Coupled-Cluster Theory. Jürgen Gauss

Transcription

1 Coupled-Cluster Theory Jürgen Gauss

2 Electron Correlation, Size Consistency, and Exponential Ansatz

3 Electron Correlation: the Helium Atom nucleus electron 2 correlation effect wave function Ψ exact (r 1,r 2 ) as function of r 1 figure from T. Helgaker, P. Jørgensen, J. Olsen Molecular Electronic-Structure Theory

4 Electron Correlation: the Helium Atom nucleus electron 2 Coulomb hole figure from T. Helgaker, P. Jørgensen, J. Olsen Molecular Electronic-Structure Theory

5 Magnitude of Correlation Energies H 2 O HCN E HF ΔE corr ΔE rel correlation energies typically < 1% of the total energies

6 Chemical Relevance of Electron Correlation Hartree ˆ= kj/mol correlation energies strongly dependent on valence electrons correlation effects always important when bonds are broken

7 Chemical Relevance of Electron Correlation example: CO à C( 3 P) + O( 3 P) dissocation energy (D e ) of CO C O CO D e in a.u. E(HF) E(corr) E(total)

8 Chemical Relevance of Electron Correlation example: CO à C( 3 P) + O( 3 P) dissocation energy (D e ) of CO C O CO D e in kj/mol E(HF) E(corr) E(total) electron-correlation effects are significant

9 Size Consistency non-interacting systems A and B: A r AB B E AB = E A + E B additivity of energy multiplicability of wavefunction

10 Size Consistency A r AB B A method is termed size consistent if the sum of energies computed for two non-interacting subsystems A and B is equal to the energy obtained for the supersystem consisting of both A and B a) individual quantum-chemical calculations for A and B à E A + E B b) one quantum-chemical calculations for A+B à E AB size consistency: E AB = E A + E B

11 Multiplicative Property of the Wavefunction non-interacting systems A and B => separation ansatz Ĥ AB = ĤA + ĤB E AB = E A + E B AB = Â A B additive multiplicative antisymmetrizer

12 Multiplicative Property of the Wavefunction H 2 molecule, minimal basis CID = HF + c D (not normalized) = (1 + ˆD) HF HF HF determinant D double excitation operator, generates a by c weighted double excitations H 2 : CID = FCI = exact solution

13 Multiplicative Property of the Wavefunction two H 2 molecule, minimal basis CID wavefunction H 2 (A) + H 2 (B) CID = (1 + ˆD,A + ˆD,B ) HF but  CID,A CID,B = Â{(1 + ˆD,A) HF,A} {(1 + ˆD,B ) HF,B} =  (1 + ˆD,A) (1 + ˆD,B ) HF,A HF,B = (1 + ˆD,A ) (1 + ˆD,B ) HF = (1 + ˆD,A + ˆD,B + ˆD,A ˆD,B ) HF quadruple excitation, missing in CID => problem

14 Multiplicative Property of the Wavefunction problem of CI: excitations are additive i=a,b,... ˆD,i required is a multiplicative property contradiction!?! i=a,b,......

15 Multiplicative Property of the Wavefunction one, two, three,... H 2 molecules 1 = (1 + ˆD,A ) HF 2 = (1 + ˆD,A ) (1 + ˆD,B ) HF 3 = (1 + ˆD,A ) (1 + ˆD,B ) (1 + ˆD,C ) HF... = i=a,b,... (1 + ˆD,i ) HF

16 Exponential Form for the Wavefunction proper form of wavefunction = i=a,b,... (1 + ˆD,i ) HF rewrite in exponential form (1 + ˆD,i ) HF exp( i=a,b,.. i=a,b,.. ˆD,i ) HF multiplicative property via additivity in exponent

17 Exponential Form for the Wavefunction exponential ansatz = exp( ˆT ) 0 excitations represented via an operator reference determinant proper multiplicative behaviour => size consistency

