Hartree-Fock symmetry breaking around conical intersections
HHartree-Fock symmetry breaking around conical intersections
Lena C. Jake
Department of Chemistry, Rice University, Houston, TX 77005-1892
Thomas M. Henderson and Gustavo E. Scuseria
Department of Chemistry, Rice University, Houston, TX 77005-1892 andDepartment of Physics and Astronomy, Rice University, Houston, TX 77005-1892 (Dated: April 10, 2018)We study the behavior of Hartree-Fock (HF) solutions in the vicinity of conical intersections.These are here understood as regions of a molecular potential energy surface characterized by de-generate or nearly-degenerate eigenfunctions with identical quantum numbers (point group, spin,and electron number). Accidental degeneracies between states with different quantum numbers areknown to induce symmetry breaking in HF. The most common closed-shell restricted HF insta-bility is related to singlet-triplet spin degeneracies that lead to collinear unrestricted HF (UHF)solutions. Adding geometric frustration to the mix usually results in noncollinear generalized HF(GHF) solutions, identified by orbitals that are linear combinations of up and down spins. Near con-ical intersections, we observe the appearance of coplanar GHF solutions that break all symmetries,including complex conjugation and time-reversal, which do not carry good quantum numbers. Wediscuss several prototypical examples taken from the conical intersection literature. Additionally,we utilize a recently introduced a magnetization diagnostic to characterize these solutions, as wellas a solution of a Jahn-Teller active geometry of H +28 . I. INTRODUCTION
For the past several years we have been interestedin the development of low-scaling computational meth-ods for dealing with near-degenerate states, the so-calledstrong correlation problem [1–3]. Accidental degenera-cies, by which we mean degeneracies between states withdifferent quantum numbers, are ubiquitous. They occur,for example, in most molecular dissociations to open-shellfragments. At the mean-field level, these degeneracies areassociated with symmetry breaking, and can be detectedby examining the eigenvalues of the symmetry adaptedmolecular orbital (MO) Hessian where they lead to well-studied instabilities of singlet, triplet and complex char-acter [4, 5]. Already, we have shown that symmetry pro-jected methods in their variation after projection version[6–8] are capable of dealing with accidental degeneraciesin a variety of practical contexts [9–11].The purpose of the present study is to extend our un-derstanding of symmetry breaking and restoration to thesituation of conical intersections (CX), where degeneratestates share all of the same quantum numbers. Theseintersections or near intersections are necessary for anynonadiabatic process, providing a means of traveling be-tween potential energy surfaces. They play an integralrole in excited state dynamics and radiationless relax-ation, explaining photochemical mechanisms for internalconversion. Though they have been little explored, to usethe words of Domcke, Yarkony, and K¨oppel, their pres-ence seems to be “the rule rather than the exception” inpolyatomic molecules [12].It is convenient to discuss these intersections in termsof how the degeneracy is lifted. At a genuine CX, onecan define two directions in which the degeneracy is liftedlinearly, together forming what is known as the branching plane. These two directions are defined by the gradientdifference (GD) and derivative coupling (DC) vectors,defined respectively as (cid:126)x = ∂ ( E − E ) ∂ (cid:126)R , (1a) (cid:126)x = (cid:104) ψ ∂ψ ∂ (cid:126)R (cid:105) , (1b)where E and E are the energies of the intersectingstates, ψ and ψ are their wave functions, and (cid:126)R arethe nuclear coordinates. The remaining 3 N − (cid:126)x can be used to optimize a conicalintersection geometry, as it will point toward the apex ofthe cone when at a nearby geometry [12, 14, 15].To the best of our knowledge, the only application ofsymmetry projected Hartree-Fock (PHF) to conical inter-sections that has been carried out so far is in ozone [16],where non-orthogonal configuration interaction (NOCI)in the Hartree-Fock basis produces a qualitatively correctdescription of the relevant states. The limited number ofdegrees of freedom in ozone permits a scan of all degreesof freedom, a luxury not afforded by the molecules exam-ined here. As a first step in this process, we explore theHartree-Fock (HF) landscape in the branching planes ofCXs optimized by the Complete Active Space Self Con-sistent Field (CASSCF) method.The description of conical intersections at the Hartree-Fock level is complicated by the tendency of HF to breaksymmetries, which