Correlated Geminal Wave Function for Molecules:An Efficient Resonating Valence Bond Approach Michele Casula, Claudio Attaccalite, and Sandro Sorella

Correlated geminal wave function for molecules:An efficient resonating valence bond approach Michele Casula, Claudio Attaccalite, and Sandro Sorella

Citation: J. Chem. Phys. 121, 7110 (2004); doi: 10.1063/1.1794632 View online: http://dx.doi.org/10.1063/1.1794632 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v121/i15 Published by the American Institute of Physics.

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Downloaded 18 Nov 2011 to 147.122.49.74. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions JOURNAL OF CHEMICAL PHYSICS VOLUME 121, NUMBER 15 15 OCTOBER 2004

Correlated geminal wave function for molecules: An efficient resonating valence bond approach Michele Casula,a) Claudio Attaccalite,b) and Sandro Sorellac) International School for Advanced Studies (SISSA) Via Beirut 2,4 34014 Trieste, Italy and INFM Democritos National Simulation Center, Trieste, Italy ͑Received 27 May 2004; accepted 27 July 2004͒ We show that a simple correlated wave function, obtained by applying a Jastrow correlation term to an antisymmetrized geminal power, based upon singlet pairs between electrons, is particularly suited for describing the electronic structure of molecules, yielding a large amount of the correlation energy. The remarkable feature of this approach is that, in principle, several resonating valence bonds can be dealt simultaneously with a single determinant, at a computational cost growing with the number of electrons similar to more conventional methods, such as Hartree-Fock or density functional theory. Moreover we describe an extension of the stochastic reconfiguration method, which was recently introduced for the energy minimization of simple atomic wave functions. Within this extension the atomic positions can be considered as further variational parameters, which can be optimized together with the remaining ones. The method is applied to several molecules from Li2 to benzene by obtaining total energies, bond lengths and binding energies comparable with much more demanding multiconfiguration schemes. © 2004 American Institute of Physics. ͓DOI: 10.1063/1.1794632͔

I. INTRODUCTION configuration interaction ͑CI͒ technique, which is able to take into account many Slater determinants, have been ͓ The comprehension of the nature of the chemical bond shown to be successful for small molecules e.g., Be2 Ref. deeply lies on quantum mechanics; since the seminal work 3͔. In these cases it is indeed feasible to enlarge the varia- 1 by Heitler and London, very large steps have been made tional basis up to the saturation, the electron correlation towards the possibility to predict the quantitative properties properties are well described and consequently all the chemi- of the chemical compounds from a theoretical point of view. cal properties can be predicted with accuracy. However, for ͑ ͒ Mean field theories, such as Hartree-Fock HF have been interesting systems with a large number of atoms this ap- successfully applied to a wide variety of interesting systems, proach is impossible with a reasonable computational time. although they fail in describing those in which the correla- Coming back to the H2 paradigm, it is straightforward to tion is crucial to characterize correctly the chemical bonds. show that a gas with N H2 molecules, in the dilute limit, can For instance, the molecular hydrogen H2 , the simplest and be dealt accurately only with 2N Slater determinants, other- first studied molecule, is poorly described by a single Slater wise one is missing important correlations due to the anti- determinant in the large distance regime, which is the para- bonding molecular orbital contributions, referred to each of digm of a strongly correlated bond; indeed, in order to avoid theNH molecules. Therefore, if the accuracy in the total expensive energy contributions—the so called ionic terms— 2 energy per atom is kept fixed, a CI-like approach does not that arise from two electrons of opposite spin surrounding scale polynomially with the number of atoms. Although the the same hydrogen atom, one needs at least two Slater deter- polynomial cost of these quantum chemistry algorithms— minants to deal with a spin singlet wave function containing ranging from N5 to N7—is not prohibitive, a loss of accu- bonding and antibonding molecular orbitals. Moreover