18 Connected and Disconnected Excitations = i=a,b,... (1 + ˆD,i ) HF = HF + ˆD,A HF ˆD,AˆD,B HF +... connected excitations disconnected excitations (products of connected excitations) CI does not differ between connected and disconnected excitations

19 Standard Coupled-Cluster Theory

20 Coupled-Cluster Theory exponential ansatz for wavefunction CC = exp(t ) 0 with excitation operator reference determinant T = T 1 + T 2 + T T 1 = i T 2 = 1 4 i,j a t a i a aa i a,b t ab ij a aa b a ja i unknown parameters: amplitudes second quantization t a i,t ab ij,... equivalent to FCI, different parameterization of wavefunction

21 CC Energy and Equations insertion into Schrödinger-Equation multiplication from left with exp(-t) H exp(t ) 0 = E exp(t ) 0 exp( T )H exp(t ) 0 = E 0 projection onto reference determinant CC energy E = 0 exp( T )H exp(t ) 0 projection onto excited determinants CC equations 0 = p exp( T )H exp(t ) 0 non-linear equations for amplitudes

22 Algebraic Expressions for CC Equations E = 0 exp( T )H exp(t ) 0 0 = p exp( T )H exp(t ) 0 Baker-Campbell-Hausdorff expansion exp( T ) H exp(t ) = H + [H, T] + 1 2! [[H, T],T] +... Commutator expressions: E = h0 H +[H, T]+ 1 [[H, T],T] 0i 2 0=h p H +[H, T]+ 1 [[H, T],T]+... 0i 2 evaluation via Slater-Condon rules, Wick s theorem, diagrammatic rules, computer algebra,...

23 Approximate CC Methods truncation of the cluster operator T: cluster operator approximation cost T=T 1 +T 2 CCSD N 6 T=T 1 +T 2 +T 3 CCSDT N 8 T=T 1 +T 2 +T 3 +T 4 CCSDTQ N 10 T=T 1 +T 2 +T 3 +T 4 +T 5 CCSDTQP N 12 T=T 1 +T 2 + +T N FCI

24 CCD Approximation truncated CC wavefunctions T = T 2 truncation in the cluster operator CCD = CC doubles CCD = 0 + T ! T ! T connected double excitations disconnected quadruple, sextuple,... excitations

25 CCD Approximation cluster operator normal-ordered Hamiltonian T = T 2 H N = p,q f pq {a pa q } p,q,r,s pq rs {a pa qa s a r } CC energy: E = 0 W N T 2 0 f N + W N 1el terms 2el terms CC equations: 0=hD W N +[f N,T 2 ]+[W N,T 2 ]+ 1 2 [[W N,T 2 ],T 2 ] 0i

26 Algebraic CCD Equations CC energy: E = 1 4 ij ab ij ab t ab ij CC equations: 0 = ab ij + P (ab) e f ae t eb ij P (ij) mn mn ij t ab mn ef ab ef t ef ij m f mi t ab mj +P (ij)p (ab) 1 2 P (ab) mn ef m e mb ej t ae im mn ef t af mnt eb ij 1 2 P (ij) mn ef mn ef t ef in tab mj mn ef mn ef t ab mnt ef ij P (ij)p (ab) mn ef mn ef t ae imt bf jn

27 Accuracy of CC versus CI Methods deviation from FCI (in mh) for CO CI CC SD SDT SDTQ SDTQP SDTQPH calculations with cc-pvdz basis, frozen core E(FCI) = H, ΔE korr = mh

28 Approximate Treatment of Higher Excitations

29 Need for Higher Excitations High-Accuracy Calculations: 1 kj/mol or better in thermochemical applications 0.1 pm accuracy in computed bond distances... require considerations of excitations beyond singles and doubles => triples, quadruples,...

30 Additional Approximations for Triples additional approximations in the amplitude equations no storage of amplitudes for triple excitations reduced scaling in computational cost CCSDT-1, CCSDT-2, CCSDT-3, CC3,...

