Tensor Networks for Ab Initio Quantum Systems
a
E.M. Stoudenmire Sep 2019 - Jülich This talk:
The many-electron problem
Tensor networks
Intro to quantum chemistry
Chemistry approaches for tensor networks The Many-Electron Problem Behavior of electrons in matter: • continuum problem • three dimensional • strong interaction (repulsion) between electrons
Credit: MARK GARLICK/SCIENCE PHOTO LIBRARY/Getty Images Can simplify various ways: • Born-Oppenheimer approximation (classical nuclei) • ignore most relativistic effects
Figure credit: iqmol.org Then problem simplifies to @ i = Hˆ
1 2 1 Hˆ = ˆ† + v(r) ˆ + u(r, r0) ˆ† ˆ† ˆ ˆ r r r r0 r0 r
u(r, r0) = two-body interaction
The "electronic structure problem" Form of the one-body potential
Z = nucleus v(r)= a r Ra
Z
R
R
Z
R
Attraction to classical point nuclei, atomic number Z Form of the two-body potential
1 = nucleus u(r, r0)= r r
r
Coulomb repulsion of electrons Accurate ground state energy of electronic structure problem extremely useful
Example: energy of two atoms, distance R
E R
equilibrium bond length
R binding energy
vibrational frequency Most techniques require discrete and finite system To achieve this can: • integrate out "core" electrons (pseudopotentials) • treat high energy states with approximations such as perturbation theory • project electron motion to certain orbitals
Credit: EVA PAVARINI Regardless of discretization, system becomes a "lattice" with four states per "site"
0, , , Four states of a site are
† † † H = tijcˆi cˆj + VVijklcˆi cˆ cˆk 0 cˆl
(r) cˆi = i(r) ˆ (r) E.g. if projecting into orbitals
{ }
(r ) (r ) (r ) (r ) V i 1 j 2 k 2 l 1 V
1:
2:
3: . .
0, , , Four states per site
N = 10, 410 ~ 106
N = 20, 420 ~ 1012
N = 30, 430 ~ 1018
For N > 130, number of amplitudes greater than number of atoms in the known universe But can "nature's computer" really work this way?
Are the amplitudes of a realistic wavefunction all different?
Or is there some relationship between them?
� Tensor Network Wavefunctions H ˆ is a 4N x 4N matrix
= wavefunction ~ has 4N components )
Hˆ ~ = E ~ · Most of these eigenvectors / wavefunctions are indeed exponentially complex
~ = But some have hidden structure that makes them tractable
~ ≃
• low-energy states • equilibrium states • short-time dynamics Wavefunction a rule, mapping configurations to numbers
0
0.1 Wavefunction a rule, mapping configurations to numbers
0
0.05 Wavefunction a rule, mapping configurations to numbers
0
-0.2 Formally a tensor with N indices
s1 s2 s3 s4 s5 s6
s s s s s s 1 2 3 4 5 6 =
4N numbers inside s1 s2 s3 s4 s5 s6
s s s s s s 1 2 3 4 5 6 =
Problem seems hopeless (maximum N ~ 20)
Physical intuition: weak correlations between distant particles
s s h i ji
i j | | Neglect correlations
s1s2s3s4s5s6 s1 s2 s3 s4 s5 s6 '
s1 s2 s3 s4 s5 s6
Expected values of individual sites ok
Missing correlations Restore correlations locally
s1s2s3s4s5s6 s1 s2 s3 s4 s5 s6 ' i1 i1 i2 i2 i3 i3 i4 i4 i5 i5
s1 s2 s3 s4 s5 s6
matrix product state (MPS)
Local expected values accurate
Exponentially decaying correlations Name matrix product state derives from
s1s2s3s4s5s6 s1 s2 s3 s4 s5 s6 ' i1 i1 i2 i2 i3 i3 i4 i4 i5 i5
s3=0
i2i3
s = 3 " = M " i2i3
s3= # = M # i2i3 i2i3
s3= "# = M"# i2i3
" # " " # M1"M2# M3" M4" M5# ⇡
0
" # # " "# M1"M2# M3# M4" M
s1s2s3s4s5 s1 s2 s3 s4 s5 = M1 M2 M3 M4 M5
For typical matrix size m m ⇥
N 2
M1# M2" M3" M4# M5"M6"
Why this rule? 1. Principled
Large enough matrices, represent any wavefunction
Proof (Hastings, 2007) that one-dimensional systems with exponentially decaying correlations* are 'close' to matrix product states
Error decreases rapidly (exponentially in practice) and error per site is constant
*(technically: gap between ground state and excited states) 2. Powerful Algorithms
Matrix product ansatz comes with sophisticated optimization techniques
DMRG algorithm White, PRL 69, 2863 (1992) Stoudenmire, White, PRB 87, 155137 (2013) "In one dimension... it is at the moment the closest to an ultimate weapon as one can dream of." – Thierry Giamarchi "Quantum Physics in One Dimension"
Very often get exact answer
Only takes as many parameters as needed 3. Extendable
Generalize to tensor networks
Apply to two-dimensional systems, infinite systems, ... Draw N-index tensor as blob with N lines
s1 s2 s3 s4 sN s s s s 1 2 3··· N = Diagrams for simple tensors
vector vj j
matrix Mij i j
3-index Tijk i k tensor j Joining lines means contraction, omit names
Mijvj i j j X
AijBjk = AB
AijBji =Tr[AB] Matrix product state in diagram notation
s1s2s3s4s5s6 s1 s2 s3 s4 s5 s6 = M↵1 M↵1↵2 M↵2↵3 M↵3↵4 M↵4↵5 M↵5 ↵ X
s1 s2 s3 s4 s5 s6 = ↵1 ↵2 ↵3 ↵4 ↵5
Suppress index names, very convenient Besides matrix product state network, other very interestingG. EVENBLY AND G. VIDAL networks are PEPS and PHYSICAL REVIEW B 79, 144108 ͑2009͒ MERA
PEPS MERA FIG. 1. ͑Color online͒ Quantum circuit corresponding to a specific realization of the MERA, namely, theC binary 1D MERA of Fig. 2.(2D In this particularsystems) example, circuit is made of gates involv- (critical systems) C ing two incoming wires and two outgoing wires, p=pin=pout=2. Some of the unitary gates in this circuit have one incoming wire in the fixed state ͉0͘ and can be replaced with an isometry w of type ͑1,2͒. By making this replacement, we obtain the isometric circuit of Fig. 2. FIG. 2. ͑Color online͒͑Top͒ Example of a binary 1D MERA for a lattice with N=16 sites. It contains two types of isometric L Evenbly,tensors, Vidal, organized PRB in 79T=4, 144108 layers. The (2009) input ͑output͒ wires of a tation of two-point correlators͒ and also leads to a much tensor are those that enter it from the top ͑leave it from the bottom͒. more convenient generalization in two dimensions. Verstraete,The top Cirac, tensor is cond-mat/0407066 of type ͑1,2͒ and the rank Х T(2004)of its upper index determines the dimension of the subspace VU ͒V represented by L II. MERA Orus, theAnn. MERA. Phys. The 349isometries, 117 w (2014)are of type ͑1,2͒ and are used to replace each block of two sites with a single effective site. Finally, Let denote a D-dimensional lattice made of N sites, the disentanglers u are of type ͑2,2͒ and are used to disentangle the where eachL site is described by a Hilbert space V of finite ͒N blocks of sites before coarse-graining. ͑Bottom͒ Under the dimension d, so that V ХV . The MERA is an ansatz used renormalization-group transformation induced by the binary 1D to describe certain pureL states ͉͒͒͘V of the lattice or, more L MERA, three-site operators are mapped into three-site operators. generally, subspaces VU ͒V . There are two useful waysL of thinking about the MERA u: V͒p V͒p, u† u = uu† = I, ͑ 1͒ that can be used to motivate its specific structure as a tensor → network, and also help understand its properties and how the where I is the identity operator in V͒p. For some gates, how- algorithms ultimately work. One way is to regard the MERA ever, one or several of the input wires are in a fixed state ͉0͘. as a quantum circuit whose output wires correspond to the In this case we can replace the unitary gate u with an iso- C 5 sites of the lattice . Alternatively, we can think of the metric gate w MERA as definingL a coarse-graining transformation that † maps into a sequence of increasingly coarser lattices, thus w: Vin Vout, w w = IV , ͑ 2͒ → in leadingL to a renormalization-group transformation.1 Next we where ͒pin is the space of the p input wires that are briefly review these two complementary interpretations. VinХV in not in a fixed state 0 and ͒pout is the space of the Then we compare several MERA schemes and discuss how ͉ ͘ VoutХV p =p output wires. We refer to w as a p ,p gate or to exploit space symmetries. out ͑ in out͒ tensor. Figure 2 shows an example of a MERA for a 1D lattice A. Quantum circuit made of N=16 sites. Its tensors are of types ͑1,2͒ and ͑2,2L͒. As a quantum circuit , the MERA for a pure state ͉͒͘ We call the ͑1,2͒ tensors isometries w and the ͑2,2͒ tensors ͒V is made of N quantumC wires, each one described by a disentanglers u for reasons that will be explained shortly, and HilbertL space V, and unitary gates u that transform the unen- refer to Fig. 2 as a binary 1D MERA, since it becomes a tangled state ͉0͒͘N into ͉͒͑͘see Fig. 1͒. binary tree when we remove the disentanglers. Most of the In a generic case, each unitary gate u in the circuit previous work for 1D lattices1,5– 7,16– 18 has been done using involves some small number p of wires, C the binary 1D MERA. However, there are many other pos-
144108-2 In addition to physics, tensor networks useful as an applied math technique
16
FIG. 13. Depiction of a multi-scale circuit (formed from com- position of depth N =2binaryunitarycircuits)withanopen boundary on the left, where a double layer of (scale depen- dent) unitary gates u(φz)andu(σz)isintroduced.Boundary wavelets gB are given by transforming the unit vector lo- cated on a boundary index as indicated. The angles φz and σz are chosen to ensure the boundary wavelets each have two vanishing moments.
