Approximate MR-CI methods · Approximate MR-CI methods P eter G. Szalay E otv os Lor and University...

Approximate MR-CI methods

Peter G. SzalayEotvos Lorand University

Institute of ChemistryH-1518 Budapest, P.O.Box 32, Hungary

[email protected]

P.G. Szalay: Approximate MR-CI methods Tianjin, October 10-14, 2016

The Multireference Problem

Example: ozone (O3)

6?

6?

6?

6?

ψ(1) ψ(2)

Both determinants are important for a qualitative description!

1



Example: ozone (O3)

6?

6?

6?

6?

ψ(1) ψ(2)


Both determinants should be used as reference in the truncated scheme!

1



Example: ozone (O3)

6?

6?

6?

6?

ψ(1) ψ(2)



ΦMR−CISD = c(1)ψ(1) + c(2)ψ(2)

1



Example: ozone (O3)

6?

6?

6?

6?

ψ(1) ψ(2)



ΦMR−CISD = c(1)ψ(1) + c(2)ψ(2)

+∑ia

c(1)aiψ(1)

ai +

∑ia

c(2)aiψ(2)

ai

1



Example: ozone (O3)

6?

6?

6?

6?

ψ(1) ψ(2)



ΦMR−CISD = c(1)ψ(1) + c(2)ψ(2)

+∑ia

c(1)aiψ(1)

ai +

∑ia

c(2)aiψ(2)

ai

+∑

i>j a>b

c(1)abij ψ(1)

abij +

∑i>j a>b

c(2)abij ψ(2)

abij

1



6?

6?

6?

6?

ψ(1) ψ(2)

reference

2



6?

6?

6?

6?

ψ(1) ψ(2)

reference

6?

6?

6?

6?

↓ ↓

double excited quadruply excited wrt ψ(1)

2



6?

6?

6?

6?

ψ(1) ψ(2)

reference

6?

6?

6?

6?

↓ ↓


MR-CISD includes higher than double excitation wrt to Hartree-Fock!!

2



6?

6?

6?

6?

ψ(1) ψ(2)

reference

6?

6?

6?

6?

↓ ↓


MR-CISD includes higher than double excitation wrt to Hartree-Fock!!

But not all, only a subset ⇒ substantial saving wrt CISDTQ!!

2



What type of correlation will be included:

correlation type size number method descriptionof the individual contributions

static/non-dynamic large small MCSCF qualitativedynamic small large CI,CC,PT,... quantitative

Therefore:

• MR-CISD, MR-PT (CASPT2), MR-CC are the method of choice if bothdynamic and non-dynamic correlations are present in the system

• Reference space should be small and describe only non-dynamiccorrelation (i.e. what Hartree-Fock can not)

3



orbital type reference CI space

frozen core double occupied double occupied

double occupied

or closed shell

double occupied reference −1,−2

active

or open shell

varying occupation reference ±1,±2

virtual empty empty, +1,+2

4


The (uncontracted) MR-CISD method

Reference space:

ψ(1), ψ(2), ..., ψ(nref)

Single excited space:

ψ(1)ai , ψ(2)

ai , ..., ψ(nref)

ai

Double excited space:

ψ(1)abij , ψ(2)

abij , ..., ψ(nref)

abij

Wave function:

ΨMR−CISD =

nref∑m

c(m)ψ(m) +∑ia

nref∑m

c(m)aiψ(m)

ai +

∑i>j a>b

nref∑m

c(m)abijψ(m)

abij

Expansion length: about nref times that of SR-CISD

5


The (uncontracted) MR-CISD method

Reference space:

ψ(1), ψ(2), ..., ψ(nref)

Single excited space:

ψ(1)ai , ψ(2)

ai , ..., ψ(nref)

ai

Double excited space:

ψ(1)abij , ψ(2)

abij , ..., ψ(nref)

abij

Wave function:

ΨMR−CISD =

nref∑m

c(m)ψ(m) +∑ia

nref∑m

c(m)aiψ(m)

ai +

∑i>j a>b

nref∑m

c(m)abijψ(m)

abij

Expansion length: about nref times that of SR-CISD

First MR-CI calculation of this type was performed by Liu [1] on H3 (1973).

