X marks the spot: accurate energies from intersecting extrapolations of continuum quantum Monte Carlo data

We explore the application of an extrapolative method that yields very accurate total and relative energies from variational and diffusion quantum Monte Carlo (VMC and DMC) results. For a trial wave function consisting of a small configuration interaction (CI) wave function obtained from full CI quantum Monte Carlo and reoptimized in the presence of a Jastrow factor and an optional backflow transformation, we find that the VMC and DMC energies are smooth functions of the sum of the squared coefficients of the initial CI wave function, and that quadratic extrapolations of the non-backflow VMC and backflow DMC energies intersect within uncertainty of the exact total energy. With adequate statistical treatment of quasi-random fluctuations, the extrapolate and intersect with polynomials of order two (XSPOT) method is shown to yield results in agreement with benchmark-quality total and relative energies for the C2, N2, CO2, and H2O molecules, as well as for the C2 molecule in its first electronic singlet excited state, using only small CI expansion sizes.


I. INTRODUCTION
Quantum Monte Carlo (QMC) methods are a broad family of stochastic wave-function-based techniques that accurately approximate the solution of the Schrödinger equation of an electronic system.The variational quantum Monte Carlo (VMC) method 1,2 obtains expectation values corresponding to an analytic trial wave function Ψ T (R) in real space and provides a framework for optimizing wave function parameters, 3,4 such as those in the multideterminant-Jastrowbackflow form, where {D I } are M Slater determinants, e J(R) is a Jastrow correlation factor, 5,6 and X(R) are backflow-transformed electronic coordinates. 7Diffusion quantum Monte Carlo (DMC) 2,8 is a real-space projection method which recovers the lowest-energy solution Φ(R) of the Schrödinger equation compatible with the fixed-node condition that Φ(R)Ψ T (R) be nonnegative everywhere.We refer to the VMC and DMC methods collectively as continuum QMC (cQMC) methods.
In the configuration interaction (CI) ansatz the solution of the Schrödinger equation is expressed as where {|D I } are all possible Slater determinants that can be constructed in a given orbital basis.Equation 2 is exact in the limit of an infinite expansion using a complete basis set, but in practical methods finite expansions and bases are used.Full CI quantum Monte Carlo (FCIQMC) 9,10 is a second-quantized projection technique in which random walkers sample the discrete Hilbert space defined by {|D I } in order to determine approximate values of the CI coefficients {c I }.
a) Electronic mail: p.lopez.rios@fkf.mpg.de The nominal computational cost of cQMC calculations scales polynomially with system size N , typically as N 2 -N 4 , and the quality of the resulting energies depends on the accuracy of the trial wave function for VMC, and on the accurate location of its nodes for DMC.The cQMC methods excel at describing explicit dynamic and long-ranged correlations, but the error incurred by the fixed-node approximation is often significant.By contrast, FCIQMC is formally an exponentially-scaling method which trivially captures static correlations, but requires a very large number of walkers to provide a good description of dynamic correlations.The complementary nature of the strengths of cQMC and FCIQMC makes combining these methods highly desirable.Several ways of combining cQMC and FCIQMC have been presented in the literature, such as using DMC to assist in the extrapolation to the thermodynamic limit of FCIQMC energies of the electron gas, 11 or using VMC-optimized Jastrow factors in FCIQMC with the transcorrelated method. 12,13Here we shall focus on the use of selected CI wave functions generated with FCIQMC to construct multideterminantal trial wave functions for cQMC calculations.
Multideterminant expansions have been used for decades in cQMC calculations of atomic and molecular systems, including ground-state energy calculations, [14][15][16][17][18][19] excitation energies, [20][21][22][23] and geometry optimizations. 22,24The use of truncated CI expansions in cQMC presents the problem that no reliable criteria exist to truncate wave functions for different systems in a consistent manner, resulting in energy differences of questionable accuracy.One possible approach is to use extremely large multideterminantal wave functions, [19][20][21][22][23][24][25][26][27] under the expectation that the fixed-node error in the total energies will become smaller than the target error.While algorithmic developments have vastly reduced the computational cost associated with the use of multideterminantal wave functions in cQMC, 25,[28][29][30][31] this remains an expensive choice.Using trial wave functions without a Jastrow factor reduces the nominal computational burden 20,21,27 at the cost of losing the accurate, compact description of dynamic correlation afforded by fully-optimized trial wave functions.By including explicit correlations, in the present paper we are able to explore the use of relatively small multideterminantal wave functions to perform an extrapolation of the cQMC total energy to the full-CI, complete orbital-basis limit.We test our method on a variety of molecular systems, obtaining total and relative energies within uncertainty of benchmark-quality results from the literature.
The rest of this paper is structured as follows.In Section II we present the methodological details of our extrapolation method, which we illustrate with calculations of the carbon dimer and the water molecule.We then apply our method to several atomic and molecular systems, and we report the results in Section III.Our conclusions and outlook are presented in Section IV.Hartree atomic units ( = |e| = m e = 4πǫ 0 = 1) are used throughout; the uncertainties and error bars we report refer to standard 68.3% (one-sigma) confidence intervals except when explicitly noted otherwise.

