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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
5
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
9
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
10
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
11
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
,
,
14
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)
15
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
16
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.
17
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.
18
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
19
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
20
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)
21
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
22
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)
23
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)
24
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.
25
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/
26
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