precludes assigning quantum numbersto states. After all, if we cannot assign quantum num-bers, we cannot meaningfully speak of degenerate states a r X i v : . [ phy s i c s . c h e m - ph ] A p r which have the same quantum numbers! Symmetry pro-jection, however, will restore these broken symmetries,and make the discussion of conical intersections meaning-ful again. For now, we are interested simply in lookingat which symmetries break and how those symmetriesbreak in the vicinity of a CASSCF conical intersection.Our motivation here is simple: our experience is that weshould deliberately break and projectively restore thosesymmetries which might break spontaneously anyway.The present work explores the HF landscape when allsymmetries are allowed to break. Future investigationswill examine the picture that emerges when they are re-stored. In the branching planes considered here, we ob-serve intersections of unrestricted Hartree-Fock (UHF)excited states that occur near the CASSCF CX geome-tries. Complex coplanar generalized Hartree-Fock (GHF)solutions are found in the vicinity of UHF degeneracies,in some cases interpolating between states of the same(conserved) quantum numbers. In our most detailed ex-ploration, the branching plane of cyclobutadiene, we findmultiple complex GHF intersections. To complementthese examples of coplanar spin we present a noncoplanarGHF solution of tetrahedral H +28 .Before we present our detailed results, however, let ustake a moment to discuss the various kinds of Hartree-Fock solutions and how we might distinguish betweenthem so we can properly decode what kinds of projectionoperators we will need in a symmetry restored treatmentof conical intersections. II. CLASSES OF HARTREE-FOCK SOLUTIONS
The restricted Hartree-Fock (RHF) wave function isan eigenfunction of both ˆ S and ˆ S z and is usually alsoan eigenfunction of time reversal ( ˆΘ), complex conju-gation ( ˆ K ), and point-group operators. Unfortunately,RHF fails for strongly correlated systems, and one ex-pects strong correlation in conical intersections as a mat-ter of course due to the degeneracy.Where RHF fails, one might use the unrestrictedHartree-Fock (UHF) formalism instead. By permitting ↑ -and ↓ -spin electrons to occupy different spatial orbitals,UHF provides better energies at the cost of ˆ S symmetry.Typically UHF solutions also break point group symme-try; they must break at least one of complex conjugationand time-reversal symmetries. Most UHF calculationsresult in real orbitals and are hence ˆ K eigenfunctions.Sometimes UHF also breaks down, and one must allowfor a generalized Hartree-Fock (GHF) approach [17] inwhich ˆ S z symmetry breaks in addition to ˆ S . By break-ing ˆ S z symmetry, GHF solutions provide noncollinearspin arrangements. As with UHF, GHF solutions alsousually break point group symmetry and must break atleast one of time-reversal and complex conjugation sym-metries. Real GHF solutions which are ˆ K eigenfunctionshave coplanar spin densities [18] while complex GHF so-lutions may have coplanar or noncoplanar spin densities. Noncoplanar spin has been seen previously in systemswhere high symmetry geometries induce spin frustration[18, 19] and in model Hamiltonians such as the Hubbardor Ising models [20, 21]. Though GHF significantly im-proves the shortcomings of RHF and UHF in stronglycorrelated systems, it has not been regularly used in thecommunity. A complete table classifying HF solutions bythe symmetries they preserve can be found in both [17]and [18], though note that this classification was first car-ried out by Fukutome [22] and has also been discussedextensively by Stuber [23].As we have alluded to earlier, one can check whetherthere is a lower-energy HF solution by considering theeigenvalues of the MO Hessian. A negative eigenvalueindicates the presence of a more stable solution in the di-rection of the corresponding eigenvector. By taking par-ticular blocks of the Hessian, one can limit one’s testingto consider instabilities to a particular symmetry block[5]. For example, one could test whether an RHF solutionis unstable toward other RHF solutions, or toward UHFsolutions, i.e. one can look for singlet or triplet insta-bilities. Similarly, one can test whether UHF solutionsare unstable toward GHF wave functions, and one cantest for instabilities toward solutions which break com-plex conjugation symmetry.Note that a degeneracy between occupied and virtualorbitals guarantees a negative diagonal element in theMO Hessian and therefore an instability, but is not nec-essary for such an instability to exist [24]. Symmetrybroken solutions in the branching planes explored hereprovide further counterexamples. Other cases includethe