Tensor product methods and entanglement optimization for ab initio quantum chemistry Szila´rd Szalay Max Pfeffer Valentin Murg Gergely Barcza ∗ † ‡ ∗ ¨ Frank Verstraete Reinhold Schneider Ors Legeza ‡ † ∗ December 19, 2014 Abstract The treatment of high-dimensional problems such as the Schr¨odinger equation can be approached by concepts of tensor product approximation. We present general tech- niques that can be used for the treatment of high-dimensional optimization tasks and time-dependent equations, and connect them to concepts already used in many-body quantum physics. Based on achievements from the past decade, entanglement-based methods, –developedfromdifferentperspectivesfordifferentpurposesindistinctcom- munities already matured to provide a variety of tools – can be combined to attack highly challenging problems in quantum chemistry. The aim of the present paper is to give a pedagogical introduction to the theoretical background of this novel field and demonstrate the underlying benefits through numerical applications on a text book example. Among the various optimization tasks we will discuss only those which are connected to a controlled manipulation of the entanglement which is in fact the key ingredient of the methods considered in the paper. The selected topics will be covered according to a series of lectures given on the topic “New wavefunction methods and entanglement optimizations in quantum chemistry” at the Workshop on Theoretical Chemistry, 18 - 21 February 2014, Mariapfarr, Austria. Contents 1 Introduction 3 1.1 Tensor product methods in quantum chemistry . . . . . . . . . . . . . . . . 4 1.2 Entanglement and quantum information entropy in quantum chemistry . . . 6 1.3 Tensor decomposition methods in mathematics . . . . . . . . . . . . . . . . . 7 Strongly correlated systems “Lendu¨let” research group, Wigner Research Centre for Physics, Konkoly- ∗ Thege Mikl´os u´t 29-33, 1121 Budapest, Hungary Fakult¨at II - Mathematik und Naturwissenschaften, Institut fu¨r Mathematik, Technische Universit¨at † Berlin, Strasse des 17. Juni 136, Berlin, Germany Fakult¨at fu¨r Physik, Universit¨at Wien, Boltzmanngasse 3, A-1090 Vienna, Austria ‡ 1 2 Quantum chemistry 8 2.1 The electronic Schro¨dinger equation . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Full configuration interaction approach and the Ritz-Galerkin approximation 9 2.3 Fock spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Occupation numbers and second quantization . . . . . . . . . . . . . . . . . 12 2.5 Example: Hartree-Fock determinant and change of one-particle basis . . . . 15 2.6 Ritz-Galerkin approximation in second quantization . . . . . . . . . . . . . . 15 2.7 Spatial orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3 Tensor product approximation 18 3.1 Tensor product parametrization . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2 Tensor networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 Subspace optimization and the Tucker format . . . . . . . . . . . . . . . . . 21 3.4 Matricization and tensor multiplication . . . . . . . . . . . . . . . . . . . . . 23 3.5 Matrix product states or the tensor train format . . . . . . . . . . . . . . . . 23 3.6 Dimension trees and the hierarchical tensor decomposition . . . . . . . . . . 28 3.7 Fixed rank manifolds and varieties . . . . . . . . . . . . . . . . . . . . . . . 29 3.8 Dirac-Frenkel variational principle or dynamical low rank approximation . . 31 3.9 The alternating least squares algorithm . . . . . . . . . . . . . . . . . . . . . 32 4 Numerical techniques 36 4.1 Basic terms and brief overview . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.1.1 The problem in the language of tensor factorization . . . . . . . . . . 37 4.1.2 Change of basis, truncation and iterative diagonalization . . . . . . . 39 4.1.3 Unitary transformation for two molecular orbitals . . . . . . . . . . . 39 4.1.4 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.1.5 Unitary transformation for d number of molecular orbitals and tensor product approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.1.6 Tensor topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.2 Entanglement and correlations . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.2.1 Singular value decomposition and entanglement . . . . . . . . . . . . 43 4.2.2 Block entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.2.3 One- and two-orbital entropy and mutual information . . . . . . . . . 