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 .

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 .

ab ab Remember: cij Kij includes the higher excitation effects.

E¨otv¨osLor´and University, Institute of Chemistry 8 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Exclusion Principle Violating (EPV) terms

An other way of writing the equations:

ab X c ab c X cd ab cd ab ab hφij |HN |φ0i + ckhφij |HN |φki + ckl hφij |HN |φkl i = cij Rij k,c k>l,c>d with ”6=ij,ab” ab ab ab ab X cd cd cij Rij = cij ∆E − cij ckl hφ0|HN |φkl i (6) k>l,c>d ”=ij,ab” ab X cd cd = cij ckl hφ0|HN |φkl i (7) k>l,c>d

These are the so called EPV terms. Contrary to the name, they are physically correct terms!

E¨otv¨osLor´and University, Institute of Chemistry 9 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Exclusion Principle Violating (EPV) terms

Why are the EPV terms physical?

”6=ij,ab” ab ab ab ab X cd cd cij Rij = cij ∆E − cij ckl hφ0|HN |φkl i (8) k>l,c>d

E¨otv¨osLor´and University, Institute of Chemistry 10 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Exclusion Principle Violating (EPV) terms

Why are the EPV terms physical?

”6=ij,ab” ab ab ab ab X cd cd cij Rij = cij ∆E − cij ckl hφ0|HN |φkl i (8) k>l,c>d Remember:

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 10 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Exclusion Principle Violating (EPV) terms

Why are the EPV terms physical?

”6=ij,ab” ab ab ab ab X cd cd cij Rij = cij ∆E − cij ckl hφ0|HN |φkl i (8) k>l,c>d Remember:

abcd ab cd Cluster condition (from coupled-cluster theory): cijkl ≈ cij ckl From Pauli principle: “k, l, c, d 6= i, j, a, b00 Thus, leaving out the EPV terms means that in the above equation we assume full cancellation (no restricted summation), i.e. rightmost term includes:

”=ij,ab” ”=ij,ab” ab X cd cd X abcd cd cij ckl hφ0|HN |φkl i ≈ cijkl hφ0|HN |φkl i (9) k>l,c>d k>l,c>d which are Pauli principle violating terms.

E¨otv¨osLor´and University, Institute of Chemistry 10 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Two equivalent forms of writing the corrected equations:

ab X c ab c X cd ab cd ab ab hφij |HN |φ0i + ckhφij |HN |φki + ckl hφij |HN |φkl i = cij Rij k,c k>l,c>d with ”=ij,ab” ab X cd cd Rij = ckl hφ0|HN |φkl i k>l,c>d and

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 with ”6=ij,ab” ab X cd cd Kij ≡ ckl hφ0|HN |φkl i k>l,c>d

E¨otv¨osLor´and University, Institute of Chemistry 11 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Approximations to the EPV terms

So called CEPA (Coupled Electron Pair Approximation) methods.

Just a short overview here, for more detail see:

• Kutzelnigg, W. . In Methods of Electronic Structure Theory; Schaefer III, H. F., Ed.; Plenum Press: New York, 1977.

• Koch, S.; Kutzelnigg, W. Theor. Chim. Acta 1981, 59, 387-411.

E¨otv¨osLor´and University, Institute of Chemistry 12 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Approximations to the EPV terms

It is usual to introduce the concept of pair energies by

X ab ab ij = cij hΦ0|HN |φij i. ab The sum of pair energies gives the correlation energy: X ij = ∆E. ij

With pair energies the EPV terms can be approximated as

”=ij,ab” “=i,j” ab X cd cd X Rij = ckl hφ0|HN |φkl i ≈ kl. k>l,c>d kl

Since an unrestricted summation over virtual indices is included in kl, EPV terms due to coincidence of virtual indices cannot be considered in this case.

E¨otv¨osLor´and University, Institute of Chemistry 13 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Approximations to the EPV terms CEPA(0) approximation

ab ab • neglects EPV terms completely: Kij = ∆E or Rij = 0.

• equivalent to LCCSD

• not exact for the two-electron problem

Kelly’s CEPA approximation[1, 2]

• uses pair energies:

“=ij” ! ab X X Rij = kl = ij + (ik + kj) . kl k

E¨otv¨osLor´and University, Institute of Chemistry 14 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Approximations to the EPV terms CEPA(2) approximation of Meyer[3, 4]

• exact for n separated electron pairs if localized orbitals are used

ab • just the “diagonal” EPV term is considered: Rij = ij

• not invariant under transformation of occupied orbitals

The CEPA(1) approximation by Meyer[3, 4]

• exact for the n separated electron pair problem even with non-localized orbitals ! 1 X Rab =  + ( +  ) . ij ij 2 ik kj k

• differs from Kelly’s approximation in just the factor of half

E¨otv¨osLor´and University, Institute of Chemistry 15 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Approximations to the EPV terms Averaged CEPA approximations

Gdanitz and Ahlrichs:1[5]

ne assuming 2 non interacting electron pairs, the averaged pair energy is:

