Building a wavefunction within the Complete-Active. Cluster with Singles and Doubles formalism: straightforward description of quasidegeneracy

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1 Building a wavefunction within the Complete-Active Active-Space Coupled-Cluster Cluster with Singles and Doubles formalism: straightforward description of quasidegeneracy Dmitry I. Lyakh (Karazin Kharkiv National University, Ukraine) Vladimir V. Ivanov (Karazin Kharkiv National University, Ukraine) and Ludwik Adamowicz (University of Arizona, USA) INT-08-a, University of Washington, Seattle, USA, 008

2 Smart CCSD Size-extensivity extensivity T explicitly correlates electron pairs T is responsible for orbital relaxation O(N 6 ) dependent upon the basis set size Excellent model for dynamic electron correlation accounting if: there is one determinant which dominates in the wavefunction expansion The reference determinant which defines Fermi vacuum + Ψ = e T T 0 0

3 Configurational QuasiDegeneracy Single Reference case: : C k <<, k k C0 HMultiReference (case: 0 0) MultiReference case: C { 0,,..., M }: M C i= 0 <<, k k (i HWeaker Condition k {0,,...,M})

4 MultiReference Phenomena Bond Breaking separation onto two open shell fragments (non-dynamic correlation) Transition States Open-shell Ground and Excited states spin and spatial orbital momentum symmetries (static correlation)

5 0 Failure of CCSD Singles Doubles Excitation Level Singles Singles Doubles Doubles Singles Singles Doubles Singles, Doubles EXPLICIT accounting; Singles, Doubles IMPLICIT accounting CCSD approximation becomes too rough when configurational quasidegeneracy is present Higher excitations become important because of propagation via dominating configurations: 0>, >, > Disbalanced configurational environment for 0>, >, >

6 Generalization of CCSD It is natural to explicitly treat all single and double excitations with respect to all important configurations Selection of important configurations is done by using the Complete-Active Active-Space concept: distributions of active electrons among the active orbitals generate the reference configurations those are assumed to dominate in the total wavefunction proper zero-order order approximation

7 Orbital Space Structure β α virtual spin-orbitals, a k ACTIVE ORBITAL SPACE Active electrons occupied spin-orbitals, i k virtual inactive spin-orbitals, a k virtual active spin-orbitals, A k occupied active spin-orbitals, I k All possible distributions of active electrons among the active orbitals form the REFERENCE SPACE: COMPLETE ACTIVE SPACE occupied inactive spin-orbitals, i k

8 The essence of MR schemes The essence of any multireference approach is to extract the most essential part of the total wavefunction and then to parametrize the rest of the wavefunction through non-linear amplitude products Singles and doubles upon all the reference configurations are assumed to form this most essential part of the total wavefunction

9 Different MR formalisms Fock Space MRCC State-Universal MRCC Single Reference Based MRCC State-Specific MRCC RMR-CCSD CASCCSD MkMRCCSD MRexpT BWMRCC COMMON FEATURE: explicit account of all single and double excitations with respect to all reference determinants

10 MRCC structure of Hilbert space P reference subspace spanned by reference configurations (usually CAS); S external subspace spanned by all single and double excitations with respect to all reference determinants (reference determinants themselves are excluded); R=P S iterative subspace. Every determinant from R is assigned its own variable (amplitude or CI coefficient); Q orthogonal complement to R H=R Q Hilbert space

11 Single Reference formalism for MultiReference problems: CASCCSD Tˆ k Tˆ k N. Oliphant and L. Adamowicz, JCP 94, 9 (99);JCP 96, 3739(99); P.Piecuch, N.Oliphant and L.Adamowicz, JCP 99,875 (993): building the MRCCSD wavefunction upon one formal reference determinant: (Tˆ + Tˆ + Tˆ 3 + Tˆ Tˆ M ) ΨOA = e 0 where = = k! k! k! k! a,..., a i,..., i k t k a... a i... i a, a,a...,a i, i,i,...,i k k k k aˆ + t aˆ + I...ˆ a a, a,a i i...i k iˆ + k k...,a iˆ k k aˆ +...ˆ i aˆ +  + - regular T operator  +... + k Î k Î k 3...Î iˆ iˆ Formal Reference determinant - Active-Space restricted T: Only two occupied/virtual indices at most are allowed to run through entire occupied/virtual range. All other indices MUST belong to the active orbital space! Some selected higher excited cluster will be present in OA ansatz All single and double excitations with respect to all references have own amplitude

