Lecture 7: Coupled-cluster theory

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 excitations to the the Hartree-Fock state and truncate that to the some excitation level

†1 †† CI<∗∗∗ 1C C L HF 2 • The expansion coefficients are solved using the variation principle, that corresponds to the solution of the eigenvalue equation HC=EC – Selected eigenvalues can be determined with iterative methods Multi-configurational SCF

• In the multi-configurational self-consistent field (MCSCF) theory both the orbital coefficients and the CI excitation amplitudes are optimized variationally at 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 will be sensible to introduce several reference configurations (reference space) to the CI wave function => 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 approach is 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 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<∗ 1ˆ HF ∋(

Excitation (cluster) Excitation operator amplitude (=S,T,D,..)

–This is the (full) coupled-cluster wave function – t is the connected, tt the disconnected amplitudes • Because ˆˆ,0< this expansion is equivalent to CC< e(Tˆ ) HF Tˆ < ˆ The coupled-cluster wave functions

• The hierarchy of CC wave function is established by truncating the operator T up to a certain level of excitation, e.g. – ˆˆ: Coupled-cluster doubles (CCD) wave function TT; 2 – ˆˆˆ: Coupled-cluster singles-and-doubles (CCSD) TTT; 12∗ wave function – ˆˆˆˆ: Coupled-cluster singles-doubles-and- TTTT; 123∗∗ triples (CCSDT) wave function • A successful approach has been to include only the most important terms from the highest excitation level as a correction term – E.g. the CCSD(T) wave function CC and CI methods compared

• 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

The error with respect to FCI of CC and CI wave functions as a function of the excitation level. Calculation for the water molecule at the equilibrium geometry in the cc-pVDZ basis. • For larger systems, CI starts to behave very badly, while the CC description is unaffected by the number of electrons CC performance

CC wave functions in description of dissociation of a water molecule (angle fixed). 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 but using the (linked) coupled-cluster equations

and the CC energy is obtained from the amplitudes that minimizes the projected equation above as

– e(,THTˆˆ )ˆ e( ) is referred to as the similarity-transformed Hamiltonian • The CC energy is not variational Solving the CC equations

• Let be the value of the CC amplitude equation with a given set of amplitudes t(n) , which can be expanded as ) where

Solve t from (t(n) + t)< No Χ Χ δ t(n+1) =t(n) + t ? Χ

Yes

{tλ} →|CC> Coupled-cluster perturbation theory

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

– These would be solved self-consistently after truncating the 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 compared

Difference to the FCI energy of various CC and MP levels of theory. Water molecule in equilibrium and stretched geometries.

O(N5) O(N6) O(N7) O(N8) About the accuracy: Bond lengths Comparison of models by cc-pVDZ cc-pVTZ cc-pVQZ the deviation from experimental molecular HF geometries of 29 small main-group element species MP2

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

CCSD

CCSD(T) About the accuracy: Bond lengths

Relationship between the calculated bond distances for the standard models (in pm)

Helgaker, Jorgensen, Olsen: Molecular Electronic-Structure Theory (Wiley 2002) About the accuracy: Reaction enthalpies

Helgaker, Jorgensen, Olsen: Error in the reaction enthalpies (kJ/mol) for 14 Molecular Electronic-Structure Theory reactions involving small main-group element (Wiley 2002) molecules 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 systematic route 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 standard CC and OCC are in practice Local correlation methods

• Electron correlation is spatially a local phenomenom • Local CC approach – 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 Formal State-of-the-art scaling Hartree-Fock O(N4) (Almost) linear scaling. Thousands of atoms. MP2 O(N5) O(N3)...O(N2) scaling, even linear(?). Hundreds of atoms. CISD O(N6)

CCSD O(N6) Local (LCC) approaches reduce the scaling. Tens of atoms feasible. CCSD(T) O(N7) Yoo (JPCL 1, 3122 (2010)): CCSD(T)

geom. optimization of (H2O)17

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 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 at http://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 chemical compounds • It is expensive but still applicable to medium-size systems – 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 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