PT -Symmetry in Hartree–Fock TheoryHugh G. A. Burton,1, a) Alex J. W. Thom,1 and Pierre-Francois Loos2, b)1)Department of Chemistry, University of Cambridge, Lens�eld Road, Cambridge, CB2 1EW, U.K.2)Laboratoire de Chimie et Physique �antiques (UMR 5626), Universite de Toulouse, CNRS, UPS, France

PT -symmetry — invariance with respect to combinedspace re�ection P and time reversal T — provides aweaker condition than (Dirac) Hermiticity for ensuringa real energy spectrum of a general non-HermitianHamiltonian. PT -symmetric Hamiltonians thereforeform an intermediate class between Hermitian andnon-Hermitian Hamiltonians. In this work, we derivethe conditions for PT -symmetry in the context ofelectronic structure theory, and speci�cally, withinthe Hartree–Fock (HF) approximation. We show thatthe HF orbitals are symmetric with respect to the PToperator if and only if the e�ective Fock Hamiltonianis PT -symmetric, and vice versa. By extension, if an optimal self-consistent solution is invariant under PT , thenits eigenvalues and corresponding HF energy must be real. Moreover, we demonstrate how one can constructexplicitly PT -symmetric Slater determinants by forming PT doublets (i.e. pairing each occupied orbital with itsPT -transformed analogue), allowing PT -symmetry to be conserved throughout the self-consistent process. Finally,considering the H2 molecule as an illustrative example, we observe PT -symmetry in the HF energy landscape and �ndthat the spatially symmetry-broken unrestricted HF wave functions (i.e. diradical con�gurations) are PT -symmetric,while the spatially symmetry-broken restricted HF wave functions (i.e. ionic con�gurations) break PT -symmetry.

I. INTRODUCTION

Symmetry is an essential concept in quantum mechan-ics for describing properties that are invariant under par-ticular transformations. Physical observables, for example,must be totally symmetric under the group of symmetryoperations corresponding to a quantum system, and theexact wave function must transform according to an irre-ducible representation of this group. However, for approx-imate self-consistent methods such as Hartree–Fock (HF)1

and Kohn–Sham density-functional theory (KS-DFT),2 oc-currences of symmetry-breaking are pervasive and appearintimately linked to the breakdown of the single-determinantmean-�eld approximation in the presence of strong correla-tion. From a chemical physicist’s perspective, the archety-pal example is the appearance of symmetry-broken HF solu-tions for internuclear distances beyond the so-called Coulson–Fischer point in H2 (R > RCF),3 where the two (antiparallel)electrons localise on opposing nuclei with equal probabilityto form a spin-density wave.4

Ensuring correct symmetries and good quantum numbersis critical in �nite systems, especially since, when lost, theirrestoration is not always a straightforward task.5–8 However,applying symmetry “constraints” reduces �exibility, leading tothe so-called symmetry dilemma between variationally lowerenergies and good quantum numbers.9 In general, approxi-mate HF wave functions preserve only some of the symmetries

a)Corresponding author; Electronic mail: hb407@cam.ac.ukb)Electronic mail: loos@irsamc.ups-tlse.fr

of the exact wave function for �nite systems (see Fig. 1).10–12

�e restricted HF (RHF) wave function, for example, forms aneigenfunction of the spin operators S2 and Sz by de�nition.Additionally, restriction of the RHF wave function to real val-ues ensures invariance with respect to time reversal T andcomplex conjugation K. By allowing the di�erent spins tooccupy di�erent spatial orbitals in the unrestricted HF (UHF)approach, the wave function can break symmetry under S2

but not Sz. Constraining the UHF wave function to real val-ues conserves K-symmetry, while the paired UHF (p-UHF)approach retains T -symmetry and the complex UHF (c-UHF)wave function can break both K- and T -symmetry. �e most�exible formulation, complex generalised HF (c-GHF), im-poses none of these constraints, although paired (p-GHF) orreal (GHF) variations maintain invariance with respect to Tor K respectively. All of these formalisms are independent ofthe point group symmetry (including the parity operator P ),although spatial symmetry may be imposed separately on theHF wave function.

However, the HF approximation is not restricted to Hermi-tian approaches. Holomorphic HF (h-HF) theory, for example,is formulated by analytically continuing real HF theory intothe complex plane without introducing the complex conjuga-tion of orbital coe�cients.13–15 �e result is a non-HermitianHamiltonian and an energy function that is complex ana-lytic with respect to the orbital coe�cients. In addition, non-Hermitian HF approaches are extensively used to study un-bound resonance phenomena where they occur in nature.16

Although initially intended as a method for extendingsymmetry-broken HF solutions beyond the Coulson–Fischerpoints at which they vanish,13 h-HF theory also providesa more �exible framework for understanding the nature of

arX

iv:1

903.

0848

9v2

[ph

ysic

s.ch

em-p

h] 2

1 M

ay 2

019

2

RHF (S2, Sz, T , K)

c-RHF (S2, Sz)UHF (Sz, K) p-UHF (Sz, T )

c-UHF (Sz)GHF (K) p-GHF (T )

c-GHF

T,KS

2 ,TS 2

,K

S2

Sz

K SzT

Sz

K T

FIG. 1. �e seven families of HF solutions, along with their de�nitionaccording to Stuber and Paldus11 and the symmetries they conserve.XHF, p-XHF and c-XHF stands for real, paired and complex XHF(where X = R, U and G). See main text for more details.

multiple HF solutions in general. For example, through thepolynomial nature of the h-HF equations, a mathematicallyrigorous upper bound for the number of real RHF solutionscan be derived for two-electron systems.15 Moreover, by scal-ing the electron-electron interactions using a complex param-eter λ, h-HF theory reveals a deeper interconnected topol-ogy of multiple HF solutions across the complex plane.17 Byslowly varying λ in a similar (yet di�erent) manner to an adia-batic connection in KS-DFT (without enforcing a density-�xedpath),18 one can then “morph” a ground-state wave functioninto an excited-state wave function of a di�erent symmetryvia a stationary path of h-HF solutions, as we have recentlydemonstrated for a very simple model19–23 in Ref. 17. In sum-mary, h-HF theory provides a more general non-Hermitianframework with which the diverse properties of the HF ap-proximation and its multiple solutions can be explored andunderstood.

