheorytheory

The de facto standard of modern ab initio quantum chemistry, plus some

special treatments of it

Configuration interaction

• Include to the wave function expansion the determinants that are obtained as certain excitationsto the the Hartree-Fock state and truncate that to the some excitation level

1æ ö

• The expansion coefficients are solved using the variation principle, that corresponds to the solutionof the eigenvalue equation HC=EC– Selected eigenvalues can be determined with iterative

methods

� � �1CI 1 HF

2a abi a i ij a b i j

ai aibj

C a a C a a aaæ ö÷ç ÷< ∗ ∗ ∗ç ÷ç ÷çè ø

å å L

Multi-configurational SCF

• In the multi-configurational self-consistent field(MCSCF) theory both the orbital coefficients and the CI excitation amplitudes are optimized variationallyat the same time

• CI is not applied to the whole reference state but the orbitals are categorized into– Inactive & secondary

• No CI, i.e. always doubly occupied or virtual

– Active• CI among them

Multireference CI

• In cases where the HF state is less pronounced, it willbe sensible to introduce several referenceconfigurations (reference space) to the CI wavefunction => Multireference CI (MRCI)

MRCISD wave functions in description of dissociation of a water molecule (angle fixed).

Shortcomings of the CI model

• The major disadvantage of the truncated CI approachis that it is not size-extensive– E.g. the CISD wave function would need certain T and Q

determinants to be size-extensive– There exists a size-extensive reformulation of the CISD – There exists a size-extensive reformulation of the CISD

model, called quadratic CISD (QCISD)

• It is not very economical: the recovered amount of correlation energy converges quite slowly w.r.t. the CI expansion– CISD is not sufficient, CISDTQ is too expensive

Cluster expansion

• Recast the FCI expansion into a product form

∋ (�CC 1 HFtm mm

tæ ö÷ç ÷< ∗ç ÷ç ÷çè øÕ

Excitation operatorExcitation (cluster)

– This is the (full) coupled-cluster wave function– tmn is the connected, tmtn the disconnected amplitudes

• Because this expansion is equivalent to

Excitation operator(m=S,T,D,..)

Excitation (cluster) amplitude

�CC exp( ) HF� �

T

T tm mm

t<

< å

� �, 0m nt té ù <ê úë û

The coupled-cluster wave functions

• The hierarchy of CC wave function is established bytruncating the operator T up to a certain level of excitation, e.g.– : Coupled-cluster doubles (CCD) wave function

2� �T T;

– : Coupled-cluster singles-and-doubles (CCSD) wave function

– : Coupled-cluster singles-doubles-and-triples (CCSDT) wave function

• A successful approach has been to include only the most important terms from the highest excitationlevel as a correction term– E.g. the CCSD(T) wave function

2T T;

1 2� � �T T T∗;

1 2 3� � � �T T T T∗ ∗;

• CC wave functions truncated at a given excitation level also contain contributions from determinants corresponding to higher-order excitations – CI wave functions truncated at the same level contain

contributions only from determinants up to this level

CC and CI methods compared

contributions only from determinants up to this level

• For larger systems, CI starts to behave very badly, while the CC description is unaffected by the number of electrons

The error with respect to FCI of CC and CI wave functions as a function ofthe excitation level. Calculation for the water molecule at the equilibrium geometry in the cc-pVDZ basis.

CC performance

CC wave functionsin description of dissociationof a water molecule (anglefixed). Full line: RHF reference state, dashed line: UHF reference state

Helgaker, Jorgensen, Olsen: Molecular Electronic-Structure Theory(Wiley 2002)

Solving the CC equations

• The CC equations are not solved variationally butusing the (linked) coupled-cluster equations

and the CC energy is obtained from the amplitudesand the CC energy is obtained from the amplitudesthat minimizes the projected equation above as

– is referred to as the similarity-transformedHamiltonian

• The CC energy is not variational

�� exp( ) exp( )T H T,

Solving the CC equations

• Let W be the value of the CC amplitude equationwith a given set of amplitudes t(n) , which can beexpanded as

where)

where

Solve Χt from W(t(n) + Χt)<δ?

