Flexible ansatz for N-body Configuration Interaction
Taewon David Kim, Ramón Alain Miranda-Quintana, Michael Richer, Paul W. Ayers
aa r X i v : . [ phy s i c s . c h e m - ph ] F e b Flexible Ansatz for N-body Configuration Interaction
Taewon D. Kim a , Ram´on Alain Miranda-Quintana b , Michael Richer a , Paul W. Ayers a, ∗ a Department of Chemistry and Chemical Biology, McMaster University, Hamilton, Ontario, Canada L8S 4M1 b Department of Chemistry, University of Florida, Gainesville, FL, 32603 USA
Abstract
We present a Flexible Ansatz for N-body Configuration Interaction (FANCI) that includes anymultideterminant wavefunction. This ansatz is a generalization of the Configuration Interaction(CI) wavefunction, where the coefficients are replaced by a specified function of certain parame-ters. By making an appropriate choice for this function, we can reproduce popular wavefunctionstructures like CI, Coupled-Cluster, Tensor Network States, and geminal-product wavefunctions.The universality of this framework suggests a programming structure that allows for the easy con-struction and optimization of arbitrary wavefunctions. Here, we will discuss the structures of theFANCI framework and its implications for wavefunction properties, particularly accuracy, cost,and size-consistency. We demonstrate the flexibility of this framework by reconstructing popularwavefunction ans¨atze and modifying them to construct novel wavefunction forms. FANCI providesa powerful framework for exploring, developing, and testing new wavefunction forms.
Keywords: multi-reference quantum chemistry; projected Schr¨odinger equation, antisymmetricgeminal product, coupled cluster, tensor network, wavefunction ans¨atzee
1. Introduction
In this paper, we focus on electronic systems, whose Hamiltonian can be written asˆ H elec = X ij h ij a † i a j + 12 X ijkl g ijkl a † i a † j a l a k (1)where h ij and g ijkl are the one- and two-electron integrals and a † i ( a i ) creates (annihilates) the i th spin-orbital. The exact solutions to the electronic Hamiltonian can be written as a linear ∗ Corresponding Author; [email protected]
Preprint submitted to Computational and Theoretical Chemistry February 3, 2021 ombination of all possible N -electron basis functions (Slater determinants) formed from the givenset of spin-orbitals. This is the Full Configuration Interaction (FCI) wavefunction[1]: | Ψ FCI i = ( KN ) X m C m | m i (2)where 2 K is the number of spin-orbitals, N is the number of electrons, and C m is the coefficient of theSlater determinant | m i . We can think of m as an occupation vector that specifies which of the 2 K spin-orbitals are occupied to construct the N -electron basis functions. The number of parametersfor the FCI wavefunction scales combinatorially with the number of orbitals and electrons, so brute- force direct calculations of the FCI wavefunctions are restricted to small systems with small basissets.Various approximations can be made to the Schr¨odinger equation to bring down its cost: (1) sim-plify the Hamiltonian, (2) find alternative algorithms, and (3) parameterize the FCI wavefunction[2,3, 4, 5, 6, 7, 8, 9, 10]. In this article, we focus on the parameterization of the FCI wavefunction.The simplest approximation to the FCI wavefunction involves explicitly selecting (or truncating) theSlater determinants that contribute to the wavefunction. Such wavefunctions are broadly termedselected Configuration Interaction (CI) wavefunctions: | Ψ CI i = X m ∈ S C m | m i (3)where S is a subset of the Slater determinants within the given basis. The Slater determinants canbe selected by seniority, such as the doubly occupied CI (DOCI) wavefunction[11, 12, 13, 14, 15,16, 17, 18, 19, 20, 21, 22], or by excitation-level relative to a reference Slater determinant, such as CI singles and doubles (CISD) wavefunction[23]. Alternatively, we can select all (or many) of theSlater determinants in a given set of orbitals. This leads to active-space methods like CASSCF[24],RASSCF [25], and MCSCF[26]. Finally, if the orbitals are localized, the Slater determinants thatembody chemically intuitive concepts can be linearly combined to construct Valence Bond (VB)structures[27, 28, 29, 30, 31]. This leads to VBCI methods. While there are many variants of the CI wavefunction, most are not size-consistent and choosing an efficient set of orbitals/determinants ismolecule and geometry dependent. For truly strongly-correlated systems, which have myriad Slaterdeterminants with small yet significant contributions, selected CI methods generally fail.Alternatively, the FCI wavefunction can be approximated by an alternative form (ansatz), suchas a nonlinear function of parameters that are not simply the coefficients of the Slater determinants.2or example, the Coupled-Cluster (CC) wavefunction parameterizes the CI wavefunction using anexponential ansatz[32, 33, 34, 35, 36, 37]: | Ψ CC i = exp X i X a t ai ˆ E ai + X i 2. Flexible Ansatz for N-particle Configuration Interaction (FANCI) The proposed wavefunction structure is quite simple and resembles the CI wavefunction (Equa-tion 3): | Ψ FANCI i = X m ∈ S m f ( m , ~P ) | m i (9)where S m is a set of allowed Slater determinants and f is a function that controls the weight ofeach Slater determinant, m , using the parameters, ~P . Since Slater determinants can be uniquelyrepresented with an excitation operator and a reference, Equation 9 can be rewritten with respectto excitation operators, ˆ E . | Ψ FANCI i = X ˆ E ∈ S ˆ E f ( ˆ E, ~P ) ˆ E | Φ ref i (10)where S ˆ E is a set of allowed excitation operators and f is a function that maps the weight of theexcitation operator from ˆ E and ~P . The S m and S ˆ E are equivalent representations of a set of Slaterdeterminants and can be used interchangeably. In this paper, we aim to demonstrate the generality, utility, and flexibility of this framework. Inthe next section, we show that the framework is general by expressing popular ans¨atze as FANCIwavefunctions. Then, in Section 4, we discuss how the choices of S , ~P , and f affect the accu-racy, cost, and size-consistency of the wavefunction. Finally, we demonstrate the flexibility of thisframework by constructing novel wavefunction structures. 3. Examples The ground-state Hartree-Fock (HF) wavefunction is the Slater determinant of orthonormalorbitals that provides the lowest energy[105, 106]. Starting from an arbitrary set of orthonormal5rbitals, created by { a † j } , the HF wavefunction can be obtained by optimizing the unitary transfor-mation that provides the lowest energy. | Ψ HF i = N Y i =1 K X j =1 a † j U ji | i = X m | U ( m ) | − | m i (11)where U is a unitary matrix, and U ( m ) is a submatrix of U obtained by selecting the rows thatcorrespond to the spin-orbitals in m . The derivation is given in the Appendix 8.1. If only N orthonormal orbitals are rotated, or alternatively, if there is no mixing of the occupied and virtualorbitals, then the HF wavefunction is obtained trivially: | Ψ HF i = N Y i =1 N X j =1 a † j U ji | i = | U ( m ) | − | m i (12)where m is the set of the occupied orbitals. With normalization, | U ( m ) | − becomes 1. In otherwords, the HF wavefunction is invariant to rotation of the occupied orbitals if there is no mixingbetween occupied and virtual orbitals [107]. The truncated CI wavefunction (Equation 3) is a linear combination of selected Slater determinants[108].Such wavefunctions can be trivially described in the proposed framework: the set of allowed Slaterdeterminants, S , is the same; the parameters, ~P , are the coefficients of the Slater determinants, ~C ;and the parameterizing function, f , simply selects the appropriate coefficient, C m , given the Slaterdeterminant, m . f ( m , ~C ) = ~e m · ~C where ~e m is a vector that gives 1 in the position of m and 0 elsewhere. Altogether, the CI wave-function is | Ψ CI i = X m ∈ S m (cid:16) ~e m · ~C (cid:17) | m i (13)If the CI wavefunction is expressed with respect to excitations on a reference, we get | Ψ CI i = X ˆ E ai ∈ S ˆ E (cid:16) ~e ˆ E ai · ~C (cid:17) ˆ E ai | Φ ref i (14)6 .3. Coupled-Cluster The CC wavefunction (Equation 4) uses the exponential operator to approximate high-orderexcitations as a product of lower-order excitations [109]. | Ψ CC i = exp( ˆ T ) | Φ HF i = ∞ X n =0 n ! ˆ T n | Φ HF i (15)where ˆ T = X ˆ E ai ∈ ˜ S ˆ E t ai ˆ E ai (16)and ˜ S ˆ E is a set of excitation operators. The Maclaurin series in Equation 15 lets one expressthe CI coefficients in terms of CC cluster amplitudes t ai . Specifically, the cluster amplitudes arecumulants of the CI coefficients [110, 111, 112, 113, 114]. The powers of ˆ T (cluster operator) givethe wavefunction access to excitations beyond those allowed ( ˜ S ˆ E ) by generating all (product-wise)combinations of the allowed excitation operators. However, an excitation can be described withdifferent combinations of excitation operators, and the cumulant can be simplified by groupingtogether terms that correspond to the same excitation (or Slater