In the present paper, we study a novel type of symmetry —known as PT -symmetry24–44 — in the context of electronicstructure theory. PT -symmetry, i.e. invariance with respectto combined space re�ection P and time reversal T , pro-vides an alternative condition to (Dirac) Hermiticity whichensures real-valued energies even for complex, non-HermitianHamiltonians.26 Signi�cantly, PT -symmetric quantum me-chanics allows the construction and study of many new typesof Hamiltonians that would previously have been ignored.30 AHermitian Hamiltonian, for example, can be analytically con-tinued into the complex plane, becoming non-Hermitian in theprocess and exposing the fundamental topology of eigenstates.Moreover, PT -symmetric Hamiltonians can be consideredas an intermediate class between Hermitian Hamiltonianscommonly describing closed systems (i.e. bound states) andnon-Hermitian Hamiltonians which are peculiar to resonancephenomena (i.e. open systems) where they naturally appear(see, for example, Ref. 16).

Despite receiving signi�cant a�ention across theoreticalphysics,44 to our knowledgePT -symmetry remains relativelyunexplored in electronic structure. In the current work, weprovide a �rst derivation of the conditions for PT -symmetryin electronic structure, and speci�cally, within HF theory forclosed systems. By doing so, we hope to bridge the gap toPT -symmetric physics, paving the way for future develop-ments in electronic structure that exploit PT -symmetry, forexample novel wave function Ansatze or unusual approximateHamiltonians. Atomic units are used throughout.

II. PT -SYMMETRIC HAMILTONIANS

A. Spinless PT -Symmetry

To ensure a real energy spectrum and conservation of prob-ability, it is commonly believed that a physically acceptableHamiltonian H must be Hermitian, i.e. H = H†, where †

denotes the combination of complex conjugation (∗) and ma-trix transposition (ᵀ). Although the condition of Hermitic-ity is su�cient to ensure these properties, it is not by anymeans necessary. In particular, as elucidated by Bender andcoworkers,24 the family of PT -symmetric Hamiltonians,24,38

de�ned such that [H,PT ] = 0 or H = HPT whereHPT = (PT )H(PT )−1, provides a new more generalclass of Hamiltonians that allows for the possibility of non-Hermitian and complex Hamiltonians while retaining a phys-ically sound quantum theory.26 Note that [P , T ] = 0 but Pand/or T may not commute with H.32

�e textbook example of a PT -symmetric Hamiltonianis44

H = p2 + ix3 (1)

which has been extensively studied by Bender andcoworkers.24–33,37–39 From the standard action of P and T ,where

P : p→ −p, x → −x, (2a)T : p→ −p, x → x, i→ −i, (2b)

it is clear that the application of the combined space-timere�ection PT , where

PT : p→ p, x → −x, i→ −i, (3)

leaves the Hamiltonian (1) unchanged. Moreover, althoughobviously complex, this Hamiltonian has a real, positive spec-trum of eigenvalues!

Generalising the Hamiltonian (1) to the more general para-metric family of PT -symmetric Hamiltonians24

H = p2 + x2(ix)ε, ε ∈ R, (4)

one discovers a more complex structure. It has beenobserved24 and proved45 that, for ε ≥ 0 and a particular setof boundary conditions (the eigenfunctions must decay expo-nentially in well-de�ned sectors known as Stokes wedges44),

3

the Hamiltonian (4) has an entirely positive and real spec-trum, while for ε < 0, there are some complex eigenvalueswhich appear as complex conjugate pairs. More speci�cally,in particular regions of parameter space, some eigenvalues co-alesce and disappear by forming a pair of complex conjugateeigenvalues. �e region where some of the eigenvalues arecomplex is called the broken PT -symmetry region (i.e. someof the eigenfunctions of H are not simultaneously eigenfunc-tions of PT ), while the region where the entire spectrumis real is referred to as the unbroken PT -symmetry region.Amazingly, these PT -symmetry phase transitions have beenobserved experimentally in electronics, microwaves, mechan-ics, acoustics, atomic systems and optics,42,43,46–61 and theparameter values where symmetry breaking occurs [ε = 0 inthe case of Hamiltonian (4)] correspond to the appearance ofexceptional points,52,60,62–70 the non-Hermitian analogues ofconical intersections.71

B. Electron PT -Symmetry

PT -symmetric systems involving particles with non-zerospin, in our case electrons, are much less studied than theirspinless counterparts. However, a number of studies havefocused on this subject in recent years.40,72–74 In what follows,we consider the spinor basis |α〉 = (1, 0)ᵀ and |β〉 = (0, 1)ᵀ.A single-particle state is then represented by the column vec-tor

φ =

(φα

φβ

), (5)

where φα and φβ are the α and β components of φ respectively.Note that, although a relativistic version of PT -symmetricquantum mechanics can be formulated,75 here we ignore ef-fects such as spin-orbit coupling and consider only the non-relativistic limit.

�e linear parity operatorP acts only on the spatial compo-nents and satis�es P2 = I , where I is the identity operator.Its action in the spinor basis can be represented by the block-diagonal matrix

Pφ =

(P 00 P

)(φα

φβ

), (6)

where P represents the action of P in the spatial basis.In contrast, deriving the action of T is a li�le more involved.

Fundamentally, T is required to be an anti-linear operator,T i = −iT .76,77 However, for systems containing particleswith non-zero spin, the reversal of spin-angular momentumunder the action of T must also be included such that, fora given spin operator s, we obtain T s = −sT . Although amore detailed discussion on the nature of T for particles ofgeneral spin is provided in Appendix A, here we focus on onlythe most relevant results.

