Without rigorous symmetry constraints, solutions to approximate electronic structure methods may artificially break symmetry. In the case of the relativistic electronic structure, if time-reversal symmetry is not enforced in calculations of molecules not subject to a magnetic field, it is possible to artificially break Kramers degeneracy in open shell systems. This leads to a description of excited states that may be qualitatively incorrect. Despite this, different electronic structure methods to incorporate correlation and excited states can partially restore Kramers degeneracy from a broken symmetry solution. For single-reference techniques, the inclusion of double and possibly triple excitations in the ground state provides much of the needed correction. Formally, however, this imbalanced treatment of the Kramers-paired spaces is a multi-reference problem, and so methods such as complete-active-space methods perform much better at recovering much of the correct symmetry by state averaging. Using multi-reference configuration interaction, any additional corrections can be obtained as the solution approaches the full configuration interaction limit. A recently proposed “Kramers contamination” value is also used to assess the magnitude of symmetry breaking.

## I. INTRODUCTION

A proper treatment and understanding of symmetry is an important consideration for electronic structure theory. Symmetry forms a powerful tool to characterize the electronic states of chemical species and understand the selection rules that govern transitions between them. Unfortunately, for approximate variational wavefunctions, the imposition of symmetry constraints can only raise the energy of the obtained solution. This means that we can get a “better” lower-energy wavefunction by ignoring these constraints and is known as Löwdin’s “symmetry dilemma.”^{1} In 1930, Kramers published a theorem that for systems with an odd number of electrons, the eigenvalues of a Hamiltonian that only includes electric fields (but not magnetic) are at least doubly degenerate.^{2} Much later, Klein proved that for all non-degenerate states of any *n*-electron system, the expectation value of the magnetic moment is zero as a corollary.^{3} However, it was Wigner who showed that the degeneracies required by Kramers theorem are due to invariance under time-reversal symmetry (TRS).^{4}

By relating the degeneracy of open-shell systems to symmetry, the problematic breaking of Kramers (or time-reversal) symmetry has frequently been likened to that of spin-contamination.^{5} As with spin-symmetry, time-reversal symmetry can be rigorously enforced by additional constraints on the form of the wavefunction or allowed to vary freely, leading to both Kramers-restricted and Kramers-unrestricted formalisms. The arguments surrounding the use or non-use of Kramers-unrestricted methods are thus very similar to those with the unrestricted Hartree–Fock (UHF) and restricted open-shell Hartree–Fock (ROHF) theories. While there are merits to both approaches, it has been found to often be of little consequence on the wavefunctions obtained at correlated levels of theory, such as coupled cluster.^{6} Indeed, methods based on both unrestricted- and restricted-open shell wavefunctions still enjoy considerable popularity today. However, within relativistic quantum chemistry, much of the prior work has focused on using a Kramers-restricted formalism. While it is possible to base *ab initio* correlated relativistic methods on a Kramers-restricted reference, it is often convenient to formulate electronic structure methods assuming an unrestricted, variational framework. This allows certain methods that require a variational ground state (such as response theory) to be applicable and a simpler development for computing molecular properties. It is this case that we now wish to explore and examine the extent to which it is also possible to recover the correct Kramers symmetry approximately in correlated wavefunction methods. In this perspective, we discuss the use of Kramers-unrestricted formalisms, including time-dependent density functional theory (TDDFT), coupled–cluster (CC) theory, and multireference configuration interaction (MRCI).

## II. TIME-REVERSAL SYMMETRY AND KRAMERS THEOREM

The time-reversal symmetry operator $\Theta ^$ is an anti-unitary operator, which has the effect of reversing linear momentum ** p** → −

**as well as angular momentum**

*p***→ −**

*j***, in essence playing time backward. The action of the time-reversal symmetry operator $\Theta ^$ on a fermionic one-electron wavefunction**

*j**ψ*can be written as follows:

Note that *ψ* may be either a spinor or bispinor wavefunction. |*ψ*⟩ and $|\psi \xaf\u2009$ represent a Kramers pair with the same energy. $\Theta ^$ also transforms *t* → −*t* for any operator *O*,

To properly address the time-reversal symmetry, the underlying wavefunction needs to be in the (bi-)spinor formalism. For electrons, a two-component spinor formalism can be used to represent the wavefunction. That is, the molecular orbitals (MOs) are allowed to be a linear combination of both the *α* and *β* spin degrees of freedom,

where the spatial functions ${\varphi k\alpha (r,t)}$ and ${\varphi k\beta (r,t)}$ are expanded in terms of a common set of basis functions {*χ*_{μ}(**r**)},

The spinor formalism gives rise to the two-component electronic structure methods, where the Hamiltonian matrix has a spin-blocked form,

