Variational optimization of the two-electron reduced-density matrix under pure-state
N-representability conditions
A. Eugene DePrince III
Citation: The Journal of Chemical Physics 145, 164109 (2016); doi: 10.1063/1.4965888
View online: http://dx.doi.org/10.1063/1.4965888
View Table of Contents: http://aip.scitation.org/toc/jcp/145/16
Published by the American Institute of Physics
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THE JOURNAL OF CHEMICAL PHYSICS 145, 164109 (2016)
Variational optimization of the two-electron reduced-density matrix under
pure-state N -representability conditions
A. Eugene DePrince IIIa)
Department of Chemistry and Biochemistry, Florida State University, Tallahassee, Florida 32306-4390, USA
(Received 20 July 2016; accepted 10 October 2016; published online 27 October 2016)
The direct variational optimization of the ground-state two-electron reduced-density matrix (2-RDM)
is typically performed under ensemble N-representability conditions. Accordingly, variationally
obtained 2-RDMs for degenerate ground states may not represent a pure state. When considering only
ground-state energetics, the ensemble nature of the 2-RDM is of little consequence. However, the use
of ensemble densities within an extended random phase approximation (ERPA) yields astonishingly
poor estimates of excitation energies, even for simple atomic systems [H. van Aggelen et al., Comput.
Theor. Chem. 1003, 50–54 (2013)]. Here, we outline an approach for the direct variational optimization of ground-state 2-RDMs that satisfy pure-state N-representability known as generalized Pauli
constraints. Within the ERPA, 2-RDMs that satisfy both ensemble conditions and the generalized
Pauli constraints yield much more reliable estimates of excitation energies than those that satisfy
only ensemble conditions. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4965888]
I. INTRODUCTION
The ground-state energy of an N-electron system is a
known linear functional of the two-electron reduced-density
matrix (2-RDM).1 Because the 2-RDM is a more compact
mathematical object than the N-electron wave function, it
seems natural to seek a procedure to directly determine
the elements of the 2-RDM without the use of the wave
function. Unfortunately, such a procedure is complicated by
the fact that the 2-RDM must be constrained to ensure that
it derives either from a single antisymmetrized N-electron
wave function or an ensemble of wave functions. A reduceddensity matrix (RDM) that satisfies this property is said to be
N-representable.2 For example, an ensemble N-representable
2-RDM (2D) can be obtained from the partial integration of
an ensemble N-electron density matrix as
( )
N
2
′ ′
D(x 1, x 2|x 1, x 2) =
dx 3 . . . dx N
2

(1)
wk Ψk (x 1, x 2, x 3, . . . , x N )Ψk∗ (x 1′ , x 2′ , x 3, . . . , x N ).
k
Here, Ψk represents a normalized and antisymmetrized Nelectron wave function, the symbol x 1 (x 2, etc.) represents the
spin and spatial coordinates of electron 1 (electron 2, etc.),
and the non-negative weights, wk , sum to one. If only one
coefficient in this expansion is non-zero, then the 2-RDM is
said to be pure-state N-representable.
Trivial conditions for the N-representability of a p-body
RDM (where p < N) include its Hermiticity and antisymmetry
with respect to the exchange of particles or holes. A hierarchy
of p-positivity conditions also requires that the various
representations of the p-RDM be positive semidefinite and
interrelated according to the anticommutation relations of
fermionic creation and annihilation operators.3 For example,
a)Electronic address: [email protected]
0021-9606/2016/145(16)/164109/6/$30.00
at the one-particle level, the one-electron RDM (1-RDM, 1D)
and the one-hole RDM (1Q) must satisfy
D ≽ 0, with 1 D ij = ⟨Ψ| âi†â j |Ψ⟩,
(2)
Q ≽ 0, with 1Q ij = ⟨Ψ| â j âi†|Ψ⟩,
(3)
D ij + 1Q ij = δ i j .
(4)
1
1
and
1
Here, it has become useful to introduce the notation of second
quantization; the symbols ↠and â represent creation and
annihilation operators, respectively, and the indices i and j
are spin orbital labels. The notation 1D ≽ 0 indicates that
all of the eigenvalues of the 1-RDM are non-negative.
