Multireference Configuration Interaction:

Methodological Aspects and Applications

Hans LischkaUniversity of Vienna

2

The concept of a reference wave function

A simple example: H2 (1Σg+)

In a minimal basis we have two orbitals - σgand σu

Ground state MO scheme

At R ≈ Rmin configuration I will be a qualitatively good representation of the exact wave function. However, for R→∞ the second orbital, σu, will become degenerate or quasi-degenerate and configuration I will not describe the system well. Within the given symmetry a second configuration σu

2 has to be taken into account.

R→∞σg

σu

I

σgσu

II

3

Generalization to more electrons

Reference wave function Ψ0:

nnΦΦΦΦ=Ψ K11HF

M doubly occupiedOrbitalsM = n

virtualorbitals

Single reference (SR) case (closed shell)

4

Construction of the SR wave function

excitation (substitution) of occupied orbitalsby virtual ones

nnii ΦΦΦΦΦΦ KK11

nnia ΦΦΦΦΦΦ KK11

ai Φ→Φ

Single-, double-, triple- … m-tuple excitations

Kabcijk

abij

ai ΨΨΨ ,,

Method of configuration interaction (CI):

K+Ψ∑+∑ Ψ+Ψ=Ψ −ab

ijbaji

abij

ai

ai

aiI ccc

,,,,00CSR

Variation principle (Ritz)Dynamical electron correlation

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Multireference (MR) case

M occupiedorbitals, 2n El.

M > n

virtualorbitals

Reference wave function Ψ0: Nondynamical electron correlationm-tuple excitations from

into the virtual orbitals creates a set of configurations {ΨI}

Dynamical electron correlationApplication of the Ritz variation principle leads to the MR-CI approach

refNΨΨK1

refi Ni K1, =Ψ

∑ Ψ=Ψ − IIcCIMR

6

MR-CI approach – general considerations

Stationary, nonrelativistic, clamped-nuclei, electronic Schrödinger equation

Ψ=Ψ EH

nnji

iji

i VghH ++= ∑∑<

∑−∇−=A

AiAii RZh 221

ijij rg 1=

∑<

=BA

ABBAnn RZZV

The linear ansatz for the trial function Ψ(Ritz variation principle)

Variational principle[ ] ΨΨΨΨ=Ψ HE

∑=

Φ=Ψn

sssc

1

7

leads to the following matrix eigenvalue problem

niE iii K1, == cHcwith

tsst HH ΦΦ=

and assuming orthonormality of the Φ’s, i.e.

stts δ=ΦΦ

General references:Shavitt 1977, Szabo 1989, Helgaker 2002

Energies are ordered as

nEEE ≤≤≤ K21

Each eigenvalue Ep is an upper bound to the corresponding exact eigenvalue. As additional terms are added to the expansion each eigenvalue Ep

n+1 of the (n+1)-term expansion satisfies the inequalities

np

np

np EEE ≤≤ +−

11

8

Davidson subspace method for diagonalization

For practical calculations we have to make several decisions concerning• the representation of the many-electron basisfunctions Φs

• Slater determinants• Configuration state functions (CSFs) –eigenfunctions of S2

• How to calculate the Hamiltonian matrix H(calculate it at all?). In most applications thedimension of H will be too large for explicitstorage.

Davisdon 1975, 1993

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Algorithm:A. If the kth eigenvalue is wanted, select a zeroth-order

orthonormal subspace v1, v2,…vl (l≥k) spanning the dominant components of the first k eigenvectors.Form Hv1 … Hvl and (vi,Hvj) = , 1 ≤ i ≤ j ≤ l. Diagonalize using a standard method, select the kth eigenvalue λk

(l) and the eigenvector αk(l).

B. Form the residual vector

M is the dimension of .

C. Form ||qM || and check for convergence.

D. Form

(and orthogonalize to the vi, i = 1…M).

