NWChem: Coupled Cluster Method (Tensor Contraction Engine)
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1 NWChem: Coupled Cluster Method (Tensor Contraction Engine)
2 Why CC is important?! Correlation effects are important!! CC is size-extensive theory: can be used to describe dissociation processes.! Higher-order effects can be approximated by products of lower rank cluster amplitudes.! Strong ties with many body perturbation theory (MBPT). Effective perturbative methods (CCSD(T)) can be constructed in order to encapsulate higher-order correlation effects.! Exact energy limit exists full coupled cluster approach.! Can be applied across energy and spatial scales: from nuclear structure theory to molecular nanosystems 2
3 What is Tensor Contraction Engine (TCE)! Symbolic manipulation & program generator! Automates the derivation of complex working equations based on a well-defined second quantized many-electron theories! Synthesizing efficient parallel computer programs on the basis of these equations.! Granularity of the parallel CC TCE codes is provided by the so-called tiles, which define the partitioning of the whole spinorbital domain. 3
4 What is Tensor Contraction Engine (TCE) Tile structure: α β α β S1 S2 S1 S2 S1 S2. S1 S2. Occupied spinorbitals unoccupied spinorbitals Tile-induced block structure of the CC tensors: CPU1 CPU2 CPU3. i a T T [ h [ p n ] ] m CPU n 4
5 CC TCE calculations! Closed- & open-shell CC calculations with RHF/ ROHF/UHF references! Many-body perturbation theory! CI methods: CISDT, CISDT,! Ground-state methodologies: CCSD, CCSD(T), CCSDT,! Excited-state methods: EOMCCSD,CC2, CR- EOMCCSD(T), EOMCCSDt, EOMCCSDT! Linear response CC methods for calculating molecular properties: static & dynamic CCSD polarizabilities, static CCSDT polarizabilities, static CCSD hyperpolarizabilities 5
6 Coupled Cluster method Ψ 0 = T e Φ cluster operator reference function (HF determinant) Φ abc... ijk Coupled cluster equations ( ) He T 0 Φ =... C CC eqs. are energy indepent Connected diagrams only: CC theory can properly describe dissociation processes energy is a sum of energies in the non-interacting system limit E = T C ( ) Φ He Φ
7 Coupled Cluster method Method CCSD (singles & doubles) CCSD(T) (perturbative triples) CCSDT (singles & doubles & triples) CCSDTQ (singles & doubles &triples & quadruples) Numerical complexity N 6 N 7 N 8 N 10 Hartree- Fock (N 4 ) 4-index transformation (N 5 ) CCSD (N 6 ) CCSD(T) (N 7 ) 7
8 How to define reference?! Three types of references can be used in singlereference TCE CC calculations: RHF, ROHF, UHF scf thresh 1.0e-10 tol2e 1.0e-10 singlet maxiter 100 rhf scf thresh 1.0e-10 tol2e 1.0e-10 doublet maxiter 100 rohf scf thresh 1.0e-10 tol2e 1.0e-10 singlet maxiter 100 uhf 8
9 How to choose 4-index transformation?! RHF/ROHF references! Default: spinorbital 4-index tarnsformation! Alternatives: orbital 4-index transformations tce... tilesize 20 2eorb 2emet 4 split 2 attilesize tce... tilesize 20 2eorb 2emet 14 split 2 attilesize # Always: tilesize <= attilesize; available in the GA version only 4-index trans. Is performed using two batches of atomic 2-electron integrals. This is more memory efficient version, attilesize defines the so-called atomic tilesize for 4-index trans.! UHF reference: default spinorbital 4-index transformation will be executed. 9
10 Local memory management in CC TCE module! Approaches based on the single and double excitations (CCSD,EOMCCSD,LR-CCSD) ~ (tilesize) 4! Perturbative CCSD(T) & CR-EOMCCSD(T) methods 2 * (tilesize) 6 # choose tilesize wisely! Iterative CCSDt,CCSDT,EOMCCSDt,EOMCCSDT methods 4 * (tilesize) 6 # choose tilesize wisely 10
