Coupled-Cluster Approaches and Quasi-degeneracy: An Historical Slice

Transcription

1 INT Symposium 50 Years of Coupled-Cluster Theory University of Washington, Seattle June 30 July 2, 2008 Coupled-Cluster Approaches and Quasi-degeneracy: An Historical Slice Josef Paldus and Xiangzhu Li Department of Applied Mathematics University of Waterloo Supported by NSERC

2 Outline Introduction A brief remark on the beginnings Quasidegeneracy, correlation types, SR vs MR approaches A road to RMR-type methods, externally vs internally corrected CCSD Single reference (SR) approaches: RMR CCSD, RMR CCSD(T) and plmr CCSD Examples: Triple-bond breaking: N 2 S-T splitting in diradicals: C 2, BN Symmetry breaking: N 3,BNB Reaction barriers, transition metal complexes, nickel carbonyls, pyridynes,etc. Multi-reference (MR) approaches : C-conditions - GMS SU CCSD NR CCSD, NR CCSD(T), and an ec version: (M,N)-CCSD Some examples (time permitting) Conclusions

3 Roy Mc Weeny 1967 Inaugural Lecture at the University of Sheffield: believe it or not, it is taken from one of the Chemistry Journals! In: A Link between Disciplines (University of Sheffield, 1967, pp. 8-9)

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6 BH 3 MBS μH μh

7 Outline Introduction A brief remark on the beginnings Quasidegeneracy, correlation types, SR vs MR approaches A road to RMR-type methods, externally vs internally corrected CCSD Single reference (SR) approaches: RMR CCSD, RMR CCSD(T) and plmr CCSD Examples: Triple-bond breaking: N 2 S-T splitting in diradicals: C 2, BN Symmetry breaking: N 3,BNB Reaction barriers, transition metal complexes, nickel carbonyls, pyridynes,etc. Multi-reference (MR) approaches : C-conditions - GMS SU CCSD NR CCSD, NR CCSD(T), and an ec version: (M,N)-CCSD Some examples (time permitting) Conclusions

8 Basic types of correlation effects Correlation type: Due to: Well described by: Dynamic Large number of excited states CC Non-dynamic Static Quasi-degeneracy of references Degeneracy of reference configurations CI

9 The role of CC amplitudes SR formalism: T 1 secondary unless non-hf MOs employed T 1 = 0 for Brueckner (max overlap) orbitals Their import growth with increasing quasi-degeneracy T 2 T 3 T 4 the most important basic approx ns CCD and CCSD n 6 generally small, but required to achive chemical accuracy of 1 kcal/mol CCSDT CCSD(T) the gold standard of QC n 8 very demanding computationally CCSDTQ, CCSD(TQ), etc. n 10 T k general scaling: n ok n v (k+2) ~ n 2(k+1) MR formalism: MR CCSD VU, SU, and SS Generally: much theory but few applications currently, most effort goes into the SS methods Grau, teurer Freund, ist alle Theorie, und grün des Lebens goldner Baum in MR CC: Theories are green, but praxis is grey

10 Internally vs externally corrected CC approaches Internally corrected (ic): fully based on CC formalism, in general fully size-extensive, but not necessarily size-consistent examples: CCSD(T), R-CCSD(T), CR-CCSD(T), CR-CC(2,n), MkMRCC, plmr CCSD, etc. Externally corrected (ec): exploit some external source of information about higher-than-pair clusters, may slightly violate size-extensivity, but size-consistent with a proper choice of the external source, e.g., VB, CAS SCF, MR CISD, etc. examples: RMR CCSD, RMR CCSD(T), (M,N)-CCSD, etc.

