Implications of pinned occupation numbers for natural orbital expansions. I: Generalizing the concept of active spaces

The concept of active spaces simplifies the description of interacting quantum many-body systems by restricting to a neighbourhood of active orbitals around the Fermi level. The respective wavefunction ansatzes which involve all possible electron configurations of active orbitals can be characterized by the saturation of a certain number of Pauli constraints $0 \leq n_i \leq 1$, identifying the occupied core orbitals ($n_i=1$) and the inactive virtual orbitals ($n_j=0$). In Part I, we generalize this crucial concept of active spaces by referring to the generalized Pauli constraints. To be more specific, we explain and illustrate that the saturation of any such constraint on fermionic occupation numbers characterizes a distinctive set of active electron configurations. A converse form of this selection rule establishes the basis for corresponding multiconfigurational wavefunction ansatzes. In Part II, we provide rigorous derivations of those findings. Moroever, we extend our results to non-fermionic multipartite quantum systems, revealing that extremal single-body information has always strong implications for the multipartite quantum state. In that sense, our work also confirms that pinned quantum systems define new physical entities and the presence of pinnings reflect the existence of (possibly hidden) ground state symmetries.


Introduction
At first sight, an accurate description of fermionic quantum many-body systems seems to be highly challenging, if not impossible: The interaction between the particles can lead to strong correlations which in principle may distribute over an exponentially large Hilbert space. Yet, realistic physical systems exhibit some additional structure. To name possibly the most important one, the particles interact only by two-body forces and the respective ground state problem can therefore be addressed in the reduced two-particle picture [1,2]. Since most subcommunities restrict to systems all characterized by the same pair interaction (for instance Coulomb interaction in quantum chemistry, contact interaction in quantum optics and Hubbard interaction in solid state physics) the ground state problem should de facto involve only the one-particle reduced density matrix. Indeed, for Hamiltonians of the form H κ (h)=h+κV, where h represents the one-particle terms and V the fixed pair interaction with coupling strength κ, the conjugate variable to H κ (h) and h, respectively, is the one-particle reduced density operator ρ 1 . The corresponding exact one-particle theory is known as reduced density matrix functional theory (RDMFT) and is based on the existence of an exact energy functional r r k r º + k   h Tr 1 1 1 There is also another less profound motivation for the description of quantum many-body systems in the one-particle picture, as governed by the one particle reduced density operator. Whenever, the coupling κ between the identical fermions vanishes the respective Hamiltonian H κ=0 (h) contains only one-particle terms and the ground state problem can be entirely discussed and solved in the much simpler one-particle picture: in a first step one needs to diagonalize the one-particle Hamiltonian h on the one-particle Hilbert space  1 , e c c = å ñá ). This emphasizes the significance of the concept of active spaces. To be more specific, it allows one to exploit significantly simplified ansatzes for Yñ | involving only configurations < ¼ < with a certain number of fully frozen (core) orbitals and some inactive virtual orbitals. In quantum chemistry such ground state ansatzes are referred to as complete active space self-consistent field (CASSCF) ansatz (see, e.g. [5][6][7][8]).
The general aim of our paper is to illustrate and prove in a mathematically rigorous way that also the saturation of the generalized Pauli constraints (GPCs) (pinning) [9][10][11][12] gives rise to specific, generalized active spaces. In that sense, our work shall provide the foundation for possible future applications of the new concept of GPCs within quantum chemistry and physics, particularly in the form of more systematic multiconfiguration self-consistent field (MCSCF) ansatzes. The paper therefore consists of two complementary parts. Part I explains and comprehensively illustrates various results in the context of fermionic quantum systems and avoids any technicalities. Quite in contrast, Part II provides rigorous derivations of our results and extends them to nonfermionic systems.
The present Part I is structured as follows. After fixing the notation and introducing the basic concepts in section 2, we illustrate in section 3 the connection of pinning of Pauli constraints and structural simplifications of the N-fermion quantum state. This link between the one-particle and N-particle picture provides in particular a solid foundation for the concept of (complete) active spaces. In section 4, we explain and illustrate how this concept of active spaces could be generalized. To be more specific, we present and illustrate our main results stating that the saturation of the GPCs implies a selection rule identifying the N-fermion configurations contributing in a respective natural orbital expansion. A converse form of this selection rule establishes the basis for corresponding multiconfigurational wavefunction ansatzes.