47 4.2.4 One- and two-orbital reduced density matrix and generalized correla- tion functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.3 Methods based on block transfromation procedures . . . . . . . . . . . . . . 52 4.3.1 Block renormalization group method (BRG) . . . . . . . . . . . . . . 52 4.3.2 Numerical renormalization group method (NRG) . . . . . . . . . . . 54 4.3.3 Density matrix renormalization group method (DMRG) . . . . . . . . 55 4.3.4 Higher dimensional network: Tree tensor network state (TTNS) . . . 59 4.3.5 Efficient factorization of the interaction terms . . . . . . . . . . . . . 64 4.4 Optimization of convergence properties . . . . . . . . . . . . . . . . . . . . . 66 4.4.1 Error sources and data sparse representation of the wavefunction . . . 67 4.4.2 Targeting several states together . . . . . . . . . . . . . . . . . . . . . 68 2 4.4.3 Optimization of the Schmidt ranks using dynamic block state selection (DBSS) approach and entropy sum rule . . . . . . . . . . . . . . . . . 69 4.4.4 Optimization of the network hierarchy (ordering) and entanglement localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.4.5 Optimization of the network topology . . . . . . . . . . . . . . . . . . 73 4.4.6 Optimization of the basis using entanglement protocols . . . . . . . . 73 4.4.7 Optimization of the network initialization based on entanglement . . 75 4.4.8 Optimization of the sparsity using symmetries . . . . . . . . . . . . . 78 4.4.9 Stability of the wavefunction . . . . . . . . . . . . . . . . . . . . . . . 80 4.4.10 Possible black-box QC-DMRG and QC-TTNS . . . . . . . . . . . . . 81 4.5 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.5.1 Simulation of real materials, geometrical optimization and excited states 82 4.5.2 Four-component density matrix renormalization group . . . . . . . . 83 4.5.3 Possible technical developments: hybrid CPU/GPU parallelization . . 85 5 Summary and outlook 86 6 Acknowledgments 87 1 Introduction For the approximation of the wave function of the electronic structure of an atomic or molecular system, any method chosen will have to compromise between the demanded ac- curacy on the one hand and the high computational complexity of the task on the other. While Density Functional Theory (DFT)47 and Coupled Cluster (CC) or Quantum Monte Carlo methods96,204,226 are in this sense standard methods for the quantitative study of large weakly correlated systems, there has been no method-of-choice solution for finding a suffi- ciently accurate, data-sparse representation of the exact many-body wave function if many electrons are strongly correlated, as, for instance, in open-shell systems as transition metal complexes15,26,30,53,94,114,148,173,194,198,199,217,261. Due to the many-electron interactions present, strongly correlated problems cannot be sufficientlydescribedbysmallperturbationsofasingleSlaterdeterminant. Forthetreatment of other many-particle systems, e.g., spin systems, alternative representations have been pro- posed, resulting in the development of so-called Matrix Product States (MPS)184,211,212,238. The MPS method represents the wavefunction of a system of d components or “sites” (cor- responding, e.g., to molecular orbitals) by forming products of d matrices, each belonging to one component of the system. The computational complexity of the task is now governed by the size of these matrices, related to the eigenvalue spectrum of the corresponding subsys- tem density matrices187 characterizing in a formal way the so-called entanglement among the different components11,64,95,125,127,193,234,241. MPS consists in a linear arrangement of the com- ponents, while more recently the approach has been generalized to so-called Tensor Network States (TNS)45,67,69,135,136,147,151,167,168,180,219,236,237,240, allowing a more flexible connection of thecomponentsoftherespectivesystem. Identical,butindependentapproachesweredevised in numerical mathematics under the term of tensor product approximation, where low-rank 3 factorization of matrices is generalized to higher order tensors82–84. In quantum chemistry, the MPS35,36,39–41,43,57,58,74,79,80,114,125–127,129,144,148,150,158,160,162–165,176,201,253,268,269,274–276 15,26,27,29,30,62,70,94,108,109,112,113,115,117,139,146,147,151,157,161,166,173,206,217,218,229,261,262,264,270 and TNS147,169,170,172 representation can be used to approximate the full-CI wave func- tion37,38,42,44,111,124,135,136,149,151,259,265. By this new concept of