Compact Wavefunctions from Compressed Imaginary Time Evolution

Compact wavefunctions from compressed imaginary time evolution

Jarrod R. McClean1 and AlánAspuru-Guzik1 1Department of Chemistry and Chemical Biology, Harvard University, Cambridge MA, 02138 Simulation of quantum systems promises to deliver physical and chemical predictions for the frontiers of technology. Unfortunately, the exact representation of these systems is plagued by the exponential growth of dimension with the number of particles, or colloquially, the curse of dimen- sionality. The success of approximation methods has hinged on the relative simplicity of physical systems with respect to the exponentially complex worst case. Exploiting this relative simplicity has required detailed knowledge of the physical system under study. In this work, we introduce a general and efficient black box method for many-body quantum systems that utilizes technology from compressed sensing to find the most compact wavefunction possible without detailed knowledge of the system. It is a Multicomponent Adaptive Greedy Iterative Compression (MAGIC) scheme. No knowledge is assumed in the structure of the problem other than correct particle statistics. This method can be applied to many quantum systems such as spins, qubits, oscillators, or electronic systems. As an application, we use this technique to compute ground state electronic wavefunctions of hydrogen fluoride and recover 98% of the basis set correlation energy or equivalently 99.996% of the total energy with 50 configurations out of a possible 107. Building from this compactness, we introduce the idea of nuclear union configuration interaction for improving the description of reaction coordinates and use it to study the dissociation of hydrogen fluoride and the helium dimer.

