On the eective reconstruction of expectation values from ab initio quantum embedding Max Nusspickel Basil Ibrahim and George H. Bootha

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On the eﬀective reconstruction of expectation values from ab initio quantum

embedding

Max Nusspickel, Basil Ibrahim, and George H. Bootha)

Department of Physics, King’s College London, Strand, London WC2R 2LS, U.K.

Quantum embedding is an appealing route to fragment a large interacting quantum system into several

smaller auxiliary ‘cluster’ problems to exploit the locality of the correlated physics. In this work we critically

review approaches to recombine these fragmented solutions in order to compute non-local expectation values,

including the total energy. Starting from the democratic partitioning of expectation values used in density

matrix embedding theory, we motivate and develop a number of alternative approaches, numerically demon-

strating their eﬃciency and improved accuracy as a function of increasing cluster size for both energetics

and non-local two-body observables in molecular and solid state systems. These approaches consider the

N-representability of the resulting expectation values via an implicit global wave function across the clus-

ters, as well as the importance of including contributions to expectation values spanning multiple fragments

simultaneously, thereby alleviating the fundamental locality approximation of the embedding. We clearly

demonstrate the value of these introduced functionals for reliable extraction of observables and robust and

systematic convergence as the cluster size increases, allowing for signiﬁcantly smaller clusters to be used for

a desired accuracy compared to traditional approaches in ab initio wave function quantum embedding.

I. INTRODUCTION

Quantum chemical methods to describe explicit corre-

lations in an ab initio many-electron system can be highly

accurate, though their applicability is often stymied by a

steep computational scaling with respect to system size

which (despite signiﬁcant recent progress) limits their

use for extended systems1–6. To combat this, the lo-

cality of this correlated physics is increasingly exploited,

enabling a reduction in scaling to be competitive com-

pared to mean-ﬁeld or density functional approaches,

whilst remaining free from empiricism7,8. The ﬁeld of

‘local correlation’ methods in quantum chemistry gener-

ally build these locality constraints in the particle-hole

excitation picture of the system, localizing each of these

spaces separately9–11. While highly related, ‘quantum

embedding’ approaches from condensed matter physics

are also increasingly coming to the fore as an alternative

paradigm and applied to quantum chemical and ab initio

systems12.

A loose (and necessarily imperfect) characterization

of a key diﬀerence in these approaches could be that

quantum embedding does not build this locality from a

particle-hole picture—rather, a fully local set of ‘atomic-

orbital-like’ degrees of freedom are chosen initially (which

will in general have neither fully occupied or unoccupied

mean-ﬁeld character), which we will call the ‘fragment’

space, though is also often called the ‘impurity’ space

for historical reasons in traditional quantum embedding

literature. A larger space is then constructed by aug-

menting these fragment orbitals with additional orbitals

(often called ‘bath’ orbitals). These are designed to re-

produce the quantum ﬂuctuations, entanglement and/or

hybridization between the fragment and the rest of the

a)Electronic mail: george.booth@kcl.ac.uk

system, as characterized by some tractable (generally

mean-ﬁeld) level of theory which can be performed on

the full system. These individual local quantum prob-

lems of the fragment and bath orbitals deﬁne a ‘cluster’,

which is then solved to provide the correlated properties

of the original fragment space, potentially with a subse-

quent self-consistency then applied to update the original

mean-ﬁeld/low-level theory on the full system.

The general algorithm in most quantum embeddings

is therefore summarized as a) fragment the system, b)

for each fragment, construct a bath space describing

the coupling to the wider system, c) solve an interact-

ing problem in the cluster space of each fragment via a

‘high-level’ correlated method, d) extract properties of

the system, e) optionally, a self-consistency is performed

to embed the correlated eﬀects from the cluster model

back into the low-level full system method to update

the coupling between fragment and environment. There

are a large number of choices and variations within this

general framework, including (but not limited to) how

the bath space is deﬁned (including the choice of ‘low-

level’ theory)13–15, how the interacting cluster Hamilto-

nian is constructed and solved16–19, and the choice of

self-consistent requirements20–22. Furthermore, the fun-

damental quantum variables by which these quantities

over the diﬀerent spaces are characterized can vary, with

dynamical mean-ﬁeld theory and its variants working

in a Green’s function (dynamical) formalism23–29, while

density matrix embedding theory (DMET) and its vari-

ants work in a wave function (static) formalism30–44, al-

though these two approaches are not fundamentally dis-

tinct, and can also be rigorously connected via a common

framework45–47.