Taken together, Eqs. (2)–(4) imply that the eigenvalues
of an N-representable 1-RDM must lie between zero and
one. Indeed, these bounds, when combined with the trace
condition, Tr(1D) = N, constitute Coleman’s necessary and
sufficient ensemble N-representability conditions for the 1RDM.2 Similarly, at the two-particle level, the 2-RDM, the
two-hole RDM, and the electron-hole RDM must be positive
semidefinite and interrelated. These necessary yet insufficient
conditions on the N-representability of the 2-RDM are the
“PQG” conditions of Garrod and Percus.4
The variational determination of the ground-state 2RDM under such ensemble N-representability conditions can
routinely be achieved via semidefinite programming (SDP)
techniques.3,5–15 On the other hand, the direct determination of
excited-state densities and energies is much more challenging.
Without the imposition of additional constraints that differentiate the ground- and excited-state 2-RDMs, variational
approaches can only yield the 2-RDM for the lowest-energy
state of a given spin symmetry. Hence, several methods
have been proposed to extract excited-state information
from ground-state 2-RDMs, including the Hermitian operator
method16–20 and the extended random-phase approximation
145, 164109-1
Published by AIP Publishing.
164109-2
A. Eugene DePrince III
(ERPA);21,22 both of these approaches seem to work well
with variationally determined 2-RDMs for simple closedshell systems. However, van Aggelen et al.22 demonstrated
a striking failure of the ERPA for open-shell second-row
atoms with degenerate ground states. In such cases, 2-RDMs
optimized under ensemble N-representability conditions do
not represent a pure state, and the ensemble nature of the
2-RDMs compromises the quality of the corresponding ERPA
spectra. This result is unfortunate, as it suggests that the utility
of the ERPA in the context of variational 2-RDM (v2RDM)
methods is limited to systems with non-degenerate ground
states.
Necessary and sufficient pure-state N-representability
conditions for the 1-RDM, also called generalized Pauli
constraints (GPC), are much more complicated than
the ensemble conditions, and, until a short while ago,
such conditions were only known for special cases.23
Recently, Altunbulak and Klyachko devised an algorithm
to systematically identify all GPC for systems composed of
an arbitrary numbers of electrons distributed among a finite
number of natural spin orbitals.24 Now that expressions for
the GPC are known, several recent studies have explored
where pure N-representable solutions to the many-electron
problem lie relative to the boundary of the polytope formed
by the constraints.25–28 There is some debate as to whether
ground states are exactly pinned or only quasi-pinned to
the boundary of the polytope, yet excited states can clearly
be buried deep within it.27 Beyond these geometric studies,
Theophilou et al.29 have explored the role of the GPC within
reduced-density-matrix functional theory (RDMFT).30 For
the ground states of several three electron systems, multiple
GPC were consistently violated, regardless of the choice of
RDMFT functional. Further, the quality the RDMFT energy
improved when the computations enforced the GPC. We
do not necessarily expect the imposition of GPC to similarly
impact the quality of the ground-state energy within a v2RDM
optimization, but the failures of the ERPA demonstrated in
Ref. 22 suggest that the constraints could play an important
role in that context.
In this work, we develop a procedure to impose the
generalized Pauli constraints within a ground-state v2RDM
optimization. For many-electron systems with degenerate
ground states, we demonstrate that the ERPA excitation
energies from variationally obtained 2-RDMs that satisfy
these pure-state N-representability conditions are in good
agreement with excitation energies obtained from complete
active space self-consistent field (CASSCF) computations.
On the other hand, the ERPA spectra for RDMs that
satisfy only ensemble N-representability conditions are
unreliable. In this case, agreement between the ERPA
and CASSCF is sometimes poor, and the ERPA spectra
vary wildly, depending on the initial v2RDM optimization
conditions.
II. SEMIDEFINITE OPTIMIZATION
WITH ENSEMBLE CONDITIONS
The ground-state electronic energy is an exact functional
of the one- and two-electron reduced-density matrices,
J. Chem. Phys. 145, 164109 (2016)
E=

1  2 ij
1 i
Dk l (ik| jl) +
D j hi j .
2 i jkl
ij
(5)
Here, the symbol (ik| jl) represents a two-electron repulsion
integral in Mulliken notation, hi j represents the sum of
kinetic and electron-nucleus potential energy integrals, and
the elements of the 2-RDM are defined in second quantized
notation as
2
Dki jl = ⟨Ψ| âi†â†j âl âk |Ψ⟩.