I. INTRODUCTION of quantum systems [13, 14], but little has been done to exploit the possibilities for many-body eigenstates, which The prediction of chemical, physical, and material are critically important in the analysis and study of phys- properties from first principles has long been the goal of ical systems. computational scientists. The Schrödingerequation con- In this work, we concisely describe a new method- tains the required information for this task, however its ology for finding compact ground state eigenfunctions exact solution remains intractable for all but the small- for quantum systems. It is a Multicomponent Adaptive est systems, due to the exponentially growing space in Greedy Iterative Compression (MAGIC) scheme. This which the solutions exist. To make progress in prediction, method is general in that it is not restricted to a spe- many approximate schemes have been developed over the cific ansatz or type of quantum system. It operates by years that treat the problem in some small part of this expanding the wavefunction with imaginary time evo- exponential space. Some of the more popular methods lution, while greedily compressing it with orthogonal in both chemistry and physics include Hartree-Fock, ap- matching pursuit [15]. As an application, we choose proximate density functional theory, valence bond the- the simplest possible ansatze for quantum chemistry, ory, perturbation theory, coupled cluster methods, multi- sums of non-orthogonal determinants, and demonstrate configurational methods, and more recently density ma- that extremely accurate solutions are possible with very trix renormalization group [1–10]. compact wavefunctions. This non-orthogonal MAGIC These methods have been successful in a wide array of scheme we refer to as NOMAGIC, and apply it electronic problems due largely to the intricate physics they com- wavefunctions in quantum chemistry. pactly encode. For example, methods which are essen- tially exact and scale only polynomial with the size of the II. COMPRESSED IMAGINARY TIME system have been developed for one-dimensional gapped EVOLUTION quantum systems [11]. However such structure is not always easy to identify or even present as the size and complexity of the systems grow. For example, some bi- Beginning with general quantum systems, an N- particle eigenfunction of a quantum Hamiltonian H, Ψ , ologically important transition metal compounds as well | i arXiv:1409.7358v1 [physics.chem-ph] 25 Sep 2014 as metal clusters lack obvious structure, and remain in- may be approximated by a trial function Ψ˜ that is the sum of many-particle component functions| iΦi , such tractable with current methods. | i The field of compressed sensing exploits a general type that of structure, namely simplicity or sparsity, which has N Xc been empirically observed and is adaptive to the prob- Ψ˜ = c Φi (1) | i i | i lem at hand. Recent developments in compressed sensing i have revived the notion that Occam’s razor is at work in physical systems. That is, the simplest feasible solution where Nc is the total number of configurations in the sum is often the correct one. Compressed sensing techniques and no relation need be assumed between Φi and Φj have had success in quantum simulation in the context for i = j. A simple example of such a component| i function| i of localized wavefunctions [12] and vibrational dynamics for a6 general quantum system is the tensor product of N 2 single particle functions φi | ji i i i i Φ = φ0 φ1 ... φN 1 (2) | i | i | i | − i and we will consider its anti-symmetric counterpart in applications to electronic systems. In this work, we adopt a state to be simple, sparse, or compact in the total space if the number of configurations Nc needed to represent a state to a desired precision is much less than the total dimension of the Hilbert space. One method for determining Ψ˜ is a direct variational approach based on the particular| parametrizationi of Φi | i and choice of Nc. This approach can plagued by issues related to the choice of initial states, difficulty of adding new states, and numerical instability of the optimization procedure if proper regularization is not applied [16–20]. We present an alternative technique that selects the important configurations in a black-box manner and is naturally regularized to prevent numerical instability. It is built through a combination of imaginary-time evolution and compressed sensing. Imaginary-time evolution can be concisely described as follows. Given a quantum FIG. 1. A schematic diagram of the MAGIC approach. At system with a time-independent Hamiltonian H and as- each iteration the wavefunction is expanded by means of the imaginary time propagator G, and subsequently com- sociated eigenvectors χi , any state of the system Ω | i | i pressed with orthogonal matching pursuit. The imaginary may be expressed in terms of those eigenvectors as time propagator corresponding to projection into the manifold at |Ψ(τ)i, denoted |δΨ(τ)i, typically prescribed by differential X i Ω = ci χ (3) time dependent variational principles is given as GVP and de- | i | i i picted to emphasize that expansion with the operator G can explore a greater part of Hilbert space. The compression is and the the evolution of the system for imaginary-time τ performed simultaneously with expansion in our implemen- is given as tation to prevent rapid growth of the wavefunction. These steps are iterated until convergence at a specified maximum Hτ X Eiτ i G Ω = e− Ω = c e− χ (4) number of component functions, at which point an optional | i | i i | i i variational relaxation may be performed. where E0 < E1 E2... EN 1 are the eigenenergies as- ≤ ≤ − sociated with χ . By evolving and renormalizing, even- the manifold spanned by linear variations in the func- | ii tually one is left with only the eigenvector associated with tion at the previous time step, sometimes referred to as the lowest eigenvalue, or ground state. Excited states Galerkin or time-dependent variational methods includ- may be obtained with a number of approaches includ- ing imaginary time MCTDH [28, 29] and DMRG in some 2 ing spectral transformations (e.g. H0 = (H λ) [21]), limits [27]. While computationally convenient, it is often − matrix deflation, or other techniques. However we will unclear how projection onto the original linear subspace concern ourselves only with ground states in this work. at every time can affect evolution with respect to the ex- Imaginary time evolution approaches may be broadly act evolution. In this work, we show that the first class grouped into two classes. The first class involves the of explicit evolution can be used on any ansatz without explicit application of the imaginary-time propagator G configuration explosions or stochastic sampling by uti- to the wavefunction. This approach typically generates lizing a technique from the field of compressed sensing, many configurations at every step, causing a rapid ex- namely orthogonal matching pursuit [15, 30]. pansion in the size of the wavefunction. As a result, The algorithm we use is diagrammed in Fig 1, and these methods have almost exclusively been restricted to proceeds iteratively as follows. The wavefunction at time Monte Carlo sampling procedures which attempt to as- τ = 0, Ψ(τ) , may be any trial wavefunction that is suage this explosion by stochastically sampling or select- not orthogonal| i to the desired eigenstate. We determine ing the most important configurations [22, 23], however the wavefunction at time τ + dτ = τ 0 greedily, fitting i the recently developed imaginary time-evolving block one configuration Φ (τ 0) at a time by maximizing the decimation also belongs to this class, performing trunca- functional | i tions after expansion along a virtual bond dimension [24– i P i j 27]. Φ (τ 0) G Ψ(τ) c (τ 0) Φ (τ 0) Φ (τ 0) h | | i − j

i 4 with respect to the parameters that determine Φ (τ 0) . 10− Such that after k iterations, the wavefunction is given| byi 1.0 × FCI 2 16 0.8 k RHF 4 24 X Ψ(˜ τ) = c (τ) Φi(τ) (6) | i i | i 0.6 i