11 General scheme of CC methods SR CCD, CCSD, CCSDT, SR formalism UGA CCSD CCSDT Q ec CCSD RMR-CCSD SS VU MR Genuine MR MR formalism VU SU SU BW GMS SU CCSD (M,N)-CCSD

12 Outline Introduction A brief remark on the beginnings Quasidegeneracy, correlation types, SR vs MR approaches A road to RMR-type methods, externally vs internally corrected CCSD Single reference (SR) approaches: RMR CCSD, RMR CCSD(T) and plmr CCSD Examples: Triple-bond breaking: N 2 S-T splitting in diradicals: C 2, BN Symmetry breaking: N 3,BNB Reaction barriers, transition metal complexes, nickel carbonyls, pyridynes,etc. Multi-reference (MR) approaches : C-conditions - GMS SU CCSD NR CCSD, NR CCSD(T), and an ec version: (M,N)-CCSD Some examples (time permitting) Conclusions

13 A (very incomplete) history of the road to RMR CCSD ~1970 UHF J. Čížek + JP ACP ACP-D45 ACCD ACPQ MR ACPQ ecccsd CASCC ecccsd RMR CCSD B.G. Adams + K. Jankowski + JP K. Jankowski + JP (also H4, P4, etc.) R.A. Chiles + C.E. Dykstra JP + M. Takahashi + J. Čížek + R.W.H. Cho P. Piecuch + JP JP + J. Planelles + X. Li L.Z. Stolarczyk P. Piecuch, R. Tobola + JP X. Li + G. Peris + J. Planelles + JP X. Li + JP X. Li + JP most recent results present talk TCCSD ncc recent related developments: T. Kinoshita + O. Hino + G.K.-L. Chan + R.J. Bartlett R.J. Bartlett + M. Musiał ACP, ACP-D45, ACCD 2CC CAS-SCF or CAS-FCI based ecccsd (for active MOs)

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15 ACP-D45

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17 1, ACP ACP-D45 ACCD ACPQ FCP, FCCD BG Adams + K Jankowski + JP, CPL 67, 144 (1979) K Jankowski + JP, IJQC 18, 1243 (1980) RA Chiles + CE Dykstra, CPL 80, 69 (1981) JP+ J Čížek + M. Takahashi, PRA 30, 2193 (1984) B. Jeziorski et al. JCP 81, 368 (1984)

18 Cyclic polyene π-electron model C N H N N = 2n = 4ν + 2 PPP Hubbard 2β HF Brueckner MOs Note: CCD CCSD

19 Percentage error in correlation energy as a function of the resonance (hopping) integral β for cyclic polyenes C N H N PPP Hubbard

20 Total correlation energy Δε as a function of β for C 22 H 22 L-CCD β crit

21 Outline Introduction A brief remark on the beginnings Quasidegeneracy, correlation types, SR vs MR approaches A road to RMR-type methods, externally vs internally corrected CCSD Single reference (SR) approaches: RMR CCSD, RMR CCSD(T) and plmr CCSD Examples: Triple-bond breaking: N 2 S-T splitting in diradicals: C 2, BN Symmetry breaking: N 3,BNB Reaction barriers, transition metal complexes, nickel carbonyls, pyridynes,etc. Multi-reference (MR) approaches : C-conditions - GMS SU CCSD NR CCSD, NR CCSD(T), and an ec version: (M,N)-CCSD Some examples (time permitting) Conclusions

22 Standard Single-Reference (SR) CC approach Wave function cluster Ansatz: Energy expression: Amplitude equations: For CCSD : CCSDT : etc.

23 Motivation for ec CCSD T 1 and T 2 clusters fully determine the energy ΔE = Φ 0 H ( T 1 + T 2 + ½T 12 ) Φ 0 Truncating the full CC chain of equations by setting T 3 = T 4 = 0 we obtain CCSD equations that determine CCSD T 1 and T 2 Φ j (1) H [ 1 + T 1 + T 2 + T 1 T 2 + T 12 /2 + T 13 /6 + T 3 ] Φ 0 C = 0 CCSD Φ j (2) H [ 1 + T 1 + T T 14 /24 + T 3 + T 1 T 3 + T 4 ] Φ 0 C = 0 Φ j (3) H [ T 1 + T T 3 + T 1 T 3 + T T 5 ] Φ 0 C = 0 etc. We can achieve a physically more appropriate decoupling by choosing a good approximation for T 3 and T 4. When we use exact T 3 and T 4, we recover the exact FCI energy!