The anti-unitary property of $\Theta ^$ means that we can represent $\Theta ^$ as a product of a unitary operator *Û* and the complex conjugation operator $K^$,

In the case of time-reversal symmetry for electrons, this is represented as $i\sigma yK^$, where *σ*_{y} is the Pauli spin matrix, given by

In the case of the two-component ansatz, it can be seen that the effect of $\Theta ^$ on a given electron is to flip the spin and take complex conjugates,

When the Hamiltonian obeys time reversal symmetry, so $[\u0124,\Theta ^]=0$, this implies that $\u0124=\Theta ^\u0124\Theta ^\u22121$, and we can determine necessary constraints,

Equation (11) shows that the requirement of time-reversal symmetry can be imposed by two constraints such that we only have two independent blocks in the Fock matrix: a same-spin block and a mixed-spin block (i.e., **F**_{αα} = **F**_{ββ} and $F\beta \alpha =\u2212F\alpha \beta *$). Note that since *Ĥ* is Hermitian, the off-diagonal mixed-spin block must also be related by complex conjugation. This structure leads to paired molecular orbitals, or “paired generalized Hartree–Fock (GHF),” which is analogous to the ROHF theory. In the case of relativistic wavefunctions, where spin is no longer a good quantum number, this is also known as a Kramers-restricted formalism.^{7–16}

Before discussing symmetry breaking and restoration, we note that Kramers theorem only applies when the molecule is not subject to magnetic fields. If a molecule is in the presence of magnetic fields, then the true solution need not obey time reversal symmetry as $[\u0124,\Theta ^]\u22600$. On the other hand, if a molecular system is only subject to electric fields, the Hamiltonian can be shown to be time-reversal invariant since the vector potential can be chosen to be zero. We emphasize that this holds true even when relativistic effects such as spin–orbit coupling are included, such as for the Dirac–Coulomb and Dirac–Coulomb–Breit Hamiltonians. To see this, it is a straightforward exercise to check how each of the different operators in the Hamiltonian are transformed under $\Theta ^$. For example, because the spin–orbit operator is of the form **L** · **s** = (**r** × **p**) · **s**, under time reversal, **r** → **r**, **p** → −**p**, and **s** → −**s**, and there is no overall change. For the remainder of our discussion, we assume that the system is not subject to magnetic fields and thus that the Hamiltonian under consideration is time-reversal invariant so that $[\u0124,\Theta ^]=0$.

## III. SYMMETRY BREAKING AND RESTORATION

For approximate electronic structure methods, there might be artificial symmetry breaking due to an unbalanced treatment of the different configurations for open-shell systems. This problem is particularly acute for single-reference methods where the reference is typically chosen to be the lowest energy single Slater determinant. When variational constraints are removed, as is the case in four- and two-component Dirac–Hartree–Fock compared to more common real-valued restricted Hartree–Fock (RHF) or unrestricted Hartree–Fock (UHF), symmetry breaking is likely to occur and is nearly guaranteed for open-shell systems.

One possible remedy is to apply a projection operator to form a solution that is a good eigenfunction of the operator and remove the offending components. This is frequently done to form good eigenfunctions of spin operators such as $S\u0303$^{2} and $S\u0303$_{z}.^{17–19} Analogous projection methods to form eigenfunctions of the *Ĵ*^{2} operator would be a very interesting development, though evaluation of ⟨*J*^{2}⟩ remains a challenging problem. Alternatively, one can use a multi-configurational approach such as “average-of-configuration,”^{7–9,12–16,20,21} as implemented in the DIRAC code.^{22} This method uses a minimal active space and fractional occupancy to build in the effect of optimizing the orbitals for multiple configurations. In this way, the Kramers-unrestricted problem becomes a Kramers-paired, multi-configurational solution.

For open shell systems, molecular orbitals (MOs), *ψ*_{p′}, in a variational solution of Hartree–Fock in spinor basis, such as the generalized Hartree–Fock (GHF) or two-component Dirac–Hartree–Fock methods, generally do not satisfy time-reversal symmetry. That is, for a given $\psi p\u2032$, there is no single MO $\psi \xafp\u2032$ that is the Kramers partner. Instead, the time-reversed spinor $\psi \xafp\u2032=\Theta ^\psi p\u2032$ is a linear combination of several other occupied and virtual orbitals. These expansion coefficients can be easily determined from the overlaps of the time-reversed spinor $\psi \xafp\u2032$ with each of the broken symmetry spinors $\psi q\u2032$,

More generally, for a given basis function space, the set of orbitals satisfying the time-reversal symmetry, *ψ*_{p} and $\psi \xafp$, can be expressed as linear combinations of the broken symmetry spinor MOs (*ψ*_{p′}),

This transformation from broken-symmetry spinor MOs to Kramers-restricted orbitals is unitary and can be equivalently expressed as orbital rotations,

where

where *λ*_{p′q′} is an element of a Hermitian matrix.

