On the energy invariance of open-shell perturbation theory with respect to unitary transformations of molecular orbitals

A number of recently proposed single-reference open-shell perturbation theories based on a spin-restricted open-shell Hartree-Fock reference function are examined, with an emphasis on a consistent formalism within which the theories may be compared. In particular, the effect of unitary transformations among the molecular orbitals on the energy is discussed. Of the seven different perturbation theories examined here, the restricted Mo ” ller–Plesset theory, open-shell perturbation theory method 1, the method of Hubac ˇ nd Čársky, Z-averaged perturbation theory, and invariant open-shell perturbation theory methods are found to be invariant to all types of rotations for which the reference wave function is unaffected, though all are invariant to transformations of a more limited nature. Explicit equations for the generalized invariant forms of each perturbation theory are presented, in order to provide working equations for extension of the theories to local correlation schemes or coupled-cluster perturbational corrections, among others. © 1996 American Institute of Physics.@S0021-9606 ~96!00327-3#


I. INTRODUCTION
For both practical and aesthetic reasons, a desirable characteristic of the wave function in most methods in electronic structure theory is that of invariance of the energy to certain types of unitary transformations among the reference molecular orbitals ͑MOs͒.For example, it has sometimes proved more computationally efficient to use a variety of definitions of MOs in order to improve the convergence of the self-consistent field ͑SCF͒ procedure [1][2][3] ͑particularly for open-shell systems͒, though at convergence each set of MOs gives the same energy.Also, localized orbitals ͑as opposed to SCF canonical orbitals͒ have been used [4][5][6] to reduce the magnitude of components contributing to certain types of correlated wave functions, such as configuration interaction ͑CI͒, coupled electron-pair approximation ͑CEPA͒, 7 and second-order Mo "ller-Plesset perturbation theory ͑MP2͒ 8 wave functions, among others.Additionally, it is well known that energy invariance can be used to simplify the construction of analytic energy gradients. 9oothaan's original papers concerning the solution of the Hartree-Fock equations for closed-shell 10 and high-spin open-shell 1 systems note the invariance characteristics of the single-determinant wave function.Specifically, a unitary transformation among the component molecular spin orbitals does not alter the wave function, apart from a trivial phase factor, and, hence, the energy and all properties remain unaffected.In terms of spin-restricted Hartree-Fock wave functions, similar properties hold for transformations among the spatial orbitals alone.For closed-shell wave functions, any rotations among the doubly occupied orbitals or among the virtual orbitals leave the energy unchanged.For high-spin open-shell wave functions, spatial orbital rotations are allowed, provided they are restricted to the doubly occupied space, the virtual space, or the singly occupied space.Rotations in the singly occupied space are allowed due to the fact that all occupied spin orbitals are associated with alpha spin functions, while the unoccupied spin orbitals are associated with beta spin functions, by convention.