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Piotr Piecuch Department of Chemistry, Michigan State NEW COUPLED-CLUSTER METHODS FOR MOLECULAR POTENTIAL ENERGY SURFACES: II. EXCITED-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 EXCITED-STATE POTENTIAL ENERGY SURFACES 2 EQUATION-OF-MOTION (OR RESPONSE) COUPLED CLUSTER THEORY (EOMCC) (R.J. Bartlett, P. Jørgensen, and others) mA mA (A) Ψ = R(A)eT Φ ; T (A) = T ; R(A) = R j Ki K j i n K K;n n=1 n=0 X X mA = N { exact theory mA < N { approximate methods mA = 2 T = T1 + T2, RK = RK;0 + RK;1 + RK;2 EOMCCSD mA = 3 T = T1 + T2 + T3, RK = RK;0 + RK;1 + RK;2 + RK;3 EOMCCSDT PROBLEMS WITH THE STANDARD EOMCC APPROXIMATIONS (A) mA (A) mA (T = n=1 Tn; R = n=0 RK;n; mA < N) P P Example: CH+ [K. Kowalski and P. Piecuch, J. Chem. Phys. 115, 2966 (2001)] −37.4 + Full CI, 1Σ Full CI, 1Π −37.5 Full CI, 1∆ EOMCCSD −37.6 −37.7 −37.8 Energy (hartree) −37.9 −38.0 −38.1 2.0 2.5 3.0 3.5 4.0 4.5 5.0 R C−H (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] EOMCC Methods with the Complete Inclusion of Higher-Than-Doubly • Excited Clusters (e.g., the Full EOMCCSDT Method of Kowalski and Piecuch; cf., also, Kucharski et al:) THE PROPOSED RESEARCH FOCUSES ON METH- ODS THAT COMBINE THE SIMPLICITY OF THE STANDARD EOMCC APPROACHES, SUCH AS EOMCCSD, EOMCCSD(T), or CC3, WITH THE EFFICIENCY WITH WHICH THE MULTI- REFERENCE METHODS DESCRIBE EXCITED- STATE POTENTIAL ENERGY SURFACES SPECIFIC GOALS New CC methods for excited-state potential energy surfaces: • { method of moments of CC equations { active-space EOMCC approaches 4 EXTENSION OF THE METHOD OF MOMENTS OF COUPLED-CLUSTER EQUATIONS (MMCC) TO EXCITED STATES H Ψ = E Ψ j Ki Kj Ki Via the State-Universal Multi-Reference CC Formalism M mA SUCC-A (A) T (p) (p) (p) Ψ = U χ = c e A Φ ; T = T (m < N) j K i j K i pK j pi A m A p=1 m=1 X X M N n (A) (p) TA (p) δK = EK EK = ΨK (e )n m Γm (mA) Φp − h j − j i p=1 n=m +1 m=m +1 X XA XA Φ χA = Ψ ΨSUCC-A (K = 1; : : : ; M) ×h pj Ki h Kj K i Γ(p)(m ) Φ =generalized moments of the SUCC equations m A j pi ( Via the Equation-of-Motion CC Formalism mA mA (A) Ψ(A) = R(A)eT Φ ; T (A) = T ; R(A) = R (m < N) j K i K j i n K K;n A n=1 n=0 X X (A) (A) (A) (A) (A) (A) (A) (A) (A) (A) T (A) T (A) (Q H¯ Q )R Φ = ! R Φ ; ! = E E ; H¯ = e− He K j i K K j i K K − 0 N n (A) EOMCC δK = EK EK = ΨK QnCn j(mA)MK;j (mA) Φ = − h j − j i n=mA+1 j=mA+1 X X (A) Ψ eT R(A) Φ h Kj K j i M EOMCC(m ) Φ =generalized moments of the EOMCC equations K;j A j i ( 5 Extension of the MMCC Theory to the EOMCC Formalism N n (A) (A) EOMCC T (A) (A) δK EK EK = ΨK Qn Cn j(mA) MK;j (mA) Φ = ΨK e RK Φ ≡ − h j − j i h j j i n=m +1 j=m +1 XA XA T (A) Cn j(mA) = (e )n j − − M EOMCC(m ) Φ = Q (H¯ (A)R(A)) Φ = (j) (m ) Φ(j) K;j A j i j K j i MK;J A j J i XJ (j) (m ) = Φ(j) (H¯ (A)R(A)) Φ generalized moments of the EOMCC equations MK;J A h J j K j i − (we only need moments with j > mA) The MMCC(mA,mB) Approaches (A) EK(mA; mB) = EK + δK(mA; mB) mB n EOMCC T (A) (A) δK(mA; mB) = ΨK Qn Cn j(mA) MK;j (mA) Φ = ΨK e RK 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) Approximation EOMCCSD EK(2; 3) = EK + δK(2; 3) CCSD δ (2; 3) = Ψ M EOMCC(2) Φ = Ψ eT RCCSD Φ K h Kj K;3 j i h Kj K j i CCSD CCSD T = T1 + T2; RK = RK;0 + RK;1 + RK;2 M EOMCC(2) Φ = abc (2) Φabc K;3 j i MK;ijk j ijki i<jX<k a<b<c abc (2) = Φabc H¯ CCSDRCCSD Φ + rCCSD Φabc H¯ CCSD Φ MK;ijk h ijkj open K;open C j i K;0 h ijkj j i ¯ ¯ ¯ H TD H TS + H TD abc (2) = Φabc (H¯ CCSDR ) Φ + Φabc [H¯ CCSD(R + R )] Φ MK;ijk h ijkj 2 K;2 Cj i h ijkj 3 K;1 K;2 Cj i + Φabc (H¯ CCSDR ) Φ + rCCSD Φabc H¯ CCSD Φ h ijkj 4 K;1 Cj i K;0 h ijkj j i ¯ ¯ H TS H T0 abC Ψ ΨCISDt = 1 + C + C + C Φ j Ki ' j K i 1 2 3 Ijk j i 7 Example: Potential Energy Surfaces for CH+ (the MMCC(2,3) Study) −37.4 + Full CI, 1Σ Full CI, 1Π −37.5 Full CI, 1∆ EOMCCSD −37.6 −37.7 −37.8 Energy (hartree) −37.9 −38.0 −38.1 2.0 2.5 3.0 3.5 4.0 4.5 5.0 R C−H (bohr) −37.4 + Full CI, 1Σ Full CI, 1Π −37.5 Full CI, 1∆ MMCC(2,3) −37.6 −37.7 −37.8 Energy (bohr) −37.9 −38.0 −38.1 2.0 2.5 3.0 3.5 4.0 4.5 5.0 R C−H (bohr) 8 + Vertical excitation energies (in eV) of CH , N2, and C2. The full CI values represent the excitation energies, whereas all remaining values are the deviations from the full CI results. Molecule State Full CI EOMCCSD CC3 EOMCCSDt EOMCCSDT MMCC(2,3) MMCC(2,4) CH+ 2 1Σ+ 8.549 0.560 0.230 0.092 0.074 0.084 0.023 3 1Σ+ 13.525 0.055 0.016 0.000 0.001 0.000 -0.001 4 1Σ+ 17.217 0.099 0.026 0.012 -0.002 0.015 0.008 1 1Π 3.230 0.031 0.012 0.003 -0.003 0.007 0.010 2 1Π 14.127 0.327 0.219 0.094 0.060 0.105 0.037 1 1∆ 6.964 0.924 0.318 0.057 0.040 0.051 0.031 2 1∆ 16.833 0.856 0.261 0.016 -0.038 0.006 0.061 1 N2 Πg 9.584 0.081 0.033 0.029 0.092 0.080 1 Σu− 10.329 0.136 0.007 -0.005 0.008 0.032 1 ∆u 10.718 0.180 0.009 0.001 0.024 0.039 1 Πu 13.609 0.400 0.177 0.090 0.246 0.085 1 C2 1 Πu 1.385 0.090 -0.068 -0.062 -0.078 -0.043 1 1 ∆g 2.293 2.054 0.859 0.269 0.130 0.011 1 + 1 Σu 5.602 0.197 -0.047 0.085 -0.032 -0.039 1 1 Πg 4.494 1.708 0.496 0.076 -0.026 0.057 Mean absolute errors in the calculated vertical excitation energies relative to the corresponding full CI values (in eV).
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