noncollinear solutions found in fullerene molecules[10, 18], where large band gaps persist as RHF succumbsto UHF and ultimately GHF.The number of zero eigenvalues can also yield infor-mation regarding symmetry breaking and stability, as aUHF or GHF solution will acquire improper zero modesas an artifact of symmetry breaking. The simplest ex-ample of this would be the dissociation of H , where be-yond the Coulson-Fischer point UHF yields a more stablesolution than the singlet RHF that is stable at equilib-rium bond length. Past this point, the lowest Hessianeigenvalue of the RHF solution becomes negative, whilethe UHF solution has two improper zero modes due tobreaking ˆ S x and ˆ S y [1]. For an equally simple example ofa triplet instability, the interested reader might examineHF solutions to the Be atom [1, 17]. III. DETERMINING COPLANARITY
It should be noted that while we have discussed thesymmetry breaking permitted by different incarnations ofHF, permitting a symmetry to break does not guaranteethat it will. That is to say, a GHF search may still arriveat a UHF solution. In such a case, the UHF solution mayeven be an eigenfunction of spin in some other direction– say, ˆ S z – and it is not immediately obvious how to FIG. 1. The cyclobutadiene CX geometry (center, top and bottom) and CX displaced by (cid:126)x (top) and (cid:126)x (bottom) with weightsof ± T of τ Characterization3 3 nonmagnetic (RHF)2 2-3 collinear (UHF)0-1 1-3 coplanar (GHF)0 0 noncoplanar (GHF) tell a noncollinear solution from a rotated UHF solution.Similarly, it is not necessarily simple to tell whether aGHF solution is coplanar or noncoplanar.In the last few years, means of differentiating betweencollinear and noncollinear solutions have emerged. Small,Sundstrom, and Head-Gordon define a test [25] whichuses the fact that if a wave function is an eigenfunctionof ˆ S ˆ n for some direction ˆ n , then (cid:104) ˆ S n (cid:105) − (cid:104) ˆ S ˆ n (cid:105) = 0. Thus,if the matrix (cid:104) ˆ S i ˆ S j (cid:105) − (cid:104) ˆ S i (cid:105)(cid:104) ˆ S j (cid:105) has any zero eigenvalues,the solution must be collinear. This revelation is integralto the diagnostic used here, but in its original formulationhas the drawback of relying on the two-particle densitymatrix. In a recent paper [18], we have shown a simplifiedtest which is identical to that of Small and coworkers forsingle determinants and which can discriminate betweencoplanar and noncoplanar solutions.The density matrix γ can be decomposed into its x , y , and z spin components as M x = γ ↑↓ + γ ↓↑ , (2a) M y = i ( γ ↑↓ − γ ↓↑ ) , (2b) M z = γ ↑↑ − γ ↓↓ . (2c)If spin rotations can make two of these components van-ish, the density matrix is collinear. If spin rotations canmake one of these components vanish, the density matrixis coplanar. We can check this possibility by diagonaliz-ing the matrix T with componentsT ij = Tr( M i S M j S ) (3)where S is the overlap matrix. Collinear determinantscorrespond to one non-zero eigenvalue of T , while fornoncollinear determinants T has two or three non-zeroeigenvalues.While this test is identical to that of Small and cowork-ers for single determinants, we can generalize it slightly totest for coplanarity. Noncoplanar density matrices meanthat all three of M x , M y , and M z must have non-zeroreal parts. Thus, we can distinguish coplanar from non-coplanar GHF solutions by diagonalizing the related ma-trix τ with components τ ij = Tr[ Re ( M i ) S Re ( M j ) S ] . (4)For coplanar GHF solutions, τ has a zero eigenvalue.Our test is summarized in Table I. Note that we ordereigenvalues so that after rotation, T zz ≥ T xx ≥ T yy , andsimilarly for eigenvalues of τ . More details about thismagnetization diagnostic can be found in [18]. −151.75−151.73−151.71−0.4 −0.2 0 0.2 0.4 E ( E h ) w m=0m=0m=1 −151.71−151.7−151.69 −0.2 −0.1 0 0.1 0.2 E ( E h ) w m=0m=0m=1UHFGHF FIG. 2. Energies for displacement of the cyclobutadiene CX by the DC vector (cid:126)x , plotted as a function of displacement weight w . Left panel: CASSCF(4,4) energies. Right panel: HF energies, with regions of noncollinear spin indicated by dashed lines.Collinear solutions are labeled by the m quantum number associated with ˆ S z . Not pictured is the stable HF solution, which iscollinear with m = 0. IV. RESULTS
In its most recent version, the
Gaussian suite of pro-grams only supports CX optimization using the CASSCFmethod. The resulting geometry and branching plane arenot necessarily the same as those defined by a PHF de-generacy, and it is not guaranteed that such a degeneracycould be classified as a CX at all. In this work, we followHF solutions in the CASSCF branching plane.Conical intersection geometries were optimized us-ing equally-weighted state-averaged CASSCF calcula-tions with no symmetry constraints, as implemented in