Notation and concepts
In the following, we fix the notation and introduce some basic concepts. To keep our work self-contained we in particular recall some concepts which were already introduced and discussed in [13,14]. In our work, we always consider a finite d-dimensional one-particle Hilbert space  1 . In the context of numerical approaches in physics and quantum chemistry, such  1 typically arises from the truncation of the full infinite-dimensional oneparticle Hilbert space of square integrable wave functions Ä ( ) by choosing a finite basis set of d spinorbitals. A prime example would be electrons in an atom, i.e. spin = s 1 2 with the underlying configuration space  given by º   3 and a basis set of d atomic spin-orbitals.

Natural orbitals and natural occupation numbers (NONs)
The crucial object of our work is the one-particle reduced density operator ρ 1 of an N-fermion quantum state There are two equivalent routes that one could follow for introducing ρ 1 . By exploiting first quantization, one naturally embeds the N-fermion Hilbert space  The partial trace in equation (2) is indeed well-defined since the choice of the N−1 factors to be traced out does not matter due to the well-defined exchange-symmetry of Yñ | . An alternative but equivalent approach to define ρ 1 is based on second quantization. After fixing some orthonormal reference basis c ñ = j j d 1 {| } for the one-particle Hilbert space  1 and introducing the respective creation and annihilation operators, ρ 1 follows from its matrix representation c r c á ñºáY Yñ Diagonalizing the Hermitian one-particle reduced density operator ρ 1 , gives rise to the NONs n j and the natural orbitals ñ j | , the corresponding eigenstates [15,16]. This terminology also motivates the normalization r = +¼+ = n n N Tr [ ] which allows us to interpret the eigenvalues of r 1 as occupation numbers, the occupancies of the natural orbitals. Moreover, for the following considerations we order the NONs decreasingly, . The natural orbitals of any N-fermion state Yñ | form an orthonormal basis = ñ =  j j d 1 1 {| } for the oneparticle Hilbert space  1 . This basis is unique (up to phases) as long as the NONs are non-degenerate. Based on the natural orbital basis  1 , we introduce a natural orbital induced operator which will play a crucial role for the compact formulating of our main results: | [ ]and let  1 be a basis of natural orbitals. For any polynomial L of d variables of degree one, we define where the particle number operators º n f f j j ĵ † refer to the natural orbitals  1 .
Since we use this concept of an orbital induced operator only with respect to the natural orbitals of a given quantum state Yñ | we refrained from extending the definition of  L 1 to arbitrary orthonormal bases  1 . We would also like to stress again that here and in the following, the natural orbitals ñ j | of Yñ | are only unique as long as the NONs are non-degenerate and their labeling resembles that of the corresponding NONs, i.e. ¼    n n n d This means that the vector n of NONs follows as the 'center of mass' for masses c i 2 | | located at positions n i in  d . ] would correspond to configuration states of particle numbers different than N and therefore will not play any role in the present work which restricts to fixed particle number N. This geometric picture is illustrated in figure 2 in section 4.4 for the Borland-Dennis setting, i.e. for the case of three fermions and a six-dimensional one-particle Hilbert space.
Lastly, we point out a geometric aspect concerning the action of operators  L 1 from definition 1 on configuration states ñ i | . Namely, for a given = å = L l n j d j j 1 it is straightforward to check that  L 1 is diagonal in the NO-basis and that its diagonal entries follow by the geometric formula Here, we use the standard notation for the dot-product of vectors, i.e. å = l n n p l