φz σz z =1 0.615479708 0.261157410 z =2 0.713724378 0.316335000 z =3 0.752040089 0.339836909 z =4 0.769266332 0.350823961 FIG. 12. Plots of dilation m =4orthogonalwaveletsfrom(a) depth N =4and(b)depthN =9quarternarycircuitswith z =5 0.777461322 0.356148400 angles θk as given in Tab. VIII. The top three panels of each z =6 0.781461114 0.358770670 group denote the (exactly symmetric) scaling sequence h+ z =7 0.783437374 0.360072087 − + − and wavelet sequences h , g , h ,whichpossessthenumber z =8 0.784419689 0.360720398 of vanishing moments (v.m.) as indicated. The bottom three 89% accuracy on Fashion MNIST data set z =9 0.784909404 0.361043958 panels of each group depict the scaling functions and wavelets in the continuum limit (windowed to include only the non- z =10 0.785153903 0.361205590 vanishingly small part of the functions). z =11 0.785276063 0.361286369 z =12 0.785337120 0.361326749
TABLE IX. Angles φz and σz parameterizing the boundary structed as follows. Let be a semi-infinite lattice of L unitary gates at scale z,seeFig.13,oftheboundaryD4 sites r = [1, 2, 3,...], i.e. with an open boundary on the Daubechies wavelets. left. We apply as much of the multi-scale circuit cor- responding to the D4 wavelets, i.e. composed of depth NEvenbly,= 2 binary unitary circuits White, with angles θ =5 π/12 Stoudenmire, 1 ing moments. Here the angles φ1 and σ1 can be found and θ2 = π/6(seeSect.IIIB),thatcanbesupported deterministically using a algorithm similar to the circuit on . Then a double layer of scale-dependent 2 2 uni- L × construction algorithm discussed in Sect. III C 2. We tary"Representation gates, parameterized by angles φ1, φ2, φ3,... andand then design apply the algorithm of iteratively, wavelets scale-by-scale, to "Learning Relevant Features of Data σ , σ , σ ,... , where subscripts here{ denote scale}z, is { 1 2 3 } fix remaining angles φ2, φ3,... and σ2, σ3,... such introduced on the boundary of the multi-scale circuit. that all boundary wavelets{ have} two vanishing{ moments.} usingStarting from the unitary lowest scale the angles circuits"φ1 and σ1 are The resulting angles are given in Tab. IX, and are with Multi-scale Tensor Networks" B fixed such that the corresponding boundary wavelet g seen to converge to the values φz = π/4andσz = at that scale, as depicted in Fig. 13(a), has two vanish- 0.361367123906708 in the large scale limit, z . →∞ Brief Note on Fermions Typical tensor network approach uses second quantization
This means:
s1s2s3s4 s =0, 1
s s s s s s s s = 1 2 3 4 (ˆc†) 1 (ˆc†) 2 (ˆc†) 3 (ˆc†) 4 0
can be all antisymmetry any tensor handled in this part
No need to antisymmetrize (or symmetrize) amplitude tensor represented by tensor network When do the signs enter in?
When using operators: • applying Hamiltonian • computing observables
s1s2s3s4 s1 s2 s3 s4 cˆ2 (ˆc1†) (ˆc2†) (ˆc3†) (ˆc4†) 0
Sign of result will depend on value of
s1 index Fermion minus signs & tensor networks
Programming approaches – 3 alternatives:
• map fermionic operators to non-local bosonic operators (Jordan-Wigner transformation); work only with these
• choose canonical, reference ordering of sites and always permute basis states to this order
• anti-commuting tensor indices (newest approach) ⚛
Quantum Chemistry
Chemistry an instance of the many-electron problem:
1 2 1 Hˆ = ˆ† + v(r) ˆ + u(r, r0) ˆ† ˆ† ˆ ˆ r r r r0 r0 r
electronic structure Hamiltonian Unlike some areas of condensed matter, mostly after energies & quantitative properties
E R
equilibrium bond length
binding energy R
vibrational frequency
But qualitative properties also important: Will two molecules or atoms bind? State of atoms during a reaction? Biggest challenges in quantum chemistry:
• continuum nature of problem
• strong correlation Tensor network methods don't suffer from strong correlation
Strong correlation is fact that some wavefunctions (e.g. stretched diatomic molecules) are sum of exponentially many Slater determinants �
Tensor networks do not use sums of Slater determinants ✅ Continuum is the bigger issue for tensor networks...
Standard approach pioneered by John Pople is to use Gaussian basis functions to approximate the continuum
Figure credit: iqmol.org Consider H2 molecule
Cartoon of Gaussian basis set:
Basis sets also include linear combinations of Gaussians:
Nn 2 ⇣n,i(r rA) bn(r)= cn,ie
And multiplicative factors: x
H = t c† c + V c† c† c c ij i j ijkl i j 0 k 0 l
Coulomb "integrals" V
The coefficients
For N orbitals, there are N4 of these....
Say N=100, then N4 = one hundred million!
Just constructing the Hamiltonian is a serious business...