31 Iterative Approximations to CCSDT triples equations 0 = T (W N T 2 ) c +(f N T 3 ) c +(W N T 3 ) c +(W N T 1 T 2 ) c +(f N T 3 T 1 ) c + 1 2! (f NT 2 2 ) c + 1 2! (W N T 2 2 ) c +(W N T 3 T 1 ) c + 1 2! (W N T 2 T 2 1 ) c +(W N T 3 T 2 ) c + 1 2! (W NT 2 2 T 1 ) c + 1 2! (W N T 3 T 2 1 ) c + 1 3! (W N T 2 T 3 1 ) c 0 CCSDT-1: CCSDT-2: + CCSDT-3: + + CC3: +

32 Non-Iterative Approximations to CCSDT two-step procedure: perform CCSD calculation T = T 1 + T 2 add perturbative corrections due to T 3 E = E(CCSD) + ΔE(T) use of converged T 1 and T 2 amplitudes CCSD+T(CCSD), CCSD(T),...

33 The CCSD(T) Approximation perturbative corrections on top of CCSD fourth-order contribution E T (4) = 1 wijk abc 2 36 i + j + k a b c i,j,k a,b,c w abc ijk = P (ij/k)p (ab/c){ e bc ek t ae ij computed with fifth-order contribution CCSD amplitudes E T (5) = 1 wijk abc wabc ijk 36 i + j + k a b c i,j,k a,b,c w abc ijk = P (ij/k)p (ab/c) bc jk t a i m mc jk t ab im} permutation operator: P (pq/r)z(pqr) =Z(pqr)+Z(qrp)+Z(rpq)

34 The CCSD(T) Approximation computational cost w abc ijk = P (ij/k)p (ab/c){ e bc ek t ae ij m mc jk t ab im} one summaton index (e or m) six target index (i,j,k,a,b,c) => cost scale with n occ3 N virt 4 CCSD(T): cost N 6 per iteration, N 7 for non-iterative step no need to store T 3 amplitudes gold standard of quantum chemistry

35 Accuracy of Approximate Treatment of Triples deviation from FCI and CCSDT (in mh) for CO Δ(FCI) Δ(CCSDT) CCSD CCSDT CCSDT CCSDT CC CCSD(T) CCSD+T calculations with cc-pvdz basis, frozen core

36 High-Accuracy Quantum Chemistry: Computational Thermochemistry

37 Computational Thermochemistry definition: (quantum-chemical) calculation of thermochemical energies via energy differences most often computed: atomization energies heats of formation calculation involves: electronic part non-electronic part

38 Chemical Accuracy accuracy required for realistic chemical predictions 1 kcal/mol ˆ= kj/mol ev 350 cm mhartree sub-chemical accuracy: 1 kj/mol and better DFT usually not accurate enough

39 Thermochemistry with Sub-Chemical Accuracy HEAT = High-accuracy Extrapolated Ab initio Thermochemistry E = E HF + E CCSD(T) + Ecc pvtz CCSDT + E rel + E SO + E ZPE + E DBOC + Ecc pvdz CCSDT(Q) A. Tajti et al., JCP 121, (2004) Y.J. Bomble et al., JCP 125, (2006) M.E. Harding et al., JCP 128, (2008)

40 Statistical Evaluation High-accuracy Extrapolated Ab-initio Thermochemistry atomization energies of N 2, H 2, F 2, CO, O 2, C 2 H 2, CCH, CH 2, CH, CH 3, CO 2, H 2 O 2, H 2 O, HCO, HF, HO 2, NO, OH mean error max. error RMS error Harding et al., J. Chem. Phys. 128, (2008)

41 Thermochemistry with Sub-Chemical Accuracy atomization energies of vinyl chloride benzene chemical accuracy 1 kcal/mol 1 kj/mol 1 kcal/mol 1.6 mh 0.4 mh