(6)
The indices i, j, k, and l run over all spin orbitals.
The minimization of the electronic energy given by Eq. (5)
subject to known N-representability conditions constitutes
a semidefinite optimization problem. Specifically, it is the
primal formulation of the problem, expressed in the language
of SDP as
Eprimal = cT · x
Ax = b
M(x) ≽ 0.
minimize
such that
and
(7)
Here, x represents the primal solution vector, Eprimal represents
the energy of the primal solution, and the matrix A and vector
b offer a compact representation of the N-representability
constraints (see below). When enforcing the two-particle
(PQG) N-representability conditions, the mapping M(x) maps
the primal solution vector onto the RDMs as
1
D
*.
.. 0
M(x) = .. 0
..
.0
,0
0
Q
0
0
0
1
0
0
2
D
0
0
0
0
0
2
Q
0
0
+
0 //
/
0 // .
/
0 //
2
G-
(8)
Equation (8) introduces two additional 2-body RDMs; the
symbols 2Q and 2G represent the two-hole RDM and the
electron-hole RDM, respectively, the elements of which are
Q ikjl = ⟨Ψ| âi â j âl†âk† |Ψ⟩
(9)
G ikjl = ⟨Ψ| âi†â j âl†âk |Ψ⟩.
(10)
2
and
2
The vector c similarly maps onto the one- and two-electron
integrals such that Eprimal and the energy given by Eq. (5) are
equivalent.
The notation M(x) ≽ 0 indicates that the eigenvalues of
each RDM are non-negative. In addition to the positivity of the
RDMs, necessary N-representability conditions require that
the blocks of M(x) correctly map to one another according
to the anticommutation relations of fermionic creation and
annihilation operators. These linear equality constraints are
represented within the SDP problem by the action of the
constraint matrix, A, on the primal solution vector: Ax = b.
For example, if the dimension of the one-electron basis is k,
then the relationship between 1D and 1Q given in Eq. (4) can
be mapped onto the action of k 2 rows of the matrix A on
x; the right-hand side of Eq. (4) can similarly be mapped
onto k 2 elements of the constraint vector b. Additional
equality constraints, including spin, trace, and contraction
conditions,31,32 can also be represented in this way.
164109-3
A. Eugene DePrince III
J. Chem. Phys. 145, 164109 (2016)
Alternatively, the semidefinite optimization problem can
be cast in terms of its dual formulation,
maximize
such that
and
Edual = bT · y
z = c − AT y
M(z) ≽ 0,
(11)
where y and z are the dual solution vectors, Edual represents
the energy of the dual solution, and M(z) must also
be non-negative. The symbols bT and AT represent the
transpose of the constraint vector and matrix described
above.
To solve the SDP problem, we use a boundary-point
optimization, similar to those outlined in Refs. 15 and 33–35,
that consists of the following iterative two-step procedure:
1. Solve AAT y = A(c − z) + µ(b − Ax) for y.
2. Update M(x) = µ−1Ω+ and M(z) = −Ω−, where Ω = M
(µx + AT y − c).
Here, µ is a parameter that drives the optimization toward
either the primal or dual solution; µ can be updated in the
course of the optimization according to the prescription in
Ref. 35. Step 1 is solved by conjugate gradient methods,
and the positive/negative components of Ω are separated
by diagonalization. Steps 1 and 2 are repeated until the
primal error ∥Ax − b∥, the dual error ∥AT y − c + z∥, and the
difference between the primal and dual energies are sufficiently
small.
III. SEMIDEFINITE OPTIMIZATION
WITH PURE-STATE CONDITIONS
(12)
These bounds are enforced by the procedure outlined
above. The pure-state N-representability conditions place
additional equality/inequality constraints on the natural orbital
occupation numbers. For example, consider a system of threeelectrons distributed among six spin orbitals. The Hilbert
space in which the electrons reside is represented by H6, and
the family of antisymmetrized three-electron wave functions
in this Hilbert space is denoted by ∧3H6. Table I provides the
GPC satisfied by a pure-state N-representable 1-RDM for the
family ∧3H6. Here, the natural orbitals are ordered such that
λ i ≥ λ i+1. Note that constraint 1 in Table I is usually expressed
as an inequality constraint (λ 4 − λ 5 − λ 6 ≤ 0), but we have
recast it as an equality constraint via the introduction of an
TABLE I. Generalized Pauli constraints for ∧3 H6.