at the bond distance it turns out that the resonance between those racy, decreasing exponentially with the number of atoms is two orbitals is important to yield accurate bond length and always implied, at least in their simplest variational formu- binding energy, as the correct rate between the ionic and lations. This is related to the loss of size consistency of a covalent character is recovered. Another route that leads to truncated CI expansion. On the other hand, this problem can the same result is to deal with an antisymmetrized geminal be overcome by coupled cluster methods, which however, in 4 power ͑AGP͒ wave function, which includes the correlation their practical realization are not variational. in the geminal expansion; Barbiellini in Ref. 2 gave an illu- An alternative approach, not limited to small molecules, minating example of the beauty of this approach solving is based on density functional theory ͑DFT͒. This theory is in principle exact, but its practical implementation requires an merely the simple problem of the H2 molecule. On the other hand the variational methods based on the approximation for the exchange and correlation functionals based on first principles, such the local density approximation ͑LDA͒ and its further gradient corrections ͑GGA͒,oron a͒Electronic mail: [email protected] b͒Electronic mail: [email protected] semiempirical approaches, such as BLYP and B3LYP. For c͒Electronic mail: [email protected] this reason, even though much effort has been made so far to

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⌿ go beyond the standard functionals, DFT is not completely B as long as the interaction between the electrons cou- reliable in those cases in which the correlation plays a crucial pling the different regions A and B can be neglected. In this role. Indeed it fails in describing HTc superconductors and limit the total energy of the wave function approaches the Mott insulators, and in predicting some transition metal com- sum of the energies corresponding to the two space-disjoint pounds properties, whenever the outermost atomic d shell is regions. This property, which is obviously valid for the exact near half-filled, as for instance, in the high potential iron many-electron ground state, is not always fulfilled by a ge- 5 proteins. Also H2 molecule in the large distance regime neric variational wave function. must be included in that list, since the large distance Born- Our variational wave function is defined by the product Oppenheimer energy surface, depending on van der Waals of two terms, namely, a Jastrow J and an antisymmetric part ⌿ϭ ⌿ forces, is not well reproduced by the standard functionals, ( J AGP). If the former is an explicit contribution to the although recently some progress has been made to include dynamic electronic correlation, the latter is able to treat the these important contributions.6 nondynamic one arising from near degenerate orbitals Quantum Monte Carlo ͑QMC͒ methods are alternative to through the geminal expansion. Therefore our wave function the previous ones and until now they have been mainly used is highly correlated and it is expected to give accurate results in two versions. especially for molecular systems. The Jastrow term is further ͑ ͒ ͑ ͒ ϭ i Variational Monte Carlo VMC applied to a wave split into a two-body and a three-body factors, J J2J3 , de- function with a Jastrow factor that fulfills the cusp conditions scribed in the following sections after the AGP part. and optimizes the convergence of the CI basis.7,8 A. Pairing determinant ͑ii͒ Diffusion Monte Carlo ͑DMC͒ algorithm used to improve, often by a large amount, the correlation energy of any As is well known, a simple Slater determinant provides given variational guess in an automatic manner.9 the exact exchange electron interaction but neglects the elec- Hereafter we want to show that a large amount of the tronic correlation, which is by definition the missing energy correlation energy can be obtained with a single determinant, contribution. In the past different strategies were proposed to using a size-consistent AGP-Jastrow ͑JAGP͒ wave function. go beyond Hartree-Fock theory. In particular a sizable Clearly our method is approximate and in some cases not yet amount of the correlation energy is obtained by applying to a satisfactory, but in a large number of interesting molecules Slater determinant a so called Jastrow term, which explicitly we