42 Thermochemistry with Sub-Chemical Accuracy E HF E CCSD(T) E cc pvtz CCSDT E cc pvdz CCSDT(Q) E rel E SO E ZPE E DBOC C 2 H 3 Cl C 6 H total exp ± ±0.5 values in kj/mol M.E. Harding et al., J. Phys. Chem. A 111, (2007)

43 Thermochemistry with Sub-Chemical Accuracy E HF E CCSD(T) E cc pvtz CCSDT E cc pvdz CCSDT(Q) E rel E SO E ZPE E DBOC C 2 H 3 Cl C 6 H total exp ± ±0.5 values in kj/mol Harding et al., JCP, 135, (2011)

44 Computational Requirements CCSD(T)/cc-pCV5Z calculation 1200 basis functions CCSDT/cc-pVTZ calculation 264 basis functions CCSDT(Q)/cc-pVDZ calculation 114 basis functions ~ 41 hours (16 node single-core 3.4 GHz Xeon ) ~ 77 hours (dual-core 3.0 GHz Xeon) ~ 32 hours (12 node AMD64) Bottleneck: large basis calculation at lower CC levels Harding, Metzroth, Auer, Gauss, JCTC 4, 64 (2008)

45 Analytic Derivatives in Coupled-Cluster Theory

46 Importance of Energy Derivatives for Molecular Properties E x 2 E x y 3 E x y z 2 E i j 2 E m N i I N i B j 2 E I N j forces on nuclei geometries, transition states force constants harmonic vibrational frequencies anharmonicities fundamental frequencies, polarizabilities nuclear magnetic shieldings NMR chemical shifts indirect spin-spin coupling constants etc.

47 Numerical vs Analytic Differentiation numerical differentation e.g. analytic differentiation E x E( x) E( x) 2 x i.e., direct evaluation using analytic expressions numerical analytic implementation easy difficult efficiency low high precision limited high imaginay perturbations complex Ψ yes frequency dependence no yes analytic differentiation preferable

48 Gradients via Straightforward Differentiation energy E = 0 exp( T )H exp(t ) 0 gradient E x = 0 exp( T ) H x exp(t ) [exp( T )H exp(t ), T x ] 0 integral derivatives perturbed amplitudes perturbed amplitudes via 0= p exp( T ) H x exp(t ) 0 + p [exp( T )H exp(t ), T x ] 0 to be solved for all x; cost similar to CC equations efficiency?

49 Gradients Using Lagrange Multipliers energy functional Lagrange multipliers Ẽ = 0 exp( T )H exp(t ) 0 + energy p p p exp( T )H exp(t ) 0 CC equations as constraints compact notation Ẽ = 0 (1 + ) exp( T )H exp(t ) 0 Λ deexcitation operator: = = i a i a a i a a 2 = 1 4 i,j a,b ij ab a i a aa j a b

50 Gradients Using Lagrange Multipliers stationarity conditions Ẽ p =0 = 0 = p exp( T )H exp(t ) 0 CC equations Ẽ t p =0 = 0 = 0 (1 + ) (exp( T )H exp(t ) E) p gradient Λ equations Ẽ x = 0 (1 + ) exp( T ) H x exp(t ) 0 no perturbed wavefunction parameters required

51 Algebraic Form of Λ Equations in the CCD Model (perturbation independent) linear equations for λ amplitudes

52 Density-Matrix Formulation of CC Gradients introduction of density matrices E x = 0 (1 + ) exp( T ) H x exp(t ) 0 = p,q D pq f pq x + p,q,r,s pqrs pq rs x reduced one-particle density matrix reduced two-particle density matrix 0 (1 + ) exp( T ){a pa q } exp(t ) 0 0 (1 + ) exp( T ){a pa qa s a r } exp(t ) 0 gradients in terms of integral derivatives and density matrices