∆E  = ne 2 “=i,j” ab X ∆E and therefore : Rij ≈ kl =  = ne/2 kl • an approximation to CEPA(2) since averaged pair energy is used • no orbital-invariance problem exists • can easily be generalized to MR case • exact for the model of non-interacting electron pairs

1For the ACPF (Averaged Coupled Pair Functional) method

E¨otv¨osLor´and University, Institute of Chemistry 16 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Approximations to the EPV terms

Meissner[6] and Szalay & Bartlett2[7, 8] The correlation energy is distributed among all possible electron pairs:

0 ∆E  =   ne 2 summing over electron pairs with common indices:       ab ne ne − 2 ∆E (ne − 2)(ne − 3) Rij = −   = 1 − ∆E. 2 2 ne ne(ne − 1) 2 • simple modification to the formula of Gdanitz and Ahlrichs; • includes some interaction of the electron pairs in the EPV term; • an averaged version of CEPA(1); • vanishes for two and three electrons

2For the AQCC (Averaged Quadratic Coupled Cluster) method

E¨otv¨osLor´and University, Institute of Chemistry 17 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Approximations to the EPV terms

“Exact CEPA”

EPV terms can be calculated exactly according to the formula:

”=ij,ab” ab ab ab X cd cd cij Rij = cij ckl hφ0|HN |φkl i k>l,c>d

One realization: The Self-Consistent-Size-Consistent CI ((SC)2CI) method of Daudey et al.[9]

E¨otv¨osLor´and University, Institute of Chemistry 18 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Multireference case

The full CI space is partitioned:

P ˆ • P: reference functions (φp ), P

Q ˆ • Q: all single and double excitations out of P (φq ), Q

R ˆ • R: all higher excitations (φr ), R

The full CI wave function:

X P P X Q Q X R R Ψfci = cp φp + cq φq + cr φr p∈P q∈Q r∈R

E¨otv¨osLor´and University, Institute of Chemistry 19 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Multireference case

To generate the function φt of the expansion space, define the operator Et:

Et|φ0i = |φti

As in the SR case use cluster condition:

R cr ≈ ctcs but now: EtEs|φ0i ∈ R is not fulfilled for all s, t ∈ {P,Q}!

Therefore so called REDUNDANCY terms also appear which must also subtracted!

E¨otv¨osLor´and University, Institute of Chemistry 20 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Multireference case

not all EiEj|φ0i is zero – EPV terms

↑ ↓ ↓ ↑ ↓ ↑ ↑ ↓ ↓ ↑ ↓ ↑ ↓ ↑ ↓ ↑

φ0 φ1 φi φj EiEjφ0 = 0

E¨otv¨osLor´and University, Institute of Chemistry 21 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Multireference case

not all EiEj|φ0 > belongs to R – redundancy terms

↑ ↓ ↓ ↑ ↓ ↓ ↑ ↓ ↑ ↓ ↓↑ ↑ ↓ ↑ ↓ ↑ ↓ ↑ ↑

φ0 φ1 φi φj EiEjφ0

E¨otv¨osLor´and University, Institute of Chemistry 22 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Approximations to the redundancy terms

• Ruttink et al.[10]: averaged according to excitation classes

• Malrieu et al.[11]: exact (MR-(SC)2-CI method)

E¨otv¨osLor´and University, Institute of Chemistry 23 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Approximations to the redundancy terms

• Ruttink et al.[10]: averaged according to excitation classes

• Malrieu et al.[11]: exact (MR-(SC)2-CI method)

X P Q P X Q Q Q cp hEs Φ0|HN |Ep Φ0i + cq hEs Φ0|HN |Eq Φ0i + csKs = ∆Ecs p∈P q∈Q with

restr X Q Ks = ct hΦ0|HN |EtΦ0i t∈Q

E¨otv¨osLor´and University, Institute of Chemistry 23 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 FORM OF THE MR WAVE FUNCTIONS

Ψ = Ψ0 + Ψc with Ψ0 ∈ P is the reference function and Ψc is the correlation correction

E¨otv¨osLor´and University, Institute of Chemistry 24 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 FORM OF THE MR WAVE FUNCTIONS

Ψ = Ψ0 + Ψc with Ψ0 ∈ P is the reference function and Ψc is the correlation correction

Two choices for Φ0:

• eigenfunction of the Hamilton in the reference space P:

PˆHˆ PˆΨ0 = E0Ψ0 (10)

• solution of an effective Hamiltonian equation:

−1 PˆHˆ Pˆ + PˆHˆ Qˆ(Qˆ(Ex − Hˆ Qˆ) Qˆ)Hˆ PˆΨ0 = EΨ0 (11)

with Ex being a constant depending on the method (see later).