Point of Gaussians is computing integrals quickly! (Especially on 1990's computers...) DMRG & MPS for Quantum Chemistry ⚛ ⚛ ⚛ ⚛ ⚛ ⚛ ⚛ ⚛ ⚛ ⚛ DMRG and MPS require system to be discrete*
Finite basis is needed
Let's briefly discuss 3 types:
1. Gaussian basis sets [standard]
2. sliced basis sets [new]
3. gausslet basis [new]
* exception are continuous MPS, but still new topic 1. Gaussian basis DMRG (MPS tensor network) a. choose a Gaussian basis set, orthogonalize basis,
& compute integrals tij, Vijkl
(r ) (r ) (r ) (r ) V = i 1 j 2 k 2 l 1 ijkl r r Zr1,r2 | 1 2| 1. Gaussian basis DMRG (MPS tensor network) b. treat orbitals as "sites" of a pseudo-1D system
H = t c† c + V c† c† c c ij i j ijkl i j 0 k 0 l
1 2 3 4 5 ARTICLES NATURE CHEMISTRY DOI: 10.1038/NCHEM.1677
‘switches’ potential energy surfaces so as to avoid high barriers36. 10 Note that although other examples of multistate reactivity often require forbidden spin transitions, both surfaces here are interest-
) ingly for the same total spin, which makes such transitions facile. –1 Second, the two surfaces, labelled Sp p* and Ss s* in Fig. 4, are associated with (changes in occupancy! of) different! frontier orbitals 1 (the highest and lowest lying s, s* and p, p* orbitals of the ∆ E (kJ mol Mn4–O4 bond, respectively). As these frontier orbitals have different symmetries, this allows the cluster to exhibit different reactivity based on the direction of substrate approach, reminiscent of the chemistry of 37,38 0.1 haem and non-haem oxygen activating iron enzymes .Evenifthe 0 1,000 2,000 3,000 distortion coordinate modelled here does not ultimately turn out to be m acoordinatedirectlyassociatedwithdi-oxygenformation(butis,for example, related to an isomerization coordinate), our low-lying Figure 2 | Error of the DMRG total energy relative to the m 5 limit potential-energy surfaces clearly demonstrate the importance of 1 (DE )versusthenumberofrenormalizedbasisstatesm kept per block. non-adiabatic effects in the chemistry of the Mn4CaO5 cluster, The m limit energy, that is, the numerically exact eigenvalue of the which hitherto have been hard to access. ¼ 1 Hamiltonian matrix with dimensions of 1 1018,wasobtainedbyan " × 26 extrapolation from a linear fit of the energies and discarded densities . Spin projections of the S2 state. We now turn from the S1 state of The total energy is converged to better than 0.16 kJ mol21. the OEC to the next stage in the Kok cycle. Although an XRD 39 structure for the S2 state has not yet been definitively obtained , are displayed for the four Mn sites (Mn1–4). We1. find that Gaussian the Mn the core topology establishedbasis by Umena DMRG and colleagues10 greatly (MPS tensor network) oxidation states in the XRD structure are Mn1III Mn2IV Mn3III facilitates the search for candidate structures. Recently, using DFT, II { 30,34 Mn4 , and are robust to different protonations of the cluster several workers studied a family of closely related S2 (Supplementary} Fig. S2). These oxidation states do not structures, and Ames and co-workers30 used experimental correspond to the widely accepted oxidation states for the S1 state, hyperfine couplings to identify promising candidates. Note that Mn1III Mn2IV Mn3IV Mn4III 4,5, and suggest that the XRD hyperfine couplings depend on the spin states of the Mn atoms, {structure is reduced by X-ray damage.} Oxygen orbital occupancies which are many-electron quantities and cannot be reliably (Supplementary Tables S1–S4) indicate that thec. cluster if has beenconverged,accessed through the single-electron obtain density functional picture.state of the art results! reduced by two or three electrons. Consequently, the XRD Consequently, the spin states were determined indirectly by structure is unlikely to correspond to the true resting biological parameterizing a simple spin model (a Heisenberg Hamiltonian) 30 structure of the S1 state. from which spin states could be obtained.PC62CH22-ChanG Ames and colleagues ARI 25 February 2011 16:11 What is the correct S1 structure? There have been several recent refinements of the XRD structures using a combination of spec- III IV a Mn1 b Mn2 troscopy (for example, extended X-ray absorption fine structure, σ* Mn1–O1,3 σ* Mn2–O1,3 EXAFS28) and DFT calculations29–34. The refined structures increase σ* Mn2–O2,3 the bond connectivity of O5 with the Mn3 and Mn4 sites. We have Error from exact soln of binding energy of N2 recomputed the many-electron wavefunction at a recent refined (0.18) structure29 (denoted QM/MM) and our calculations for this (0.20) III eg (0.13) geometry reproduce the equilibrium S1 Mn redox states Mn1 eg 0.8 IV IV III { Mn2 Mn3 Mn4 (Fig. 3). Thus, our wavefunction calculations t2g 2 O1,3 Mn2 } 2p O1,3→Mn1 t p → strongly support an increased connectivity of O5 with Mn3 and 2g 2p O2,3→Mn2 Mn4 in the physiological S1 state geometry, as has been (1.84) (1.88) suggested recently. (1.82) The importance of the coupling between cluster electronic state 0.6 and structure, as illustrated by the changed connectivity between σ Mn1–O1,3 the XRD and refined structures, has been highlighted by Umena σ Mn2–O1,3 σ Mn2–O2,3 and colleagues10 and is no doubt central to the functioning of the IV Mn4III OEC. Indeed, there is some evidence that the Mn4CaO5 cluster geo- cdMn3 0.4 metry can even be somewhat fluxional, exhibiting potentially facile σ* Mn3–O5 σ* Mn3–O2,4 σ* Mn4–O4 UCCSDTQPH isomerization of O5 between Mn1 and Mn432,35. One way to under- stand the effect of structural distortions on the electronic properties (0.12) (0.22) is to compute the potential energy surfaces associated with the (0.20) Error from FCI (mH) distortion. These surfaces can be directly accessed through the e eg 0.2 g M = 2,000 many-electron wavefunctions that we use, but are difficult, if not 2p O5 Mn3 t → impossible, to obtain through a single-electron formalism such as 2g t2g 2p O2,4→Mn3 2p O4→Mn4 DFT. We calculated singlet potential energy surfaces along a coordi- (1.85) M = 4,000 nate connecting the XRD structure and the refined (QM/MM) (1.90) (1.81) 0 structure. Intriguingly, we find that the ground states at the 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 QM/MM structure and XRD structure are not adiabatically con- Bond length (bohr) σ Mn4–O4 nected (Fig. 4). Instead, the two surfaces cross at a conical intersec- σ Mn3–O5 σ Mn3–O2,4 tion midway along the reaction coordinate, at a Mn3–Mn4 distance Figure 4 of 2.87 Å. Several features of the potential energy surfaces connect Figure 3 | Orbital energy levels and electron occupancies of four Mn sites Errors from frozen-core full configuration interaction (FCI) for the nitrogen binding curve. The large the distortion of the structure to electronic reactivity. First, easy for the refined structure. a–d, Orbital energy levels and electron M 4,000 density matrix renormalization group (DMRG) calculation is nearly exact. However, even when access to a second low-lying surface along the distortion coordinate occupancies for Mn1 (a), Mn2 (b), Mn3 (c)andMn4(d). The layout and = enables multistate reactivity. This is a phenomenon where a reaction labelling follows Fig. 1 (XRD model). we halve the number of states, the M 2,000 DMRG calculation remains parallel to the exact result, = demonstrating the balanced treatment of strong correlation in the method. By contrast, the unrestricted 662 NATURE CHEMISTRY | VOL 5 | AUGUST 2013 | www.nature.com/naturechemistry coupled cluster with hextuple excitations (UCCSDTQPH) theory gives excellent results near equilibrium, Kurashige, Chan, Yanai, Nat. Chem. (2013) but the error grows very rapidlyChan, as theSharma, bond is stretched. Ann. Rev. Figure Phys. adapted Chem. from Reference (2011) 28. Study of photosystem-II Review: DMRG in Quantum Chemistry The more promising domain of application of the DMRG method in molecules is to solve active-space strong correlation or multireference problems, which do not benefit from a bias toward the Hartree-Fock reference, particularly in cases in which the spaces are too large for an FCI treatment. One study that demonstrated this possibility was a calculation of the nitrogen binding curve (28) (Figure 4). As seen in the figure, the DMRG curves exhibit errors that are reasonably parallel, indicating that they achieve a balanced description of the correlation across the potential energy curve. By contrast, unrestricted coupled cluster with hextuple excitations theory, although extremely accurate at the equilibrium geometry, shows a very rapid increase in error as the bond is stretched. Given the strength of the DMRG for large multireference electronic problems, a clear do- main of application is complicated transition metal problems. A number of preliminary stud- by University of California - Irvine on 03/20/12. For personal use only. ies have been carried out in this area on molecules with one or two metal atoms, particularly by the group of Reiher (18, 37–40) and recently by Kurashige & Yanai (21). One common
Annu. Rev. Phys. Chem. 2011.62:465-481. Downloaded from www.annualreviews.org finding, as pointed out in studies of Cu2O2 models of the tyrosinase core (21, 40), is that, although the valence active spaces involved in many transition metal problems may be very large—e.g., 50 orbitals or more—in many cases one can exploit the balance of the DMRG method to obtain converged energy differences at much smaller M than is required to con-
verge the total energy. (The Cu2O2 problem is described further in the next section.) Recently, the DMRG was used to obtain a near-exact solution of the active-space electronic structure in
the Cr2 molecule, with a 24-electron, 30-orbital active space (21). These results are shown in Table 2. They show the importance of high-body correlations in this molecule: CCSDTQ the-
ory is still off by 14 mEh. This study illustrates the potential applicability of the DMRG for quite large active spaces even in the absence of a true one-dimensional topological structure to the correlations.
474 Chan Sharma · 1. Gaussian basis DMRG (MPS tensor network)
Drawbacks of this approach?
Gaussian basis functions overlap significantly (especially after orthogonalization)
Must keep all N4 Coulomb integrals (can't truncate)
Hamiltonian non-local; DMRG scales poorly (large MPS bond dimension required) Alternatives to Gaussian basis set approach? Consider 1D particles in a box:
n =3
ˆ n =2 Approach 1: basis set cn = n(x) (x) x Z n =1
1 @2 H = ˆ†(x) ˆ(x) H = t c† c 2 @x2 ! nm n m x nm Z X
- Loss of locality - Must compute integrals + Variational Consider 1D particles in a box:
ˆ Approach 2: grid approximation cj = pa (xj)
1 @2 H = ˆ†(x) ˆ(x) 2 @x2 Zx ! 1 H (c†c 2n + c† c ) ⇡ 2a2 j j+1 j j+1 j j X + (a2) O + Local / short range + No integrals to compute! - Not variational Possible to mix basis set and grid approaches?