No.
Constraint
1
2
3
4
g 1 + λ4 − λ5 − λ6 = 0
λ1 + λ6 = 1
λ2 + λ5 = 1
λ3 + λ4 = 1
1
D
*.
.. 0
.0
M(x) = ...
.. 0
.. 0
,0
0
Q
0
0
0
0
1
0
0
2
D
0
0
0
0
0
0
2
Q
0
0
0 0
+
0 0//
/
0 0//
/,
0 0//
/
2
G 0//
0 g-
(13)
where g represents a diagonal matrix with g p on the diagonal.
The GPC in Table I are expressed in terms of the natural
orbital occupation numbers. However, v2RDM optimizations
in general are not performed in the natural orbital basis.
For example, the orbitals in a v2RDM-driven CASSCF
computation are chosen to minimize the electronic energy,
rather than to diagonalize the 1-RDM. We instead express the
GPC in terms of the nondiagonal 1-RDM and its eigenvectors.
Then, for example, constraint 1 in Table I becomes

g1 +
[Ui4U j4 − Ui5U j5 − Ui6U j6]1 D ij = 0,
(14)
ij
The ensemble N-representability conditions require that
the natural orbital occupation numbers (the eigenvalues of the
1-RDM, λ i ) lie between zero and one,
0 ≤ λ i ≤ 1.
additional variable quantity, g1, which must be non-negative.
For a system composed of a larger numbers of electrons
or described by a Hilbert space of a larger rank, the GPC
include many more inequality constraints. Hence, the general
case requires that we introduce many g p , where p indicates
a particular constraint. The constraints can be written such
that each g p must be non-negative, and, in the course of
the optimization, the positivity of g p can be enforced along
with that of the RDMs. That is, we can define the modified
mapping
where U is the unitary matrix that diagonalizes 1D. As linear
equality constraints, the GPC can now be incorporated into a
modified boundary-point semidefinite optimization in which
the constraint matrix, A, changes as the natural orbitals evolve.
For a given 1-RDM, we determine the natural orbitals and
solve for the dual solution according to the expression in Step 1
above. The primal solution is updated in Step 2, new natural
orbitals are determined, and the optimization continues. We
have incorporated the GPC into a development version of
our SDP solver for the ground-state v2RDM problem.15 The
solver is implemented as a plugin to the P4 electronic
structure package.36
IV. EXTENDED RANDOM PHASE APPROXIMATION
The semidefinite optimization procedure outlined above
yields 1- and 2-RDMs for the ground state of a given spin
symmetry. Excited-state information can then be determined
using an extended random phase approximation. Within the
ERPA, the nth excited electronic state, |Ψn ⟩, is defined by
single excitations out of the ground state:

|Ψn ⟩ =
cinj â†j âi |Ψ⟩.
(15)
ij
The expansion coefficients, cinj , and the excitation energies,
ω n , are determined by solving the generalized eigenvalue
problem
164109-4
A. Eugene DePrince III

J. Chem. Phys. 145, 164109 (2016)
cinj ⟨Ψ|[âk† âl , [ Ĥ, â†j âi ]]|Ψ⟩
ij
= ωn

cinj ⟨Ψ|[âk† âl , â†j âi ]|Ψ⟩.
(16)
ij
The left- and right-hand sides of Eq. (16) can expressed
in terms of the ground-state 1- and 2-RDM, which can be
obtained from a v2RDM computation as described above.
In all computations presented below, the summation over
spin-orbitals i and j is restricted to include only those orbitals
that are active in the underlying v2RDM optimization. We
have implemented the ERPA as a plugin to P4.
