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The Size-Consistency Problem in Configuration Interaction Calculation

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The Size-Consistency Problem in Configuration Interaction Calculation The Size-Consistency Problem in Configuration Interaction calculation P´eter G. Szalay E¨otv¨osLor´and University Institute of Chemistry H-1518 Budapest, P.O.Box 32, Hungary [email protected] P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Size-consistency Consider two subsystems at infinite separation. We have two choices: • treat the two system separately; • consider only a super-system. Provided that there is no interaction between the two systems, the two treatments should give the same result, a basic physical requirement. E¨otv¨osLor´and University, Institute of Chemistry 1 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Size-consistency Let us use the CID wave function to describe this system! For the super − system we have : ΨCID = ΦHF + ΦD (1) ΦD is the sum all double excitations out of ΦHF (including coefficients). For the subsystems we can write: A A A ΨCID = ΦHF + ΦD (2) B B B ΨCID = ΦHF + ΦD (3) The product of these two wave functions gives the other choice for the wave function of the super-system: A+B A B ΨCID = ΨCID ΨCID (4) A B A B B A A B = ΦHF ΦHF + ΦHF ΦD + ΦHF ΦD + ΦD ΦD A B = ΦHF + ΦD + ΦD ΦD E¨otv¨osLor´and University, Institute of Chemistry 2 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Size-consistency This simple model enables us to identify the origin of the size-consistency error: The difference of the two super-system wave functions: A B A B ΨCID ΨCID − ΨCID = ΦD ΦD (5) i.e. simultaneous double excitations on the subsystems are missing from the CI wave function. E¨otv¨osLor´and University, Institute of Chemistry 3 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Size-consistency This simple model enables us to identify the origin of the size-consistency error: The difference of the two super-system wave functions: A B A B ΨCID ΨCID − ΨCID = ΦD ΦD (5) i.e. simultaneous double excitations on the subsystems are missing from the CI wave function. This error is present also if there is an interaction between A and B, but we cannot quantify it by two calculations E¨otv¨osLor´and University, Institute of Chemistry 3 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Size-consistency This simple model enables us to identify the origin of the size-consistency error: The difference of the two super-system wave functions: A B A B ΨCID ΨCID − ΨCID = ΦD ΦD (5) i.e. simultaneous double excitations on the subsystems are missing from the CI wave function. This error is present also if there is an interaction between A and B, but we cannot quantify it by two calculations ⇓ lack of size-extensivity E¨otv¨osLor´and University, Institute of Chemistry 3 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Definitions Size-consistency Property of a computational method which ensures that the calculation of the energy for molecule AB at infinite separation gives the same result as the sum of the energy from the separate calculations on A and B. (This also implies that the wave function of AB can be write as a product of the wave functions of A and B). The term size-consistency is strongly related to the term ’size-extensivity’ but the latter is more mathematically motivated and can be defined also for interacting systems. Size-extensivity Property of a computational method which ensures that the energy of the system scales properly with its size. A rigorous definition can be given within the Many Body theory (e.g. CC theory, Perturbation Theory). The term size-extensivity is strongly related to “size-consistency” but the latter ensures the scalability only for non-interacting systems while the former is meaningful at any distance. E¨otv¨osLor´and University, Institute of Chemistry 4 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Origin of the size-consistency error Consider first the full CI wave function: X