Yes... 2. sliced-basis DMRG [new]
Slice 3D basis sets along z-direction:
Stoudenmire, White, arxiv:1702.03650 2. sliced-basis DMRG [new]
Slice 3D basis sets along z-direction:
Map to 1D 'chain' with 1000's of sites
(small
• Leverage ability of DMRG to scale to long systems • Can reach high (chemical) accuracy for a . 0.1 • Scalable to 1000's of atoms Stoudenmire, White, arxiv:1702.03650 2. sliced-basis DMRG [new]
Slices roughly equivalent to using basis set of "functions":
1 (r)= 2 (z n a) ' (x, y)
n = 1 2 3 4 5 6 7 8 9
Stoudenmire, White, arxiv:1702.03650 2. sliced-basis DMRG [new]
1 (r)= 2 (z n a) ' (x, y)
(r ) (r ) (r ) (r ) V = i 1 j 2 k 2 l 1 ijkl r r Zr1,r2 | 1 2|
But treat slices as orthogonal. Then V ijkl non-zero only if i, l on same slice and j, k on same slice
N 4 N 2N 4 Number of terms:
Stoudenmire, White, arxiv:1702.03650 13
Algorithm 2 A modified ISVD method for robust construction of the MPO repre- sentation N N Input: V R ⇥ ,M 2 1: S1 ( V(1, 2:N) ) k kT (N 1) 1 2: W1 (V(1, 2:N) / V(1, 2:N) ) R ⇥ k k 2 3: a (1) 1 4: b (V(1, 2)) 2 5: for p = 1:N 2 do 6: Q, R QR factorization of W (2:N p, :) p 7: x argmin W (2:N p, :)x V(p+1,p+2:N)T xk p k 8: r V(p+1,p+2:N)T W (2:N p, :)x p 9: if r > tol then k k T SpR 0 10: X , S , Y SVD of keep M singular values p+1 p+1 p+1 xT RT r { } ✓ k k ◆ 11: W Qr/ r Y p+1 k k p+1 12: else T SpR 13: X , S , Y SVD of keep M singular values p+1 p+1 p+1 xT RT { } ✓ ◆ 14: W QY p+1 p+1 15: end if 16: b S WT (:, 1) p+2 p+1 p+1 17: if p = N 1 then 6 18: a X (p+1, :)T , X˜ X (1:M,:) p+1 M+1 p+1 p+1 19: end if 20: end for Output: a , b , X˜ for 1 i N 1, 2 j N,2 k N 1, and MPO i j k according to (3.10).2. sliced-basis DMRG [new]
3.3. A modified practical algorithm. Despite that Algorithm 1 computes a, b, X˜ directly, in practice it can still become unstable when N becomes large. This is because the singular values[Technical of V˜ grows exponentially Slides] with – respect Important to N.Thenin for Practitioners Algorithm 1, the SVD step (3.7) also increasingly ill-conditioned as p increases.
2 2 (r) (r0) V = | i | | j | ij r r
Fig. 3.1. The purple and red rectangles are matrix blocks V(1:p, ⌦p) with p =3and 4, for which we are going to compute approximate truncated SVDs in Algorithm 2. Diagonal entries are marked with blue dots. Sliced Coulomb integrals have upper-triangular-low- We now further modify the ISVD algorithm as follows. Instead of trying to ap- rank property (SVD of upper triangles compresses well)
Lin, Tong, arxiv:1909.02206 Stoudenmire, White, arxiv:1702.03650 13
Algorithm 2 A modified ISVD method for robust construction of the MPO repre- sentation N N Input: V R ⇥ ,M 2 1: S1 ( V(1, 2:N) ) k kT (N 1) 1 2: W1 (V(1, 2:N) / V(1, 2:N) ) R ⇥ k k 2 3: a (1) 1 4: b (V(1, 2)) 2 5: for p = 1:N 2 do 6: Q, R QR factorization of W (2:N p, :) p 7: x argmin W (2:N p, :)x V(p+1,p+2:N)T xk p k 8: r V(p+1,p+2:N)T W (2:N p, :)x p 9: if r > tol then k k T SpR 0 10: X , S , Y SVD of keep M singular values p+1 p+1 p+1 xT RT r { } ✓ k k ◆ 11: W Qr/ r Y p+1 k k p+1 12: else T SpR 13: X , S , Y SVD of keep M singular values p+1 p+1 p+1 xT RT { } ✓ ◆ 14: W QY p+1 p+1 15: end if 16: b S WT (:, 1) p+2 p+1 p+1 17: if p = N 1 then 6 18: a X (p+1, :)T , X˜ X (1:M,:) p+1 M+1 p+1 p+1 19: end if 20: end for Output: a , b , X˜ for 1 i N 1, 2 j N,2 k N 1, and MPO i j k according to (3.10).2. sliced-basis DMRG [new]
3.3. A modified practical algorithm. Despite that Algorithm 1 computes a, b, X˜ directly, in practice it can still become unstable when N becomes large. This is because the singular values[Technical of V˜ grows exponentially Slides] with – respect Important to N.Thenin for Practitioners Algorithm 1, the SVD step (3.7) also increasingly ill-conditioned as p increases.
SVD
Fig. 3.1. The purple and red rectangles are matrix blocks V(1:p, ⌦p) with p =3and 4, for which we are going to computeFrom approximate SVD's truncated of SVDs all in Algorithm upper 2. Diagonal triangle entries are blocks, marked with blue dots. construct efficient matrix product operator (MPO) We now further modify the ISVD algorithm as follows. Instead of trying to ap- of Coulomb Hamiltonian terms
Lowers scaling all the way to linear in number of z slices (number of atoms) Lin, Tong, arxiv:1909.02206 Stoudenmire, White, arxiv:1702.03650 Possible to do better? 3. multi-sliced gausslet basis [new]
Drawback of sliced bases include:
• require very many slices to resolve z-direction
• problems with Gaussians in x,y directions
n = 1 2 3 4 5 6 7 8 9 3. multi-sliced gausslet basis [new]
Seek functions which are local (compact), orthogonal, and sums of them can represent any smooth function
Can findQuantum such chemistry DMRGfunctions using theory of wavelets8.21
4 1 (a) (b) 2 (x) ( x )
i 0
j 0.5 u(x) G G
-2
0 -4
-10 0 10 -4 -2 0 2 4 x x
1.5 Fig. 4: One-dimensional array of gaussletc functions with a length scale(d of) 1.0. The gausslet centered at the origin is highlighted in( solid) black to emphasize details. 1 Nucleus x ➞ ( x ) about the details of constructingj them, see Ref. [28] which proposes and constructs gausslets, ˜ G 0.5 building upon the development in Ref. [30] of compact and symmetric families of orthogonal y ➞ Gaussletswavelets. = wavelet (scaling functions) built from Gaussians Importantly, after constructing a grid0 of gausslets like in Fig. 4 and using them to discretize quantum chemistry Hamiltonians, one can obtain very accurate continuum results using a gaus- -10 0 10 slet spacing of about 1.0, in contrast to grid discretizationx which requires about an order of magnitude smaller spacing to obtain similar accuracy [28]. Better yet, we will see next that by adapting the grid on which the gausslets are centered, one can even better resolve the continuum using small numbers of gausslet functions.