II. METHODOLOGY
Let M gen be the number of determinants occupied at a given point in an equilibrated FCIQMC calculation, representing the CI wave function where c I is obtained as the sum of the signed weights of the walkers occupying the Ith determinant.The values of the first few coefficients {c I } I≪Mgen converge relatively quickly in FCIQMC calculations and can be expected to be reasonably close to their full CI (FCI) values.This makes FCIQMC an ideal method for quickly generating good-quality selected CI wave functions of moderate sizes -studying the suitability of other CI solvers for this purpose is beyond the scope of this paper.
Let us consider the wave function obtained by truncating Eq. 3 to size M ≪ M gen .The sum of the squares of the coefficients of the resulting wave function relative to that at size M gen is which goes to 1 as M → M gen .This CI wave function of size M can be combined with a Jastrow factor, and optionally with a backflow transformation, to produce a multideterminant-Jastrow(-backflow) trial wave function for cQMC, as given in Eq. 1.The wave function parameters can be (re-)optimized in the context of VMC, producing a trial wave function with which to compute VMC and DMC energies.Repeating this procedure by truncating the original CI wave function to different sizes yields a set of VMC and DMC energies that can be plotted as a function of w.Plots of this kind, albeit using other CI solvers, can be found in the literature; see Fig. 3  example.The present work is in fact inspired by the observation that the VMC and DMC curves in these plots appear to be smooth and would seem to be about to intersect just off the right-hand side of the graph.In Fig. 1 we plot the VMC and DMC energies we obtain for the ground state of the allelectron C 2 molecule, see Table I, using Hartree-Fock orbitals expanded in the cc-pCVTZ basis set, 32,33 along with quadratic fits to the data of the form where a, b, and c are fit parameters.w FIG. 1. VMC and DMC total ground-state energy of the carbon dimer using multideterminant-Jastrow trial wave functions as a function of w, using Hartree-Fock orbitals expanded in the cc-pCVTZ basis set.Quadratic fits to the data are extended beyond w = 1 to show their intersection, which is in good agreement with the estimated exact nonrelativistic total energy of the system. 34e fits to the VMC and DMC data intersect at w = 1.031, corresponding to a total energy of −75.9287Ha, not far off the exact nonrelativistic total energy estimate of −75.9265Ha given in Ref. 34.We refer to this way of estimating the total energy of a system as the extrapolate and intersect with polynomials of order two (XSPOT) method.
In what follows we develop the methodology to enable the application of the XSPOT method in practice using as test systems the C, N, and O atoms, the ground-state C 2 , N 2 , H 2 O, and CO 2 molecules, and the C 2 molecule in its lowest-lying singlet electronic excited state, which we refer to simply as C 2 * .These atoms and molecules are simulated as all-electron, both in the sense that no effective-core potentials are used and that excitations from "core" orbitals are allowed in the CI wave function.In Table I we give the states and geometries we have used for these systems.