In the bosonic case, a basis can always be found in whichT is represented simply as T = K.72 Here K is the dis-tributive anti-linear complex-conjugation operator which actsonly to the right by convention and does not have a matrix

representation.76 Applying K in algebraic manipulations canlead to some non-intuitive results, and particular care must beexercised. In contrast, the representation of T for electrons(i.e. spin- 1

2 particles) is given by T = iσyK.72 To see whythis must be the case, consider expressing a spin operator s ina basis of the Pauli spin matrices

(σx, σy, σz

), where

σx =

(0 11 0

), σy =

(0 −ii 0

), σz =

(1 00 −1

), (7)

and s =(σx, σy, σz

). Simply taking T = K (as in the

bosonic case) yields

T s = K(σx, σy, σz

)=(σx,−σy, σz

)K, (8)

which clearly does not give the desired outcome. In contrast,taking the form T = iσyK yields72

T s = iσyK(σx, σy, σz

)= −

(σx, σy, σz

)iσyK, (9)

therefore satisfying the correct behaviour T s = −sT . Finally,consider also the behaviour of T 2, for which

T 2 =(iσyK

)(iσyK

)= σyσyi2K2 = −σyσy = −I .

(10)Signi�cantly, in fermionic systems, T must be applied fourtimes to return to the original state, leading to the action oftime-reversal in fermionic systems being classi�ed as odd.72

Overall, in the spinor basis, the action of T on φ can berepresented by

T φ = iσyKφ =

(0 1−1 0

)K(

φα

φβ

)=

(φ∗β−φ∗α

). (11)

To �nd the representation of the combined PT operator, wesimply combine the results of Eqs. (6) and (11) to obtain

PT φ = φPT = P iσyKφ

=

(0 P−P 0

)K(

φα

φβ

)=

(Pφ∗β−Pφ∗α

).

(12)

C. PT -doublet

A direct result of the odd character under T is that it isimpossible to �nd a single fermionic state φ that is invariantunder the PT operator. Instead, the closest analogue is a pairof states assembled into a PT -doublet72 of the form(

φ −φPT)=

(φα −Pφ∗βφβ Pφ∗α

), (13)

where φ and φPT are both eigenvectors of a PT -symmetricHamiltonian. �e action of PT on a PT -doublet is thengiven by

PT(φ −φPT

)=

(0 P−P 0

)K(

φα −Pφ∗βφβ Pφ∗α

)

=

(Pφ∗β φα

−Pφ∗α φβ

)=(φPT φ

),

(14)

4

where the pair of eigenvectors have been simply swappedalong with the introduction of a single minus sign. Weshall see later that the use of Slater determinants as anti-symmetric many-electron wave functions enables strict PT -invariance. Note that invariance under PT implies that theenergies of φ and φPT are related by complex conjugation,while the additional assumption of unbroken PT -symmetryimplies that φ and φPT must form degenerate pairs withreal energies.72 Finally we note the inverse relationshipsP−1 = P and (iσy)−1 = −iσy = iσᵀy which, in combi-nation, yield (PT )−1 = −iσyKP .

III. PT -SYMMETRY IN HARTREE–FOCK

A. Hartree–Fock in practice

In the HF approximation, the wave function ΨHF for a sys-tem of n electrons is represented by a single Slater determinantconstructed from a set of n occupied one-electron molecularorbitals φi as

ΨHF = A(φ1φ2 . . . φn), (15)

where A is the anti-symmetrising operator.1 �e single-particle orbitals φi are expanded in a �nite-size direct productspace of N (one-electron) real spatial atomic orbital basis{χ1, . . . , χN} and the spinor basis { |α〉 , |β〉} as

φi =N

∑µ=1

Cαµiχµ |α〉+

N

∑µ=1

Cβµiχµ |β〉 = φiα |α〉+ φiβ |β〉 ,

(16)where φiα and φiβ represent the α and β components of φi

respectively. �e coe�cients Cαµi and Cβ

µi are used to de�nethe Slater determinant, and can be considered as componentsof a (2N × n) matrix C with the form

C =

(Cα

Cβ

), (17)

where Cα and Cβ are (N × n) sub-matrices representing theexpansions of φiα and φiβ.

In general, the atomic orbital basis set is not required tobe orthogonal, although a real matrix X can always be foundsuch that XᵀSX = I, where S is the overlap matrix be-tween atomic orbitals. One particularly convenient choice isX = S−1/2, but other choices are possible.1 Without loss ofgenerality, we assume in the following that we are workingin an orthogonal basis.

As an approximate wave function, ΨHF does not form aneigenfunction of the true electronic Hamiltonian H. Instead,ΨHF is identi�ed by optimising the HF energy EHF de�ned bythe expectation value for a given inner product 〈·|·〉 as

EHF =

⟨ΨHF

∣∣H ΨHF⟩

〈ΨHF|ΨHF〉. (18)

�e optimal set of HF molecular orbital coe�cients C aredetermined using a self-consistent procedure. On each iter-ation k, an e�ective one-electron “Fock” Hamiltonian F(k)

is constructed using the current occupied set of orbitalsC(k), such that F(k) = h + D(k)G, where h and G are theone- and two-electron parts of the Fock matrix and D(k) isthe density matrix at the kth iteration. �e new optimalmolecular orbitals C(k+1) are then obtained by diagonalisingF(k), i.e. F(k)C(k+1) = C(k+1)ε(k+1) where ε is a diagonalmatrix of the orbital energies, and the process is repeateduntil self-consistency is reached.1 At convergence we �ndFD− DF = 0, demonstrating that, only at self-consistency,the Fock and density matrices commute. (We drop the in-dex k for converged quantities.) Note that F is linear withrespect to D, and that h and G are iteration independent andpre-computed at the start of the calculation.