4.2.2 Adapted grid of gausslets
Using an even-spaced grid of gausslet functions, as in Fig. 4 to discretize quantum chemistry Hamiltonians is more efficient that using a simple grid, yet still requires more functions than are actually needed. The reason is that while high resolution is required to capture details of the electronic wavefunction near atomic nuclei, much less resolution is needed away from nuclei. A straightforward way to reduce the number of functions needed while preserving high resolution near nuclei is to perform a coordinate mapping on the gausslet functions so that they form an adapted grid, with a finer spacing near nuclei and a coarser spacing otherwise. Such a coordinate mapping may be defined via a function x(u) which maps from a fictitious space u where the gausslets are defined to have a regular grid spacing into the actual space x used for the quantum chemistry calculation. Let u(x) be defined as the inverse of the mapping x(u). For the case of a single atom, a sensible coordinate mapping is
1 1 u(x)= sinh (x/a) (31) s defined by a scale parameter s and a core cutoff parameter a. Figure 5(a) shows how this mapping takes an evenly spaced grid along the y direction of the plot into a variable spaced grid along the x direction. 2
-0.498 -1.130 4 1 (a) b (a) ( ) s=0.9 a =0.6s, Full H (b) 0 H 3. multi-sliced gausslet basis [new] 2 -1.132 2 a0=0.6s, Diag Exact H cc-pVDZ R=2 Diagonal Approx (x) ( x ) i j 0.5 0 cc-pVTZ G G u(x) -0.499 cc-pVQZ 1 (a) 4 b s=0.8 -1.134 ( ) δ corrected; a0=s
By scale-adapting and "multi-slicing"-2 E 2 E CBS
(x) 0 ( x )
i 0
j 0.5 u(x) -1.136 G s=0.7 can representG smooth 3D functions-4 -2 s=0.6 -4 -2 0Quantum 2 chemistry 4 DMRG-10 0 10 8.23 0 x -0.5 8.22 E. Miles Stoudenmire-4 x s=0.5 -1.138 -10 0 10 -41.5 -2 0 2 4 x x d 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.4 0.5 0.6 0.7 0.8 0.9 (a) (b) 1.5 c ( ) a s 4 ( ) 0 1 (a) b (d) Nucleus ( ) (c)1 2 1 Nucleus x ➞ x ➞ ( x ) j ( x ) j (x) ˜ ( x ) ˜ i 0 j 0.5 FIG. 2. (a)Energies of a hydrogen atom in an MSG basis as u(x) G G G
G 0.5 0.5 y ➞ y ➞ -2 a function of a and s, in Hartrees, using the full Hamiltonian
0 0 -4 (“exact H”) and using the integral diagonal approximation. 0 -10 0 10 -10 0 10 -4 -2 0 2 4 x The exact energy is 1/2. (b) Energy of a hydrogen molecule x x 1.5 Fig. 5: Illustration of (a) a coordinate mapping Eq. (31) which-10 maps a regularly0 spaced grid 10 with separation R = 2 in standard Gaussian and MSG bases. in the u coordinatesd to an adapted grid in the x coordinates. The resulting adapted gausslets c ( ) Fig. 6: Illustration of a fixed-z cut through a multi-sliced grid, showing the primary non-zero ( ) shown in (b) remain orthonormal, but achieve a higher resolution near the origin.x One gausslet 1 Nucleus support of selected gausslet functions as colored rectangles. The vertical lines are selected is shown with a bold line toscale-adaptation highlight details. multi-slicing x ➞ x-slices and points are centers of adapted gausslets. The position of a nucleus is shown, illus- ( x ) j ˜
G trating how the multi-sliced grid bunches more gausslets of a smaller size nearby. 0.5 Having chosen a coordinate mapping, they ➞ transformationFIG. 1. of (a) the gausslets Array which of moves gausslets, their with the gausslet centered at For 3D basis functions, a coordinate-product form centers onto the adapted, variable-spacing grid while preserving their orthogonality and other the origin emphasizedadapted to grid show of discrete detail.z values z , k (b)=1, 2 Coordinate,...,N which are more trans- closely spaced when- 0 good properties is k z (x, y, z)=f(x)g(y)h(z) greatly simplifies evaluation of ever planes z = zk pass nearby atomic nuclei. This transformation defines planes z = zk with G˜ formation function u(x) for a single atom, with a = s =0.7 G -10 0 10 j(x)=Gj(u(x)) u0(x) (32) ˜ adapted functions Gk(z) centered on them, forming what is called a z-slice. Now within eachintegrals defining the Hamiltonian. To keep this form, x p where Gj(u) is the gausslet centered at thein integer Eq. grid (2),point j in to the giveu space. gausslets Figurez-slice, 5(b) a coordinate variable transformation resolution. is applied to the y (c)coordinates, Distorted defining an adapted grid ˜ y we apply coordinate transformations to each coordinate shows the adapted gausslets Gj(x) resultinggausslet from using the coordinate basis mapping basedu(throughx on) defined the a function transformationu (y) which yields discrete ofy values (b),ykj whichwith j =1 is, 2,...,N or-y. The trans- in Eq. (31) above. One of the adapted gausslets is highlighted with a bolder line, and you y formation u (y) is also chosen to make the ykj values more closely bunched whenever linesseparately, in a method we call “multislicing”. The co- can observe that it takes a distorted shape comparedthonormal to the unadapted and gausslets allows in Fig.of fixed 4. a The(y diagonalkj,zk) pass nearby approximation. an atomic nucleus. This second One step defines of they-slices as lines of placement of more and finer-sized gausslets near the origin gives better resolution there. ˜ ˜ functions is emphasized.fixed (ykj,z (d)k) with Schematic functions Gkj(y) G representationj(z) centered on them. Finally of mul- discrete x points ordinatexkji directions are sliced up sequentially, z (which Note that when adapting gausslets for systems of multiple atoms, there are modifications of the x with i =1, 2,...,Nx are defined through a transformation u (x) such that points (xkji,ykjruns,zk) along the chain), then y,thenx. A first coordi- transformation Eq. (31) which make it bettertislicing suited for treating in molecules. 2D. The The supplementalare vertical more densely spacedlines the represent closer they are to nuclei. slices, All these with transformations three taken together information of Ref. [29] discusses such multi-atom coordinate transformations. ˜ ˜ ˜ z shown in detail. Eachdefine dot 3D functions is theGkji center(x) Gkj(y) G ofj(z) acentered basis on points function,(xkji,ykj,zk) and. To make the basisnate transformation u (z) determines a set of z-values finite, only functions whose centers lie within a certain distance of at least one of the atoms z 4.2.3 Multi-sliced grid the shaded rectanglesare kept illustrate in the basis. By the construction principle these final support basis functions region maintain a product of formz andk (k =1, 2,...), with u (zk)=k, at which are centered To apply the above ideas of gausslet basis setssome to 3D systems, of the the most straightforwardfunctions,orthonormality, approach although while being they adapted have for higher smooth resolution near nuclei. tails For well more technicaldistorted dis- 1D gausslets G˜ (z). The plane z = z and func- cussion on how to make a good choice for the uz, uy, and uz coordinate transformations, see k k is to define basis functions as products Gi(x)beyondGj(y) Gj(z) of 1Dthe gausslets. rectangles. But how to maintain The multicolored shaded rectangles this product form while also adapting the gausslet spacing near atomic nuclei is less obvious;the supplemental for information section of Ref. [29]. Essentially these coordinate transformationtion G˜k(z) together define a z-slice. Next we slice up each example, performing the coordinate transformationsrepresent in a radially long, symmetric thin way destroysfunctions basis the are functions chosen to have a form which like that of one Eq. (31), would but with the wants and a parameters to varying product form of the 3D functions, resulting incontract integrals which atare too a costly later when constructing stage.according to the 3D distance of a particular z or y slice from the nearest atomic nucleus. z-slice in the y direction, with a coordinate transforma- the discrete Hamiltonian. Figure 6 shows a 2D cut through a multi-sliced grid, with boxes illustrating the non-zero supporttion unique to k, uy(y), which defines a set of y-values Fortunately, there is a simple way around this problem that only incurs a modest overheadof selected in the gausslet functions. From the figure, one can observe that a downside of multi-slicing k total number of functions needed. This workaround is called multi-slicing and is justis thethat idea it results in many long and thin functions far away from any nucleus which provideykj.Ay “slice” (or “subslice” of a “parent” z-slice) is of performing the coordinate transformation sequentially: first in the z direction (themore direction resolution than is actually needed. But by using an approximate wavefunction as a guide, along the greatest extent of the system), thenits in the inversey direction, andu finally(x) the define in x direction.such as a a wavefunction 1D smooth from a Hartree-Fock one-to-one calculation, coordinateone can take the additional stepthe of line z = zk, y = ykj, with associated 2D function In more detail, one starts by first defining a coordinate transformation uz(z) to determinecombining an such redundant functions together into single functions to reduce the basis size. mapping, which will be used to make the grid narrow and G˜k(z)G˜kj(y). Finally, for each y-slice, define a unique x closely spaced near nuclei, and wide and sparse far away. coordinate transformation ukj(x), determining a set of x First consider a 1D arrangement, with just one atom at values xkji, and 3D basis functions G˜k(z)G˜kj(y)G˜kji(x). x = 0. Define the gausslets on a uniform grid in the The key point in using this successive procedure is to u space and then map to x-space, inserting a Jacobian use of the knowledge of where a slice is, relative to the factor to preserve orthonormality. If Gj(u) is a gausslet nuclei, to make subsequent transformations with the low- centered at integer j,define est density of functions. This is illustrated schematically in 2D in Fig. 1(d). Preserving the product form via mul- ˜ Gj(x)=Gj(u(x)) u (x) . (1) tislicing means that some basis functions are long and thin; however, at a later stage on can devise methods The G˜ are orthonormal if the