24 Basic tenets of ec CCSD E = E (T 1, T 2 ) Complementarity of CI and CC: MR CI : static and nondynamic correlations SR CC : dynamic correlation Simple relationship between CI and CC Ansätze C n = T n + T 1 T n-1 +. and an easy transformation: MR CISD SR CISDTQ T 3 and T 4 extracted from MR CISD automatically account for T 5, T 6, etc. A modest size MR CISD involves a very small number of t 3 and t 4 amplitudes In the limit FCI FCC

25 What is an optimal source for external corrections (ec)? Desirable attributes size consistency universality affordability Preferred choice: modest size MR CISD Earlier choices UHF, PUHF VB CAS SCF CAS FCI Motivation: Complementarity of CI and CC Size-extensivity: A proper scaling with a particle number N as N When AB A + B, then E(A+B) = E(A) + E(B) Size-consistency: Adequate reference space enabling a proper description of the studied dissociation process, i.e., of the inherent degeneracy or quasi-degeneracy N 2

26 ecccsd [RMR CCSD and RMR CCSD(T)] Cluster Ansatz : Steps: from an external source (for RMR CCSD we use a modest size MR CISD) (i) MR CISD SR CISDTQ (ii) Use cluster analysis to extract terms (iii) Compute and terms and correct an absolute term in CCSD equations (iv) Solve standard CCSD equations and compute the energy RMR CCSD(T) : amplitudes that are not involved in are corrected in a standard perturbative way

27 Basic strategies: Comprehensive choice: based on the CAS concept Simple & straightforward: implied by the choice of active orbitals (A ve Os) but 1) A ve Os are not necessarily frontier orbitals, and 2) Dimension of the model space rapidly increase with the number of A ve Os Pragmatic choice: based on a suitable test function for RMR CCSD(T) we use SR CCSD amplitudes: Choose a threshold T (usually 0.1 or 0.05) if all t i < T SR CCSD case if a given t i T include it in a model space

28 Summary of RMR-type methods SR CI S D T Q H ,6, SR CCSD SR CCSDT MR CISD REF 1 st order IS SR cluster analysis of MR CISD 0 S D T Q H RMR CCSD RMR CCSD(T) (T) tr d MR CISD SR cluster analysis of tr d MR CISD REF IS T Q H tr d RMR CCSD tr d RMR CCSD(T) (T )

29 Outline Introduction A brief remark on the beginnings Quasidegeneracy, correlation types, SR vs MR approaches A road to RMR-type methods, externally vs internally corrected CCSD Single reference (SR) approaches: RMR CCSD, RMR CCSD(T) and plmr CCSD Examples: Triple-bond breaking: N 2 S-T splitting in diradicals: C 2, BN Symmetry breaking: N 3,BNB Reaction barriers, transition metal complexes, nickel carbonyls, pyridynes,etc. Multi-reference (MR) approaches : C-conditions - GMS SU CCSD NR CCSD, NR CCSD(T), and an ec version: (M,N)-CCSD Some examples (time permitting) Conclusions

30 2

31 Model (reference) spaces for N 2 b 1u b 2u, b 3u σ π x *, π y antibonding b 2g, b 3g a 1g π x, π y σ bonding Φ 0 = Φ 1 = σ 2 π x2 π y 2 σ 2 π y2 π 2 x Φ σ 2 π 2 2 = x Φ 3 = σ 2 π 2 x π 2 y π 2 y Φ 4 = Φ 5 = Φ 6 = Φ 7 = π x2 π y 2 π y2 π 2 x π 2π 2 x y σ 2 σ 2 σ 2 π 2 σ 2 x π 2 y (6 electrons /6 orbitals) space spanned by determinants with M S = 0 and belonging to the a 1g symmetry species 4R 8R 56R

32 E (a.u.) E(a.u.) R RMR(T) N FCI CCSD(T) DZ basis R RMR CCSD(T) CR-CC(2,3) CCSD R R(a.u.)

33 N 2 cc-pvtz basis RKR 8R RMR SR CCSD experiment (RKR) and 8R RMR CCSD

34 Experiment-based and theoretical vibrational term values for 14 N 2 isotope (in cm -1, relative to Anal. ) R.J. LeRoy et al., JCP 125, (2006) cc-pvqz cc-pvtz cc-pvqz cbs

35 3-parameter vs 2-parameter extrapolation

36 Experimentally-determined and theoretical PECs for the ground state of N 2-1 E(cm ) RKR levels Expt. 56R RMR 56R RMR(T) 8R RMR n=19 n=15 8R RMR(T) CCSD(T) o R(A)