͑This statement also holds, of course, for the so-called symmetric spin orbitals basis, 11 in which different spin functions are used for the open shells.This will be discussed later in this work.͒These properties of the SCF wave function have been pointed out by Bobrowicz and Goddard. 12orrelated wave functions constructed as a sum of determinants produced through substitution of occupied molecular orbitals from the SCF ͑reference͒ wave function by unoccupied orbitals may also, in general, be shown to have similar invariance properties. 9This includes coupled-cluster ͑CC͒ and configuration interaction ͑CI͒ wave functions, though it has been recently pointed out 13 that the use of the so-called first-order interacting space 14 in the construction of CI wave functions for high-spin open-shell systems is necessary in order to maintain the general invariance of the energy.On the other hand, coupled-pair functional ͑CPF͒ wave functions are not invariant with respect to such orbital rotations. 15In general, wave functions constructed via manybody perturbation theory ͑MBPT͒ also exhibit these invariance properties, although it is sometimes necessary to exercise some care in identifying an appropriate partitioning of the electronic Hamiltonian.Such partitioning requires ͑implicitly or explicitly͒ some definition of canonical molecular orbitals. 16n the last fifteen years, and especially since 1991, a number of methods for the construction of high-spin openshell perturbation theory wave functions based on spin-a͒ Electronic mail: crawdad@zopyros.ccqc.uga.edu8][19][20][21][22][23] These include: ͑1͒ the method of Hubac ˇand C ˇa ´rsky 17 ͑here referred to as HCPT͒, ͑2͒ restricted open-shell Mo "ller-Plesset theory ͑ROMP͒, 18 ͑3͒ restricted Mo "ller-Plesset theory ͑RMP͒ 19 or ͑ROHF-MBPT͒ 20 ͑independently developed but identical methods͒, ͑4͒ open-shell perturbation theory method 1 ͑OPT1͒, 21 ͑5͒ open-shell perturbation theory method 2 ͑OPT2͒, 21 ͑6͒ Z-averaged perturbation theory ͑ZAPT͒, 22 and ͑7͒ invariant open-shell perturbation theory ͑IOPT͒. 23These methods demonstrate varying convergence properties, and careful analyses have been presented to account for the differences among them. 24,25However, the general invariance properties of these approaches have not yet been examined in detail. 26n this paper we will explicitly examine the effect of unitary transformations among the SCF reference MOs on the energy given by each of the recently presented singlereference open-shell perturbation theories.In section II, we will present a discussion of the invariance properties of closed-shell MP2 energies with respect to such rotations.We will then provide generalized perturbation theory equations, which do not require the zeroth-order Hamiltonian to be diagonal in the N-electron expansion basis.In section III, the partitioning of the Hamiltonian will be given for each method, and iterative equations for the first-order wave function and expressions for the second-order energy will be presented.From these equations, the allowed rotations of the reference MOs may be ascertained.The generalized invariant forms of the zeroth-order Hamiltonian and perturbation presented here may be used as working equations for, for example, the construction of local correlation schemes, such as those developed previously for Mo "ller-Plesset wave functions, 5,6 or for new perturbational corrections to coupledcluster wave functions, such as the well-known (T) correction. 27,28

