Second quantisation for unrestricted references: formalism and quasi-spin-adaptation of excitation and spin-flip operators

A second quantised formulation for the evaluation of spin-dependent properties in the UHF basis will be introduced. The configuration interaction singles ansatz will be used as an illustration of the practical usage of this formulation and the results will be compared to spin-restricted cases of the same ansatz. On a more tentative basis, we will also discuss the notion of quasi-spin-adaptation which ensures that the ansatz becomes spin-adapted as the unrestricted orbitals approach the restricted ones. GRAPHICAL ABSTRACT


Introduction
Second quantisation is a technique originally conceived of by Dirac [1] and further developed by Jordan [2,3] in the late 1920s and by Fock [4] in the early 1930s. The first mathematical analysis of it is due to Cook [5] two decades later. In field theory, second quantisation is introduced when the wavefunction represented as a field operator is expanded in a one electron basis [6]. In chemistry, a somewhat simpler approach usually traced back to Longuet-Higgins [7] is pursued and is the subject of several textbook level treatments [8][9][10][11][12][13]. It is particularly useful in defining ansätze with respect to a given reference and evaluating reduced density and transition density matrices [10,14], but it has also given rise to a family of Fock space methods [15,16]. It is also at the foundations of emerging applications of quantum computing to chemistry, since current approaches mostly start from the second quantised Hamiltonian and apply mappings such as the one originally proposed by Jordan and Wigner [3]  to transform them into a form that acts on qubits, the fundamental logical units of quantum computers. In the current context, it is also of interest to note that the preparation of spin-eigenstates on quantum computers has also been described recently [17,18].
Usually, quantum chemical ansätze are defined by choosing a reference function and an operator that constructs a better approximate solution from the reference. As a reference, usually Hartree-Fock (HF) or some multiconfigurational wavefunction is chosen; here we will restrict ourselves to the former case. The operator defining the ansatz is usually built from excitation, ionisation or spin-flip operators that can be easily defined in the second quantised language. A particularly important question concerning any approximate wavefunction is whether it is an eigenfunction of the total-spin-squared operator. In the HF case, this leads to a distinction between restricted HF (RHF) [19] and restricted open shell HF (ROHF) [20] on the one hand, and unrestrcited HF (UHF) [21] on the other. The difference between the two is particularly significant in the open shell case: while the UHF wavefunction consists of a single determinant in which different spatial orbitals are assigned to different spins, in typical cases ROHF is a multideterminantal reference function in which the expansion coefficients are fixed by spin symmetry and which uses identical spatial orbitals for different spins. This implies that the linear combination of determinants in low-spin ROHF in most cases [22] gives a single configuration state function (CSF) as reference in which the expansion coefficients of determinants need not be optimised, i.e. ROHF is still single reference in the basis of CSFs. Unfortunately, it is quite difficult to work out the expansion coefficients for arbitrary spin states [22]. In this regard, the UHF wavefunction is much simpler, albeit its major disadvantage is that the total spin is no longer a good quantum number. The ROHF wavefunction also has issues with obeying Koopmans' theorem [23] and the ordering of orbital energies [24]. As UHF calculations are considerably easier to perform, it was already proposed by Löwdin that a projection operator might be applied to the UHF solution to recover a total spin eigenstate [25], which might be simplified to remove nearest-lying spin components [26]. The projection technique may be applied after the self consistent field (SCF) procedure has converged, or, as this leads to artefacts, as a starting point of the variational procedure [27]. Unfortunately, the latter extended HF method does not yield size-consistent results. A more practical approach consists of choosing a set of 'quasi-restricted' orbitals based on various criteria [28,29]. More recently, the ROHF solution has also been obtained from a self-consistent UHF procedure relying on the projection technique, in which the spindensity eigenvalues are constrained [30]. As the main interest of this study is to develop a framework for evaluating the action of UHF based excitation operators, we will not consider the properties of the reference function any further, although ideally, it should at least be close to a spin-eigenstate.
Second quantisation approaches in quantum chemistry are either spin-orbital or spatial orbital based. The first one is applied with success in many areas in chemistry, although it is not trivial to ensure spin-adaptation using it. Coupled cluster (CC) theory is especially complicated in this regard, due to the non-linearity of the ansatz. Szalay et al. solved this problem by introducing an additional set of spin-equations that leads to exact spin-eigenstates if all CSFs are included. While this is usually not possible, the spin equations can be solved to yield approximate eigenstates in a truncated CSF manifold. In this study, we will focus on spatial orbital or spin-free approaches that ultimately put constraints on the commutator of the ansatz operator with the total-spin-squared operator. Already in the 1960s Matsen called for a spin-free formulation of quantum chemistry [31] based on the observation that spin-adapting the spin part of the wavefunction using a two dimensional special unitary group, SU(2), approach is equivalent to constructing the spatial part using irreps of the n dimensional symmetric group, S N describing permutation symmetry [32]. But since the Hamiltonian can also be expressed using the generators of the unitary group, it is also possible to give a spin-free representation using representations of the n dimensional unitary group, U(n) [32], of spatial orbitals which can be constructed using second quantisation. Unfortunately, this formulation leads to non-commuting cluster operators for open shells and linear dependencies in the more highly excited manifolds. How this problem can be overcome for CC theory is discussed elsewhere [33], here, we will limit ourselves to the simplest ansatz: configuration interaction singles (CIS) [34], which is a useful starting point for excited state calculations. Beyond restricted CIS (RCIS), CIS has also been extended to open shell systems [35] both in an unrestricted (UCIS) and restricted open shell (ROCIS) fashion. In fact, ROCIS has extended formulations that are not only spin-adapted, but spin-complete (span the entire spin space) [36], explore different multiplets using appropriately constructed flip operators [37,38] and also ionised states [39]. For a deeper discussion of most spin adaptation related issues, see the book of Pauncz [40].
As mentioned before, the primary goal of this study is to provide a practical tool for evaluating UHF based quantities. To achieve this, we will develop a spatial orbital approach resembling second quantisation for non-orthonormal orbitals [12], but with the simplifying condition that the two sets of orbitals are orthogonal within themselves. We begin by reviewing the basic formulation of second quantisation, proceed to discuss the various restricted approaches on the example of the CIS wavefunction, and extend to formulation so that the unrestricted expressions, especially expectation values of spin-operators can also be handled. As spin operators are the most difficult to construct using existing second quantisation approaches, we will also take the analysis one step further and introduce the concept of quasi-spin-adaptation. While in the process the conceptual simplicity of UHF is lost, since a multideterminantal expansion is created, the resulting wavefunction is only spin-contaminated as a result of spin-polarisation and the different sets of orbitals for α and β spins. Such quasispin-adapted wavefunctions recover the ROHF CSFs as the overlap between the two orbital sets approaches unity.