37 Analytic PECs for N 2 in the critical region of highly-stretched geometries vs computed ab initio values (relative to dissociation energy D e ) 0 E(cm -1) EMO 2 (6) MLJ 1(6) EMO (7) 3 RMR/cc-pVQZ RMR/cbs RMR(T)/cc-pVQZ MLJ (7) 2 RMR(T)/cbs MLJ (8) 3 MLR (8) MLR (6,8) 4 EMO Expanded Morse oscillator MLJ Morse/Lennard-Jones MLR Morse/long range o R(A) LeRoy et al., JCP 125 (2006)

38 Theoretical and experimental spectroscopic parameters Y kl for the ground state of the 14 N 2 molecule Most recent value (2005) f Accounting for the core effect All values in cm -1 except for R e (in Å) and the dissociation energy D e (in ev)

39 Dependence of basic equilibrium spectroscopic constants for the ground state of 14 N 2 on the size of the basis set

40 2 and

41 C 2 G. Herzberg Molecular Spectra and Molecular Structure. I. Spectra of Diatomic Molecules, p a 1 Σ g + X 3 Π u

42 R e and ω e for the X 1 Σ and a 3 Π states of C 2 A-type RMR: 4R for singlet, SR for triplet [ as implied by (2,2) active space] B-type RMR: 4R for singlet, 2R for triplet

43 R e and ω e for the X 3 Π and a 1 Σ states of BN ACPF 461 A-type RMR: 4R for singlet, SR for triplet [ as implied by (2,2) active space] B-type RMR: 4R for singlet, 2R for triplet

44 BN T e (shifted) = T e T e (cc-pv5z) T (shifted) (cm -1 e ) T e (shifted) (cm -1 ) Shifted singlet-triplet splittings T e (shifted) relative to their value for the cc-pv5z basis CCSD RMRCCSD,A RMRCCSD,B CCSD(T) CR-CC(2,3) RMRCCSD(T),A RMRCCSD(T),B CR-CC(2,4),A cc-pvxz cc-pvxz

45 3

46 Cuts of the ROHF and UHF ground state PES of N 3 R (Å) symmetric stretch R 1 = R e + R R 2 = R e + R asymmetric stretch R 1 = R e + Δ R 2 = R e Δ Δ (Å)

47 Equilibrium bond length R e (in Å) and fundamental vibrational frequencies (in cm -1 ) for the GS of N 3 with cc-pvtz basis set Method R e ν 1 ν 2 (y) ν 2 (x) ν 3 CCSD UCCSD R RMR CCSD CCSD(T) UCCSD(T) R RMR CCSD(T) Exp

48 PECs for the ground state of N 3 along the asymmetric stretching coordinate Δ E (cm -1 ) CCSD v = 4 CCSD(T) Harmonic PEC v = 3 v = 2 v = 1 Harmonic PEC corresponds to experimental asymmetric stretching frequency of cm -1 v = 0 Δ (Å)

49 PECs for the ground state of N 3 along the asymmetric stretching coordinate Δ v = 4 E (cm -1 ) v = 3 UCCSD v = 2 v = 1 UCCSD(T) v = 0 Δ (Å)

50 Vibrational energies ΔE(n 3 ) (in cm -1 ) relative to E(0) [given in parentheses] for asymmetric stretching mode ν 3 of the GS of N 3 with cc-pvtz basis n 3 CCSD CCSD(T) UCCSD UCCSD(T) (786) (949) (887) (885) RMR CCSD 0 (836) RMR CCSD(T) 0 (974) In UHF-based potentials the singularity was replaced by a constant value

51 Vibrational energies (in cm -1 ) for the first 5 levels of the asymmetric stretching mode ν 3 in the GS of N 3 with cc-pvqz basis E(n 3 ) E(0) E(n 3 ) E(n 3 1) n 3 CCSD CCSD(T) Exp. CCSD CCSD(T) 0 0 (792) 0 (947) E(0) is given in parentheses