II. GENERAL THEORY
Throughout this discussion, we will use a number of notational conventions.Spin orbitals will be indicated by lowercase letters and spatial orbitals by uppercase letters.Orbital indices p, q, r, and s will refer to general spin orbitals, while i, j, and k ͑a, b, and c͒ will refer to spin orbitals occupied ͑unoccupied͒ in the reference wave function ͑with no distinction between the doubly occupied and singly occupied spaces͒.The indices l, m, and n will refer to spin orbitals in the doubly occupied space, d, e, and f to spin orbitals in the ''doubly unoccupied'' space, and t, u, and v to spin orbitals in the singly occupied space.The uppercase versions of all of the above will apply to spatial orbitals.
In standard closed-shell second-order Mo "ller-Plesset perturbation theory ͑MP2͒, 8 the zeroth-order Hamiltonian, H ˆ͑0͒ , is taken to be the spin orbital Fock operator, which is diagonal in the basis of molecular spin orbitals corresponding to the SCF canonical set.Hence, in second quantization

͑2͒
where ⑀ p is the pth eigenvalue of the spin orbital Fock matrix, f pp .Throughout this discussion, we will make use of normal-ordered strings of annihilation and creation operators, indicated by ͕͖ as in Eq. ͑2͒. 29A diagonal zeroth-order Hamiltonian allows the use of the standard Rayleigh-Schro ¨dinger perturbation theory ͑RSPT͒ expressions, which expand the nth-order wave function as a linear combination of the zeroth-order eigenstates, i.e. the set of determinants constructed from the SCF canonical spin orbitals.This leads to the well-known expression for the second-order MP2 energy where ͗i j͉͉ab͘ is an antisymmetrized two-electron integral in Dirac's notation.Any non-trivial unitary transformation among the occupied orbitals or the virtual orbitals ͑including those that mix only the alpha spin orbitals while leaving the beta spin orbitals alone, thereby destroying any spin restriction which may have been imposed on the original orbitals͒ will result in different values for the ⑀ p in the denominator of Eq. ͑3͒.The zeroth-order Hamiltonian ͑i.e., the Fock matrix͒ is no longer diagonal in the expansion basis, and thus the standard RSPT expressions do not apply.This does not imply that the MP2 energy is not invariant to such rotations; only that Eq. ͑3͒ is valid only for canonical SCF orbitals.In order to construct a set of equations which do not depend on the orbitals, all components of the zeroth-order Hamiltonian must be included in the derivation of the nth-order wave function.
The general zeroth-and nth-order ''Schro ¨dinger equations'' resulting from a many-body perturbation expansion of the wave function and energy may be written respectively, where V ˆis the perturbation and ͉⌿ 0 (n) ͘ is the nth-order wave function.Expansion of ͉⌿ 0 (n) ͘ in the complete set of substituted determinants, ͉⌿ ͑0͒ ͘, and left projection of Eq. ͑5͒ by ͗⌿ 0 ͑0͒ ͉ and ͗⌿ ͑0͒ ͉ gives the nth-order energy and wave function equations respectively, where the a (n) are the nth-order expansion coefficients.It should be noted that the nth-order wave function must be determined from Eq. ͑7͒ via an iterative procedure if off-diagonal elements of H ˆ͑0͒ are present.
These generalized perturbation theory equations will be used in the next section as a framework for analyzing the open-shell formalisms mentioned in section I. Specifically, we examine the partitioning of the Hamiltonian and the iterative expressions for the first-order wave functions in order to determine the invariance properties associated with each method.

III. INVARIANCE OF OPEN-SHELL PERTURBATION THEORIES
As discussed in section II, the generalized perturbation theory of Eqs.͑6͒ and ͑7͒ may be used in order to correctly determine the invariance properties of the perturbed wave functions given a particular partitioning of the Hamiltonian,

͑8͒
The subscript N here refers to the normal-ordered form of this second-quantized operator.For many perturbation theories, the MBPT expansion is carried out by first defining H ˆ͑0͒ as a sum of certain diagonal blocks of some one-electron operator 30 which provides a convenient definition of molecular orbitals,

͑9͒
where O ˆis some one-electron operator and p and q ͑r and s͒ span a subspace of the total orbital space that does not contain r and s ͑p and q͒, i.e., the orbital subspaces are disjoint.For example, p and q might be within the doubly occupied space and r and s might be within the singly occupied space.The partitioning is defined, then, by subtracting the diagonal blocks of the operator out of the Hamiltonian, The fundamental concept here is that a perturbation theory based on this partitioning will be invariant to a rotation of the orbitals within a certain subspace if that rotation leaves the partitioning unchanged.Hence, examination of the partitioned Hamiltonian alone will be sufficient to determine the orbital invariance properties of the method, provided the partitioning is written in a sufficiently general form.Therefore, if the rotation

͑12͒
since O ˆis assumed to be Hermitian.Because the transformation is unitary,

͑13͒
Hence, the spectrum and trace of the zeroth-order Hamiltonian remain unchanged so long as the rotation occurs only within the subspace for which the operator has been defined.Similar arguments may be used for the components of V ˆ.If, on the other hand, this rotation mixes orbitals in separate subspaces, e.g., the rotation where p and r lie in different subspaces, such a rotation will move components from the perturbation into the zeroth-order Hamiltonian and vice versa.Explicitly, the above rotation affects the first term on the right hand side of Eq. ͑9͒ as Hence, components of the perturbation appear in the zerothorder Hamiltonian, thereby altering the partitioning, and the spectrum and trace of the operator.Therefore, the perturbed energies will not be invariant to rotations of this type.These general concepts will be useful in the following sections as we examine each of the open-shell perturbation theories in detail.The order in which each method will be presented has been chosen strictly for pedagogical reasons.