E¨otv¨osLor´and University, Institute of Chemistry 24 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 FORM OF THE MR WAVE FUNCTIONS

Ψ = Ψ0 + Ψc with Ψ0 ∈ P is the reference function and Ψc is the correlation correction

Two choices for Φc:

• only Q space contribute:

Ψc = ΨQ

• also P space contribute:

0 Ψc = ΨQ + ΨP = ΨQ + (1 − |Ψ0ihΨ0|)ΨP

E¨otv¨osLor´and University, Institute of Chemistry 24 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Size-consistency corrections

According to the above derivations there are two basic ways to make size-consistency corrections:

• a posteriori – correct the energy after solving the equation

• a priori – modify the equations

These two ways lead to:

• Davidson-type corrections

• CEPA-type methods

E¨otv¨osLor´and University, Institute of Chemistry 25 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 A posteriori or Davidson corrections

Above we have derived the following corrected equations:

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

0ab This means, the corrected coefficient (c ij ) expressed with the original CISD ab coefficient (cij ) reads:

ab 0ab ab cij ab c ij = cij − ab ab Kij hφij |HN |φij i

Using this, the first order corrected correlation energy reads:

0 X cd 2 cd ∆E = ∆E + (ckl ) Kkl k>l,c>d

E¨otv¨osLor´and University, Institute of Chemistry 26 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 The original Davidson correction[15, 16]

ab For the CEPA(0) approximation (Kij = ∆E), the corrected correlation energy becomes:

0 X cd 2 ∆E = ∆E + (ckl ) ∆E k>l,c>d

Using the normalization condition (assuming CID):

2 X cd 2 c0 + (ckl ) = 1 k>l,c>d the correction of the CI energy will be:

2 EDC = (1 − c0)∆E

E¨otv¨osLor´and University, Institute of Chemistry 27 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Bruckner correction[17]

We have made a mistake: Energy is given in intermediate normalization, ab while normalization to unity has been used to obtain c0 from cij ’s. Doing this correctly we have:

X 1 − c2 (ccd)2 = 0 kl c2 k>l,c>d 0 and therefore the correction becomes:

2 (1 − c0) EBC = 2 ∆E c0

This formula was in fact given much before Davidson’s paper by Bruckner[17]. Also known as the “renormalized Davidson correction”

E¨otv¨osLor´and University, Institute of Chemistry 28 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Davidson-Silver[18] or Siegbahn correction[19]

Remember the equation for the corrected coefficients:

ab 0ab ab cij ab c ij = cij − ab ab Kij hφij |HN |φij i We can replace c by the corrected coefficient c0:

0ab 00ab 0ab c ij ab c ij = c ij − ab ab Kij hφij |HN |φij i ab Doing this recursively, using Kij = ∆E, inserting into the energy expression, 2 P ab 2 1−c0 replacing (cij ) by 2 , summing the series we get: c0

2 (1 − (c0) ) EDS = 2 ∆E 2(c0) − 1

E¨otv¨osLor´and University, Institute of Chemistry 29 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Pople correction[20]

pn2 + 2n tan2(2θ) − n E = e e e∆E PC 2(sec(2θ) − 1) with cos(θ) = c0.

This formula includes EPV terms in the first time since (assuming c0 ∼ 1):

2   (1 − (c0) ) 2 EPC ∼ 2 ∆E 1 − . (c0) ne which can be easily derived by considering the following averaged CEPA approximation:   ab 1 Kij = 1 − ∆E ne/2

E¨otv¨osLor´and University, Institute of Chemistry 30 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Pople correction

Advantages:

• vanishes for two electrons (CISD is exact for two electrons!!)

• beyond CEPA(0) approximation

E¨otv¨osLor´and University, Institute of Chemistry 31 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Meissner correction[6]

The correction proposed by Meissner uses the other averaged formula:   ab (ne − 2)(ne − 3) Kij = 1 − ∆E ne(ne − 1) which leads to:

2   (1 − (c0) ) (ne − 2)(ne − 3) EMC = 2 ∆E (c0) ne(ne − 1)

Additional advantage:

• vanishes also for three electrons (there is no EPV contribution for three electrons!)

E¨otv¨osLor´and University, Institute of Chemistry 32 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Davidson-type corrections for the multireference case

Ad hoc suggestion by Bruna, Peyerimhoff and Buenker[21]:   X 2 EBPB = 1 − cp (EMR−CISD − EMCSCF ) (10) p∈P i.e. c0 of the Davidson correction has been replaced by the norm in the reference space and E0 = EMCSCF More rigorous derivation by Meissner[6] According to this, any of the above single-reference approximations can be used by the same manner. Other definitions by Shepard[22]:

hΦMR−CISD|PˆHˆ Pˆ|ΦMR−CISDi c0 = hΦMCSCF |ΦMR−CISDi E0 = hΦMR−CISD|ΦMR−CISDi

E¨otv¨osLor´and University, Institute of Chemistry 33 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Davidson-type corrections for the multireference case

Some more involved corrections using MR-CC argumentation in the derivation has been introduced by Duch and Diercksen [23], by Meissner and coworkers [24, 25] and by Hubaˇcet al. [26].