E. v(M+1) = d(M+1)/||d(M+1)||.

F. Form Hv(M+1)

G. Form

H. Diagonalize and return to step B with

ijH~

H~

( ) ∑∑==

−=M

ii

Mk

Mki

M

ii

MkiM

1

)()(,

1

)(, vHvq λαα

H~

( ) NIqHd MIIIMkMI K1,,

1)()1(, =−λ=

−+

( ) .11,,~)1(1, +== ++ MiH MiMi KHvv

H~

.and )1()1( ++ λα Mk

Mk

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The main computational work is the matrix-vector multiplication w = Hv.Options:• Compute H, store it on disk, and compute

Hvi afterwards – conventional CI• We do not need to construct H, we only

need wi = Hvi! This is the basis of the “direct CI” (Roos 1972, Roos 1977)

Matrix elements

Orbital integrals hij and gijkl are defined byand

∑∑ +=ΦΦijkl

ijklstijkl

ijij

stijts gbhaH

)( jiij hh ϕϕ= 1

)( lkjiijkl gg ϕϕϕϕ= 12

∑∑ ∑∑+=t ij t ijkl

tijklstijkltij

stijs vgbvhaw

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The coupling elements are simple (-1 or +1) in case of single determinants (Slater-Condon rules). For CSFs several possibilities exist (Shavitt 1977):• Spin projection• Symmetric group• Unitary group

(Graphical) Unitary Group Approach (G)UGAWe are using Gelfand statesHamiltonian:

where and are spinorbital creation and annihilation operatorsand the sum is over the spin.

)(∑∑ +=ijkl

klijij

ijij eklijEhH ,

∑σ

σ+σ= jiij XXE +

σiX σjX

βα=σ ,

ilkjklijijkl EEEe δ−=

12

A matrix element of H between two Gelfandstates and is given as

Thus, the matrix elements of the one-body and two-body unitary group operators are the coupling coefficients of the corresponding one- and two-electron integrals, respectively.

Structure of the UGA graph for a MR-CISD calculation

(Siegbahn 1979, 1980)The orbitals are arranged in groups (from top to bottom): active orbitals, closed-shell orbitals, virtual orbitals.

m′ m

)(∑∑ ′+′=

′=′

ijklklij

ijijij

mm

memklijmEmh

mHmH

,

,

13

Shavitt 1981

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The internal (active + closed shell) part of the graph is complicated, but relatively small in comparison to the virtual (external) space. The graph for the external is space is very simple due to the fact that we allow only double excitations. Moreover, its structure is independent of the internal part. Respective loops (coupling elements) can be computed once and for all. The interface between internal and external space is given by the vertices Z (0-excitations), Y (single-excitations), X (double excitations, triplet coupling) and W (double excitations, singlet coupling)

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The loops are split into an internal part and into an external one. The internal part is computed explicitly and either stored on a file (formula file) or recomputed every Davidson iteration. The external part is added “on-the-fly” when the total coupling elements are computed.The two-electron integrals are sorted according to the number of internal indices: all (four)-internal, three-internal, two-internal, one internal and all-external

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Algorithm for w=Hv

∑∑ ∑∑+=t ij t ijkl

tijklstijkltij

stijs vgbvhaw

In this approach (COLUMBUS program, Lischka 1981, Shepard 1988, Lischka 2001, 2003) the computational scheme is driven by the integral indices

Loop over types of two-electron integralsFor each set

Loop over the quadruple of indicesGet coupling elements, integrals and vCompute the respective contribution to wUpdate w

End loop indicesEnd loop integral types

The remaining steps of the Davidson iteration are simple.

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The update of w can be performed via matrix and vector operations if the whole virtual space if the same virtual orbital space is used for excitations from all reference configurations. In this case selection schemes can only be applied at the level of the reference configurations.

Individual selection schemes specific for each reference configurations (e.g. multireference double excitation CI (MRD-CI) Buenker1968, 1974, 1975, Hanrath 1997) do not have this restriction.

