NEW COUPLED-CLUSTER METHODS FOR MOLECULAR POTENTIAL ENERGY SURFACES: I. GROUND-STATE APPROACHES Piotr Piecuch Department of Chemistry, Michigan State University, East Lansing, Michigan 48824 P. Piecuch and K. Kowalski, in: Computational Chemistry: Reviews of Current Trends, edited by J. Leszczynski´ (World Scientific, Singapore, 2000), Vol. 5, pp. 1-104; K. Kowalski and P. Piecuch, J. Chem. Phys. 113, 18-35 (2000); 113, 5644-5652 (2000); J. Molec. Struct.: THEOCHEM 547, 191-208 (2001); Chem. Phys. Lett. 344, 165-175 (2001); P. Piecuch, S.A. Kucharski, and K. Kowalski, Chem. Phys. Lett. 344, 176-184 (2001); P. Piecuch, S.A. Kucharski, V. Spirkˇ o, and K. Kowalski, J. Chem. Phys. 115, 5796-5804 (2001); P. Piecuch, K. Kowalski, and I.S.O. Pimienta, Int. J. Mol. Sci., submitted (2001); P. Piecuch and K. Kowalski, Int. J. Mol. Sci., submitted (2001); K. Kowalski and P. Piecuch, J. Chem. Phys. 115, 2966-2978 (2001); P. Piecuch, K. Kowalski, I.S.O. Pimienta, and S.A. Kucharski, in: Accurate Description of Low-Lying Electronic States and Potential Energy Surfaces, ACS Symposium Series, Vol. XXX, edited by M.R. Hoffmann and K.G. Dyall (ACS, Washington, D.C., 2002), pp. XXX-XXXX; K. Kowalski and P. Piecuch, J. Chem. Phys., submitted (2001); P. Piecuch, S.A. Kucharski, and V. Spirkˇ o, J. Chem. Phys. 111, 6679-6692 (1999); K. Kowalski and P. Piecuch, J. Chem. Phys. 113, 8490-8502 (2000); 115, 643-651 (2001); K. Kowalski and P. Piecuch, Chem. Phys. Lett. 347, 237-246 (2001). 1 The \holy grail" of the ab initio electronic structure theory: The development of simple, \black-box," and affordable methods that can pro- vide highly accurate ( spectroscopic) description of ENTIRE GROUND- AND ∼ EXCITED-STATE POTENTIAL ENERGY SURFACES 14000 12000 10 NaF+H 10000 8 8000 V/cm-1 6000 6 Na *+ FH 4000 Energy/eV 4 2000 Na .. FH 0 Na+HF 2 1.5 Na...FH Na + FH + - 2.5 0.6 Na F + H 2 2.5 0.8 0 3.5 3 3.5 1 o R(F-H)/bohr o 4 1.4 1.2 4 6 8 4.5 R(Na-F)/A 4.5 R(F-H)/A 10 12 14 R(Na-F)/bohr Examples of applications: dynamics of reactive collisions • highly excited and metastable ro-vibrational states of molecules • rate constant calculations • collisional quenching of electronically excited molecular species • Motivation: elementary processes that occur in combustion (e.g., reactions involving • OH and NxOy) collisional quenching of the OH and other radical species • IN THIS PRESENTATION, WE FOCUS ON NEW \BLACK-BOX" COUPLED-CLUSTER METHODS FOR GROUND-STATE POTENTIAL ENERGY SURFACES 2 SINGLE-REFERENCE COUPLED-CLUSTER (CC) THEORY (J. Cˇ´ıˇzek, 1966, 1969; J. Cˇ´ıˇzek and J. Paldus, 1971) mA (A) Ψ = eT Φ ; T (A) = T j i j i k Xk=1 i a ij ab i1i2:::ik a1a2:::ak T1 Φ = t Φ ; T2Φ = t