12 CASCCSD excitations Singles Doubles Excitation Level Singles Singles Doubles Doubles Singles Singles Doubles Now ALL Singles and Doubles are explicitly accounted for! Selected higher excited clusters are included into the wave operator The ansatz obtained is symmetry-broken, that is, there is one specific reference determinant to be chosen as a FORMAL REFERENCE In order to achieve the best convergence with respect to the excitation level the formal reference MUST be the dominating determinant in the total wave function

13 CASCCSD method Balanced description of dynamic and non- dynamic electron correlation: ext ext ext ext ext ext ext Ψ = exp(tˆ + Tˆ + Tˆ + Tˆ +...Tˆ + Tˆ + + Tˆ + CASCCSD ( + Ĉ int + Ĉ int Ĉ int M ) M M M ) Spans the reference space. Describes non-dynamic electron correlation effects. Builds all single and double excitations upon all reference determinants. Describes dynamic electron correlation effects.

14 Solving CASCCSD equations ext ext ext ext ext ext ext int int int Ψ = CASCCSD exp(tˆ Tˆ Tˆ 3 Tˆ 4...Tˆ M Tˆ M+ Tˆ M+ ) ( + Ĉ + Ĉ ĈM ) 0 C C C C 3 4 = t = t = t = t t! + t 3 t t t + + 3!! t t 3 +! t t + 4! t 4 Projections on all configurations from the Iterative Subspace Ψ = CASCCSD exp(tˆ Tˆ Tˆ 3 Tˆ 4...Tˆ M Tˆ M Tˆ M+ + ) 0 Use of SRCC diagrams with Active Space Restrictions: Active Space Restrictions are built directly into the computer code which evaluates diagram values

15 SRCC diagrams SRCC cost CASCCSD cost CCSD 48 O(o v 4 ) O(o v 4 ) CCSDT 0 O(o v 5 ) O(o v 4 ) CCSDTQ 83 O(o v 6 ) O(o v 4 ) CCSDTQP 89 O(o v 7 ) O(o v 4 ) CCSDTQPS 43 O(o v 8 ) O(o v 3 ) CCSDTQPSS 607 O(o v 9 ) O(o v 3 ) CCSDTQPSSO 89 O(o v ) O(o v 3 ) In our computer program the formulae/code generation is automated

16 Some History CASCCD approach (V. Ivanov and L. Adamowicz, JCP, 958 (000); JCP 3, 8503 (000)): Tˆ + Tˆ 4 Ψ = e ( + Ĉ) 0 J. Olsen, JCP 3, 740 (000) general Active-Space CC via general active space CI technique. Necessity of inclusion of higher excited clusters forced development of automated formulae/code generation procedures: - M.Kallay, P.R.Surjan, JCP 5, 945 (00); M.Kallay, P.G.Szalay, P.R.Surjan, JCP 7, 980 (00). - V. V. Ivanov, D. I. Lyakh, and L.Adamowicz, Kharkiv University Bulletin, Chemical Series 549, 5 (00) (in Russian). - D. I. Lyakh, V. V. Ivanov, and L. Adamowicz, JCPJ, 0408 (005). - S. Hirata, Journal of Physical Chemistry A, 07, 9887 (003).

17 Choosing a wavefunction It necessary to estimate importance of different excitation levels for description of a certain problem It necessary to select a proper minimal active space which is sufficient for extracting the most essential part of the wavefunction

18 Excitation Level Diagnostics Level weight index: τ = W N () ( μ ) ( W = (C μ corr Shannon Indices (informational content): ( μ ) ( μ ) (Ck ) (Ck ) - Total: IC = log k,μ N N k k ) ) N N = corr μ= 0 ( W μ) - Excitation Level: I μ = k ( μ ) (Ck ) N log ( μ ) (Ck ) N - Inter-level: I W = μ W N (μ) log W N (μ) Every CC approach contains all determinants of all excitation levels!