a a X ab ab X abc abc ΨFCI = φ0 + ci φi + cij φij + cijkφijk + ... i,a i>j,a>b i>j>k,a>b>c Notation: • φ0: reference determinant • i, j, k...: occupied orbitals; a, b, c...: virtual orbitals • E0: reference energy; HN = Hˆ − E0: correlation energy operator E¨otv¨osLor´and University, Institute of Chemistry 5 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 The correlation energy is: X cd cd ∆E = ckl hφ0|HN |φkl i k>l,c>d ab Equation for a double excited coefficent cij : ab X c ab c X cd ab cd hφij |HN |φ0i + ckhφij |HN |φki + ckl hφij |HN |φkl i k,c k>l,c>d X cde ab cde X cdef ab cdef + cklmhφij |HN |φklmi + cklmnhφij |HN |φklmni k > l > m k > l > m > n c > d > e c > d > e > f ab = cij ∆E E¨otv¨osLor´and University, Institute of Chemistry 6 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Slater rule: ab cdef cd hφij |HN |φklmni = hφ0|HN |φkl iδ{ij,mn}δ{ab,ef} E¨otv¨osLor´and University, Institute of Chemistry 7 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Slater rule: ab cdef cd hφij |HN |φklmni = hφ0|HN |φkl iδ{ij,mn}δ{ab,ef} abcd ab cd Cluster condition (from coupled-cluster theory): cijkl ≈ cij ckl E¨otv¨osLor´and University, Institute of Chemistry 7 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Slater rule: ab cdef cd hφij |HN |φklmni = hφ0|HN |φkl iδ{ij,mn}δ{ab,ef} abcd ab cd Cluster condition (from coupled-cluster theory): cijkl ≈ cij ckl From Pauli principle: “k, l, c, d 6= i, j, a, b00 E¨otv¨osLor´and University, Institute of Chemistry 7 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Slater rule: ab cdef cd hφij |HN |φklmni = hφ0|HN |φkl iδ{ij,mn}δ{ab,ef} abcd ab cd Cluster condition (from coupled-cluster theory): cijkl ≈ cij ckl From Pauli principle: “k, l, c, d 6= i, j, a, b00 Using all these: ab X c ab c X cd ab cd hφij |HN |φ0i + ckhφij |HN |φki + ckl hφij |HN |φkl i k,c k>l,c>d ”6=ij,ab” X ab cd cd ab X cd cd + cij ckl hφ0|HN |φkl i = cij ckl hφ0|HN |φkl i k>l,c>d k>l,c>d Note: triple excitations have been neglected for the sake of simplicity E¨otv¨osLor´and University, Institute of Chemistry 7 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Slater rule: ab cdef cd hφij |HN |φklmni = hφ0|HN |φkl iδ{ij,mn}δ{ab,ef} abcd ab cd Cluster condition (from coupled-cluster theory): cijkl ≈ cij ckl From Pauli principle: “k, l, c, d 6= i, j, a, b00 Using all these: ab X c ab c X cd ab cd hφij |HN |φ0i + ckhφij |HN |φki + ckl hφij |HN |φkl i k,c k>l,c>d ”6=ij,ab” X ab cd cd ab X cd cd + cij ckl hφ0|HN |φkl i = cij ckl hφ0|HN |φkl i k>l,c>d k>l,c>d E¨otv¨osLor´and University, Institute of Chemistry 7 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Slater rule: ab cdef cd hφij |HN |φklmni = hφ0|HN |φkl iδ{ij,mn}δ{ab,ef} abcd ab cd Cluster condition (from coupled-cluster theory): cijkl ≈ cij ckl From Pauli principle: “k, l, c, d 6= i, j, a, b00 Using all these: ab X c ab c X cd ab cd hφij |HN |φ0i + ckhφij |HN |φki + ckl hφij |HN |φkl i k,c k>l,c>d ”6=ij,ab” X ab cd cd ab X cd cd + cij ckl hφ0|HN |φkl i = cij ckl hφ0|HN |φkl i k>l,c>d k>l,c>d The cancellation is not complete because of the restricted summation!! E¨otv¨osLor´and University, Institute of Chemistry 7 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Therefore the red term on the left hand side must be considered: ”6=ij,ab” ab X cd cd ab ab cij ckl hφ0|HN |φkl i ≡ cij Kij k>l,c>d ab which is the definition of Kij . The equation becomes: ab X c ab c X cd ab cd hφij |HN |φ0i + ckhφij |HN |φki + ckl hφij |HN |φkl i k,c k>l,c>d ab ab ab +cij Kij = cij ∆E E¨otv¨osLor´and University, Institute of Chemistry 8 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Therefore the red term on the left hand side must be considered: ”6=ij,ab” ab X cd cd ab ab cij ckl hφ0|HN |φkl i ≡ cij Kij k>l,c>d ab which is the definition of Kij . The equation becomes: ab X c ab c X cd ab cd hφij |HN |φ0i + ckhφij |HN |φki + ckl hφij |HN |φkl i k,c k>l,c>d ab ab ab +cij Kij = cij ∆E ab ab Conclusion: CISD equation should be corrected by the new term cij Kij .
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