Much recent progress has been made in these various

quantum embeddings and their application to ab initio

systems, including the use of quantum computation as

a high-level solver48–52. The key point in all of these

embedding approaches however, is that the scaling with

arXiv:2210.14561v2 [cond-mat.str-el] 22 Jan 2023

respect to the full system size is deﬁned by only the scal-

ing of the low-level (often mean-ﬁeld) method, given the

local nature of the auxiliary cluster problems. Further-

more, the choice of high-level solver and arbitrary atomic-

orbital-like fragmentation allows for spaces which capture

strong (albeit still local) correlation eﬀects, beyond the

traditional constraints of the particle-hole picture of most

local quantum chemistry.

In this work we focus on a critical aspect of quan-

tum embedding which we believe has received less at-

tention, but which has substantial ramiﬁcations for its

accuracy and applicability. This concerns how non-local

properties (including the total energy) of the full sys-

tem can be reconstructed from the independent cluster

solutions of each fragment. We will assess the eﬀect of

the inherent locality approximation of quantum embed-

ding on the convergence of diﬀerent functionals of these

non-local expectation values, and motivate and demon-

strate new approaches which substantially accelerate con-

vergence with respect to fragment and bath size in the

embedding. While this is quite a technical work, the out-

come is that general design principles by which diﬀerent

functionals can be devised become clear, including the

N-representability of these estimates.

Here, we focus on wave function-based quantum em-

bedding (we believe that the ability and approach for

constructing appropriate functionals in a Green’s func-

tion perspective is clearer). We start from the density

matrix embedding theory as the parent wave function ap-

proach to quantum embedding30,33. We always consider

fragments consisting of a single atom only and, where we

seek to systematically improve expectation values, en-

large the cluster spaces by adding additional interact-

ing bath orbitals. We believe that this is an eﬃcient

and ‘black-box’ approach, and avoids the ambiguities and

non-monotonic improvements in the alternative of deﬁn-

ing a systematic expansion in the fragment sizes for ab

initio systems, which also can suﬀer from reducing the

symmetry of the problem53.

The expansion of the bath space is deﬁned from the ap-

proximate interacting density matrix (or instantaneous

hybridization) between the fragment and environment

at a simple approximate second-order perturbation the-

ory (MP2), and is controlled by a single cutoﬀ parame-

ter as detailed in Ref. 53. This provides cluster-speciﬁc

bath natural orbitals (BNOs) as a controllable, reli-

able and well-deﬁned expansion of the bath space (and

hence overall cluster space) of a given fragment. Fur-

thermore, in this work we will neglect considerations of

self-consistency of the original mean-ﬁeld (beyond where

self-consistency is required for meaningful extraction and

comparison of expectation values, e.g. to control to-

tal electron number). More extensive self-consistency to

qualitatively change the original full-system mean-ﬁeld

will be considered in future work20,21, but is unlikely to

change the conclusions of this work, especially as conver-

gence with cluster size (either via expansion of the frag-

ment or, as in this work, an interacting bath) obviates

the eﬀect of self-consistency.

We start the paper recapitulating the original DMET

‘democratic partitioning’ for expectation values, which

can be computed via fragment-projection of the reduced

density matrices of each fragment33. We then describe

an improved approach for the total energy based around

a cumulant functional for the two-body eﬀects. We move

on to an approach based around direct projection of wave

function amplitudes, rather than density matrices, for-

mally satisfying N-representability conditions (not sat-

isﬁed in the aforementioned density-matrix approaches)

and substantially improving expectation values. Finally,

practical approaches and further approximations to these

will be described to deﬁne an eﬃcient protocol for arbi-

trary expectation values and high-level wave function de-

scriptions. These diﬀerent approaches are benchmarked

for energetics and other non-local properties (spin corre-

lation functions) on both molecular systems via the W4-

11 test set54, and periodic systems, ﬁnding an eﬃcient

approach for reconstruction of non-local observables in

quantum embedding of general fragmented systems.