5


MR-CISD: Some important things to consider

• Some of the excited CSFs can be produced from several referencefunctions → redundancy

– in CI we can simply leave them out– for Coupled Cluster this is a big problem

• Some of the excited CSFs do not have non-vanishing matrix elementswith any of the reference functions → one can select just the first orderinteracting space [2]

– not a problem if excitation level defined wrt spinorbitals like in CC

• Single-double excitations out of non-symmetrical reference functionsmight have the right symmetry → these should be included if non-sym-metrical regions of the potential energy surface are also of interest.

• Orbitals: usually from MCSCF using the reference space, but this is notnecessary: it can be larger or even smaller (technical issue).

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Shavitt graph for MR-CISD wavefunction:

• Thick lines: reference CSFs

– run together in the double occupiedand virtual space

– diverge in the “active” space

• both double occupied and virtual partof the graph has simple structure

7


Internally-contracted MR-CISDMeyer [3], Siegbahn [4], Werner and Reinisch [5]

Reference space (with a(m) fixed, e.g. from MCSCF calculation):

{ψ(1), ψ(2), ..., ψ(nref)} → ψ0 =

nref∑m

a(m)ψ(m)

Single excited functions:

ψai ≡ a

+a aiψ0 =

nref∑m

a(m)ψ(m)ai

Double excited functions:

ψabij ≡ a

+a a

+b aiajψ0 =

nref∑m

a(m)ψ(m)abij

Wave function – same expansion length as SR-CISD:

Ψic−MR−CISD = c0ψ0 +

∑ia

cai ψ

ai +

∑i>j a>b

cabij ψ

abij

8


MR-CI ansatze

Expansion functions and their number in different CI ansatze:

Reference Singles Doubles

SR ψ0

ψai ψ

abij

1 nocc × nvrt(nocc2

)×(nvrt2

)

uc-MR a(1)ψ(1) + a(2)ψ(2) ψ(1)ai , ψ(2)

ai ψ(1)

abij , ψ(2)

abij

nref nref × nocc × nvrt nref ×(nocc2

)×(nvrt2

)

ic-MR ψ0 = a(1)ψ(1) + a(2)ψ(2) ψai ψ

abij

1 nocc × nvrt(nocc2

)×(nvrt2

)uc: uncontracted, ic: internally contracted

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ic-MR-CISD by Werner and Knowles [6, 7]

Reference space:

ψ(1), ψ(2), ..., ψ(nref)

Single excited space – not contracted:

ψ(1)ai , ψ(2)

ai , ..., ψ(nref)

ai

Double excited functions:

ψabij ≡ a

+a a

+b ajaiψ0 =

nref∑m

a(m)ψ(m)abij

Wave function:

Ψic−MR−CISD =

nref∑m

c(m)ψ(m) +∑ia

nref∑m

c(m)aiψ(m)

ai +

∑i>j a>b

cabij ψ

abij

Expansion length: not substantially larger than SR-CISD

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Internally-contracted MR-CISD – newest developments:

• more complete contraction by Shamasundar, Knizia, and Werner[8]: alsothose single excitations are contracted which include active orbitals.

• GUGA based ic implementation by Wang, Han, Lei, Suo, Zhu, Song, andZhenyi Wen[9]

• DMRG-based internally contracted MRCI by Saitow, Kurashige, Yanai[10, 11]

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Internally-contracted MR-CISD

What are the advantages?

• the size of the expansion is practically independent of the number ofreference functions

• the contracted configurations span exactly the first order interactingspace of the reference function

• much cheaper than uncontracted version

What are the complications?

• the contracted configurations are, in general, not orthogonal

• structure for the contracted configurations is extremely complex

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What are the approximations?

• much less variational parameter

• coefficients are approximated by product: c(m)abij = a(m) cabij

• semi internal excitations: if {i, j} or {a, b} refer to active orbitals, notall reference can be excited.

What are the problems?

• problem for surface crossing: degeneracy point will be different for thereference space and for the full space.

• no way to systematically estimate the error

• very often not considered as approximation

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Internal contraction (ic) vs. none-contraction (uc):ozone ground stateQUANTUM CHEMICAL CALCULATIONS ON THE POTENTIAL ENERGY SURFACE OF OZONE

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Figure 2: C2v Symmetric Cut and MEP-Cut were calculated with MR-AQCC and ic-MR-AQCC and a cc-pVTZ-basis set. The energy within the C2v Symmetric Cuts is relative to the ozone energy at r1 = 2.4 a.u.,r2 = 2.4 a.u. and ! = 117!, within the MEP Cut relative to r1 = 2.275 a.u., r2 = 3.4 a.u. and ! = 117!.