A. Restricted Mo " ller-Plesset theory (RMP)
RMP theory 19,20 is perhaps the most straightforward of the open-shell theories investigated here.The zeroth-order Hamiltonian is defined to be the occupied/occupied and virtual/virtual blocks of the spin orbital Fock operator

͑16͒
According to this definition, the perturbation consists of the remaining terms in the Hamiltonian ͓Eq.͑8͔͒

͑17͒
Examination of Eqs.͑16͒ and ͑11͒-͑13͒ shows that a unitary transformation of the occupied spin orbitals among themselves or of the unoccupied spin orbitals among themselves will change neither f ˆoo nor f ˆvv .Similar statements may be made for V ˆ.Thus, such rotations leave the energy unchanged.On the other hand, rotations which mix occupied and virtual spaces shift components from V ˆinto H ˆ͑0͒ and vice versa, as is clear from Eqs. ͑14͒ to ͑15͒.Thus, the partitioning changes, and subsequently, the calculated energy changes.Rotations of this type, however, do not leave the reference wave function itself unchanged.Therefore, RMP theory is invariant to all rotations allowed for the ROHF reference wave function.
In the usual application of this method, a diagonal H ˆ͑0͒ is obtained by diagonalizing the spin orbital Fock matrix in the occupied and virtual subspaces, separately.The resulting orbitals are referred to as ''semi-canonical.''If such orbitals are not used, Eqs.͑6͒ and ͑7͒ must be used to construct iterative expressions for the nth-order energy and wave function.For example, the first-order RMP wave function is constructed iteratively from single and double substitutions via the two identities where a i a (1) and a i j ab (1) are coefficients of the first-order wave function.The second-order energy may then be determined from ͗i j͉͉ab͘a i j ab͑1͒ .͑20͒ These equations are the same as those presented by Lauderdale et al., 31 who derived them from the coupled-cluster singles and doubles ͑CCSD͒ equations.Lauderdale et al. also make note of the invariance properties of this method.Diagonalization of H ˆ͑0͒ results in a shift of the spatial orbitals, such that different orbitals are associated with different spins ͑DODS͒.This orbital set more closely resembles a spin-unrestricted Hartree-Fock wave function ͑UHF͒ in structure, though the SCF energy is unchanged.Additionally, H ˆ͑0͒ is spin dependent, resulting in spin contaminated perturbed wave functions.However, it has been shown 19,25 that the perturbed energies are spin projected, eliminating direct but not indirect spin contamination. 32 Restricted open-shell Mo " ller-Plesset theory (ROMP) ROMP 18 theory is closely related to RMP 19,20 theory.The zeroth-order Hamiltonian here is defined as the doubly occupied/doubly occupied, singly occupied/singly occupied, and virtual/virtual blocks of the spin orbital Fock operator

͑21͒
The perturbation therefore contains the remaining terms of the Hamiltonian In this case, more limited rotations leave the perturbed wave functions unaffected.A unitary transformation which mixes the spin orbitals of the doubly occupied, singly occupied, and virtual spaces independently will not change the partitioning.However, unitary transformations which mix spin orbitals in the doubly occupied space together with spin orbitals in the singly occupied space ͑a rotation of the ␣ spin orbitals, which does not change the reference wave function͒ will alter the partitioning of the Hamiltonian via the coupling component f ˆds .This result follows directly from the analysis given at the beginning of section III.Therefore, ROMP theory is not invariant to general rotations among the occupied spin orbitals.This lack of invariance will not generally affect the calculated perturbed energies, since the ROHF orbitals are constructed with spin-restriction imposed prior to the limited diagonalization of the spin-orbital Fock matrix.However, this can affect the construction of a local correlation scheme for ROMP theory, for example, if the local orbital definition used were based on a spin-dependent ͑DODS͒ representation.Implementation of ROMP theory usually requires first that H ˆ͑0͒ be diagonalized in the three subspaces separately.This diagonalization results in DODS, just as in RMP theory.Additionally, because the H ˆ͑0͒ is spin-dependent, the perturbed wave functions are not eigenfunctions of S ˆ2, 24,25 though the perturbed energies are spin projected. 32f H ˆ͑0͒ is not diagonalized, then iterative expressions based on Eqs.͑6͒ and ͑7͒ must be solved.For ROMP, the equations for the single and double substitution contributions to the first-order wave function are