) 0.4 h The coefficients in this expansion, ci(τ 0) are solved for simultaneously after each iteration by orthogonal projec- 0.2 tion, which after simplification reduces to the following ∆ E (E 0.0 linear system for the coefficient vector c 0.2 Sc = v (7) − 0.4 i j i − where Sij = Φ (τ 0) Φ (τ 0) and vi = Φ (τ 0) G Ψ(τ) . h | i h | | i 0.6 Together, the fit and orthogonal projection step is equiv- − 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 alent to orthogonal matching pursuit [15, 30] applied to the signal generated by the imaginary time evolution of R(A)˚ the state at each time step G Ψ(τ) . The expansion- compression procedure is advanced| i to the next imagi- FIG. 2. The bond dissociation curve of the helium dimer in nary time step either when some accuracy convergence the aug-cc-pVDZ basis showing rapid and consistent conver- criteria is met, or when some pre-set maximum number gence in the number of non-orthogonal determinants. These of components N is reached, and the total simulation represent the nuclear union curves constructed from a number c of local determinants at each nuclear point given by the line is terminated when the wavefunction converges between label, and are sampled at a spacing of 0.04 A.˚ The curves have imaginary-time steps. We provide additional details of been shifted by the tail values in order to allow comparison the numerical procedure in the supporting information of the features for this sensitive bond, and the 24 determi- for interested readers. nant curve is indistinguishable from the FCI solution in the Note that one is free to choose a convenient form for graphic. At a point near the equilibrium geometry, R = 3.01 the propagator G. In this work we use the linearized A,˚ the 24 determinant curve with the nuclear union config- propagator G (1 dτH), which is both easy to im- uration interaction technique recovers 99.9899% of the basis plement and provably≈ − free of bias in the final result for set correlation energy, or equivalently 99.9999% of the total finite single particle basis sets given some restrictions on energy. dτ [31]. Orthogonal matching pursuit attempts to find the sparsest solution to the problem of state reconstruc- standard procedure of expanding the linear state space tion [15, 32], and thus is ideal for keeping the number by excitation known as configuration interaction (CI), of configurations minimal throughout the imaginary time which can eventually yield the numerically exact solution evolution. However, while the solution is sparsest in the within a basis when the full state space has been covered. limit of total reconstruction and naturally regularized This is referred to as full configuration interaction (FCI) against configuration collinearity, for very severe trunca- and is the standard to which we compare. Comparison tions of the wavefunction, the sparsifying conditions gen- to methods considering explicit correlation beyond that erate a solution which is not variationally optimal for the covered by a specific traditional Gaussian basis, such as given number of configurations. For this reason, we fin- explicitly correlated f12 type wavefunctions, are not yet ish the computation with a total variational relaxation of within the scope of this work. the expectation value of the energy with respect to both In the context of our approach, the indistinguishability coefficients and states. This retains both the benefits of of electrons necessitates handling of anti-symmetry. The imaginary time evolution in avoiding local energetic min- simplest way to include anti-symmetry into the wave- ima and of variational optimality in the final solution. function is by utilizing anti-symmetric component ten- sors Φi . The most common anti-symmetric component function| i is the Slater determinant, such that we express III. APPLICATION TO CHEMICAL SYSTEMS the wavefunction as

N Xc The method we have outlined may be readily applied Ψ = c Φi (8) | i i | i to any quantum system, such as spins or oscillators, how- i ever as a first application we consider ground-state electronic wavefunctions of molecules. We will take the ap- where Φi are Slater determinants with no fixed rela- proach that is conventional to the field of quantum chem- tions between| i Φi and Φj for i = j. While this sim- istry, and solve the problem in a basis of Gaussian-type ple form lacks| extensivityi | [33],i it is6 attractive for other functions [8]. After a basis has been selected, there is a reasons. Namely the quality of description and rate of 4