A. Summary of ﬁndings

FIG. 1. Binding energy of the Chlorine dimer in a STO-6G

basis compared to FCI, with E[γx,Γx] corresponding to the

democratically partitioned energy expression of traditional

DMET33. Diﬀerent energies correspond to diﬀerent total

energy functionals, constructed from the same solutions to

the two DMET embedding problems (with fragment spaces

of the (L¨owdin-orthogonalized) core and valence orbitals of

each Chlorine atom). The two cluster spaces are deﬁned by

10 orbitals (9 fragment and 1 bath), with the full space com-

prising 18 orbitals. The motivation and deﬁnition of these

diﬀerent energy functionals from the embedded wave func-

tions are given in the rest of this work, with more details on

this system (and others like it) discussed in section IV.

In Figure 1, we show results from the diﬀerent total

energy functionals described in this work, as a represen-

tative illustration of the signiﬁcant diﬀerence that this

choice can make in a simple system (Chlorine dimer, min-

imal basis, two atomic fragments). For all the diﬀerent

total energy functionals shown in Fig. 1, the same iden-

tical fragment and DMET cluster space is used, and the

same (exact) solver and wave function are found from

each cluster30,31. The only diﬀerence is the choice of en-

ergy functional to reconstruct the total energy from the

two cluster solutions, with E[γx,Γx] representing the tra-

ditional ‘democratically partitioned’ DMET energy used

primarily in the literature to date33.

As a summary regarding the reconstruction of expec-

tation values from cluster solutions in this work, we ﬁnd

the main conclusions to be the following:

1. It is advantageous to separate factorizable products

of lower-rank contributions to expectation values

where possible. In this way, we can construct e.g.

the two-electron energy from the one-body density

matrix and two-body cumulant, rather than the

two-body density matrix directly. We demonstrate

that this improves estimators via the inclusion of

implicit cross-cluster contributions to the expecta-

tion values. Many of the improvements in this work

arise from implicitly building in contributions to ex-

pectation values (or the wave function itself) from

products of diﬀerent cluster contributions, coupling

the individual cluster solutions in a non-local fash-

ion, and minimizing the local approximations in-

herent to the embedding framework.

2. N-representability of estimators can be used as an

eﬀective guiding principle, i.e. that they can be

derived from a valid ‘global’ wave function from

the combined cluster wave function solutions. We

show how this can be achieved exactly. Deﬁning

this ‘global’ wave function ensures conservation of

many quantum numbers e.g. total electron number,

which removes the necessity for costly (and some-

times ill-deﬁned) self-consistency conditions, and

can allow for variational estimators40. Beyond this,

enabling convergence with respect to bath size of

each fragment can render self-consistency entirely

unnecessary.

3. It is possible to construct estimators from this well-

deﬁned global wave function by casting the expec-

tation value as a functional of the wave function

amplitudes of each cluster, rather than combining

density matrices or observables directly. Each fac-

tor of the wave function amplitudes in the expec-

tation value has their occupied indices (symmetri-

cally) projected onto the fragment spaces of each

cluster to avoid double-counting, and a sum over all

cluster solutions for each factor of the wave func-

tion parameters is introduced.

4. Along with dramatically improved estimates, this

wave function approach furthermore avoids the re-

quirement for optimizations of chemical potentials,

as the global electron number is strictly conserved,

and the condition of the union of the fragment

spaces is only that they span the occupied, rather

than the full orbital space of the original system

in order to converge to exact results (e.g. as the

(interacting) bath is expanded to completeness for

each cluster), vastly reducing the burden of large

fragment spaces spanning virtual orbitals for cal-

culations in realistic basis sets.