A. Theoretical justification
The extrapolation shown in Fig. 1 might seem simplistic from a quantum chemical perspective, given that all calculations involved have been performed with the same, finite orbital basis, so one would expect an orbital-basis dependent result which should itself be extrapolated to the complete-basis System Geometry 34,35 State Atoms and molecules considered in this work, along with their electronic states and geometries.
limit.For instance, the FCIQMC energy tends to a basis-set dependent FCI limit as the number of walkers tends to infinity, and this must in turn be extrapolated to the basis-set limit in order to obtain the exact energy of the system.
In what follows we will conceptually combine the choice of molecular orbitals (e.g., Hartree-Fock, natural orbitals, . . . ) with the choice of basis set (e.g., cc-pCVDZ, cc-aug-pVTZ, . . .), so we shall discuss the completeness of the (molecular) orbital basis instead of that of the basis set alone to emphasize this point.
The XSPOT extrapolation procedure can be easily justified in the hypothetical case of using an infinite, "complete" orbital basis.The FCIQMC energy with this "complete" orbital basis would tend to the exact total energy of the system E 0 in the infinite walker-number limit, and the sum of the squared CI coefficients would also tend to that of the exact wave function, w 0 .The exact wave function has no dynamic correlation left to recover, so the Jastrow factor and backflow displacement in the cQMC trial wave function would optimize to zero, and the VMC and DMC energies would both coincide with E 0 .At finite expansion sizes, w < w 0 , the VMC and DMC methods yield variational energies satisfying E VMC ≥ E DMC ≥ E 0 , which, assuming these to be smooth functions of w, validates the XSPOT method with the "complete" orbital basis.
We note that in a truncated CI wave function, the infinite, "complete" orbital basis is effectively finite, since a finite number of determinants can only contain a finite number of distinct orbitals.Conversely, a sufficiently small selected CI wave function with a finite orbital basis is indistinguishable from a CI wave function of the same size with the "complete" orbital basis -assuming the finite basis contains the first few orbitals in the "complete" orbital basis.
As the orbitals in a finite basis get used up, the cQMC energies can be expected to plateau as a function of w as they tend to their orbital-basis dependent limit.We refer to this phenomenon as "orbital-basis exhaustion", and to the onset of this plateau as the exhaustion limit w exh. .Note that orbital bases such as natural orbitals can be constructed so as to compactly describe the system with fewer orbitals, which has the side effect of reducing the value of w exh. .We discuss this aspect further in Section II C.
As a proxy for the degree of orbital-basis exhaustion, in Table II we show the fraction of orbitals used in CI wave functions of the same size for Hartree-Fock orbitals expanded in four different basis sets in the cc-pVxZ and cc-pCVxZ families 32,33  numbers, we use the cc-pCVTZ basis throughout this paper to ensure we have enough leeway to increase the multideterminant wave function size before hitting the exhaustion limit.
We provide an a posteriori assessment of this choice in Section III.Finite-orbital-basis FCIQMC and cQMC calculations performed at w < w exh.behave as if one were using the "complete" orbital basis.Therefore it is legitimate to expect that the extrapolation of quadratic fits to these VMC and DMC data intersect at w = w 0 and E = E 0 , provided that the VMC and DMC energies are smooth functions of w representable by a second-order polynomial for w h.o.< w < w exh., where w h.o. is a threshold below which higher-order polynomials would be needed.
Note that the initial FCIQMC wave function with M gen determinants is not required to be below the exhaustion limit since it simply serves to construct selected CI wave functions of size M ≪ M gen , which are required to be below the exhaustion limit, and to define the arbitrary point at which w = 1 in the plots; w = 1 has no special significance in this method.
In our calculations we choose CI wave function sizes so that the points are more or less evenly spaced in the w axis, and we make sure that different points correspond to wave functions containing a different number of distinct spatial orbitals so as to capture the effect of simultaneously growing the CI expansion and the orbital basis.

