Recent Advances in the MRexpT approach

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1 Recent Advances in the MRexpT approach Michael Hanrath University of Cologne Institute for Theoretical Chemistry July 8, 008 Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, 008 / 50

2 Outline The MRCC problem Applications Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, 008 / 50

3 Outline The MRCC problem MRCI MRCC Ansätze Applications H 4, ground state, higher excitations P 4, ground state, higher excitations BeH, C v insertion problem, avoided crossing, higher excitations N LiF, potential energy surface, avoided crossing Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

4 MRCI The Multireference Case Introduce generalization according to 0 = µ c µ µ 0 : } particle space V a, b,... } active space A p, q,... } hole space O i, j,... Determinants: λ, µ,...: Reference determinants, P-space α, β,...: Excited determinants, Q-space ρ, σ,...: Determinants from P Q-space Orbitals: i, j,... O: inactive p, q,... A: active a, b,... V: virtual α ˆτ λ α λ Q(λ) ˆτ λ β P β Q ˆτ µ β µ Q(µ) ˆτ µ γ δ Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

5 MRCI The Key Problems with MRCC Ambiguity of genealogy of determinants / connectivity Connectivity Ambiguity of genealogy Single-reference (SR) coupled-cluster (CC) method Ψ CC = e ˆT 0 Cluster operator ˆT = X i t iˆτ i crucial: association amplitude excitation Reformulate CI and MRCI by means of Ĉ = P i ciˆτi Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

6 MRCC Ansätze MRCC History 960 SRCC VUMRCC SUMRCC 990 SRMRCC 000 c SUMRCC MkMRCC BWMRCC MRexpT Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

7 MRCC Ansätze Single Reference Formalism MRCC Ψ = e ˆT 0 ˆT = Xˆτ i T tˆτ i ˆτ i ( T) 0! = ls ( Ψ MRCI ) c ip c iq ip iq ˆτ ip ap ˆτ ip ia ˆτ iq ia ˆτ iq aq MRCI ap ia aq c ap c ia c aq ip t i aˆτ i a t p aˆτ p a t p qˆτ p q t ip aqˆτ ip aq ap t i a ia t p a iq t p q t i aˆτ i a SR-MRCC aq t ip aq + t p qt i a + 3 = ord({t j}) = ord(q) = 3: State selective, no intruder states Not Fermi vacuum invariant (symmetry broken solutions, one particular reference) N. Oliphant and L. Adamowicz, J. Chem. Phys. 95, 6645 (99), P. Piecuch, N. O., and L. A., ibid. 99, 875 (993) Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

8 MRCC Ansätze State Universal (Hilbert space) MRCC Ψ ν = X cνµe ˆT (µ) µ µ X ˆT (µ) = ˆτ i (µ) T µ tˆτi (µ)ˆτ i(µ) c ip c iq ap c ap ap ˆτ ip ap ti a( ip )ˆτi a cip ip ip ˆτ ip ia tp a( ip )ˆτp a ia c ia ia ˆτ iq ia tq a( iq )ˆτq a iq ciq iq ˆτ iq aq ti a( iq )ˆτi a aq c aq aq MRCI SU-MRCC cipti a( ip ) ciptp a( ip ) + ciqtq a( iq ) ciqti a( iq ) + Fermi vacuum invariant (there is no particular reference) 4 = ord({t j(µ)}) > ord(q) = 3: Too few equations, too many parameters (underdetermined), to be state selective use Bloch equation intruder states often poor performance B. Jeziorski and H. J. Monkhorst, Phys. Rev. A 4, 668 (98) Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

9 MRCC Ansätze Mukherjee s MRCC Now consider projections onto Q µ : (assume Q µ = {α, α }, Q µ = {α, α 3 }) α ˆτ λ α λ Q(λ) ˆτ λ β P β Q ˆτ µ β µ Q(µ) ˆτ µ γ δ In general: The solution of the set of MkMRCC working equations does strictly correspond to the solution of the Schrödinger equation projected onto µ Q µ only. For MRCCSD... n (=FCI) it is Q µ = µ Q µ As approaching the full cluster operator MkMRCC will become exact Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

10 MRCC Ansätze MRexpT Ansatz: Determinant based amplitude indexing Ψ = X µ cµe ˆT (µ) µ ˆT (µ) = ϕ(c µ) Xˆτ i (µ) T µ tˆτi (µ) µ ˆτ i(µ) (ϕ(z) = e i arg(z) ) c ip ip ˆτ ip ap ap c ap ap t ap ˆτ i a c ip ip c ip t ap c iq iq ˆτ ip ia ˆτ iq ia ˆτ iq aq ia aq c ia c aq t ib ˆτ p b ib t ib ˆτ q b c iq iq t aq ˆτ i a aq cip t ib + ciq t ib c iq t aq MRCI MRexpT + 3 = ord({t j(µ)}) = ord(q) = 3: State selective + Fermi vacuum invariant + Size consistent (Algebraic proof, numerical check) +/ Core connected (Algebraic proof, numerical check), not valence extensive M. Hanrath, J. Chem. Phys. 3 (005) 840, M. Hanrath, Chem. Phys. Lett. 40 (006) 46 M. Hanrath, Theo. Chem. Acc. accepted Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