19 Active Space Selection Natural Orbital Expansion (Occupation Numbers) Particular Orbital Contributions: - Energy POC (POCE): E ε i ε a ab a b i = fia + ij ab + i ij ) a ab ε t (t ti t ε j a = ti fia + ij ab (tij + a j, a, b = = i i a a i, j, b a b t ) i t j Show which orbitals contribute the most to the energy - Configurational POC (POCC): γ p = { k} p С k Show which orbitals contribute the most to the WF

20 BH dissociation problem Weight BH, Weight Indices 3 Re W CCSD W CCSD W3 CCSD W4 CCSD W CCSDT W CCSDT W3 CCSDT W4 CCSDT W CCSDTQ W CCSDTQ W3 CCSDTQ W4 CCSDTQ BH, Shannon Index I0 CCSD I0 CCSDT Im and Iw indices Value Re I0 CCSDTQ I CCSD I CCSDT I CCSDTQ I CCSD I CCSDT I CCSDTQ I3 CCSD I3 CCSDT I3 CCSDTQ I4 CCSD I4 CCSDT I4 CCSDTQ Value Re Ic CCSD Ic CCSDT Ic CCSDTQ Iw CCSD Iw CCSDT Iw CCSDTQ

21 BH dissociation problem Re POC indices: Configurational POC MO# E-0 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+00 MO# E+0 MO# E+0 MO# E+0 MO# E+00 MO# E+00 Occupied POCs: Energy POC Unoccupied POCs: POC distribution is smooth: no multireferencity

22 BH dissociation problem Re POC indices: Configurational POC MO# E-0 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+00 MO# E+00 Occupied POCs: Energy POC Unoccupied POCs:

23 BH dissociation problem 3Re POC indices: Configurational POC Energy POC MO# - MO# - MO# 3 - MO# 4 - MO# 5 - MO# 6 - MO# 7 - MO# 8 - MO# 9 - MO# 0 - MO# E E E E E E E E E E E+00 Occupied POCs: Unoccupied POCs:

24 BH dissociation problem BH dissociation curve: differences in mhartree with respect to Full CI are given Method Re Re 3Re CCSD (N 6 ) CCSDT (N 8 ) CCSDTQ (N 0 ) CAS(4,3)CCSD (N 6 )

25 FH dissociation problem FH, Weight Indices W CCSD W CCSDT W CCSDTQ 0.8 W CCSD 0.6 W CCSDT Weight W CCSDTQ W3 CCSD W3 CCSDT 0 3 W3 CCSDTQ W4 CCSD Re W4 CCSDT W4 CCSDTQ Value Shannon index 3 Re I0 CCSD I0 CCSDT I0 CCSDTQ I CCSD I CCSDT I CCSDTQ I CCSD I CCSDT I CCSDTQ I3 CCSD I3 CCSDT I3 CCSDTQ I4 CCSD I4 CCSDT I4 CCSDTQ Value Im and Iw indices 3 Re Ic CCSD Ic CCSDT Ic CCSDTQ Iw CCSD Iw CCSDT Iw CCSDTQ

26 Re POC indices: FH dissociation problem Configurational POC MO# E-0 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+00 MO# E+00 Occupied POCs: Energy POC Unoccupied POCs:

27 Re POC indices: FH dissociation problem Configurational POC MO# E-0 MO# E+00 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+00 MO# E+0 MO# E+0 MO# E+0 MO# E+00 Occupied POCs: Energy POC Unoccupied POCs:

28 3Re POC indices: FH dissociation problem Configurational POC Energy POC MO# E-0 MO# E+00 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+00 MO# E+0 MO# E+0 MO# E+0 MO# E+00 Occupied POCs: Unoccupied POCs:

29 FH dissociation problem FH dissociation curve: differences in mhartree with respect to Full CI are given Method Re Re 3Re CCSD (N 6 ) CCSDT (N 8 ) CCSDTQ (N 0 ) CAS(,)CCSD (N 6 )

30 H O: symmetrical bond stretching Weight HO, Wieght Indices 3 Re W CCSD W CCSDT W CCSDTQ W CCSD W CCSDT W CCSDTQ W3 CCSD W3 CCSDT W3 CCSDTQ W4 CCSD W4 CCSDT W4 CCSDTQ HO, Shannon I0 CCSD I0 CCSDT HO, Im and Iw Value Re I0 CCSDTQ I CCSD I CCSDT I CCSDTQ I CCSD I CCSDT I CCSDTQ I3 CCSD I3 CCSDT I3 CCSDTQ I4 CCSD I4 CCSDT I4 CCSDTQ Value Re Ic CCSD Ic CCSDT Ic CCSDTQ Iw CCSD Iw CCSDT Iw CCSDTQ

31 H O: symmetrical bond stretching Re POC indices: Configurational POC MO# E-0 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+00 MO# E+0 MO# E+00 MO# E+00 Occupied POCs: Energy POC Unoccupied POCs:

32 H O: symmetrical bond stretching Re POC indices: Configurational POC MO# E-0 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+00 MO# E+00 MO# E+00 Occupied POCs: Energy POC Unoccupied POCs:

33 H O: symmetrical bond stretching 3Re POC indices: Configurational POC MO# E-0 MO# E+00 MO# E+0 MO# E+03 MO# E+03 MO# E+03 MO# E+0 MO# E+0 MO# E+0 MO# 0-0.0E+0 MO# E+00 MO# E+00 MO# E+00 Occupied POCs: Energy POC Unoccupied POCs:

34 H O: symmetrical bond stretching H O symmetrical OH bond stretching. Differences in mhartree with respect to Full CI are given Method Re Re 3Re CCSD (N 6 ) CCSDT (N 8 ) CCSDTQ (N 0 ) CAS(4,4)CCSD_RHF (N 7 ) CAS(4,4)CCSD_CASSCF (N 7 )

35 H 8 model system Strong QuasiDegeneracy near D 4h symmetry Weights of different excitation levels CCSDTQPSSO CCSD Level

36 H 8 model system Configurational POC Method ENERGY, Hartree Error, mh MO# - MO# - MO# 3 - MO# 4 - MO# 5 - MO# 6 - MO# 7 - MO# E E E E E E E E+0 RHF CCS CCSD CCSDT CCSDTQ CCSDTQP CCSDTQPS CCSDTQPSS Triples are DANGEROUS! CCSDTQPSSO CAS(,)CCSD

37 N dissociation Pair excitations (, 4, 6!) Value.40E+0.0E+0.00E E E E+00.00E E+00 N, TZ basis set, Weights.5 Bohr 3 8 R, Bohr W CCSD W CASCCSD W CCSD W CASCCSD W3 CCSD W3 CASCCSD W4 CCSD W4 CASCCSD W5 CCSD W5 CASCCSD W6 CCSD W6 CASCCSD W7 CCSD W7 CASCCSD W8 CCSD W8 CASCCSD

38 N dissociation TZ basis set; Configurational POC indices for CCSD (RHF reference) MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+00 MO# E+00 MO# E+00 MO# E+00 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+00 MO# E+00 MO# E+00 MO# E+00 MO# E+00 MO# E+00 MO# E+00 MO# E-0 MO# E-0 R=.5 Bohr R=3.00 Bohr R=8.00 Bohr MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+00 MO# E+00 MO# E+00 MO# E+00 MO# E+00 MO# E+00 MO# E+00 MO# E+00 MO# E+00 MO# E+00 MO# E+00 MO# E+00 MO# E+00 MO# E-0 MO# E-0 MO# E+00 MO# E+00 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+0 MO# E+00 MO# E+00 MO# E+00 MO# E+00 MO# E+00 MO# E+00 MO# E+00 MO# E+00 MO# E-0 MO# E-0 MO# E-0 MO# E-0 MO# E-0 MO# E-0 MO# E-0 MO# E-0

39 N dissociation TZ basis set; frozen core approx. Differences in mh from Full CI Method.5 Bohr 3 8 CAS(6,6)CCSD8_CASSCF CCSD (6,6)R SNA -RMR-CCSD cc-pvdz basis set; frozen core approx. Differences in mh from Full CI Method.8 Bohr CCSD CCSDT CCSDTQ CAS(6,6)CISD CAS(6,6)CCSD8_CASSCF CAS(6,6)CCSD6_CASSCF

40 R / H ratio Norm of the Iterative Subspace Norm of Hilbert Space * System Active Space Configuration Max Excitation in R; (Number of non-zero configurations) Max Excitation in H; (Number of non-zero configurations) R norm H norm Ratio BH (4,3) 4(396) 6(7056) FH (,) 4(99) 6(43763) H O (4,4) 6(875) 6(50783) H8 (,) 4(430) 8(33) * H here corresponds to the subspace of Hilbert space spanned by all determinants up to a given excitation level

41 Excited States Problem: XCASCCSD extension The CASCCSD ansatz is not invariant with respect to the choice of the formal reference, 0> If the wave function is dominated by a pair of degenerate (symmetrical) determinants and one of them is taken as the formal reference, then the wavefunction will be symmetry broken

42 Degenerate formal references Symmetrical (degenerate) configurations belong to a different excitation level that breaks the spin or spatial orbital momentum symmetry of the total CASCCSD wave function