B. Obtaining statistically meaningful results
In reality cQMC energies do not exactly follow smooth curves, but it is reasonable to assume that a smooth underlying trend E(w) exists, and that the cQMC energy E i deviates from it by a a quasi-random amount q i .Considering also the statistical uncertainty ∆E i , the ith point in a set of cQMC energies can then be modelled as where ζ i is a random number drawn from the standard normal distribution.In order to make this generic model for the quasi-random fluctuations useful in practice, we make the approximation that q i = ξ i α, where ξ i is a random number drawn from the standard normal distribution and α is a constant amplitude independent of w.In Fig. 2 we illustrate our model of cQMC energy data as a function of w.We estimate the value of α by performing a preliminary least-squares fit to the bare data, Ẽ(w), and evaluating i.e., we obtain α as the root-mean-square deviation of the data from the fit value not accounted for by statistical uncertainty alone.For this procedure to produce a meaningful result, the number of data points in each curve must be significantly greater than the number of parameters in the quadratic fit function; we use at least 7 data points for all fits reported in this paper.
In order to account for the statistical uncertainty and quasirandom fluctuations in the XSPOT method, we use a Monte Carlo resampling technique in which we generate 100 000 instances of each VMC and DMC dataset in which a random amount proportional to α 2 + ∆E i 2 is added to the original energy values.We then perform fits to these shifted data and find the intersection point for each such instance, and obtain the final result by averaging over instances; see Fig. 3 for an illustration of this process.This procedure provides meaningful uncertainties on the intersection energies which account for both the cQMC statistical uncertainty and quasi-random deviations from the smooth trend.
We demonstrate the full statistical procedure of the XSPOT method on multideterminant-Jastrow data for the carbon dimer in Fig. 4. Notice that the distribution of intersection points shown in the inset of Fig. 4 has a tail extending towards low E and large w.These tails become more problematic the more parallel the two intersecting curves are, eventually preventing the evaluation of an intersection point at all.It is therefore important to try to apply the XSPOT method to curves which are as close to perpendicular as possible.
In Fig. 5 we include additional VMC and DMC data using an inhomogeneous backflow transformation ("bVMC" and "bDMC") for the carbon dimer.We list the intersections between pairs of curves in Table III, all of which are within un-  the best-resolved results, which is to be expected since these curves intersect at the widest angle among all pairs of curves, as can be seen in Fig. 5.By contrast, the DMC and bVMC curves intersect at a narrow angle and incur a small but nonzero fraction of "missed" intersections, i.e., random instances of the data whose fits fail to intersect at w > 1, which signals the presence of heavy tails in the intersection distribution, resulting in a large uncertainty on the intersection energy.
For the four curves in Fig. 5, the estimated amplitude of the quasi-random fluctuations ranges from α = 0.17 to 0.41 mHa.Throughout this paper we converge the cQMC calculations so that the uncertainties satisfy (∆E i ) 2 < α 2 , i.e., so that quasirandom fluctuations represent the main contribution to the uncertainty on the fits and intersections; see the supplementary material for the list of values of α obtained.The statistical uncertainties on the cQMC energies can thus be neglected for all practical purposes.

C. Choice of orbital basis
As alluded to in Section II A, the choice of molecular orbitals plays a crucial role in the behavior of the cQMC energies as a function of w, modifying the point w exh. at which the effects of exhaustion start to become noticeable.In Fig. 6 we demonstrate this for the H 2 O molecule by comparing the cQMC energies obtained using Hartree-Fock orbitals expanded in the cc-pCVTZ basis set and natural orbitals expanded in the same basis set constructed so as to diagonalize the one-body density matrix in coupled cluster singles and doubles (CCSD).While CCSD natural orbitals produce lower cQMC energies and correspond to larger values of w at fixed expansion sizes, the cQMC energies we obtain using Hartree-Fock orbitals follow the quadratic trend throughout the whole w range considered, while those obtained with natural orbitals plateau very early on, preventing their meaningful extrapolation.

A. Calculation details
In our calculations we use Hartree-Fock orbitals expanded in the cc-pCVTZ Gaussian basis set 32,33 obtained using MOLPRO.36 We perform a small-scale FCIQMC calculation using the NECI package 37 with configuration state functions (CSFs) instead of determinants as walker sites, 38 which reduces the number of FCIQMC walkers required to accurately represent the wave function; note that the use of CSFs is not a requirement of the XSPOT method.The FCIQMC population is grown to 10 6 walkers and equilibrated, and the coefficients of the M gen = 10 000 most-occupied CSFs are recorded from a population snapshot.From this information we build CI expansions with the M CSFs with largest absolute coefficients, where M ≤ 1500 ≪ M gen .In our cQMC calculations the orbitals are corrected to obey the electron-nucleus cusp condition. 39he CSF coefficients are reoptimized in the presence of a Jastrow factor of the Drummond-Towler-Needs form, 5,6 and of an optional inhomogeneous backflow transformation including electron-electron, electron-nucleus, and electronelectron-nucleus terms. 7We do not optimize any of the parameters in the molecular orbitals, which provide degrees of freedom that overlap significantly with those in the backflow transformation.Note that even though CSFs are used, the presence of the Jastrow factor and of the backflow transformation prevents the cQMC trial wave function from formally being an exact spin state. 40We optimize our wave function parameters using linear least-squares energy minimization 3,4 with 10 6 statistically independent VMC-generated electronic configurations, a number large enough that the optimized cQMC energy can be assumed to lie reasonably close to its variational minimum; note that any remaining optimization error can be considered to be absorbed into the quasirandom error.
The resulting trial wave function is then used to run two DMC calculations with time steps 0.001 and 0.004 a.u. and target populations of 2048 and 512 configurations, respectively, except for bDMC runs on CO 2 at 500 and 1000 CSFs for which we use 65536 and 16384 configurations.These energies are then linearly extrapolated to the zero time-step, infinite-population limit. 2,41e use the CASINO package 2 to run the cQMC calculations, and use multi-determinant compression 30 to reduce the computational expense of evaluating the trial wave function.We perform the fits to the data and find their intersections using our custom POLYFIT tool. 42The cQMC energies obtained for all systems can be found in the supplementary material.