11 MRCC Ansätze Genealogy and Working Equations The full graph including products ap t ap ˆτ i a c ip ip t ib ˆτ p b t ib ˆτ p b t ib ˆτ q b ib t ap ˆτ i a t aq ˆτ i a ab c ip t ap t ib + c iq t aq t ib c iq iq t ib ˆτ q b t aq ˆτ i a aq Project Schrödinger equation onto ρ P Q X c µ ρ (Ĥ E)e ˆT (µ) µ = 0 µ = Missing equation from normalization P µ c µ = System of equations linear in c µ, non-linear t α Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, 008 / 50

12 MRCC Ansätze Cluster Operator Perturbation Expansion: Denominators SRCC, SRMRCC ˆT = α e ˆT ˆV e ˆT 0 ˆτ 0 α ɛ 0 ɛ α SUMRCC ˆTλ = α Q(λ) MkMRCC ˆTλ = MRexpT α e ˆT λ ˆV e ˆTλ λ ˆτ λ α ɛ λ ɛ α µ λ α Q(λ) α ˆV λ c λ (E α Ĥα ) ˆτ λ α e ˆT λ e ˆT µ ˆT λ = ϕ(c λ ) c µ [ α e ˆT µ ˆV e ˆTµ µ B µ + C µ ] µ c µ (ɛ µ ɛ α ) α Q(λ) µ P α vanishes core for ˆT λ {}}{ ˆT λ = ϕ(c λ ) c µ [ α e ˆT µ ˆV e ˆTµ µ B µ + C µ ] µ cµ (ɛ µ ɛ α ) ˆτ λ α ˆτ λ α Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, 008 / 50

13 MRCC Ansätze Problems... MkMRCC MRexpT connectivity fully connected core connected only projected Schrödinger eq. within µ Q µ only within Q = µ Q µ active unitary invariance no no problems c µ 0 with yes yes Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

14 MRCC Ansätze Complexity and Cost... α... Ĥ { e ˆT µ ( + Ĉµ) } µ =... µ method bra projection ˆT / Ĉ excitation reference multiplicator SRCC, SRCI Q n MRCI SRMRCC µ Q µ n µ Q µ n + n act SUMRCC BWMRCC MkMRCC MRexpT Q µ, µ n m µ Q µ n m Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

15 Outline The MRCC problem MRCI MRCC Ansätze Applications H 4, ground state, higher excitations P 4, ground state, higher excitations BeH, C v insertion problem, avoided crossing, higher excitations N LiF, potential energy surface, avoided crossing Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

16 H 4, ground state, higher excitations H 4, Higher Excitations System FCI potential energy surface δ = πα δ a.u. E(FCI) in a.u α Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

17 H 4, ground state, higher excitations H 4, SD E(...) E(FCI) in a.u SRCC SRMRCC a a SRMRCC a b SUCC apbwcc BWCC MkCC MRexpT SRMRCC, MRexpT: This work, others: F. A. Evangelista, et al., J. Chem. Phys (006) Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

18 H 4, ground state, higher excitations H 4, SD, close up SRMRCC a a SRMRCC a b MRexpT E(...) E(FCI) in a.u α Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

19 H 4, ground state, higher excitations H 4, SDT E(...) E(FCI) in a.u e SRCC SRMRCC a a SRMRCC a b SUCC apbwcc BWCC MkCC MRexpT SRMRCC, MRexpT: This work, others: F. A. Evangelista, et al., J. Chem. Phys (006) Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, /

20 H 4, ground state, higher excitations H 4, SDT, close up e-05 SRMRCC a a SRMRCC a b MkCC MRexpT E(...) E(FCI) in a.u. 6e-05 4e-05 e e α Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

21 P 4, ground state, higher excitations P 4, Higher Excitations System FCI potential energy surface a.u. α E(FCI) in a.u α Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, 008 / 50

22 P 4, ground state, higher excitations P 4, SD E(...) E(FCI) in a.u SRCC SRMRCC a b SRMRCC a 3u b u SUCC apbwcc BWCC MkCC MRexpT α Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, 008 / 50

23 P 4, ground state, higher excitations P 4, SDT E(...) E(FCI) in a.u. e SRCC SRMRCC a b SRMRCC a 3u b u SUCC apbwcc BWCC MkCC MRexpT SRMRCC, MRexpT: This work, others: F. A. Evangelista, et al., J. Chem. Phys (006).5.5 Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, /