obtain results comparable and even better than multide- takes into account the pairwise interaction between electrons. terminants schemes based on few Slater determinants per QMC allows to deal with this term in an efficient way.11 On atom that are affordable by QMC only for rather small mol- the other hand, within the quantum chemistry community ecules. AGP is a well known ansatz to improve the HF theory, be- Moreover, we have extended the standard stochastic re- cause it implicitly includes most of the double excitations of configuration ͑SR͒ method to treat the atomic positions as a HF state. further variational parameters. This improvement, together Recently we proposed a trial function for atoms, which with the possibility to work with a single determinant, has includes both the terms. Only the interplay between them allowed us to perform a structural optimization in a non- yields in some cases, such as Be or Mg, an extremely accu- trivial molecule such as the benzene radical cation, reaching rate description of the correlation energy. In this work we the chemical accuracy with an all-electron and feasible varia- extend this promising approach to a number of small mo- tional approach. lecular systems with known experimental properties, which The paper is organized as follows: In Sec. II we intro- are commonly used for testing new numerical techniques. duce the variational wave function, which is expanded over a The major advantage of this approach is the inclusion of set of nonorthogonal atomic orbitals both in the determinan- many CI expansion terms with the computational cost of a tal AGP and the Jastrow part. This basis set is consistently single determinant, which allow us to extend the calculation optimized using the method described in Sec. III that, as with a full structural optimization up to benzene, without a mentioned before, allows also the geometry optimization. particularly heavy computational effort on a single processor Results and discussions are presented in the remaining sec- machine. For an unpolarized system containing N electrons tions. ͑the first N/2 coordinates are referred to the up spin elec- trons͒ the AGP wave function is a N/2 ϫN/2 pairing matrix II. FUNCTIONAL FORM OF THE WAVE FUNCTION determinant, which reads ⌿ ͒ϭ ⌽ ͒ ͑ ͒ In this paper we are going to extend the application of AGP͑r1 ,...,rN det͓ AGP͑ri ,rjϩN/2 ͔, 1 the JAGP wave function, already used to study some atomic systems.10 We generalize its functional form in order to de- and the geminal function is expanded over an atomic basis scribe the electronic structure of a generic cluster containing ⌽ ͑ ↑ ↓͒ϭ ␭l,m␾ ͑ ↑͒␾ ͑ ↓͒ ͑ ͒ several nuclei. With the aim to determine a variational wave AGP r ,r ͚ a,b a,l r b,m r , 2 l,m,a,b function, suitable for a complex electronic system, it is important to satisfy, as we require in the forthcoming chapters, where indices l,m span different orbitals centered on atoms the ‘‘size consistency’’ property: if we smoothly increase the a,b, and i, j are coordinates of spin up and down electrons, distance between two regions A and B each containing a respectively. The geminal functions may be viewed as an given number of atoms, the many-electron wave function ⌿ extension of the simple HF wave function, based on molecu- ⌿ϭ⌿ factorizes into the product of space-disjoint terms A lar orbitals, and in fact the geminal function coincide with

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HF only when the number M of nonzero eigenvalues of the (1s,2s) ␭ ͑ ͒ ␭ j, jЈ␾ˆ Ј ͑ ϭ ͒ϭϪ ␭ j, jЈ␾ ͑ ϭ ͒ ͑ ͒ matrix is equal to N/2. Indeed the general function 2 can ͚ a,b a, j r Ra Za͚ c,b c, j r Ra , 5 be written in diagonal form after an appropriate transforma- j c, j tion for all b and jЈ; in the left-hand side the caret denotes the