Gaussian16 [26]. The more affordable spin-free Hartree-Waller determinants were used in CASSCF calculations,and therefore where singlets and triplets are shown to-gether it should be noted they come from separate state-averaged calculations. Active spaces were defined asonly the π orbitals and electrons, and all calculations,CASSCF or HF, were carried out in the STO-3G ba-sis. This minimal basis set was used in an effort toavoid smearing static correlation effects with those ofdynamic correlation. The CXs of aromatic and antiaro-matic molecules are well documented in computationalorganic chemistry literature [12, 27–30], providing start-ing points for geometry optimizations.A variety of initial guesses yielded many UHF solu-tions, which were in turn followed as the geometry wasdisplaced by a range of weights of the branching planevectors. Where we refer to singlet or triplet UHF solu-tions, we should clarify that this is in reference to the m quantum number associated with ˆ S z , rather than the s quantum number associated with ˆ S . To find GHF so-lutions, we destroyed ˆ S z symmetry with application of aFermi-Contact perturbation to the converged UHF andhalted the resulting calculation after several iterations.This symmetry broken initial guess served as a starting point for a GHF calculation with no perturbation. Di-rections for the perturbation were selected from linearcombinations of branching plane vectors, with the mo-tivation that the directions that lift degeneracy shouldalso make convergence to a new, symmetry broken solu-tion more likely than convergence back to either of theintersecting collinear surfaces.A modified development version of Gaussian [31] car-ried out the collinearity test of Small and coworkers.Once a solution was identified as noncollinear, in-housecode was used to determine coplanarity by diagonaliza-tion of τ of Eq. 4. Another modified development versionof Gaussian [31] calculated the GHF Hessian to deter-mine stability. The number of Hessian zero modes, in ad-dition to reflecting the symmetry breaking we show withthe magnetization diagnostic, will also be used to de-tect degeneracies, near degeneracies, or symmetric invari-ances. Where we present molecular geometries, these fig-ures have been created using the X-Window CrystallineStructures and Densities software [32]. Below, we discussHF in the branching plane of four different CXs, with afocus on that of cyclobutadiene. We observe intersectingUHF states near the CASSCF CX in all cases, and incyclobutadiene we see complex coplanar GHF solutionscross as well. In each branching plane we converged tocomplex coplanar GHF for geometries around UHF in-tersections.
A. Cyclobutadiene
A CX was optimized between the first two singlets ofcyclobutadiene, resulting in a loosely defined C s geom-etry (Fig. 1) where the ring has been bent to a 25 ◦ degree dihedral angle and the hydrogen atoms pulled outof plane [27]. At this geometry, the CASSCF energy dif-ference between these two states is 0.2 kcal/mol. τ µµ w µ =x µ =y µ =z < S > m=0m=0m=1UHFGHF H O M O − L U M O G a p ( e V ) m=0m=0m=1UHFGHF FIG. 3. HF band gap, (cid:104) ˆ S (cid:105) , and eigenvalues of τ (Eq. 4) fordisplacement of the cyclobutadiene CX by the DC vector (cid:126)x ,plotted as a function of displacement weight w . Line styleand color scheme are consistent with Fig. 2. Motion along the DC vector (cid:126)x corresponds to shorten-ing and lengthening of alternate C-C bonds, resulting indissociation into different C H fragments in the positiveand negative directions (Fig. 1). Along (cid:126)x , the CASSCFexcitation energy remains linear until weights of about ± (cid:126)x , displacementfrom the CX geometry results in a more bent dihedralangle in the four carbons, and an increase in alternatingbond angles of the ring (Fig. 1). Motions in the positiveand negative directions have less symmetric effects onthe CASSCF energy than seen along the DC vector, andthe excitation energy becomes nonlinear before weightsof ± (cid:126)x (Fig. 2),two singlet UHF states intersect at a geometry very nearthe CASSCF CX, with a triplet UHF solution about 2kcal/mol above them. Complex coplanar GHF solutionsinterpolate between each of the singlet states and thetriplet, intersecting 0.3 kcal/mol below the UHF singlets. −0.3−0.15 0 0.15 0.3w −0.3 −0.15 0 0.15 0.3w −10−5 0 5 10 15 20 E − E i n t ( k c a l / m o l ) FIG. 4. Energies of two intersecting UHF solutions in thebranching plane of cyclobutadiene, plotted as the differencefrom the energy at their intersection. Color scheme is consis-tent with that of Fig. 2 and 3.