determinant). Each combinationcorresponds to a subset of ˜ S ˆ E , such that the set of all Slater determinants in the CC wavefunctioncan be described in terms of all possible subsets of ˜ S ˆ E : S ˆ E = Y ˆ E k ∈ T ˆ E k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T ⊆ ˜ S ˆ E (17)Then, the wavefunction can be written as a sum over all possible Slater determinants and a sumover all possible combinations of excitation operators that produce the given Slater determinant. f ( ˆ E ai , t ) = X { ˆ E a i ... ˆ E a n i n }⊆ ˜ S ˆ E sgn Q nk =1 ˆ E a k i k = ˆ E ai sgn( σ ˆ E a i ... ˆ E a n i n ) 1 n ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t a i . . . t a n i n ... . . . ... t a i . . . t a n i n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + = X { ˆ E a i ... ˆ E a n i n }⊆ ˜ S ˆ E sgn Q nk =1 ˆ E a k i k = ˆ E ai sgn( σ ˆ E a i ... ˆ E a n i n ) n Y k =1 t a k i k (18)7here n is the dimension of the subset { ˆ E a i . . . ˆ E a n i n } . The sum can be interpreted as a sum overall possible partitions of a given excitation operator, ˆ E ai , into excitations from the given set. Thesignature of the permutation, sgn( σ ˆ E a i ... ˆ E a n i n ), results from reordering the creation and annihilationoperators of the lower-order excitations to the same order as the given excitation operator:ˆ E ai = sgn( σ ˆ E a i ... ˆ E a n i n ) ˆ E a i · · · ˆ E a n i n (19)The permanent, | A | + , accounts for all possible orderings within a given set of excitation operators.Altogether, the CC wavefunction can be reformulated as | Ψ CC i = X ˆ E ai ∈ S ˆ E X { ˆ E a i ... ˆ E a n i n }⊆ ˜ S ˆ E sgn Q nk =1 ˆ E a k i k = ˆ E ai sgn( σ ˆ E a i ... ˆ E a n i n ) n Y k =1 t a k i k ˆ E ai | Φ HF i (20)More details are provided in the Appendix 8.4. The TPS (and MPS) wavefunction (Equation 5) determines the weight of a Slater determinantby tensor (and matrix) contractions, where each shared index corresponds to a correlation betweenorbitals[42, 9]: | Ψ TPS i = X n ...n K X i ...i K i ...i K ... i K − K ( M ) n i ...i K ( M ) n i i ...i K . . . ( M K ) n K i K ...i K − K | n . . . n K i Each spatial orbital, k , is associated with a tensor, M k , and each tensor is associated with the occupation of its spatial orbital (i.e. its state), n k , and with other tensors using its auxiliaryindices, { i k . . . i k − k i k k +1 . . . i kK } . Then, the coefficient associated with the Slater determinant,represented by { n . . . n K } , is approximated by tensor-contraction. Many variants of TPS, includingMPS, impose some structure on the tensor product so that the evaluation and optimization of thewavefunction are computationally tractable. Since { n . . . n K } is yet another representation of the Slater determinant, we can describe thewavefunction with respect to n : n = { n . . . n K } | Ψ TPS i = X n ∈ S FCI K K k =1 ( M k ) n k | n i (21)where K is the number of spatial orbitals, and J describes the specific tensor-contraction used inthe wavefunction. While it is not common to do so, the TPS wavefunctions can be equivalentlyexpressed with respect to spin-orbitals. If each state of the TPS wavefunction corresponds to theoccupation of a spin-orbital, m k , then the same notation can be used as in Equation 9 | Ψ TPS i = X m ∈ S FCI K K k =1 ( M k ) m k | m i (22)where M k is the tensor associated with spin-orbital k . Just as the HF wavefunction is constructed as an antisymmetrized product of one-electronwavefunctions (orbitals), the APG wavefunction is constructed as an antisymmetrized product oftwo-electron wavefunctions (geminals)[66, 68, 69, 74]: | Ψ APG i = N/ Y p =1 G † p | i = N/ Y p =1 2 K X ij C p ; ij a † i a † j | i = N/ Y p =1 X m k C p ; m k A † m k | i where the m k denotes a set of a pair of indices and A † m k denotes the creation operator thatcorresponds to m k . Similar to the CC wavefunction, the product of sums can be expanded out as asum over each Slater determinant and a sum over the different combinations of electron pairs thatcreate the Slater determinant. In the HF wavefunction, the product of sums results in a determinantdue to the antisymmetry with respect to the interchange of electrons. In the APG wavefunction,however, the interchange of electron pairs is symmetric, and the product of sums results in apermanent. Given the set of all possible two-electron creation operators, ˜ S , a subset of exactly N two-electron creators, { A † m . . . A † m N/ } , is needed to construct a given Slater determinant, m ,where the number of electrons, N , is even. Since an orbital cannot be occupied more than once9nd all the orbitals are necessary to construct a given Slater determinant, any selection of orbitalpairs, { m . . . m N/ } , must be disjoint and exhaustive. f ( m , C ) = X { m ... m N/ }⊆ ˜ S N/ S k =1 m k = mm p ∩ m q = ∅ ∀ p = q sgn( σ m ... m N/ ) | C ( m , . . . , m N/ ) | + = X { m ... m N/ }⊆ ˜ S sgn A † m ...A † m N/ | i = | m i sgn( σ m ... m N/ ) | C ( m , . . . , m N/ ) | + (23)where the orbital pairs, { m . . . m N/ } , are selected such that they result in the given Slater deter-minant, m , without duplicate orbitals. Similar to the CC wavefunction (Equation 18), the sum canbe interpreted as a sum over all allowed partitions of the given Slater determinant into the elec-tron pairs. The signature of the permutation, sgn( σ m ... m N/ ), results from reordering the creationoperators in the electron pairs to the same order as in the given Slater determinant: Y i ∈ m a † i = sgn( σ m ... m N/ ) N/ Y p =1 A † m p (24)Altogether, the APG wavefunction is reformulated as | Ψ APG i = X m ∈ S FCI m X { m ... m N/ }⊆ ˜ S sgn A † m ...A † m N/ | i = | m i sgn( σ m ... m N/ ) | C ( m , . . . , m N/ ) | + | m i (25)where S FCI m is a set of all possible Slater determinants (i.e. Slater determinants of a FCI wavefunc-tion). The derivation is given in the Appendix 8.2.The Antisymmetrized Product of Interacting Geminals (APIG) is a special case of the APGwavefunction such that only the electron pairs within the same spatial orbital, i.e. doubly occupiedspatial orbitals, are used to build the wavefunction[102]. The sum over the partitions reduces to asingle element because there is only one way to construct a given (seniority-zero) Slater determinantfrom electron pairs of doubly occupied orbitals. | Ψ APIG i = X m ∈ S DOCI m | C ( m ) | + | m i (26)10here S DOCI m is the set of all seniority-zero (no unpaired electrons) Slater determinants, and | C ( m ) | + is a permanent of the parameters that correspond to the spatial orbitals used to construct m . TheAPIG wavefunction can be further simplified by imposing structures onto the permanent: theAntisymmetrized Product of 1-reference Orbitals Geminals (AP1roG) wavefunction assumes thata large portion of the coefficient matrix is an identity matrix [103]; and the AntisymmetrizedProduct of rank-2 Geminals (APr2G) wavefunction assumes that the coefficient matrix is a Cauchy matrix[104]. APr2G reduces the cost of evaluating a permanent ( O ( n !)) to that of a determinant( O ( n )). AP1roG has the cost of O ( m !) where m is the order of excitation with respect to thereference Slater determinant. It is cheap to evaluate the overlap of the AP1roG wavefunction withlow-order excitations of the reference determinant. The CI, CC, TPS, and APG wavefunctions and their variants can be expressed within theFANCI framework using different S , ~P , and f . We can define a multideterminant wavefunction asa function that has a well-defined overlap with a set of orthonormal Slater determinants. Providedthat the wavefunction exists within the space spanned by the Slater determinants, the wavefunctioncan be re-expressed as a linear combination of Slater determinants via a projection: | Ψ( ~P ) i = X m ∈ S m | m i h m | Ψ( ~P ) i = X m ∈ S m f ( m , ~P ) | m i (27)where f ( m , ~P ) = D m (cid:12)(cid:12)(cid:12) Ψ( ~P ) E (28)Therefore, all multideterminant wavefunctions, as defined above, can be expressed within the frame-work of Equation 9: S is the minimal set of Slater determinants required to fully describe thewavefunction; ~P is the parameters of the wavefunction; and f is the overlap of the wavefunctionwith the Slater determinant, m . 