The dissociation energy predicted at uc-MR-AQCC level is larger by 86 meV to 95 meV than its inter-nally contracted counterpart (see Table 8). In fact, uc-MR-AQCC dissociation energy with a cc-pV5Zeven overestimates the experimental value by 0.030 eV.The results suggest, that the barrier is an artifact of the internal contraction which is also in line withthe experimental finding of mass-independent isotope enrichment effect in ozone. In addition, this mayindicate that the good agreement of the ic-MR-AQCC dissociation energy at CBS limit 1.126 eV andthe experimental value 1.143 eV is owed to error compensation. On the other hand, overestimating theexperimental dissociation energy at the CBS limit with the uc-MR-AQCC by 88 meV is larger than tobe expected at this level of theory.

3.3 (ic)-MR-CISD

The energies calculated with MR-CISD+QP and ic-MR-CISD+QP almost reproduce the respectiveMR-AQCC data (Fig. 4).The a posteriori Pople size consistency correction decreases the barrier height, lowers the van der Waalsminimum and shifts the barrier maximum below the dissociation limit.

Also the MR-CISD energies a posteriorily Pople corrected are in agreement with the observed trendsfor MR-AQCC: the dissociation energies derived from the internally contracted variants only slightlyunderestimate the experimental value and the size-extensivity correction amounts to 0.06 eV at theCBS limit. The uncontracted calculations yield at cc-pV5Z level a higher dissociation energy by0.062 eV (MR-CISD) compared to the ic calculation, a larger with the Pople correction (0.083 eV)also consistently leading to an overestimate of the experimental dissociation energies as compared touc-MR-AQCC.Within the framework of the ic-MR-CISD method it is possible to increasingly betterapproximate the uc-MR-CISD method by increasing the number of reference states. To investigatethe correctness of the uncontracted methods, the energies of r2 = 3.9 a.u. of the MEP (barrier top)

72

• uc curve is less steep

• barrier is disappearing

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ic vs. uc: reef in ozone dissociation channel disappearsQUANTUM CHEMICAL CALCULATIONS ON THE POTENTIAL ENERGY SURFACE OF OZONE

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Figure 4: MEP-Cut was calculated with MR-CISD+QP and ic-MR-CISD+QP, and with MR-CISD and MR-CISD+QP. The energies of the MEP Cut are relative to ozone at r1 = 2.275 a.u., r2 = 3.4 a.u. and ! = 117!.

Table 9: Comparison of the dissociation energies calculated with ic-MR-CISD methods with and without sizeconsistency correction

basis set ic-MR-CISD ( eV) ic-MR-CISD+QD ( eV) ic-MR-CISD+QP ( eV) ic-MR-AQCC ( eV)

cc-pVTZ 0.857 0.895 0.893 0.887cc-pVQZ 0.982 1.032 1.030 1.022cc-pV5Z 1.032 1.089 1.087 1.078cc-pV6Z 1.051 1.111 1.109 1.099

(Q,5) 1.077 1.142 1.139 1.129(5,6) 1.075 1.138 1.136 1.126

Table 10: Comparison of the dissociation energies calculated with MR-CISD methods with and wihtout sizeconsistency correction

basis set MR-CISD (eV) MR-CISD+QD (eV) MR-CISD+QP (eV) MR-AQCC (eV)

cc-pVTZ 0.916 0.984 0.983 0.973cc-pVQZ 1.040 1.124 1.123 1.111cc-pV5Z 1.094 1.191 1.177 1.173

(Q,5) 1.140 1.235 1.234 1.231

Table 11: Relative energies of the barrier top and the van der Waals minimum calculated with ic-MR-CISD+QP,a cc-VTZ basis set and a number of reference wave functions and compared with the MR-CISD+QP-energy

Number of reference wave functions Relative energies ( cm!1)barrier top van der Waals minimum

r2 = 3.9 a.u. r2 = 5.0 a.u.