At the point of intersection, other properties of the GHFsolutions coincide as well (Fig. 3). Another coplanarGHF, not pictured in Fig. 2 or 3, branches off of theUHF triplet and vanishes before rejoining the UHF sin-glet. A fourth complex coplanar GHF solution, plottedin green, exists only for a small range of w around theCX geometry, another 0.3 kcal/mol below the GHF in-tersection. While this appears at first glance to interpo-late between the singlet UHF states, it vanishes beforereaching either. In all cases where these GHF disappear,spin properties change dramatically as this geometry isapproached, seeming to signal the convergence failure tocome. None of the Hartree-Fock states described thus faris the ground state; the stable solution is collinear with m = 0 and 16 kcal/mol lower, giving all states at leastone negative Hessian eigenvalue.It is worth noting that for a range of weights the low-est energy GHF has four Hessian zero modes. Three canbe attributed to symmetry breaking, while the fourth in-dicates a quasi-symmetric invariance that we have notbeen able to fully identify. This unaccounted for zero-mode emphasizes the need for future work investigatingHF around this CX. Another interesting trait of this GHFsolution is its MO structure. For each of the UHF solu-tions, orbital energies occur in degenerate pairs aroundthe CX. While the intersecting GHF solutions do not re-flect the loose C s symmetry of the CX geometry in thesame way, this lowest energy GHF does.While the CASSCF degeneracy is lifted along the GDvector, both the UHF and GHF solutions seen inter-secting along (cid:126)x remain nearly degenerate as they arefollowed along (cid:126)x (Fig. 4), a clear deviation from thebranching plane behavior we would expect. Rather thanrestoring ˆ S z symmetry for negative w , the GHF solu- FIG. 5. Left to right: The benzene, styrene, and fulvene CX geometries. tions remain noncoplanar. The GHF solutions that van-ish along (cid:126)x also vanish for negative displacements along (cid:126)x , at a weight of just over w = − .