53 Algebraic Expressions for CCD Density Matrices one-particle density matrix D ij = 1 2 m e,f jm ef tef im D ab = 1 2 m e,f mn ae t be mn two-particle density matrix ijkl = 1 8 e,f kl ef t ef ij abcd = 1 8 m,n mn ab t cd mn ajib = 1 4 m e im ae t be jm abij = 1 4 ij ab mn ef t ef ij tab mn mn ef t ae imt bf jm ijab = 1 4 tab ij mn, e,f 1 4 m,n e,f mn ef t eb mnt af ij 1 4 m,n e,f m,n, e,f mn ef t ef mj tab in

54 CC Gradients with Orbital Relaxation extended energy functional CC energy + constraints Ẽ = 0 (1 + ) exp( T )H exp(t ) 0 CC energy and CC equations +2 a D ai f ai i + p,q I pq Lagrange multipliers c µp S µ c q µ, qp Brillouin condition orthonormality of MOs stationarity conditions elimination of MO derivatives

55 CC Gradients with Orbital Relaxation parameterization of orbital changes c µp = q c µq T qp matrix T stationarity requirements general (non-unitary) transformation (1) (2) (3) (4) Ẽ = 0 = I ab T ab Ẽ = 0 = I ai + I ia T ia Ẽ = 0 = Z vector equations for D ai T ai Ẽ = 0 = I ij T ij

56 Z-Vector Equations for CC Gradients Z-vector equations e e D em [ ae im + am ie + ae im ( a i )] = X ai with X ai = p,q,r ( ipqr ap qr + qrip qr ap ) p,q,r ( apqr ip qr + qrap qr ip ) p,q D pq ( pa qi + pi qa ) linear equations for orbital relaxation contribution to D ai

57 CC Gradients with Orbital Relaxation differentiation of extended energy functional contains no derivative of wavefunction parameters H x : Hamiltonian with AO derivative integrals

58 AO Formulation of CC Gradients differentiation of extended energy functional integral derivatives contain no MO derivative contributions reformulation in AO representation

59 Implementation of CC Gradients required steps for gradients CC and Λ equations ( N 6,N 8, ) CC density matrices ( N 6,N 8, ) MO AO transformation of density matrices ( N 5 ) contraction with integral derivatives (no storage, N 4 ) computational cost do not scale with N pert available for all CC models (CCSD, CCSD(T), CCSDT, CCSDTQ,, FCI)

60 Accuracy of CI and CC Geometrical Parameters calculated r(oh) for H 2 O (in Å) CI CC SD SD(T) SDT SDTQ SDTQP SDTQPH FCI calculations with cc-pvdz basis Kállay et al., J. Chem. Phys. 119, 2991 (2003)

61 Equilibrium Structure of Ferrocene (2) 108.0(7) CCSD(T)/cc-pwCVTZ (3) 672 basis functions 96 correlated electrons r(fec5): pm, <(C5H): 0.52 o r(fec5): pm, <(C5H): 3.7 o gas-phase electron diffraction

62 Accuracy of CC Equilibrium Geometries 12 closed-shell molecules CH 2, CH 4, NH 3, H 2 O, HF, N 2, C O, HC N, HN C, HC CH, O=C=O, F 2 wave function models CCSD(T), CCSDT, CCSDTQ basis sets cc-p(c)vxz, X = T,Q,5,6 + additivity assumptions empirical equilibrium geometries Heckert et al., Mol. Phys. 103, 2109 (2005)

63 Statistical Evaluation mean error pm standard deviation pm CCSD(T)/cc-pV6Z -0,01-0,008-0,006-0,004-0, ,002 0,004 0,006 0,008 0,01

64 Statistical Evaluation mean error pm standard deviation pm CCSD(T)+core CCSD(T)/cc-pV6Z -0,01-0,008-0,006-0,004-0, ,002 0,004 0,006 0,008 0,01

65 Statistical Evaluation mean error pm standard deviation pm CCSDT CCSD(T)+core CCSD(T)/cc-pV6Z -0,01-0,008-0,006-0,004-0, ,002 0,004 0,006 0,008 0,01