4. Characteristics In the formulation of Equation 9, a multideterminant wavefunction is defined using only a spec-ified (sub)set of Slater determinants, S , wavefunction parameters, ~P , and function f . Since the11haracteristics of a wavefunction ansatz depend on its structure, all characteristics of a multideter-minant wavefunction can be deduced from the specified S , ~P , and f . Designing a wavefunction withdesirable characteristics, therefore, merely requires selecting S , ~P , and f . We propose to approach method development in electronic structure theory as a search for S , ~P , and f that induce thedesired wavefunction features. While many features can be considered, and we shall consider addi-tional features in future work, here we shall address just three important characteristics: accuracy,cost, and size-consistency. Ultimately, the FANCI wavefunction models the FCI wavefunction by parameterizing the weightsof each Slater determinant. If there are Slater determinants absent from the FANCI wavefunction,i.e. S ⊂ S FCI , then the omitted Slater determinants are assumed to have no contributions to theFCI wavefunction. The effects of S can be viewed as a modification of the parameterizing function. | Ψ FANCI i = X m ∈ S FCI g ( m , ~P ) | m i where g ( m , ~P ) = f ( m , ~P ) ; m ∈ S m S Alternatively, the FANCI wavefunction can be viewed as a model for the CI wavefunction builtusing the same restricted set of Slater determinants. In either case, preventing Slater determinantsfrom contributing to the wavefunction will cause deviations from the FCI wavefunction.As with any parameterization (or fitting) problem, it becomes easier to find a function thataccurately describes each weight as the number of parameters increase. FANCI wavefunctions cannot be exact, in general, unless the number of parameters is greater than or equal to the numberof parameters in the Hamiltonian which, in the case of electronic structure theory, means thereshould be at least as many parameters as there are two-electron integrals[115, 116]. Methods withmany fewer parameters are, typically, static correlation methods. On the other hand, appropriatelyconstructed FANCI ans¨atze should approach the FCI limit as the number of parameters approaches the number of Slater determinants. However, the cost associated with optimizing the wavefunctiontypically increases superlinearly as the number of parameters increases.12 .2. Cost The cost associated with a wavefunction can be divided into the cost of its storage, evaluation,and optimization, all of which are intricately linked. The cost of storage is associated with thenumber of parameters needed to describe the wavefunction. The cost of evaluating the wavefunctiondepends on the cost of evaluating f and on the number of times f needs to be evaluated. Forexample, in order to evaluate the norm of a wavefunction, f must be evaluated for every Slaterdeterminant in S . h Ψ | Ψ i = X m ∈ S X n ∈ S f ∗ ( m ) h m | n i f ( n )= X m ∈ S f ∗ ( m ) f ( m )Upon optimization, a new set of parameters are found such that the wavefunction satisfies theSchr¨odinger equation. ˆ H | Ψ i = E | Ψ i (29)Equation 29 is often rewritten in its variational form or its projected form to make it easier to solvenumerically. The optimization procedure and the associated costs depend on the equations that are being solved.The variational Schr¨odinger equation involves integrating both sides of Equation 29 with thewavefunction[1]. h Ψ | ˆ H | Ψ i = E h Ψ | Ψ i X m , n ∈ S f ∗ ( m , ~P ) h m | ˆ H | n i f ( n , ~P ) = E X m ∈ S f ∗ ( m , ~P ) f ( m , ~P ) (30)If the number of Slater determinants in S is comparable to those in the FCI wavefunction, evensetting up Equation 30 will require far too many evaluations of f to be computationally tractable.During the optimization, all terms need to be evaluated at each step, where the number of stepsneeded for convergence varies depending on the system and the optimization algorithm. The projected Schr¨odinger equation can be obtained from Equation 29 using the resolution of13dentity: ˆ H | Ψ i = E | Ψ i X m ∈ S FCI | m i h m | ! ˆ H | Ψ i = E X m ∈ S FCI | m i h m | ! | Ψ i X m ∈ S FCI | m i (cid:16) h m | ˆ H | Ψ i − E h m | Ψ i (cid:17) = 0 (31)Since the Slater determinants are linearly independent, this equation will hold only if h m | ˆ H | Ψ i = E h m | Ψ i for every Slater determinant m . These equations are expressed as a system of equations: h m | ˆ H | Ψ i − E h m | Ψ i = 0... h m M | ˆ H | Ψ i − E h m M | Ψ i = 0 (32)Essentially, the Schr¨odinger equation (Equation 29) is broken apart into separate equations foreach contributing Slater determinant. If the projection operator is not complete (i.e. contributionsfrom certain Slater determinants are discarded) then the equation (or system of equations) will bean approximation of the original Equation 29.In general, the Schr¨odinger equation can be expressed with respect to arbitrary function, Φ[85]. h Φ | ˆ H | Ψ i = E h Φ | Ψ i (33)If Φ is not Ψ, then certain components of Ψ may be projected out, imposing additional structure on the wavefunction through the optimization process. We can express this projection explicitly witha projection operator onto a set of basis functions. When this basis set is complete, the projectedSchr¨odinger equation is equivalent to the variational formulation.Therefore, we can interpret the projected Schr¨odinger equation as an approximation to the vari-ational formulation that reduces it to a set of functions that capture the important characteristicsof the wavefunction. The cost of evaluating Equation 32 and 33 depends on the functions ontowhich the Schr¨odinger equation is projected. Some wavefunction structures have special functionssuch that Equation 32 or 33 can be evaluated cheaply. For example, the CC wavefunctions areoften projected against h Φ HF | exp( − ˆ T ) because h Φ HF | exp( − ˆ T ) ˆ H exp( ˆ T ) | Φ HF i , can be simplifiedusing the Baker-Campbell-Hausdorff expansion[117]. For a general FANCI wavefunction, however,14t is convenient to project onto a set of Slater determinants, { m . . . m M } , obtaining a system of(generally nonlinear) equations to solve: X m ∈ S f ( m , ~P ) h m | ˆ H | m i = Ef ( m , ~P )... X m ∈ S f ( m , ~P ) h m M | ˆ H | m i = Ef ( m M , ~P ) (34)In order to find a solution, the number of equations in the system of equations must be greater thanthe number of unknowns. Since the number of possible Slater determinants grows exponentially, there will not be a shortage of equations (Slater determinants), and the number of equations willalmost always be greater than the number of unknowns. If the least-squares solution of the nonlinearequations is found, then the residual can be used to measure the error associated with the optimizedwavefunction (and energy).Unless there is a special algorithm that limits the number of evaluated terms in the variational Schr¨odinger equation (Equation 30) or a function Φ that allows cheap integration of the Schr¨odingerequation (Equation 33), a wavefunction should be evaluated using the projected Schr¨odinger equa-tion (Equation 32) to control the optimization process. Both the cost and accuracy of the wavefunc-tion can be controlled; as the number of projections increases, both accuracy and cost increases.In addition, we can impose symmetry on the wavefunction by projecting the Schr¨odinger equation onto a space that satisfies a particular symmetry [99, 8, 118, 119, 120]. For example, we can rein-troduce particle number symmetry onto a number-symmetry broken wavefunction by projecting itonto Slater determinants with the selected particle number. Let there be a system, AB , composed of two non-interacting subsystems, A and B . Then, awavefunction is size-consistent if the energy of the wavefunction for AB is the sum of the energiesof the wavefunctions for A and B , [117] i.e. H AB | Ψ AB i = E AB | Ψ AB i = ( E A + E B ) | Ψ AB i (35)15here H A | Ψ A i = E A | Ψ A i H B | Ψ B i = E B | Ψ B i (36)Since subsystems A and B are non-interacting, there are no nonzero terms in the Hamiltonian thatcouple A and B , i.e. H AB = H A + H B . Then, the (partially symmetry broken) wavefunction Ψ AB can be written as a product of the (orthogonal) subsystem wavefunctions, Ψ A and Ψ B . | Ψ AB i = | Ψ A i | Ψ B i Similarly, a FANCI wavefunction is size-consistent if the weight function is multiplicativelyseparable: | Ψ AB i = X m ∈ S AB f AB ( m , ~P ) | m i = X m A ∈ S A X m B ∈ S B f A ( m A , ~P A ) f B ( m B , ~P B ) | m A i | m B i = X m A ∈ S A f A ( m A , ~P A ) | m A i X m B ∈ S B f B ( m B , ~P B ) | m B i = | Ψ A i | Ψ B i (37)where subscripts A and B designate that the quantity belongs only to subsystem A and B , re- spectively. This not only requires that f be multiplicatively separable f , i.e. f AB = f A f B , butalso that the Slater determinants, m , and the parameters, ~P , must be divisible into two disjointparts, { m A , m B } and { ~P A , ~P B } respectively. To separate each Slater determinant into the sub-systems, m must be expressed using orbitals localized to each subsystem. Since each m can havevarying contribution from the subsystems A and B , S A and S B contain Slater determinants with different numbers of electrons. However, we can impose the particle number symmetry duringthe optimization process. Similarly, the parameters must represent quantities that are specific toeach subsystem. Notice that f AB = f A f B is true when f is a determinant (Hartree-Fock), ex-ponential (Coupled-Cluster), or permanent (geminals). Note that this is a simplified approach tosize-consistency and a more rigorous approach requires ensuring that the systems have the correct symmetries upon dissociation. 