Crucially, although the true n-electron Hamiltonian H isalways Hermitian, the process of dressing H using the HForbitals can lead to a non-Hermitian e�ective one-electronHamiltonian. In fact, the symmetry of D(k) and F(k) can de-pend of the speci�c choice of the inner product in Eq. (18).For example, the most common choice is the Dirac Hermitianinner product 〈x|y〉H = x†y, leading to Hermitian densityD(k) = C(k)(C(k))† and Fock matrices F(k) = (F(k))†, andexplicitly enforcing real energies. Alternatively, the complex-symmetric inner product 〈x|y〉C = xᵀy requires complex-symmetric density D(k) = C(k)(C(k))ᵀ and Fock matricesF(k) = (F(k))ᵀ, with energies that are complex in general. Incontrast to complex-Hermitian HF, the complex-symmetricvariant provides the unique analytic continuation of real HFfor complex orbital coe�cients. �is non-Hermitian formula-tion is used in h-HF theory to ensure solutions exist over allgeometries,13–15,17 and for describing resonance phenomenathrough non-Hermitian approaches.16

In what follows, we employ the complex-symmetric in-ner product 〈.|.〉 ≡ 〈.|.〉C to explore the conditions for PT -symmetry under the HF approximation and understand underwhat circumstances EHF is real. In particular, we make useof the non-Hermitian h-HF formulation since this providesthe natural mathematical extension of real h-HF for complexorbital coe�cients.15 We note that rigorous formulations ofPT -symmetric quantum mechanics introduce an additionallinear operator C and the CPT inner product to de�ne apositive-de�nite inner product and ensure conservation ofprobability, although identifying C is o�en non-trivial.27,44

However, as an inherently approximate approach, HF theoryrequires only a well-de�ned inner product. In our case, sincethe Fock matrix is explicitly complex-symmetric, its eigenvec-tors naturally form an orthonormal set under 〈.|.〉C withoutneeding to introduce the CPT inner product.

B. One-electron picture

We turn now to the behaviour of the one-electron den-sity, Fock matrices, and orbital energies under the PT -operator. First consider the relationship between the complex-

5

symmetric density matrix D = CCᵀ and the equivalentdensity matrix constructed using the PT -transformed coef-�cients denoted CPT = PT C. �e combined PT operatorcan be represented as the product PT = UK, where U is a(2N × 2N) real (linear) unitary matrix.72 Remembering thatK only acts on everything to the right, we subsequently �nd(

CPT︸︷︷︸UKC

)(CPT

)ᵀ= (UC∗)(UC∗)ᵀ

= UK︸︷︷︸PT

CCᵀ KUᵀ︸ ︷︷ ︸(PT )−1

= DPT .(19)

where DPT = (PT )D(PT )−1. As a result, the densitymatrix constructed using the PT -transformed coe�cientsis a PT -similarity transformation of the density matrix con-structed using the original set of coe�cients. Consequently, ifa set of coe�cients is PT -symmetric, then the density matrixmust be as well, and vice versa.

Next consider the symmetry of the Fock matrixF[D] = h + DG, which, due to its dependence onD, inherits the symmetry of the density used to construct it.Assuming that D is PT -symmetric, i.e. D = DPT , we �nd

F[D] = h + DPT G = F[DPT ]. (20)

�is result is trivial since F is linear with respect to D.Moreover, since the one- and two-electron parts of the Fockmatrix are PT -symmetric, i.e. h = (PT )h(PT )−1 andG = (PT )G(PT )−1, we �nd

(PT )F[D](PT )−1 = F[DPT ]. (21)

By equating Eqs. (20) and (21) we see that, if D is PT -symmetric, then F is also PT -symmetric. As a result, thesymmetry of F(k) on a given iteration k is dictated by thesymmetry of the electron density from the current iterationD(k). By extension, the symmetry of the new molecular or-bitals C(k+1) is controlled by the symmetry of F(k) and, if onestarts with a PT -symmetric guess D(0), then PT -symmetrycan be conserved throughout the self-consistent process. Fur-thermore, since C(k+1) is PT -symmetric if and only if thee�ective Fock Hamiltonian F(k) is PT -symmetric (and viceversa), the existence of PT -symmetry in HF can be identi�edby considering only the symmetry of the density itself.

Self-consistency of the HF equations requires the eigenvec-tors, which satisfy F[D]C = Cε, to be equivalent to the coe�-cients used to build D itself. In other words, F and D commuteand share the same set of eigenvectors. Acting on the le� withPT and exploiting the fact that (PT )−1PT = I yields

PT F[D]C = PT F[D](PT )−1PT C = UK(Cε)

=⇒ F[DPT ](CPT ) = (UC∗︸︷︷︸CPT

)ε∗, (22)

where we have used the result of Eq. (21) and the propertythat T is both anti-linear and distributive (see above) suchthat KCε = C∗ε∗. Combining with the result of Eq. (19),

we can draw two conclusions. Firstly, if a given set of or-bital coe�cients C represents an optimised self-consistentHF solution with eigenvalues ε, then its PT -transformedcounterpart CPT must also be a self-consistent solution witheigenvalues ε∗. Secondly, and by extension, if an optimal self-consistent solution is invariant under PT (i.e. C = CPT ),then its eigenvalues must be real.

C. Many-electron picture

We turn now to the symmetry of the full HF Slater determi-nant ΨHF and its associated energy EHF. In the many-electronpicture, the PT -operator for an n-electron system is givenas a product of one-electron operators,

PT =n⊗

i=1

π(i)τ(i), (23)

where π(i) and τ(i) are the parity and time-reversal operatorsacting only on the single-particle orbital occupied by electroni. From the determinantal form of ΨHF [see Eq. (15)], itssymmetry under PT can be extracted as a product of itsconstituent orbitals symmetries.