Φ ; Tk Φ = t Φ j i aj i i abj ij i j i a1a2:::ak j i1i2:::ik i i i > j i1 > i2 > > i · · · k Xa Xa > b a1 > a2X> > a · · · k mA = N { exact theory mA < N { approximate methods 2 4 2 2 mA = 2 T = T1 + T2 CCSD nonu (nonu) 3 5 3 3 mA = 3 T = T1 + T2 + T3 CCSDT nonu (nonu) 4 6 4 4 mA = 4 T = T1 + T2 + T3 + T4 CCSDTQ nonu (nonu) PROBLEMS WITH THE STANDARD CC APPROXIMATIONS (A) mA (T = k=1 Tk; mA < N) P Example: N2 [K. Kowalski and P. Piecuch, J. Chem. Phys. 113, 5644 (2000); K. Kowalski and P. Piecuch, Chem. Phys. Lett. 344, 165 (2001); P. Piecuch, S.A. Kucharski, and K. Kowalski, ibid. 344, 176 (2001)] −199.00 CCSD CCSD CCSD(T) −108.6 CCSD(T) CCSD(TQf) CCSD(TQf) −199.02 CCSDT (practically exact) −108.7 CCSDT(Qf) Full CI CCSDT −199.04 −108.8 −108.9 −199.06 −109.0 Energy (hartree) −199.08 Energy (hartree) −109.1 −199.10 −109.2 −199.12 −109.3 2.0 3.0 4.0 5.0 6.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 R F−F (bohr) R N−N (bohr) 3 Existing solutions Multi-Reference CC Methods (Jeziorski, Monkhorst, Paldus, Piecuch, • Bartlett, Mukherjee, Lindgren, Kaldor et al:) [also, Multi-Reference CI and MBPT Approaches; cf. the presentation by Professor Mark S. Gordon] State-Selective, Active-Space CC Methods (e.g., the CCSDt and CCSDtq • approaches of Piecuch et al:) Externally-Corrected CC Methods (e.g., the RMRCCSD approach of Pal- • dus and Li) THE PROPOSED RESEARCH FOCUSES ON METHODS THAT COMBINE THE SIMPLICITY OF THE STANDARD SINGLE-REFERENCE CC APPROACHES, SUCH AS CCSD(T), WITH THE EFFICIENCY WITH WHICH THE MULTI-REFERENCE METHODS DESCRIBE GROUND- STATE POTENTIAL ENERGY SURFACES SPECIFIC GOALS New CC methods for ground-state potential energy surfaces: • { method of moments of CC equations { renormalized CC approaches 4 METHOD OF MOMENTS OF COUPLED-CLUSTER EQUATIONS: A NEW APPROACH TO THE MANY-ELECTRON CORRELATION PROBLEM A new relationship in quantum-mechanical theory of many-fermion (many-electron) systems δ = E E(A) = Λ[Ψ; (j)(m ); j > m ] − MJ A A E(A) { the electronic energy obtained using the approximate coupled- cluster calculations (e.g., CCSD) E { the exact energy (the exact eigenvalue of the electronic Hamil- tonian) Ψ { the exact wave function (j)(m ) - the generalized moments of coupled-cluster equations MJ A EMMCC = E(A) + δMMCC MMCC - the method of moments of coupled-cluster equations - provides us with fundamentally new ways of performing electronic structure calculations and renormalizing the failing standard tools of quantum chemistry. 