For the configuration interaction (CI) method, we can conveniently write the electronic wavefunction $M\u2032$ as a linear combination of configurations (single Slater determinants), Φ_{j′}, constructed with broken-symmetry spinor MOs,

Any excited configurations constructed from spinor MOs with broken symmetry will not have Kramers symmetry either. Similar to orbital rotation, the variation of expansion coefficients *C*_{j′M′} can be expressed as a unitary transformation,

where

Equations (14) and (16) suggest that the two types of rotations are redundant in the full CI limit, and indeed, the full CI result does not depend on the particular choice of basis functions, only the span. In other words, Kramers symmetry can be recovered either in the orbital rotations of Hartree–Fock or in the CI framework. In the full CI limit, $[\u0124,\Theta ^]=0$, meaning that the eigenfunction of the Hamiltonian is also an eigenfunction of the time-reversal symmetry operator. Therefore, minimization of the energy function in the CI framework also recovers the time-reversal symmetry even if the reference spinor orbitals do not have Kramers symmetry. For truncated CI, the configurational space generated from excitation operators may not be large enough to create a complete redundancy of orbital rotation. This means that when any truncated configuration space is used, TRS may not be completely recovered if the reference is symmetry-broken.

The complete active space self-consistent field (CASSCF) procedure is rather unique in that it includes both types of rotations, obtaining both orbital rotations in the SCF portion and CI-type rotations in configuration interaction of the active space orbitals. As a result, the CASSCF procedure should also be able to recover TRS, despite limiting the CI expansion to the active space. Additionally, we note that the state averaged (SA) CASSCF procedure can force a solution where the average energy of a set of degenerate states is equal and thus treat all Kramers pairs equally. The flexibility to recover TRS in two different ways can be even further exploited in the multi-reference configuration interaction (MRCI) method built on top of a CASSCF wavefunction.

We note that the results from the SA-CASSCF procedure and that of the average of configuration (AOC) SCF approach are similar, though not identical. While AOC enforces symmetry via constraints on the *orbitals*, SA-CASSCF recovers symmetry by averaging over *states*. In AOC-SCF, a set of open shell orbitals (*S*) is defined, and the average energy of a set of CI states in this orbital space is recast in terms of orbitals with fractional occupation. The average energy can be written as

where *f*_{S} is the fractional occupation in shell *S*, *M*_{S} and *N*_{S} are the number of orbitals and electrons in shell *S*, respectively, and the two-electron contribution is *Q*^{S} = *f*_{S} ∑_{r∈S}⟨*pr*||*qr*⟩.

This energy is minimized in an SCF procedure and the results in orbitals that have TRS. In the non-relativistic case, this can be used to enforce degeneracy of, for example, open shell *p* orbitals. In the relativistic case, TRS symmetry of the orbitals can be recovered yielding the correct symmetry of *p*_{1/2} and *p*_{3/2} orbitals. Complete open shell CI (COSCI)^{21} using the AOC-SCF orbitals as a reference gives a set of CI states that also obey TRS.

In contrast to AOC-SCF enforcing TRS of the orbitals, SA-CASSCF may use broken-symmetry orbitals to recover TRS of the orbitals and states. Additionally, the AOC-SCF/COSCI approach is applicable to the open-shell orbitals only, while in SA-CASSCF, extension of the active space can build in correlation during the orbital optimization procedure. That is, SA-CASSCF allows for a flexible choice of which orbitals are included and which states are averaged. While averaging over orbitals or states is equivalent in the one-electron case, for multi-electron systems, these choices can lead to different results. In both cases, it is necessary to use explicit multi-reference techniques to describe excited states.

To quickly assess spin-contamination, it is commonplace to calculate ⟨*S*^{2}⟩ and see how it deviates from the ideal values. In analogy with spin-contamination, it has also been recently proposed to calculate “Kramers contamination” by deviations in the expected value of an operator that is the generator of the many-electron time-reversal operator.^{5,23–25} That is, one can evaluate the quantity

where the many-electron time-reversal generator Θ_{+} is given by a sum of the one-electron time-reversal operators,

For true multi-configurational wavefunctions, it has been found that $\u2009\Theta +2\u2009=\u2212k2$, where *k* is the number of open-shell electrons. However, for a single determinant wavefunction, $\u2009\Theta +2\u2009=\u2212k$.