B. Results
In this section we test the XSPOT method on all eight systems under consideration to assess the different aspects discussed in Section II and to determine the broader applicability of the method.The VMC and bDMC energies and fits we obtain for the eight systems are shown in Fig. 7; additional plots containing the bVMC and DMC energies can be found in the supplementary material.
All the curves in Fig. 7 are relatively smooth and provide a well-defined intersection.The apparent non-monotonicity of the bDMC curve for the carbon atom is an artifact of the use of a fit function which formally allows non-monotonic behavior, and should be interpreted accordingly: E(w) can be regarded to approach the intersection with negligible slope, and the region w > w 0 should be ignored since E(w) does not have a physical meaning there.All of the other fits appear to be monotonic in the range shown.
The fraction of orbitals used, a proxy for the degree of orbital-basis exhaustion, is plotted for each of the systems in Fig. 8.We do not use up all of the orbitals in the basis in any of our calculations, and the curves in Fig. 7 do not seem to exhibit symptoms of orbital-basis exhaustion.The bVMC energies do seem to plateau somewhat, which we discuss briefly in the supplementary material; note that we do not use the bVMC data to obtain our final results.
In Table IV we compare the total energies obtained from applying the XSPOT method to VMC and bDMC energy data with benchmark-quality estimates of the exact nonrelativisitic energies of the systems from the literature, along with our best bDMC result and prior cQMC results for reference.The atomization energies of the ground-state molecules are shown in Table V.For excited-state C 2 * we compare the vertical excitation energy with that calculated with internally-contracted multi-reference coupled cluster theory (ic-MRCC); [44][45][46] we have computed the total energy of C 2 * shown in Table IV by adding the ic-MRCC excitation energy to the estimated ground-state energy of C 2 from Ref. 34.
All of the total energies reported in Table IV are within statistical uncertainty of their corresponding benchmark values.An important observation is that our individual cQMC energies are not lower than those from prior cQMC calculations, implying that we incur a lower computational cost, but our XSPOT results are in general closer to the benchmarks than cQMC results from prior studies.Our XSPOT energies are on average 1.1 standard errors above the benchmark, with a rootmean-square deviation of 1.9 standard errors.These results are compatible with the XSPOT method being exact when the method's assumptions are satisfied.The relative energies are likewise in agreement with the benchmark values.
We find that the magnitude α of the quasi-random fluctuations of the cQMC energies is of up to 0.7 mHa.These fluctuations are particularly visible in the VMC data for N, O, and H 2 O in Fig. 7, for example; in the supplementary material we give the values of α we have obtained for each of the curves.The magnitude of the quasirandom fluctuations does not seem to increase too rapidly with system size, but their effect on the extrapolated energy becomes more pronounced the further the cQMC data are from the intersection in the plots.This increasing uncertainty on the XSPOT total energies, reaching 4 mHa for the CO 2 molecule, hints at a limitation of the methodology: the cQMC energies and values of w ob-  IV) as a dotted line with a shaded area of ±1 kcal/mol around it, and the intersection point between the VMC and bDMC curves.The insets show the statistical distributions of intersection points as color maps with overlaid contour curves.
tained using modest-sized multideterminant expansions with a fixed basis set move away from the intersection point with increasing system size, which in turn exacerbates the effects of quasirandom noise on the uncertainty of the XSPOT energy; TABLE IV.Total energies in Ha obtained with the XSPOT method using the VMC and bDMC data for the various atoms and molecules considered in this work, along with results from prior multi-determinant cQMC studies, our best individual bDMC energy for each system, and benchmark-quality nonrelativistic total energies from the literature.one would have to use bigger basis sets and larger multideterminantal expansions to get data closer to the intersection in order to reduce this uncertainty, increasing the computational cost of the approach.