11 Example: CCSD calculation Example: h2o_dimer_ccsd_aug_cc_pvdz.nw 11 scf thresh 1.0e-10 tol2e 1.0e-10 singlet rhf tce freeze atomic ccsd maxiter 100 tilesize 15 diis 5 thresh 1.0d-5 2eorb 2emet 13 attilesize 40 task tce energy max. number of iterations length of the diis cycle conv. threshold default value 40
12 Examples: open-shell CCSD(T) calculation Example: cnh2o_ccsd_t_cc_pvdz.nw scf thresh 1.0e-10 tol2e 1.0e-10 doublet rohf tce freeze atomic ccsd(t) maxiter 100 tilesize 15 diis 5 lshift 0.2 thresh 1.0d-5 2eorb 2emet 13 attilesize 40 CCSD(T) calculation will be performed Level shifting may be helpful in converging open-shell CCSD equations 12
13 Challenging situations bond breaking processes: renormalized methods Example: tce_cr_ccsd_t_ozone_pol1.nw # in single-bond breaking/forming processes renormalized methods may provide better description of ground-state potential energy surfaces tce freeze atomic 2eorb 2emet 13 cr-ccsd(t) tilesize 15 thresh 1.0d-5 completely renormalized CCSD(T) method is invoked 13
14 Examples: CCSDT calculations Example: h2o_dimer_ccsdt_cc_pvdz.nw tce freeze atomic ccsdt maxiter 100 tilesize 10 diis 3 thresh 1.0d-2 2eorb 2emet 13 attilesize 40 CCSDT theory is invoked. To reduce memory requirements one can make diis cycle length smaller very relaxed conv. threshold task tce energy 14
15 Examples: MBPT calculations Example: mbpt2_h2o.nw scf thresh 1.0e-10 tol2e 1.0e-10 singlet rhf tce mbpt2 task tce energy MBPT(n) approaches are byproducts of various order CC implementations. For example, second order MBPT correction can be restored in the first iteration of the CCD method when initial guesses For cluster amplitudes are set equal to zero. 15
16 Examples: MBPT calculations towards higher orders Example: mbpt4sdq_h2o.nw MBPT(2) < MBPT(3) < MBPT(4,SDQ)< MBPT(4) tce mbpt4(sdq) task tce energy MBPT(3) + selected MBPT(4) contributions; an approximate method to include the effect of quadruply excited configurations 16
17 Excited-state EOMCC calculations cluster operator Ψ K = R K e T Φ reference function (HF determinant) excitation operator HR K Φ = E K R K Φ H = e T He T (Equation of Motion Coupled Cluster Equations ) 17
18 Excited-state EOMCC calculations EOMCCSD < CR-EOMCCSD(T) < EOMCCSDT < EOMCCSDTQ Method Numerical complexity EOMCCSD (singles & doubles) N 6 Excitation energies of singly Excited states CR-EOMCCSD(T) (perturbative triples) EOMCCSDT (singles & doubles & triples) EOMCCSDTQ (singles & doubles & triples & quadruples) N 7 N 8 N 10 } Excited-state potential energy surfaces, doubly excited state 18
19 19 Excited-state EOMCC calculations
20 20 Excited-state EOMCC calculations
21 Excited-state EOMCC calculations Towards experimental accuracy Providi Errors of the EOMCCSD (blue) and CR-EOMCCSD(T) (green) vertical excitation energies with respect to the experimental data. Modeling complex excited state processes Charge transfer processes in open-shell systems TiO 2 EOMCCSd 3.69 ev N- doped TiO 2 EOMCCSd 2.51 ev 21
22 Excited-state EOMCC calculations L a state POL1 basis set Excitation energy (ev) Expt. EOMCCSD CR-EOMCCSD(T) Number of rings 22
23 Excited-state calculations: EOMCCSD Example: h2o_dimer_eomccsd_aug_cc_pvdz.nw tce freeze atomic ccsd tilesize 20 diis 5 thresh 1.0d-5 2eorb 2emet 13 nroots 2 eomsol 1 task tce energy number of roots EOM-CCSD right-hand side iterations Residuum Omega / hartree Omega / ev Cpu Wall Iteration 29 using 48 trial vectors Iterations converged largest EOMCCSD amplitudes: R1 and R2 Singles contributions 11a' (alpha) --- 8a' (alpha) a' (alpha) --- 9a' (alpha) a' (alpha) --- 9a' (alpha) a' (alpha) --- 9a' (alpha) a' (alpha) --- 9a' (alpha) a' (alpha) --- 8a' (alpha) a' (alpha) --- 9a' (alpha) a' (alpha) --- 9a' (alpha) Doubles contributions... old eigensolvers (default option) - requires more memory but works for doubly excited states