M spherical average of the orbital gradient. The system can be ⌽ ↑ ↓͒ϭ ␭k␾˜ ↑͒␾˜ ↓͒ ͑ ͒ solved iteratively during the optimization processes, but if AGP͑r ,r ͚ k͑r k͑r , 3 k we impose that the orbitals satisfy the single atomic cusp conditions, it reduces to ␾˜ ϭ͚ ␮ ␾ where k(r) j,a k, j,a j,a(r) are just the molecular orbitals of the HF theory whenever MϭN/2. Notice that with ␭ j, jЈ␾ ͑ ͒ϭ ͑ ͒ ͚ c,b c, j Ra 0, 6 respect to our previous pairing function formulation also off- c(Þa),j diagonal elements are now included in the ␭ matrix, which and because of the exponential orbital damping, if the nuclei must be symmetric in order to define a singlet spin orbital are not close together, each term in the previous equations is state. Moreover it allows one to easily fulfill other system very small, of the order of exp(Ϫ͉R ϪR ͉). Therefore, with symmetries by setting the appropriate equalities among dif- a c the aim of making the optimization faster, we have chosen to ferent ␭ . For instance, in homo-nuclear diatomic mol- l,m use 1s and 2s orbitals satisfying the atomic cusp conditions ecules, the invariance under reflection in the plane perpen- and to disregard the sum ͑6͒ in Eq. ͑5͒. In this way, once the dicular to the molecular axis yields the following relation: energy minimum is reached, also the molecular cusp condi- ϩ tions ͑5͒ are rather well satisfied. ␭a,b ϭ͑Ϫ ͒pm pn␭b,a ͑ ͒ m,n 1 m,n , 4 where pm is the parity under reflection of the mth orbital. Another important property of this formalism is the pos- B. Two-body Jastrow term sibility to describe resonating bonds present in many struc- As it is well known the Jastrow term plays a crucial role tures, such as benzene. A ␭a,b different from zero represents m,n in treating many-body correlation effects. One of the most a chemical bond formed by the linear combination of the mth important correlation contributions arises from the electron- and nth orbitals belonging to ath and bth nuclei. It turns out electron interaction. Therefore it is worth using at least a that resonating bonds can be well described through the two-body Jastrow factor in the trial wave function. Indeed geminal expansion switching on the appropriate ␭a,b coeffi- m,n this term reduces the electron coalescence probability, and so cients: the relative weight of each bond is related to the am- decreases the average value of the repulsive interaction. The plitude of its ␭. two-body Jastrow function reads Also the spin polarized molecules can be treated within this framework, by using the spin generalized version of the N ͑ ͒ϭ ͩ ͑ ͒ͪ ͑ ͒ AGP, in which the unpaired orbitals are expanded as well as J2 r1 ,...,rN exp ͚ u rij , 7 the paired ones over the same atomic basis employed in the iϽ j 12 geminal. As already mentioned in the Introduction of this ϭ͉ where u(rij) depends only on the relative distance rij ri paper, the size consistency is an appealing feature of the Ϫr ͉ between two electrons and allows to fulfill the cusp AGP term. Strictly speaking, the AGP wave function is cer- j conditions for opposite spin electrons as long as u(rij) tainly size consistent when both the compound and the sepa- → rij/2 for small electron-electron distance. The pair corre- rated fragments have the minimum possible total spin, be- lation function u can be parametrized successfully by few cause the geminal expansion contains both bonding and variational parameters. We have adopted two main functional antibonding contributions, which can mutually cancel the forms. The first is similar to the one given by Ceperley,13 ionic term arising in the asymptotically separate regime. Moreover the size consistency of the AGP, as well as the one F u͑r͒ϭ ͑1ϪeϪr/F͒, ͑8͒ of the Hartree-Fock state, holds in all cases in which the spin 2 of the compound is the sum of the spin of the fragments. However, similarly to other approaches,4 for spin polarized with F being a free variational parameter. This form for u is systems the size consistency does not generally hold, and, in particularly convenient whenever atoms are very far apart at such cases, it may be important go beyond a single AGP distances much larger than F, as it allows to obtain good size wave function. Nevertheless we have experienced that a consistent energies, approximately equal to the sum of the single reference AGP state is able to describe accurately the atomic contributions, without changing the other parts of the electronic structure of the compound around the Born- wave function with an expensive optimization. Within the Oppenheimer minimum even in the mentioned polarized functional form ͑8͒, it is assumed that the long range part of cases, such as in the oxygen dimer. the Jastrow, decaying as a power of the distance between The last part of this section is devoted to the nuclear atoms, is included in the three-body Jastrow term described cusp condition implementation. A straightforward calculation in the following section. The second form of the pair func- shows that the AGP wave function fulfills the cusp condition u, particularly convenient at the chemical bond distance, tions around the nucleus a if the following linear system is where we performed most of the calculations, is the one used satisfied: by Fahy, Wang, and Louie:7

