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Use of Density Functional Theory Orbitals in the GVVPT2 Variant of
Second-Order Multistate Multireference Perturbation Theory
Mark R. Hoffmann*,† and Trygve Helgaker*,‡
†
Chemistry Department, University of North Dakota, Grand Forks, North Dakota 58202, United States
Centre for Theoretical and Computational Chemistry, University of Oslo, N-0315 Oslo, Norway
‡
ABSTRACT: A new variation of the second-order generalized van Vleck
perturbation theory (GVVPT2) for molecular electronic structure is suggested. In
contrast to the established procedure, in which CASSCF or MCSCF orbitals are first
obtained and subsequently used to define a many-electron model (or reference) space,
the use of an orbital space obtained from the local density approximation (LDA)
variant of density functional theory is considered. Through a final, noniterative
diagonalization of an average Fock matrix within orbital subspaces, quasicanonical
orbitals that are otherwise indistinguishable from quasicanonical orbitals obtained
from a CASSCF or MCSCF calculation are obtained. Consequently, all advantages of
the GVVPT2 method are retained, including use of macroconfigurations to define
incomplete active spaces and rigorous avoidance of intruder states. The suggested
variant is vetted on three well-known model problems: the symmetric stretching of
the O−H bonds in water, the dissociation of N2, and the stretching of ground and
excited states C2 to more than twice the equilibrium bond length of the ground state. It is observed that the LDA-based GVVPT2
calculations yield good results, of comparable quality to conventional CASSCF-based calculations. This is true even for the C2
model problem, in which the orbital space for each state was defined by the LDA orbitals. These results suggest that GVVPT2
can be applied to much larger problems than previously accessible.
I. INTRODUCTION
The description of electronic structure in molecules in which
both dynamic correlation and nondynamic (or static)
correlation are significant remains a subject in quantum
chemistry that continues to see active development. In recent
years, significant advances have been made in the frameworks
of multireference perturbation theory (MRPT), especially in
low orders (see refs 1 and 2 for recent comparative studies) and
multireference configuration-interaction (MRCI) theory, including methods with size-extensivity corrections such as the
coupled electron-pair approximation (CEPA) (see ref 3 for a
recent review). There have also been substantial theoretical
advances in multireference coupled-cluster (MRCC) theory,
although fully general computational realizations remain elusive
(see refs 4 and 5 for recent reviews). Each of these frameworks
has particular challenges, in addition to some shared issues.
MRPT methods are particularly challenged by the sizes of
the reference (or model) spaces. Precisely because perturbation
theories are most effective in low-order (e.g., consider the
utility of single-reference CCSD(T) with only the triples
treated perturbatively vis-à-vis MP4 theory with singles,
doubles, and triples treated perturbatively; see the discussion
in Chapter 14 of the monograph ref 6), a significant amount of
the correlation energy in a multireference approach needs to be
captured in the part of the method that is (pseudo)variational.
In contrast, MRCI and MRCC theories tend to have more
modest model spaces and rely more heavily on the corrections.
The necessity of MRPTs (or quasi-degenerate perturbation
© 2014 American Chemical Society
theory (QDPTs), and we will not quibble about distinctions,
real or otherwise) to use large model spaces then translates into
a large number of one-electron molecular orbitals (MOs) with
variable occupancies in the model space (i.e., be considered as
active).
In most MRPT implementations, the model space is of the
complete varietyin other words, the configurational structure
generated by the active orbitals is fully variational (i.e., full
configuration-interaction (FCI) theory). Generally, this approach is referred to as complete-active-space configurationinteraction (CASCI) theory, and with the usual concomitant
optimization of orbitals it is called complete-active-space selfconsistent-field (CASSCF) theory. Of course, the number of
variational parameters to be optimized in a CASSCF calculation
grows exponentially (more precisely, doubly exponentially)
with the number of orbitals. Despite the relatively simple
structures of FCI wave functions, and significant theoretical and
computational advances, calculations with large CASSCF
spaces remain problematic (see, e.g., ref 6) and often are the
limiting steps of use of MRPTs.