E¨otv¨osLor´and University, Institute of Chemistry 34 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Comparison of the different MR-CEPA methods Method EPV Redundancy Ref. space gradient implementation MR-LCCMa no no unrelaxed yes Columbus MR-CEPA(0)b) no no relaxed yes Columbus QDVPT no no eff. hamilt. no Meldf MRCEPA no averaged relaxed no local MRCPA(4) no exactc eff. hamilt. no local MR-CEPA-1 CEPA(1) yesd unrelaxed no N/A MR-CEPA-2 CEPA(2) yesd unrelaxed no N/A MC-CEPA Kelly’s CEPA no relaxed no local MR-ACPF av. CEPA(2) no relaxed yes Columbus, Molproe) QDVPT-APC av. CEPA(2) no eff. hamilt. no Meldf MR-AQCC av. CEPA(1) no relaxed yes Columbus, Molproe) MR-AQCC-v av. exact CEPA no relaxed yes Columbuse) MR-AQCC-mc av. CEPA(1) averaged relaxed no Columbusf) MR-(SC)2SCI exact CEPA exact relaxed no local a) Equivalent to MR-CEPA-0 (see text); b) Equivalent to VPT, UCEPA and MRCPA(2) (see text); c) fourth order version of the scheme termed as “exact” in this article; d) non-iterative; e) internally contracted and no gradients are available; f) special version.

E¨otv¨osLor´and University, Institute of Chemistry 35 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 MR-CEPA-type methods

X P Q P X Q Q Q cp hEs Φ0|HN |Ep Φ0i + cq hEs Φ0|HN |Eq Φ0i + csKs = ∆Ecs p∈P q∈Q

Correction Ks introduced into the equations

E¨otv¨osLor´and University, Institute of Chemistry 36 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 MR-LCCM [27, 28] and MR-CEPA-0[29] methods

Multireference Linearized Coupled-Cluster Method by Bartlett et al. [27, 28] uses the CEPA(0) approximation and neglects redundancy effects:

Ks = ∆E.

For the wave function, this method uses configurations and the orthogonal complement of the reference space is excluded.

The simplest version of MR-CEPA-n series by Fulde and Stoll [29] referred to as MR-CEPA-0 is equivalent to MR-LCCM as acknowledged already by Fulde and Stoll [29].

E¨otv¨osLor´and University, Institute of Chemistry 37 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 MR-CEPA(0) [5], UCEPA [30], VPT [31] methods

• MR-CEPA(0) by Gdanitz and Ahlrichs [5] was just a by-product of MR-ACPF (see later),

• Unitary CEPA (UCEPA) of Hoffmann and Simons [30] was based on unitary CC ansatz [32]

• Variational Perturbation Theory (VPT) method of Cave and Davidson [31] uses perturbation theory

All four methods are equivalent! CEPA(0) approximation and no redundancy effects:

Ks = ∆E.

Orthogonal complement of the reference space to Φ0 included (only difference to MRLCCM!)

E¨otv¨osLor´and University, Institute of Chemistry 38 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 QDVPT[33] and MRCPA(2)[34, 35] methods

Shortly after VPT [31], Cave and Davidson proposed a method called Quasi-Degenerate Variational Perturbation Theory (QDVPT) method [33].

It still uses CEPA(0) approximation

Ks = ∆E. but the effective Hamiltonian equation with Ex = E0 is used to define Φ0. MRCPA(2) (MR Coupled Pair Approximation) method of Tanaka et al.[34, 35] is equivalent to QDVPT.

E¨otv¨osLor´and University, Institute of Chemistry 39 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 MRCEPA method [10]

MRCEPA method of Ruttink et al. [10] considered redundancy effects for the first time.

EPV terms are completely neglected, i.e. CEPA(0) approximation is used.

Redundancy effects considered in an averaged way:

X Ks = (1 − M[(ks, ls), (kt, lt)])(kt, lt); φs ∈ (ks, ls)

(ks,l2) where M[(ks, ls), (kt, lt)] is one for redundant terms and zero otherwise. The original derivation used determinental formulation, but application of the formulea on spin-adapted configuration instead should not have large effect on the results [10, 36].

E¨otv¨osLor´and University, Institute of Chemistry 40 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 MR-CEPA-n (n=0,1,2) methods [29]

Fulde and Stoll [29] derived a series of methods which are analogues to the single reference CEPA(n) (n=0,1,2) methods. The reference function is from MCSCF calculation and not relaxed later on. The correction in MR-CEPA-2 is given by:

ab X cd ab Rij = ij + cij hΦ0|φij i. cd

In case of MR-CEPA-1 method: 1 X X 1 X Rab =  + ( +  ) + ccdhΦ |φabi + (ccdhΦ |φcdi + ccdhΦ |φcdi) ij ij 2 ik kj ij 0 ij 2 ik 0 ik ki 0 ki k cd kcd

Redundancy effects are not considered in the equations but through the energy expression. No implementation has been reported.