16 . Ans¨atze In the formulation of Equation 9, a multideterminant wavefunction can be entirely expressedwith a set of Slater determinants, S , parameters, ~P , and a weight function, f . Altering the S , ~P ,or f of a given ansatz will effectively result in a new ansatz. Additionally, the optimization method can be modified to produce an “ansatz” with a different accuracy and cost. For example, DMRG isan algorithm for optimizing MPS[121]. Here, we modify the FANCI forms of the CC (Equation 20),TPS (Equation 21), and APG (Equation 25) wavefunctions to construct several new wavefunctionstructures. Just as the TPS and APG wavefunctions are expressed with respect to creation operators, wecan replace the excitation operators in the CC wavefunction with creation operators. | Ψ CC i = exp X b i C b i ˆ A † b i + X f i C f i ˆ A † f i ! | i (38)where A † b i is a creation operator of even number of electrons (denoted as even-electron), A † f i is acreation operator of odd number of electrons (denoted as odd-electron), b i is the set of orbitalscreated by A † b i , and f i is the set of orbitals created by A † f i . For a consistent notation, we define˜ S b and ˜ S f as the set of allowed creation operators. Since each creation operator can create adifferent number of electrons, the total number of operators needed for m can vary depending onthe selection of creation operators. For a given Slater determinant, let n b be the number of even-electron creators and n f be the number of odd-electron creators. Similar to the APG wavefunction,there can be multiple combinations of creation operators that give the same Slater determinant.We represent these combinations as a sum over { b . . . b n b f . . . f n f } such that the product of theassociated creators results in the given Slater determinant: Y i ∈ m a † i = sgn (cid:16) σ ( ˆ A † b . . . ˆ A † b nb ˆ A † f . . . ˆ A † f nf ) (cid:17) ˆ A † b . . . ˆ A † b nb ˆ A † f . . . ˆ A † f nf (39)Similar to the signature in the CC wavefunction, sgn (cid:16) σ ( ˆ A † b . . . ˆ A † b nb ˆ A † f . . . ˆ A † f nf ) (cid:17) is the signa-ture resulting from reordering the one-electron creators into the same order as the given Slaterdeterminant. 17n the CC wavefunction, all of the excitation operators commute with one another. Accountingfor all possible orderings of the operators results in a permanent of the parameters with identicalrows, i.e. (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t a i . . . t a N i N ... . . . ... t a i . . . t a N i N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + Similarly, if all of the creation operators commute with one another, i.e. they are all even-electroncreators, then | Ψ i = X m ∈ S m X { b ... b nb }⊆ ˜ S b sgn ˆ A † b ... ˆ A † b nb | i = | m i sgn (cid:16) σ ( ˆ A † b . . . ˆ A † b nb ) (cid:17) n b Y i =1 C b i | m i (40)where S m = Y ˆ A † k ∈ T ˆ A † k | i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T ⊆ ˜ S b For systems with an odd number of electrons, there must be at least one odd-electron creator.The anticommutation between these creators results in a determinant. Unlike the permanent, thedeterminant of a matrix with identical rows is zero. Therefore, there must be one odd-electroncreator within a set of creation operators that construct m . | Ψ i = X m ∈ S m X { b ... b nb }⊆ ˜ S b , f ∈ ˜ S f sgn ˆ A † b ... ˆ A † b nb ˆ A † f | i = | m i sgn (cid:16) σ ( ˆ A † b . . . ˆ A † b nb ˆ A † f ) (cid:17) n b Y i =1 C b i ! C f | m i (41)The derivation is provided in the Appendix 8.5.In the case where only two-electron creation operators are used, this wavefunction reduces to theAntisymmetrized Geminal Power (AGP)[91], HF-Bogoliubov[122], or the BCS superconducting[123]18avefunction (Equation 42). | Ψ i = exp X ij c ij a † i a † j | i = X m ∈ S m X { m ... m N/ }⊆ ˜ S sgn A † m ...A † m N/ | i = | m i sgn( σ m ... m N/ ) N/ Y k =1 C m k | m i (42)Notice that this type of coupled-cluster wavefunction is not size consistent and that it breaks particle number symmetry (unless it is restored with a projection onto correct particle number). In the TPS wavefunction (Equation 22), each parameter in ( M k ) n k describes the correlationbetween a spatial orbital, k , and all of the other orbitals. In the APG wavefunction (Equation 7)and CC-motivated quasiparticle wavefunction (Equation 41), each parameter is associated with acluster of spatial orbitals (quasiparticle). Then, we should be able to build a TPS-like wavefunctionusing creation operators of an arbitrary number of electrons, rather than the one-electron creationoperators. | Ψ i = X m ∈ S m X { m ... m n }⊆ ˜ S sgn A † m ...A † m n | i = | m i sgn( σ m ... m n ) K k ∈ ˜ S ( M k ) δ ( k , { m ... m n } ) | m i (43)where S m = Y ˆ A † k ∈ T ˆ A † k | i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T ⊆ ˜ S M k is a tensor that corresponds to the creation operator ˆ A † k , and δ is a function that checks if thecreator ˆ A † k is in a set of creators, { m . . . m n } . δ ( k , { m . . . m n } ) = k ∈ { m . . . m n } k 6∈ { m . . . m n } M k ) and ( M k ) are tensors that correspond to the absence and presence of the creator ˆ A † k ,respectively. Just as the CC wavefunction can be rebuilt with creation operators, we can rebuild the TPSwavefunction with excitation operators. Each tensor, t ˆ E ai , can be associated with ˆ E ai , and hasauxiliary indices that describe the correlation between operators. | Ψ i = X ˆ E ai ∈ S ˆ E X { ˆ E a i ... ˆ E a n i n }⊆ ˜ S ˆ E sgn Q nk =1 ˆ E a k i k = ˆ E ai sgn( σ ˆ E a i ... ˆ E a n i n ) K ˆ E k ∈ ˜ S ˆ E ( t ˆ E k ) δ ( ˆ E k , { ˆ E a i ... ˆ E a n i n } ) ˆ E ai | Φ HF i (44)where S ˆ E = Y ˆ E k ∈ T ˆ E k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T ⊆ ˜ S ˆ E and δ describes the presence of an excitation operator, ˆ E k , in the given set, { ˆ E a i . . . ˆ E a n i n } δ ( ˆ E k , { ˆ E a i . . . ˆ E a n i n } ) = E k ∈ { ˆ E a i . . . ˆ E a n i n } E k 6∈ { ˆ E a i . . . ˆ E a n i n } (45)Comparing Equation 44 with Equation 20, the CC wavefunction can be considered a specialcase of this wavefunction, where the tensor ( t ˆ E k ) δ k is 1 if δ k = 0 and a variable scalar value if δ k = 1. Just as the TPS wavefunction can be built with excitation operators (Equation 44), we canrewrite the APG wavefunction with excitation operators. | Ψ i = n Y p =1 X ˆ E k ∈ ˜ S ˆ E t p ; ˆ E k ˆ E k | Φ HF i (46)where n is the number of excitation operators that will be multiplied together. Unlike the CCwavefunction, which can generate all possible combinations of the excitation operators, this wave-function can only account for the combinations of n excitation operators. The corresponding weight20unction in the FANCI notation is | Ψ i = X ˆ E ai ∈ S ˆ E X { ˆ E a i ... ˆ E a n i n }⊆ ˜ S ˆ E sgn Q nk =1 ˆ E k = ˆ E ai sgn( σ ˆ E a i ... ˆ E a n i n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t 1; ˆ E a i . . . t 1; ˆ E a n i n ... . . . ... t n ; ˆ E a i . . . t n ; ˆ E a n i n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + ˆ E ai | Φ HF i (47)where S ˆ E = Y ˆ E k ∈ T ˆ E k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T ⊆ ˜ S ˆ E and sgn( σ ˆ E a i ... ˆ E a n i n ) is the signature of the permutation of the one-electron creation and annihilationoperators from the product of excitation operators, ˆ E a i · · · ˆ E a n i n , to the given excitation operator,i.e. ˆ E ai = sgn( σ ˆ E a i ... ˆ E a n i n ) ˆ E a i · · · ˆ E a n i n (48)Other combinations of excitation operators can be included into Equation 46 via the summation. | Ψ i = X n ∈ P n ! n Y p =1 X ˆ E k ∈ ˜ S ˆ E t p ; ˆ E k ˆ E k | Φ HF i (49)where P is the allowed number of excitation operators that can be combined. Recall that removingan index (so that all the geminals are identical) in the traditional APG wavefunction form leads tothe AGP wavefunction. Similarly, when one removes the index, p , from the excitation-based APGwavefunction form and allows all possible combinations of excitation operators, one obtains the CCwavefunction: | Ψ CC i = ∞ X n =0 n ! n Y p =1 X ˆ E k ∈ ˜ S ˆ E t ˆ E k ˆ E k | Φ HF i = ∞ X n =0 n ! X ˆ E k ∈ ˜ S ˆ E t ˆ E k ˆ E k n | Φ HF i = exp X ˆ E k ∈ ˜ S ˆ E t ˆ E k ˆ E k | Φ HF i (50) In the Equation 41, any creation operator can be used (within the sets ˜ S b and ˜ S f ) to construct aSlater determinant. Similarly, we can generalize the APG wavefunction to include all even-electron21reation operators. | Ψ i = n Y p =1 X ˆ A † k ∈ ˜ S b C p ; m k ˆ A † k | i (51)where n is the number of quasiparticles in the wavefunction (i.e. number of operators used toconstruct a Slater determinant), and ˜ S b is a set of even-electron creation operators. In the FANCIformulation, | Ψ i = X m ∈ S m X { m ... m n }⊆ ˜ S b sgn A † m ...A † m n | i = | m i sgn( σ m ... m n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C m . . . C m n ... . . . ... C n ; m . . . C n ; m n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + | m i (52)where ˜ S b is a set of allowed even-electron creation operators, S m = Y ˆ A † k ∈ T ˆ A † k | i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T ⊆ ˜ S b and sgn( σ m ... m n ) is the signature