## IV. COMPUTATIONAL RESULTS

For this perspective, we use relativistic two-component time-dependent Hartree-Fock (TDHF), equation-of-motion coupled-cluster theory (EOM-CCSD), CASSCF, and MRCI to examine how these different approaches recover time-reversal symmetry. Specifically, we use the one-electron exact-two-component (X2C)^{26–39} transformation in all implementations of relativistic methods. Our goal is to see how well the different Kramers-unrestricted methods obtain the correct degeneracies of both ground and excited states for several small open-shell systems. The systems used in our consideration here were chosen primarily to be computationally feasible to do larger configuration interaction spaces such as triples and quadruples but also to have reliable experimental data with which to compare. However, we do admit that these results may not be representative of larger molecular systems where covalency and reduced point group symmetry effects play a larger role. Calculations with X2C-EOM-CCSD^{40} were run in Chronus Quantum,^{41} while the X2C-TDHF,^{36} X2C-CASSCF,^{42} and X2C-MRCI^{43} calculations were run in a locally modified development version of the Gaussian software package.^{44} Two-component molecular orbitals are plotted, as described in Ref. 45.

### A. Kramers symmetry in *p*-shell—sodium doublet

The first system we discuss is the neutral sodium atom, which has a doubly degenerate ground state and a well-known doublet transition from 3*s* → 3*p*. Spin–orbit coupling splits the *p*-manifold of the lowest excited state into two sets of sublevels based on their total angular momentum *J*: 3*p*_{1/2} and 3*p*_{3/2} with degeneracies of two and four, respectively, if the Kramers symmetry is preserved.

The orbitals obtained from the SCF procedure using X2C-HF with the 6-31G basis set show symmetry breaking. As shown in the spinor MO diagram in Fig. 1, the two 3*s* spinor orbitals, which should form a Kramers pair, are separated by around 5 eV. We note that this large value is not necessarily indicative of extreme symmetry breaking as the energies of spin-orbitals are greatly influenced by occupation in the Hartree–Fock approximation. As a result of the spin polarization of the unpaired electron in the 3*s* orbital, the 3*p* orbitals, which would ideally have degeneracies of two and four, are found in a degeneracy pattern of three and three. Spin–orbit coupling further splits these orbitals on the order of meV; however, the spin polarization effect dominates. When calculating the Kramers contamination using Eq. (20), a value of −1.000 286 is obtained, rather than the ideal value of −1. Although the deviation from the exact Kramers symmetry seems small, the manifestation on the energies and degeneracies is significant. As the 3*s* electron *should* be formally unpaired, the increased contamination value is due to the lack of ideal pairing of spinors in the ten electrons in the nominal 1*s*^{2}2*s*^{2}2*p*^{6} closed-shell configuration.

It is evident that spinor orbitals from variational X2C-HF calculations for an open-shell system do not satisfy the Kramers symmetry. Discussions in Sec. III suggest that procedures that allow either orbital rotations or expansion in a configuration interaction or coupled-cluster framework could recover the time-reversal symmetry. We start off with calculations in the 6-31G basis using single reference techniques that include orbital transitions between occupied and virtual spinors. Table I lists eigenvalues from solving X2C-CIS, X2C-TDHF, X2C-CISD, and X2C-EOM-CCSD, respectively. The first eigenroots from X2C-TDHF and X2C-EOM-CCSD are zeros, suggesting that the ground state time-reversal symmetry is properly recovered. By constrast, neither X2C-CIS or X2C-CISD fully recovers the degeneracy of the ground state. In fact, in the X2C-CISD calculation, the degeneracy is worse due to the additional correlation included in the reference determinant that is not in the corresponding Kramers-paired determinant. In X2C-TDHF, this “excitation” involves transitions from the occupied 3*s* spinor to the two virtual MOs that compose the time-reversed spinor. The relative contributions of 91.2% and 8.8% are the linear combination coefficients of the broken symmetry MOs and match exactly those determined using Eq. (12). Compared to X2C-HF spinor orbital energies, the time-reversal symmetry in the *J* = 1/2 and *J* = 3/2 manifolds are largely recovered, showing the expected two- and four-fold degeneracy. Although the linear response X2C-TDHF method allows only single excitations/de-excitations, the energy of a Kramers pair is recovered to within a few meV error. Including double excitations in X2C-EOM-CCSD, all Kramers pairs are obtained within a 20 *μ*eV error.