24 Excited-state calculations: EOMCCSD Example: h2o_dimer_eomccsd_aug_cc_pvdz_eomsol2.nw tce freeze atomic ccsd tilesize 20 diis 10 thresh 1.0d-5 2eorb 2emet 13 nroots 1 eomsol 2 symmetry targetsym a' task tce energy new EOMCCSD solver with improved memory management should be used for singly excited states only; initial starts taken from the CIS calculations } states of a symmetry will be calculated 24
25 Excited-state calculation: EOMCCSDT Example: tce_h2o_eomccsdt_cc-pvdz.nw # CCSDT/EOMCCSDT methods are much more expensive than # the CCSD/EOMCCSD formalisms tce freeze core atomic ccsdt dipole thresh 1.0d-6 nroots 1 task tce energy calculates excited-state dipole moments and transition moments 25
26 Excited-state calculations: active-space EOMCCSDT methods (EOMCCSDt) Example: tce_active_ccsdt_be3.nw # EOMCCSDt uses selected set of triply excited amplitudes # it makes it less expensive than the full EOMCCSDT approach! tce freeze atomic ccsdta tilesize 12 thresh 1.0d-4 active_oa 3 active_ob 3 active_va 9 active_vb 9 t3a_lvl 1 nroots 1 symmetry targetsym a1 } Definition task tce energy of the active space: active occupied alpha spinorbitals active occupied beta spinorbitals active virtual alpha spinorbitals Aactive virtual beta spinorbitals Only ABc t ijk amplitudes included 26
27 Excited-state calculations: CR-EOMCCSD(T) Example: tce_cr_eom_t_ozone_pol1.nw # Excitation energies accuracy: # EOMCCSD < CR-EOMCCSD(T) < EOMCCSDT tce freeze atomic 2eorb } 2emet 13 creomsd(t) tilesize 15 thresh 1.0d-4 nroots 1 symmetry targetsym b2 RHF reference is employed, orbital from of 2-electron integrals can be used CR-EOMCCSD(T) calculation is composed of several steps: (1) CCSD calculation (2) EOMCCSD calculation (3) calculation of the CR-EOMCCSD(T) non-iterative correction task tce energy 27
28 Linear response CC calculations! Property calculations with the CC method: The cluster operator T(t) is expanded in order of timedepent perturbation T( t) (0) (1) (2) = T + T ( t) + T ( t) +...! Dipole polarizabilities can be calculated as a linear response function α ( ω) << µ µ >> ij = i ; j ω CC eq. Λ-CC eq. first-order LR-CC eq. 28
29 Linear response CC calculations! Accurate calculations for dipole polarizabilities with NWChem:! Important correlation effects can be included in the CCSD and CCSDT approximations! Various propertytailored basis sets should be used in order to attain the experimental accuracy 29
30 Property calculations: CCSD dipole polarizabilities Example: tce_polar_ccsd_n2_aug_cc_pvdz.nw tce scf ccsd io ga 2eorb linear response CCSD calculations will be performed only x and z components of the first-order response cluster amplitudes will be calculated (i.e. α(xx) α(zz) α(xz) will be available) set tce:lineresp T set tce:respaxis T F T set tce:afreq task tce energy List of required ω values 30
31 Property calculations: CCSD dipole polarizabilities benzene Example: tce_polar_ccsd_benzene.nw scf singlet rhf thresh 1.0e-10 tce scf freeze core ccsd io ga 2eorb CCSD Linear Response polarizability / au Frequency = / au Wavelength = Inf / nm atomic units (bohr^3) angstroms^3 X Y Z X Y Z X Y Z Eigs = Isotropic = Anisotropic = set tce:lineresp T set tce:respaxis T T T set tce:afreq 0.0 task tce energy static case 31
32 32 Questions?
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