Pauli constraints and concept of active spaces
The properties and the behavior of fermionic quantum systems strongly rely on Pauli's exclusion principle [18]. This principle defines a constraint on the one-particle picture as governed by the one-particle reduced density operator r 1 . For any N-fermion state Yñ | the occupancies of one-particle states jñ | are restricted, áY Yñ requires the smallest NONs to be arbitrarily small for large or even infinite basis set size d N  . In the following, we would like to formalize the concept of active spaces by relating their structure in the Nparticle picture to the possible saturation of multiple Pauli constraints concerning the one-particle picture. For this, we express the family of Pauli constraints (15) in a more compact form. For any pair r s , we define the constraints (see also [14]) on the non-increasingly ordered NONs n. The family of those constraints is equivalent to the Pauli constraints in their original form (15). From the geometric point of view (recall section 2.3), all vectors n of non-increasingly ordered NONs obeying the Pauli exclusion principle form a specific polytope in  d , the Pauli simplex Σ, This implies a selection rule on the expansion coefficients in the sense that only those configuration states To be more precise, this means where n i is the unordered spectrum of the configuration state ñ i | as introduced in equation (11).
Proof. Since = áY Yñ . Since the smallest eigenvalue is zero, implies that the whole weight of Yñ | needs to lie in the zero eigenspace. The Selection Rule (19) follows then immediately by plugging in the expansion (8) into (18) and using again the fact that  S r s , The proof of theorem 2 and the derivation of the consequences of pinning by the Pauli constraints, respectively, was rather elementary. This is due to the fact that the natural orbital induced operator  S r s , 1 ( ) has no negative eigenvalues, i.e. it is positive semi-definite. Therefore, whenever could not be canceled out by contributions from eigenspaces with negative eigenvalues. This will be different when we discuss in the following the consequences of pinning of GPCs, = n D 0 ( ) , since their respective natural orbital induced operators  D 1 have both negative and positive eigenvalues.

Generalized Pauli constraints
Despite the remarkable significance of Pauli's exclusion principle (15), (16) on all physical length scales, it has conclusively been shown only recently [11,12,21] that the fermionic exchange symmetry implies even greater restrictions on the one-particle picture. To be more specific, as illustrated in figure 1, the set of pure Nrepresentable vectors n of (non-increasingly ordered) NONs form a polytope, a proper subset of the Pauli simplex Σ (17). For each setting of N fermions and a d-dimensional one-particle Hilbert space  1 , this polytope  is described by a finite family of linear inequalities, the GPCs , the respective coefficients k i j ( ) can be chosen as integers. In particular, by referring to the canonical choice of minimal integers, the l 1 -distance of n to the hyperplane defined by º D 0 i follows as n D j ( ) up to a prefactor (for more details see [22]). While the GPCs for the smaller settings with  N d , 7have already been derived several decades ago by some brute force approach [9,23,24], it was Klyachko's breakthrough [11,21] on how to find a systematic procedure which allows one to determine for all settings N d , ( ), at least in principle, the family of GPCs. Yet, it is still an ongoing challenging to develop more efficient algorithms for determining the GPCs and in particular to approximate them (see, e.g. [25]). Before we briefly discuss the potential physical relevance of the GPCs, we would like to present them for the first non-trivial setting, = N d , 3,6 ( ) ( ), and comment on their triviality for the smallest few settings.
First, due to the particle hole duality on the fermionic Fock space we can restrict ourselves without loss of generality to  N d 2. Indeed, one has (see, e.g. [21]) ) of d−N fermions and a d-dimensional one-particle Hilbert space follows from those of N d , ( ) by just replacing Second, as summarized by example 4, the GPCs for all settings with only one or two fermions (and according to the particle-hole duality, lemma 3, also those with one or two holes) are trivial [26]. The first non-trivial setting is thus the Borland-Dennis setting, i.e. - We remind the reader that the NONs are always ordered non-increasingly, . Notice that the inequality  n D 0 ( ) is more restrictive than Pauli's exclusion principle, which just states implies - . The incidence of GPCs taking the form of equalities (instead of inequalities) as those in (21) is rather unique since this happens only for the Borland-Dennis setting and the settings with at most two fermions or at most two holes.