Term . | Experiment^{46}
. | X2C-CIS . | X2C-TDHF . | X2C-CISD . | X2C-EOM-CCSD . |
---|---|---|---|---|---|

^{2}S_{1/2} | 0.0000 | 0.0000 | 0.0000 | 0.0000 | |

0.0018 | 0.0000 | 0.0314 | 0.0000 | ||

^{2}P_{1/2} | 2.1023 | 1.9949 | 1.9956 | 2.0282 | 1.9971 |

1.9954 | 1.9959 | 2.0282 | 1.9971 | ||

^{2}P_{3/2} | 2.1044 | 1.9964 | 1.9972 | 2.0299 | 1.9989 |

1.9974 | 1.9976 | 2.0299 | 1.9989 | ||

1.9978 | 1.9979 | 2.0300 | 1.9989 | ||

1.9981 | 1.9981 | 2.0300 | 1.9989 |

Term . | Experiment^{46}
. | X2C-CIS . | X2C-TDHF . | X2C-CISD . | X2C-EOM-CCSD . |
---|---|---|---|---|---|

^{2}S_{1/2} | 0.0000 | 0.0000 | 0.0000 | 0.0000 | |

0.0018 | 0.0000 | 0.0314 | 0.0000 | ||

^{2}P_{1/2} | 2.1023 | 1.9949 | 1.9956 | 2.0282 | 1.9971 |

1.9954 | 1.9959 | 2.0282 | 1.9971 | ||

^{2}P_{3/2} | 2.1044 | 1.9964 | 1.9972 | 2.0299 | 1.9989 |

1.9974 | 1.9976 | 2.0299 | 1.9989 | ||

1.9978 | 1.9979 | 2.0300 | 1.9989 | ||

1.9981 | 1.9981 | 2.0300 | 1.9989 |

These same trends mirror the two-component GHF non-relativistic analogs in which there is spin contamination rather than Kramers contamination. The values are given in Table II. However, instead of two nearly degenerate sets of two and four states each, there is a single set of six nearly degenerate states. Indeed, these calculations show that just as unrestricted calculations are still commonplace in the non-relativistic regime, the use of Kramers-unrestricted calculations perform similarly to their non-relativistic counterparts.

Term . | CIS . | TDHF . | CISD . | EOM-CCSD . |
---|---|---|---|---|

^{2}S | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

0.0018 | 0.0000 | 0.0318 | 0.0000 | |

^{2}P | 1.9919 | 1.9911 | 2.0250 | 1.9926 |

1.9919 | 1.9911 | 2.0250 | 1.9926 | |

1.9919 | 1.9911 | 2.0250 | 1.9926 | |

1.9928 | 1.9928 | 2.0252 | 1.9926 | |

1.9928 | 1.9928 | 2.0252 | 1.9926 | |

1.9928 | 1.9928 | 2.0252 | 1.9926 |

Term . | CIS . | TDHF . | CISD . | EOM-CCSD . |
---|---|---|---|---|

^{2}S | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

0.0018 | 0.0000 | 0.0318 | 0.0000 | |

^{2}P | 1.9919 | 1.9911 | 2.0250 | 1.9926 |

1.9919 | 1.9911 | 2.0250 | 1.9926 | |

1.9919 | 1.9911 | 2.0250 | 1.9926 | |

1.9928 | 1.9928 | 2.0252 | 1.9926 | |

1.9928 | 1.9928 | 2.0252 | 1.9926 | |

1.9928 | 1.9928 | 2.0252 | 1.9926 |

As discussed previously, another way to recover the appropriate Kramers degeneracy is through the use of multi-reference methods such as CASSCF where the orbitals are reoptimized and the wavefunction takes on the multi-determinantal character of the true ground state. In order to recover the Kramers symmetry, state-averaged CASSCF treatment must be used as optimization for only a single state will favor the symmetry-broken solution.^{42} This allows for a balanced treatment of all Kramers pairs such that each orbital manifold feels the same mean-field potential. For the following calculations, we include all the relevant states to the terms. That is, for a minimal active space of SA-CASSCF(1,2), we state average over both microstates in the ^{2}S_{1/2} term, and in the larger SA-CASSCF(1,8) calculation, we average over all eight states in the relevant ^{2}S_{1/2}, ^{2}P_{1/2}, and ^{2}P_{3/2} terms. The Kramers contamination for the state-averaged CASSCF wavefunction is reduced by an order of magnitude from the HF solution to −1.000 03 for both SA-CASSCF(1,2) and SA-CASSCF(1,8).