5 The MMCC Energy Formula (the Ground-State Problem) N n (A) T (A) δ = E E = Ψ Qn Cn j(mA) Mj(mA) Φ = Ψ e Φ − h j − j i h j j i n=m +1 j=m +1 XA XA T (A) T (A) Mj(mA) = HN e ; Cn j(mA) = e C;j − n j − T (A) (j) (j) Mj(mA) Φ = Qj HN e Φ = J (mA) ΦJ j i C j i M j i XJ (j) (j) T (A) J (mA) = ΦJ HN e Φ generalized moments of the CC equations M h j C j i − Approximate MMCC Methods: The MMCC(mA,mB) Approaches mB n (A) T (A) E(mA; mB) = E + Ψ Qn Cn j(mA) Mj(mA) Φ = Ψ e Φ h j − j i h j j i n=m +1 j=m +1 XA XA Various approximate forms of Ψ lead to different classes of MMCC(m ; m ) schemes. j i A B 6 The MMCC(2,3), MMCC(2,4), and MMCC(3,4) Methods E(2; 3) = ECCSD + Ψ Q M (2) Φ = Ψ eT1+T2 Φ h j 3 3 j i h j j i E(2; 4) = ECCSD + Ψ Q M (2) + Q [M (2) + T M (2)] Φ = Ψ eT1+T2 Φ h j 3 3 4 4 1 3 j i h j j i E(3; 4) = ECCSDT + Ψ Q M (3) Φ = Ψ eT1+T2+T3 Φ h j 4 4 j i h j j i M (2) Φ = Q H eT1+T2 Φ = abc(2) Φabc 3 j i 3 N C j i Mijk j ijki i<jX<k a<b<c M (2) Φ = Q H eT1+T2 Φ = abcd(2) Φabcd 4 j i 4 N C j i Mijkl j ijkl i i<jX<k<l a<b<c<d M (3) Φ = Q H eT1+T2+T3 Φ = abcd(3) Φabcd 4 j i 4 N C j i Mijkl j ijkl i i<jX<k<l a<b<c<d abc(2) = Φabc H eT1+T2 Φ Mijk h ijkj N C j i abcd(2) = Φabcd H eT1+T2 Φ Mijkl h ijkl j N C j i abcd(3) = Φabcd H eT1+T2+T3 Φ Mijkl h ijkl j N C j i 7 The MMCC(2,3)/CISDT, MMCC(2,3)/CISDt, MMCC(2,4)/CISDTQ, and MMCC(2,4)/CISDtq Methods MMCC(2,3)/CISDT ΨCISDT = (1 + C + C + C ) Φ j i 1 2 3 j i MMCC(2,3)/CISDt (CISDt = active-space CISDT) abC ΨCISDt = 1 + C + C + C Φ j i 1 2 3 Ijk j i MMCC(2,4)/CISDTQ ΨCISDTQ = (1 + C + C + C + C ) Φ j i 1 2 3 4 j i MMCC(2,4)/CISDtq (CISDtq = active-space CISDTQ) abC abCD ΨCISDtq = 1 + C + C + C + C Φ j i 1 2 3 Ijk 4 IJkl j i 8 The Renormalized and Completely Renormalized CCSD[T], CCSD(T), CCSD(TQ), and CCSDT(Q) Methods MBPT(2)-like choices of Ψ lead to the renormalized (R) and completely renormalized (CR) CCSD[T], CCSD(T), CCSD(TQ), CCSDT(Q), etc. approaches. Here are some examples of these methods: CR-CCSD(T) completely renormalized CCSD(T) ΨCCSD(T) = 1 + T + T + R(3)(V T ) + R(3)V T Φ j i 1 2 0 N 2 C 0 N 1 j i CR CCSD(T) CCSD CCSD(T) CCSD(T) T +T E − = E + Ψ Q M (2) Φ = Ψ e 1 2 Φ h j 3 3 j i h j j i R-CCSD(T) renormalized CCSD(T) M (2) (V T ) 3 ≈ N 2 C;3 R CCSD(T) CCSD CCSD(T) CCSD(T) T +T E − = E + Ψ Q (V T ) Φ = Ψ e 1 2 Φ h j 3 N 2 Cj i h j j i # 1.0 CCSD(T) ! CR-CCSD(TQ) ΨCCSD(TQ) = ΨCCSD(T) + 1T 2 Φ j i j i 2 2 j i CR CCSD(TQ) CCSD CCSD(TQ) E − = E + Ψ Q M (2) h jf 3 3 + Q [M (2) + T M (2)] Φ = ΨCCSD(TQ) eT1+T2 Φ 4 4 1 3 gj i h j j i 9 CR-CCSDT(Q) ΨCCSDT(Q) = 1 + T + T + T + T T + 1T 2 Φ j i 1 2 3 1 2 2 2 j i CR CCSDT(Q) CCSDT CCSDT(Q) CCSDT(Q) T +T +T E − = E + Ψ Q M (3) Φ = Ψ e 1 2 3 Φ h j 4 4 j i h j j i The (C)R-CCSD[T], (C)R-CCSD(T), (C)R-CCSD(TQ), and (C)R-CCSDT(Q) methods can be viewed as the extensions of the standard CCSD[T], CCSD(T), CCSD(TQf), and CCSDT(Qf) methods, respectively.