Potential physical relevance of the GPCs
In complete analogy to Pauli's exclusion principle, the physical significance of the GPCs is primarily be based on their possible (approximate) saturation in concrete systems. In an analytical study [27] of the ground state of three harmonically interacting fermions in a one-dimensional harmonic trap it has been shown that the GPCs are not fully saturated. Yet, given this it is quite remarkable that the vector n of NONs has just a tiny distance to the polytope boundary given by the eighth power of the coupling strength, k µ D 8 . A succeeding comprehensive and conclusive study of harmonic trap systems [22,[28][29][30][31][32] has confirmed that such quasipinning represents a genuine physical effect whose origin is the universal conflict between energy minimization and fermionic exchange symmetry in systems of confined fermions [30]. The presence of such quasipinning (or even pinning if the system's chosen Hilbert space is quite small) has been verified also in smaller atoms and molecules [33][34][35][36][37][38][39][40][41][42][43][44][45] ). A comment is in order concerning the non-triviality of such (quasi)pinning by the GPCs. Since at least some NONs in most realistic ground states are close to one, the vector Î  n of NONs is typically close to the boundary of the surrounding Pauli simplex Σ (17) and consequently (recall Ì S  and see figure 1) it is also close to the boundary of the polytope . The more crucial question is therefore whether the (quasi)pinning by the GPCs is non-trivial in the sense that it does not already follow from (quasi)pinning by the Pauli constraints, or in other words, whether the GPCs have any significance beyond the Pauli constraints (16). This also necessitates a systematic treatment of systems with symmetries, since symmetries are known to favor the occurrence of (quasi)pinning [38,41,42]. A more systematic recent analysis based on the so-called Q-parameter [14] has shown that the quasipinning by the GPCs is indeed non-trivial [14,32,45].
It has been speculated and suggested that such (quasi)pinning would reduce the complexity of the system's quantum state and would define 'a new physical entity with its own dynamics and kinematics' [33] (see also [13,46,47]). Based on this expected implication of (quasi)pinning as an effect in the one-particle picture on the structure of the N-fermion quantum states, variational ansatzes for ground states have been proposed as part of an ongoing development [47][48][49][50][51]. Moreover, general investigations and deeper insights into the structure of quantum states suggest that taking the GPCs into account may help to turn RDMFT into a more competitive method [52,53] (for more specific results see [50,54,55]). In particular, it has been shown [56] for all translationally invariant one-band lattice systems (regardless of their dimensionality, size and interactions) that the gradient of the exact universal functional diverges repulsively on the polytope boundary ¶. It is exactly this latter result and the suggested implications of (quasi)pinning which motivate us to explore and rigorously derive here the implications of pinning on the respective N-fermion quantum state.

Borland-Dennis setting: implications of pinned occupation numbers
We first discuss the implications of pinning within the specific Borland-Dennis setting, i.e. for = N d , 3,6 ( ) ( ). This in particular also allows us to understand how those implications may look like in the case of degenerate NONs.
In the following, we use n n n , ,

Degenerate NONs
In general, understanding the implications of pinning for degenerate NONs turns out to be rather challenging. There are two reason for this: first, there is no unique natural orbital basis anymore and it is therefore not clear whether a selection rule of the form (30) may refer to all possible natural orbital bases or to just one of them.
Second, the saturation of some GPC  D 0 and an additional ordering constraint may automatically enforce the saturation of additional GPCs. In that case, the corresponding selection rule for the saturation º D 0 might be more restrictive than in the case of non-degenerate NONs. The latter happens in the Borland-Dennis setting in case of a degeneracy = n n From the left side we can infer again that the GPCs are more restrictive than Pauli's exclusion principle constraints since the respective polytope  is a proper subset of the Pauli simplex Σ (given by the polytope together with an extension shown in red). On the right side, the geometric picture as introduced in section 2.3 is In case of non-degenerate NONs, only the configurations i may contribute in the self-consistent expansion (8) according to (30) whose unordered spectra n i lie on the hyperplane corresponding to pinning (shown in blue). The same is still true in case of degeneracies = n n 4 5 or = n n 5 6 . For a degeneracy = n n 3 4 (i.e. = n 4 1 2 ) and a generic choice of the natural orbitals in the = n n 3 4 subspace also the configurations i whose vectors n i lie on the light blue hyperplane may contribute. This latter hyperplane is given by the swapping « n n 3 4 of the blue hyperplane. Yet, according to (33) there exists at least one basis  1 of natural orbitals with respect to which the weights on the light blue hyperplane are transformed away and would lie solely on the blue hyperplane.
The analysis of the Borland-Dennis setting suggests the following implications of pinning by a GPC > D 0 in a general setting N d , ( ): In case of non-degenerate NONs there is no ambiguity since the natural orbitals are unique and only those configurations i may contribute to Yñ | whose unordered spectra n i (recall equation (12)) lie on the respective hyperplane corresponding to pinning, º D 0. In case of degenerate NONs there exist at least one basis  1 of natural orbitals with respect to which the original selection rule for non-degenerate NONs applies.
Although those main results of our work (see theorems 6, 10 and corollaries 7, 11 below) could be presented for both cases of non-degenerate and degenerate NONs together, we split them. This has the advantage that at least the results for non-degenerate NONs can be stated in a less technical form, namely not involving the ambiguity of natural orbital bases. For the proofs of various results we refer the reader to Part II.