52 Vibrational energies ΔE(n 3 ) = E(n 3 ) E(0) for the symmetric stretching mode of N 3 obtained with cc-pvtz basis n 3 CCSD CCSD(T) RMR CCSD RMR CCSD(T) Exp. 0 0 (696) 0 (662) 0 (688) 0 (661) ΔE E(0) given in parentheses ΔE = (1/5) m = 1 [E(m) E(m -1)]

53 Typical HF PESs for linear ABA species (a) (b) (c) SA BS R symmetric stretching mode Δ asymmetric stretching mode (a) UHF solution for N 3 (b) ROHF solution for N 3 (c) BNB case: SA - symmetry adapted ROHF solution BS - broken symmetry ROHF solution

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55 BNB ROHF UHF 100 mh 50 mh FCI 10 mh

56 Energy differences ΔE (in mhartree) between the BNB ground state having linear symmetric (R 1 = R 2 = 1.34 Å) and linear asymmetric (R 1 = 1.26, R 2 = 1.42 Å) geometries as obtained with various methods and STO-3G basis set a Iterative and b non-iterative account of the T 1 T 3 term

57 PECs for antisymmetric stretching mode of the BNB ground state E (cm 1 ) 2R, cc-pvdz 2R, cc-pvtz 4R, cc-pvtz 2R, cc-pvqz Δ (Å) Δ = ½ (R 1 R 2 )

58 Equilibrium bond lengths R e, R 1e, and R 2e (in Å) and ground state energies E (in a.u) for the symmetric and asymmetric structures of BNB basis R e E+104 R 1e R 2e E+104 ΔE cc-pcvdz cc-pvdz cc-pvtz cc-pvqz R RMR CCSD(T) method and cationic QRHF MOs used throughout All electrons were correlated for the cc-pcvdz basis; otherwise only valence electrons were correlated Symmetry-breaking effects ΔE are in cm 1

59 Harmonic frequencies (in cm 1 ) for the symmetric stretching, bending, and antisymmetric stretching modes of BNB frequency cc-pvdz cc-pvtz cc-pvqz experiment v 1 (symm c stretch) v 2 (bend) v 3 (antisymm c stretch) a, 1143±40 b a, 855±40 b 2R RMR CCSD(T) method with cc-pvxz (X = D, T, and Q) basis sets a From L. Andrews et al., J. Chem. Phys. 98, 922 (1993), where these values were assigned to the stretching fundamentals of the cyclic B 2 N radical with the C 2v symmetry. These assignments were revised in Ref. b for the linear BNB geometry. b From K. R. Asmis et al., J. Chem. Phys. 111, 8838 (1999).

60 PECs along the antisymmetric stretching coordinate Δ as obtained with 2R RMR CCSD(T) cc-pvqz R 1 = R 1e + Δ R 2 = R 2e Δ E (cm 1 ) cc-pvdz cc-pvdz cc-pvqz R 1e R 2e Δ (Å)

61 Calculated and experimentally determined vibrational energies ΔE(ν) relative to the zero point energy E(0) [given in parentheses] for the first seven levels of the antisymmetric stretching mode ν 3 of BNB BD(T)/cc-pVDZ a experiment b RMR CCSD(T) c v A B IR(mtx) PES cc-pvdz cc-pvqz (332) (497) (?) d (?) d (±40 cm 1 ) 0 (353) (370) a Based on a BD(T)/cc-pVDZ potential whose singular behavior at Q 3 =0 was replaced by a constant potential interconnecting two broken-symmetry minima (case A) or via a polynomial interpolation resulting in a central hump (case B) see Asmis et al. paper. b Matrix IR spectrum (as interpreted by Asmis et al.) and anion photoelectron spectrum. c Computed with the LEVEL codes using RMR CCSD(T) potentials (see figure). d Question mark indicates an uncertain assignment.