convergence in Nc are invariant to invertible local trans- 1 10− formations of the state (i.e. atomic orbitals vs. natural 5 × orbitals) [19], and the mathematical machinery related to the use and extension of such a wavefunction is already 4 well developed [34–37] 3 While the method we use for determinant selection is unique, non-orthogonal Slater determinants have been 2 ) used successfully in valence bond theory [4, 5] as well h 1 as more recent symmetry breaking and projection methods [38, 39]. Unconstrained non-orthogonal Slater deter- E (E 0 minants have been utilized before, but in a purely variational context [16, 20]. Using this machinery yields ex- 1 − plicit gradients that we utilize in the optimization of de- 2 FCI 2 8 terminants [37]. The scaling of these constructions with − RHF 4 32 2 2 3 current algorithms is O(Nc max(M ,Ne )) [33] where Nc 3 − is the number of determinants and Ne is the number of 1.0 1.5 2.0 2.5 3.0 3.5 4.0 electrons, however development of approximations in this R(A)˚ area have received comparatively less attention with respect to orthogonal reference wavefunction methods and FIG. 3. The bond dissociation curve of hydrogen fluoride in there may be ways to improve upon this scaling. the cc-pVDZ basis showing rapid convergence in the number We introduce an additional enhancement for the study of non-orthogonal determinants. These are the nuclear union of chemical reactions that is greatly facilitated by the curves constructed from a number of local determinants at compactness of our expansions. Namely, when consider- each nuclear point given by the line label, and are sampled at a spacing of 0.04 A.˚ The 32 determinant curve is nearly indis- ing a full reaction coordinate, such as that for a bond tinguishable from the FCI curve in this graphic. At a point dissociation, we perform an additional linear variational near the equilibrium geometry, R = 0.93 A,˚ the 32 determi- calculation in the space of components (determinants) nant curve with the nuclear union configuration interaction found locally at neighbouring nuclear configurations. As technique recovers 98.6% of the basis set correlation energy, a proof of principle, we include configurations from the or equivalently 99.997% of the total energy. entire curves in the following examples, but more economical truncations can be used as well. We refer to this additional step, as the nuclear union configuration with rapid convergence to a quantitative approximation interaction method and describe it in more depth in the by 32 determinants. supplemental information. In Fig. 4 we select two points on the HF dissociation As a first application, we consider He2 in the aug-cc- curve, and study the convergence of the energy as a func- pVDZ basis [40]. The helium dimer is unbound in the tion of the number of determinants in the NOMAGIC case of a single determinant with restricted Hartree Fock method and a traditional CI expansion with the canon- and is not held together by a covalent bond, but rather ical Hartree-Fock orbitals. In particular, we study both dispersive forces and dynamical correlation only. In Fig. a point near the equilibrium bond length (R = 0.93 A)˚ 2, we consider the dissociation of this molecule under where traditional CI expansions perform relatively well different numbers of non-orthogonal determinants. De- and a more stretched geometry (R = 1.73 A)˚ where tra- spite the sensitive nature of this bond, it is qualitatively ditional CI expansions perform more poorly. We see that captured with as few as 4 local determinants and quan- in both cases, if one considers a fixed level of accuracy in titatively captured with as few as 24 determinants. The the energy, the NOMAGIC method is considerably more dimension of the space of this molecule is on the order of compact. For example, to achieve a level of accuracy su- 104 when reduced by considerations of point group sym- perior to the CISDT expansion that uses 36021 determi- metry. The NOMAGIC approach does not yet utilize any nants, NOMAGIC requires only 24 determinants at both symmetry other than the spin symmetry enforced by the geometries. That is, for equivalent accuracy, the NO- parameterization of the wavefunction. MAGIC wavefunction is roughly 1500 times more com- As a second example, the dissociation of hydrogen flu- pact in the space of Slater determinants. By 50 determi- 7 oride in a cc-pVDZ basis [41] is studied. The total config- nants out of a possible 10 in the NOMAGIC wavefunc- uration space for this molecule is on the order of 107 and tion, we recover 98% of the basis set correlation energy it involves a homolytic bond breaking of a covalent bond or equivalently 99.996% of the total energy. in the gas phase. Considering the results in Fig. 3, one can see that while restricted Hartree Fock (RHF) yields an unphysical dissociation solution, as few as 2 deter- IV. CONCLUSIONS minants are sufficient to fix the solution in a qualitative sense. Beyond this, the addition of more determinants In this work, we introduced a general method to find represents a monotonically increasing degree of accuracy, compact representations of quantum eigenfunctions by 5

1 with imaginary-time evolution in quantum systems. 10− 3.0 Given a quantum state Ψ(τ 0) that one wishes to recon- × | i CI-Eq NOMAGIC-Eq struct, orthogonal matching pursuit is a greedy decom- 2.5 CI-St NOMAGIC-St position algorithm that approximates the sparse problem of ﬁnding Ψ(˜ τ 0) such that | i 2.0 2 ) min Ψ(τ 0) Ψ(˜ τ 0) h 2 Ψ(˜ τ 0) || | i − | i || | i 1.5 subject to Ψ(˜ τ 0) < N (9) || | i ||0