24 P 4, ground state, higher excitations P 4, SDT, close up E(...) E(FCI) in a.u. 5e e SRCC SRMRCC a b 3u SRMRCC a b u SUCC BWCC MkCC MRexpT.8.9 α Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, /

25 BeH, C v insertion problem, avoided crossing, higher excitations BeH, Higher Excitations System FCI potential energy surface -5.6 H Be E(FCI) in a.u H α Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

26 BeH, C v insertion problem, avoided crossing, higher excitations BeH, SD E(...) E(FCI) in a.u SRCC a SRCC a a b a 3a SUCC BWCC MkCC MRexpT x Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

27 BeH, C v insertion problem, avoided crossing, higher excitations BeH, SDT 6 E(...) E(FCI) in a.u SRCC a SRCC a a b a 3a SUCC BWCC MkCC MRexpT x Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

28 BeH, C v insertion problem, avoided crossing, higher excitations BeH, SDTQ 7 8e-06 6e-06 E(...) E(FCI) in a.u. 4e-06 e e-06-4e-06-6e-06-8e-06 SRCC a SRCC a a b a 3a SUCC BWCC MkCC MRexpT.5 Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50.5 x

29 N N, FCI Potential Energy Surface E(FCI) in a.u N( D 0 )+N( D 0 ) N( 4 S 0 )+N( P 0 ) N( 4 S 0 )+N( D 0 ) N( 4 S 0 )+N( 4 S 0 ) Σ + 3 g Σ + 5 u Σ + 7 g Σ + 3 u Π g 5 Π u 3 u 3 Σ u Σ u Π g u Σ u R in a.u Σ + g Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT 3 Σ + 5 u Σ + July 8, /

30 N N, FCI Potential Energy Surface, Avoided Crossing E(FCI) in a.u r in a.u B u 3 B u A u 4 A u B u A u 3 A u Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

31 N N, Table of States State D h irrep Dominating determinant a X Σ + g A g... 3σg πux πuy 3 Σ + u B u... 3σg π ux πuy πgy 5 Σ + g A g... 3σg π ux π uy π gxπ gy 7 Σ + u B u... 3 σ g π ux π uy 3σ uπ gxπ gy 3 Π g B g (B 3g)... 3 σ g πux πuy π gx 5 Π u { B u (B 3u)... 3 σ g π 3 B u... 3σg πux π uy π gy u A u... 3σg πux π uy π gx ux πuy πgxπgy 3 Σ u A u... 3σ g π ux π uy π gx Σ u A u... 3σ π π uy π gx g ux Π g B g (B 3g) {... 3 σ g πux π uy πgx u { B u A u... 3σg πux π uy π gy b {... 3 σ g π uxπ uy3 σ uπ gxπ gy c... 3σg πux π uy π gx b {... 3 σ gπux 3 σuπgxπgy c Σ u A u... 3σg πux π gx πgy e... πux π uyπ gxπgy d a core orbitals σg σg σu σu... R < 4.5 bohr R 4.5 bohr R < 3.5 bohr R 3.5 bohr Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

32 N Energy Differences 0.00 MRCI MRexpT SRMRCC R in a.u. R in a.u. R in a.u. A. Engels-Putzka, M. Hanrath, submitted to J. Chem. Phys. Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

33 N Benchmarking wavefunction properties at full exp. level MRCC: Usually energetic benchmarks only Wavefunction benchmarks missing (important for many properties) Implementation now capable of full exponential function evaluation (up to algebraic termination) Application to Spin projection errors State overlap errors Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

34 N Spin projection errors At non-relativistic level it is [Ĥ, Ŝ ] = 0 we should get spin eigenfunctions Possible choices for the cluster operator substitutions Spin averaged Ê + [Ê, Ŝ ] = 0 Lacks to span spin space when applied to open shell refs Possible solution: Overlapping annihilator and creator sets problem: Non-commuting algebra Spin specific ê + Spans spin space + Commutes [Ê, Ŝ ] 0 possible spin contamination At untruncated cluster limit: FCI (spin adapted) How to check for spin contamination? Expectation value: Ψ CC Ŝ Ψ CC = S( S + ) S = S not sufficient for Ψ CC to be spin eigenfunction Projection onto CSFs: CSFs S Ψ CC = δ δ = 0 Ψ CC is spin eigenfunction Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

35 N Spin projection errors Space SOMRCI MRexpT SRMRCC 0. Ref. exact exact e-04 e MRSD A. Engels-Putzka, M. Hanrath, submitted to J. Chem. Phys. e-04 e-04 e-04 e e e exp e-04 e-04 e e Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