Implications of non-degenerate pinned occupation numbers
In case of non-degenerate NONs the structural implications of pinning can be stated as It is worth noticing that theorem 6 applies to various saturated GPC simultaneously.
Theorem 6 implies immediate structural simplifications for the state Yñ | which are particularly wellpronounced in the self-consistent expansion (8) as already illustrated above: | [ ]be an N-fermion quantum state whose non-degenerate NONs n saturate a GPC, = n D 0 ( ) . Then, only those configurations i may contribute in the self-consistent expansion (8) of Yñ | whose unordered spectra n i (recall equation (11)) lie on the hyperplane defined by = n D 0 i ( ) . In other words, for each configuration i we have We present an example which illustrates theorem 6 and the corresponding selection rule, corollary 7: ⎛ ⎝ ⎞ ⎠ configurations to just 9 highlights the remarkable implications of pinning as an effect in the one-particle picture on the structure of the corresponding many-fermion quantum state.

Implications of degenerate pinned occupation numbers
Based on the analysis of pinning by degenerate NONs in the Borland-Dennis setting (section 4.3) one may expect the following generalization of theorem 6 to degenerate NONs: | [ ]be an N-fermion quantum state whose degenerate NONs n saturate some (possibly several) GPCs. Then, there exists an orthonormal basis  1 of natural orbitals such that Yñ | lies in the zero-eigenspace of the respective  D 1 -operators of various saturated GPCs (recall definition 1), i.e.
There are actually a number of reasons (highlighted in Part II which presents various mathematical proofs) why the generalization of theorem 6 to non-degenerate NONs and its proof are quite involved.
In the following we present a weaker extension of theorem 6 to degenerate NONs. It refers to the saturation of exactly one GPC. Its proof requires in addition the validity of a technical assumption (presented as assumption 13 in Part II) which we could verify for all GPCs known so far. Hence, there is little doubt that the assumption is always valid and the corresponding addition to the following theorem might be unnecessary. | [ ]be an N-fermion quantum state whose degenerate NONs n saturate exactly one GPC, = n D 0 ( ) and assume that the technical assumption 13 from Part II is met. Then, there exists an orthonormal basis  1 of natural orbitals such that Yñ | lies in the zero-eigenspace of the respective  D 1 -operator (recall definition 1), i.e.
Despite the ambiguity of the natural orbital basis  1 it is worth recalling that the natural orbitals ñ j {| }are still referring to the non-increasingly ordered NONs (see also (4)).
In complete analogy to theorem 6 and corollary 7, theorem 10 implies immediately a corresponding selection rule identifying all configurations which may contribute to Yñ | in case of pinning: Corollary 11 (Selection rule for degenerate NONs). Let Yñ Î   N 1 | [ ]whose degenerate NONs n saturate exactly one GPC, = n D 0 ( ) and assume that the technical Assumption 13 from Part II is met. Then, there exists an orthonormal basis  1 of natural orbitals such that only configurations i may contribute to the self-consistent expansion (8) of Yñ | whose unordered spectra n i (recall equation (11)) lie on the the hyperplane º D 0, i.e. | [ ]whose NONs map to the face F, Minimizing the energy expectation value of a given Hamiltonian H of a system of interacting fermions over  F then defines a variational scheme associated with the face F with a corresponding variational energy º áY Yñ From a qualitative point of view, one can say that the higher dimensional the face F, the higher dimensional the corresponding state manifold  F and thus the more computationally demanding the respective ansatz. Some well-known examples for such polytope face-associated variational schemes are the CASSCF ansatzes (see, e.g. [5][6][7][8]). Indeed, according to theorem 2 they can be characterized by the saturation of a certain number of Pauli exclusion principle constraints. Our main