Multiconfigurational Perturbation Theory With Grimme Scaling

Multiconfigurational perturbation theory with separately scaled energy terms ´ P´eter Nagy and Agnes Szabados Laboratory of Theoretical Chemistry, E¨otv¨os University,Budapest, Hungary [email protected] Short content

Introduction

• Feenberg-scaling in perturbation theory (PT)

A widely applied approach in quantum chemistry to obtain wavefunctions and energies is perturbation theory (PT). There exist numerous multireference function based formulations which provide perturbative corrections. Multiconfigurational PT (MCPT) is one of these[1]. To improve the second order Møller-Plesset PT a special scaling philosophy was introduced by Grimme [4] in the single-reference framework. This scaling is based on the systematic separation of the different contributions of doubly excited states. It was shown[2] that a similar scaling can be obtained by a system-dependent optimization procedure, as suggested by Feenberg[3]. In this work scaling methods are investigated within MCPT, following the ideas of Feenberg and Grimme. We introduce various novel scaling strategies suited for covalent bond dissociation. The basic theory of the applied methods and pilot applications on small model systems are presented.

• Grimme-scaling in Møller-Plesset PT • Multiconfigurational PT (MCPT) • Grimme-scaling as a two parameter Feenberg-scaling • Extension of the theories to MCPT • Results and conclusion

Feenberg-scaling in PT

Multiconfigurational PT

ˆ =H ˆ (0) + W ˆ • Initial partition: H µ 1 (0) ˆ − ˆ ˆ ˆ (0) µ ∈ ℜ • Repartition: H = H +W H 1 {z −µ {z } | } |1 − µ ˆ (0)′ H

ˆ′ W

3 ∂ P • Determine µ from: ∂µ E (i)′ = 0

(0)

• Reference function: Ψ0

P

k6=0

ck |ki

|0i: principal determinant, |ki: excited determinant

• Separate the energy into terms BG and AG,H: B and A:

• Bi-orthogonal basis set in the CI space: (0)

(2) 2

)

• The new energy: E (2)′ = E (2)−E (3) and E (3)′ = 0

Grimme’s spin-component scaling (SCS)

(2)

EMCPT =

+ − − • Separate doubles: parallel (’triplet’, |Tk i = b+ σ aσ iσ jσ |0i) + − − and anti-parallel (’singlet’, e.g. |Sk i = b+ a σ σ iσ jσ |0i)

(3)

EMCPT =

−

˜W ˆ |kihk| ˆ |Ψ(0) 1 h0|W 0 i c0 ∆k

k6=0 1,2× P P 1−4× k6=0 l6=0

E (3)(p) =

=

P

k6=0

bk

• System independent scaling parameters: pS = 1.2; pT = 13 2× 2× P |h0|H|Sk i|2 P |h0|H|Tk i|2 (2) − pS ESCS-MP2 = −pT ∆k ∆k Sk Tk Parameters pS and pT can be determined following Feenberg, but the concept of two parameters requires generalization.

k

groups

bk =

groups

P

pGpH

P

pGBG

G

in H in G P P

k6=0 l6=0

akl =

˜W ˆ |kihk| ˆ |lih˜ ˆ |Ψ(0) h0|W l|W 0 i = c0∆l ∆k

(pGpH(AGH + AHG) − δGHpGBG)

G,H

P

k,l6=0

• Substitution to (2): A p = B provide the paramerters

akl

Specification of the partitioning • (Møller–Plesset: nondiagonal H (0) in MRPT) • (Epstein-Nesbet: degeneracy leads often to divergence)

◦ S(TLJ): two groups, S and the others, two optimized param. ◦ Opt-Fee: one group, no separation, one optimized param. ◦ Grimme: scaling the S and T groups with Grimme‘s param. ◦ STLJ: four groups, four optimized param.

|Tki

in G P

G,H

P

• Grouping variants: |Sk i

pG

groups

• Singles (L) , triples and quadruples (J) also appear, at difference with ordinary MP2 and MP3

|0i

P G

• Bi-orthogonal MCPT formulae (definition of b and a): 1,2× P

groups

E (2)(p) =

|Ψ0 i and |ki ; k = 0, . . . , NF CI ˜ = hk| − ck h0| ˜ (0)| = 1 h0| and hk| hΨ 0 c0 c0