Error (E 1.0 This is done by considering an overcomplete dictionary i Φ (τ 0) that can express Ψ˜ as {| i} | i 0.5 X i Ψ(˜ τ 0) = c (τ 0) Φ (τ 0) (10) | i i | i 0.0 i 0 1 2 3 4 5 6 7 8 i Log N Determinants and at each stage selecting selecting the Φ (τ 0) which 10 maximizes the overlap with the residual with| respecti to the target signal Ψ(τ 0) , FIG. 4. A curve of the energetic error with respect to full CI | i for HF bond dissociation in the cc-pVDZ basis as a function i P i j Φ (τ 0) Ψ(τ 0) cj(τ 0) Φ (τ 0) Φ (τ 0) of the log of the number of determinants included for both max |h | i − j

We thank Prof. John Parkhill and Dr. Dmitrij Rap- i j for the coefficient vector c, where Sij = Φ (τ 0) Φ (τ 0) , poport for their valuable comments on the manuscript. i h | i vi = Φ (τ 0) Ψ(τ 0) , and ci = ci(τ 0). J.M is supported by the Department of Energy Computa- Throughouth | thisi procedure, one also has a choice of tional Science Graduate Fellowship under grant number how to represent the target signal Ψ(τ 0) . In some cases, DE-FG02-97ER25308. A.A.G thanks the National Sci- | i it is feasible to construct Ψ(τ 0) explicitly from a previ- ence Foundation for support under award CHE-1152291. ous time step and imaginary| timei propagator G, and do- ing so could potentially facilitate the optimization procedure by examining properties of the state. However, ex- VI. SUPPLEMENTAL INFORMATION act expansion of the state Ψ(τ 0) using G can have many terms for even modestly sized| quantumi systems, negating A. Orthogonal Matching Pursuit the potential benefits of compressing the wavefunction. In practice, we found that a much better approach is to In this section, we offer some additional details on directly with G Ψ(τ) without first expanding the wave- the implementation of Orthogonal Matching Pursuit [15] function explicitly.| Wheni using the linearized propagator 6

G (1 dτH), this means that Hamiltonian and over- This yields a convenient construction for the overlap be- lap≈ matrix− elements and their derivatives are sufficient tween two component functions for the implementation of the procedure. K L K L Φ Φ = M = det (V ) = det T †ST (18) In principle, at any time step, one may continue to h | i KL KL i add elements Φ (τ 0) until an arbitrary convergence tol- One quantity of convenience is the so-called transition | i ˜ erance is reached, i.e. Ψ(τ 0) Ψ(τ 0) 2 < for some density matrix defined between determinants K and L > 0. However, as only|| | thei final − | statei || in the large τ KL K L K 1 L limit is of interest, and any state that is not completely P = T T †ST − T † (19) orthogonal to this state will eventually converge to it, some errors in intermediate steps are permissible. Thus Hamiltonian matrix elements may be written as a more economical approach, is to terminate the addi- h i 1 i H = M Tr P KLhˆ + Tr P KLGKL (20) tion of states Φ at intermediate time steps according KL KL 2 to some proxy,| suchi as sufficient decrease in the energy E˜(τ 0) = Ψ(˜ τ 0) H Ψ(˜ τ 0) from the previous time step. ˆ h | | i where h are the single electron integrals, Z 2 ! r X Zi h = dσ φ∗ (σ) ∇ φ (σ) (21) B. Electronic Wavefunction Parameterization µν µ − 2 − R r ν i | i − | (22) Here we detail the electronic wavefunction parametrization used in this work, as well as the where σ = (r, s) denotes electronic spatial and spin vari- expressions used for the implementation of orthogonal ables and the nuclear positions and charges are Ri and KL matching pursuit and variational relaxation in electronic Zi. G is given by systems. ! In quantum chemistry, frequently one first chooses a X GKL = P KL(g g ) (23) suitable single particle spin-orbital basis for the descrip- µν λσ µνλσ − µλνσ tion of the electrons, which we denote φ . This basis λσ {| ii} typically consists of atom-centered contracted Gaussian with the two electron integrals gµνλσ type functions with a spin function, and are in general Z non-orthogonal such that they have an overlap matrix φµ∗ (σ1)φν (σ1)φλ∗ (σ2)φσ(σ2) gµνλσ = dσ1 dσ2 (24) defined by r1 r2 | − | From the description of orthogonal matching pursuit, S = φ φ (13) ij h i| ji we see that to utilize non-linear optimization of the component functions Φk with analytic first derivatives, one Linear combinations of these atomic orbitals are used to | i form molecular orbital functions needs the variations of HKL and MKL with respect to T K . Allowing variations δT K , the required expressions X χ = ci φ (14) in the non-orthogonal spin orbital basis are as follows: | mi m | ii i L KL 1 K δMKL = MKL Tr ST (V )− δT † (25) KL KL K KL 1 L which have an inner product δP = [1 P S]δT (V †)− T † (26) − ! X i i X i j KL X KL χm χn = cm∗cn φi φj = cm∗cnSij (15) δG = δP (g g ) (27) h | i h | i µν λσ µνλσ − µλνσ i,j i λσ A = Tr P KLGKL (28) In our implementation, the N electron component wave- KL − KL KL L KL 1 K functions may be formed from the anti-symmetrized δA = Tr (1 SP †)G †T (V )− δT † KL − N fold product of molecular orbital functions, also (29) known− as Slater determinants. One must take care in implementing this expression, as k k k k it is a special case of the adjugate relations that is only Φ = χ0 χ1 ... χN 1 (16) | i A | i | i | − i strictly valid when V KL is non-singular. To use this ex- where is the anti-symmetrizing operator. A convenient pression in evaluating cases when V KL is singular, tech- computationalA representation of an anti-symmetric com- niques developed elsewhere utilizing the singular value ponent function Φk is given by the coefficient matrix decomposition of V KL and exact interpolation can be | i used [36]. Note also that numerical simplifications are K K K K T = c0 c1 ... cN 1 (17) possible by explicitly considering spin (α, β) and noting | | | − that T K = T Kα T Kβ. These reductions of the above which denotes an M N matrix whose m’th column are equations are straightforward⊕ and we do not give them the coefficients defining× the m’th molecular orbital χk . here. | mi 7