data-sparse representation, an accurate representation of the electronic structure will then be possible in polynomial time if the exact wave function can be approximated to a sufficient extent by moderately entangled TNS representations. The underlying Molecular Orbital (MO) basis can be optimized by well known techniques from multi-configurational methods96 as, e.g., Multi Configuration Self Consistent Field (MCSCF) method, which constitutes a tensor approximation method as well at the level of first quantization. Entanglement-based methods, – developed from different perspectives for different pur- poses in distinct communities, already matured to provide a variety of tools – can be com- bined to attack highly challenging problems in quantum chemistry1–5. A very promising direction is, especially, to develop and implement an efficient Quantum Chemistry algorithm based on Tree Tensor Network States (QC-TTNS), in particular enabling the treatment of problems in quantum chemistry that are intractable by standard techniques as DFT or CC135,169,170,172. The aim of the present paper is to give a pedagogical introduction to the theoretical background of this novel field and demonstrate the underlying benefits through numerical applications on a text book example. We give a technical introduction to low rank tensor factorization and do not intend to present a detailed review of the field. Only some selected topics will be covered according to lectures given on the topic “New wave function meth- ods and entanglement optimizations in quantum chemistry” at the Workshop on Theoretical Chemistry, 18 - 21 February 2014, Mariapfarr, Austria6. In accordance with this, the orga- nization of the present paper is as follows. In Secs. 2 and 3 a very detailed description of the theory follows so that those interested readers who just entered in the field could follow recent developments. A brief summary in order to highlight the most important concepts used in numerics is presented in Sec. 4 together with numerical applications by outlining ideas and existing algorithmic structures that have been used to arrive at an efficient imple- mentation. At this stage among the various optimization tasks only those will be analyzed which are connected directly to the manipulation of entanglement, which is in fact the key ingredient of the methods presented in the paper. To unify notations, in what follows, we mainly use terms and notations as common in physics and chemistry. 1.1 Tensor product methods in quantum chemistry Multi-particle Schr¨odinger-type equations constitute an important example of problems posed on high-dimensional tensor spaces. Numerical approximation of solutions of these problems suffers from the curse of dimensionality, i.e., the computational complexity scales exponentially with the dimension of the space. Circumventing this problem is a challenging topic in modern numerical analysis with a variety of applications, covering aside from the electronic and nuclear Schro¨dinger equation e.g., the Fokker-Planck equation and the chem- ical master equation141. Considerable progress in the treatment of such problems has been made by concepts of tensor product approximation81–83. 4 In the year 1992, S. R. White introduced a very powerful numerical method, the Density- Matrix Renormalisation Group (DMRG)248–250. It allows us to determine the physical prop- erties of low-dimensional correlated systems such as quantum spin chains or chains of inter- acting itinerant electrons to unprecedented accuracy61,92,93,178,186,211. Further success of the DMRG method in quantum physics motivated its application to Quantum Chemical problems (QC-DMRG)39,125,159,162,253. In the treatment of problems where large active spaces are mandatory to obtain reliable and accurate results, it has proven capable of going well beyond the limits of present day quantum chemistry methods and even reach the full-CI limit37,38,42,44,111,124,135,136,149,151,259,260. In the past decade, the method has gone through major algorithmic developments by various groups38,111,124,150,150,211,265. For example, the two-body reduced density matrices calculated with the DMRG method39,125,275 can be used in the standard orbital optimization procedure96. Resulting methods are the DMRG Complete Active Space Self Consistent Field (DMRG-CASSCF) or DMRG Self Consistent Field (DMRG-SCF)74,268,274,275. Another di- rection is the post-DMRG treatment of dynamic correlation. DMRG as it can be