͑24͒
The notation, px indicates that the term will not contribute unless the index p corresponds to an orbital in the subspace of orbitals x, where x represents the doubly occupied (d), singly occupied (s), or virtual (v) subspaces.The secondorder energy is then calculated from Note first that no single substitutions involving singly occupied spin orbitals contribute to the second-order energy for ROMP.As a result, the first-order single substitutions in Eq. ͑23͒ look very similar to those of RMP theory, though the summations in ROMP theory involve only doubly occupied and virtual orbitals.Note that the subspace restrictions indicated in Eq. ͑24͒ always eliminate two of the summations, depending on the spaces of i, j, a, and b.As a result, the first-order double substitution equations are also similar to those of RMP theory.

C. Open-shell perturbation theory method 1 (OPT1)
In both RMP and ROMP theory, diagonalization of the spin orbital Fock operator fragments in restricted orbital sub-spaces results in a DODS spin orbital set.OPT1 was introduced as an open-shell perturbation theory that maintains spin restriction on the spatial orbitals. 21In OPT1, the zerothorder Hamiltonian is constructed based on an averaged Fock operator, which is defined in terms of spatial orbitals to be F ˆav ϭh ˆϩ2J ˆcϪK ˆcϩJ ˆoϪ 1 2 K ˆo, ͑26͒ where h ˆis the usual one-electron Hamiltonian, J ˆc and K ˆc are the Coulomb and exchange operators, respectively, including only the closed-shell orbitals, and J ˆo and K ˆo are these operators including only the singly occupied orbitals.Spin orbital forms of F ˆav allow direct comparison to the original spin orbital Fock operator and subsequent partitioning of the Hamiltonian.
In OPT1, the zeroth-order Hamiltonian is then constructed by writing F ˆav in the spin orbital representation in the doubly occupied, singly occupied, and virtual subspaces,

͑27͒
The perturbation then consists of the remaining terms from the Hamiltonian Based on the arguments given in section III earlier, it is clear that this perturbation theory is invariant to rotations which mix the doubly occupied orbitals, the singly occupied orbitals, or the virtual orbitals independently, even if those rotations fail to maintain spin restriction.However, because the zeroth-order Hamiltonian is independent of spin, the nthorder wave function and energy may be written in terms of spatial orbitals alone.Indeed, one of the goals of OPT1 is to construct perturbed wave functions which are eigenfunctions of spin.Hence, spin orbital rotations are not an issue for OPT1 from a practical perspective, since the spin free implementation prevents loss of spin restriction.Therefore, OPT1 is invariant to all spatial orbital rotations which are allowed for the ROHF reference wave function.
Implementation of OPT1 usually requires diagonalization of F ˆav in the three standard subspaces to obtain orbital energies.These energies in the singly occupied space have been described as averages of electron affinities and ionization energies. 21If this diagonalization is not performed, iterative expressions based on Eq. ͑7͒ for the nth-order wave function must be solved to obtain the correct energy.For OPT1, the equations for the single and double substitution contributions to the first-order wave function are respectively.The second-order energy is then calculated from ͗i jʈab͘a i j ab͑1͒ .͑31͒ Equations ͑29͒-͑31͒ show strong similarities to the analogous equations for ROMP ͑Eqs.͑23͒-͑25͒͒.The second-order energy expression is in fact identical for the two approaches, though the first-order coefficients are defined differently.This is reasonable since both methods partition the orbital space similarly.The primary difference lies in the use of F ˆav in OPT1.