25. All GHFsolutions continue in the positive direction and restorecollinearity before w = 0 .
5. It seems that while degen-eracies of HF states occur very near the CASSCF CXgeometry, the motions corresponding to the CASSCFbranching plane vectors do not lift the degeneracies ofHF states in quite the same way. This suggests that theCASSCF and projected Hartree-Fock branching planeswill be distinct. τ µµ w µ =x µ =y µ =z −227.8−227.7−227.6 H F E ( E h ) m=0m=0GHFUHF FIG. 6. HF energy and eigenvalues of τ (Eq. 4), for fulvenealong the GD vector (cid:126)x , plotted as a function of displacementweight w . B. Benzene, Fulvene, and Styrene
CXs between singlets in benzene[28, 33] andstyrene[29, 33] share similar geometries with loose C s symmetry, such that one atom of the ring is pushed outof plane to a pre-fulvene-like puckered ring (Fig. 5).For each, displacement along the DC vector (cid:126)x resultsin pushing the out of plane moiety either further out ofor into the plane of the ring, depending on the directionof displacement. Neither CX is the ground state, eachbeing less than 30 kcal/mol above the stable CASSCFtriplet.Exploring Hartree-Fock along this vector reveals in-tersecting UHF singlets above a complex coplanar GHFground state in each case. In benzene, the GHF solutioninterpolates between the UHF singlets, while in styrenethe GHF solution interpolates between one of the UHFsinglets and a UHF triplet that crosses the singlets nearby(Fig. 7). Attempts to follow this UHF singlet fail after asudden change in spin properties, as seen in in the van-ishing solutions of cyclobutadiene.The CX geometry in fulvene (Fig. 5) is achieved bytwisting the methylene group approximately 30 ◦ [30, 33]from the planar ground state geometry. Displacementalong the GD vector (cid:126)x results in pulling the two carbonsopposite the ring’s substituent close together and pullingthe methylene group away from the ring. UHF singletsintersect near the CASSCF CX, one of which fails toconverge for larger positive weights. A complex coplanarGHF interpolates between these, though along the otherbranching plane vector the solution never joins UHF andremains noncollinear. C. T d H +28 While the GHF solutions found in the branching planesdiscussed here break all symmetries of the Hamiltonian,spin remains coplanar in all cases. To observe noncopla-nar spin, we turn to the Jahn-Teller active H +28 (Fig. 8).Here, we have taken the tetrahedral H model and deco-rated each surface of the tetrahedron with an additionalhydrogen atom, resulting in a structure that is also tetra- τ µµ w µ =x µ =y µ =z −227.74−227.72−227.7−227.68−227.66 H F E ( E h ) m=0m=0UHFGHF τ µµ w µ =x µ =y µ =z −303.7−303.66−303.62−303.58 H F E ( E h ) m=0m=0m=1GHFUHF FIG. 7. HF energy and eigenvalues of τ (Eq. 4) for displacement of the CX by the DC vector (cid:126)x , plotted as a function ofdisplacement weight w . Left panel: benzene. Right panel: styrene. hedral. Examining the MO structure of the lowest energyreal RHF solution, the neutral species has a triply degen-erate HOMO due to point group symmetry. Removingtwo electrons results in a ground state degeneracy thatis eliminated upon distortion to lower symmetry pointgroups.Thus, there is a Jahn-Teller mandated CX in this tetra- FIG. 8. Geometry of tetrahedral H . TABLE II. Eigenvalues of τ for the stable GHF of H +28 .Element Value τ xx τ yy τ zz hedral H for every H-H bond length. Unlike our pre-vious examples, the stable solution here is complex andnoncoplanar. The high symmetry leads to a density ma-trix structured such that M x = M y = M z . This isreflected in the eigenvalues of τ for a H-H bond length of1.67 ˚A, seen in Tab. II. While removing just one electronwould also result in a Jahn-Teller active ion, for H +8 thestable solution is collinear. It seems that having a differ-ent number of ↑ - and ↓ -spin electrons interferes with thespin frustration introduced by the tetrahedral geometry. V. DISCUSSION