As noticed from X2C-TDHF and X2C-EOM-CCSD results, single- and double-excitations contribute the most to the recovery of Kramers symmetry. One could use additional excitations from the CAS reference to recover the Kramers symmetry, giving rise to the multi-reference configuration interaction (MRCI) method.^{43} In MRCI, additional electron correlation is included naturally accompanied with the additional recovery of correct degeneracy. As seen in Fig. 2, the balanced treatment of the smaller SA-CASSCF(1,2) performs very similarly to the larger SA-CASSCF(1,8), indicating that only the minimal active space for the ground state is necessary to recover symmetry. The microstates in the ^{2}S, ^{2}P_{1/2}, and ^{2}P_{3/2} terms computed using MRCI on both the state-averaged X2C-CASSCF(1,2) and X2C-CASSCF(1,8) references are degenerate to within the convergence tolerance (10^{−6} hartree). In Fig. 2, the states are grouped together according to the assigned terms as in Table I, ^{2}S, ^{2}P_{1/2}, and ^{2}P_{3/2}. Since formally the states within these terms should be degenerate, the difference between the highest and lowest energies in the group is used as a measure of the symmetry breaking. Comparing the CI results using HF and SA-CASSF references, it is clear that using the less contaminated reference (SA-CASSCF) leads to better degeneracy recovery, with the energy range of states in the same term decreased by several orders of magnitude. In particular, it can be seen that at the quadruple level, there is little effective difference between the choice of references as the error is primarily due to the numerical precision in solving for the eigenvectors and eigenvalues when iteratively diagonalizing the CI matrix. This is because the correlation space now includes the Kramers-paired space, so the resulting wavefunctions will have the expected symmetry.

As discussed in Sec. III, both orbital rotation and CI expansion can recover the Kramers symmetry. An interesting aspect to investigate is which approach is more effective. Since the SA-CASSCF can be converged to a given accuracy, a loosely converged solution can be thought of as only partially optimizing the orbitals to recover the correct symmetry. A natural question then is if this has any impact on how quickly symmetry is recovered in the MRCI framework. That is, does the amount of contamination in the reference affect how well the degeneracy is recovered? Surprisingly, for Na, the rate at which degeneracy is recovered for the SA-CASSCF(1,2) and SA-CASSCF(1,8) systems is insensitive to how well the reference was converged. As seen in Fig. 3, this difference in initial contamination remains even at the triples level. Only in the calculations with quadruple excitations where there is almost no remaining symmetry breaking do the calculations provide indistinguishable results. This suggests that in the MRCI framework, it is more efficient to recover symmetry through orbital rotation in the reference than through the CI excitations, which require high order that is often not affordable.

### B. Kramers symmetry in *d*-shell—Sc^{2+}

To examine how the recovery of Kramers degeneracy might be affected by more pronounced relativistic effects, we continue down the periodic table to Sc^{2+} ion, which has a ground state electronic configuration of 3*p*^{6}3*d*. The unpaired electron now has the entire 3*d* manifold as well as the low-lying 4*s*, creating a much more complicated multireference problem. The MO diagram for the X2C-HF reference is given in Fig. 4 and shows the 4*s* orbitals in-between the occupied 3*d* orbital and the higher lying unoccupied 3*d* orbitals. When calculating the Kramers contamination using Eq. (20), a value of −1.001 21 is obtained, rather than the ideal value of −1. The difference is an order of magnitude worse than for Na.

The results for the single-reference X2C-TDHF and multireference X2C-SA-CASSCF are given in Table III. The active space for the X2C-SA-CASSCF(1,12) system included both the 4*s* and 3*d* orbitals, and state-averaging was performed over the 12 relevant states (^{2}D_{3/2}, ^{2}D_{5/2}, and ^{2}S_{1/2}). While the effect of symmetry breaking is relatively small for the 3*d* → 4*s* transition for the ^{2}S_{1/2} term, the fine structure within the *d*-manifold is unable to be meaningfully captured using the single reference X2C-TDHF. This is because the spin–orbit splitting and Kramers symmetry breaking are of similar magnitude. Only one “excitation” has a zero eigenvalue using X2C-TDHF corresponding to the Kramers paired orbital of the occupied 3*d*. The other two excitations in the ^{2}D_{3/2} manifold involve transitions to 3*d* orbitals from a different Kramers pair, and so these orbitals are not able to fully relax and recover the time-reversal symmetry. As expected, the X2C-SA-CASSCF is able to recover the correct degeneracies to within the convergence tolerance.

Term . | Degeneracy . | X2C-TDHF . | X2C-SA-CASSCF . | Experiment^{47,48}
. |
---|---|---|---|---|

^{2}D_{3/2} | 4 | 0.0000 | 0.0000 | 0.0 |

0.0000 | 0.0000 | |||

0.0315 | 0.0000 | |||

0.0365 | 0.0000 | |||

^{2}D_{5/2} | 6 | 0.0401 | 0.0283 | 0.024 504 |

0.0422 | 0.0283 | |||

0.0468 | 0.0283 | |||

0.0481 | 0.0283 | |||

0.0612 | 0.0283 | |||

0.0758 | 0.0283 | |||

^{2}S_{1/2} | 2 | 3.2629 | 3.1842 | 3.166 472 |

3.2689 | 3.1842 |

Term . | Degeneracy . | X2C-TDHF . | X2C-SA-CASSCF . | Experiment^{47,48}
. |
---|---|---|---|---|