{tλ} →|CC>

Yes

Not(n+1) =t(n) + Χt

Coupled-cluster perturbation theory

• Partition the Hamiltonian as in MPPT and use it in the context of CC equations, leading to amplitudeequations

– These would be solved self-consistently after truncatingthe cluster operator to some excitation level

• Can be used to determine correction terms– E.g. (T) in CCSD(T); MP2 = CC(D)

• There are succesful fully iterative CCPT schemes– CC2 approximates CCSD– CC3 approximates CCSDT

CC and MP hierarchies comparedDifference to the FCI energy of various CC and MP levels of theory. Water moleculein equilibrium and stretched geometries.

O(N5) O(N6) O(N7) O(N8)

About the accuracy: Bond lengths

HF

cc-pVDZ cc-pVTZ cc-pVQZ

MP2

Comparison of models bythe deviation fromexperimental moleculargeometries of 29small main-group elementspecies


CCSD

CCSD(T)

About the accuracy: Bond lengths

Relationship between thecalculated bond distancesfor the standard models(in pm)


About the accuracy: Reaction enthalpies


Error in the reaction enthalpies (kJ/mol) for 14 reactions involving small main-group elementmolecules

Special formulations of the CC theory

• Equation-of-motion coupled-cluster (EOM-CC)– CI with the similiarity-transformed Hamiltonian– The cluster operator provides a good description for

electron correlation, the CI formalism offers a systematicroute for the excitation structureroute for the excitation structure

• Orbital-optimized coupled-cluster (OCC)– Optimize the orbital coefficients in each iteration together

with the cluster amplitudes• Compare with MCSCF

– It is surprising how small the differences between standardCC and OCC are in practice

Local correlation methods

• Electron correlation is spatially a localphenomenom

• Local CC approach– Localize occupied orbitals– Localize occupied orbitals– Use a semi-local projected AO space for virtual orbitals– Restrict the excitations basing on spatial thresholds

• Projected AO basis approach– Write everything in the (projected) AO space– Amplitudes decay now exponentially with respect to the

system size => prescreening possible

Computational cost considerations

Model Formalscaling

State-of-the-art

Hartree-Fock O(N4) (Almost) linear scaling. Thousandsof atoms.

MP2 O(N5) O(N3)...O(N2) scaling, evenMP2 O(N ) O(N )...O(N ) scaling, evenlinear(?). Hundreds of atoms.

CISD O(N6)

CCSD O(N6) Local (LCC) approaches reduce the scaling. Tens of atoms feasible. Yoo (JPCL 1, 3122 (2010)): CCSD(T) geom. optimization of (H2O)17

CCSD(T) O(N7)

MCSCF Exp(Nact)

Some quantum chemistry software packages

Name Models Basis Periodic License

HF DFT CC CI MC

ACES III GTO No GPL

ADF STO Yes Commercial

CP2K GTO/PW Yes GPLCP2K GTO/PW Yes GPL

Dalton GTO No Academic

Gaussian GTO Yes Commercial

Molpro GTO No Commercial

NWChem GTO,PW No/Yes ECL

Turbomole GTO No Commercial

See a comprehensive list athttp://en.wikipedia.org/wiki/Quantum_chemistry_computer_programs

Concluding remarks

• Coupled-cluster theory is the de-facto standard of modern ab initio quantum chemistry– ”The right answer for the right reason”: Able to accurately

reproduce most chemical properties of most chemicalcompoundscompounds

• It is expensive but still applicable to medium-sizesystems– Local correlation approaches extend the applicability of CC

theory

Things to think about & homework

• Using the Baker-Campbell-Hausdorff expansion, show that the similarity-transformed Hamiltonian is no higher than quartic in the amplitudes

• Set up the CCSD amplitude equationsSet up the CCSD amplitude equations• Study the Chapter 5, have a glance at 6.1 and 6.2• Read the review article on CC theory:

O. Christiansen, Theor. Chim. Acta 116, 106 (2006)• Exercise session on Friday

– Starting time correlated with the PhD defence of Suvi Ikäläinen

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Lecture 7: Coupled-cluster theory - Helsinki