(1)

i=0

(E

Generalized Feenberg scaling ˆ =H ˆ (0)′(p) + W ˆ ′(p) • Partitioning: H   ∂ • Determine (p) from (1): ∂p E (2)(p) + E (3)(p) = 0

model space

= c0|0i +

• Davidson-Kapuy: DK DK ˆ ˆ ∆DK k = Ek − E0 = hk|F |ki − h0|F |0i

• Denominator as a difference of IPs and EAs [5]: occ. virt. P P ∆IP/EA = ǫ⊕ ǫ⊖ a k i − ǫ⊕ i =

Averaging method

i i (0) (0) ˆ (0) ˆ |Ψ(0)i hΨ |a+Ha hΨ0 |H|Ψ 0 i − 0 (0) i + i (0)0 (ǫ⊖ (0) (0) a analogously) hΨ0 |Ψ0 i hΨ0 |ai ai|Ψ0 i

• Averaging before determining the scaling factors, • sum for certain principals, |ki in the model space • weight them with their importance: BGAV =

Scaling in SR case

model Pspace kP

c2k BGk

c2k

and analogusly for AAV GH

Scaling in DK partitioning

Bond fission of HF molecule, 3−21G basis, IP/EA partitioning

Bond fission of HF molecule, 3−21G basis

0.1 0.005

CAS 2x2 ref. SRPT2 SRPT3 S(TLJ)−SRPT3 Opt−Fee−SRPT3 0.06 Grimme−SRPT2 MCPT3

0.012

Consequences

0.009 0.006

0.04

0.003

0.02

−0.003

∆E = EPT−EFCI / a. u.

∆E = EPT−EFCI / a. u.

0.08

• Essential, non-obvious factors: the partitioning, the grouping strategy and the choice of the principal determinant

0 0.8

1

1.2

Summary

1.4

0

0

−0.005

MCPT2 MCPT3 S(TLJ)−MCPT3 Opt−Fee−MCPT3 Grimme−MCPT2 STLJ−MCPT3

−0.01

• Feenberg-scaling is generalizable for the MR case 1

2

3 4 Bond length / Å

5

6

• Scaled SRPT does not compete with MCPT upon dissociation

1

• We explored several partitionings, grouping variants and we averaged for principal determinants • There are cases where scaling may provide better results

2

3 4 Bond length / Å

5

6

• Some scaling variants ameliorate MCPT, e.g. Opt–Fee, Grimme • Grouping matters a lot (especially scaling of non-doubles)

Further plans Scaling in IP/EA partitioning

Scaling, averaging and damping

• Find a generally applicable scaling strategy

Bond fission of HF molecule, 3−21G basis

Bond fission of HF molecule, 3−21G basis

• Reduce the calculation cost by finding Grimmetype system independent scaling parameters

0.01

AV−MCPT3 AV−S(TLJ)−MCPT3 AV−Opt−Fee−MC AV−STLJ−MCPT3 NONAV−MCPT3 NONAV−S(TLJ)−MCPT3 NONAV−STLJ−MCPT3

References

0.004 0.002 0

´ Szabados, Z. Rolik, G. T´oth, and P. R. [1] A. Surj´an. J. Chem. Phys., 122, 114104, (2005). ´ Szabados. J. Chem. Phys., 125, 214105, [2] A. (2006).

MCPT2 MCPT3 S(TLJ)−MCPT3 Opt−Fee−MCPT3 Grimme−MCPT2 STLJ−MCPT3 for comparison: DK−MCPT3

−0.002 −0.004 −0.006 1

2

3 4 Bond length / Å

5

[3] E. Feenberg. Phys. Rev., 103, 1116, (1956).

0.006

0.004

0.002

[4] S. Grimme. J. Chem. Phys., 118, 9095, (2003). 6

• IP/EA denominators remove irregular behaviour at large distance • Scaling may improve MCPT, but the grouping is still an issue

0.008 ∆E = EPT−EFCI / a. u.

∆E = EPT−EFCI / a. u.

0.006

[5] A. Zaitsevskii, JP. Malrieu. Chem. Phys. Lett., 223, 597, (1995).

0

1

2

3 4 Bond length / Å

• Averaging and damping improves parallelity

5

6

(2)