66 Statistical Evaluation mean error pm standard deviation pm CCSDTQ CCSDT CCSD(T)+core CCSD(T)/cc-pV6Z -0,01-0,008-0,006-0,004-0, ,002 0,004 0,006 0,008 0,01

67 Statistical Evaluation CCSDTQ CCSDT CCSD(T)+core CCSD(T)/cc-pV6Z -0,01-0,008-0,006-0,004-0, ,002 0,004 0,006 0,008 0,01 full CCSDT no improvement over CCSD(T) CCSDTQ significantly more accurate than CCSD(T)

68 High-Accuracy Quantum Chemistry Rotational Spectroscopy and Equilibrium Geometries

69 Rotational Spectrum of H 2 Si=S investigation of ten isotopologues McCarthy et al. (Harvard University)

70 Spectroscopic Parameters of H 2 Si=S calc. exp. A (14) B (6) C (6) all values in MHz fc-ccsd(t)/cc-pv Z + ΔT/cc-pVTZ+ΔQ/cc-pVDZ+Δcore/cc-pCV5Z plus CCSD(T)/cc-pV(Q+d)Z vibrational corrections plus CCSD(T)/aug-cc-pVQZ electronic contributions Thorwirth et al., Chem. Comm (2008)

71 Equilibrium Structure of H 2 Si=S H Si S H experimental structure using CCSD(T)/cc-pV(Q+d)Z vibrational corrections fc-ccsd(t)/cc-pv Z + ΔT/cc-pVTZ+ΔQ/cc-pVDZ+Δcore/cc-pCV5Z Thorwirth et al., Chem. Comm (2008)

72 Multireference Coupled-Cluster Theory: Recent Developments and Challenges

73 Coupled-Cluster Theory exponential ansatz for wavefunction i = exp(t ) 0i exact parameterization of Ψ truncation of T required for applicable schemes standard approach for high-accuracy calculations

74 Multiconfigurational Theories linear combination of determinants i = X µ µi c µ non-dynamical/static correlation large coefficients for several determinants MCSCF, MRCI,... needed for bond breaking, potential energy surfaces,...

75 Multireference Coupled-Cluster Theory coupled-cluster ansatz multiconfigurational ansatz i = exp(t ) 0i i = X µ µi c µ what is the marriage?

76 Wish List for MRCC size extensivity orbital invariance low computational cost good accuracy efficient implementation multiplet problem spin adaptation molecular properties excitation energies

77 Multireference Coupled-Cluster Theory coupled-cluster ansatz multiconfigurational ansatz i = exp(t ) 0i i = X µ µi c µ possible solution: i = exp(t ) X µ µi c µ internally contracted MRCC

78 Internally Contracted MRCC ansatz i = exp(t ) X µ µi c µ cluster operator multideterminantal reference T = X q t q q = nx T k core + active k T k = 1 (k!) 2 C+A X ij... A+V X ab... t ab... ij... active + virtual a aa b...a ja i à non-commuting operators

79 Internally Contracted MRCC det 1 det 2 virtual active T 1 T 2 core à redundancies in T linear dependencies to be eliminated

80 Internally Contracted MRCC H exp(t ) i = exp(t ) i amplitude equations via projection 0 = h apple P exp( T ) H exp(t ) i energy eigenvalue problem H eff c = E c H eff µ = h µ exp( T ) H exp(t ) i elimination of redundancies via singular-value decomposition

81 Accuracy of Internally Contracted MRCC HF dissociation curve, deviation from FCI, CAS(2,2), DVZ F.A. Evangelista, J. Gauss, J. Chem. Phys. 134, (2011)

82 Internally Contracted MRCC HF dissociation curve, deviation from FCI, CAS(2,2), DVZ Early work by: Mukherjee et al. (1975), Banerjee & Simons (1981), Kong (2009) Recent progress by: Evangelista & Gauss (2011, FCI based implementation) Hanauer & Köhn (2011, automatized implementation)