1 0 02 -75.11 -52.524 -133.11 -82.835 -143.14 -89.35

10 -191.73 -108.36

MR-CISD+QP -3175.07 -2813.97

74

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Internal contraction is also used in:

• MC-CEPA

• CASPT2

• certain MR-CC ansatze

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Externally contracted MR-CI

Introduced in 1983 by Siegbahn [12].

Expansion space consists of the following double excited functions:

ψ(m)ij =∑a>b

c(m)abij ψ(m)abij

i.e. the configuration with the same hole orbitals are contracted.

The contraction coefficients are determined from first-order PT:

cabij =〈ψ0|H|ψ(m)abij 〉

E0 − 〈ψ(m)abij |H|ψ(m)abij 〉

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If the virtual orbitals are placed at the bottom of the Shavitt graph thanthe number of configurations equals the number of the internal walks.

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Computational effort:

• comparable with a single iteration of an uncontracted MRCI calculation.

Applicable:

• for low-lying valence states, but not highly excited states

Not popular nowadays (but see Wang, Ga, Su, Wen [13] or certain PTmethods).

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Doubly Contracted MRCI

Introduced in 2004 by:

• Yubin Wang, Binbin Suo, Gaohong Zhai, Zhenyi Wen[14]

Simultaneous use of internal and external contraction.

Suggested for:

• potential energy surface calculation

Although the initial results were very promising, only a few applicationshave been reported.

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Graphical Contraction Function method

The GCF method of Shepard and coworkers is also a special contracted CImethod which uses the Shavitt-graph.

For details see Ron’s talk.

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Individual selection

Both internal and external contractions make the CI expansion space smaller,however, treat kind of average of the individual expansion functions.

One can also estimate the contribution of individual CSF’s by perturbationtheory, and leave out those with small contribution from the expansionspace.

Two steps of the selection procedure:

• estimation of the contribution

• correction for the missing contribution

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Assume a multiconfigurational reference function:

Ψ0 =

k∑i

aiΦi

Two strategies are used to calculate the individual contribution of a CSF(Φj, j > k):

First, the energy of adding Φj to the space {Φi i = 1, k} can beestimated by perturbation theory (Ak procedure by Gershgorn and Shavitt[15]):

εj =|〈Φj|H|Ψ0〉|2

E0 −Hjj

with E0 = 〈Ψ0|H|Ψ0〉 being the reference energy.

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The second is termed as the Bk procedure byGershgorn and Shavitt [15]:

– diagonalize the matrix which is complete inthe reference space, but only diagonal elementsare present for the external space

– the resulting coefficients cj are used to

calculate the contribution εj =(E−Hjj)c2j

1−c2j

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The selection can be again performed in twoways:

• Threshold selection: if εj is smaller than a given number (usually10−4 − 10−5 hartree), the configuration is not included in the ansatz.

• Cumulative selection: CSF’s are excluded one by one in order ofincreasing contribution until the sum of the contributions of the excludedconfigurations exceeds a threshold (usually 10−1 − 10−2 hartree).

The later shows an advantage for potential surfaces calculation!

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Correction for the neglected configurations: trivially, the εj estimates canbe added to the total energy of the CI procedure.

Disadvantage: perturbation theory overestimate the effect.

Extrapolation procedure by Buenker and Peyerimhoff [16]:

E(0) = E(T ) + λ∆E(T )

E(T ) is the energy with threshold T , ∆E(T ) is the corresponding correction(sum of the excluded energy contributions). λ will be fitted, its value issmaller than 1.

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In praxis:

• MRD-CI by Peyerimhoff and Buenker [17]

• CIPSI by Malrieu et al. [18]

• MELDF by Davidson et al. [19]

• Diesel CI by Hanrath and Engels et al. [20]

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Approximate MR-CI ansatze

• contracted CI (see above)

• individual selection (see above)

• local approaches

– first idea by Saebo and Pulay [21]– more elaborate implementation by Carter et al. [22]

• pseudospectral methods by Friesner [23] and Martinez and Carter [24]

• reduced virtual space methods

– PNO (Pair Natural Orbitals) [25] – also in MC-CEPA

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Inclusion of connected higher excitations

No routine technique yet, but there are several attempts:

• via dressed Hamiltonian by Malrieu et al. [26]

• using Coupled-Cluster equations by Meissner [27]

• including selected higher excitations by Sherrill and Schaefer [28]

• Sychrovsky and Carsky [29] used the so called Bk approximation

• Correlation Energy Extrapolation by Intrinsic Scaling by Bytautas andRuedenberg [30]

• MRCISD(TQ), a hybrid variational-perturbational approach by Khait,Hoffman et al. [31]

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References

[1] B. Liu, J. Chem. Phys. 58, 1925 (1973).