Use of incomplete model spaces in MRPTs is one solution,
although this is not without its own problems. In contrast to
Special Issue: 25th Austin Symposium on Molecular Structure and
Dynamics
Received: July 27, 2014
Revised: September 16, 2014
Published: September 17, 2014
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been shown to compensate for variations in accuracies of model
spaces with geometry.
In the present study, we explore the use of LDA MOs in
GVVPT2, restricting ourselves to CASCI model spaces,
although extension to incomplete model spaces should be
possible. After we provide a description of the method in
section II, several well-known model problems are considered
numerically in section III. Section IV summarizes the work.
the cost of CASSCF and CASCI calculations, the cost of
general multiconfiguration self-consistent-field (MCSCF) calculations has polynomial growth with the number of orbitals.
However, the number of active orbitals remains large, so that
the intruder-state problem still needs attention. Also, control of
size-extensivity errors becomes more difficult and, more
generally, there is a significant burden on the user to be
“expert.” Moreover, convergence of general MCSCF wave
functions is significantly more difficult than that of CASSCF
wave functions. Whereas first-order methods are usually
sufficient to converge a CASSCF wave function, general
MCSCF wave functions require second-order methods or
advanced first-order methods that employ whole or partial
spectral decompositions.7 Finally, whereas CASSCF wave
functions have sufficiently simple structures to readily allow
the use of internal contractions for the perturbation
corrections,8 general MCSCF wave functions tend to have
structures that do not allow this simplificationinstead,
efficient perturbation treatments must be made with elaborate
layers of screening such as the use of macroconfigurations.9
For both complete and incomplete model spaces, generation
of MOs in which the many-electron functions (determinants or
configuration-state functions (CSFs)) are expanded can
become the computationally limiting step of MRPT calculations. Few alternatives have been suggested, and fewer yet
have been shown to lead to practicable MRPT treatments. One
of the most successful schemes is the improved orbital scheme
of Chaudhuri, Freed, and co-workers,10 which uses a constrained Fock operator to generate the low occupancy orbitals
and follows from the long-term developments of Freed and coworkers to use a sequence of Fock operators. Connections to
earlier work by Huzinaga11 and by Morokuma and Iwata12 can
be seen. The method has been used successfully in the context
of the effective valence-shell Hamiltonian variant of MRPT and
in the state-specific MRPT (SS-MRPT) of Mukherjee and coworkers.13−15 A philosophical concern with this approach is the
arbitrariness of the orbital from which an electron is removed to
generate the Fock operator used to construct low occupancy
orbitals. Practitioners of this approach have developed robust
rules for its implementation, and it does not seem to be a
practical problem, at least for ground states. More recently, the
emergence of so-called spin-flip methods, in which MOs are
determined from a high-spin, single-determinant Hartree−Fock
(HF) calculation16−19 have been shown to be effective in a
variety of post-HF methods, including MRPTs.20−23 The
concern in this case is the ability to identify a globally relevant
single determinant HF wave function.
In this paper, we suggest and explore an alternative protocol
for obtaining MOs for MRPT calculations. Besides removing
the onus of selecting a special orbital among the highoccupancy orbitals, the suggested approach has a natural
connection with orbitals obtained from CASSCF calculations.