Notation
Orbitals are partitioned into subsets of doubly occupied molecular orbitals (DOMO) denoted as i, j, . . ., singly occupied molecular orbitals (SOMO) labelled as t, u, . . . and virtual molecular orbitals (VMO) with labels a, b, . . .. The labels p, q, . . . refer to arbitrary orbitals. The Greek letters σ , τ . . . will be used to refer to either of the spin functions α, β, andσ will denote the opposite spin with respect to σ . A spin-orbital φ p (x) is a function of x = (r, ν), with r being the spatial and ν the spin coordinate of the electron. Furthermore, φ p σ (x) can be written as a product of a spatial orbital ϕ p σ (r) and a spin function . Thus, as a shorthand, we will refer to this product as p σ σ . Since in UHF, there are different spatial orbitals for different spins, we refer to these by the subscript σ in p σ . In the RHF/ROHF case, p α = p β ≡ p, and hence a spin-orbital is of the form pσ in the shorthand notation. As in spinsummed operators even the spin function disappears from the notation, it becomes hard to distinguish when p refers to a spin or spatial orbital. Thus, when necessary, spatial orbitals will be capitalised as P to distinguish them from spin-orbitals p. For creation and annihilation operators, we will adopt Kutzelnigg's notation [14], i.e. an annihilator is denoted by a lower orbital index,â p and a creator by an upper one,â p ≡â † p . Repeated indices will also be summed over.