considered as a CAS Configuration Interaction (CAS-CI) technique can recover static correlation, and, depending on the size of the active space, one can afford also some portion of the dynamic correlation. Quite recently, various advanced methods accounting for dynamic correlation on top of the DMRG framework have been developed115,139,176,206,251,269,270. The first im- plementation of the relativistic quantum chemical two- and four-component density matrix renormalization group algorithm (2c- and 4c-DMRG) has also been presented109. TheDMRGmethodcanbeusedtocalculategroundaswellasexcitedstates. Thiscanbe achievedeitherbytargetingseveraloftheminastateaveragefashion57,74,125,126,139,165,170,218,264 or alternatively based on the MPS tangent vectors88,89,173,190,263. Since the DMRG method is very flexible it can be used even in such situations when the wave function character changes dramatically29,126,165,170,264. Additionally, the ansatz is size consistent by construction and symmetries as particle number, spin projection253, spin reflection symmetries137, Abelian point group symmetries41,126,127 and even non-Abelian symmetries can be factored out ex- plicitly123,152–155,192,205,217,220–223,230,231,245,245,262,264,276. Quiterecently,MPSandfurthertensor product approximations have been applied in post Hartree-Fock (post-HF) methods to the decomposition of the two electron integrals, the AO-MO (Atomic Orbital-Molecular Orbital) transformation, and the Møller-Plesset perturbation theory (MP2) energy expression22. In the MPS like methods the computational complexity of the task is governed by the size of the matrices used to approximate the wavefunction, which can, however, be controlled based on various truncation criteria to achieve a priory set error margin99,125. In a system withidenticalsites,thisfeatureisdirectlyconnectedtothescalingofentanglementwhensub- systemsincludelargerandlargerportionofthetotalsystem,alsocalledasarealaw64,95,193,234. The situation is more complicated in quantum chemical applications since the ranks of the matrices also depend strongly on the ordering of the matrices15,30,125–127, thus different or- derings lead to better or worse results if the ranks are kept fixed39,114,146,158,161,162,201,264,269. Another main aspect that effects the performance of the method is the optimization of the basis74,133,146,169,268,274 and initialization of the network15,39,127,159,163,264. Even though the significant efforts dedicated to the various optimization tasks, it remains an open question to determine the minimum of computational effort to obtain results with a given accuracy threshold. 5 Shortly after DMRG was introduced, it was found that DMRG may also be phrased in terms of MPS184, first formulated for special spin systems as the Affleck-Kennedy-Lieb- Tasaki (AKLT) model8. More recently, the Higher Order Singular Value Decomposition (HOSVD)237,238,243 have made MPS the basis of variational frameworks and revealed a pro- found connection to quantum information theory127,212,241. In this context, it became appar- ent that MPS is only one of a more general set of formats: while MPS corresponds to an arrangement of orbitals in a linear topology, quantum states may more generally be arranged as more complex topologies, leading to tensor network states TNS147,169,170,172. For applica- tionstosmallersystems, prototypicaltensor-networkstateapproachestoquantumchemistry have already been developed, including the so called Complete Graph Tensor Network State (CGTNS) approach147, and the Tree Tensor Network State (TTNS) approach169,170,172. The QC-TTNS combines a number of favorable features that suggest it might represent a novel, flexible approach in quantum chemistry: the more general concept of data-sparsity inherent in the TNS representation allows for the efficient representation of a much bigger class of wave functions than accessible by state-of-the-art methods. The desired accuracy may be adjusted, so that the ansatz in principle permeates the whole full-CI space. These developments foster the hope that with their help some of the major questions in quantum chemistry and condensed matter physics may be solved. The concept of MPS and tree structured tensor network states has been rediscovered independently in numerical mathematics for tensor product approximation81,183. 