Recall that CASSCF orbitals are determined for use in
describing some amount electron correlation, although the
variation in that “amount” with geometry can itself be
problematic. We suggest that the span of the active orbitals
be obtained as MOs from Kohn−Sham density functional
theory (DFT) in the local-density approximation (LDA).24,25
Such MOs are simple to obtain and unambiguous. Suitability
with nuclear geometrical variation is not a principal concern as
the MRPT of focus for this initial study is the second-order
generalized van Vleck perturbation theory (GVVPT2),26,27
which is rigorously continuous (actually differentiable) and has
II. METHODOLOGY
The GVVPT2 method is a subspace-specific MRPT, in which
the perturbed reference functions are allowed to interact with
the unperturbed functions complementary to the unperturbed
reference functions in the domain of the model space. In the
language of intermediate Hamiltonians, the perturbed primary
functions and the unperturbed secondary functions form a basis
of the effective Hamiltonian,
1
+
Heff
(HPQ X QP + X QP
HQP)
PP = HPP +
(1a)
2
Heff
SP = HSQ X QP
(1b)
Heff
SS = HSS
(1c)
where XQP is the matrix representation of the first-order wave
function. P is the subspace of primary functions, p ∈ P; S
denotes the complement of P in a subspace spanned by the
determinants or CSFs that form a basis of the model space, M,
i.e., P ⊕ S = M; and Q is the complement of M in the full
Hilbert space. Because the electronic Hamiltonian is composed
of only rank-one and rank-two excitation operators, the manyelectron functions that form a basis for Q, q ∈ Q, can be
restricted for the purpose of expressing the first-order wave
function, without approximation, to those functions that are no
more than doubly excited relative to any function in M. An
element of the matrix XQP has the explicit form
̃ (0p) − E(0)
Xqp = tanh(Hqq
p )
Hqp
E(0)
p
̃ (0p)
− Hqq
(2)
where the superscript 0 signifies that the quantities are
unperturbed. The somewhat complicated form of the
coefficients of the first-order wave function ensures rigorous
continuity and preservation of sensible limits. The GVVPT2
has been described in detail in previous publications;26,27 only
salient features relevant to this study as described here.
The model space from which the primary functions are
constructed is spanned by a basis of antisymmetrized manyelectron functions (determinants or CSFs) generated from
sums of products of one-electron functions obtained from a
preceding calculation. The preceding calculation is in fact
arbitrary, although the choice made in all preceding
descriptions of the method has been a CASSCF or general
MCSCF calculation. In this work, the characteristics of
GVVPT2 calculations constructed using MOs derived from a
preceding Kohn−Sham calculation are investigated. As is wellknown, the effective potential is the same for all orbitals
(occupied or virtual) in Kohn−Sham theory, whereas, in
Hartree−Fock theory, the potential is different for different
orbitals. In particular, whereas an electron in an occupied
Hartree−Fock orbital experiences the mean field generated by
the other N − 1 electrons in the system, an electron in a virtual
Hartree−Fock orbital experiences the mean field generated by
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1.0, 1.5, and 2.0 Re).37 In this study, comparison is made with
results obtained in the cc-pVDZ basis.38
The symmetric stretch of O−H bonds in water is principally
of interest because of the strong variation in the amount of
nondynamical correlation: near equilibrium geometries, the
wave function is heavily dominated by a single determinant,
whereas at the longer bond lengths, the wave function is
profoundly multiconfigurational. Contrarily, the dynamic
correlation is larger at the equilibrium geometry, because of
the limited extent of the electron distribution function.
Consequently, the ability of a method to balance the increasing
nondynamic correlation with bond length with the decreasing
dynamic correlation is tested.
A valence orbital space is used to construct the model space.
Specifically, the eight valence electrons were distributed over
the orbitals dominated by the 2s and 2p orbitals of oxygen and
the 1s orbitals of hydrogen; this resulted in a valence orbital
space of 3 a1, 1 b1 and 1 b2 MOs. All CSFs that transformed as
1
A1 defined the model space. The O(1s)-dominated lowestenergy orbital was kept doubly occupied in construction of the
model space but had variable occupancy in the GVVPT2
calculation.
The agreement of GVVPT2 energies, whether based on
CASSCF orbitals or on LDA orbitals, is sufficiently close to FCI
results that a plot is not revealing. Instead, Figure 1 presents the
all N electrons in the system. Consequently, Hartree−Fock
virtual orbitals are more diffuse than Kohn−Sham orbitals and
less suited to describe correlation. For a discussion of the
suitability of Kohn−Sham orbitals for perturbation and CI
treatments, see Gritsenko, Schipper, and Baerends.28,29
Over the years, many approximate exchange−correlation
functionals have been developed for Kohn−Sham calculations.30,31 In the simplest local-density approximation (LDA),
the exchange−correlation functional is obtained by applying
locally relations that are valid globally for the uniform electron
gas.24,25,30 The LDA model provides a surprisingly good
description of molecular electronic systems. In particular, its
exchange−correlation hole remains localized upon dissociation
of H2, suggesting that the LDA model provides a balanced
treatment of exchange and static correlation, which is important
for dissociation. In the generalized gradient approximation
(GGA), where density-gradient corrections are introduced in
the exchange−correlation functional, thermochemistry and
molecular equilibrium properties are typically better described
than with LDA. On the other hand, GGA calculations are more
strongly affected by triplet instabilities than are the LDA
calculations. This is even truer for hybrid DFT, where some
proportion of exact (orbital-dependent) DFT is introduced in
the description, thereby destroying a favorable error cancellation between the description of exchange and correlation in
pure DFT.28,32,33 For this reason, all calculations performed
here use the simplest and most robust LDA functional.