k i C. Nuclear Union Configuration Interaction H(R0) as ΦR0 = Φ where i is now an index set vari- able that runs| i over| alli the component functions at all the geometries being considered. This could be a whole reac- In this section we give some of the details of the nuclear tion coordinate, or simply neighbouring points depending union configuration interaction method used to improve on computational restrictions or chemical/physical con- the description of reaction coordinates. In the study of a siderations. At each nuclear configuration R we find new set of related problems, such as set of electronic Hamil- coefficients ci(R) by solving tonians differing only by the positions of the nuclei, one would like to describe each configuration with an equiv- H(R)C = SCE (30) alent amount of accuracy, to get the best relative features possible. In multi-reference methods, this is often for its ground state eigenvector, and we define done by selecting the same active space at each configu- H(R) = Φi H(R) Φj (31) ration, and rotating the orbitals and coefficients at each ij h | | i i j geometry accordingly. In the nuclear union configuration Sij = Φ Φ (32) interaction method, we propose each reuse of the compo- h | i nents(determinants) found locally at other geometries to Note that the overlap matrix may become singular, as give a totally identical variational space for all nuclear configurations from nearby geometries are often very sim- configurations. As the wavefunctions produced by the ilar. This can be handled either through canonical or- NOMAGIC method are especially compact, this intro- thogonalization [8] or by removing redundant configura- duces little extra overhead to the method as a whole. tions before attempting the diagonalization procedure. Moreover, one might expect that additional compression Specifically, denote the component functions found at is possible in this space, and this is the subject of current nuclear configuration R0 with corresponding Hamiltonian research.