of the permutation needed to reorder the one-electron creationoperators from the product of quasiparticle creation operators to the given Slater determinant. Y i ∈ m a † i = sgn( σ m ... m n ) n Y k =1 Y i ∈ m k a † i (53)Note that the zero-electron creation operator can be considered as an even-electron creation oper-ator.Similarly, the wavefunction constructed using only odd-electron creation operator can be ex-pressed using a determinant. | Ψ i = X m ∈ S m X { m ... m n }⊆ ˜ S f sgn A † m ...A † m n | i = | m i sgn( σ m ... m n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C m . . . C m n ... . . . ... C n ; m . . . C n ; m n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − | m i (54)where S m = Y ˆ A † k ∈ T ˆ A † k | i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T ⊆ ˜ S f and ˜ S f is a set of selected odd-electron creation operators.22hen both even and odd-electron creation operators are present in ˜ S , the interchange of creatorscommutes or anticommutes depending on the pair of creators, and additional structure is necessaryto account for this behaviour. First, we distinguish between the set of even and odd orbitalswith b i and f j , respectively. A given Slater determinant m is constructed with n = n b + n f creation operators, where n b is the number of even-electron creators and n f is the number ofodd-electron creators. A permanent is needed to account for all possible ordering of the even-electron creators and the resulting commutations. A determinant is needed to account for allpossible ordering of the odd-electron creators and the resulting anticommutations. Finally, thecommutation between an even-electron and an odd-electron creator can be accounted for by a sumover all possible selections of the n b even-electron creators from n positions, i.e. { i b . . . i bn b } ⊆{ b . . . b n b f . . . f n f } . The positions of the odd-electron creators are the remaining n f positions, i.e. { i f . . . i fn f } = { b . . . b n b f . . . f n f }\{ i b . . . i bn b } . Then, we can construct the generalized quasiparticlewavefunction (Equation 51) in which any set of creators can be used. | Ψ i = X m X { b ... b nb }⊆ ˜ S b , { f ... f nf }⊆ ˜ S f sgn ˆ A † b ... ˆ A † b nb ˆ A † f ... ˆ A † f nf | i = | m i sgn( σ ˆ A † b ... ˆ A † b nb ˆ A † f ... ˆ A † f nf ) X { i b ...i bnb }⊆{ b ... b nb , f ... f nf }{ i f ...i fnf } = { b ... b nb , f ... f nf }\{ i b ...i bnb } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C b i b . . . C b i bnb ... . . . ... C bn b i b . . . C bn b i bnb (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C f i f . . . C f i fnf ... . . . ... C fn f i f . . . C fn f i fnf (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − | m i (55)where S m = Y ˆ A † k ∈ T ˆ A † k | i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T ⊆ ˜ S b ∪ ˜ S f Y i ∈ m a † i = sgn( σ ˆ A † b ... ˆ A † b nb ˆ A † f ... ˆ A † f nf ) n b Y k =1 A † b k n f Y l =1 A † f l and C is a n × dim( ˜ S b ∪ ˜ S f ) matrix. Note that the set of orbitals, b i and f j , correspond to a column in C and thus are used as a column index. Here, the column indices are made explicit using theindex i .The derivation and more details are in the Appendix 8.6. Often times, the algorithm for optimizing the parameters is synonymous with the wavefuncton ansatz. For example, DMRG is often associated with MPS wavefunctions, and Quantum Monte-23arlo (QMC) is often associated with FCI wavefunctions. Using different algorithms to optimizethe parameters will change the cost and the reliability of the wavefunction. Certain algorithms canbe applied to a wide range of wavefunction structures and systems; whereas specialized algorithmsare cheaper, but are often limited to specific wavefunction structures and systems. For example, the DMRG algorithm with the MPS wavefunction provides variational results andis effective in describing linear systems. The algorithm is specific to the MPS wavefunction, whereinthe orbitals are ordered such that adjacent orbitals are more correlated than the rest. As a result,DMRG must be extended beyond MPS to more general TPS forms, but these algorithms do notseem to be as elegant or as computationally efficient. Solving the TPS wavefunction as a projected Schr¨odinger equation mitigates certain complications that are present in DMRG, such as orderingof the orbitals and generalizing a one-dimensional algorithm to multiple dimensions. It is true,however, that the tensor structures that are most efficiently optimized in a variational ansatz are,at least in all cases we have considered, the same as those that are most efficiently optimized usingthe projected Schr¨odinger equation. Above, we constructed new wavefunction structures by constructing the CC wavefunction withcreation operators and the TPS and APG wavefunctions using excitation operators. Effectively,the parameters are changed from the contributions of creation operators to those of excitationoperators, and vice versa. Since both operators originate from orbitals, the wavefunctions are size-consistent if f is selected appropriately, provided the orbitals are localized. In Equation 41,52, 54, and 55, particle number symmetry is not conserved, which is not a problem since thewavefunction can be optimized and projected onto the appropriate particle number. Then, just asexcitation operators change the state of a reference Slater determinant, products of an arbitrarynumber of creation and annihilation operators can be used to explore different particle numbers with respect to the reference. For example, the product of creators and annihilators, where thenumber of creators exceeds the number of annihilators, will ionize a Slater determinant. Expressingthe wavefunction with respect to different operators will result in different sets of parameters, evenif these operators make no reference to the Slater determinants or creation operators. For example,grid-based wavefunctions will have parameters for each point in space. So far, we have seen weight functions, f , in various forms: the CC wavefunction (Equation 20)24ses a cumulant; the TPS wavefunction (Equation 21) uses a tensor product; the even-electronquasiparticle wavefunction (Equation 54) uses a permanent; the odd-electron quasiparticle wave-function (Equation 52) uses a determinant; and the generalized quasiparticle wavefunction (Equa-tion 55) uses some mix of a permanent and determinant (an immanent). In Equations 25, 41, 43, and 55, creators of arbitrary number of electrons are combined in fairly complicated manner.Since creators have distinct commutative (and anticommutative) relations with one another, thecorresponding f must be symmetric (and antisymmetric) with respect to interchange of differentcreation operators (or parameters). Grouping together all the terms that correspond to the sameset of creation operators results in a sum over all combinations of products of parameters. These combinatorial variants of product functions seem to be useful for size-consistency since they aremultiplicatively separable by construction. Therefore, we can construct novel wavefunctions withquasiparticle origins using different symmetric (or antisymmetric) polynomial functions, includingbut not limited to determinant, permanent, immanent[124], pfaffian [125], hafnian[126, 127], hy-perdeterminant [128], multidimensional permanent[129], hyperpfaffian[130], hyperhafnian[131], and mixed discriminant[132]. Unfortunately, among all the aforementioned size-consistent combinatoricforms, only the determinant can be evaluated in polynomial time. If we disregard size-consistencyand do not require any quasiparticle structure, then any function can be chosen. One alterna-tive, however, is to use a ratio of a product of determinants as the overlap; this form retainssize-consistency and computational feasibility, and generalizes a single Slater determinant (one de- terminant in the numerator, none in the denominator) and the APr2G wavefunction (a special casewith one structure determinant in the denominator and the determinant of its element-wise squarein the numerator). If the total number of determinants is odd, this wavefunction describes a normalfermionic wavefunction, while if the number of determinants is even, this wavefunction representsa bosonic seniority-zero structure. A great deal of flexibility is available within the proposed wavefunction framework: the set ofSlater determinants, S , can be any set of orthonormal (for convenience) Slater determinants; theparameters, ~P , can be any set of numbers that describe the wavefunction or the operators withwhich it is built; the parameterizing function, f , can be any function that maps the parametersto a coefficient for a given Slater determinant. We hope to find the combination that effectively models the optimized coefficients of the FCI wavefunction (for accuracy) using the minimal numberof parameters (for cost) for as many systems as possible (for generality).25 . Conclusion In the proposed framework, any multideterminant