III. RESULTS AND DISCUSSION
Two sets of calculations were performed. In the first set, an
LDA Kohn−Sham calculation was carried out to generate an
orbital basis of the occupied valence orbitals and the lowest
lying unoccupied orbitals, consistent with the irreducible
representations generated by the atomic valence orbitals (i.e.,
the 2s and 2p subshells of nitrogen in dinitrogen, N2, generate 4
a1, 2 b1, and 2 b2 orbitals in C2v symmetry). Then, a CASCI
calculation was performed in this space and the average Fock
matrix constructed and diagonalized. Quasicanonical orbitals
and orbital energies were obtained as eigenvectors of the
rotationally invariant blocks of the average Fock matrix.
GVVPT2 calculations were performed using model and
external spaces spanned by antisymmetrized many-electron
functions constructed from these final orbitals. In the second
set of calculations, a CASSCF wave function of the same
specification as the CASCI wave function described above was
obtained, followed by the identical procedure for determining
final orbitals and subsequent GVVPT2 calculation.
Calculations were performed using the UNDMOL suite of
molecular electronic structure programs.34 The current codes
cannot make use of non-Abelian point groups, so that full use
of D∞h was not possible for C2 and N2. To allow for spin
polarization of the LDA orbitals in the N2 study, the C2v
subgroup of D2h was used. In all correlated calculations based
on LDA orbitals, the spatial parts of the alpha spin−orbitals
were used as MOs.
III.A. Symmetrically Stretched Water. Description of the
symmetric stretch of the two O−H bonds in water has been a
staple for the assessment of new methodology, especially since
the pioneering FCI results of Handy and co-workers using a
DZ basis.35,36 More recent comparisons tend to use the (allelectron) FCI/cc-pVDZ results, which also consider a greater
geometric variation (i.e., including 2.5 Re and 3.0 Re to results at
Figure 1. Energy differences (mH) of GVVPT2 calculation based on
MCSCF orbitals (black squares) and based on LDA orbitals (red
circles) from full CI results (data from ref 37).
difference between the GVVPT2 and FCI results. As can be
seen, the variation in geometry is quite small, GVVPT2(CAS)
values varying between about 11 and 5 mH and the
GVVPT2(LDA) results varying between about 7 and 3 mH.
More precisely, the nonparallelity error (NPE) (i.e., the
difference between the maximum deviation and minimum
deviation) is 6.0 mH for the GVVPT2(CAS) calculation and
3.7 mH for the GVVPT2(LDA) calculation. Moreover, neither
curve showed sharp or irregular features. The GVVPT2(LDA)
curve lies below the GVVPT2(CAS) curve and is closer to the
FCI curve; however, this is not a general feature of the
GVVPT2(LDA) model as the situation is the reversed for N2 in
a larger basis, as discussed in the next subsection.