V. RESULTS
Consider the ground state of a boron atom (2P). This
state is triply degenerate, so a v2RDM optimization under
ensemble N-representability conditions is unlikely to yield
pure-state RDMs. As discussed in Ref. 22, the ensemble
nature of the RDMs has a significant impact on the quality
of excitation energies extracted using the ERPA; the error
in the 4P← 2P excitation energy, relative to that from full
configuration interaction, was reported in that work to be
2.674 eV. Here, we explore the role of the GPC in the ability of
the ERPA to describe this transition in a series of five-electron
atoms and ions. We compare the 4P← 2P excitation energies
computed by the v2RDM/ERPA approach to those obtained
at the state-averaged (SA) CASSCF level of theory. Both the
2
P and 4P states are triply degenerate, so the SA-CASSCF
procedure involved all six states. In the CASSCF and v2RDM
optimizations, the 1ag , 2ag , 1b1u , 1b2u , and 1b3u orbitals
were active; the boron isoelectronic series thus belongs to
the ∧5H10 family. All computations were performed within
the cc-pVQZ basis set, and the v2RDM procedure used
SA-CASSCF-optimized orbitals. The v2RDM optimizations
enforce the ensemble two-particle (PQG) N-representability
conditions,4 as well as constraints on the expectation value of
Ŝ 2.31,32
Figure 1 illustrates violations in the 160 GPC for ∧5H10
systems in units of electrons; expressions for these constraints
can be found in the supplementary material of Ref. 24. The
data were generated using two different initial choices for the
1- and 2-RDM. First, the v2RDM optimizations were seeded
with Hartree-Fock RDMs [Fig. 1(a)]. For all species, many
GPC are violated, and the greatest violations range from 0.3
electrons for Be− to 5.9 electrons for Ne5+. Despite these large
violations, in all cases, the v2RDM ground-state energies
agree with those from SA-CASSCF to within 0.6 mEh . The
violations in the GPC are even more apparent when seeding
the v2RDM optimization with random positive semidefinite
RDMs [Fig. 1(b)]. For all systems, some GPC are violated by
at least 3.0 electrons, and the greatest error for Ne5+ is almost
7.0 electrons. It would be interesting to consider what physical
meaning can be ascribed to GPC that are consistently violated
here. However, the form of the GPC is so complicated that it
is difficult to attach any physical meaning to them whatsoever.
For example, the constraint that is violated to the greatest
degree in Fig. 1(b) takes the form
FIG. 1. Violations (in units of electrons) in the generalized Pauli constraints
for ∧5 H10 when seeding an ensemble v2RDM optimization with (a) HartreeFock and (b) random positive semidefinite RDMs. Constraints that are satisfied appear as black.
− 7λ 1 + 3λ 2 + 13λ 3 + 13λ 4 + 23λ 5 + 33λ 6
− 7λ 7 − 27λ 8 − 27λ 9 − 17λ 10 ≤ 45.
(17)
It is not immediately clear what useful physical meaning, if
any, such a complex expression contains.
Despite the sizeable differences in the degree to
which the GPC are violated when using different initial
conditions, ground-state energies obtained from v2RDM
optimizations seeded with Hartree-Fock or random RDMs
agree to within 0.01 mEh , which is consistent with the
energy convergence criterion used throughout this work
(|Eprimal − Edual| < 0.1 mEh ). At first pass, it may come
as a surprise that such strikingly different 1-RDMs can
be associated with essentially identical energies, and it is
tempting to attribute the differences to an incomplete or
unsuccessful optimization. However, the differences simply
reflect the ensemble nature of the optimized 1-RDMs. Recall
that an ensemble N-representable 1-RDM can be obtained
from the partial integration of an ensemble N-electron density
matrix, similar to the integration given in Eq. (1). In the case
of the triply degenerate 2P ground state for the boron series,
an ensemble N-electron density matrix and the corresponding
1-RDM contain contributions from all three states. Changes
to the initial conditions in the v2RDM optimization alter the
relative weights of the states that contribute to the optimized
ensemble 1-RDM.
Table II illustrates the 4P← 2P excitation energies for the
boron series computed at the SA-CASSCF and v2RDM/ERPA
levels of theory, using the same set of active orbitals utilized
in the ground-state computations. In describing the 4P states,
the SA-CASSCF procedure considers the low-spin (MS = 21 )
4
P states, while the ERPA considers the high-spin (MS = 23 )
4
P states. That is, the ERPA expansion of the wave function
in Eq. (15) was restricted to include only spin-forbidden
excitations that increase the value of MS for the wave function.
The rationale behind this discrepancy is practical. The
164109-5
A. Eugene DePrince III
J. Chem. Phys. 145, 164109 (2016)
TABLE II. 4P← 2P excitation energies (eV) for the boron isoelectronic series.