IV. CONCLUSIONS
We have presented an empirical extrapolation strategy for cQMC energies as a function of the sum of the squared multideterminant coefficients in the initial selected CI wave function from which the trial wave function is constructed.This approach is made possible by the smoothness of the energies as a function of the CI expansion size, and we have presented a simple statistical procedure to handle the quasi-random nonsmoothness in the data, which we show to work very well in practice.We find that Hartree-Fock orbitals expanded in standard basis sets provide the type of gradual convergence required for the XSPOT method to work well.The results from the tests we have conducted are compatible with the XSPOT method being capable of obtaining exact total energies, with the caveat that trial wave function complexity must increase with system size in order to control the uncertainty on the results.

SUPPLEMENTARY MATERIAL
See the supplementary material for the cQMC data used in this paper, a table of the magnitude of the quasirandom fluctuations encountered, and discussion of connected extrapolation approaches.The supplementary material additionally cites Refs.1-5.
Supplemental information for "X marks the spot: accurate energies from intersecting extrapolations of continuum quantum Monte Carlo data"

S1. VMC AND DMC DATA
Tables S1-S8 contain the cQMC energies we have obtained for our manuscript.These energies are plotted in Fig. S1, as Fig. 4 of the manuscript with added backflow VMC and non-backflow DMC curves.
The bVMC energy curves in Fig. S1 would appear to plateau somewhat at large w, a potential symptom of orbital-basis exhaustion.We hypothesize that the backflow transformation makes some of the information contained in the orbitals redundant, which effectively reduces the value of w exh.with respect to the non-backflow data.Judging by the quality of the intersections, this issue affects bVMC more than it does bDMC, which would imply that this redundancy does not involve the location of the nodes of the trial wave function to the same degree as its values away from the nodes.This is in any case a tentative explanation; we do not use the bVMC data to obtain our final results.S2.VMC, DMC, bVMC, and bDMC energies for the N atom obtained for our manuscript, in Hartree atomic units.nCSF is the number of CSFs in the wave function, n det is the number of (not necessarily unique) determinants, and n orb is the number of orbitals from the 43-orbital basis used in the wave function.4 of the manuscript) as a dotted line with a shaded area of ±1 kcal/mol around it, and the intersection point between the VMC and bDMC curves.

S2. VALUES OF α
Table S9 shows the magnitude of the quasirandom fluctuations α estimated for each of the curves we have obtained, as defined in Eq. 7 of the manuscript.Note that all the individual cQMC calculations reported in Tables S1-S8 have been run for long enough that the statistical uncertainties on them are smaller then the corresponding value of α.