E¨otv¨osLor´and University, Institute of Chemistry 41 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 The MR-ACPF method[5]

MR Averaged Coupled Pair Functional method of Gdanitz and Ahlrichs uses:

 2  Rs = 1 − ∆E. ne

• an approximation to CEPA(2) since correction is equal to the averaged pair energy

• Rs is proportional to ∆E and is the same for all s and therefore a functional exists

• no orbital-invariance problem

• not strictly size-consistent though exact for the model of non-interacting electron pairs

E¨otv¨osLor´and University, Institute of Chemistry 42 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 The MR-AQCC method[7, 8]

The MR Averaged Quadratic CC method of Szalay and Bartlett is a slight modification to MR-ACPF:

  (ne − 2)(ne − 3) Rs = 1 − ∆E. ne(ne − 1)

• an averaged version of CEPA(1)

• invariant under transformation of occupied orbitals

• an averaged interaction of the electron pairs is introduced

• not strictly extensive and not exact for non-interacting electron pairs

• exact for three electrons as it should

E¨otv¨osLor´and University, Institute of Chemistry 43 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 The MR-AQCC-mc and MR-ACPF-mc methods[36]

F¨usti-Moln´arand Szalay included the redundancy effects into MR-AQCC similarly to MRCEPA:

The elements of M are given by:

    No − a na − f(b − a) c f(d − c) M([a, b]; [c, d]) = 1 −     No na c f(d − c) where No is the number of double occupied orbitals, na is the number of active electrons, f(x) = x if x ≥ 0 and f(x) = 0 otherwise.

MR-ACPF version has also been defined[36]

E¨otv¨osLor´and University, Institute of Chemistry 44 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 QDVPT-APC method [37]

Murray et al. [37] extended the QDVPT method [33] to include EPV terms. Using 2 Ex = E0 + ∆E ne an ACPF version of QDVPT has been defined.

This method is called QDVPT-APC (QDVPT with Averaged Pair Correction) and it can also be viewed as a modification of MR-ACPF via the effective Hamiltonian equation.

QDVPT type generalization of MR-AQCC can trivially be given:

  (ne − 3)(ne − 2) Ex = E0 + 1 − ∆E ne(ne − 1)

E¨otv¨osLor´and University, Institute of Chemistry 45 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 MCCEPA method [38]

The Multi-Configurational reference CEPA method of Fink and Staemmler [38] can be described by:

X  1 1  R = 1 − (|Eˆ φ |2 + |Eˆ φ |2)c hΦ |Hˆ |φ i i 21 + δ i j j i j 0 N j j ij which reduces to Kelly’s formula [1, 2] in the single reference case [38].

Non-orthogonality of functions produced by products of operators are considered by the norms.

Redundancies are neglected.

Size-consistency is not discussed in the paper of Fink and Staemmler [38], but in our opinion it is not fulfilled since redundancies are neglected.

E¨otv¨osLor´and University, Institute of Chemistry 46 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 MRCPA methods[34, 39, 35, 40]

Several variants of the so called Multireference Coupled Pair Approximation (MRCPA) have been worked out by Tanaka and coworkers [34, 39, 35, 40].

These methods use the effective Hamiltonian formalism like QDVPT.

There are two levels of approximations:

MRCPA(2): uses a CEPA(0) approximation[35, 40] and as such, it is equivalent to MR-LCCM [40]. (formerly know as MRCPA(0)[34, 39])

MRCPA(4): [35] includes redundancy effects (a slight modification of the variant formerly know as MRCPA(2) [34, 39])

The method is claimed to fulfill the size-consistency criteria for non- interacting electron pairs[35].

E¨otv¨osLor´and University, Institute of Chemistry 47 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 MR-(SC)2CI method [11]

The Multireference Size-Consistent Self-Consistent CI method of Malrieu et al. [11] includes both EPV and redundancy terms exactly:

ciRi = −∆Eci+  

X X X X  ca ci  hφa|H|φji − hφa|H|Ejφai − hφa|H|Ejφai   ρia a∈P j∈Q j∈Q j∈Q Ejφa∈Q Ejφa=0

The method is proved to be extensive [41]

Determinental formulation was presented in the original papers [11, 41].

The only disadvantage of this method is that no functional can be found for it. (Approximate functional has been defined [42], but this is useless for gradient calculations).

E¨otv¨osLor´and University, Institute of Chemistry 48 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Numerical tests

BeH2

Reference:

2 2 2 2x2 CAS: 1a12a1(3a11b2)

Basis: 3s, 1p for Be and 2s for H

FCI results: Purvis et al.

E¨otv¨osLor´and University, Institute of Chemistry 49 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Numerical tests

a) Be/H2 system using 2x2 CAS reference : Davidson corrections r(BeH2)/r(H2)(a.u.) 2.50/2.78 2.75/2.55 3.00/2.32 error max. MR-CI 0.84 2.01 3.08 1.98 3.08 Davidson correction -2.39 -3.40 -3.26 -3.01 -3.40 Bruckner correction -2.60 -3.90 -3.95 -3.45 -3.95 Davidson-Silver correction -2.85 -4.49 -4.83 -4.05 -4.83 Pople correction (EPC) -1.50 -2.04 -1.78 -1.77 -2.04 Pople correction (E’PC) -1.46 -1.93 -1.61 -1.67 -1.93 Meissner correction -0.54 -0.35 0.26 -0.21 -0.54 b) Pople correction (EPC) -0.88 -0.93 -0.43 -0.74 -0.93 b) Pople correction (E’PC) -0.88 -0.94 -0.44 -0.75 -0.94 Meissner correctionb) 0.27 1.03 1.90 1.06 1.90 a) b) Energies relative to full CI in mhartree; ne=4, i.e. number of valence electrons