5. If the approach above results in an intractable scal-

ing with respect to number of fragments in the sys-

tem, then a principled approximation can be made,

which we show can fortuitously lead to eﬀective

cancellation of errors and an even faster conver-

gence of the ground state energy to the exact global

expectation values.

We motivate and evidence these conclusions in the fol-

lowing sections, resulting in our ﬁnal recommendation

for an approach to one- and two-body expectation val-

ues from DMET and related wave function quantum em-

beddings in ab initio systems, while retaining at most

an O(N3) scaling with system size in the evaluation of

these observables. We detail the practical implementa-

tion of these schemes for both an exact high-level solver,

and an (arbitrary order) coupled-cluster framework. All

results can be reproduced from our recently released

Vayesta codebase for quantum embedding55, which in-

terfaces with the PySCF code56,57, with scripts to gener-

ate many of the results of this work (including the input

required for Fig. 1) found in the Supplementary Informa-

tion (SI).

II. GLOBAL EXPECTATION VALUES FROM CLUSTER

DENSITY MATRICES

A. Democratic partitioning of density matrices

Expectation values in DMET derived from opera-

tors which span more than one fragment are calcu-

lated from ‘democratically partitioned’ reduced density-

matrices (RDMs)33. These can be written as

γpq =

Nfrag

(ˆ

Pxγx)pq (1)

Γpqrs =

Nfrag

(ˆ

PxΓx)pqrs,(2)

where γand Γ are one- and two-body reduced density

matrices formed from the high-level solution of each clus-

ter problem, as

γx

pq =hΨx|ˆc†

pˆcq|Ψxi,(3)

Γx

pqrs =hΨx|ˆc†

pˆc†

rˆcsˆcq|Ψxi,(4)

where the high-level cluster wave function, |Ψxi, includes

the contribution from the doubly occupied environmen-

tal orbitals of each cluster. ˆ

Pxis an operator, which

introduces a projection onto the fragment subspace of

the cluster xin a symmetrically averaged fashion, e.g.

(ˆ

Pxγx)pq =1

Nmo

(Px

ptγx

tq +Px

qtγx

pt),(5)

(ˆ

PxΓx)pqrs =1

Nmo

(Px

ptΓx

tqrs +Px

qtΓx

ptrs (6)

+Px

rtΓx

pqts +Px

stΓx

pqrt).

In these, xlabels both the Nfrag fragments and clusters,

for which there is an unambiguous one-to-one correspon-

dence, and Px

pq is the fragment projection matrix acting

over the whole molecular orbital (MO) space, as

pq =

xCTSCx

fpx CTSCx

fqx ,(7)

with Crepresenting the MO coeﬃcients, Sthe atomic

orbital (AO) overlap matrix, and Cx

fthe coeﬃcients of

the fragment orbitals in cluster x. Note that a post mean-

ﬁeld cluster solver will only modify the 1-RDM in the

cluster–cluster part, whereas the 2-RDM also acquires

contributions in oﬀ-diagonal cluster–environment parts.

As a result, the projection in Eq. (1) can be performed

in the respective cluster spaces, whereas in Eq. (2) it has

to be performed in the full system space, in order to take

these changes in the oﬀ-diagonal parts into account.

The projection onto the fragment space of each clus-

ter is required since the bath spaces overlap signiﬁcantly

between diﬀerent clusters, and we must project out any

double counting arising from contributions from overlap-

ping bath spaces. In contrast, the partitioning of the sys-

tem into fragment spaces is considered to be a disjoint

fragmentation, where the set of all fragment orbitals are

an orthonormal set with no overlapping fragment spaces

in diﬀerent clusters. From the democratically partitioned

density-matrices, the total energy can be calculated as

Etot =Enuc +

Nmo

hpqγqp +1

Nmo

pqrs

(pq|rs)Γpqrs.(8)