D. Perturbation theory of Hubac ˇand C ˇa ´rsky (HCPT)
HCPT was the first of the open-shell perturbation theories based on a spin restricted reference wave function, 17 and is closely related to OPT1 in that orbitals are defined in a way as to maintain this spin restriction.However, HCPT uses different operators from those of OPT1 to define the doubly occupied and virtual orbitals, namely and where F ˆav and K ˆo are the same as defined earlier in section III C. The F ˆav operator is used to define the singly occupied orbitals, as in OPT1.The orbitals constructed from these operators are the same as those originally proposed by Roothaan for high-spin open-shell SCF calculations. 1pin orbital forms of the three operators allow comparison to the original spin orbital Fock matrix.Thus, the zerothorder Hamiltonian is defined as

͑34͒
The perturbation then consists of the remaining terms from the Hamiltonian Similarly to OPT1, HCPT is invariant only to independent rotations of the doubly occupied spin orbitals, singly occupied spin orbitals, or the virtual spin orbitals.However, the fact that H ˆ͑0͒ is spin independent means that a spin free implementation of the method is possible.Hence, by the same arguments as for OPT1, HCPT is invariant to all spatial orbital rotations which are allowed for the ROHF reference wave function.
Orbital energies are obtained in HCPT by diagonalization of F ˆD, F ˆav , and F ˆV in the three standard subspaces.If this diagonalization is not carried out, the iterative expres-sions based on Eq. ͑7͒ must be solved to obtain the perturbed energy.For HCPT, the equations for the single and double substitution components of the first-order wave function are respectively.The second-order energy is then calculated from ͗i jʈab͘a i j ab͑1͒ .͑38͒ Equations ͑36͒-͑38͒ are nearly identical to the analogous expressions for OPT1 ͓Eqs.͑29͒-͑31͔͒.The only differences between the two come from the use of F ˆD and F ˆV for the doubly occupied and virtual orbital subspaces, respectively, in HCPT.As a result, the two methods are identical in terms of computational expense.

E. Z-averaged perturbation theory (ZAPT)
Lee and Jayatilaka introduced ZAPT 22 as an attempt to produce nth-order coefficients with higher symmetry than those of RMP and ROMP, and hence, to reduce the computational cost of those approaches.ZAPT requires the redefinition of the spin orbital basis: 11 for each doubly occupied spatial orbital and each unoccupied spatial orbital, the usual ␣ and ␤ spin functions are used, but for the singly occupied orbitals, new spin functions, are used.ϩ functions are, by convention, associated with occupied spin orbitals, and Ϫ functions with unoccupied spin orbitals.This spin basis is referred to as the symmetric spin basis.In this basis the spin orbital Fock operator is reconstructed.In schematic form, the matrix is

͑41͒
This form illustrates some of the symmetry of the Fock matrix in the symmetric spin basis.Certain elements of this matrix have been set to zero due to the ROHF convergence conditions or to the orthogonality of the component spin functions. 22It is interesting to note that elements such as F ˆL␤ M ␣ are not zero, in general, since the ␣ and ␤ spin functions are not orthogonal to the ϩ and Ϫ spin functions.Additionally, the block diagonal components in the doubly occupied and virtual spaces ͑e.g.F ˆL␣ M ␣ ͒ are identical to F ˆav after spin integration of the former.For this reason, Lee and Jayatilaka chose to use the same orbital definition as that of the OPT1 method.H ˆ͑0͒ is thus defined to include only the block diagonal elements of the spin orbital Fock operator in the doubly occupied and virtual subspaces.In the singly occupied space, however, only the diagonal elements of F ˆZAPT are included in zeroth-order.Therefore, and the perturbation is, therefore,