83 Large-Scale Application of icmrcc singlet-triplet splittings (in kcal/mol) in p-benzyne Basis set icmrccsd cc-pvdz 3.72 cc-pvtz 3.73 cc-pvqz 3.74 extrapol ZPVE +0.3 Experiment 3.8± 0.3 icmrccsd/cc-pvqz calculation (two commutators) F. A. Evangelista, M. Hanauer, A. Köhn, J. Gauss, J. Chem. Phys. 136, (2012)

84 Checklist for icmrcc size extensivity orbital invariance scaling with N ref good accuracy efficient implementation spin adaptation molecular properties excitation energies (yes) yes no yes??? in progress numerically in progress main developers: M. Hanauer, A. Köhn (Mainz)

85 Multireference Coupled-Cluster Theory coupled-cluster ansatz multiconfigurational ansatz i = exp(t ) 0i i = X µ µi c µ possible solution: i = X µ exp(t µ ) µi c µ Jeziorski-Monkhorst ansatz

86 Jeziorski-Monkhorst Ansatz Jeziorski-Monkhorst ansatz for wavefunction dx i = µ=1 sum over references exp( ˆT µ ) µi c µ reference-specific cluster operator weight of each reference insertion into Schrödinger eq. and projection onto references X H e µ c = E c µ non-hermitian eigenvalue problem H e µ = h µ H exp(t ) i CMS = h µ exp( T )H exp(t ) i

87 Redundancy Problem det 1 det 2 virtual active T 1 T 2 core commuting operators for each reference redundancies in JM ansatz

88 Mk-MRCC Theory X µ n H exp(t µ ) µi c µ E exp( ˆT o µ ) µi c µ insert = 0 1 = exp(t µ )(P + Q) exp( T µ ) projector on reference space projector on excitation manifold Mahapatra, Datta, Mukherjee, J. Chem. Phys. 110, 6171 (1999)

89 Mk-MRCC Theory => X µ exp(t µ ) Q exp( T µ ) H exp(t µ ) µi c µ + X X µ exp(t ) µi H eff µ c = E X µ exp(t µ ) µ i c µ interchange µ and ν in second term exp(t µ ) Q exp( T µ ) H exp(t µ ) µi c µ + X exp(t ) µi H eff µ c = E exp(t µ ) µ i c µ lift sum over µ => sufficiency conditions Mahapatra, Datta, Mukherjee, J. Chem. Phys. 110, 6171 (1999)

90 Mk-MRCC Theory exp(t µ ) Q exp( T µ ) H exp(t µ ) µi c µ + X exp(t ) µi H eff µ c = E exp(t µ ) µ i c µ multiplication with exp(-t µ ) and projection on Φ q (µ) h q (µ) exp( T µ ) H exp(t µ ) µi c µ + X (6=µ) h q (µ) exp( T µ )exp(t ) µi H e µ c = 0 Mk-MRCC amplitude equations Mahapatra, Datta, Mukherjee, J. Chem. Phys. 110, 6171 (1999)

91 Mk-MRCCSD and Mk-MRCCSDT Mk-MRCCSD and Mk-MRCCSDT in various program packages

92 Orbital Invariance of Mk-MRCC BeH 2 model system, CAS(2,2), DVZ basis F. A. Evangelista, J. Gauss, J. Chem. Phys. 134, (2010)

93 Tools for Calculating Properties in MRCC C N C equilibrium geometry vibrational frequencies C C C excitation energies... 2,6-pyridyne two reference determinants analytic Mk-MRCC gradients Mk-MRCC response theory