Creators and annihilators
Second quantisation relies on operations that place or remove electrons in orbitals. When acting on a basis state |n containing n electrons, a creator produces (up to a sign),â where the occupation number n p is either 1 or 0, depending on whether p is occupied or not. An annihilator, on the other hand, yields (up to a sign), where |n is zero for negative values of n by definition. Even more important than these relationships are the anticommutation rules, Paving the way to a spin-free formulation, the spatial (p σ ) and spin (σ ) parts of orbitals can be explicitly considered. Assuming for now a spin-restricted reference, i.e. p α = p β ≡ p, a spin-orbital is the product pσ , with the anticommutation rules remaining essentially unchanged, and the anticommutators between operators of the same type still yielding zero. If the spatial orbital basis is not orthonormal, then this rule becomes where S p q is the orbital overlap matrix. The other two rules again yield zero, and may only change under a transformation that mixes creators and annihilators, e.g. a Bogoliubov transformation [41,42].

Parametrizing a state
Post-Hartree-Fock methods construct the wavefunction | k of the kth state using the HF state |0 as reference, It is usually desirable that k should satisfŷ for the operatorsŜ z ,Ŝ 2 andN , which are the spin z component, the total spin squared and the number operators, respectively. The eigenvalues O k will then be M S , S(S + 1), and N in terms of the total spin magnetic quantum number (M S ), total spin quantum number (S) and the particle number (N ) of the kth state. The left hand side of the above equation can be written in the form If the reference is an eigenstate ofÔ with eigenvalue O, then commutator yields the change due to the action of is also an eigenfunction ofÔ. The RHF, ROHF and UHF reference states are all eigenstates ofŜ z andN , i.e. the sum and difference of the number of electrons with alpha and beta spin are well-defined. If the excitation operator commutes with these operators, then the M S and N values are not changed byR k . Nonzero integer values of O correspond to transition to a different magnetic sublevel or ionisation. The RHF and ROHF reference states are also eigenfunctions ofŜ 2 by construction. While the same is not true for UHF, there are projection techniques to recover a total spin eigenstate from UHF. The commutator has a similar significance as it does for RHF and ROHF: if the excitation operator commutes withŜ 2 , it does not change the total spin. For the RHF and ROHF references, and for UHF if it is at least close to be an eigenstate, requiring the appropriate change in S(S + 1) ensures that the action ofR k yields another spin eigenstate. Thus, it is said that an excitation operatorR k is spin adapted with respect to a reference |0 if it satisfies for the total spin numbers of the final state S and the initial state S. Note the significance of the projection to |0 , as this equation is much simpler to satisfy than an unprojected operator equation.

Excitation and spin operators
Let us now define our elementary excitation operators from whichR k might be built. The spin-traced excitation operator is defined aŝ Its trace in the spatial orbital space yields the number operatorN The triplet operatorsT are given as with the M S values indicated as +, 0, −, see the more standard notation in textbooks [9]. The latter also have orbital traced versions, The total spin squared operator is then given bŷ Here, the quantities h p q and g sq pr are the one and two electron integrals, in physicist's notation, h p q = p|ĥ|q , whereĥ contains the kinetic energy operator of the electrons and the nuclear-electron potential term, while g qs pr = pr|qs are the electron-electron repulsion integrals.