S3. CONNECTION WITH SELECTED CI AND DMC VARIANCE EXTRAPOLATION
In selected CI methods one usually obtains the full-CI limit by extrapolates the total energy to the limit where the difference between the total energy and the variational energy (which typically amounts to a perturbation-theory correction) is zero.S1,S2 In the same spirit, one could consider extrapolating the DMC energy to the limit where the VMC-DMC energy difference is zero, which, by analogy with the selected CI approach, would effectively treat DMC as a perturbative correction of sorts on top of VMC.
Note that E VMC − E DMC is proportional to the DMC variance, S3 so extrapolating to E VMC − E DMC → 0 amounts to DMC variance extrapolation, analogous to VMC variance extrapolation schemes which have been used over the years, recent examples of which include Refs.S5.
One potential advantage of the DMC variance extrapolation approach over XSPOT is that the "exact" limit corresponds to a predefined value (zero) of the independent variable E VMC − E DMC , instead of an unknown value w 0 , so one does not need to explicitly intersect curves.
Clearly E VMC − E DMC = 0 implies that the VMC and DMC energy curves intersect as a function of w.From XSPOT, one can in fact derive that the DMC energy can be approximated by a second-order polynomial in E VMC − E DMC , but since w is eliminated in this process, the fit form for DMC variance extrapolation expresses no explicit assumptions on the dependence of the cQMC energies on the wave function.On the one hand, this is conceptually simpler, and could in principle avoid the orbital-basis exhaustion issue of the XSPOT method, since if both the VMC and DMC curves exhibit plateaus as functions of w then the DMC variance extrapolation curve may stall but need not deform.On the other hand, the explicit assumptions made by XSPOT could result in useful restrictions that guide the fits and reduce the uncertainty on the final results, an improvement that DMC variance extrapolation would miss on.
We have applied DMC variance extrapolation to our non-backflow and backflow data using the same fitting methodology we have used for XSPOT and in Fig. S2 we show the resulting fits.The energy data exhibit significant curvature, justifying the need for the quadratic fitting function, and out quasirandom fluctuation analysis produces values of α of the same order of magnitude as with XSPOT.We find that, for the most part, DMC variance extrapolation works, but yields much larger uncertainties than XSPOT.This is likely the case due to the data points being less evenly-spaced and often clustering together, resulting in several fits with non-monotonic mean values.It is noteworthy that the non-backflow and backflow curves do not line up in any of the cases, which one might expect them to since the extrapolation method is in principle agnostic to the specifics of the wave function.We have also applied DMC variance extrapolation to H 2 O with natural orbitals, see Fig. S3, to see if the w-independent character of this approach gets around the orbital-basis exhaustion problem encountered by XSPOT, as mentioned above.We find that, in contrast with XSPOT which fails to produce an intersection as shown in Fig. 6 of the manuscript, DMC variance extrapolation does yield an energy estimate, but it is affected by a significant uncertainty and misses the benchmark by just over three sigma, which is overall a worse-quality result than when Hartree-Fock orbitals are used.While DMC variance extrapolation has its merits, we believe that the XSPOT method we present in our manuscript is a superior option when applicable.
of Ref. 4 or Fig. 4 of Ref. 29, for

FIG. 2 .
FIG.2.Illustration of the expected behavior of cQMC energies as a function of w.The cQMC energies (circles and squares) deviate from the underlying smooth trend (lines) by a quasirandom amount (of amplitude represented by the width of the shaded area around the lines).The smooth trend can be represented by a quadratic function, E(w) (dashed line), for w h.o.< w < w exh.(shaded middle region), while for w < w h.o.higher-order contributions become important, and for w > w exh.orbital-basis exhaustion sets in.At the value of w corresponding to the exact wave function the quadratic function gives the exact energy, E0 = E(w0).

EFIG. 4 .
FIG.3.Illustration of the Monte Carlo resampling scheme used to compute statistics on the intersection between two curves.For each curve, having obtained an estimate of α (width of the shaded region) from the original energy data (not shown), we create a synthetic instance of the dataset by shifting the original points by a random amount proportional to α 2 + ∆Ei 2 (squares of same color saturation), perform a quadratic fit (line), and find the intersection between both fits (circled diamond).This process is repeated over the random instances (three shown in the illustration), from which statistics on the intersection are obtained.
FIG. 5. VMC, DMC, bVMC, and bDMC energies of the groundstate C2 molecule as a function of w.Mean values of the fits to the data are shown as lines, and the translucent areas around them represent 95.5% (two-sigma) confidence intervals.Also shown is the estimated exact nonrelativistic energy 34 as a dotted line with a shaded area of ±1 kcal/mol around it, and the intersection points between each of the six possible pairs of curves; error bars on these are only shown in the inset for clarity.

FIG. 6 .
FIG.6.VMC and DMC energies of the H2O molecule as a function of w, both using Hartree-Fock orbitals (left) and CCSD natural orbitals (right) expanded in the cc-pCVTZ basis set.In both cases the same numbers of CSFs are used.Mean values of the fits to the data are shown as lines, and the translucent areas around them represent 95.5% (two-sigma) confidence intervals.Also shown is the estimated exact nonrelativistic energy35 as a dotted line with a shaded area of ±1 kcal/mol around it, and the intersection point between the VMC and DMC curves in the left panel.The cQMC energies obtained with natural orbitals plateau with w, preventing the quadratic extrapolations from reaching the exact energy.

2 FIG. 7 .
FIG. 7. VMC and bDMC energies of the atoms and molecules considered in this work as a function of w.Mean values of the fits to the data are shown as lines, and the translucent areas around them represent 95.5% (two-sigma) confidence intervals.Also shown in each plot is the relevant benchmark energy (see details in text and TableIV) as a dotted line with a shaded area of ±1 kcal/mol around it, and the intersection point between the VMC and bDMC curves.The insets show the statistical distributions of intersection points as color maps with overlaid contour curves.