Note that in practice one can avoid forming the full sys-

tem density-matrices to calculate the total energy and

instead calculate energy contributions directly from the

individual cluster density-matrices over purely the clus-

ter degrees of freedom. The contribution from the doubly

occupied environmental orbitals can be integrated out, by

forming the Coulomb- and exchange potential of the un-

entangled occupied orbitals of each DMET cluster via an

eﬀective one-electron potential. This leads to the more

common expression for the DMET energy33, equivalent

to Eq. (8) via construction of democratically partitioned

density matrices of Eq. (5), as

E[γx,Γx] = Enuc+

Nfrag





pq(ˆ

Pxγx)qp

pqrs

(pq|rs)( ˆ

PxΓx)pqrs

,

(9)

where γxand Γxrefer to the cluster reduced density ma-

trices, with the projector purely acting in this cluster

space, and Nx

cl denoting the number of orbitals in the

cluster x. The energetic eﬀect of these states (static

Coulomb and exchange contributions) is then included

via the construction of the ˜

hx, which includes the poten-

tial from these unentangled states to the one-body hamil-

tonian as 1

2Pmn(pq||mn)γx

core,mn where γx

core is the den-

sity matrix from these core states. We can exploit frag-

ments that are (by symmetry) in identical chemical en-

vironments by only computing the cluster solutions and

energy contributions of symmetry-unique fragments. In

the rest of this work, the expression E[γx,Γx] will de-

note this democratically partitioned energy functional,

shorthand for E[{γx},{Γx}], denoting that the energy is

computed from the set of individually constructed cluster

one- and two-body RDMs.

This energy expression is exact when two conditions

are satisﬁed. First, it requires that the fragmentation

of the full system is complete, i.e. that the union of

the fragment spaces spans all degrees of freedom of the

system. This condition ensures that the trace of the sum

of the diﬀerent fragment projectors is exactly equal to the

total number of orbitals in the system, or alternatively,

that

Nfrag

Px=ˆ

1.(10)

While it is a relatively mild condition to ensure that

the combined fragment spaces span the generally local-

izable occupied space, ensuring that they span the (gen-

erally much larger) high-energy virtual space is harder

to achieve and leads to much larger fragment spaces.

This has required DMET simulations in realistic basis

sets to augment fragment spaces with projected atomic

orbitals (PAOs)58 to ensure this condition is fulﬁlled for

reasonable results35. The second criteria which must be

fulﬁlled, is that the individual cluster density matrices

must be exact, which in general will require the clusters

of each fragment themselves to be enlarged to complete-

ness, either by increasing the size of the fragment or (in-

teracting) bath space. This ensures that |Ψxi→|Ψiand

the projected density matrices of the clusters (Eq. 5) are

equivalent to the projections of the exact density matrix

over the whole system. Combined with the complete-

ness of the projector (Eq. 10), this will lead to the exact

energy from Eq. (9).

Away from this exact limit, there are a number of

drawbacks to this approach to compute properties from

the DMET solutions. Firstly, the reconstructed full-

system density matrices (Eqs. (1) and (2)) from the pro-

jected cluster solutions are not N-representable, meaning

that they cannot be derived from a valid wave function.

This can be seen as the democratically partitioned 1-

RDM of Eq. (1),

γpq =

Nfrag

PxhΨx|c†

pcq|Ψxi,(11)

cannot be rewritten as a simple expectation

value h˜

Ψ|c†

pcq|˜

Ψiof some wave function |˜

Ψi. As a

speciﬁc consequence, this can result in eigenvalues

(occupation numbers) becoming negative or greater than

two (in a restricted basis), violating the Pauli principle,

and removing any variational guiding principle in the

method40. Furthermore, conserved quantities and good

quantum numbers in the individual cluster solutions

such as electron number (N), spin and its z-projection

(Sz,S2) and other symmetries are not maintained in

these composite full system descriptions.