results, theorems 6, 10 and the respective selection rules, Corollaries 7, 11, highlight that even more elaborated variational ansatzes can be introduced by referring not only to the saturation of Pauli constraints but to extremal one-fermion information in general, i.e. pinned NONs. The motivation for proposing such generalizations of CASSCF ansatzes is twofold. On the one hand, the study of smaller atoms [45] has revealed that the GPCs have an additional significance for ground states beyond the one of the Pauli exclusion principle constraints, as quantified by the Q-parameter [14]. On the other hand, not all configurations i within a complete active space are relevant and it would be preferable to identify only the most significant ones. The gain in computational time could be used to increase the basis set size, allowing one to recover more of the dynamic correlation. A comment is in order concerning the practical implementation of such variational schemes. After having fixed F, i.e. the corresponding family of contributing configurations i, one would minimize both the respective expansion coefficients c i and the involved natural orbitals ñ = j j d 1 {| } . Such variational approaches are known in quantum chemistry as MCSCF-ansatzes (see, e.g. the textbook [57]). Yet, the stringent use of pinning-based variational ansatzes in the form (44) would be quite challenging and not particularly efficient. This is due to the fact that the selection rule 7 defines  F by referring to the self-consistent expansion (8), i.e. rather involved selfconsistency conditions on the expansion coefficients c i would need to be imposed. From a converse point of view, an arbitrary superposition of all allowed configurations i is typically not self-consistent. Hence, its relation to the face F seems to be rather loose, since its vector n of non-increasingly ordered NONs lies actually in the interior of the polytope  rather than on the face F. To illustrate this, let us revisit example 8. We pick a random , i.e. Î  n lies far away from the polytope facet defined by º D 0. This is actually quite different for the vector ¢ n obtained by permuting the NONs according to some specific permutation π. Of course, ¢ n does not lie in the polytope  anymore since its entries are not properly ordered. Yet, by extending the face º D 0 of  to a hyperplane in the space of all occupation number vectors (including the ones which are not decreasingly ordered), ¢ n turns out to lie on that hyperplane, . This is rather astonishing in particular since the one-particle reduced density matrices of such arbitrary superpositions are not diagonal in the original reference basis anymore. For instance, one finds for the superposition above r á ñ = - {| } of  1 we define i.e. the vector space of all superpositions of configurations i fulfilling the selection rule 7 with respect to the basis  1 for all GPCs D k with Î k K F . Then, for any Yñ Î   F 1 | ( ) there exists a basis ¢  1 of (possibly wrongly ordered) natural orbitals of Yñ | such that all configurations Î ¢ Yñ  i Supp 1(| ) also fulfill the selection rules = n D 0 i k ( ) for all Î k K F . In particular, the corresponding vector n of (possibly wrongly ordered) NONs saturates  D 0 k for all Î k K F , i.e. n lies on the hyperplane obtained by extending the face F to non-decreasingly ordered occupation number vectors.
To illustrate the first part of this theorem we revisit example 8. Let º ñ =  i i Such ansatzes are indeed MCSCF ansatzes in a strict sense: In a first step, one identifies (via the choice of a face F) a specific set of configurations i contributing to Yñ | . Then, in a second step one minimizes the energy expectation value with respect to various expansion coefficients (without any additional constraints on them) and all possible orbital choices  1 . The corresponding variational energy is at least as good as the original one (  E F ) and the computational effort is significantly reduced by omitting the quadratic self-consistency conditions required in the characterization of  D .