Closed shell CIS
The closed shell RHF wavefunction can be written as with N c (= N α = N β ) being the number of closed shell orbitals. In configuration interaction singles (CIS) ansatz, the excitation operator can be simply given aŝ This operator is manifestly spin-adapted in the sense of Equation (12) since even without projection to |0 . This means that the E operators will not change the ground state S or M S labels. The triplet operators do not in general commute withŜ 2 , S z , but the projection of the commutators to the ground state reference can be evaluated, It follows that all these operators raise the S value with one unit producing different M S values. Note finally, that the spin sublevels can be changed without affecting S values, by using the ladder operators in conjunction with any of the operatorsQ =Ê, 0T , +T , −T as follows, In practice, it is enough to generate a single sublevel of a multiplet, since matrix elements containing the other M S values can be obtained using the Wigner-Eckart theorem [43,44]. Thus, a spin-flip excitation operator  (14), +T a i , 0T a i and −T a i in Equation (16). may be defined asR The action of these operators is illustrated via a simple example in Figure 1.

Open shell CIS
In the following, we will assume a high spin ROHF wave function as a reference (|0 ), We further assume that the SOMO electrons have alpha spin (N β < N α ). A spin adaptedR k containing only a singleÊ operator can be written aŝ AsÊ p q commutes withŜ 2 , this operator yields a total spin eigenstate, i.e. it is spin adapted. This is ensured by the fact thatÊ p q is spin-summed, as in Equation (14). However, excitation classes in terms ofÊ p q do not span the full spin space, i.e. they are not spin complete. The latter can also be achieved by adding excitations of the typê E a i − 2Ê a tÊ t i which cause spin flip in the SOMO space. For our purposes, the above spin-adapted form ofR k will be sufficient. See Figure 2 for a simple example.
As theÊ p q operators commute withŜ 2 , there is also no need to assume a special structure for the reference state. For an operator that changes the S value compared to that of the reference, the commutator expressions withŜ 2 can be considerably simplified by projection to the reference, as will be seen below. Here, we only consider the high spin case for which the reference is a single determinant containing all alpha SOMOs. In low spin cases, there is merit in treating the SOMOs explicitly and considering only the closed shell part as the reference |0 , as this is shared by all terms in the sum of determinants constituting a spineigenstate. In the language of many-body physics, this corresponds to a different choice of the particle-hole vacuum state, and it would influence the way normal order is used in the evaluation of second quantised operator strings [14]. For our purposes, considering Equation (29) as the reference is perfectly sufficient.
We may now generalise our treatment for spin flip states. Starting with +T , this operator turns out to be spin adapted with respect to the chosen reference, and it will produce an S + 1 function from one with S. Furthermore, the possible ranges of p and q are constrained by the projection, resulting in the excitation operatorR For the sake of completeness, let us evaluate the remaining possibilities. For 0T p q , which is not spin adapted. Following a procedure outlined in the supplementary material (SI), we get with the property The excitation operator than takes the form Note that theŜ − +T p q term only survives inX a i . This class does not change the S value, but it produces a different combination of determinants than theÊ operators, see Figure 2.
Finally, for −T p q , The procedure described in the SI yields a new operator The excitation operator again has the general form remembering that not all terms inŶ survive projection to the reference. Furthermore, the operators presented so far still need to be normalised.

Reference and anticommutation rules
In the UHF case, we need to distinguish between the spin part of the spin-orbital, and the spin label of the spatial part which specifies the set of SCF equations the orbital is a solution of. Thus, we have in principle p α α and p β β, where the spatial spin label is in the lower index. Part of the UHF problem is that it is possible to get spin-orbitals of the type p β α and p α β under the action of spin operators, which is the reason why the UHF wavefunction is not a spin eigenstate. Assuming that in the open shell case there are more alpha electrons than beta ones (N β ≤ N α ), the UHF determinant has the structure When acting on this reference, the creators and annihilators behave aŝ Since α and β orbitals are separately optimised, the spatial orbitals corresponding toâ i α andâ i β are not orthonormal, while the spin-orbitalsâ i α α andâ i β β still are because the spin parts are orthonormal. Thus, we prefer to stay in the spin-orbital basis, for which we have the following anti-commutator rules Here, S is the spatial overlap defined as S q τ p σ = p σ | q τ . So long as alpha and beta orbitals are orthonormal among themselves and they span the same space, S has to be unitary. This is a convenience, since S serves as the metric and its inverse can then be assumed to be its transposed. This also means that which establishes the conversion between α and β spatial orbitals. Note the inversion (transposition) of the metric.