E¨otv¨osLor´and University, Institute of Chemistry 50 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 a) Be/H2 system using 2x2 CAS reference : MR-CEPA methods r(BeH2)/r(H2)(a.u.) 2.50/2.78 2.75/2.55 3.00/2.32 av.error max. MR-CI 0.84 2.01 3.08 1.98 3.08 MRLCCM -2.62 -2.40 -5.50 -3.50 -5.50 MR-CEPA(0) -3.28 4.30 -5.50 -1.45 -5.50 QDVPT -2.9 -4.7 -5.5 -4.36 -5.5 MRCEPAb) -1.54 -1.94 -4.81 -2.76 -4.81 MRCEPAc) -1.65 -2.55 -5.88 -3.36 -5.88 MRCPA(4) -1.9 -3.1 -3.6 -2.87 -5.5 MR-ACPF -0.90 -0.90 -0.53 -0.78 -0.90 QDVPT+ACP -0.91 -1.03 -0.53 -0.82 -1.03 MR-AQCC 0.29 1.11 1.98 1.13 1.98 MCCEPA 0.20 0.15 0.54 0.30 0.54 MR-(SC)2-CI 0.54 1.43 0.66 0.88 1.43 MR-ACPF-mc -0.93 -0.76 -2.20 -1.30 -2.20 MR-AQCC-mc -0.19 0.57 0.36 0.25 0.57 a) Energies relative to full CI in mhartree; b) CSF basis; c) Determinental basis

E¨otv¨osLor´and University, Institute of Chemistry 51 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005

BeH2: CONCLUSIONS

• CEPA(0) methods give larger error of the same magnitude → going beyond CEPA(0) is essential

• other corrections (i.e. different description of the relaxation in reference space or correction of the CEPA(0) approximation for the multi-reference case) are not very important compared to the error of the CEPA(0) approximation

• Comparing MRCEPA(0) and QDVPT or ACPF and QDVPT+ACP shows that the different ways of considering the relaxation in the reference space is not important compared to other errors

• All methods but MR-AQCC and MCCEPA overestimate the error of MR-CI giving a lower energy than full CI

• MR-AQCC is inaccurate for the third geometry, MR-AQCC-mc corrects the error

E¨otv¨osLor´and University, Institute of Chemistry 52 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Numerical tests WATER

Reference:

2 2 4 4x4 CAS: 1a12a11b1(3a14a11b22b2)

FCI results: Olsen et al. with cc-PVDZ basis

E¨otv¨osLor´and University, Institute of Chemistry 53 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Numerical tests Davidson corrections: water

5 MR-CI EDC EBC 4 EDS EPC EPC(10el) E'PC 3 E'PC(10el) ) EMC e

e EMC(10el) r t r a

H 2 m ( I C F

o 1 t r o r r E 0

-1

-2 1.0 1.5 2.0 2.5 3.0 O-H distance (in units of re)

E¨otv¨osLor´and University, Institute of Chemistry 54 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Numerical tests MR-CEPA methods: water

5 MR-CI MR-AQCC MR-AQCC-mc 4 MR-ACPF MR-ACPF-mc MR-(SC)2CI 3 MRCEPA MR-CEPA(0)

) MR-AQCC(10el) e

e MR-ACPF(10el) r t

r 2 a H m (

I 1 C F o t

r 0 o r r E -1

-2

-3 1.0 1.5 2.0 2.5 3.0 O-H distance (in units of re)

E¨otv¨osLor´and University, Institute of Chemistry 55 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Numerical tests: water, conclusions

• Davidson-corrections and MR-CEPA methods perform similarly for this case

• the CEPA(0) type methods considerably overestimate the effect of higher excitations

• Pople correction and ACPF give the smallest absolute error: however this correction should not lead to FCI!!!!

• Meissner correction and MR-AQCC gives systematically higher energy than FCI, but the surfaces are very parallel to the FCI one

• The absolute energy of MR-AQCC-mc is very close to MR-(SC)2-CI, which is the “exact CEPA”

• surface of MR-CEPA is nicely parallel, too, showing that the redundancies are important

E¨otv¨osLor´and University, Institute of Chemistry 56 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Test of size-consistency