[1] J. C. Slater, Phys. Rev. 81, 385 (1951). [19] W. Hackbusch, Tensor spaces and numerical tensor cal- [2] E. Baerends, D. Ellis, and P. Ros, Chem. Phys. 2, 41 culus, Vol. 42 (Springer, 2012). (1973). [20] H. Goto, M. Kojo, A. Sasaki, and K. Hirose, Nanoscale [3] R. G. Parr and W. Yang, Density-functional theory of Res. Lett. 8, 1 (2013). atoms and molecules, Vol. 16 (Oxford University Press, [21] J. K. L. MacDonald, Phys. Rev. 46, 828 (1934). 1989). [22] W. A. Lester, B. Hammond, and P. J. Reynolds, Monte [4] W. J. Hunt, P. J. Hay, and W. A. Goddard, J. Chem. Carlo methods in ab initio quantum chemistry (World Phys. 57, 738 (1972). Scientific, 1994). [5] W. A. Goddard III, T. H. Dunning Jr, W. J. Hunt, and [23] G. H. Booth, A. J. Thom, and A. Alavi, J. Chem. Phys. P. J. Hay, Acc. Chem. Res. 6, 368 (1973). 131, 054106 (2009). [6] C. Möllerand M. S. Plesset, Phys. Rev. 46, 618 (1934). [24] G. Vidal, Phys. Rev. Lett. 93 (2004). [7] R. J. Bartlett, Ann. Rev. Phys. Chem. 32, 359 (1981). [25] U. Schollwöck, Rev. Mod. Phys. 77, 259 (2005). [8] T. Helgaker, P. Jorgensen, and J. Olsen, Molecular Elec- [26] B. K. Clark and H. J. Changlani, ArXiv e-prints (2014), tronic Structure Theory (Wiley, Sussex, 2002). arXiv:1404.2296 [cond-mat.str-el]. [9] S. R. White and R. L. Martin, J. Chem. Phys. 110, 4127 [27] J. Haegeman, C. Lubich, I. Oseledets, B. Vander- (1999). eycken, and F. Verstraete, ArXiv e-prints (2014), [10] G. K.-L. Chan and S. Sharma, Ann. Rev. Phys. Chem. arXiv:1408.5056 [quant-ph]. 62, 465 (2011). [28] M. H. Beck, A. Jäckle, G. Worth, and H.-D. Meyer, [11] Z. Landau, U. Vazirani, and T. Vidick, ArXiv e-prints Phys. Rep. 324, 1 (2000). (2013), arXiv:1307.5143 [quant-ph]. [29] M. Nest, T. Klamroth, and P. Saalfrank, J. Chem. Phys. [12] V. Ozoli, R. Lai, R. Caflisch, and S. Osher, Proc. Natl. 122, 124102 (2005). Acad. Sci. U.S.A. 110, 18368 (2013). [30] Y. C. Pati, R. Rezaiifar, and P. Krishnaprasad, in Sig- [13] Y. Wu and V. S. Batista, J. Chem. Phys. 118, 6720 nals, Systems and Computers, 1993. 1993 Conference (2003). Record of The Twenty-Seventh Asilomar Conference on [14] X. Chen and V. S. Batista, J. Chem. Phys. 125, 124313 (IEEE, 1993) pp. 40–44. (2006). [31] N. Trivedi and D. M. Ceperley, Phys. Rev. B 41, 4552 [15] J. A. Tropp and A. C. Gilbert, IEEE Trans. Inf. Theory (1990). 53, 4655 (2007). [32] D. Needell and R. Vershynin, Found. Comput. Math. 9, [16] H. Koch and E. Dalgaard, Chem. Phys. Lett. 212, 193 317 (2009). (1993). [33] E. J. Sundstrom and M. Head-Gordon, J. Chem. Phys. [17] T. G. Kolda and B. W. Bader, SIAM review 51, 455 140, 114103 (2014). (2009). [34] M. Head-Gordon, P. E. Maslen, and C. A. White, J. [18] M. Espig and W. Hackbusch, Numerische Mathematik Chem. Phys. 108, 616 (1998). 122, 489 (2012). [35] P.-O. Löwdin,Phys. Rev. 97, 1474 (1955). 8

[36] C. Amovilli, Quantum systems in chemistry and Scuseria, J. Chem. Phys. 139, 204102 (2013). physics. trends in methods and applications, edited by [39] L. Bytautas, C. A. Jiménez-Hoyos, R. Rodr´ıguez- R. McWeeny, J. Maruani, Y. Smeyers, and S. Wil- Guzmán, and G. E. Scuseria, Mol. Phys. 112, 1938 son, Topics in Molecular Organization and Engineering, (2014). Vol. 16 (Springer Netherlands, 1997) pp. 343–347. [40] D. E. Woon and T. H. Dunning, J. Chem. Phys. 100, [37] L. Song, J. Song, Y. Mo, and W. Wu, J. Comp. Chem. 2975 (1994). 30, 399 (2009). [41] T. H. Dunning, J. Chem. Phys. 90, 1007 (1989). [38] C. A. Jiménez-Hoyos, R. Rodr´ıguez-Guzmán, and G. E.