Implementation of the pinning-based MCSCF ansatzes
It will be one of the future challenges to implement and test our proposed pinning-based MCSCF ansatzes (for a proof of concept see [48]). In particular, one may worry about two potential obstacles. First, the GPCs are known so far only for one-particle Hilbert spaces of dimension  d 11. Choosing an underlying active space from the very beginning involving only eleven specific (spin-)orbitals would then restrict the scope of our approach to rather small systems which could even by exactly diagonalized. Second, it is not clear which polytope face F one should employ in the ansatz (49).
Concerning the first potential obstacle, we would like to stress that the algorithm used so far for calculating the GPCs is not particularly efficient. We are therefore optimistic that one could calculate the GPCs (or approximations of them as proposed in [25]) for larger settings N d , ( ) in the nearer future. Moreover, it is crucial to notice that the limitation  d 11 concerning the GPCs does not require us to restrict such pinningbased ansatzes to basis sets of size up to eleven. Quite in contrast, it merely means that we populate up to = d 11 orbitals j j ñ ¼ ñ , , d 1 | | which shall then be optimized within a huge dimensional one-particle Hilbert space of dimension 100-1000. This is in analogy to the Hartree-Fock scheme in which the N populated orbitals are optimized within a huge-dimensional one-particle Hilbert space. Compared to methods such as exact diagonalization ( full configuration interaction) employing rigid, fixed active spaces, our method will benefit a lot form its flexibility, namely the fact that we can determine the best possible occupied orbitals within a large orbital space.
Identifying the most suitable polytope faces F for the MCSCF ansatz (49) is a crucial challenge. Let us first recall that one encounters the same problem in the traditional CASSCF as well: it is not clear at the beginning which complete active space is the most appropriate one. What would be appropriate numbers r of frozen electrons and s of virtual inactive orbitals? Or in more mathematical terms, how large can we choose r s , such that the respective Pauli constraint  n S 0 r s , ( ) ( ) 16 is still approximately saturated? The answer to this crucial question can be found in practice only after trying out different active space sizes and comparing the corresponding CASSCF energies. The same applies in principle to our generalized definition of the concept of active spaces. Yet, we also would like to stress that the geometrical structure of the polytope, to be more precise its so-called 'face lattice' leads to a natural hierarchy of generalized active spaces which could be exploited in practice. Working it out in a comprehensive way and illustrating its effectiveness for determining ground state energies would go beyond the scope of the present work though. Nonetheless, let us briefly outline this hierarchical scheme: The higher dimensional a face of the polytope the more accurate and more expensive is the respective MCSCF ansatz (49). One therefore may start with the Hartree-Fock ansatz, corresponding to the zero-dimensional face which consists only of the Hartree-Fock point. Then, one continues by considering all one-dimensional faces (edges) which contain the Hartree-Fock point. By running all respective MCSCF ansatzes, one identifies the one among them which leads to the lowest MCSCF energy. The crucial idea is then to proceed by considering only those two-dimensional faces which contain the best one-dimensional one. Continuing with this procedure would allow one to systematically identify step-by-step the most important electron configurations to the ground state. Hence, this renormalization group-inspired scheme has the potential of improving significantly the rigid CASSCF scheme since it offers more flexibility in superposing carefully chosen electron-configurations. It is also worth noticing that this MCSCF hierarchy contains all CASSCF ansatzes as special cases since for any pair  r s , 0the constraint º S 0 r s , ( ) (16) defines a polytope face.

Presence of pinning reveals symmetries of quantum states
According to theorem 6 and its generalization including the case of degenerate NONs, theorem 10, pinning = n D 0 ( ) implies that the N-fermion quantum state Yñ | lies in the zero-eigenspace of the corresponding natural orbital induced operator  D 1 . This means nothing else than that  D 1 is the generator of a continuous symmetry of Yñ | ,