III.B. Dinitrogen. The dissociation curve of the ground state
of dinitrogen, N2, is problematic for a number of methods,
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the CASSCF, CASCI, and GVVPT2(CAS) curves have
asymptotes that cluster around 0.33−0.34 H, whereas the
GVVPT2(LDA) asymptote at 0.36 H is a little higher. It can be
seen that, although the GVVPT2(CAS) curve is parallel to the
MRCISD curve at essentially all bond lengths (including the
difficult crossover region, see insert), the GVVPT2(LDA)
curve, though still satisfactory, deviates somewhat more at
longer bond lengths. The MRCISD dissociation energy at 0.348
H lies between the two values, but slightly closer to the
GVVPT2(CAS) limit of 0.341 H than to the GVVPT2(LDA)
limit of 0.360 H. It is tempting to speculate that the higher
energy of the GVVPT2(LDA) method relative to the
GVVPT2(CAS) method is related to the performance of the
LDA method itself, which greatly overestimates the energy in
the dissociation limit; see ref 38. However, we also note that
the MRCISD energy is not size extensive and that a sizeextensivity correction would increase the MRCISD dissociation
energy, making it difficult to draw firm conclusions regarding
the relative performance of the GVVPT2(LDA) and GVVPT2(CAS) methods
III.C. Dicarbon. The ground and two lowest-lying excited
spin-singlet states of C2 are problematic because of the strong
geometry-dependent proximities of curves, which undergo an
avoided crossing between 11Σ+g and 21Σ+g , an actual crossing
between 11Σ+g and 11Δg within the interval of 1.5−2.0 Å and a
crossing between 21Σ+g and 11Δg at a shorter bond length; see
Figure 3. The FCI calculations of Abrams and Sherrill, using a
primarily because of the strong interbond correlation, with
concerted motion of at least three electrons. Consequently, the
CCSD method39 fails and, perhaps not surprisingly, so does the
CCSD(T) method.40 In contrast, even rather modest multiconfigurational methods obtain qualitatively correct answers.
Thus, the dinitrogen model problem is of interest partly
because the MOs are not obtained with the benefit of selfconsistency, thus putting the onus of orbital relaxation on
GVVPT2. This model problem is also of interest because a
relatively large, flexible basis set, namely, the aug-cc-pVTZ
basis,41 has been used. Precisely because of the diffuse basis
functions, the lowest lying unoccupied orbitals of a Hartree−
Fock calculation would be entirely unsuitable to describe
correlation. Indeed, practical experience shows that such
orbitals have sufficiently small overlaps with the final correlating
orbitals that they are not even good starting orbitals for
CASSCF calculations. Consequently, the ability of the Kohn−
Sham LDA method to obtain good low occupancy orbitals lying
above the usual occupied orbitals is tested here.
Calculations were performed at 44 bond lengths, with higher
density near the experimental equilibrium geometry (Re =
1.0975131 Å). The CAS model space was generated by
distributing the 10 valence electrons among the 8 valence
orbitals, in all ways consistent with a spin singlet and the A1
irrep of the C2v point group. The 1s-dominated orbitals were
held doubly occupied in all calculations.
There are six potential energy curves in Figure 2, calculated
with the LDA method, with the CASSCF method and the
Figure 3. Potential energy curves of the 1,21Σ+g (black and blue,
respectively) and 11Δg (red) states of C2. Solid lines are Bézier
interpolations of data from ref 42; points are results of GVVPT2(LDA) calculations.
Figure 2. Potential energy curves of the ground state of N2 as a
function of internuclear distance (Å) relative to the minimum energy
of a particular method: LDA results in gray; CASCI in orange;
CASSCF in dark cyan; GVVPT2, based on CASSCF orbitals, in red;
GVVPT2 based on LDA orbitals in blue; MRCISD in black. The inset
shows an expanded view near the crossover from molecular to atomic
regimes.
DZP basis,42 have provided a useful model problem on which
methods have been assessed. This model problem is particularly
challenging for LDA-derived orbitals, because only ground-state
LDA orbitals were used to define the model space.
Calculations were performed at all geometries reported by
Abrams and Sherrill,42 except the shortest, and three additional
points were calculated near the 11Σ+g and 11Δg crossing (at 1.55,
1.65, and 1.75 Å). Following the FCI study, the two low-lying
C 1s dominated MOs were kept doubly occupied in all
calculations. LDA calculations were performed on the ground
electronic state at all geometries. Quasi-canonical orbitals were
generated from an average Fock matrix obtained from a one-
MRCISD method in the same CAS space, with the CASCI
method using LDA orbitals, and with the GVVPT2(LDA) and
GVVPT2(CAS) methods. The energy curves are virtually
identical near the equilibrium distance, staying close to one
another as the distance increases to about 1.7−1.8 Å. At that
distance, which corresponds to the beginning of the transition
to atomic-like orbitals, the curves gradually spread until the
onset of the asymptotic region near 2.3 Å. It is interesting that
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Notes
particle reduced density matrix with a (1:1:1) weighting of
density matrices from CI calculations in the space of model
space LDA orbitals.