2 FIG. S1 .
FIG. S1.VMC, DMC, bVMC, and bDMC energies of the atoms and molecules considered in this work as a function of w.Mean values of the fits to the data are shown as lines, and the translucent areas around them represent 95.5% (two-sigma) confidence intervals.Also shown in each plot is the relevant benchmark energy (see details in text and Table4of the manuscript) as a dotted line with a shaded area of ±1 kcal/mol around it, and the intersection point between the VMC and bDMC curves.

2 FIG. S2 .
FIG. S2.Non-backflow (blue circles) and backflow (red triangles) DMC energy as a function of the corresponding VMC-DMC energy difference for the atoms and molecules considered in this work.Mean values of the fits to the data are shown as lines, and the translucent areas around them represent 95.5% (two-sigma) confidence intervals.Also shown in each plot is the relevant benchmark energy as a dotted line with a shaded area of ±1 kcal/mol around it.

H 2
FIG.S3.DMC energy as a function of the VMC-DMC energy difference for H2O using CCSD natural orbitals.The mean value of the fit to the data is shown as a line, and the translucent area around it represents the 95.5% (two-sigma) confidence interval.Also shown is the benchmark energy as a dotted line with a shaded area of ±1 kcal/mol around it.
In this simple example, for the all-electron carbon dimer.Based on these

TABLE III
. Location of all six pairwise intersections of the VMC, DMC, bVMC, and bDMC curves shown in Fig.5for the C2 molecule."Missed intersections" refer to random instances of the curves that do not intersect at w > 1 in the Monte Carlo resampling procedure.

TABLE V .
Ref. 34; b Ref. 35; c Refs. 44-46.Atomization and excitation energies in mHa of the various molecules considered in this work corresponding to the total energies in TableIVobtained from the XSPOT method, along with benchmark-quality nonrelativistic relative energies from the literature. *

TABLE S1 .
VMC, DMC, bVMC, and bDMC energies for the C atom obtained for our manuscript, in Hartree atomic units.nCSF is the number of CSFs in the wave function, n det is the number of (not necessarily unique) determinants, and n orb is the number of orbitals from the 43-orbital basis used in the wave function.

TABLE S3 .
VMC, DMC, bVMC, and bDMC energies for the O atom obtained for our manuscript, in Hartree atomic units.nCSF is the number of CSFs in the wave function, n det is the number of (not necessarily unique) determinants, and n orb is the number of orbitals from the 43-orbital basis used in the wave function.

TABLE S4 .
VMC, DMC, bVMC, and bDMC energies for the H2O molecule obtained for our manuscript, in Hartree atomic units.nCSF is the number of CSFs in the wave function, n det is the number of (not necessarily unique) determinants, and n orb is the number of orbitals from the 71-orbital basis used in the wave function.

TABLE S5 .
VMC, DMC, bVMC, and bDMC energies for the ground-state C2 molecule obtained for our manuscript, in Hartree atomic units.nCSF is the number of CSFs in the wave function, n det is the number of (not necessarily unique) determinants, and n orb is the number of orbitals from the 86-orbital basis used in the wave function.

TABLE S6 .
VMC, DMC, bVMC, and bDMC energies for the excited-state C2 * molecule obtained for our manuscript, in Hartree atomic units.nCSF is the number of CSFs in the wave function, n det is the number of (not necessarily unique) determinants, and n orb is the number of orbitals from the 86-orbital basis used in the wave function.

TABLE S7 .
VMC, DMC, bVMC, and bDMC energies for the N2 molecule obtained for our manuscript, in Hartree atomic units.nCSF is the number of CSFs in the wave function, n det is the number of (not necessarily unique) determinants, and n orb is the number of orbitals from the 86-orbital basis used in the wave function.
TABLE S8.VMC, DMC, bVMC, and bDMC energies for the CO2 molecule obtained for our manuscript, in Hartree atomic units.nCSF is the number of CSFs in the wave function, n det is the number of (not necessarily unique) determinants, and n orb is the number of orbitals from the 129-orbital basis used in the wave function.

TABLE S9 .
Magnitude of the quasirandom error α for the VMC, DMC, bVMC, and bDMC curves used in our manuscript, in mHa.