Now, let us consider the relationship between the total HFenergies of the two coe�cient matrices C and CPT . Notingthat UᵀU = UUᵀ = I and exploiting the invariance of thetrace to cyclic permutations, i.e. Tr(ABC) = Tr(CAB), we�nd

EHF[C] =12

Tr{D(h + F[D])}

=12

Tr{UDUᵀU(h + F[D])Uᵀ}.(24)

Note that K2 = I and, since K acts only to the right, itsapplication on the far right-hand side has no e�ect. �erefore,by applying K to both sides and inserting K2 = I in themiddle, we �nd explicitly

EHF[C]∗ = K Tr{

UDUᵀK2U(h + F[D])Uᵀ}K (25)

Exploiting the distributive nature ofK over the matrix productwithin the trace, and since the reality of U provides KU =UK, we can migrate the K operators to �nd

EHF[C]∗ = Tr{

UK︸︷︷︸PT

D KUᵀ︸ ︷︷ ︸(PT )−1

UK︸︷︷︸PT

(h + F[D]) KUᵀ︸ ︷︷ ︸(PT )−1

}= Tr

{DPT

(h + (PT )F[D](PT )−1

)}= Tr

{DPT

(h + F[DPT ]

)}= EHF[CPT ],

(26)

where we employ the result of Eq. (21) and remember that h isPT -symmetric. Overall we conclude that the respective HFenergies corresponding to the coe�cient matrices C and CPTare related by complex-conjugation. Clearly by extension theHF energy of a PT -symmetric set of orbital coe�cients mustbe real.

6

D. Hartree–Fock PT -doublet

To construct a set of occupied orbitals in the structure ofa PT -doublet [see Eq. (13)], we require an explicit form ofthe matrix U . �e linear parity operator P acts only on thespatial basis and can be represented in the full direct productspace by the Kronecker product I2 ⊗ P, giving

PC =

(P 00 P

)(Cα

Cβ

), (27)

where P is a real (N × N) matrix representation of P in thespatial basis, satisfying P2 = IN . (IN denotes the identitymatrix of size N.) As a result, the combined PT operator canbe represented by the (2N × 2N) matrix constructed fromthe Kronecker product U = (iσy)⊗ P, such that

PT C = UKC =

(0 P−P 0

)K(

Cα

Cβ

)=

(PC∗β−PC∗α

). (28)

In the coe�cient matrix representation [see Eq. (17)], aPT -doublet can then be constructed by pairing each occupiedorbital with its PT -transformed analogue, giving

C = (c −PT c) =

(cα −Pc∗βcβ Pc∗α

), (29)

where c and −PT c form (2N × n/2) sub-matrices repre-senting the paired orbitals of the PT -doublet. �e action ofPT on a PT -doublet is then

PT (c −PT c)

= UK(

cα −Pc∗βcβ Pc∗α

)=

(Pc∗β cα

−Pc∗α cβ

)= (PT c c) = (c −PT c) ,

(30)

where the last line exploits the anti-symmetry of a determi-nantal wave function under the permutation of two columnsin C. Moreover, since the many-electron representation of(PT )2 is given by

(PT )2 =n⊗

i=1

π(i)2τ(i)2 = (−1)nI , (31)

we see that it is only possible to de�ne a PT -symmetric statein systems with ms = 0 (i.e. nα = nβ = n/2) where theoccupied orbitals are paired in the structure of a PT -doubletof the form given by Eq. (29).

�e behaviour of a PT -doublet can be illustrated by con-sidering a simple two-electron Slater determinant constructedfrom the orbitals (φ,−φPT )

Ψ =1√2

∣∣∣∣φ(1) −φPT (1)φ(2) −φPT (2)

∣∣∣∣=−φ(1)φPT (2) + φPT (1)φ(2)√

2.

(32)

�anks to the linearity and antisymmetry properties of deter-minants, Eq. (30) immediately yields

PT Ψ =1√2

∣∣∣∣φPT (1) φ(1)φPT (2) φ(2)

∣∣∣∣ = Ψ. (33)

IV. EXAMPLE OF H2

We now turn our a�ention to the didactic example of theH2 molecule in a minimal molecular orbital (orthogonal) basis

σg = (χL + χR)/√

1 + 2S, (34a)

σu = (χL − χR)/√

1− 2S, (34b)

where χL and χR are the le� and right atomic orbitals andS = 〈χL|χR〉 de�nes their overlap. Without loss of generality,this paradigmatic two-electron system can be considered as aone-dimensional system, and the spatial representation of theparity operator in the (σg, σu) basis is given by

P =

(1 00 −1

). (35)

In the following, all calculations are performed with the STO-3G (minimal) atomic basis. For the sake of simplicity we focuson the ms = 0 spin manifold.

A. Real Orbital Coe�icients

1 2 3 4

Bond Length / A

−1.5

−1.0

−0.5

0.0

0.5

En

ergy

/E

h

RHF

sb-RHF

h-RHF

UHF

sb-UHF

h-UHF

FIG. 2. h-HF energy for the multiple solutions of H2. �e spatiallysymmetry-pure solutions, i.e., the lowest RHF, UHF and highest RHFsolutions correspond to the σ2

g , σgσu and σ2u con�gurations, respec-

tively. In the dissociation limit the spatially symmetry-broken UHF(sb-UHF) states and spatially symmetry-broken RHF (sb-RHF) statescorrespond to diradical con�gurations (�H – H� and �H – H�) andionic con�gurations (H+ – H– and H– – H+), respectively. At shorterbond lengths, the sb-RHF and sb-UHF states coalesce with the spa-tially symmetry-pure RHF solutions and extend into the complexplane as h-RHF and h-UHF states, respectively. �e holomorphic en-ergy, 〈ΨHF|H|ΨHF〉C, of the h-RHF and h-UHF solutions, however,remains real.

7

−0.50 −0.25 0.00 0.25 0.50

θα/π

−0.50

−0.25

0.00

0.25

0.50θ β/π

−1.0

−0.8

−0.6

−0.4

En

ergy/E

h

FIG. 3. �e UHF energy of H2 at 4 A bond length as a functionof θα and θβ showing spin-�ip symmetry (dashed cyan line) andPT -symmetry (dashed red line). �e parity (site-�ip) operationcorresponds to the mapping (θα, θβ)→ (−θα,−θβ), as indicated byinversion through the black star. Spatial symmetry-pure stationarypoints are indicated by red (RHF) and blue (UHF) circles, whilestationary points breaking spatial symmetry are illustrated by cyan(sb-RHF) and purple (sb-UHF) diamonds.