As can be seen from Figure 3, the GVVPT2(LDA) results for
the 11Σ+g state (black triangles) are close to the FCI results
(shown as a continuous black curve). Indeed, the largest
deviation is only 22 mH, and the NPE over the considered
abscissae is 7 mH. Furthermore, and perhaps more interesting,
is the close agreement of the GVVPT2(LDA) curves for the
21Σ+g and 11Δg states and the FCI curves, using an orbital space
f rom ground-state LDA calculations. For 21Σ+g , the maximum
deviation from the FCI curve is slightly larger than 22 mH and
the NPE is 8 mH; similarly, the maximum deviation for the
11Δg state is 22 mH, whereas the NPE is here only 3 mH.
Consequently, the GVVPT2(LDA) curves are well behaved in
the crossing and avoided crossing regions.
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS
This work was supported by the Norwegian Research Council
through the CoE Centre for Theoretical and Computational
Chemistry (CTCC) Grant Nos. 179568/V30 and 171185/V30
and through the European Research Council under the
European Union Seventh Framework Program through the
Advanced Grant ABACUS, ERC Grant Agreement No. 267683.
M.R.H. is grateful to the NSF (Grant No. EPS-0814442) for
additional support.
■
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IV. CONCLUSIONS
The use of MOs derived from Kohn−Sham LDA calculations
for use in GVVPT2 calculations has been suggested and
demonstrated to be efficacious for three widely studied
representative problems. Each of the three model problems,
the symmetric stretching of the O−H bonds in H2O, the
dissociation curve of N2 and the stretching of the bond in C2 to
more than twice its equilibrium value, represent situations with
strongly varying amounts of quasidegeneracy (with geometry),
so that the usual GVVPT2 paradigm of nondynamic correlation
being treated by the underlying MCSCF expansion is clearly
challenged. In the case of the H2O model problem, which uses
a cc-pVDZ basis, the maximum energy difference is 7.1 mH,
with an NPE of 3.4 mH. Moreover, the difference curves are
very smooth. The GVVPT2 relative energy curves for N2, using
an aug-cc-pVTZ basis, based on an LDA orbital space resulted
in an NPE of 18.6 mH (relative to best available results, which
was MRCISD results based on CASSCF orbitals). It is
noteworthy that the presence of diffuse basis functions does
not invalidate the suggested protocol, although it should be
noted that the unoccupied LDA orbitals in N2 were still of
valence character. The performance of the suggested protocol
when yet larger basis sets are used, and specifically when the
unoccupied orbitals acquire greater Rydberg character, remains
to be tested. In many ways, the most remarkable result was for
the C2 molecule, with DZP basis. It was shown that GVVPT2
based on LDA orbitals for the ground state was able to describe
not only the 11Σ+g ground state (8 mH NPE) but also the 21Σ+g
and 11Δg excited states (8 and 3 mH NPE, respectively).
The use of ground-state LDA orbital spaces for subsequent
GVVPT2 calculations appears to be robust for situations in
which the LDA Kohn−Sham method itself is not expected to
be accurate. This study extends the usefulness of GVVPT2,
because the potentially time-consuming (and arguably user
biased) optimization of MOs in a preceding CASSCF (or
MCSCF) step appears unnecessary. As a result, we expect that
larger molecules will become accessible to GVVPT2. One may
also suspect that LDA orbital spaces would provide efficacious
support not only for GVVPT2 (and MRCISD) but also for
other effective Hamiltonian methods.
■
REFERENCES
AUTHOR INFORMATION
Corresponding Authors
*M. R. Hoffmann. E-mail: mark.hoff[email protected].
*T. Helgaker. E-mail: [email protected].
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