62 Reaction barriers

63 Forward and reverse barrier heights (in kcal/mol) for heavy atom transfer reactions reaction CCSD RMR CCSD CCSD(T) RMR CCSD(T) best estimate

64 Forward and reverse barrier heights (in kcal/mol) for nucleophilic substitution reactions reaction CCSD RMR CCSD CCSD(T) RMR CCSD(T) best estimate

65 Forward and reverse barrier heights (in kcal/mol) for unimolecular and association reactions reaction CCSD RMR CCSD CCSD(T) RMR CCSD(T) best estimate

66 Effect of external corrections at the standard and triple-corrected levels system reference number config n type ΔE 1 ΔE 2 ΔE 1 = E CCSD(T) E RMR CCSD(T) ΔE 2 = E CCSD E RMR CCSD (in kcal/mol) TS = transition state

67 MR effect for BHs (in kcal/mol) reaction ΔBH(SD) a ΔBH[SD(T)] b V f V r V f V r a Difference between the RMR CCSD and CCSD BHs b Difference between the RMR CCSD(T) and CCSD(T) BHs

68 Modified final estimate of BHs reaction V f V r reaction V f V r Same as in Zhao + Gonzales-Garcia + Truhlar, J. Phys. Chem. A 109, 2012 (2005)

69 DZ model for reaction H + F 2 HF + F CCSD(T) RMR CCSD(T) FCI freezing 1s and 2s electrons on F Energies in a.u. and barrier heights in kcal/mol Single reference for HF and F, and four references for F 2 and HF 2 (TS)

70 Binding in transition metal compounds

71 Binding energies of transition metal compounds MCH 2 + (kcal/mol) (No corrections for zero-point energy, spin-orbital effects and basis set superposition error) System Corr d. El s. CCSD RMR CCSD CCSD(T) RMR CCSD(T) Exp. ScCH 2 + A B ± 5.3 TiCH 2 + A B ± 3.5 CrCH 2 + A B ± 1.9 FeCH 2 + A B ± 4.0 Correlated electrons: A : 3d, 4s B : 3s, 3p, 3d, 4s Basis: Wachters + f [8s6p4d1f] for M; cc-pvtz for C and H Note: For Sc and Ti, 3s and 3p els significantly contribute, not so for other M; for Sc, Ti, V, Cr, and Mn RMR CCSD(T) > CCSD(T); for Fe, Co, Ni, and Cu, this is other way around.

72 Binding energies of transition metal compounds MCH 2 + (kcal/mol) (Continued) System Corr d. El s. CCSD RMR CCSD CCSD(T) RMR CCSD(T) Exp. : VCH 2 + MnCH 2 + CoCH 2 + CuCH 2 + NiCH 2 + A B A B A B A B A B Correlated electrons: ± ± ± ± ± 1.8 A : 3d, 4s B : 3s, 3p, 3d, 4s Basis: Wachters + f [8s6p4d1f] for M; cc-pvtz for C and H

73 Nickel carbonyls

74 Structure and binding in Ni(CO) n (n=1,2,4) Binding energies (kcal/mol) n=1 n=2 n=4 CCSD CCSD(T) R(4R) RMR CCSD(T) Exp. 29±15 83± ±42 35± ± ± ±1.2

75 Geometry of NiCO (all values in Å) Method R(Ni C) R(C O) scaled R(C O) CCSD CCSD(T) R RMR CCSD(T) R RMR CCSD(T) For CO CCSD(T)/SVP CCSD(T)/cc-pVQZ R RMR CCSD(T) Exp scaling factor: 0.979

76 Geometry of Ni(CO) 2 (R in Å) C Ni C Method angle ( ) R(Ni C) R(C O) CCSD CCSD(T) R RMR CCSD(T) Exp. ~145 Ni(CO) 2 was thought to be linear. Recent experimental data, as well as RMR CCSD(T) and CCSD(T) results, show that it is bent: Computed angle is 153º; rough experimental value is 145º

77 Geometry of Ni(CO) 4 ( in Å) Method CCSD(T) t-4r RMR CCSD(T) Exp. R (Ni C) ± ±0.002 Assumptions: tetrahedral geometry frozen C O bond length at Å Leading reference CSFs are (t i ) 2 (t i *) 2, i = 1,2,3 and their 4R RMR amplitudes are about 0.06 strong SR character

78 MR CC methods

79 Multi-Reference Coupled Cluster Methods Valence Universal or Fock space (VU) State Universal or Hilbert space (SU) State Selective or State Specific (SS)

80 Multi-Reference Coupled Cluster Methods Valence Universal or Fock space (VU) State Universal or Hilbert space (SU) State Selective or State Specific (SS) General Model Space version General Model Space version

81 C-conditions: General case If the action of an excitation operator on a given reference configuration produces another reference, then the principal cluster amplitude that is associated with such an internal excitation operator is required to cancel the contributions from products of cluster amplitudes arising from all possible disconnected clusters that are associated with the given principle cluster component. These constraints are referred to as the C-conditions, since they ensure the connectedness of the GMS-based SU CC theory and the consistency of the SU CC Ansatz with the FCI expansion.