^{2}D_{3/2} | 4 | 0.0000 | 0.0000 | 0.0 |

0.0000 | 0.0000 | |||

0.0315 | 0.0000 | |||

0.0365 | 0.0000 | |||

^{2}D_{5/2} | 6 | 0.0401 | 0.0283 | 0.024 504 |

0.0422 | 0.0283 | |||

0.0468 | 0.0283 | |||

0.0481 | 0.0283 | |||

0.0612 | 0.0283 | |||

0.0758 | 0.0283 | |||

^{2}S_{1/2} | 2 | 3.2629 | 3.1842 | 3.166 472 |

3.2689 | 3.1842 |

When using X2C-MRCI, it is again evident that the remaining amount of symmetry breaking in the reference is retained for the first several excitation levels. In Fig. 5, the X2C-MRCI results are shown with different SA-CASSCF convergence tolerances. All calculations used a frozen core of the 1*s*, 2*s*, and 2*p* levels. In contrast to the previous results with Na, however, it is also apparent that the restoration of symmetry is not uniform, particularly, for the states in the ^{2}D_{3/2} and ^{2}D_{5/2} manifolds. These converge to similar solutions when triples are included. The seeming inability of the calculations to recover Kramers degeneracy to a greater degree is an artifact of using a frozen core, so not all possible excited determinants are included in the correlation space. That is, the neglect of determinants that are in the Kramers-paired space of the frozen core will lead to some symmetry breaking that cannot be recovered. This problem appears to be less severe for the ^{2}S_{1/2} state, which has the nominal configuration of 3*p*^{6}4*s*. Since neither of the 4*s* orbitals are included to a significant degree in the ground state reference, there is little energetic benefit to cause symmetry breaking in these orbitals. The results shown here demonstrate that comparatively small correlation spaces are necessary to recover most symmetry. This intuitively makes sense because Kramers pairs that are either always both occupied or both unoccupied do not cause significant symmetry breaking as there is little energetic benefit to be gained.

### C. Kramers symmetry in *f*-shell—uranium (V) ion

The uranium (V) ion is another interesting system since the valence shell is dominated by strong spin–orbit coupling and contains only a single outermost electron in the 5*f* shell. Unlike the previous examples where spin is still approximately a good quantum number, for U(V), it is imperative to consider the degeneracy in the *J* manifold. The Kramers contamination value is calculated to be −1.009 12 rather than the ideal value of −1.

While X2C-TDHF was shown in the previous examples to be able to partially recover broken Kramers symmetry of state with low angular momentum, its ability in the high angular momentum case is more limited despite having a similar Kramers contamination value. As shown in Table IV, the energy range of states that belong to the same term is ∼0.3 eV for both ^{2}F_{5/2} and ^{2}F_{7/2} terms. However, the deviation is less than the spin–orbit coupling, which splits the ^{2}F_{5/2} and ^{2}F_{7/2} terms by nearly an eV. Therefore, even though the symmetry breaking in U^{5+} might be more severe than in Sc^{2+} due to the increased degeneracy of the ground state, the overall energetic groupings of the microstates are more clear since spin–orbit coupling plays a dominant role. Interestingly, the symmetry breaking is decreased when using X2C-TDDFT compared to X2C-TDHF. While a full investigation of how Kramers symmetry breaking is manifested in DFT calculations has not been conducted, our initial results suggest that this could be due to the exchange–correlation potential acting to smear out the density over the multiple configurations within the 5*f* manifold. Table IV shows that the number of microstates in each term is correctly predicted by X2C-TDHF and X2C-TDDFT, as expected, since all are generated by single excitations from the reference. In contrast, X2C-SA-CASSCF(1,14) is able to completely recover the broken Kramers symmetry through orbital rotation and state-averaging over all 14 microstates.

Term . | Degeneracy . | X2C-TDHF . | X2C-TDDFT . | X2C-SA-CASSCF . | Experiment^{51}
. |
---|---|---|---|---|---|