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Overall CI scheme

• AO integrals• Determination of MOs by means of a SCF

or MCSCF calculation• Transformation of the two-electron

integrals into the MO basis• Sorting of the integrals• Davidson diagonalization• Properties• Energy gradient and nonadiabatic

couplings• Geometry optimization routines

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Overview of different CI approaches

• Selection at the reference configuration level– Uncontracted CI: single- and double

excitations are constructed from each reference CSF and the unique set of configurations (ΨI, I = 1…NCI) is taken. The CI coefficient of each CSF is included in the variational procedure.

– Contracted MRCI (IC-MRCI) (Meyer 1977, Werner 1988):

(p = ±1 for external singlet and triplet pairs)

The configurations span exactly the first-order interacting space

( ) 0,,21 Ψ+=Ψ ajbibjaiab

ijp pEE

∑µ

µµΨ=Ψ

RR

Ra0

abijpΨ

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Uncontracted vs. contracted CI

Uncontracted CIAdvantages:• Larger flexibility of the wave function, full variation, Excited states!• Computation of analytic gradient relatively easy (available in COLUMBUS)

Disadvantage:Much larger computational effort

Contracted CIAdvantages:• High

computational efficiency

Disadvantages:• Relaxation effects

for Ψ0

• Potential energy surfaces for excited states, avoided crossings,…

• Analytic gradient difficult (not available)

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Overview of MRD-CI

Original work: Buenker 1968, 1974, 1975• Selection scheme based on perturbation

theory – energy threshold T• Extrapolation scheme T → 0Conventional CI program• Hanrath 1997:

DIESEL-MR-CI (direct internal external separated individually selecting MR-CI):direct, division of internal/external space

Advantage of MRD-CI: great flexibility, relatively large molecules

Disadvantage: Extrapolation, gradient for extrapolated energy

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When do we use which method?

• MR-CI, MR-CI+Q, MR-AQCC (Peter Szalay) in multireference cases, otherwise use single reference method

• When geometry optimization at the CI level is not required or when pointwiseoptimization is possible – IC-MRCI or MRD-CI

• Difficult cases: robust method is required for description of large portions of the energy surfaces – uncontracted MR-CI or MR-AQCCSpecial feature: analytic energy gradients and nonadiabatic couplings (conical intersections)

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Characterization of Internal Orbital spaces

frozen core

Reference doublyoccupied

active orbitals

auxiliary orbitals

Active orbitals: usually as CASAuxiliary orbitals: for the description of Rydberg states (single excitations or individual configurations)

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Selection of CSF space

Three options:1. a) The reference configurations are

chosen to have the same symmetry as the state symmetryb) Only those singles and doubles are constructed, which have a nonzero matrix element with one of the references (interacting space restriction (Bunge1970)

2. Restriction 1b is lifted3. All possible reference configurations are

constructed within the specified orbital-occupation restrictions. Singles and doubles from reference with ”wrong”symmetry can have the correct symmetry.

Condition 3 allows consistent calculationsusing different subgroups of the actualmolecular symmetry.

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COLUMBUS project

• Set of programs for high-level ab initiocalculations

• Methods: MCSCF, MR-CISD, MR-ACPF/AQCC, Spin-orbit CI

• Focus: multireference calculations on ground and excited states

• Recent achievements: analytic MR-CI gradient, nonadiabatic couplings, parallel CI

• Authors: R. Shepard, I. Shavitt, H.L.– Vienna: Th. Müller (now Jülich), M.

Dallos (now industry), M. Barbatti– Budapest: P. G. Szalay

• Web page: http://www.univie.ac.at/columbus/

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COLUMBUS program package

• http://www.univie.ac.at/columbus/• Recent review: H. Lischka, R. Shepard, R.

M. Pitzer, I. Shavitt, M. Dallos, Th. Müller, P. G. Szalay, M. Seth, G. S. Kedziora, S. Yabushita, and Z. Zhang 2001, Phys. Chem. Chem. Phys. 3, 664

• User-friendly interface• Public domain – free of charge• Distribution of source code