Gpare. j j to contract such functions with their neighbors, reducing The coordinate mapping we choose for a single atom unnecessary degrees of freedom. The details of the co- is given by ordinate transformations in the multisliced case are dis- 1 u(x)=sinh (x/a)/s (2) cussed in the Supplementary Material. Each basis function has a well defined center where the parameter s,thescale, sets or adjusts the over- (xkji,ykj,zk), and we can make a simple rule for which all gausslet spacing, and a,thecore cuto sets the range functions to keep: if the basis function is within a dis- in x over which we stop decreasing the gausslet spacing. tance b of an atom, we keep it. Here b = 9 a.u. proved The smallest gausslet spacing at the nucleus is about very accurate (< 0.1 mH errors compared to larger b) a s. This transformation is shown in Fig. 1(b), with except for R = 1 for H10,whereweusedb = 13. the· resulting 1D functions shown in Fig. 1(c). In the Figure 2(a) shows energies for a single hydrogen atom Supplemental Material, we discuss the motivation for the for various a and s, using both the standard Hamilto- above form of the transformation u(x), as well as a modi- nian and one where a diagonal approximation is made for fied form of the transformation better suited for multiple the single particle potential [1]. Since there are only N 2 atoms. single particle terms, using this diagonal approximation Results of sliced-basis and multi-sliced gausslets Hydrogen chains make good benchmark systems • continuum limit • strong correlation • many-atom (thermodynamic) limit • treatable by most methods
+
Density cross- section of H10 hydrogen chain: Quantum chemistrySliced-basis DMRG results for H10: 8.19
Quantum chemistry DMRG 8.19
Fig. 2: Ground-state energies of H10 chains as a function of inter-atomic spacing R calculated Fig. 2: Ground-state energiesusing DMRG of Hwithin10 standardchains Gaussian as basis a functionsets (dashed curves) of and inter-atomic sliced basis sets (solid spacing R calculated using DMRG within standardcurves and Gaussian points) using a uniform basis grid spacing sets of (dasheda =0.1 atomic units curves) [26]. and sliced basis sets (solid curves and points) using a uniform grid-0.482 spacing of a =0.1 atomic units [26]. STO-6G
-0.484 4000 (sec) sweep
-0.482 t 2000 STO-6G-0.486 E / N (Hartree)
0 -0.488 0 500 1000 SB-STO-6G Natom -0.484 4000 0 0.02 0.04 0.06 0.08 0.1 1/Natom (sec) Fig. 3: Scaling with number of atoms of sliced-basis calculations up to 1000 hydrogen atoms. The inter-atomic spacing is fixed to R =3.6 and a sliced basis derived from the STO-6G sweep
t 2000 Gaussian-0.486 basis was used. The outer plot shows the ground state energy from DMRG using the standard STO-6G basis and the sliced version (SB-STO-6G). The inset shows the average time per DMRG sweep, taking a bond dimension of m =100. E / N (Hartree)
4.2 Approach 2: multi-sliced0 gausslet basis The-0.488 sliced basis approach to discretizing0 the electronic500 structure Hamiltonian1000 demonstrates a successful marriageSB-STO-6G of the grid and basis set approachesN toatom quantum chemistry. Counterintu- itively, it demonstrates that using more functions to represent the Hamiltonian can result in a more affordable calculation overall, by choosing the functions to be local (at least along one direction), so that the DMRG algorithm and MPO methods for compressing the Hamiltonian (Section 4.3.2)0 can perform 0.02 to their full 0.04 potential. 0.06 0.08 0.1 1/Natom
Fig. 3: Scaling with number of atoms of sliced-basis calculations up to 1000 hydrogen atoms. The inter-atomic spacing is fixed to R =3.6 and a sliced basis derived from the STO-6G Gaussian basis was used. The outer plot shows the ground state energy from DMRG using the standard STO-6G basis and the sliced version (SB-STO-6G). The inset shows the average time per DMRG sweep, taking a bond dimension of m =100.
4.2 Approach 2: multi-sliced gausslet basis
The sliced basis approach to discretizing the electronic structure Hamiltonian demonstrates a successful marriage of the grid and basis set approaches to quantum chemistry. Counterintu- itively, it demonstrates that using more functions to represent the Hamiltonian can result in a more affordable calculation overall, by choosing the functions to be local (at least along one direction), so that the DMRG algorithm and MPO methods for compressing the Hamiltonian (Section 4.3.2) can perform to their full potential. Quantum chemistry DMRG 8.19
Quantum chemistry DMRG 8.19 Fig. 2: Ground-state energies of H10 chains as a function of inter-atomic spacing R calculated using DMRG within standard Gaussian basis sets (dashed curves) and sliced basis sets (solid curves andSliced-basis points) using results a uniform for grid H spacing1000: of a =0.1 atomic units [26].
-0.482 STO-6G
-0.484 4000 Fig. 2: Ground-state energies of H10 chains as a function of inter-atomic spacing R calculated using DMRG within standard Gaussian basis sets (dashed curves) and sliced basis sets (solid
curves and points) using a(sec) uniform grid spacing of a =0.1 atomic units [26]. sweep
t 2000 -0.486 -0.482 STO-6G E / N (Hartree) -0.484 4000
0 (sec) sweep -0.488 -0.486 0 t 2000 500 1000
SB-STO-6GE / N (Hartree) Natom 0 -0.488 0 500 1000 SB-STO-6G Natom 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 1/N1/N atomatom
Fig. 3: Scaling with number of atoms of sliced-basis calculations up to 1000 hydrogen atoms. The inter-atomic spacing is fixed to R =3.6 and a sliced basis derived from the STO-6G Fig. 3: Scaling with numberGaussian of basis atoms was used. of The sliced-basis outer plot shows the ground calculations state energy from up DMRG to using 1000 the hydrogen atoms. The inter-atomic spacingstandard is fixed STO-6G basis to R and the=3 sliced. version6 and (SB-STO-6G). a sliced The inset basis shows the derived average time from the STO-6G per DMRG sweep, taking a bond dimension of m =100. Gaussian basis was used. The outer plot shows the ground state energy from DMRG using the standard STO-6G basis and4.2 Approach the sliced 2: multi-sliced version gausslet (SB-STO-6G). basis The inset shows the average time per DMRG sweep, takingThe a sliced bond basis dimension approach to discretizing of m the=100 electronic structure. Hamiltonian demonstrates a successful marriage of the grid and basis set approaches to quantum chemistry. Counterintu- itively, it demonstrates that using more functions to represent the Hamiltonian can result in a more affordable calculation overall, by choosing the functions to be local (at least along one direction), so that the DMRG algorithm and MPO methods for compressing the Hamiltonian 4.2 Approach 2: multi-sliced(Section 4.3.2) can perform gausslet to their full potential. basis The sliced basis approach to discretizing the electronic structure Hamiltonian demonstrates a successful marriage of the grid and basis set approaches to quantum chemistry. Counterintu- itively, it demonstrates that using more functions to represent the Hamiltonian can result in a more affordable calculation overall, by choosing the functions to be local (at least along one direction), so that the DMRG algorithm and MPO methods for compressing the Hamiltonian (Section 4.3.2) can perform to their full potential. 3
8.26 E. Miles Stoudenmire barely improves computational e ciency, but one would -4.145 expect this approximation to mimic the performance of R=1 the more important two-particle diagonal approximation. MSG (HF) 1 -4.15 The diagonal approximation is sensitive to the singular- QZ HF Extrap(D-5) ity in the potential at the nucleus, but increasing the E -4.155 basis function density near the nucleus by decreasing a, Extrap(T-5) 5Z (mH) 0.5 for fixed s, nearly eliminates the diagonal approximation
atom -4.16 error. A simple procedure to systematically converge to 0.4 0.6 0.8 )/N s the ground state for this system would be to fix a/s to 5Z Fig.0 7: Hartree-Fock energies (in units of Hartree) of 10-atom hydrogen chains with inter- be a constant, say 0.5-0.6, and then decrease s. atomic spacing R =1. The points are results from multi-sliced gausslet (MSG) bases with Figure 2(b) shows the energy for a hydrogen molecule, (E-E varying scale parameter controlling the spacing betweens=0.8 gausslets. The lines labeled QZ and Multi-sliced-gausslet5Z are energies obtained with the cc-pVQZ(MSG) and results cc-pv5Zs=0.7 Gaussian for H basis10: sets, while the lines compared to standard basis sets cc-pVxZ, where x=D, labeled Extrap are extrapolations to the complete basis set limit using either double-zeta or triple-zeta up through 5-zeta. s=0.6 T, and Q, and also compared to the exact energy from a -0.5 s=0.5 4 MSG-CBS treatment in special coordinates [9]. A diagonal approx- ence plots were made relative to MRCI+Q for smaller Chemical accuracy systems and AFQMC for larger ones. DMRG based on imation for the two particle interaction is used here and 1 1.5 2 2.5 3 3.5 standard Gaussian basis sets could not be done beyond in all subsequent MSG bases, since calculations would R the TZ level, so no CBS results were available. Here, we 1 UCCSD(T) find systematic convergence of MSG-DMRG to energies MRCI+Q
not be practical with the standard V form. All re- (mH) ijkl AFQMC agreeing with the LR-DMC-AGP method. Agreement sults shown are exact (full CI) given the approximate FIG. 3. Hartree Fock energiesatom per atom ofSBDMRG H versus R,was poorer at small R with a DMC method based on an Comparison s=0.910 LDA trial function. There are systematic errors in DMC )/N Hamiltonian. The MSG bases systematically converge relative to theto diffusion Gaussian basis set cc-pV5Z (5Z)[4].s=0.8 TheMSG-DMRG con-stemming from the fixed node approximation, which are