In the branching planes of conical intersections we ob-serve HF solutions that break all symmetries, includingthose not represented by quantum numbers ( ˆ K and ˆΘ).Use of a recently developed magnetization diagnostic re-vealed that all GHF solutions found in these branch-ing planes are coplanar. The same diagnostic identifieda noncoplanar GHF solution in the Jahn-Teller activetetrahedral geometry of H +28 . It seems that while thespin frustration introduced by this highly symmetric ge-ometry will lead to noncoplanar spin in HF, the strongcorrelation around a CX will not. Our work here suggeststhat we will need to deliberately break and projectivelyrestore both ˆ S and ˆ S z symmetries as well as point groupand complex conjugation or time reversal.While the spontaneous symmetry breaking seen hereprecludes the use of Hartree-Fock in the descriptionof conical intersections, it is encouraging for projectedHartree-Fock methods. Even the simple projection aftervariation formalism will restore good symmetries, andshould allow for the description of conical intersectionsreasonably well. Even better is to use the variation afterprojection approach, in which the mean-field determinantis optimized in the presence of the symmetry projectorrather than in its absence. Either way, symmetry pro-jection by means of a NOCI will lead to multireferencewave functions obtained with loosely mean-field compu-tational cost in the vicinity of a CX. This seems a logicalconsequence of the breakdown of the Born-Oppenheimerapproximation and our mean-field attempt to describedynamics on multiple potential energy surfaces.While the present results show some of the qualitativefeatures of the CASSCF branching plane reflected in HFpotential energy surfaces, there are some inconsistencies.Namely, it seems our CASSCF definition of the branchingplane in cyclobutadiene only lifts HF degeneracy alongone of its defining vectors. This suggests but does notguarantee that we will see a different branching plane atthe PHF level. If symmetry breaking and restoration isto be considered an affordable alternative to the CASSCFor Full Configuration Interaction (FCI) levels of theory,a necessary step is to confirm that HF excited states will exhibit the same phenomena, such as CXs, that we areable to observe these higher levels of theory. Throughout,we have been cautious in the language used to describeHF degeneracies.As our symmetry broken solutions do not have goodquantum numbers, they cannot define a CX or CX seam.The symmetry restored solutions, however, could poten-tially be optimized to CX geometry. Characteristic of aCX is the appearance of an observable geometric phaseknown as the Berry phase, whose existence is reliant onpreservation of time reversal ˆΘ. This effect is not exclu-sive to conical intersections; it emerges in any situationwhere there is coupling to variables, in this case nucleardegrees of freedom, that have been excluded from theHilbert space of the eigenvalue problem. It is nonlocaland can be observed in any wavefunction that traversesa closed loop containing the CX [20, 21, 34]. Projectionof ˆΘ and calculation of this observable for a PHF CXprovides an interesting future direction of this work.There is clearly work to be done, but the current resultsare promising: it may be possible to tap into the potentialof symmetry breaking and restoration in HF as an FCIalternative with mean-field computational scaling. ACKNOWLEDGMENTS
This work was supported by the National ScienceFoundation under Award No. CHE-1462434. G.E.S. is aWelch Foundation Chair (No. C-0036). We would like tothank Irek Bulik and Yao Cui for development of
Gaus-sian links for GHF Hessian diagonalization, and furtherthank Irek Bulik for development of an additional
Gaus-sian link for testing the collinearity of GHF solutions. [1] Y. Cui, I. W. Bulik, C. A. Jimen´ez-Hoyos, M. H. Hen-derson, and G. E. Scuseria, J. Chem. Phys , 154107(2013).[2] G. E. Scuseria, C. A. Jim´enez-Hoyos, T. M. Henderson,K. Samanta, and J. K. Ellis, J. Chem. Phys , 124108(2011).[3] C. A. Jim´enez-Hoyos, T. M. Henderson, and G. E. Scuse-ria, J. Chem. Phys. , 164109 (2012).[4] T. D. Crawford, E. Kraka, J. F. Stanton, and D. Cremer,J. Chem. Phys , 10638 (2001).[5] R. Seeger and J. Pople, J. Chem. Phys. , 3045 (1977).[6] C. A. Jimen´ez-Hoyos, R. Rodr´ıquez-Guzm´an, and G. E.Scuseria, J. Chem. Phys , 204102 (2013).[7] C. A. Jimen´ez-Hoyos, R. Rodr´ıquez-Guzm´an, and G. E.Scuseria, J. Chem. Phys. , 224110 (2013).[8] R. Rodr´ıquez-Guzm´an, C. A. Jimen´ez-Hoyos, and G. E.Scuseria, Phys. Rev. B. , 195109 (2014).[9] P. Rivero, C. A. Jimen´ez-Hoyos, and S. G. E., J. Phys.Chem. , 8073 (2013).