r For the s-wave orbitals we have found energetically con- u͑r͒ϭ , ͑9͒ Ϫ 2͑1ϩbr͒ venient to add a finite constant cl /(N 1). As shown in the Appendix B, a nonzero value of the constant cl for such with a different variational parameter b. ␺ orbitals a,l is equivalent to include in the wave function a In both functional forms the cusp condition for antipar- size consistent one body term. As pointed out in Ref. 15, it is allel spin electrons is satisfied, whereas the one for parallel easier to optimize a one body term implicitly present in the spins is neglected in order to avoid the spin contamination. three-body Jastrow factor, rather than including more orbitals This allows to remove the singularities of the local energy in the determinantal basis set. due to the collision of two opposite spin electrons, yielding a The chosen form for the three-body Jastrow ͑10͒ is simi- smaller variance and a more efficient VMC calculation. lar to one used by Prendergast, Bevan, and Fahy16 and has Moreover, due to the Jastrow correlation, an exact property very appealing features: it easily allows including the sym- of the ground state wave function is recovered without using a,b metries of the system by imposing them on the matrix gl,m many Slater determinants, thus considerably simplifying the exactly as it is possible for the pairing part ͓e.g., by replacing variational parametrization of a correlated wave function. ␭a,b a,b ͑ ͔͒ m,n with gm,n in Eq. 4 . It is size consistent, namely, the atomic limit can be smoothly recovered with the same trial a,b Þ C. Three-body Jastrow term function when the matrix terms gl,m for a b approach zero a,b In order to describe well the correlation between elec- in this limit. Notice that a small nonzero value of gl,m for Þ trons the simple Jastrow factor is not sufficient. Indeed it a b acting on p-wave orbitals can correctly describe a weak takes into account only the electron-electron separation and interaction between electrons such as the the van der Waals forces. not the individual electronic position ri and rj . It is expected that close to nuclei the electron correlation is not accurately described by a translationally invariant Jastrow, as shown by III. OPTIMIZATION METHOD different authors, see for instance, Ref. 14. For this reason We have used the SR method already described in Ref. ͑ we introduce a factor, often called three-body electron- 17, which allows to minimize the energy expectation value ͒ electron nucleus Jastrow, which explicitly depends on both of a variational wave function containing many variational the electronic positions ri and rj . The three-body Jastrow is parameters in an arbitrary functional form. The basic ingre- chosen to satisfy the following requirements. dient for the stochastic minimization of the wave function ⌿ ͑A͒ The cusp conditions set up by the two-body Jastrow ͕␣0͖ determined by p variational parameters k kϭ1,...,p , is the term and by the AGP are preserved. solution of the linear system ͑B͒ It does not distinguish the electronic spins otherwise p causing spin contamination. ⌬␣ ϭ ⌿͉ ͑⌳ Ϫ ͉͒⌿ ͑ ͒ ͑ ͒ ͚ s j,k k ͗ Ok I H ͘, 12 C Whenever the atomic distances are large it factorizes kϭ0 into a product of independent contributions located near each atom, an important requirement to satisfy the size consis- where the operators Ok are defined on each N electron con- ϭ͕ ͖ tency of the variational wave function. figuration x r1 ,...,rN as the logarithmic derivatives with ␣ Analogously to the pairing trial function in Eq. ͑2͒ we respect to the parameters k : ץ define a three-body factor as k͑ ͒ϭ ⌿͑ ͒ Ͼ ͑ ͒ ln x for k 0, 13 ␣ץ O x k J ͑r ,...,r ͒ϭexpͩ ͚ ⌽ ͑r ,r ͒ͪ , 3 1 N J i j Ͻ ϭ i j while for k 0 Ok is the identity operator equal to one on all ͑ ͒ ϩ ϫ ϩ 10 the configurations. The (p 1) (p 1) matrix sk, j is easily ⌽ ͑ ͒ϭ a,b␺ ͑ ͒␺ ͑ ͒ J ri ,rj ͚ gl,m a,l ri b,m rj , expressed in terms of these operators: l,m,a,b ͗⌿͉O O ͉⌿͘ ϭ j k ͑ ͒ where indices l and m indicate different orbitals located s j,k ͗⌿͉⌿͘ , 14 around the atoms a and b, respectively. Each Jastrow orbital ␺ a,l(r) is centered on the corresponding atomic position Ra . and is calculated at each iteration through a standard varia- We have used Gaussian and exponential orbitals multiplied tional Monte Carlo sampling; the single iteration constitutes by appropriate polynomials of the electronic coordinates, re- a small simulation that will be referred in the following as lated to different spherical harmonics with given angular mo- ‘‘bin.’’ After each bin the wave function parameters are itera- ␣ →␣ ϩ⌬␣ ⌬␣ mentum, as in the usual Slater basis. Analogously to the tively updated ( k k k / 0), and the method is ⌽ ⌳ geminal function AGP , whenever the one particle basis set convergent to an energy minimum for large enough .Of ␺ ͑ ͒ ⌿ ͕ a,i͖ is complete the expansion 10 is also complete for the course for particularly simple functional form of (x), con- ⌽ generic two particle function J(r,rЈ). In the latter case, taining, e.g., only linear CI coefficients, much more efficients however, the one particle orbitals have to behave smoothly optimization schemes do exist.18 close to the corresponding nuclei, namely, as SR is similar to a standard steepest descent ͑SD͒ calculation, where the expectation value of the energy E(␣ ) ␺ ͑r͒Ϫ␺ ͑R ͒Ӎ͉rϪR ͉2, ͑11͒ k a,i a,i a a ϭ ͗⌿͉H͉⌿͘/͗⌿͉⌿͘ is optimized by iteratively changing the ␣ or with larger power, in order to preserve the nuclear cusp parameters i according to the corresponding derivatives of conditions ͑5͒. the energy ͑generalized forces͒

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⌿͘ scribed distance. The fundamental difference between the SR͉͒ ץ⌿͉ ϩ ϩ͑͗ E OkH HOk ␣ Hץ ϭϪ ϭϪ k ⌿͉⌿͘ minimization and the standard steepest descent is just related͗ ␣ץ f k k to the definition of this distance. For the SD it is the usual 2 ͗⌿͉O ͉⌿͗͘⌿͉H͉⌿͘ one defined by the Cartesian metric ⌬␣ϭ͚ ͉␣ЈϪ␣ ͉ , in- ϩ k ͑ ͒ k k k 2 ͗⌿͉⌿͘2 , 15 stead the SR works correctly in the physical Hilbert space ⌿ ⌬ ϭ͚ ␣Ј metric of the wave function , yielding ␣ i, j¯si, j( i namely Ϫ␣ ␣ЈϪ␣ i)( j j), namely, the square distance between the two ␣ →␣ ϩ⌬tf . ͑16͒ normalized wave functions corresponding to the two differ- k k k ␣ ␣ ent sets of variational parameters ͕ Ј͖ and ͕ k͖. Therefore, ⌬t is a suitable small time step, which can be taken fixed or from the knowledge of the generalized forces f k , the most determined at each iteration by minimizing the energy expec- convenient change of the variational parameters minimizes tation value. Indeed the variation of the total energy ⌬E at the functional ⌬Eϩ⌳¯ ⌬␣ , where ⌬E is the linear change in each step is easily shown to be negative for small enough ⌬t the energy ⌬EϭϪ͚ f (␣ЈϪ␣ ) and ⌳¯ is a Lagrange mul- because, in this limit i i i i tiplier that allows a stable minimization with small change ⌬␣ of the wave function ⌿. The final iteration ͑17͒ is then ⌬ ϭϪ⌬ 2ϩ ͑⌬ 2͒ E t͚ f i O t . i easily obtained. The advantage of SR compared with SD is obvious be- Thus the method certainly converges at the minimum when cause sometimes a small change of the variational param- all the forces vanish. Notice that in the definition of the gen- eters correspond to a large change of the wave function, and ͑ ͒ eralized forces 15 we have generally assumed that the the SR takes into account this effect through the Eq. ͑17͒.In variational parameters may appear also in the Hamiltonian. particular, the method is useful when a nonorthogonal basis This is particularly important for the structural optimization set is used as we have done in this work. Indeed by using the since the atomic positions that minimize the energy enter reduced matrix ¯s it is also possible to remove from the cal- both in the wave function and in the potential. culation those parameters that imply some redundancy in the In the following we will show that similar considerations variational space. As shown in Appendix A, a more efficient hold for the SR method, which can be therefore extended to change in the wave function can be obtained by updating the optimization of the geometry. Indeed, by eliminating the only the variational parameters that remain independent ϭ ͑ ͒ equation with index k 0 from the linear system 12 , the SR within a prescribed tolerance, and therefore, by removing the iteration can be written in a form similar to the steepest de- parameters that linearly depend on the others. A more stable scent: minimization is obtained without spoiling the accuracy of the calculation. A weak tolerance criterium ⑀Ӎ10Ϫ3, provides a ␣ →␣ ϩ⌬ Ϫ1 ͑ ͒ i i t͚ ¯si,k f k , 17 very stable algorithm even when the dimension of the varia- k tional space is large. For a small atomic basis set, by an where the reduced pϫp matrix ¯s is appropriate