͑43͒
In Eq. ͑42͒, ⑀ T ϩ is the diagonal element of F ˆT ϩ U ϩ and ⑀ T Ϫ that of F ˆT Ϫ U Ϫ , and these same terms have been removed from Eq. ͑43͒.
Based on our previous analyses and, in particular, comparison to Eq. ͑15͒, ZAPT is not invariant to rotations which mix doubly occupied spin orbitals and singly occupied spin orbitals together, similarly to ROMP theory.However, these rotations do not, in general, have practical implications for ZAPT, due to the orbital canonicalization chosen by Lee and Jayatilaka. 22Specifically, since the F ˆav operator is used to define orbitals in all three spaces ͑just as for OPT1͒, spin restriction of the orbitals is maintained.In a truly spin restricted implementation of the method, then, rotations of doubly occupied and singly occupied spatial orbitals are not allowed to occur since they alter the ROHF reference wave function itself.
Additionally, ZAPT is not generally invariant to rotations which mix singly occupied spin orbitals together, since only the diagonal elements of F ˆZAPT in this block have been used to define H ˆ͑0͒ .However, as pointed out by Lee and Jayatilaka, in many practical applications, such mixings cannot occur since the singly occupied orbitals are of different spatial symmetries.Hence, the singly occupied block of F ˆZAPT will be diagonal regardless of the chosen canonicalization conditions.Therefore, it is reasonable to say that from a practical perspective, ZAPT is invariant to all spatial orbital rotations which are allowed for the ROHF reference wave function, though this is not true for all conceivable cases.
The iterative expressions for the single and double substitution contributions to the first-order wave function for ZAPT, which must be solved in the case that the F ˆav operator is not diagonalized in the three standard subspaces, are The second-order energy is then calculated from ͗i jʈab͘a i j ab͑1͒ .͑47͒ Equations ͑44͒-͑47͒ appear to be significantly different from both ROMP ͓Eqs.͑23͒-͑25͔͒ and OPT1 ͓Eqs.͑29͒-͑31͔͒.However, the complication here is primarily notational, as we have illustrated the necessary spin combinationsin order to make the relationship to the symmetric spin basis clear.Additionally, Lee and Jayatilaka have pointed out 22 that first-order interacting-space arguments, 14 when applied to the symmetric spin basis, suggest that spin-flip substitutions such as that indicated by a L ␣ D ␤ (1) should be considered double substitutions since their matrix element with the reference wave function through the Hamiltonian contains no one-electron contributions. 11Hence, the symmetric spin orbitals obey a form of Brillouin's theorem and the secondorder ZAPT energy contains no contributions from singly substituted determinants, similar to the MP2 energy.

F. Open-shell perturbation theory method 2 (OPT2)
OPT2 was introduced by Murray and Davidson 21 as an improvement to the OPT1 method; it is designed to have better convergence.OPT2 chooses the same orbital definition as OPT1, i.e., the eigenfunctions of F ˆav .The primary distinction between OPT1 and OPT2 comes in the addition of a two-electron component to the zeroth-order Hamiltonian of OPT1: where K ˆTT o is the Tth diagonal spatial orbital component of the usual exchange operator defined over the singly occupied orbitals only, and n ˆP is the operator Thus, in normal-ordered operator notation, the OPT2 zerothorder Hamiltonian becomes,

͑50͒
The purpose of this correction term is to shift the singly occupied orbital energies by a factor of 1/2K TT o depending on whether an excitation is occurring into or out of the orbital.That is, if the excitation occurs into the orbital, the orbital energy will resemble an electron affinity, while if the excitation occurs out of the orbital the orbital energy will resemble an ionization potential.
The inclusion of individual components of W ˆN in the last four terms in the above equation differentiates OPT2 from all of the other perturbation theories examined so far.It is clear from the structure of H ˆOPT2 ͑0͒ that the energy will be invariant to orbital rotations within the doubly occupied and virtual spaces separately, just as for OPT1.However, an analysis similar to that presented earlier for one-electron operators indicates that rotations of orbitals in the singly occupied space will alter the partitioning of the Hamiltonian, and, therefore, the energy.A practical implication is that different geometric representations ͑such as rotation of the molecule relative to a space-fixed axis system͒ can result in different energies; the OPT2 energy expression can be a multivalued function of nuclear geometry even though the mathematical definition of the spatial orbitals is maintained.This has been shown to occur in triplet twisted ethylene, 25 where rotations within the degenerate pair of singly occupied orbitals caused energy variations on the order of 9 kcal/mol.