Excitation and spin operators
To define methods beyond the HF level, we need excitation operators of various kind. Unlike in the RHF case, these are not spin-traced, but remain in the spin-orbital basis,Ê where σ denotes either α or β without implying summation over spin labels. It is of course still possible to define the number operator usingÊ p σ σ p σ σ for the two spins. Consequently,Ŝ z is defined similarly as in the closed shell case,Ŝ and it is also easy to see that the UHF reference function is still an eigenfunction ofŜ z . For the spin ladder operators, we haveŜ These can then be used to defineŜ 2 via Equation (18).
Here, there is some ambiguity at the second quantised level, since once an alpha electron can be placed on a beta spatial orbital (p β α) and vice versa, subsequent spin operators could also annihilate such electrons, and thus it seems that more combinations of the creators and annihilators could contribute to the above product (e.g. a p α αâ p α β toŜ + ). However, evaluating p σ σ |Ŝ ± |q τ τ from first quantisation confirms that the above representation suffices in the space of UHF spin-orbitals p σ σ . Nevertheless, spin operators do give rise to orbitals of the type p σ τ . See Figure 3 for an illustration about the action ofŜ + on the UHF reference, Here, a summation over a α is made explicit to avoid confusion, although in the example in Figure 3, only a single virtual orbital is present. Note that a string like i α α i β α does not amount to zero, as in the closed shell case where i α = i β , rather, a i β α = S i β p α a p α α . Since strings like i α α i α α are zero even in the UHF case, p can only be a virtual label to avoid repetition. In order to demonstrate how to handle these operators, let us consider the simple example of evaluating i α α i β β|Ŝ −Ŝ+ |i α α i β β step-by-step, sinceâ i α α commutes withŜ + and S i β p β = δ i p by Equation (43). Next, where the second term survives by Equation (43) and Since the second term is just the original determinant, we have Applying Equation (43) again, we finally have which agrees with the textbook result [8]. A general proof proceeds along the same lines, but it is more involved. Finally, in order that the above machinery be applicable, it should be ensured that the second quantised representation ofŜ −Ŝ+ can be represented as the product of the two second quantised operatorsŜ − and S + . This turns out to be true in the (extended) basis of UHF spin orbitals p σ σ , since p σ σ |Ŝ −Ŝ+ |q τ τ − r κ κ p σ σ |Ŝ − |r κ κ r κ κ |Ŝ + |q τ τ = 0, which can be proven in a similar fashion as above.
The representation of the Hamiltonian in the UHF basis can be easily obtained starting from the spin-orbital form,Ĥ = h q pâ pâ and making the substitution p → p σ σ for every orbital label. Then the summation over spin labels can be simplified since only integrals of the type h q σ p σ and g q σ s τ p σ r τ survive spin integration. Finally, using the anti-commutation rules in Equation (43) the creators and annihilators can be brought to the form of Equation (45). With all these preparations, the Hamiltonian can be written aŝ The various methods, such as CIS can now be derived using this definition, and that of the excitation operators. So far we have only dealt with the spin conserving ones. In the RHF case, theŜ ± andŜ z operators are closely related to the spin flip exitation operators. In the UHF case, more care is needed when defining such operators. The lowering and raising operators can be defined as The adjoint relation +T † = −T still holds. While 0T might be defined, it has the same form asÊ, since there are no plus and minus combinations of these operators in the UHF case. A conventional UHF calculation may converge to either of these combinations, or something that does not respect spin symmetry at all. So far, normalisation factors and sign conventions have been avoided in the definitions of these operators, but these can be introduced when needed. In practice, spin-flip operators are usually defined using −T .