choice of the Jastrow and Slater orbitals, the re- ϭ Ϫ ͑ ͒ duced matrix ¯s is always very well conditioned even for the ¯s j,k s j,k s j,0s0,k 18 largest system studied, and the above stabilization technique and the ⌬t value is given by is not required. Instead the described method is particularly important for the extension of QMC to complex systems 1 ⌬tϭ . ͑19͒ with large number of atoms and/or higher level of accuracy, ͗⌿͉H͉⌿͘ ͩ ⌳Ϫ Ϫ͚ ⌬␣ ͪ because in this case it is very difficult to select—e.g., by trial 2 Ͼ s ͗⌿͉⌿͘ k 0 k k,0 and error—the relevant variational parameters, which allow a well conditioned matrix ¯s for a stable inversion in Eq. ͑17͒. From the latter equation the value of ⌬t changes during the simulation and remains small for large enough energy shift A. Setting the SR parameters ⌳ . However, using the analogy with the steepest descent, In this work, instead of setting the constant ⌳, we have convergence to the energy minimum is reached also when equivalently chosen to determine ⌬t by verifying the stabil- ⌬ the value of t is sufficiently small and is kept constant for ity and the convergence of the algorithm at fixed ⌬t value, each iteration. Indeed the energy variation for a small change which can be easily understood as an inverse energy scale. of the parameters is ⌬ Ͼ⌳ ⌳ The simulation is stable whenever 1/ t cut , where cut is an energy cutoff that is strongly dependent on the chosen ⌬ ϭϪ⌬ Ϫ1 E t͚ ¯si, j f i f j . wave function and is generally weakly dependent on the bin i, j length. Whenever the wave function is too much detailed, It is easily verified that the above term is always negative namely, has a lot of variational freedom, especially for the because the reduced matrix ¯s, as well as ¯sϪ1, is positive high energy components of the core electrons, the value of ⌳ definite, s being an overlap matrix with all positive eigenval- cut becomes exceedingly large and too many iterations are ues. required for obtaining a converged variational wave function. For a stable iterative method, such as the SR or the SD In fact a rough estimate of the corresponding number of it- one, a basic ingredient is that at each iteration the new pa- erations P is given by P⌬tӷ1/G, where G is the typical rameters ␣Ј are close to the previous ␣ according to a pre- energy gap of the system, of the order of few electron volts

FIG. 1. Example of the convergence of the SR method for the variational parameters of the Be atom, as a function of the number of stochastic iterations. In the upper͑lower͒ panel the Ja- strow ͑geminal͒ parameters are shown. For each iteration, a variational Monte Carlo calculation is employed with a bin containing 15 000 samples of the energy, yielding at the equilibrium a standard deviation of Ӎ0.0018H.For the ﬁrst 200 iteration ⌬t ϭ0.00125HϪ1, for the further 200 iterations ⌬tϭ0.0025HϪ1, whereas for the remaining ones ⌬tϭ0.005HϪ1.

in small atoms and molecules. Within the SR method it is the minimization. As shown in Fig. 2 the Euler conditions therefore extremely important to work with a bin length are fulfilled within statistical accuracy even when the bin rather small, so that many iterations can be performed with- used for the minimization is much smaller than the overall out much effort. simulation. On the other hand, if the bin used is too small, as In a Monte Carlo optimization framework the forces f k we have already pointed out, the averaging of the parameters ␩ are always determined with some statistical noise k , and by is affected by a sizable bias. iterating the procedure several times with a fixed bin length Whenever it is possible to use a relatively small bin in the variational parameters will fluctuate around their mean the minimization, the apparently large number of iterations values. These statistical fluctuations are similar to the ther- required for equilibration does not really matter, because a mal noise of a standard Langevin equation comparable amount of time has to be spent in the averaging of the variational parameters, as shown in Fig. 1. The com- ϭ f ϩ␩ , ͑20͒ ␣ ץ t k k k parison shown in Ref. 19 about the number of the iterations where required, though is clearly relevant for a deterministic ␩ ͒␩ ͒ ϭ ␦ Ϫ ͒␦ ͑ ͒ method, is certainly incomplete for a statistical method, be- ͗ k͑t kЈ͑tЈ ͘ 2Tnoise ͑t tЈ k,kЈ . 