[2] A. Bunge, J. Chem. Phys. 53, 20 (1970).

[3] W. Meyer, in Methods of Electronic Structure Theory, edited by H. F. Schaefer III,

page 413, New York, 1977, Plenum Press.

[4] P. E. M. Siegbahn, Int. J. Quantum Chem. 18, 1229 (1980).

[5] H.-J. Werner and E. A. Reinisch, J. Chem. Phys. 76, 3144 (1982).

[6] H.-J. Werner and P. J. Knowles, J. Chem. Phys. 89, 5803 (1988).

[7] P. J. Knowles and H.-J. Werner, Chem. Phys. Lett. 145, 514 (1988).

[8] K. R. Shamasundar, G. Knizia, and H.-J. Werner, J. Chem. Phys. 135, 054101

(2011).

[9] Y. Wang, H. Han, Y. Lei, B. Suo, H. Zhu, Q. Song, and Z. Wen, J. Chem. Phys.

141, 164114 (2014).

[10] M. Saitow, Y. Kurashige, and Y. T, J. Chem. Phys. 139, 044118 (2013).

[11] M. Saitow, Y. Kurashige, and Y. T, J. Chem. Theory Comput. 11, 5120 (2015).

[12] P. E. M. Siegbahn, Int. J. Quantum Chem. 23, 1869 (1983).

[13] Y. Wang, Z. Gan, K. Su, and Z. Wen, Chem. Phys. Lett. 312, 277 (1999).

[14] Y. Wang, B. Suo, G. Zhai, and Z. Wen, Chem. Phys. Lett. 389, 315 (2004).

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[15] Z. Gershgorn and I. Shavitt, Int. J. Quantum Chem. 2, 751 (1968).

[16] R. J. Buenker and S. D. Peyerimhoff, Theor. Chim. Acta 39, 217 (1975).

[17] R. J. Buenker and S. D. Peyerimhoff, in New Horizons In Quantum Chemistry, edited

by P.-O. Lovdin and B. Pullman, page Dodrecht, 1983, Reidel.

[18] S. Evangelisti, J.-P. Daudey, and J.-P. Malrieu, Chem. Phys. 75, 91 (1983).

[19] Meldf electronic structure codes was developed by l.e. mcmurchie, s.t. elbert, s.r.

langhof, e.r. davidson and d. feller and was extensively modified by d.c. rowlings and

r.j. cave.

[20] M. Hanrath and B. Engels, Chem. Phys. 225, 197 (1997).

[21] S. Saebo and P. Pulay, Annu. Rev. Phys. Chem. 44, 213 (1993).

[22] D. Walter, A. Venkatnathan, and E. A. Carter, J. Chem. Phys. 118, 8127 (2003).

[23] R. Friesner, Annu. Rev. Phys. Chem. 42, 341 (1991).

[24] T. J. Martinez and E. A. Carter, J. Chem. Phys. 102, 7564 (1995).

[25] R. Ahlrichs, F. Driessler, H. Lischka, V. Staemmler, and W. Kutzelnigg, J. Chem.

Phys. 62, 1235 (1975).

[26] J. Sanchez-Marin, I. Nebotgil, D. Maynau, and J.-P. Malrieu, Theor. Chim. Acta 92,

241 (1995).

[27] L. Meissner, Int. J. Quantum Chem. 108, 2199 (2008).

[28] C. Sherrill and H. F. Schaefer, J. Phys. Chem. US 100, 6069 (1996).

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[29] V. Sychrovsky and P. Carsky, Mol. Phys. 88, 1137 (1996).

[30] L. Bytautas and K. Ruedenberg, J. Chem. Phys. 121, 10905 (2004).

[31] Y. G. Khait, H. Song, and M. R. Hoffmann, Chem. Phys. Lett. 372, 674 (2003).

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Approximate MR-CI methods · Approximate MR-CI methods P eter G. Szalay E otv os Lor and University...

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