^{2}F_{5/2} | 6 | 0.000 00 | 0.000 00 | 0.000 00 | 0.0 |

0.000 00 | 0.000 08 | 0.000 00 | |||

0.168 61 | 0.000 18 | 0.000 00 | |||

0.232 76 | 0.000 82 | 0.000 00 | |||

0.285 33 | 0.153 07 | 0.000 00 | |||

0.326 59 | 0.395 50 | 0.000 00 | |||

^{2}F_{7/2} | 8 | 0.942 00 | 0.942 34 | 0.971 94 | 0.943 35 |

1.015 22 | 0.955 43 | 0.971 94 | |||

1.066 19 | 1.003 24 | 0.971 94 | |||

1.183 74 | 1.005 19 | 0.971 94 | |||

1.188 78 | 1.012 23 | 0.971 94 | |||

1.261 54 | 1.029 83 | 0.971 94 | |||

1.271 05 | 1.072 09 | 0.971 94 | |||

1.292 11 | 1.097 73 | 0.971 94 |

Term . | Degeneracy . | X2C-TDHF . | X2C-TDDFT . | X2C-SA-CASSCF . | Experiment^{51}
. |
---|---|---|---|---|---|

^{2}F_{5/2} | 6 | 0.000 00 | 0.000 00 | 0.000 00 | 0.0 |

0.000 00 | 0.000 08 | 0.000 00 | |||

0.168 61 | 0.000 18 | 0.000 00 | |||

0.232 76 | 0.000 82 | 0.000 00 | |||

0.285 33 | 0.153 07 | 0.000 00 | |||

0.326 59 | 0.395 50 | 0.000 00 | |||

^{2}F_{7/2} | 8 | 0.942 00 | 0.942 34 | 0.971 94 | 0.943 35 |

1.015 22 | 0.955 43 | 0.971 94 | |||

1.066 19 | 1.003 24 | 0.971 94 | |||

1.183 74 | 1.005 19 | 0.971 94 | |||

1.188 78 | 1.012 23 | 0.971 94 | |||

1.261 54 | 1.029 83 | 0.971 94 | |||

1.271 05 | 1.072 09 | 0.971 94 | |||

1.292 11 | 1.097 73 | 0.971 94 |

## V. CONCLUSION

The restoration of correct symmetry is important for spectroscopic or other physical interpretation of electronic states. In relativistic calculations, time-reversal, or Kramers symmetry, gives rise to well-known degeneracy patterns. For calculations of open-shell molecules, Hartree–Fock or other reference wavefunctions that break Kramers symmetry can have this symmetry partially restored through the inclusion of excitation operators, or through reoptimization of orbitals as in a state-averaged CASSCF.

X2C-TDHF has been shown to be able to partially recover the broken Kramers symmetry. While it works well for low-angular momentum orbitals such as the *s*- and *p*-shells, its effectiveness is limited for the *d*- and *f*-shells, where a higher degree of degeneracy limits the description within a single-determinant framework. Nevertheless, the use of DFT seems to have a mediating effect on symmetry breaking, although much further study in this area would be required to see if this is only true for some functionals. Including higher excitation operators in coupled-cluster enables a better symmetry recovery. In contrast, multi-reference techniques, such as X2C-CASSCF and X2C-MRCI, are much better able to recover the Kramers symmetry through orbital rotation and state-averaging. This is the “average-of-configuration” approach. The MRCI framework allows one to shift the cost of restoring symmetry between the orbital rotation and including additional excitations in an aid to performance and can allow for the use of smaller active spaces in a Kramers-unrestricted framework.

Formalisms that can utilize broken symmetry solutions effectively allows for flexibility in method development for molecular response properties and extension to regimes where Kramers symmetry need not hold. While considerable work has been done comparing the unrestricted- and restricted-open shell frameworks in non-relativistic calculations, much less work has been done comparing Kramers-unrestricted and Kramers-restricted formalisms in relativistic calculations. Most calculations utilize either a Kramers-restricted formalism or a pseudo-restricted formalism such as average-of-configuration. The recently proposed “Kramers contamination” value does indeed capture the contamination in the broken symmetry reference, though it remains unclear how much deviation in the number of unpaired electrons would be considered problematic, since the size of the open-shell is also crucial. We believe that future studies exploring how methods using a Kramers-symmetry broken reference such as relativistic coupled-cluster theory will be important to new advances in relativistic calculations and magnetic properties. Understanding the methodological performance in recovering Kramers symmetry will aid in differentiating artificial from physical symmetry breaking.

## AUTHORS’ CONTRIBUTIONS

J.M.K. and A.J.J. contributed equally to this work.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## ACKNOWLEDGMENTS

The development of relativistic electronic density functional theory is funded by the U.S. Department of Energy (Grant No. DE-SC0006863). The development of relativistic multi-reference method is supported by the U.S. Department of Energy (Grant No. DE-SC0021100). The development of the computational spectroscopic method is supported by the National Science Foundation (Grant No. CHE-1856210). The development of the Chronus Quantum open source software package is supported by the National Science Foundation (Grant No. OAC-1663636). Computations were facilitated through the use of advanced computational, storage, and networking infrastructure provided by the Hyak supercomputer system at the University of Washington, funded by the Student Technology Fee.

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