[10] C. A. Jim´enez-Hoyos, R. Rod´ıguez-Guzm´an, and G. E.Scuseria, J. Chem. Phys. A , 9925 (2014).[11] R. Rodr´ıquez-Guzm´an, C. A. Jimen´ez-Hoyos, and G. E.Scuseria, Phys. Rev. B. , 195110 (2014).[12] W. Domcke, D. R. Yarkony, and Horst, Conical Inter- sections: Electronic Structure, Dynamics & Spectroscopy (World Scientific Publishing Co. Pte. Ltd., 5 Toh TuckLink, Singapore 596224, 2004).[13] G. Herzberg and H. C. Longuet-Higgins, Faraday Soc. , 77 (1963).[14] M. A. Robb, “Advances in physical organic chemistry,”(Elsevier Ltd.: Academic Press, 2014) Chap. 3, pp. 189–228.[15] F. Sicilia, L. Blancafort, M. J. Bearpark, and M. A.Robb, J. Chem. Phys. , 2182 (2007).[16] A. J. W. Thom and M. Head-Gordon, J. Chem. Phys. , 124113 (2009).[17] C. A. Jim´enez-Hoyos, T. M. Henderson, and G. E. Scuse-ria, J. Chem. Theor Comput. , 2667 (2011).[18] T. M. Henderson, C. A. Jim´enez-Hoyos, and G. E. Scuse-ria, J. Chem. Theory Comput. (2017).[19] J. E. Peralta, G. E. Scuseria, and M. J. Frisch, Phys.Rev. B. , 125119 (2007).[20] R. Resta, J. Phys. Condens. Matter , R107 (2000).[21] C. D. Batista, S. Lin, S. Hayami, and Y. Kamiya, Re-ports on Progress in Physics , 084504 (2016).[22] H. Fukutome, Int. J. Quantum Chem. , 9551065(1981).[23] J. L. Stuber and J. Paldus, “Symmetry breaking in the independent particle model,” in Fundamental World ofQuantum Chemistry: A Tribute Volume to the Memoryof Per-Olov L¨owdin , Vol. 1, edited by E. J. Br¨andas andE. S. Kryachko (Kluwer Academic Publishers, Dordrecht,The Netherlands, 2003) Chap. 4, pp. 67–139.[24] T. Yamada and S. Hirata, J. Chem. Phys , 114112(2015).[25] D. W. Small, E. J. Sundstrom, and M. Head-Gordon, J.Chem. Phys. , 094112 (2015).[26] M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E.Scuseria, M. A. Robb, J. R. Cheeseman, G. Scal-mani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li,M. Caricato, A. V. Marenich, J. Bloino, B. G. Janesko,R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Ortiz,A. F. Izmaylov, J. L. Sonnenberg, D. Williams-Young,F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng,A. Petrone, T. Henderson, D. Ranasinghe, V. G. Za-krzewski, J. Gao, N. Rega, G. Zheng, W. Liang, M. Hada,M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida,T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven,K. Throssell, J. Montgomery, J. E. Peralta, F. Ogliaro,M. J. Bearpark, J. J. Heyd, E. N. Brothers, K. N. Kudin,V. N. Staroverov, T. A. Keith, R. Kobayashi, J. Nor-mand, K. Raghavachari, A. P. Rendell, J. C. Burant, S. S.Iyengar, J. Tomasi, M. Cossi, J. M. Millam, M. Klene,C. Adamo, R. Cammi, J. W. Ochterski, R. L. Martin,K. Morokuma, O. Farkas, J. B. Foresman, and D. J.Fox, “Gaussian16 Revision A.03,” (2016), gaussian Inc.Wallingford CT.[27] M. Sumita and K. Saito, J. Chem. Phys. , 30 (2010).[28] I. J. Palmer, I. N. Ragazos, F. Bernardi, M. Olivucci, and M. A. Robb, J. Am. Chem. Soc. , 673 (1993).[29] Y. Amatatsu, J. Comp. Chem. , 950 (2002).[30] O. Deeb, S. Cogan, and Z. S., J. Chem. Phys. , 251(2006).[31] M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E.Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani,V. Barone, B. Mennucci, G. A. Petersson, H. Nakat-suji, M. Caricato, X. Li, H. P. Hratchian, A. F. Iz-maylov, J. Bloino, G. Zheng, J. L. Sonnenberg, W. Liang,M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa,M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai,T. Vreven, J. A. Montgomery, Jr., J. E. Peralta,F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N.Kudin, V. N. Staroverov, T. Keith, R. Kobayashi, J. Nor-mand, K. Raghavachari, A. Rendell, J. C. Burant, S. S.Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam,M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo,J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev,A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski,R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A.Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, P. V.Parandekar, N. J. Mayhall, A. D. Daniels, O. Farkas,J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J.Fox, “Gaussian developmental version, Revision H.21,”(2010), gaussian Inc. Wallingford CT.[32] A. Kokalj, Comp. Mater. Sci. , 155 (2003).[33] W. Domcke and D. R. Yarkony, Annu. Rev. Phys. Chem. , 325 (2012).[34] M. V. Berry, Proc. R. Soc. A392