wavefunction can be expressed with respectto the components S , ~P , and f . A wavefunction can be characterized using only these compo- nents and different combinations will result in different wavefunction characteristics. Then, wecan systematically develop new structures by simply finding novel combinations. While the pro-posed wavefunctions are not necessarily cheap to compute, the general ansatz does not need to becheap to be effective. For example, the CC wavefunction with all orders of excitations, the TPSwavefunction with infinitely large tensors, and the quasiparticle wavefunctions with N electron quasiparticles are no less expensive than the FCI wavefunction. However, different wavefunctionstructures inspire different approximations and new algorithms that reduce the computational costto a tractable level. The greatest advantage to this perspective is that it is pragmatic: approxima-tions and algorithms that were restricted to one wavefunction can be generalized to others usingthe FANCI framework; and many different methods can be implemented computationally using a common framework. Though we will defer detailed discussions on dynamical correlation correc-tions and orbital optimization to future papers, these elaborations to FANCI are clearly possibleand are performed using similar techniques to what one would use in traditional selected CI andCC methods. Many of the proposed methods have been implemented in an open-source librarycalled Fanpy , which will be presented elsewhere. 7. Acknowledgements The authors thank NSERC, Compute Canada, McMaster University, and the Canada ResearchChairs for funding. RAMQ acknowledges financial support from the University of Florida in theform of a start-up grant. 26 eferences [1] S. Boys, Electronic Wave Functions. I. A General Method of Calculation for the StationaryStates of Any Molecular System, Proceedings of the Royal Society of London A 200 (1063)(1950) 542–554. doi:10.1098/rspa.1950.0036 .[2] T. Helgaker, P. Jørgensen, J. Olsen, Modern electronic structure theory, Wiley, Chichester,2000. [3] T. Helgaker, P. Jørgensen, J. Olsen, Modern electronic structure theory, Journal of PhysicalChemistry 100 (1996) 13213–13225.[4] K. Raghavachari, J. Anderson, Electron correlation effects in molecules, Journal of PhysicalChemistry 100 (1996) 12960–12973.[5] K. Marti, M. Reiher, New electron correlation theories for transition metal chemistry, Physical Chemistry Chemical Physics 13 (2011) 6750–6759.[6] T. Helgaker, S. Coriani, P. Jørgensen, K. Kristensen, J. Olsen, K. Ruud, Recent Advancesin Wave Function-Based Methods of Molecular-Property Calculations, Chemical Reviews 112(2012) 543–631.[7] C. Sherrill, Frontiers in electronic structure theory, Journal of Chemical Physics 132 (2010) doi:10.1063/1.3643338 .[9] G. Chan, S. Sharma, The Density Matrix Renormalization Group In Quantum Chemistry, Annu. Rev. Phys. Chem. 62 (2011) 465–81. doi:10.1146/annurev-physchem-032210-103338 .[10] D. Small, M. Head-Gordon, Post-modern valence bond theory for strongly corre-lated electron spins, Physical Chemistry Chemical Physics 13 (2011) 19285–19297. doi:10.1039/C1CP21832H . 143 (2015) 094105.[13] M. Van Raemdonck, D. Alcoba, W. Poelmans, S. De Baerdemacker, A. Torre, L. Lain, G. Mas-saccesi, D. Van Neck, P. Bultinck, Polynomial scaling approximations and dynamic correlationcorrections to doubly occupied configuration interaction wave functions, Journal of ChemicalPhysics 143 (2015) 104106. [14] D. Alcoba, A. Torre, L. Luis, O. O˜na, P. Capuzzi, M. Van Raemdonck, P. Bultinck,D. Van Neck, A hybrid configuration interaction treatment based on seniority number andexcitation schemes, Journal of Chemical Physics 141.[15] D. Alcoba, A. Torre, L. Luis, G. Massaccesi, O. O˜na, P. Ayers, M. Van Raemdonck, P. Bult-inck, D. Van Neck, Performance of Shannon-entropy compacted N-electron wave functions for configuration interaction methods, Theoretical Chemistry Accounts 135 (2016) 153.[16] R. Carbo, J. Hernandez, General multiconfigurational paired excitation self-consistent field-theory (MC PE SCF), Chemical Physics Letters 47 (1977) 85–91.[17] C. Kollmar, B. Hess, A new approach to density matrix functional theory, Journal of ChemicalPhysics 119 (2003) 4655–4661. [18] C. Kollmar, A size extensive energy functional derived from a double configuration interactionapproach: The role of N representability conditions, Journal of Chemical Physics 125.[19] D. Cook, Doubly-occupied orbital MCSCF methods, Molecular Physics 30 (1975) 733–743.[20] A. Veillard, E. Clementi, Complete multi-configuration self-consistent field theory, TheoreticaChimica Acta 7 (1967) 134–143. [21] C. Roothaan, J. Detrich, D. Hopper, An improved MCSCF method, International Journal ofQuantum Chemistry S13 (1979) 93–101. 2822] F. Weinhold, E. Wilson Jr, Reduced Density Matrices of Atoms and Molecules. I. The 2Matrix of Double-Occupancy, Configuration-Interaction Wavefunctions for Singlet States, TheJournal of Chemical Physics 46 (7) (1967) 2752–58. doi:10.1063/1.1841109 . [23] J. Pople, R. Seeger, R. Krishnan, Variational Configuration Interaction Methods and Com-parison with Perturbation Theory, International Journal of Quantum Chemistry: QuantumChemistry Symposium 11 (1977) 149–163.[24] B. Roos, P. Taylor, P. Siegbahn, A complete active space SCF method (CASSCF) usinga density matrix formulated super-CI approach, Chemical Physics 48 (2) (1980) 157–173. doi:10.1016/0301-0104(80)80045-0 .[25] J. Olsen, B. Roos, Determinant based configuration interaction algorithms for complete andrestricted configuration interaction spaces, The Journal of Chemical Physics 89 (4) (1988)2185–92. doi:10.1063/1.455063 .[26] M. Schmidt, M. Gordon, The construction and interpretation of MCSCF wavefunctions, Annual Review of Physical Chemistry 49 (1998) 233–266. doi:10.1146/annurev.physchem.49.1.233 .[27] G. Gallup, Valence bond methods: Theory and applications, Cambridge UP, Cambridge,2002.[28] S. Shaik, P. Hiberty, A chemist’s guide to valence bond theory, Wiley, Hoboken, 2008. [29] W. Wu, P. Su, S. Shaik, P. Hiberty, Classical Valence Bond Approach by Modern Methods,Chemical Reviews 111 (2011) 7557–7593.[30] W. Goddard III, T. Dunning Jr., W. Hunt, P. Hay, Generalized valence bond description ofbonding in low-lying states of molecules, Accounts of Chemical Research 6 (1973) 368–376.[31] R. McWeeny, The valence bond theory of molecular structure I. Orbital theories and the valence-bond method, Proceedings of Royal Society A 223 (1152) (1954) 63–79. doi:0.1098/rspa.1954.0100 .[32] J. Paldus, X. Li, Critical assessment of coupled cluster method in quantum chemistry, in:I. Prigogine, S. Rice (Eds.), Advances in Chemical Physics, Vol. 110, 1999, pp. 1–175.2933] J. Cizek, On the Correlation Problem in Atomic and Molecular Systems. Calculation of Wave- function Components in Ursell-Type Expansion Using Quantum-Field Theoretical Methods,Journal of Chemical Physics 45 (1966) 4256–4266.[34] I. Shavitt, R. Bartlett, Many-body methods in chemistry and physics: MBPT and coupled-cluster theory, Cambridge, Cambridge, 2009.[35] R. Bartlett, M. Musia l, Coupled-cluster theory in quantum chemistry, Reviews of Modern Physics 79 (1) (2007) 291–352. doi:10.1103/RevModPhys.79.291 .[36] F. A. Evangelista, G. K. L. Chan, G. E. Scuseria, Exact parameterization of fermionic wavefunctions via unitary coupled cluster theory, Journal of Chemical Physics 151 (2019) 244112.[37] F. A. Evangelista, Alternative single-reference coupled cluster approaches for multireferenceproblems: The simpler, the better, Journal of Chemical Physics 134 (2011) 224102. [38] V. Murg, F. Verstraete, R. Schneider, P. Nagy, O. Legeza, Tree Tensor Network State withVariable Tensor Order: An Efficient Multireference Method for Strongly Correlated Systems,Journal of Chemical Theory and Computation 11 (2015) 1027–1036.[39] O. Buerschaper, J. Mombelli, M. Christandl, M. Aguado, A hierarchy of topological tensornetwork states, Journal of Mathematical Physics 54 (2013) 012201. [40] N. Nakatani, G. Chan, Efficient tree tensor network states (TTNS) for quantum chemistry:Generalizations of the density matrix renormalization group algorithm, Journal of ChemicalPhysics 138 (2013) 134113.[41] H. Changlani, J. Kinder, C. Umrigar, G. Chan, Approximating strongly correlated wavefunctions with correlator product states, Physical Review B 80 (2009) 245116. [42] R. Ornus, A practical introduction to tensor networks: Matrix product statesand projected entangled pair states, Annals of Physics 349 (2014) 117–58. doi:10.1016/j.aop.2014.06.013 .[43] K. Gunst, F. Verstraete, S. Wouter, O. Legeza, D. Van Neck, T3ns: three-legged tree tensornetwork states, Journal of Chemical Theory and Computation 14 (2018) 2026–2033. arXiv:0407066 .[45] G. Vidal, Entanglement renormalization, Physical Review Letters 99 (2007) 220405.[46] V. Murg, F. Verstraete, O. Legeza, R. M. Noack, Simulating strongly correlated quantumsystems with tree tensor networks, Physical Review B 82 (2010) 205105. [47] N. Nakatani, G. K. L. Chan, Efficient tree tensor network states (ttns) for quantum chemistry:Generalizations of the density matrix renormalization group algorithm, Journal of ChemicalPhysics 138 (2013) 134113–134113.