Spin-conserving excitations
The UHF-CIS excitation operator can be defined in analogy to the restricted case aŝ To make our treatment more general, we introduce another operator,L and write to most general matrix elements of some oper-atorÔ in the form 0|LÔR |0 . Apart fromŜ ± , the operators defined this way behave pretty much the way their spin-traced counterparts do in RHF. The one body reduced densities, as well as the variousŜ 2 expectation values along with the necessary commutation rules to evaluate them are given in the SI. We will next consider the change in the expectation value ofŜ 2 and compare it with the ROHF case. The transition from UHF to ROHF can be understood as one that transforms the UHF overlap matrix between α and β spatial orbitals into the unit matrix, S p α q β → 1 p q . In this limit, the overlap becomes a projector to the various ROHF orbital subspaces. For a mapping between the UHF and RHF orbital labels, see Figure 4. It is worth noting that the S a α i β block of the overlap approaches zero in the ROHF limit. Using the rules discussed, the expectation value is found to be Figure 4. The structure of the UHF overlap matrix between α and β spatial orbitals. In the limit where the overlap is unity, the ROHF solution is recovered. The mapping between the UHF orbitals i α , a α , i β , a β and their ROHF equivalents denoted here with capital letters I, T, A is also shown.
Applying the map from UHF to ROHF orbitals and taking the limit of unit overlap, the resulting terms cancel each other exactly. This is expected since the ROHF excitation operators commute withŜ 2 .

Spin-flip excitations
As mentioned before, the spin-flip up and spin-flip down operators can also be defined. In the former case, Here, it is important to remember that there are more alpha than beta electrons in Equation (41), i.e. the set of i β orbitals correspond to the ROHF orbitals I, while the set of a α to A (see Figure 4). This implies that the operator above corresponds to the ROHF excitation class I → T. Thus, the corresponding expectation value is SinceL andR are biorthonormal, their inner product is 1 or 0 (δ LR ). Thus, as the overlap goes to unity, the change in the expectation value ofŜ 2 approaches 2(S + 1), which is the expected from Equation (13) in the spin-adapted ROHF case, if S = S + 1. In contrast, for where the set of i α includes those of I and T orbitals in the ROHF limit, and similarly, a β corresponds to the sets of T and A orbitals (see Figure 4). Thus, in this case, the ROHF equivalent of the excitation operator above contains the classes I → T, I → A, T → U and T → A. Consequently, the UHF and ROHF expectation values are which indicates that even in the ROHF limit, the UHF state is not spin-adapted. This again is expected since − T is not in itself spin-adapted when it acts on the reference Equation (41) in which all the SOMOs are assumed to have alpha spin.

Quasi-spin-conserving excitations
The results of the previous section suggest that the amount of deviation from a spin-eigenvalue (spin contamination) comes from two interdependent sources: (a) the non-orthogonality of the UHF spatial orbitals, (b) the excitation operators not generating the necessary combination of determinants. The former can be recovered if the overlap is set to unity, the latter cannot. We will show in this section that it is also possible to ensure in the UHF basis that the appropriate determinants are generated by the excitation operators. It will be convenient to define quasi-spin-adapted operators in analogy with Equation (13) as In such operators, spin contamination only arises from the non-orthonormality of the orbitals and the appropriate spin states are fully recovered in the ROHF limit.
In the SI, the most general excitation operator is defined asĜ Using this notation, we may define a special excitation operator in analogy ofÊ p q by summing over spin labelŝ with the property that following from the commutation rules derived in the SI. The spin labels σ , τ may now be fixed to define an excitation operator similar to the regular UHF one, When acting on the UHF reference determinant, this operator can be rewritten in terms of the usual UHF excitation operators aŝ with the appearance of a constant term and transformed singles amplitudes These formulae amount to a modification of the usual UHF equations.