Ensemble v2RDM optimizations were seeded with either (1) Hartree-Fock
or (2) random positive semidefinite RDMs. Pure-state v2RDM optimizations
were seeded with ensemble RDMs initially optimized from (3) Hartree-Fock
or (4) random positive semidefinite RDMs.
ERPA
Ensemble
Pure
1
2
3
4
SA-CASSCF
Be−
1.409
1.409
1.552
1.622
1.384
1.384
1.384
1.384
1.352
B
3.107
3.107
3.312
3.413
3.039
3.039
3.030
3.030
2.991
C+
4.889
4.889
5.126
5.228
4.735
4.735
4.747
4.747
4.694
N2+
6.704
6.704
6.950
7.047
6.453
6.453
6.466
6.466
6.410
O3+
8.555
8.555
8.791
8.870
8.178
8.179
8.191
8.191
8.131
F4+
10.445
10.445
10.641
10.705
9.901
9.902
9.912
9.912
9.849
Ne5+
12.386
12.386
12.522
12.544
11.621
11.621
11.620
11.620
11.564
0.822
0.980
0.057
0.063
Maximum error
SA-CASSCF procedure in P4 requires that each state has
the same MS value. However, the low-spin 4P states will not
be correctly described by the ERPA when considering only
single excitations out of the ground state; the low-spin 4P
states have significant double-excitation character. Similarly,
even for the high-spin 4P states, we can only expect the
ERPA to properly describe two of the three degenerate states.
Consider the Hartree-Fock reference configuration for the
b1u -symmetry 2P state of boron [(1ag )2(2ag )2(1b1u )1]. The
b2g - and b3g -symmetry 4P states can be generated by single
excitations out of this reference, but the b1g -symmetry 4P state
differs from the reference by two electrons. Hence, Table II
includes only two excitation energies from v2RDM/ERPA.
The ERPA excitation energies were generated from four
different solutions to the ground-state v2RDM problem. First,
the 1- and 2-RDM were optimized under the usual PQG
N-representability conditions with an initial guess for the
RDMs of either the Hartree-Fock solution or random positive
semidefinite matrices (labeled “ensemble”). The RDMs were
also optimized subject to the PQG conditions and the
160 GPC for ∧5H10 starting from either of the ensemble
solutions (labeled “pure”). Ensemble RDMs optimized from
a Hartree-Fock guess yield excitation energies that differ
substantially (by up to 0.814 eV) from those from SACASSCF, but the ERPA at least correctly predicts that the
4
P states are degenerate. These large errors are troubling
in and of themselves, but what is more problematic is
the degree to which the excitation energies change when
seeding the v2RDM optimization with a different guess.
In the case of a random seed, the errors are as large
as 0.98 eV, and the ERPA no longer predicts that the
4
P states are degenerate. This result is not too surprising,
considering the degree to which violations of the GPC
increase with the random initial seed [see Fig. 1(b)]. With
the application of pure-state N-representability conditions,
the errors in the ERPA excitation energies, relative to those
from SA-CASSCF, fall to only 0.007-0.057 eV, and the 4P
states appear as degenerate states in the spectra (to within
1 meV, in two cases). The ERPA excitation energies from
pure-state v2RDM optimizations seeded with different initial
RDMs differ by at most 13 meV (0.5 mEh ); this difference
can be traced to the convergence thresholds in the groundstate v2RDM optimization. Throughout this work, v2RDM
optimizations were considered converged when ∥Ax − b∥
< 10−4, ∥AT y − c + z∥ < 10−4, and |Eprimal − Edual| < 10−4Eh .
With these thresholds, the 4P← 2P excitation energies for
boron using the ERPA with pure-state RDMs differ by
8 meV, depending on the initial optimization conditions.
By decreasing the primal and dual convergence thresholds to
10−5 and the primal/dual energy gap to 10−5Eh , this difference
reduces to only 1 meV.
We have also explored the role of the GPC in describing
the 5S← 3P excitation in the carbon (six-electron) isoelectronic
series. Table III provides excitation energies for the series
computed at the SA-CASSCF and v2RDM/ERPA levels of
theory, using the cc-pVQZ basis set. The 3P state is triply
degenerate, the 5S state is nondegenerate, and the SA-CASSCF
procedure involved all four states. Again, the 1ag , 2ag , 1b1u ,
1b2u , and 1b3u orbitals were active, and the v2RDM procedure
used SA-CASSCF-optimized orbitals. As with the boron
series, the ERPA considers the high-spin (MS = 2) 5S state,
while the SA-CASSCF procedure considers the MS = 1 state.