To mitigate some of these eﬀects, a global chemical

potential (or potentially a fragment-speciﬁc chemical po-

tential) is almost universally optimized in DMET to en-

sure that at least an exact, integer number of electrons

is recovered in these democratically partitioned density

matrices33,59. This can move electrons between the frag-

ment and bath of each correlated cluster solution, such

that the known global electron number is maintained as

a constraint. While this corrects one known global quan-

tum number in the density matrices, it does not correct

others, and does not in general restore N-representability

of the full system RDMs. Furthermore, this necessitates

costly additional self-consistent loops over the high-level

calculations. Furthermore, this requirement of a chemi-

cal potential optimization to a known total electron num-

ber further underlines the importance of the constraint

of Eq. (10), ensuring that the full (occupied and virtual)

space is spanned by the fragments, as all can be par-

tially occupied in the correlated state and the democrat-

ically partitioned density matrices must trace to the cor-

rect electron number. Numerical demonstration of the

breaking of these N−representability constraints in the

democratically partitioned RDMs will be given in Sec. IV

(with and without a global chemical potential optimiza-

tion), also showing the deleterious eﬀect on properties

and computed energetics of the system that result.

B. Democratic partitioning of cumulants

In the following, we propose a simple alternative for

the construction of democratically partitioned two-body

density-matrices from DMET clusters, from which ex-

pectation values such as the energy can be calculated.

Instead of partitioning the 2-RDM directly as in Eq. (2),

we partition the two-body cumulant, ˜

K, deﬁned (in a

restricted basis)60 via

Γpqrs =γpqγrs −1

2γprγsq +˜

Kpqrs.(12)

The non-cumulant (disconnected) contributions to the 2-

RDM can then be reconstructed from the democratically-

partitioned one-body density-matrix, given by Eq. (1),

such that Eq. (2) is replaced by

Γpqrs =γpqγrs −1

2γprγsq +

Nfrag

(ˆ

Px˜

Kx)pqrs.(13)

The diﬀerence between Eq. (2) and (13) lies purely in

the non-cumulant contribution to the two-body density

matrix, which can be written as the product of demo-

cratically partitioned 1-RDMs. In the standard DMET

partitioning of Eq. (2) these are taken from a single em-

bedding problem at a time, whereas the partitioning of

Eq. (13) contains ‘cross-cluster’ contributions, which can

be seen by inserting Eq. (1) into the ﬁrst term of Eq. (13):

γpqγrs =

Nfrag

(ˆ

Pxγx)pq(ˆ

Pyγy)rs.(14)

In this way, the non-local (correlated) one-body physics

of two distinct clusters, x6=y, contribute to global

two body expectation values; the same is not possible in

Eq. (2). This is expected to be important in cases where

the orbitals pand qare far from the orbitals rand s,

and are not spanned together in any single DMET clus-

ter. In this case, the conventional DMET partitioning of

Eq. (2) will not account for the relaxation of the exter-

nal Coulomb- and exchange potential of a fragment due

to the (potentially correlation-induced) density changes

within the other orbital set. In contrast, this will be im-

plicitly included in the partitioning according to Eq. (13).

This is the key physics where we expect a partitioning

of cumulants to be superior for two-body physics to the

traditional democratic partitioning approach of Sec. II A

for non-local expectation values. We will denote any to-

tal energies resulting from the democratically partitioned

cumulant approach described in this section as E[γx, Kx]

in the rest of this work.

We present a simple example showing the diﬀerence

between the partitioned density matrices and the parti-

tioned cumulant approach in Fig. 2, for a representative

system of N2in a minimal basis set. The fragment space

consists of the ﬁve symmetrically (L¨owdin) orthogonal-

ized atomic orbitals of a single atom (the 1s, 2s, 2px,

2py, 2pzspaces) with the DMET bath consisting of an

additional three bath orbitals consistent with the bond

order of the dimer in a minimal basis. The DMET clus-

ter in this example therefore contains 10 electrons in 8

orbitals, which is compared to the full system of 14 elec-

trons and 10 orbitals, and is solved with exact diago-

nalization (FCI). It is found that regardless of whether

the fragment chemical potential is optimized or not, the

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OntheeectivereconstructionofexpectationvaluesfromabinitioquantumembeddingMaxNusspickel,BasilIbrahim,andGeorgeH.Bootha)DepartmentofPhysics,King'sCollegeLondon,Strand,LondonWC2R2LS,U.K.Quantumembeddingisanappealingroutetofragmentalargeinteractingquantumsystemintoseveralsmallerauxiliary`cluster'proble...

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