DMC s=0.7 Monte Carlo unusually small in this 1D system, but hard to quantify. to the exact results, and the diagonal approximation for nected symbols are MSG-HF0 results at constant s =}a,as (DMC) (E-E Since the nature of errors in DMC and MSG-DMRG are the single particle potential closely approximates the full labeled. For small R, they converge to a small but notice-completely di erent, and since the MSG-DMRG energies ably di erent result from cc-pV5Z. The inset shows MSG-HFconverge systematically with a control parameter, we can Hamiltonian, particularly for smaller s. be rather sure that MSG-DMRG and DMC are both get- data versus s = a,atR = 1. Horizontal lines show Gaussianting the most accurate energies. Also shown in Fig. 2(b) is a basis with a special delta- 1 1.5 2 2.5 3 3.5 results, along with two exponential extrapolations,R based on The MSG-DMRG errors for fixed s = a are biggest function correction for the nuclear cusp. Increasing the (TZ,QZ,5Z) (labeled T-5), and on (DZ,TZ,QZ,5Z) (labeledat small R. This is expected; at small R, it would be Fig. 8: Energies of 10-atom hydrogen chains computed by various methods relative tomore those natural to scale s with R, keeping the number of FIG. 4. Complete basis set energies per atom of H versus R, resolution near nuclei by using a small a is ine cient, D-5). Theobtained MSG-HF by diffusion agrees Monte for Carlo small (EDMC).s Energywith differences the T-5 are10 shown extrapo- in milli-Hartree.basis function more nearly constant. The smallest grid Results are fromrelative Ref. [27]. to a di usion Monte Carlo method, for MSG-DMRG leading to many basis functions. For example, for the lation, although the(labeled D-5 by extrapolations) versus various approaches was used from Ref. in 4.Ref. [4]. spacings are about a s, or about 0.5 for s =0.7. Small R is challenging to the· Gaussian basis set methods because hydrogen atom of Fig. 2(a), taking a =0.3, s =0.6 from competitive quantum chemistry methods including coupled cluster (UCCSD(T)),the multi- basis functions become linearly dependent. produced 1179 functions, which resulted in an error of reference configuration interaction (MRCI+Q), auxiliary field quantum Monte Carlo (AFQMC),In summary, even in this first implementation of the DMRG usingabout a sliced-basis 3000-4000 (SBDMRG), basis functions. and finally diffusion (In contrast, quantum standard Monte Carlo (DMC),MSG-DMRG method, for the strongly-correlated H DMRG in a Gaussian basis—with no diagonal approxi- 10 0.13 mH. Our correction consists of adding a single- As shownwhich in the is used inset, as the reference at R energy= for 1 the the figure. Gaussians These results were converge first obtained andsystem dis- we surpass the best Gaussian basis approaches mation and no compression—is limited to about 100-200 in the high-accuracy CBS regime. We believe larger Z particle potential at each atom of the form v ( r r ). slowly, and di erentactive extrapolations basis functions.) We find give that dithe DMRGerent performs results. systems, not just in the linear geometry of H10, could be The parameter v is set by “turning o ” all nuclear elec- As a rough comparisonvery well. of For the the calculational very high accuracy results e ort shown for thesebe- treated straightforwardly using pseudopotentials. How- tron potentials for atoms other than (yet keeping the low, we generally only needed to keep about 200 states ever, we believe our approach can also be improved so very high accuracyfor larger calculations:R, and up to for 400-500R for=R 1,=1(duetoitsa = s =0.5,that resorting to pseudopotentials is not necessary for same set of functions to be used for the entire system), the MSG basis hasmore just metallic over character). 13, 000 This basis excellent functions; performance is themoderate Z. For example, one could add some Gaus- due to the high locality of the basis, which DMRG and sians from a standard basis to a gausslet basis, orthog- and adjusting v so that the one-electron ground state number of two-electronother tensor integrals network methods is the [10–12] square strongly of prefer. this, oronalizing the Gaussians to the gausslets, to better rep- energy is the exact hydrogen atom energy 1/2. The 8 We find that the correlation energy converges faster resent core orbitals. This is very simple to do in prin- 1.7 10 . The 5Zwith basiss than has the 550 HF energy. functions, This is notbut surprising: the number the 4 10 ciple, but we would also like to find diagonal approxi- errors associated with choosing a too large are localized of integrals (N ,representation ignoring symmetry) of the nuclear cusp is is 9 poor.2 with10 a coarse.Themations involving the Gaussians, or develop convenient near the nuclei; the delta function potential alters the gausslet basis, which is is primarily a single particle e ect. partially-diagonal approximations, where the number of calculation time ofTherefore, our UHF to get total algorithm, energies we which use the HF takes energy ad-non-diagonal terms is not too big. The delta correction terms in the Hamiltonian only for the basis functions vantage of the diagonalwith very nature small s, and of add the to Hamiltonian, it the correlation energy scaleswould likely be eliminated in any of these approaches. obtained with a larger s, where the correlation energy is Another way to improve fitting core orbitals would in- overlapping with a nucleus. Most importantly, v 0 as N 2N ,whereN is the number of electrons, with the ! e definede by subtracting the unrestricted HF energy from volve adapting the gausslets during the slicing to fit 1D as a 0 or s 0, so this correction does not change dominant part comingthe total fromenergy for a the Davidson same basis set. diagonalization,Gaussians taken from a standard basis. Regarding how ! ! In Fig. 4, we show a comparison of total energies one uses MSG bases, in the hydrogen chains studied here, what the basis converges to, only how fast it converges, for Ne eigenvectors,for several of the methods Fock [4] matrix. and our MSG-DMRG for var- the linear geometry makes DMRG especially powerful. accelerating the convergence. In Fig. 2 and for the rest For correlated calculations,ious s = a. All methods to decrease attempt to the reach number the CBS ofFor less linear molecules or solids, one might couple mul- limit; for all but the MSG-DMRG and DMC methods, tisliced gausslets with tensor network states [10–12] or of the results, we set a = s and use the delta correction. basis functions, onethis involved can use an extrapolation the HF occupied in the basis set. orbitals The en- toquantum Monte Carlo. We now turn to a more challenging system, a linear contract the MSGergy basis di erences to smaller here are generally size. This well below can chemical be done We acknowledge useful conversations with Ryan Bab- accuracy. Often such high accuracy is unnecessary, but bush, Jarrod McClean, Shiwei Zhang, Mario Motta, Gar- chain of hydrogen atoms spaced R apart. Hydrogen chain in a way that maintainsstudying the the high diagonal accuracy limit form is an of excellent the interac- way to net Chan, and Sandro Sorella. We acknowledge support demonstrate the usefulness of MSG-DMRG. The ener- from the Simons Foundation through the Many-Electron systems were the subject of a recent benchmark study tions. One can alsogies are extrapolate measured relative in to a one cuto of the dithat usion controls Monte Collaboration, and from the U.S. Department of Energy, which compared more than a dozen methods in their this contraction,Carlo to obtain methods, LR-DMC-AGP results for (or the DMC). uncontracted In Ref. [4], at O ce of Science, Basic Energy Sciences under award this level of accuracy, none of the best available methods #DE-SC008696. The Flatiron Institute is a division of ability to reach the combined limit of exact correlation, basis. The largestagreed, systems so it was not needed known which for was a extrapolation best, and refer- the Simons Foundation. complete basis set, and infinite number of atoms [4]. We are still about a factor of 2 or 3 smaller than the uncon- first consider unrestricted Hartree Fock (UHF) on H10, tracted basis, and the results below follow this procedure, shown in Fig. 3. The plot shows HF energy di erences which is described in the Supplementary Material. relative to those of a large Gaussian basis, cc-pV5Z. The We now turn to MSG-DMRG calculations for H10.Our convergence of the MSG basis is irregular because the DMRG implementation uses the matrix product opera- centers of the gausslets are not aligned with the nuclei; tor compression of our earlier sliced basis DMRG (SB- but it is easy to get very accurate results and judge the DMRG) approach [3]. This compression makes the cal- accuracy. At small R, the Gaussian basis sets have trou- culation time for fixed accuracy per atom scale linearly ble due to linear dependence [4], leading to a small but in the number of atoms in a hydrogen chain both in SB- noticeable discrepancy between the 5Z and MSG results. DMRG and MSG-DMRG. We are currently limited to Future Directions & Conclusions A lot of creativity is possible in going beyond standard Gaussian approach to quantum chemistry
Sliced-basis and gausslet basis are just two of many ideas to be tried Quantum chemistry DMRG 8.21
4 1 (a) (b) 2 (x) ( x )
i 0
j 0.5 u(x) G G
-2
0 -4
-10 0 10 -4 -2 0 2 4 x n = 1 2 3 4 5 6 7 8 9 x 1.5 Fig. 4: One-dimensional array of gaussletc functions with a length scale(d of) 1.0. The gausslet centered at the origin is highlighted in( solid) black to emphasize details. 1 Nucleus x ➞ ( x ) about the details of constructingj them, see Ref. [28] which proposes and constructs gausslets, ˜ G 0.5 building upon the development in Ref. [30] of compact and symmetric families of orthogonal y ➞ wavelets. Importantly, after constructing a grid0 of gausslets like in Fig. 4 and using them to discretize quantum chemistry Hamiltonians, one can obtain very accurate continuum results using a gaus- -10 0 10 slet spacing of about 1.0, in contrast to grid discretizationx which requires about an order of magnitude smaller spacing to obtain similar accuracy [28]. Better yet, we will see next that by adapting the grid on which the gausslets are centered, one can even better resolve the continuum using small numbers of gausslet functions.