1.2 Entanglement and quantum information entropy in quantum chemistry Inquantumsystems, correlationshavingnocounterpartinclassicalphysicsarise. Purestates showing these strange kinds of correlations are called entangled ones11,64,95,103,193,195,227,234, and the existence of these states has so deep and important consequences21,48,63 that Schr¨o- dinger has identified entanglement to be the characteristic trait of quantum mechanics214,215. The QC-DMRG and QC-TTNS algorithms approximate a composite system with strong in- teractionsbetweenmanypairsoforbitals,anditturnedoutthattheresultsofquantuminfor- mationtheory177,257 canbeusedtounderstandthecriteriaoftheirconvergence.127,211,212,238,241. Recently, quantum information theory has also appeared in quantum chemistry giving a fresh impetus to the development of methods in electronic structure theory127,129,171,174,201,277 9,15,17,26,28–30,62,70,105,109,112,145,166. The amount of contribution of an orbital to the total cor- relation can be characterized, for example, by the single-orbital entropy127, and the the sum of all single-orbital entropies gives the amount of total correlation encoded in the wave function129,133. This quantity can be used to monitor changes in entanglement as system parameters are adjusted, for example, changing bond length or other geometrical proper- ties62,70,170. A useful quantity to numerically characterize the correlations (classical and quantum together) between pairs of orbitals is the mutual information15,130,201 and it to- gether with the orbital entropy provides chemical information about the system, especially about bond formation and nature of static and dynamic correlation15,29,30,62. Thetwo-orbitalmutualinformationalsoyieldsaweightedgraphoftheoveralltwo-orbital correlation of both classical and quantum origin reflecting the entanglement topology of the 6 molecules. Therefore, this quantity can also be used to carry out optimization tasks based on theentanglementbetweenthedifferentcomponents–itselfdeterminingthecomplexityofthe computation – since it depends strongly on the chosen network topology and is in principle unknown for a given system. To promote the efficiency of tensor product methods various entanglement-basedapproachesareusedtodetermine,forexample,theappropriateordering, network topology, optimal basis, and efficient network initialization. These important and non-trivial tasks will be considered in Sec. 4. 1.3 Tensor decomposition methods in mathematics Approximation of quantities in high dimensional spaces is a hard problem with a variety of applications, and the development of generic methods that circumvent the enormous com- plexity of this task have recently, independent of the developments in the study of quantum systems, gained significant interest in numerical mathematics82. A recent analysis shows that, beyond the matrix case (corresponding to tensors of order 2), almost all tensor prob- lems, even that of finding the best rank-1 approximation of a given tensor, are in general NP hard97. Although this shows that tensor product approximation in principle is an ex- tremely difficult task, a variety of generic concepts for the approximation of solutions of certain problem classes have recently been proposed83,84, some of which56,78,82,140,141,203,232 bear a surprising similarity to methods used to treat problems in quantum physics86,87. The classical Tucker format attains sparsity via a subspace approximation. Multi- configurational methods like MCSCF or CASSCF are in fact a Tucker approximation in the framework of antisymmetry. Its unfavorable scaling has recently been circumvented by a multilevel or hierarchical subspace approximation framework named Hierarchical Tucker format81,82, interestingly corresponding to the TTNS. A similar format called Tensor Trains, developed independently181,182,208, is a formal version of the MPS with open boundary condi- tions. Investigation of the theoretical properties of TNS and MPS in a generic context have shown that they inherit desirable properties of matrix factorization. E.g., closedness of the set of tensors of fixed block size82 implies the existence of minimizers in these sets for convex optimization problems. Also, these sets possess a manifold structure that helps to remove re- dundancy in the parametrization by the introduction of so-called gauge conditions100. They can be used to set up variational frameworks for the treatment of optimization problems135 and of time-dependent problems56,87,140,143, bearing again a close connection to approaches in the quantum physics community86. In this general context, the