36 N State Overlaps Hermitian operators non-degenerate eigenvalues } orthogonal eigenfunctions In case of CI: trivial (orthogonality from linear algebra) CC: Not trivial Only reason for orthogonality with CC: Ψ CC should be approximation of eigenfunction Attention: Σ u, Σ u Same D h irrep Same multiplicity Same dissociation channel degenerate eigenvalues for R Σ u Σ u arbitrary Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

37 N State Overlaps MRexpT SRMRCC States R = 3 a.u. R = 0 a.u. R = 3 a.u. R = 0 a.u. 5 Σ + g X Σ + g Σ + u u u u u 3 Σ + u Σ + u u Σ + u 3 Σ + u Σ + u 3 u Π g Π g u Σ u Σ u Σ u Σ u u u Σ u u u u Σ u Σ u Σ u Σ u u Σ u Σ u Σ u 3 u A. Engels-Putzka, M. Hanrath, submitted to J. Chem. Phys. Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

38 N State Overlaps MRexpT SRMRCC MRexpT state overlap Ψi Ψj, i j e-05 e-0 e-5 SRMRCC state overlap Ψi Ψj, i j e-05 e-0 e-5 e R e-0 Overlaps within same D h irrep shown only MRexpT overlaps very good with few exceptions SRMRCC worse, especially for R, no cure by spin adaption possible R Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

39 N N, -State degeneracy errors e-05 MRexpT state degeneracy error in a.u. 0 -e-05-4e-05-6e-05-8e R in a.u. 7 u 3 u Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

40 N N, Incomplete Model Space (closed shell refs only) E(...) E(FCI) in a.u R in a.u MRCI MRexpT SRMRCC Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

41 LiF, potential energy surface, avoided crossing LiF: Ground/excited state CAS avg. Orbitals E(FCI) in a.u R in a.u X Σ + Σ + 3 Σ + Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

42 LiF, potential energy surface, avoided crossing LiF: Ground/excited state, avoided crossing region CAS avg. Orbitals E(FCI) in a.u X Σ +, FCI X Σ +, MRCI X Σ +, SRMRCC X Σ +, MRexpT Σ +, FCI Σ +, MRCI Σ +, SRMRCC Σ +, MRexpT R in a.u Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

43 LiF, potential energy surface, avoided crossing LiF: Errors with respect to FCI CAS avg. Orbitals E(FCI) E(...) in a.u X Σ +, MRCI X Σ +, SRMRCC X Σ +, MRexpT Σ +, MRCI Σ +, SRMRCC Σ +, MRexpT R in a.u Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

44 LiF, potential energy surface, avoided crossing LiF: Dipole Moments CAS avg. Orbitals µ in a.u R in a.u. 3 4 X Σ +, FCI X Σ +, MRCI X Σ +, SRMRCC X Σ +, MRexpT Σ +, FCI Σ +, MRCI Σ +, SRMRCC Σ +, MRexpT X Σ +, FCI X Σ +, MRCI Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT + July 8, / 50

45 LiF, potential energy surface, avoided crossing Reference Coefficients in Crossing Region CAS avg. Orbitals SOMRCI MRexpT FCI c σ π σ 5σ π r in a.u r in a.u r in a.u Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

46 LiF, potential energy surface, avoided crossing LiF: Ground/excited state, avoided crossing region CAS avg. Orbitals E(FCI) in a.u X Σ +, FCI X Σ +, MRCI X Σ +, SRMRCC X Σ +, MRexpT Σ +, FCI Σ +, MRCI Σ +, SRMRCC Σ +, MRexpT R in a.u..5 Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

47 LiF, potential energy surface, avoided crossing LiF: Ground/excited state, avoided crossing region SCF Orbitals E(FCI) in a.u X Σ +, FCI X Σ +, MRCI X Σ +, SRMRCC X Σ +, MRexpT Σ +, FCI Σ +, MRCI Σ +, SRMRCC Σ +, MRexpT R in a.u Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

48 LiF, potential energy surface, avoided crossing LiF: Errors with respect to FCI SCF Orbitals E(FCI) E(...) in a.u X Σ +, MRCI X Σ +, SRMRCC X Σ +, MRexpT Σ +, MRCI Σ +, SRMRCC Σ +, MRexpT R in a.u Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

49 . Conclusion No ultimate MRCC so far Major bugs: SRMRCC Symmetry broken FSMRCC Closed shell parent state mandatory SUMRCC Intruder state problems BWMRCC Not extensive MkMRCC Solves projected Schrödinger equation within µ Q µ only MRexpT Core connected only MRexpT: Efficient and general algebraic engine Full exponential wavefunction analysis carried out MRexpT energetically as accurate as SRMRCC but no inherent symmetry breaking Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

50 . Acknowledgements Anna Engels-Putzka: Term simplification, matrix contraction DFG SPP 45 Grant HA 56/- Michael Hanrath, Univ. of Cologne, Theor. Chemistry Advances MRexpT July 8, / 50

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