4.2.2 Adapted grid of gausslets
Using an even-spaced grid of gausslet functions, as in Fig. 4 to discretize quantum chemistry Hamiltonians is more efficient that using a simple grid, yet still requires more functions than are actually needed. The reason is that while high resolution is required to capture details of the electronic wavefunction near atomic nuclei, much less resolution is needed away from nuclei. A straightforward way to reduce the number of functions needed while preserving high resolution near nuclei is to perform a coordinate mapping on the gausslet functions so that they form an adapted grid, with a finer spacing near nuclei and a coarser spacing otherwise. Such a coordinate mapping may be defined via a function x(u) which maps from a fictitious space u where the gausslets are defined to have a regular grid spacing into the actual space x used for the quantum chemistry calculation. Let u(x) be defined as the inverse of the mapping x(u). For the case of a single atom, a sensible coordinate mapping is
1 1 u(x)= sinh (x/a) (31) s defined by a scale parameter s and a core cutoff parameter a. Figure 5(a) shows how this mapping takes an evenly spaced grid along the y direction of the plot into a variable spaced grid along the x direction. min
Tensor networks are real-space oriented, don't use Slater determinants
Require re-thinking conventional choices to get best performance Recent work by Lin and Tong for compressing Coulomb interactions into "PEPO" tensor networks 17
15 4 3 2 1
V
(a) HODLR (b) -matrixFig. 4.1. A PEPO(c) constructed Type-0 from and the type-1 snake-shaped interac- MPO by adding bonds with bond dimen- Hsion 1 (blue dotted lines).tions
ˆ Fig. 4.2. AmatrixsatisfyingtheHODLRformat(a)andthewherea ˆi and bj are local-matrix operators format defined (b). on sites (c)i showsand j, and as a result they the two types of interactions at level 4 for -matrixcommute format. with each In other. redH frames This operator are type-0 can be interactions represented by an MPO with bond and in blue frames type-1 interactions. In (a)(b)(c)H dimension The 3,numbers since we showcan treat the it level as a special of each case submatrix. of the interaction in (2.1) in which = 0. The notion of rank-one operators can be easily generalized to a 2D system, for operators in the form Note only interactions allowed in the interaction list is covered on this level, which is ˆ an important fact that enables us to construct MPOs with a small bondaˆibj, dimension. i j Then we can decompose the tensor as X where is some order assigned to the system. This can be represented by a snake- shapedL MPO [24] with bond dimension 3. The coe cient matrix is a UTLR matrix Vˆ = of rankVˆ ( 1.`). We add some bonds with bond dimension 1 to make it a PEPO. The bond dimension of the PEPO is still 3. A graphical illustration of the PEPO is provided in X`Fig.=1 4.1. ˆ Sometimes the operator acts trivially everywhere except for a given region (de- This sum covers all the interactions in V .scribed by an index set ) in the system, we may order the sites in such a way that all I ˆ ˆ ˆ Because of the hierarchical low-ranksites structure, in are labeled we before can the yield rest of some the sites. approximate Then the operator is O = O I c .In I ˆ ˆ ˆI ⌦ I ˆ such casesT we call the set the support of operator O. Now suppose O1, O2, , Om low-rank decomposition V( `;2↵, `;2↵+1) AB .Then I ··· I I are⇡ m rank-one operators in a 1D system. Integer intervals 1, 2, , m are the supports of each operator respectively. If = , then theI operatorsI ··· IOˆ , Oˆ are M i j i j non-overlapping. The sum of these operatorsI \ canI be expressed; as an MPO with bond V(i, j)ˆninˆj dimension 5. We give(A the(i, MPO r)ˆni rules)(B in(j, Appendix r)ˆnj). A. ⇡ This notion of two operators being non-overlapping can be easily extended to 2D i `;2↵ j `;2↵+1 r=1 i `;2↵ j `;2↵+1 2XI 2IX X 2systems.XI 2 InsteadIX of intervals the supports are replaced by boxes. The sum of these Note that on the right-hand side each r indexesoperators acan rank-one be expressed operator. as a PEPO with The bond support dimension of 5. We will also give the PEPO rules in Appendix B. As mentioned before, in 2D we need to define an order each rank-one operator is `;2↵ `;2↵+1 =for` the1;↵ rank-one. operators. However when we consider a sum of non-overlapping I [ I ` rank-one 1 I operators, the order only needs to be defined locally within each support, Therefore, for each ↵ =0, 1, , 2 1, we have M rank-one operators. For i.e. we do not need to impose a global ordering valid for all rank-one operators. each ↵, we pick one from these M···operators correspond to a given index r, and sum 4.2. Hierarchical low-rank matrix. In this section we introduce the hierar- them up to get, chical low-rank matrix, and its use to construct MPO and PEPO representation of tensor network operators with long-range interactions. Consider the 1D system first. ˆ (`) Vr = (A(i, r)ˆni)(B(j, r)ˆnj), ↵ i j X 2XI`;2↵ 2IX`;2↵+1 and then we have M ˆ (`) ˆ (`) V = Vr . r=1 X (`) Hence all the interactions at level ` can be collected into M operators Vˆr , each one is a sum of non-overlapping rank-one operators. From this procedure, we have collected all interactions at level ` into a sum of M MPOs each with bond dimension 5. M can be chosen to be (log(N/✏)) for the coe cient matrix to have a 2-norm error of ✏. The system containsO L levels so in total we need ML = (log(N) log(N/✏)) MPOs, each with a bounded rank. O 15
⌃
Fig. 4.1. A PEPO constructed from the snake-shaped MPO by adding bonds with bond dimen- sion 1 (blue dotted lines). Combined with recent progress in optimizing PEPS*, wherea ˆi and ˆbj are local operators defined on sites i and j, and as a result they commute with each other. This operator can be represented by an MPO with bond dimension 3,could since we can treatsoon it as a special see case ofPEPS the interaction quantum in (2.1) in which chemistry! = 0. The notion of rank-one operators can be easily generalized to a 2D system, for operators in the form
ˆ In principle scalableaˆibj, to huge 2D planes of atoms, i j X where iscontrolled some order assigned to the & system. accurate, This can be represented handling by a snake- strong correlation shaped MPO [24] with bond dimension 3. The coe cient matrix is a UTLR matrix of rank 1. We add some bonds with bond dimension 1 to make it a PEPO. The bond dimension of the PEPO is still 3. A graphical illustration of the PEPO is provided in Fig. 4.1. * Liao, Liu, Wang, Zhang, arxiv:1903.09650 Sometimes the operator acts trivially everywhere except for a given region (de- scribed by an index set ) in the system, we may order the sites in such a way that all I sites in are labeled before the rest of the sites. Then the operator is Oˆ = Oˆ Iˆ c .InZaletel, Pollmann, 1902.05100 I I ⌦ I such cases we call the set the support of operator Oˆ. Now suppose Oˆ1, Oˆ2, , Oˆm are m rank-one operatorsI in a 1D system. Integer intervals , , , ···are theHaghshenas, O'Rourke, Chan, 1903.03843 I1 I2 ··· Im supports of each operator respectively. If i j = , then the operators Oˆi, Oˆj are non-overlapping. The sum of these operatorsI \ canI be expressed; as an MPO with bondLiu, Huang, Gong, Gu, 1908.09359 dimension 5. We give the MPO rules in Appendix A. This notion of two operators being non-overlapping can be easily extended to 2DHyatt, Stoudenmire, 1908.08833 systems. Instead of intervals the supports are replaced by boxes. The sum of these operators can be expressed as a PEPO with bond dimension 5. We will also give the PEPO rules in Appendix B. As mentioned before, in 2D we need to define an order for the rank-one operators. However when we consider a sum of non-overlapping rank-one operators, the order only needs to be defined locally within each support, i.e. we do not need to impose a global ordering valid for all rank-one operators. 4.2. Hierarchical low-rank matrix. In this section we introduce the hierar- chical low-rank matrix, and its use to construct MPO and PEPO representation of tensor network operators with long-range interactions. Consider the 1D system first.