robustness and quasi- best approximation of the HOSVD, studied in the mathematics community76,82,83, and of the (one site) DMRG252 as simple and very efficient numerical methods for the treatment of optimization problems are now well-understood82,84. These fundamental properties establish MPS and TTNS as advantageous concepts in tensor product approximation. It is important to note that all these properties are no longer valid in general if the tensor networks contains closed loops, as in case of the projected entangled pair states (PEPS)236 and the multi- scale entanglement renormalization ansatz (MERA)239. It is still widely unexplored under which conditions the favorable properties of tree structured TNS can be extended to general TNS. In mathematics the phrases hierarchical tensor representation or Hierarchical Tucker format as well as Tensor Trains instead of MPS are used since there the focus is not only on quantum mechanical systems, but rather on universal tools to handle high-dimensional 7 approximations. Many of the recent developments in mathematics parallel others in quan- tum computations on a more formal, generic level, often faced with similar experiences and similar problems. 2 Quantum chemistry 2.1 The electronic Schro¨dinger equation A quantum mechanical system of N non-relativistic electrons is completely described by a state-function Ψ depending on 3N spatial variables r R3, a = 1,...,N, together with N a ∈ discrete spin variables s 1 , a = 1,...,N, a ∈ {±2} (cid:110) 1(cid:111)N (cid:16) (cid:110) 1(cid:111)(cid:17)N Ψ : R3N = R3 C ⊗ ±2 ∼ ⊗ ±2 −→ (r ,s ;...;r ,s ) Ψ(r ,s ;...;r ,s ). (1) 1 1 N N 1 1 N N (cid:55)−→ The function Ψ belongs to the Hilbert space L (cid:0)(R3 1 )N(cid:1) having the standard inner 2 ×{±2} product (cid:90) (cid:88) Ψ,Φ = Ψ(r ,s ;...;r ,s )Φ(r ,s ;...;r ,s )dr ...dr , (2) 1 1 N N 1 1 N N 1 N (cid:104) (cid:105) R3N si=±21 (cid:112) and the norm Ψ = Ψ,Ψ . The Pauli antisymmetry principle states that the wave (cid:107) (cid:107) (cid:104) (cid:105) function of fermions, in particular electrons, must be antisymmetric with respect to the permutation of variables, i.e., for a = b (cid:54) Ψ(...;r ,s ;...;r ,s ;...) = Ψ(...;r ,s ;...;r ,s ;...). (3) a a b b b b a a − Such wave-functions are the elements of the antisymmetric tensor subspace (cid:86)N L (cid:0)R3 (cid:1) i=1 2 × 1 . The Pauli exclusion principle immediately follows: Ψ must vanish for the points of ({R±32} 1 )N which have the coordinates r = r and s = s for some a = b fermions47,197. ×{±2} a b a b (cid:54) In quantum mechanics, we are usually interested in wave-functions having definite ener- gies. This is expressed by the stationary Schr¨odinger equation, HΨ = EΨ, (4) i.e., the wave function is an eigenfunction of a differential operator, namely the Hamilton operator H, and the eigenvalue E R is the energy of the corresponding state Ψ. One of ∈ the most important quantities is the ground state energy E , which is the lowest eigenvalue. 0 The well known Born-Oppenheimer-approximation considers a nonrelativistic quantum me- chanical system of N electrons in an exterior field generated by the K nuclei. In this case H is as follows H = H +H , H = H +H , (5a) kin pot pot ext int N N K N (cid:88) 1 (cid:88)(cid:88) Z 1 (cid:88) 1 c H = ∆ , H = , H = . (5b) kin a ext int −2 − R r 2 r r c a b a a=1 a=1 c=1 | − | a,b=1 | − | b(cid:54)=a 8 Since the Hamilton operator is a linear second order differential operator, the analysis for the electronic Schr¨odinger equation has been already established to a certain extent. We would like to briefly summarize some basic results and refer to the literature47,197. The Sobolev spaces Hm := Hm(cid:0)(R3 1 )N(cid:1), m N are defined as the spaces of functions for which all derivatives up to orde×r m{±a2re}in H0 :=∈L0(cid:0)(R3 1 )N(cid:1). Consequently, the oper- 2 ×{±2} ator H maps the Sobolev space H1 continuously into its dual space H−1, i.e., H : H1 H−1 → boundedly47,197,271. The potential operator H maps the Sobolev spaces H1 continuously pot into H0, i.e., H : H1 H0 = L boundedly197,271. The electronic Schr¨odinger operator pot 2 → admits a rather complicated spectrum. We are interested mainly in the ground state energy E . If (cid:80)K Z N, in particular for electrical neutral systems, it is known197,272 that 0 c=1 c ≥ E is an eigenvalue of finite multiplicity of the operator H : H2 H0 below the essential 0 → spectrum σ (H) of H, i.e., < E < infσ (H). Summing up, the energy space