The carbon series belongs to the ∧6H10 family. By particlehole equivalence the constraints for ∧4H10 were used, except
that, in this case, the GPC apply to the eigenvalues of the onehole RDM, 1Q, rather than those of the 1-RDM. Expressions
for the 124 GPC for ∧4H10 were taken from the supplementary
material of Ref. 24. As in the boron series, ensemble RDMs
display large errors, relative to SA-CASSCF (as large as
1.560 eV), and the quality of the excitation energies varies
dramatically depending on the initial optimization conditions.
TABLE III. 5S← 3P excitation energies (eV) for the carbon isoelectronic
series. Ensemble v2RDM optimizations were seeded with either (1) HartreeFock or (2) random positive semidefinite RDMs. Pure-state v2RDM optimizations were seeded with ensemble RDMs initially optimized from (3)
Hartree-Fock or (4) random positive semidefinite RDMs.
ERPA
Ensemble
Be2−
B−
C
N+
O2+
F3+
Ne4+
Maximum error
Pure
1
2
3
4
SA-CASSCF
0.530
1.731
3.149
4.772
6.510
8.331
10.238
0.809
2.217
3.795
5.486
7.244
9.043
10.864
0.485
1.628
2.956
4.465
6.050
7.670
9.304
0.485
1.628
2.956
4.464
6.050
7.669
9.304
0.485
1.628
2.956
4.463
6.049
7.669
9.303
0.934
1.560
0.001
0.001
164109-6
A. Eugene DePrince III
This failure is particularly striking, considering the simplicity
of the structure of the high-spin 5S state; with only 10 active
spin orbitals, the wave function is a single determinant. When
the RDMs are optimized under pure-state N-representability
conditions, the errors in the ERPA excitation energies fall to
0.001 eV or less.
VI. CONCLUSIONS
The groundbreaking work of Altunbulak and Klyachko24
defined a systematic procedure to derive pure-state Nrepresentability conditions for arbitrary many-electron systems. This breakthrough has far reaching implications for
quantum chemistry, facilitating fundamental investigations
into the nature of electron correlation and providing, for
the first time, some guidance as to how one could directly
determine 1- and 2-RDMs for degenerate states that satisfy
pure-state N-representability conditions. We have presented a
procedure that incorporates these generalized Pauli constraints
into the variational optimization of the ground-state 2-RDM,
and the utility of pure RDMs over ensemble ones is clearly
visible in the context of the ERPA. For many-electron
systems with degenerate ground states, excitation energies
obtained from ensemble RDMs are quite poor. However,
when enforcing the generalized Pauli constraints, the resulting
pure-state N-representable RDMs can safely be used within
the ERPA to obtain reliable excited-state information.
The present work constitutes numerical proof that it
is indeed possible to enforce the GPC within a v2RDM
optimization. However, two extensions are necessary before
the GPC can be applied to general systems. The GPC have
only been tabulated for systems with 10 or fewer active
orbitals;24 future work in this area must include the generation
and tabulation of GPC for other systems with larger numbers
of active electrons and orbitals. The number of GPC will
increase sharply for larger systems, but this number will be
significantly less than the number of constraints associated
with other necessary N-representability conditions, such as
the two-particle PQG conditions. Second, the application of
the GPC results in a substantial increase in the complexity and
time-to-solution of the present SDP algorithm. On average,
the number of macroiterations (cycles of Steps 1 and 2
outlined in Sec. II) increases by a factor of 24 for the boron
series (∧5H10) and 8 for the carbon series (∧6H10). This
increase in the time-to-solution is not consistent across either
series. For example, the total number of macroiterations for
the Be− atom increased by a factor of ≈100, whereas the
number of iterations remained essentially constant for Ne5+.
Clearly, the cost of application of pure-state conditions could
be prohibitive for more complex systems. Future work must
consider strategies to enhance the convergence of the v2RDM
J. Chem. Phys. 145, 164109 (2016)
optimization procedure. One possible strategy would be to
decouple the enforcement of the ensemble PQG and pure-state
conditions.
ACKNOWLEDGMENTS
This material is based upon work supported by the
National Science Foundation under Grant No. CHE-1554354.
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