43 XCASCCSD extension Instead of using T-amplitudes T as variables we use in the XCASCCSD approach CI-coefficients as independent variables Then degenerate (symmetrical) configurations are assigned with only one variable an appropriate CI-coefficient This scheme guarantees proper spin or spatial symmetry for the major part (iterative subspace) of the total CASCCSD wave function

44 XCASCCSD extension To allow the use of the degenerate formal references we exploit CI-coefficient -> > T-amplitude T and back conversions during solving the CASCCSD equations: C C C C 3 4 = = = = T T T T ! T 3 T T T T + + 3!! T T 3 T +! T + 4! T 4 O(o v ) asymptotic computational cost for all CASCCSD approaches

45 The scheme of the XCASCCSD solver D.I.Lyakh, V.V.Ivanov, and L.Adamowicz. JCP 8, 0740(008)

46 FH: ground and excited states NPE (first row) and the maximal absolute deviation (second row) in mh for the PESs of the five states of the FH molecule. DZV basis set Method X Σ CAS(,)CISD X Sc CAS(,)CCSD X Sr CAS(,)CCSD CR-EOM-CCSD(T),III EOM-CCSD Σ Σ 3 Π Π For open shell singlet states regular CASCCSD gave approx. 0. spin for 5 Bohr!

47 C molecule ground and excited states PES The lowest excited states of the C are dominated by double excitations from the primary closed shell configuration There is an avoided crossing between the ground X Σ and excited B Σ states and real crossing between the ground X Σ and excited B Δ states

48 C molecule X Σ ground state PES: 6-3G*, XCAS(6,7)CCSD METHOD NPE, mh Max deviation, mh Full CI 0 0 CAS(6,7)CISD XscCAS(6,7)SD8_MC(8,8) XsrCAS(6,7)SD8_MC(8,8) XfCAS(6,7)SD8_MC(8,8) XscCAS(6,7)SD6_MC(8,8) XsrCAS(6,7)SD6_MC(8,8) CR-EOMCCSD(T),III EOM-CCSD CASPT

49 C molecule B Δ excited state PES: 6-3G*, XCAS(6,7)CCSD METHOD NPE, mh Max deviation, mh Full CI 0 0 CAS(6,7)CISD XscCAS(6,7)SD8_MC(8,8) XsrCAS(6,7)SD8_MC(8,8) XfCAS(6,7)SD8_MC(8,8) XscCAS(6,7)SD6_MC(8,8) XsrCAS(6,7)SD6_MC(8,8) CR-EOMCCSD(T),III EOM-CCSD CASPT

50 C molecule B Σ excited state PES: 6-3G*, 6 XCAS(6,7)CCSD METHOD NPE, mh Max deviation, mh Full CI 0 0 CAS(6,7)CISD XscCAS(6,7)SD8_MC(8,8) XsrCAS(6,7)SD8_MC(8,8) XfCAS(6,7)SD8_MC(8,8) XscCAS(6,7)SD6_MC(8,8) XsrCAS(6,7)SD6_MC(8,8) CR-EOMCCSD(T),III EOM-CCSD CASPT.747.6

51 HCOH vertical excitation energies cc-pvtz basis set; equilibrium geometry; energies in ev Method XA 3(n-pi pi) (n-pi pi) 3(pi pi-pi) (pi pi-pi) (sigma sigma-pi) CAS(,)SCF XcCAS(,)CCSD CAS(,)CISD EOM-CCSD CR-EOM EOM-CCSD(T),III CIS MRMP Experiment

52 Conclusions Concept of explicit accounting for all single and double excitations with respect to all reference determinants appears to be very efficient in treating the problems with quasidegeneracy: the iterative subspace formed in this way usually recovers most of the total wavefunction Different formalisms may be used for obtaining the values of variables

53 Conclusions The most straightforward approach CASCCSD seems to be efficient as well. Formal computational cost O(N 6 ) or O(N 7 ) Well-defined algorithms for active space selection can be established (POC, NOE) XCASCCSD extension of the CASCCSD approach allows treating different excited states with the required symmetry preserved

54 Conclusions However the problem of change of the formal reference along the PES needs further investigation Also dependence of a size of the minimal active space upon the basis set size should be established R/H ratio for bigger basis sets should be studied

55 Acknowledgements We would like to thank the Organizers for giving me a possibility to present our coupled-cluster cluster results to the most outstanding coupled-cluster cluster community! Thanks for your attention!