for the ess 0 ess −∞ electronic Schro¨dinger equation is N (cid:16)(cid:16) (cid:110) 1(cid:111)(cid:17)N(cid:17) (cid:94) (cid:16) (cid:110) 1(cid:111)(cid:17) = H1 R3 L R3 . (6) N 2 V × ±2 ∩ × ±2 i=1 This situation will be considered in the sequel. For the sake of simplicity, we will also always assume that E is a simple eigenvalue, i.e., 0 of multiplicity one. In the case we deal with here, i.e., the stationary electronic Schro¨dinger equation in non-relativistic and Born-Openheimer setting, we can assume without the loss of generality that the wave function is real valued. (This does not hold for linear response theory or time-dependent problems, as well as for the relativistic regime, where complex phases play an important role.) According to the well known mini-max principle197, the ground state energy and the corresponding wave function satisfies the Rayleigh-Ritz varia- tional principle197,271, i.e., the lowest eigenvalue is the minimum of the Rayleigh quotient (cid:104)Ψ,HΨ(cid:105), or equivalently, (cid:104)Ψ,Ψ(cid:105) (cid:8) (cid:9) E = min Ψ,HΨ : Ψ,Ψ = 1,Ψ , (7a) 0 N (cid:104) (cid:105) (cid:104) (cid:105) ∈ V (cid:8) (cid:9) Ψ = argmin Ψ,HΨ : Ψ,Ψ = 1,Ψ . (7b) 0 N (cid:104) (cid:105) (cid:104) (cid:105) ∈ V Since the Hamilton operator maps H : ( )∗ boundedly, we will put the eigenvalue N N V → V problem into the following weak formulation271, to find the normalized Ψ , satisfying 0 N ∈ V Φ,(H E )Ψ = 0, Ψ ,Ψ = 1, Φ . (8) 0 0 0 0 N (cid:104) − (cid:105) (cid:104) (cid:105) ∀ ∈ V We will consider the above framework47,272 throughout the present paper. 2.2 Full configuration interaction approach and the Ritz-Galerkin approximation A convenient way to approximate the wave function is to use an anti-symmetric tensor product of basis functions depending only on single particle variables (r ,s ), which can be a a realized by determinants. To this end, let us consider a finite subset of an orthonormal set 9 of basis functions ϕ : (r,s) ϕ (r,s) in H1(R3 1 ), that is, i (cid:55)→ i ×{±2} (cid:16) (cid:110) 1(cid:111)(cid:17) Bd := (cid:8)ϕ : i = 1,...,d(cid:9) B := (cid:8)ϕ : i N(cid:9) H1 R3 , i i ⊆ ∈ ⊆ × ±2 (cid:16) (cid:110) 1(cid:111)(cid:17) d := SpanBd := SpanB = H1 R3 , V ⊆ V × ±2 where (cid:90) (cid:88) ϕ ,ϕ := ϕ (r,s)ϕ (r,s)dr = δ . (9) i j i j i,j (cid:104) (cid:105) R3 s=±1 2 (For simplicity of notation, we will use the same brackets , for designating inner products (cid:104)· ·(cid:105) in Hilbert spaces, independent of the underlying Hilbert space.) In quantum chemistry these functions are called spin orbitals, because they depend on the spin variable s = 1 and the spatial variable r R3. In the sequel, we will first confine ourselves to spin±o2rbital ∈ formulations. How we go from spin orbitals to spatial orbitals will be explained later. We build Slater determinants of an N-electron system, by selecting N different indices, for example i for a = 1,...,N, out of the set 1,...,d . By this we have chosen N ortho- a { } normal spin orbitals ϕ , a = 1,...,N, to define the Slater determinant47,226 ia 1 (cid:0) (cid:1)N Φ (r ,s ;...;r ,s ) = det ϕ (r ,s ) [i1,...,iN] 1 1 N N √N! ia b b a,b=1 (10) 1 (cid:88) (cid:0) (cid:1) = P(σ) ϕ ϕ (r ,s ;...;r ,s ), i i 1 1 N N √N! σ(1) ⊗···⊗ σ(N) σ∈SN where the summation goes for all σ permutations of N elements, and P(σ) is the parity of the permutation. To fix the sign of the determinant, we suppose e.g. that i < i for a = a a+1 1,...,N 1; i.e., the indices are ordered increasingly. Therefore the Slater determinants are − uniquely defined by referring to the orbital functions ϕ , respectively indices i 1,...,d , ia a ∈ { } which are contained in the determinant. It is easy to check that the Slater determinants constructed in this way by the ortonor- malized spin-orbitals ϕ d are also orthonormalized. We define the Full Configuration i ∈ V Interaction (FCI) space for an N-electron system96,226 as the finite dimensional space d VN spanned by the Slater-determinants (cid:8) (cid:9) (cid:8) (cid:9) Bd := Φ : 1 i < i d B := Φ : 1 i < i , N [i1,...,iN] ≤ a a+1 ≤ ⊆ N [i1,...,iN] ≤ a a+1 ⊆ VN d := SpanBd := SpanB . VN N ⊆ VN N The dimension of d is VN (cid:18) (cid:19) d d! dim d = = (dN). (11) VN N N!(d N)! ∼ O − To obtain an approximate solution of the electronic Schro¨dinger equation, one may apply the Ritz-Galerkin method using the finite dimensional subspace d . I.e., consider the VN ⊂ VN solution of the finite dimensional eigenvalue problem Πd HΨ = EΨ, Ψ d, (12) N ∈ VN 10