94 Mk-MRCC Gradients: Example 2,6-pyridyne N C C active orbitals MOLDEN MOLDEN MOLDEN defaults used Edge = Space = Psi = 20 M de E C C C distance of radical centers method orbitals distance (Å) SR-CCSD RHF Mk-MRCCSD RHF Mk-MRCCSD TCSCF Jagau, Prochnow, Evangelista, Gauss, J. Chem. Phys. 132, (2010)

95 Mk-MRCC Gradients: Example 2,6-pyridyne N C C active orbitals MOLDEN MOLDEN MOLDEN defaults used Edge = Space = Psi = 20 M de E C C C coefficients of reference determinants method c 1 c 2 TCSCF Mk-MRCCSD Jagau, Prochnow, Evangelista, Gauss, J. Chem. Phys. 132, (2010)

96 Mk-MRCCSDT Gradients method ΔE A CCSD CCSDT 7.62 MK-MRCCSD 7.79 Mk-MRCCSDT 7.87 TCSCF orbitals, cc-pvdz automerization barrier ΔE A of cyclobutadiene E. Prochnow, J. Gauss, to be published

97 Spin-Orbit Splittings via Mk-MRCC OH radical Π y Π x nonrelativistic treatment Π x and Π y degenerate single-reference case

98 Spin-Orbit Splittings via Mk-MRCC OH radical Π y Π x relativistic treatment 1/2, 3/2 = 1 p 2 ( x ± i y ) multideterminantal case

99 Spin-Orbit Splittings via Mk-MRCC OH radical Π y Π x relativistic treatment E Π 1/2 Π x, Π y Π 3/2 ΔE SO

100 Spin-Orbit Splittings via Mk-MRCC treat SO effects using degenerate perturbation theory zeroth-order wavefunctions Π 1/2 and Π 3/2 (Mk-MRCC) SO splittings E SO = 2 h 1/2 Ĥ SO 1/2 i = 2 de( 1/2) d SO Mk-MRCC gradients spatially averaged π orbitals & mean-field SO treatment L.A. Mück and J. Gauss, J. Chem. Phys. 136, (2012)

101 Spin-Orbit Splittings via Mk-MRCC SO splittings (in cm -1 ): Species Mk-MRCC Exp. OH SH SeH ClO BrO Mk-MRCCSD/cc-pCVQZ(-g) calculations L.A. Mück and J. Gauss, J. Chem. Phys. 136, (2012)

102 Dynamical Polarizability of CH 2 40 CCSD Mk-MRCCSD 30 αyy ω/e h Mk-MRCC has wrong pole structure T.-C. Jagau, J. Gauss, J. Chem. Phys. 137, (2012)

103 Dynamical Polarizability of p-benzyne 80 CCSD αzz ω/e h aug-cc-pcvdz basis set at Mk-MRCCSD/cc-pCVTZ geometry T.-C. Jagau, J. Gauss, J. Chem. Phys. 137, (2012)

104 Response Theory for Excited States dispersion for α(ω) ω exc α(ω) ω exact theory pole position at excitation frequency

105 Mk-MRCCSD Excitation Energies vertical excitation energies in ev cc-pcvdz basis, Mk-MRCCSD/cc-pCVDZ geometry CCSD MkMRCCSD 1 3 B 3u B 3u A g B 2u B 2u T.-C. Jagau, J. Gauss, J. Chem. Phys. 137, (2012)

106 defaults used single point Mk-MRCCSD Excitation Energies: SiS 2 E/Eh ground state A B B A artificial solutions MOLDEN MOLDEN MOLDE (SSiS)

107 Checklist for Mk-MRCC size extensivity orbital invariance scaling with N ref good accuracy efficient implementation spin adaptation molecular properties excitation energies yes no yes??? yes (yes) gradients wrong pole structure

108 Conclusions and Outlook CC theory: main tool for high-accuracy calculations à computational thermochemistry, gas-phase spectroscopy current challenges: large molecules à local correlation, pair natural orbitals,... basis-set convergence à explicitly correlated CC heavy-elements à relativistic four- and two-component CC strong correlation à multi-reference CC