In addition to the spatially symmetry-pure con�gurationsσ2

g , σ2u and σgσu (corresponding to two RHF and a doubly

degenerate pair of UHF solutions), it is well known that, inthe dissociation limit, a pair of degenerate spatial symmetry-broken UHF (sb-UHF) solutions develop (dashed purple linein Fig. 2) in which the electrons localise on opposite atoms.1�ese solutions have a form given by the parameterisation

ΨUHF(r1, r2) =1√2

∣∣∣∣φUHF(r1)α(1) φUHF(−r1)β(1)φUHF(r2)α(2) φUHF(−r2)β(2)

∣∣∣∣ ,

(36)where φUHF represents the optimised spatial orbital corre-sponding to the UHF solution, and r1 and r2 are the spatialcoordinates of electrons 1 and 2 respectively. In the disso-ciation limit, the optimal UHF solutions can be representedschematically as the diradical con�gurations (spin-densitywaves)

•� •� and •� •� . (37)

However, apart from the chemically intuitive idea of electroncorrelation, the justi�cation for solutions existing with thisparticular form is not obvious.

Similarly (although less studied), a pair of degenerate spatialsymmetry-broken RHF (sb-RHF) solutions develop (dashedcyan line in Fig. 2) with a form given by the parameterisation

ΨRHF(r1, r2) = φRHF(r1)φRHF(r2)α(1)β(2)− β(1)α(2)√

2,

(38)

where φRHF represents the optimised spatial orbital corre-sponding to the RHF solution. In the dissociation limit, the sb-RHF solution corresponds to the localisation of both electronson the same atom to produce ionic con�gurations (charge-density waves)

•�� • and • •�� . (39)

Both sb-RHF and sb-UHF solutions are extrema of the HFequations. However, instead of being minima like the sb-UHFsolutions, the sb-RHF states correspond to maxima of the HFequations (see Fig. 2).

Rather than considering the parameterisations (36) and (38),we instead consider the full UHF space using two molecularorbitals

φα = σg cos θα + σu sin θα, (40a)φβ = σg cos θβ + σu sin θβ, (40b)

where θα and θβ are rotation angles controlling the degree oforbital mixing. �e occupied orbital coe�cient matrix in thecombined spatial and spinor direct product basis is thereforegiven by

C =

(Cα

Cβ

)=

cos θα 0sin θα 0

0 cos θβ

0 sin θβ

. (41)

Considering the complex-symmetric density matrix with theblock form

D = CCᵀ =(

Dαα 00 Dββ

), (42)

the condition for PT -symmetry [see Eq. (19)] becomes(Dαα 0

0 Dββ

)=

(PD∗ββP 0

0 PD∗ααP

). (43)

Using the parameterisation (41), this condition reduces to(cos2 θ∗β − 1

2 sin 2θ∗β− 1

2 sin 2θ∗β sin2 θ∗β

)=

(cos2 θα

12 sin 2θα

12 sin 2θα sin2 θα

),

(44)

which is satis�ed when

tan(θα) = − tan(θβ

)∗= tan(−θ∗β). (45)

Clearly for the case of real UHF, i.e. (θα, θβ) ∈ R, Eq. (45)is satis�ed only when θα + θβ = mπ for m ∈ Z, upon whichwe obtain the constrained molecular orbitals

φα = σg cos θα + σu sin θα, (46a)φβ = σg cos θα − σu sin θα. (46b)

In fact, the condition for PT -symmetry in real UHF aligns ex-actly with the parameterisation provided by Eq. (36), as shownin Fig. 3. Furthermore, it is relatively simple to understand

8

−0.4

−0.2

0.0

0.2

0.4

Im[θ

]/π

−0.50 −0.25 0.00 0.25 0.50

Re[θ]/π

−0.4

−0.2

0.0

0.2

0.4

Im[θ

]/π

−5

−4

−3

−2

−1

0

1

2

3

4

En

ergy

/E

h

FIG. 4. �e real (top) and imaginary (bo�om) components of the h-RHF energy of H2 at 0.75 A bond length for θα = θβ = θ as a functionof Re(θ) and Im(θ). �e parity (site-�ip) operator produces the inversion symmetry θ → −θ, as indicated by inversion through the blackstar. Lines of symmetry in the vertical direction (dashed red lines) along Re(θ) = 0 and Re(θ) = ±π/2 coincide with the condition (50) forPT -symmetry. �e energy on either side of this PT -symmetry line are related by complex conjugation, while the energy along the lineitself is real. Spatial symmetry-pure stationary points are indicated by red (RHF) circles, while complex holomorphic stationary points areillustrated by green (h-RHF) diamonds. Note the energy is also real along the line Im(θ) = 0 (dashed green line), since this is the line ofK-symmetry along which the orbital coe�cients are all real.

why this symmetry must exist. Since the orbital coe�cients(and by extension the density) are all real, the action of Tsimply interconverts the two spin states (spin-�ip), while thespatial (parity) operator corresponds to a site-�ip. �e spin-�ip already gives rise to the symmetry plane θα = θβ, alongwhich all RHF solutions lie (cyan line in Fig. 3). �e combinedaction of spin- and site-�ip then essentially gives rise to thePT -symmetry operation for real orbitals.

Schematically, the PT operation can be depicted as

•� •�P−−−→

site-�ip•� •�

T−−−−→spin-�ip

•� •� 3 (47)

�erefore, in the minimal basis considered here, the (real)sb-UHF solutions (which can be labelled as diradical con�gu-rations) are PT -symmetric. In contrast, following a similarargument, the sb-RHF solutions (38) (corresponding to ioniccon�gurations) are de�nitely not PT -symmetric:

•�� • P−−−→site-�ip

• •��T−−−−→

spin-�ip• •�� 7 (48)

However, appropriate linear combinations of these ionic con-�gurations (i.e. multideterminant expansions) can be madeto satisfy PT -symmetry. We note that the spatial symmetry-pure σ2

g and σ2u states both also satisfy PT -symmetry, while

the σgσu solutions are PT -symmetry broken and are inter-converted by the PT operator.