Quasi-spin-flip excitations
In a similar sense, we can construct generalised spin operators inspired by the closed shell theory, As in the ROHF case, the analogues ofX a i andŶ a i should be defined to get a (quasi-)spin-adapted result. One possible way to do this might bê since since It only remains to construct the operatorsR = r i α a αX a α Unfortunately, these operators have the property that 0|X a i = 0 and 0|Ŷ a i = 0, which makes evaluating expressions with them significantly more involved than in ROCIS. Fortunately, as the ROHF limit is reached, the action of these operators on the bra also approaches zero.

Beyond CIS
So far, we have used the simple CIS ansatz to demonstrate the utility of the formalism presented here. For higher excitation levels, including doubles (CISD) and beyond, the matrix elements of the Hamiltonian in Equation (56) should be obtained in a similar way and for suitably generalised versions of the operators in Equations (45) and (57) they should also yield the same results as other ways of deriving these expressions. This is because our formulation only differs substantially from the conventional spin-orbital based one if the spin-operatorsŜ + , S − orŜ 2 are present. The same can be said about CC: the application of this formulation yields the unrestricted CC equations for spin-conserving and flip operators at any excitation level. The main advantage of the formulation presented here is that it provides an automatic way of deriving the expectation values ofŜ 2 for both linearly (CI) and exponentially (CC) parametrised wave functions, since the usual spin-orbital formulation does not keep track of spin labels on which such operators act. Thus, even if the formulation is somewhat cumbersome for human application, it can be automated for the purpose of calculating spin contamination at various levels of theory, a non-trivial task, which was the main motivation behind this work. In a more experimental spirit, this formulation can also be used to construct combinations of UHF determinants that imitate configuration state functions in the spin restricted basis, and in fact turn into them as the spatial orbitals associated with alpha and beta spins become identical. While this formulation, termed 'quasi-spin-adaptation' here, can also be extended beyond CIS, it would face the same or even worse issues of complexity as the ROHF formulation for open shells. In the exponential CC ansatz in particular, the commutator expansions would have similar truncation issues [45], not to mention that such quasispin-adapted operators do not always produce zero when acting on the reference in the same way their ROHF counterparts do. Thus, whether this latter aspect has any utility remains to be seen.

Conclusions
In this work, we have reviewed the basic formulations of second quantisation and their usage in defining wavefunction ansaetze. We have focussed on the simple example of CIS in the RHF, ROHF and UHF bases. The most important result of this work is the introduction of a second quantised formalism that can be used to evaluate the action of spin-operators on UHF references. We hope that it will be useful in evaluating, e.g. S 2 -expectation values for complex wavefunctions. On a more tentative basis, we also considered the notion of quasi-spin-adaptation which ensures that an UHF excitation operator becomes spin-adapted as the overlap between α and β spin-orbitals reaches unity (in the RHF/ROHF limit). The practical usefulness of this latter approach remains to be seen, but it could make a difference in situation where some spin-polarisation of the orbitals is desired but where the lack of CSFs also introduces problems.

Dedication
This work is dedicated to the celebration of the 60th birthday of Prof. Péter G. Szalay in recognition of his outstanding contributions to quantum chemistry. I met Péter for the first time more than 15 years ago at a student conference where I presented my thesis work. I will always remember the question he asked me after my talk as the only one that directed attention to the merits of my work rather than to its deficiencies. During the years since then he remained an influential figure in coupled cluster theory and excited state chemistry, many of his pioneering efforts also influenced my own work in those fields. I am looking forward to the continuation of his inspirational work in the years to come.

Disclosure statement
No potential conflict of interest was reported by the author.