82 Generalization of RMR CCSD to MR SU CCSD RMR CCSD: with T 3 (0) and T 4 (0) from NR MR CCSD (N,M) SU CCSD: Special cases: T 3 (0) (i) and T 4 (0) (i) clusters from M NR MR CISD wave functions are employed in MR SU CCSD, where N M. (0,M) CCSD MR SU CCSD (N,1) CCSD NR RMR CCSD (M,M) CCSD; i.e. M = N, if no serious intruders are present In general: N > M

83 Some ad hoc MR CCSD results

84 Energy differences relative to a large CI (i.e., CISDTPQP) for the GS (1 st row, in mh) and for VEEs (in ev) for the 1,3 A 1 states of H 2 O as obtained with a cc-pvdz basis set

85 VEEs (in ev) for the 1,3 A 2 states of H 2 O as obtained with the cc-pvdz basis set. CI CISDTPQ, XR SU XR SU CCSD a) All SU VEEs were calculated relative to the 9R SU CCSD GS b) The 6R SU CCSD results could not be converged to a higher accuracy

86 A schematic representation of low-lying PECs of H 2 O

87 C 1 B 1 X 1 A 1 transition in water

88 Comparison of GMS SU CCSD (SUCC) and MR CI (TPBP) geometries, total energies, and equilibrium (T e ) as well as experimental 0-0 (T 0 ) singlet transition energies of water *) TPBP G. Theodorakopoulos et al., CPL 105, 253 (1984). GH G. Herzberg, Electronic Spectra of Polyatomic Molecules *) Corresponds to the 1 st diffuse band; origin may be much lower!

89 Energy differences relative to FCI for the GS (1 st row, in mh) and for VEEs (in ev) for the 1,3 Σ and 1,3 Δ states of CH

90 VEEs (in ev) of the 2,4 B 1 states of NH 2 as obtained with the cc-pvdz basis set The GS energies are reported as E+55. The 1R SU CCSD ( SR CCSD) energy equals a.u.

91 VEEs (in ev) for the N 2 molecule (relative to the SR CCSD ground state)

92 VEEs (ev) for the low-lying states of N 2 (cc-pvtz basis) notation as in previous slide

93 VEEs (ev) for the low-lying states of N 2 (aug-cc-pvtz basis) A u component is used for Δ states

94 VEEs (in ev) for the CO molecule (relative to the SR CCSD ground state)

95 VEEs (ev) for the low-lying states of CO (cc-pvtz basis) a) Relative to 5R CCSD GS energy of a.u. b) Relative to (5,5)-CCSD GS energy of a.u. c) Relative to 5R CCSD(T) GS energy of a.u.

96 VEEs (ev) for the low-lying states of CO (aug-cc-pvtz basis) a) Relative to 5R CCSD GS energy of a.u. b) Relative to (5,5)-CCSD GS energy of a.u. c) Relative to 5R CCSD(T) GS energy of a.u. d) EOM-D EOM-CCSD; EOM-T EOM-CCSDT (Jorgensen et.al.)

97 VEEs (in ev) for the low-lying states of C 2 A cc-pvtz; B aug-cc-pvqz; C aug*-cc-pvdz NR NR SU CCSD; NR(T) NR SU CCSD(T)

98 Conclusions ec CCSD methods Combine the advantages of CC and CI (or CAS SCF, etc.) Helps to overcome quasi-degeneracy in SR [RMR CCSD] and intruders in MR CC approaches [(M,N)-CCSD] Gives highly accurate potentials by accounting for a small but important subset of triples and quadruples (T)-corrected methods: RMR CCSD(T) and NR CCSD(T) Account for the remaining triples (dynamic correlation) More accurate than CCSD(T) or CR CCSD(T) for all geometries Useful for highly accurate potentials For excitation energies: a right balance in both states is desirable Computationally no more demanding than standard CCSD(T)

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101 Thank you!

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