G. Invariant open-shell perturbation theory (IOPT)
IOPT was introduced by Kozlowski and Davidson 23 as an adjustment to OPT2 to ensure invariance with respect to rotations among the singly occupied orbitals.This involved altering the correction function such that singly occupied or-bital energies could be shifted by a rotation-independent term, unlike that in OPT2.The zeroth-order Hamiltonian of IOPT is where N s is the number of open-shell electrons in the reference wave function and k is a constant given by Therefore, as the molecular orbitals are rotated in the singly occupied space, the number operators occurring in the final term of the zeroth-order Hamiltonian remain unaffected, as does the value of k.As a result, IOPT is invariant to all spatial orbital rotations allowed for the reference wave function, just as OPT1 is.The IOPT perturbation is simply the OPT1 perturbation, Eq. ͑28͒, less the last term shown in the equation above.The iterative equations for IOPT will therefore be exactly those presented in equations ͑29͒ and ͑30͒, with orbital-independent correction terms used to shift the diagonal elements of F pq av .Because of their similarity to the iterative expressions for OPT1 already presented, we will omit these equations here.It should be pointed out that the IOPT approach has been shown to contain size-extensivity errors, 33 though the significance of these errors has been questioned. 34

IV. CONCLUSIONS
Using a consistent notation and formalism, we have explicitly examined the effects of unitary transformations of the reference molecular orbitals on the energy and perturbed wave functions given by a number of spin-restricted singlereference open-shell perturbation theories.Of the seven methods examined here, the RMP, OPT1, HCPT, ZAPT, and IOPT approaches are found to be invariant to all types of rotations which do not alter the reference wave function.The ROMP method, which is implemented in a spin-dependent representation, and the OPT2 method, which includes twoelectron components in the zeroth-order Hamiltonian, are found to be invariant to more limited transformations.While this will not necessarily affect the practical application of ROMP ͑except perhaps in its extension to local correlation schemes͒, the consequences are more severe for OPT2, whose energy functional may be multivalued for a single nuclear framework.
Additionally, we have presented generalized invariant equations for the partitioned Hamiltonian as well as for the first-order wave function and second-order energy.These equations may be applied with any convenient set of reference molecular orbitals, subject to the constraint that the chosen orbitals are related to the particular method's socalled canonical orbitals by an allowed unitary transformation.It is clear that the standard equations associated with a particular perturbation theory ͑that is, a particular partitioning of the Hamiltonian͒ are usually valid only for a specific set of canonical orbitals ͑such as the semicanonical orbitals associated with RMP theory͒ and for no other.The use of other orbitals in the canonical orbital expressions will result in a different nth-order energy, since the zeroth-order Hamiltonian is not diagonal in the new orbital basis.The equations presented here, however, are valid for any set of molecular orbitals which are simply a ͑perhaps limited͒ unitary transformation away from the canonical orbitals defined for the perturbation theory in question.Such orbitals require, in general, an iterative construction of the perturbed wave functions.Additionally, the generalized equations presented here may serve as working forms for the extension of these methods to local correlation schemes or perturbational corrections to coupled-cluster energies, such as the well-known (T) correction.This latter example is interesting because the two most widely used approaches for open-shell systems [35][36][37] at present suffer either from a lack of invariance 35 or the use of a less convenient set of orbitals. 36,38Construction of a new correction, based on ZAPT, for example, would overcome these problems.