[48] V. Murg, F. Verstraete, R. Schneider, P. Nagy, O. Legeza, Tree tensor network state withvariable tensor order: an efficient multireference method for strongly correlated systems, Journal of Chemical Theory and Computation 11 (2015) 1027–1036.[49] K. H. Marti, B. Bauer, M. Reiher, M. Troyer, F. Verstraete, Complete-graph tensor networkstates: a new fermionic wave function ansatz for molecules, New Journal of Physics 12 (2010)103008.[50] A. Kovyrshin, M. Reiher, Self-adaptive tensor network states with multi-site correlators, Journal of Chemical Physics 147 (2017) 214111.[51] U. Schollw¨ock, The density-matrix renormalization group in the age of matrix product states,Annals of Physics 326 (2011) 96–192. doi:10.1016/j.aop.2010.09.012 .[52] J. Parks, R. Parr, Theory of Separated Electron Pairs, Journal of Chemical Physics 28 (1958)335–345. [53] T. Allen, H. Shull, Electron pairs in the Beryllium atom, Journal of Physical Chemistry 66(1962) 2281–2283.[54] P. Surjan, A. Szabados, P. Jeszenszki, T. Zoboki, Strongly orthogonal geminals: size-extensiveand variational reference states, Journal of Mathematical Chemistry 50 (2012) 534–551.[55] G. Chan, A. Keselman, N. Nakatani, Z. Li, S. White, Approximating strongly correlated wave functions with correlator product states, Journal of Chemical Physics 145 (2016) 014102.3156] S. Daul, I. Ciofini, C. Daul, S. White, Full-CI quantum chemistry using the density matrixrenormalization group, International Journal of Quantum Chemistry 79 (2000) 331–342.[57] S. White, R. Martin, Ab initio quantum chemistry using the density matrix renormalizationgroup, Journal of Chemical Physics 110 (1999) 4127–4130. [58] S. White, Density-Matrix Algorithms for Quantum Renormalization-Groups, Physical ReviewB 48 (1993) 10345–10356.[59] K. Marti, M. Reiher, The Density Matrix Renormalization Group Algorithm in QuantumChemistry, Zeitschrift Fur Physikalische Chemie-International Journal of Research in PhysicalChemistry & Chemical Physics 224 (2010) 583–599. [60] G. Chan, An algorithm for large scale density matrix renormalization group calculations,Journal of Chemical Physics 120 (2004) 3172–3178.[61] G. Chan, M. Head-Gordon, Highly correlated calculations with a polynomial cost algorithm:A study of the density matrix renormalization group, Journal of Chemical Physics 116 (2002)4462–4476. [62] S. Wouters, W. Poelmans, S. De Baerdemacker, P. Ayers, D. Van Neck, CHEMPS2: ImprovedDMRG-SCF routine and correlation functions, Computer Physics Communications 191 (2015)235–237.[63] S. Wouters, W. Poelmans, P. Ayers, D. Van Neck, CheMPS2: A free open-source spin-adaptedimplementation of the density matrix renormalization group for ab initio quantum chemistry, Computer Physics Communications 185 (2014) 1501–1514.[64] D. Zgid, M. Nooijen, The density matrix renormalization group self-consistent field method:Orbital optimization with the density matrix renormalization group method in the activespace, Journal of Chemical Physics 128 (2008) 114116.[65] G. K. L. Chan, J. J. Dorando, J. Hachmann, E. Neuscamman, H. Wang, T. Yanai, An introduction to the density matrix renormalization group ansatz in quantum chemistry, in:S. Wilson, P. J. Glout, G. Delgado-Barrio, P. Piecuch (Eds.), Frontiers in Quantum Systemsin Chemistry and Physics, Vol. 18 of Progress in Theoretical Chemistry and Physics, Springer,Dordrecht, 2008, pp. 49–65. 3266] A. Hurley, J. Lennard-Jones, J. Pople, The molecular orbital theory of chemical valency XVI, A theory of paired-electrons in polyatomic molecules Proceedings of the Royal Society ofLondon Series A 220 (1953) 446–455.[67] R. Parr, F. Ellison, P. Lykos, Generalized antisymmetric product wave functions for atomsand molecules, Journal of Chemical Physics 24 (1956) 1106.[68] R. McWeeny, B. Sutcliffe, The density matrix in many-electron quantum mechancs III. Gen- eralized product functions for Beryllium and Four-Electron Ions, Proceedings of the RoyalSociety of London Series A 273 (1963) 103–116.[69] P. Surjan, An introduction to the theory of geminals, in: P. Surjan (Ed.), Correlation andLocalization, 1999, pp. 63–88.[70] P. Tecmer, K. Boguslawski, P. Johnson, P. Limacher, M. Chan, T. Verstraelen, P. Ayers, Assessing the Accuracy of New Geminal-Based Approaches, Journal of Physical ChemistryA 118 (2014) 9058–9068.[71] J. Paldus, J. Cizek, S. Sengupta, Geminal Localization in the Separated-Pair π -ElectronicModel of Benzene, Journal of Chemical Physics 55 (1971) 2452–2462.[72] J. Paldus, S. Sengupta, J. Cizek, Diagrammatical Method for Geminals. II. Applications, Journal of Chemical Physics 57 (1972) 652–666.[73] H. Shull, Natural Spin Oribtal Analysis of Hydrogen Molecule Wave Functions, The Journalof Chemical Physics 30 (6) (1959) 1405–13. doi:10.1063/1.1730212 .[74] W. Kutzelnigg, Direct Determination of Natural Orbitals and Natural Expansion Coefficientsof Many-Electron Wavefunctions. I. Natural Orbitals in the Geminal Product Approximation, The Journal of Chemical Physics 40 (12) (1964) 3640–47. doi:10.1063/1.1730212 .[75] V. Rassolov, A geminal model chemistry, Journal of Chemical Physics 117 (2002) 5978–5987.[76] V. Rassolov, F. Xu, S. Garashchuk, Geminal model chemistry II. Perturbative corrections,Journal of Chemical Physics 120 (2004) 10385–10394.[77] V. Rassolov, F. Xu, Geminal model chemistry III: Partial spin restriction, Journal of Chemical Physics 126 (2007) 234112. 3378] P. Cassam-Chena¨ı, The electronic mean-field configuration interaction method. I. Theory andintegral formulas, Journal of Chemical Physics 124 (2006) 194109.[79] P. Cassam-Chena¨ı, V. Rassolov, The electronic mean field configuration interaction method:III - the p -orthogonality constraint, Chemical Physics Letters 487 (2010) 147–152. [80] P. Cassam-Chena¨ı, A. Ilmane, Frequently asked questions on the mean field configurationinteraction method. I-distinguishable degrees of freedom, Journal of Mathematical Chemistry50 (2012) 652–667.[81] T. Stein, T. Henderson, G. Scuseria, Seniority zero pair coupled cluster doubles theory, Jour-nal of Chemical Physics 140 (2014) 214113. [82] T. Henderson, G. Scuseria, J. Dukelsky, A. Signoracci, T. Duguet, Quasiparticle coupledcluster theory for pairing interactions, Physical Review C 89 (2014) 054305.[83] T. Henderson, I. Bulik, T. Stein, G. Scuseria, Seniority-based coupled cluster theory, Journalof Chemical Physics 141 (2014) 244104.[84] I. Bulik, T. Henderson, G. Scuseria, Can Single-Reference Coupled Cluster Theory Describe Static Correlation?, Journal of Chemical Theory and Computation 11 (2015) 3171–3179.[85] J. Cullen, Generalized valence bond solutions from a constrained coupled cluster method,Chemical Physics 202 (1996) 217–229. doi:10.1016/0301-0104(95)00321-5 .[86] K. Miller, K. Ruedenberg, Electron Correlation and Electron-Pair Wavefunction for the Beryl-lium Atom, Journal of Chemical Physics 43 (1965) S88–S90. [87] K. Miller, K. Ruedenberg, Electron Correlation and Separated-Pair Approximation. An Ap-plication to Berylliumlike Atomic Systems, Journal of Chemical Physics 48 (1968) 3414–3443.[88] D. Silver, E. Mehler, K. Ruedenberg, Electron Correlation and Separated Pair Approximationin Diatomic Molecules. I. Theory, Journal of Chemical Physics 52 (1970) 1174–1180.[89] E. Mehler, K. Ruedenberg, D. Silver, Electron Correlation and Separated Pair Approximation in Diatomic Molecules. II. Lithium Hydride and Boron Hydride, Journal of Chemical Physics52 (1970) 1181–1205. 3490] D. Silver, K. Ruedenberg, E. Mehler, Electron Correlation and Separated Pair Approximationin Diatomic Molecules. III. Imidogen, Journal of Chemical Physics 52 (1970) 1206–1227.[91] A. Coleman, Structure of Fermion Density Matrices. II. Antisymmeterized Geminal Powers, Journal of Mathematical Physics 6 (9) (1965) 1425–1431. doi:10.1063/1.1704794 .[92] A. Coleman, The AGP model for fermion systems, International Journal of Quantum Chem-istry 63 (1997) 23–30.[93] M. Bajdich, G. Drobn´y, L. Wagner, K. Schmidt, Pfaffian pairing wave functions in electronic-structure quantum Monte Carlo simulations, Physical Review Letters 96 (2006) 130201. [94] M. Bajdich, L. Mitas, L. Wagner, K. Schmidt, Pfaffian pairing and backflow wavefunctionsfor electronic structure quantum Monte Carlo methods, Physical Review B 77 (2008) 115112.[95] K. Pernal, Intergeminal Correction to the Antisymmetrized Product of Strongly OrthogonalGeminals Derived from the Extended Random Phase Approximation, Journal of ChemicalTheory and Computation 10 (2014) 4332–4341. [96] E. Pastorczak, K. Pernal, ERPA-APSG: a computationally efficient geminal-based methodfor accurate description of chemical systems, Physical Chemistry Chemical Physics 17 (2015)8622–8626.[97] P. Limacher, T. Kim, P. Ayers, P. Johnson, S. De Baerdemacker, D. Van Neck, The influenceof orbital rotation on the energy of closed-shell wavefunctions, Molecular Physics 112 (2014) wavefunctions: Open shells and beyond, Computational and Theoretical Chemistry 1116(2017) 207–219.