CH2...nHe system: SR and 2x2 CAS (on CH2) was used with DZP basis

N2: a 6x6 CAS was used (correct dissociation) and DZP basis

O2: a 4x4 CAS was used (correct dissociation) and DZ basis

E¨otv¨osLor´and University, Institute of Chemistry 57 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Test of size-consistency

a) Size-consistency test with the CH2...nHe system n 0 1 3 5 MR-CI 5.03 6.09 8.42 11.03 MR-LCCM -1.27 -1.31 -1.41 -1.51 MR-CEPA(0) -1.27 -1.32 -1.42 -1.51 MR-CEPA -1.19 -1.24 -1.34 -1.43 QDVPT -1.3 -1.3 -1.4 -1.6 MR-ACPF 1.01 0.72 0.44 0.29 MR-ACPF-mc 0.61 0.39 0.17 0.05 MR-AQCC 2.69 2.26 2.03 1.90 MR-AQCC-mc 2.51 2.14 1.85 1.77 a) relative energies with respect to full CI in mhartree, 2x2 reference space

E¨otv¨osLor´and University, Institute of Chemistry 58 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Test of size-consistency

Method monomer dimer ∆ N2 molecule, 6x6 CAS reference MCSCF -54.398748 -108.797497 0.0 MRCI -54.504190 -109.001611 6.770e-3 MR-AQCC -54.505350 -109.010536 1.64e-4 MR-ACPF -54.506569 -109.013429 -2.92e-4 O2 molecule, 4x4 CAS reference MCSCF -74.8005391687 -149.6010783374 0.0 MR-CI -74.8678727888 -149.7330924781 1.32e-3 MRLCCM -74.8693367683 -149.7386664107 3.6e-6 MRCEPA(0) -74.8693367683 -149.7386664101 3.6e-6 MR-ACPF -74.8688312368 -149.7376552841 3.6e-6 MR-AQCC -74.8684397999 -149.7367673283 5.6e-5 MRCEPA -74.85817560 -149.716354256 3.1e-6

E¨otv¨osLor´and University, Institute of Chemistry 59 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Test of size-consistency

CONCLUSIONS

• for CH2...nHe: – all methods perform similarly – error gets smaller with the increasing size of the system!

• for O2 all but AQCC methods give a small extensivity error; but AQCC’s error is two order of magnitudes smaller than CI’s

• for N2 the extensivity error of ACPF and AQCC is larger than for O2, but still smaller than that of CI. ACPF over-compensate the error of CI, AQCC does not All these methods can be classified as ’approximately size-consistent’

E¨otv¨osLor´and University, Institute of Chemistry 60 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Conclusion on Davidson-corrections

Disadvantages:

• relaxation of the coefficients is not included

• energy derivatives are not available

• “redundancy terms” are not included

Advantages:

• Simple a posteriori correction to the energy

• Several variants can be calculated at the same time

E¨otv¨osLor´and University, Institute of Chemistry 61 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Conclusion on Davidson corrections Which Davidson correction is the best? CEPA(0) approximation:

• original Davidson correction

• renormalized Davidson correction (Bruckner correction)

• Davidson-Silver correction (Siegbahn correction)

These usually overestimate higher excitation effects!! Include EPV terms (averaged way):

• Pople correction

• Meissner correction

Use them with ne equals to the number of valence electrons!!!

E¨otv¨osLor´and University, Institute of Chemistry 62 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Conclusion on CEPA-type methods

Advantages:

• relaxation of the coefficients

• energy derivatives

Disadvantages:

• possible convergence problems (non-linear equations)

E¨otv¨osLor´and University, Institute of Chemistry 63 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Conclusion on CEPA-type methods Which MR-CEPA is the best?

Definitely not suggested:

• CEPA(0) methods: MR-CEPA(0), MRLCCM, QDVPT, UCEPA, MRCEPA, MRCPA

According to terms included (theoretical view):

• MR-(SC)2CI, MR-AQCC-mc, (MR-ACPF-mc)

Existence of the functional (point of view of gradients):

• MR-ACPF, MR-AQCC

E¨otv¨osLor´and University, Institute of Chemistry 64 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Conclusion on CEPA-type methods Which MR-CEPA is the best?

According to numerical tests:

• MR-(SC)2CI

• MR-AQCC-mc

• MR-AQCC (near equilibrium)

• MR-ACPF (sometimes fails badly!)

• (MC-CEPA not too much data)

For more details about the whole talk: P.G. Szalay in Encyclopedia Computational Chemistry article cn0066: Configuration Interaction: Correction for Size-Consistency [43]

E¨otv¨osLor´and University, Institute of Chemistry 65 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 References

[1] H. P. Kelly and A. M. Sessler, Phys. Rev. 132, 2091 (1963). [2] H. P. Kelly, Phys. Rev. A. 134, 1450 (1964). [3] W. Meyer, J. Chem. Phys. 58, 1017 (1973). [4] W. Meyer, Theor. Chim. Acta 35, 277 (1974). [5] R. J. Gdanitz and R. Ahlrichs, Chem. Phys. Lett. 143, 413 (1988). [6] L. Meissner, Chem. Phys. Lett. 146, 204 (1988). [7] P. G. Szalay and R. J. Bartlett, Chem. Phys. Lett. 214, 481 (1993). [8] P. G. Szalay and R. J. Bartlett, J. Chem. Phys. 103, 3600 (1995). [9] J.-P. Daudey, J.-L. Heully, and J.-P. Malrieu, J. Chem. Phys. 99, 1240 (1993). [10] P. J. A. Ruttink, J. H. van Lenthe, R. Zwaans, and G. C. Groenenboom, J. Chem. Phys. 94, 7212 (1991). [11] J. P. Malrieu, J. P. Daudey, and R. Caballol, J. Chem. Phys. 101, 8908 (1994). [12] J. L. Heully and J. P. Malrieu, Chem. Phys. Lett. 199, 545 (1992). [13] P. G. Szalay, Towards state-specific formulation of multireference coupled-cluster theory: Coupled electron pair approximations (CEPA) leading to multireference configuration interaction (MR-ci) type equations, in Modern Ideas in Coupled-Cluster Methods, edited by R. Bartlett, pages 81–123, Singapore, 1997, World Scientific.