In summary, the real UHF energy surface shows PT -symmetry along the line coinciding with Eq. (36), justifyingthe use of this parameterisation for locating sb-UHF solutions.

As real orbital coe�cients lead to purely real energies, statesinterconverted by this symmetry are strictly degenerate ratherthan being related by complex conjugation. �e well-knownsb-UHF states, resembling diradical con�gurations, lie on thisline and are PT -symmetric solutions with their occupiedorbitals forming a PT -doublet.

B. Complex Orbital Coe�icients

Next we turn to the case of complex orbital coe�cients with(θα, θβ) ∈ C. �e constraint (45) can then be decomposedinto real and imaginary parts

Re(θα) + Re(θβ) = mπ, Im(θα) = Im(θβ). (49)

Along these lines of symmetry we expect the holomorphicHF energy to be real, while we expect the energies of densitymatrices interconverted by the PT operator to be related bycomplex conjugation.

To visualise this symmetry, we �rst consider the h-RHF casewhere θα = θβ = θ and for which we expect PT -symmetricdensities when

θ = mπ/2 + iϑ, (50)

for ϑ ∈ R. Since the holomorphic energy is complex ingeneral, we plot the real and imaginary parts of the energyseparately at a bond length of 0.75 A as functions of Re(θ)and Im(θ) in Fig. 4. As expected, lines of PT -symmetry existalong the values of θ satisfying Eq. (50), where the energy

9

−0.50 −0.25 0.00 0.25 0.50

Re[θα]/π

−0.50

−0.25

0.00

0.25

0.50

Re[θ β

]/π

−0.50 −0.25 0.00 0.25 0.50

Re[θα]/π

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

En

ergy/E

h

FIG. 5. �e real (le�) and imaginary (right) components of the h-UHF energy of H2 at 0.75 A bond length for the illustrative case of �xedimaginary components Im(θα) = Im(θβ) = π/8. Two lines of symmetry exist along Re(θα) = Re(θβ) and Re(θα) + Re(θβ) = π,corresponding to spin-�ip (dashed cyan line) and PT -symmetry (dashed red line) respectively. �e energy along the PT -symmetry line isreal, as predicted, while the energies for states related by PT -symmetry form complex-conjugate pairs. We also note that on this illustrativelandscape representing only a slice of the full holomorphic HF energy surface, there are no holomorphic HF stationary points.

either side is related by complex conjugation and the energyalong the line of symmetry is real. More explicitly, the (verti-cal) red lines in Fig. 4 along Re(θ) = 0 and Re(θ) = ±π/2coincide with the condition (50) for PT -symmetry and cor-respond to real energies. �e energy is also real along theline Im(θ) = 0 (green line), since this corresponds to theK-symmetry line along which the orbital coe�cients are allreal. As illustrated in Fig. 4, the h-RHF stationary solutions(green diamonds) lie on the red line, and the h-RHF statesare therefore PT -symmetric. Since the h-RHF solutions areall spin-�ip symmetric, we can justify PT -symmetry in theenergy landscape as the combination of site-�ip (centre ofinversion at θ = 0) and complex conjugation.

Finally, we consider the complex h-UHF case. As the h-UHFenergy is a function of four real variables (real and imaginaryparts of θα and θβ), we illustrate the general symmetry usingthe speci�c case Im(θα) = Im(θβ) = π/8 and visualise theenergy for H2 at a bond length of 0.75 A in Fig. 5. We �ndthe lines of spin-�ip (cyan) and PT -symmetry (red) occur asin Fig. 3, where now the complex-conjugation of energies oneither side of the line ofPT -symmetry is explicitly observable.Since the site-�ip operation takes (θα, θβ) → (−θα,−θβ),we cannot observe its e�ect on the energy landscape underthe constraint Im(θα) = Im(θβ) = π/8, and we note thatthere are no h-UHF stationary points in Fig. 3. Moreover,inspection of the stationary points corresponding to the h-UHF solutions in Fig. 2 reveals that they follow the form(θα, θβ) = (iϑ,−iϑ), for ϑ ∈ R, leading to the conclusionthat the h-UHF solutions in H2 are not PT -symmetric.

In summary, the PT -symmetric stationary h-HF solutionsfor H2 are the σ2

g , σ2u , sb-UHF and h-RHF states, although

the e�ect of PT -symmetry can be observed throughout theholomorphic HF energy landscape. For these PT -symmetricsolutions, the molecular orbital coe�cients possess the PT -

doublet form [see Eq. (29)]. Stationary points that do notcorrespond to PT -symmetric states (including the sb-RHFand h-UHF solutions) occur in pairs which are interconvertedby the action ofPT , and the onset ofPT -symmetry breakingcoincides with the disappearance of the h-RHF or the sb-UHFsolutions at Coulson–Fischer (quasi-exceptional) points17 ina similar manner to other types of symmetry-breaking in HFtheory.10–12

V. CONCLUDING REMARKS

In this work, we have outlined the conditions for PT -symmetry — a weaker condition than Hermiticity which en-sures real energies — in electronic structure, and speci�callythe HF approximation. In particular, we have explored theexistence of PT -symmetry in the non-Hermitian h-HF for-mulation that forms the rigorous analytic continuation of realHF. Our most important results are:

1. A set of molecular orbitals is PT -symmetric if and onlyif the e�ective Fock Hamiltonian is PT -symmetric,and vice versa.

2. Starting with a PT -symmetric guess density matrix,PT -symmetry can be conserved throughout the self-consistent process.

3. If an optimal self-consistent solution is invariant underPT , then its eigenvalues and corresponding HF energymust be real.

4. PT -symmetry can be explicitly satis�ed by construct-ing the molecular orbitals coe�cients in the structureof a so-called PT -doublet, i.e. pairing each occupiedorbital with its PT -transformed analogue.