[100] K. Boguslawski, P. Tecmer, P. Ayers, P. Bultinck, S. De Baerdemacker, D. Van Neck, Efficientdescription of strongly correlated electrons with mean-field cost, Physical Review B 89 (2014)201106. doi:10.1063/1.1671025 . [103] P. Limacher, P. Ayers, P. Johnson, S. De Baerdemacker, D. Van Neck, P. Bultinck, A NewMean-Field Method Suitable for Strongly Correlated Electrons: Computationally Facile An-tisymmetric Products of Nonorthogonal Geminals, Journal of Chemical Theory and Compu-tation 9 (3) (2013) 1394–1401. doi:10.1021/ct300902c .[104] P. Johnson, P. Ayers, P. Limacher, S. De Baerdemacker, D. Van Neck, P. Bultinck, A size- consistent approach to strongly correlated systems using a generalized antisymmetrized prod-uct of nonorthogonal geminals, Computational and Theoretical Chemistry 1003 (2013) 101–13. doi:10.1016/j.comptc.2012.09.030 .[105] J. Slater, The self-consistent field and the structure of atoms, Physical Review 32 (1928)339–348. [106] V. Fock, N¨aherungsmethode zur L¨osung des quantenmechanischen Mehrk¨orperproblems,Zeitschrift f¨ur Physik 61 (1-2) (1930) 126–148. doi:10.1007/BF01340294 .[107] P. L¨owdin, Quantum Theory of Many-Particle Systems. II. Study of the Or-dinary Hartree-Fock Approximation, Physical Review 97 (6) (1955) 1490–1508. doi:10.1103/PhysRev.97.1490 . [108] R. Nesbet, Configuration interaction in orbital theories, Proceedings of the Royal Society A230 (1182) (1955) 312–321. doi:10.1098/rspa.1955.0134 .[109] G. Purvis, B. R.J., A full coupled-cluster singles and doubles model: The inclu-sion of disconnected triples, The Journal of Chemical Physics 76 (4) (1982) 1910–18. doi:10.1063/1.443164 . [110] W. Kutzelnigg, D. Mukherjee, Normal order and extended Wick theorem for a multiconfigu-ration reference wave function, Journal of Chemical Physics 107 (1997) 432–449.36111] W. Kutzelnigg, Density-cumulant functional theory, Journal of Chemical Physics 125.[112] D. Sinha, R. Maitra, D. Mukherjee, Generalized antisymmetric ordered products, generalizednormal ordered products, ordered and ordinary cumulants and their use in many electron correlation problem, Computational and Theoretical Chemistry 1003 (2013) 62–70.[113] T. Crawford, H. Schaefer, An introduction to coupled cluster theory for computationalchemists, in: K. Lipkowitz, D. Boyd (Eds.), Reviews in Computational Chemistry, Vol. 14,2000, pp. 33–136.[114] K. Becker, M. Vojta, Cumulant approach and coupled-cluster method for many-particle sys- tems, Molecular Physics 94 (1998) 217–223.[115] W. Kutzelnigg, Separation of strong (bond-breaking) from weak (dynamical) correlation,Chemical Physics 401 (2012) 119–124.[116] W. Kutzelnigg, D. Mukherjee, Minimal parametrization of an n-electron state, Physical Re-view A 71 (2005) 022502. [117] M. Nooijen*, K. Shamasundar, D. Mukherjee, Reflections on size-extensivity, size-consistencyand generalized extensivity in many-body theory, Molecular Physics 103 (15-16) (2005) 2277–2298.[118] Y. Qiu, T. M. Henderson, G. E. Scuseria, Projected hartree-fock theory as a polynomial ofparticle-hole excitations and its combination with variational coupled cluster theory, Journal of Chemical Physics 146 (2017) 184105.[119] Y. Qiu, T. M. Henderson, G. E. Scuseria, Communication: Projected hartree fock theoryas a polynomial similarity transformation theory of single excitations, Journal of ChemicalPhysics 145 (2016) 111102.[120] C. A. Jim´enez-Hoyos, T. M. Henderson, T. Tsuchimochi, G. E. Scuseria, Projected hartree- fock theory, Journal of Chemical Physics 136 (2012) 164109.[121] S. Ostlund, S. Rommer, Thermodynamic Limit of Density-Matrix Renormalization, PhysicalReview Letters 75 (1995) 3537–3540. 37122] J. Valatin, Generalizd Hartree-Fock Method, Physical Review 122 (1961) 1012. doi:10.1103/PhysRev.122.1012 . [123] M. Piris, R. Cruz, A BCS Approach to Molecular Correlation, International Journal of Quan-tum Chemistry 53 (1995) 353–359. doi:10.1002/qua.560530402 .[124] D. Littlewood, R. A.R., Group Characters and Algebra, Philosophical Transactions of theRoyal Society A 233 (1934) 99–141. doi:10.1098/rsta.1934.0015 .[125] J. Halton, A Combinatorial Proof of Cayley’s Theorem on Pfaffians, Journal of Combinatorial Theory 1 (2) (1966) 224–232. doi:10.1016/S0021-9800(66)80029-7 .[126] M. Rudelson, A. Samorodnitsky, O. Zeitouni, Hafnians, Perfect Matchings and GaussianMatrices, The Annals of Probability 44 (4) (2016) 2858–2888. doi:10.1214/15-AOP1036 .[127] M. Ishikawa, H. Kawamuko, S. Okada, A Pfaffian-Hafnian Analogue of Borchardt’s Identity,The Electronic Journal of Combintorics 12 (2005) N9. [128] I. Gelfand, M. Kapranov, A. Zelevinsky, Hyperdeterminants, Advances in Mathematics 96 (2)(1992) 226–263. doi:10.1016/0001-8708(92)90056-Q .[129] A. Taranenko, Multidimensional Permanents and an Upper Bound on the Number ofTransversals in Latin Squares, Journal of Combinatorial Design 23 (7) (2015) 305–320. doi:10.1002/jcd.21413 . [130] A. Barvinok, New Algorithms for Linear k-Matroid Intersection and Matroid k-Parity Prob-lems, Mathematical Programming 69 (1-3) (1995) 449–470. doi:10.1007/BF01585571 .[131] D. Redelmeier, Hyperpfaffian in Algebraic Combinatorics, Ph.D. thesis, University of Water-loo (2006).URL http://hdl.handle.net/10012/1055 [132] L. Gurvits, A. Samorodnitsky, A Deterministic Algorithm for Approximating the Mixed Dis-criminant and Mixed Volume, and a Combinatorial Corollary, Discrete Computational Ge-ometry 27 (2002) 531–550. doi:10.1007/s00454-001-0083-2 .38 . Appendix | Ψ HF i = N Y i =1 K X j =1 a † j U ji | i = K X j =1 a † j U j K X j =1 a † j U j . . . K X j N =1 a † j N U j N N | i = K X j =1 2 K X j =1 · · · K X j N =1 a † j a † j . . . a † j N U j U j . . . U j N N | i (56)We can group together the terms that result in the same Slater determinant, defined here by the setof creation operators, { a † j a † j . . . a † j N } . Since the order of operators only affects the sign of the Slaterdeterminant, we can split the sum over all indices into a sum over the set of creation operators anda sum over the different orderings of the given set of creation operators. | Ψ HF i = K X j 1. In contrast, we can46ake the permanent of a matrix with repeating rows: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a . . . a n ... . . . ... a . . . a n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + = X σ ∈ S n Y i =1 a i = n ! Y i =1 a i (76)Therefore, we get | Ψ i = X m X { b ... b nb }⊆ ˜ S b , f ∈ ˜ S f sgn ˆ A † b ... ˆ A † b nb ˆ A † f | i = | m i sgn (cid:16) σ ( ˆ A † b . . . ˆ A † b nb ˆ A † f ) (cid:17) (cid:18) n b Q i =1 C b i (cid:19) C f + X { b ... b nb }⊆ ˜ S b sgn ˆ A † b ... ˆ A † b nb | i = | m i sgn (cid:16) σ ( ˆ A † b . . . ˆ A † b nb ) (cid:17) n b Q i =1 C b i | m i (77)Since the sum over { b . . . b n b f } only occurs when m has an odd number of electrons and the sumover { b . . . b n b } only occurs when m has an even number of electrons, we can separate these twocases if the desired number of electrons in the system, N , is either even or odd. | Ψ i = X m X { b ... b nb }⊆ ˜ S b , f ∈ ˜ S f sgn ˆ A † b ... ˆ A † b nb ˆ A † f | i = | m i sgn (cid:16) σ ( ˆ A † b . . . ˆ A † b nb ˆ A † f ) (cid:17) (cid:18) n b Q i =1 C b i (cid:19) C f | m i if N is odd X m X { b ... b nb }⊆ ˜ S b sgn ˆ A † b ... ˆ A † b nb | i = | m i sgn (cid:16) σ ( ˆ A † b . . . ˆ A † b nb ) (cid:17) n b Q i =1 C b i | m i if N is even (78) Just as in Section 8.5, let b i be a set of an even number of orbitals, f i be a set of an oddnumber of orbitals, ˆ A † b i and ˆ A † f i be the creation operators that create the associated orbitals. Wecan construct a quasiparticle as a linear combination of these creation operators. The desiredwavefunction is a product of these quasiparticles | Ψ i = n Y p =1 X b i C p ; b i ˆ A † b i + X f i C p ; f i ˆ A † f i ! | i (79)where C p ; b i and C p ; f i are coefficients of the creation operators ˆ A † b i and ˆ A † f i in the construction ofthe p th quasiparticle. 47aking the same approach as Equation 65, we have operators, { ˆ Q j . . . ˆ Q j n } , that are creationoperators, { ˆ A b . . . ˆ A b nb ˆ A f . . . ˆ A f nf } , and the sum over the permutation is given by Equation 70. | Ψ i = X m X { b ... b nb }⊆ ˜ S b , { f ... f nf }⊆ ˜ S f sgn ˆ A † b ... ˆ A † b nb ˆ A † f ... ˆ A † f nf | i = | m i sgn (cid:16) σ ( ˆ A † b . . . ˆ A † b nb ˆ A † f . . . ˆ A † f nf ) (cid:17) X { i b ...i bnb }⊆{ b ... b nb f ... f nf }{ i f ...i fnf } = { b ... b nb f ... f nf }\{ i b ...i bnb } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C b i b . . . C b i bnb ... . . . ... C bn b i b . . . C bn b i bnb (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C f i f . . . C f i fnf ... . . . ... C fn f i f . . . C fn f i fnf (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − | m i (80)where C b is the submatrix of C composed of the first n b rows and C f is composed of the remaining rows. It was assumed that the first n bb