E¨otv¨osLor´and University, Institute of Chemistry 66 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 [14] P. Pulay, Int. J. Quantum Chem. S17, 257 (1983). [15] E. R. Davidson, in The World of Quantum Chemistry, edited by R. Daudel and B. Pullman, Dodrecht, 1974, Reidel. [16] S. Langhoff and E. Davidson, Int. J. Quantum Chem. 8, 61 (1974). [17] K. Bruckner, Phys. Rew. 100, 36 (1955). [18] E. Davidson and D. Silver, Chem. Phys. Lett. 52, 403 (1977). [19] P. Siegbahn, Chem. Phys. Lett. 55, 386 (1978). [20] J. Pople, R. Seeger, and R. Krishnan, Int. J. Quantum Chem. S11, 149 (1977). [21] P. Bruna, S. Peyerimhoff, and R. Buenker, Chem. Phys. Lett. 72, 278 (1980). [22] R. Shepard, in Modern Electronic Structure Theory, edited by R. Yarkony, volume 1, page 345, World Scientific, Singapore, 1995. [23] W. Duch and G. Diercksen, J. Chem. Phys. 101, 3018 (1994). [24] L. Meissner and I. Grabowski, Chem. Phys. Lett. 300, 53 (1999). [25] L. Meissner and M. Nooijen, Chem. Phys. Lett. 316, 501 (2000). [26] I. Hubaˇc, P. Mach, and S. Wilson, Int. J. Quantum Chem. 89, 198 (2002). [27] W. D. Laidig and R. J. Bartlett, Chem. Phys. Lett. 104, 424 (1984). [28] W. D. Laidig, P. Saxe, and R. J. Bartlett, J. Chem. Phys. 86, 887 (1987). [29] P. Fulde and H. Stoll, J. Chem. Phys. 97, 4185 (1992). [30] M. R. Hoffmann and J. Simons, J. Chem. Phys. 90, 3671 (1989).

E¨otv¨osLor´and University, Institute of Chemistry 67 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 [31] R. J. Cave and E. R. Davidson, J. Chem. Phys. 88, 5770 (1988). [32] W. Kutzelnigg, in Methods of Electronic Structure Theory, edited by H. F. Schfaeffer III, pages 129–188, New York, 1977, Plenum Press. [33] R. J. Cave and E. R. Davidson, J. Chem. Phys. 89, 6798 (1988). [34] K. Tanaka, T. Sakai, and H. Terashima, Theor. Chim. Acta 76, 213 (1989). [35] K. Tanaka, T. K. Ghosh, and T. Sakai, Int. J. Quantum Chem. 74, 661 (1999). [36] L. F¨usti-Moln´arand P. G. Szalay, J. Phys. Chem. 100, 6288 (1996). [37] C. Murray, S. C. Racine, and E. R. Davidson, Int. J. Quantum Chem. 42, 273 (1992). [38] R. Fink and V. Staemmler, Theor. Chim. Acta 87, 129 (1993). [39] T. Sakai and K. Tanaka, Theor. Chim. Acta 83, 451 (1993). [40] K. Tanaka, T. Sakai, and Y. Mochizuki, Multireference coupled pair approximation: a state-universal approach of a cepa type variant of mrsdci, in Recent Advances in Multireference Methods, edited by K. Hirao, pages 95–130, Singapore, 1999, World Scientific. [41] J. Meller, J. P. Malrieu, and J. L. Heully, Chem. Phys. Lett. 244, 440 (1995). [42] J. Meller, J. P. Malrieu, and J. L. Heully, Mol. Pys. 101, 2029 (2003). [43] P.G. Szalay, Configuration Interaction: Correction for Size-Consistency. In onfiguration Interaction: Corrections for Size-Consistency. In von Ragu Schleyer P, Schaefer III HF, Schreiner PR, Jorgensen WL, Thiel W, Glen RC, editors. Encyclopedia of

E¨otv¨osLor´and University, Institute of Chemistry 68 P.G. Szalay: Size-consistency corrected CI Rio de Janeiro, Nov. 27 - Dec. 2, 2005 Computational Chemistry (online edition). John Wiley & Sons, Ltd, Chichester, UK. Article online posting date: 15th February 2005. DOI: 10.1002/0470845015.cn0066

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