10

5. Slater determinants built from PT -doublets lead toPT -symmetric many-electron wave functions.

PT -symmetry provides a novel intrinsic symmetry in theHF energy landscape, where the energies of densities inter-converted by PT are related by complex conjugation. Forreal HF, this symmetry corresponds to the combination of theparity and spin-�ip operations. As an illustrative example, wehave considered the H2 molecule in a minimal basis, where wehave observed the e�ects of PT -symmetry on the HF energylandscape. In particular, we have found that the sb-UHF andh-RHF wave functions are PT -symmetric, while the sb-RHFand h-UHF wave functions break PT -symmetry but occurin complex conjugate pairs related by the PT operator. �etransitions between broken and unbroken PT -symmetry re-gions coincide with the disappearance of the h-RHF or thesb-UHF solutions at Coulson–Fischer points.

By demonstrating the existence of PT -symmetric solu-tions with real energies in the HF approximation, we removethe rigorous condition of Hermiticity that is usually applied inelectronic structure theory. We are currently working on theimplementation of restricted, unrestricted and generalised HFself-consistent approaches that explicitly enforce this symme-try. Ultimately, by bridging the gap between PT -symmetricphysics and quantum chemistry, we hope to pave the way forthe development of new classes of non-Hermitian Hamiltoni-ans with real eigenvalues in electronic structure theory.

ACKNOWLEDGMENTS

H.G.A.B. thanks the Cambridge Trust for a studentship andA.J.W.T. thanks the Royal Society for a University ResearchFellowship (UF110161). We also thank Bang Huynh for in-sightful conversations throughout the development of thiswork.

Appendix A: T -Symmetry for General Spins

We loosely follow Weinberg’s discussion on the natureof T in Ref. 77, although we use notation more familiar toelectronic structure. Since the action of T reverses angularmomentum, we require spin-angular momentum operators sto satisfy

sT = −T s, s2T = T s2. (A1)

Considering a general spin state |s, ms〉, we can then show

sz[T |s, ms〉] = −T sz |s, ms〉 = −ms[T |s, ms〉], (A2)

s2[T |s, ms〉] = T s2 |s, ms〉 = s(s + 1)[T |s, ms〉]. (A3)

In combination, these results imply

T |s, ms〉 = γ(s, ms) |s,−ms〉 , (A4)

for some complex value γ(s, ms).

To identify the functional form of γ(s, ms), we use theladder operators s± = sx ± isy which, due to the anti-linearcharacter of T , satisfy

s±T = −T s∓. (A5)

From the standard ladder operator relationship

s± = ξ±(s, ms) |s, ms ± 1〉 , (A6)

where ξ±(s, ms) =√

s(s + 1)−ms(ms ± 1), we �nd

s±T |s, ms〉 = −T s∓ |s, ms〉= −ξ∓(s, ms)T |s, ms ∓ 1〉 .

(A7)

Alternatively, from the result of Eq. (A4) we can explicitlyidentify

s±T |s, ms〉 = γ(s, ms)ξ±(s,−ms) |s,−ms ± 1〉 , (A8)

and

T |s, ms ∓ 1〉 = γ(s, ms ∓ 1) |s,−ms ± 1〉 . (A9)

Inserting Eq. (A8) and Eq. (A9) into the LHS and RHSof Eq. (A7) respectively, and noting that ξ±(s,−ms) =ξ∓(s, ms), we �nd

γ(s, ms) = −γ(s, ms ∓ 1) = (−1)s−ms γ(s, s). (A10)

Here γ(s, s) is an arbitrary complex value that is convention-ally set to unity. As a result, the action of T on a function φ, ex-pressed in the basis {|s, ms〉} with dimension ns = (2s + 1),is given by

T φ = ZKφ, (A11)

where Z is an (ns × ns) orthogonal matrix representing theaction of T on the spin eigenfunctions, given explicitly as

Zij =

{(−1)2s+1−i, if i + j = 2(s + 1),0, otherwise.

(A12)

Next, consider the speci�c Z matrices for various spin cases:

s = 0 : Z = (1) ,

s =12

: Z =

(0 1−1 0

),

s = 1 : Z =

0 0 10 −1 01 0 0

,

s =32

: Z =

0 0 0 10 0 −1 00 1 0 0−1 0 0 0

,

s = 2 : Z =

0 0 0 0 10 0 0 −1 00 0 1 0 00 −1 0 0 01 0 0 0 0

.

...

11

Signi�cantly, for the bosonic (integer s) case, Z is symmetric,i.e. Z = Zᵀ, while in the fermionic (half-integer s) case, Zbecomes skew-symmetric, i.e. Z = −Zᵀ. �is observationleads to two key results. First, considering the operationT 2φ = Z2K2φ, we �nd Z2 = ZᵀZ = I in the bosoniccase and Z2 = −ZᵀZ = −I for the fermionic case. Whencombined with the relationship K2 = I , this result leadsdirectly to the even and odd character of T for bosons andfermions respectively. Secondly, since Z is symmetric andorthogonal in the bosonic case, it can be decomposed into theform Z = VΣVᵀ, where V is orthogonal and Σ is a diagonalmatrix containing the eigenvalues of Z, each equal to −1 or+1. Taking Σ = λλᵀ, we �nd

T φ = ZKφ = VλλᵀVᵀKφ = (Vλ)K(Vλ)†φ. (A13)

Consequently, for the bosonic case, one can always �nd atransformation (Vλ) into a basis under which the action oftime-reversal reduces to T = K, described in Ref. 72 as a“canonical” bosonic basis. Similarly, in the fermionic case, theproperties of skew-symmetric orthogonal matrices allow Zto be decomposed into the form Z = QΛQᵀ,78 where Q isorthogonal and Λ takes the form

Λ =

iσy. . .

iσy

. (A14)

As a result, the action of T can be expressed as

T φ = ZKφ = QΛQᵀKφ = (Q)ΛK(Q)†φ, (A15)

and thus, for fermionic systems, it is always possible to �nd atransformation Q into a canonical fermionic basis in whichthe action of time-reversal reduces to T = ΛK.72

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