21 cause in the latter case an iteration can be performed in prin- ␣ The variational parameters k , averaged over the Langevin ciple in a very short time, namely, with a rather small bin. Ϫ simulation time ͑as for instance, in Fig. 1 for tϾ2H 1), will It is indeed possible that for high enough accuracy the be close to the true energy minimum, but the corresponding number of iterations needed for the equilibration becomes E will be affected by a bias that scales to ␣ץforces f ϭϪ k k negligible from the computational point of view. In fact in zero with the thermal noise Tnoise , due to the presence of order to reduce, e.g., by a factor 10, the accuracy in the nonquadratic terms in the energy landscape. variational parameters, a bin ten times larger is required for Within a QMC scheme, one needs to estimate Tnoise ,by decreasing the thermal noise Tnoise by the same factor, ϰ increasing the bin length as clearly Tnoise 1/ bin length, because the statistical fluctuations of the forces, obviously decreasing by increasing the bin length, are related to the thermal noise by Eq. ͑21͒. Thus there is an optimal value for the bin length, which guarantees a fast convergence and avoid the forces to be biased within the statistical accuracy of the sampling. An example is shown in Fig. 1 for the optimization of the Be atom, using a DZ basis both for the geminal and the three-body Jastrow part. The convergence is reached in about 1000 iteration with ⌬tϭ0.005HϪ1. However, in this case it is possible to use a small bin length, yielding a statistical accuracy in the energy much poorer than the final accuracy of about 0.05 mH. This is obtained by averaging the variational parameters in the last 1000 iterations, when they fluc- FIG. 2. Calculation of the derivative of the energy with respect to the tuate around a mean value, allowing a very accurate deter- second Z in the 2p orbital of the geminal function for the Be atom. The mination of the energy minimum which satisfies the Euler calculation of the force was obtained, at fixed variational parameters, by 7 conditions, namely, with f ϭ0 for all parameters. Those averaging over 10 samples, allowing, e.g., a statistical accuracy in the total k energy of 0.07 mH. The variational parameters have been obtained by a SR conditions have been tested by an independent Monte Carlo minimization with fixed bin length shown in the x label. The parameter simulation about 600 times longer than the bin used during considered has the largest deviation from the Euler conditions.

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whereas a statistical average 100 times longer is indeed nec- on the nuclear positions. Now let us apply the space-warp essary to reduce the statistical errors of the variational pa- transformation to the integral involved in the calculation; the rameters by the same ratio. This means that the fraction of expectation value reads time spent for equilibration becomes ten times smaller com- 2 ⌬R ͐drJ⌬ ͑r͒⌿⌬ ͓¯r͑r͔͒E ͓¯r͑r͔͒ pared with the less accurate simulation. ͑ ϩ⌬ ͒ϭ R R L ͑ ͒ E R R ͐ ͑ ͒⌿2 ͓ ͑ ͔͒ , 26 drJ⌬R r ⌬R ¯r r where J is the Jacobian of the transformation and here and B. Structural optimization henceforth we avoid for simplicity to use the atomic subin- dex a. The importance of the space warp in reducing the In the last few years remarkable progresses have been variance of the force is easily understood for the case of an made to develop QMC techniques which are able in principle isolated atom a. Here the force acting on the atom is obvi- to perform structural optimization of molecules and complex ously zero, but only after the space warp transformation with systems.20–22 Within the Born-Oppenheimer approximation ␻ ϭ1 the integrand of expression ͑26͒ will be independent the nuclear positions R can be considered as further varia- a i of ⌬R, providing an estimator of the force with zero vari- tional parameters included in the set ͕␣ ͖ used for the SR i ance. minimization ͑17͒ of the energy expectation value. For clar- Starting from Eq. ͑26͒, it is straightforward to explicitly ity, in order to distinguish the conventional variational pa- derive a finite difference differential expression for the force rameters from the ionic positions, in this section we indicate ͕ ͖ estimator, which is related to the gradient of the previous with ci the former ones, and with Ri the latter ones. It is ⌬ ␯ϭ␣ quantity with respect to R, in the limit of the displacement understood that Ri k , where a particular index k of the ␣ tending to zero: whole set of parameters ͕ i͖ corresponds to a given spatial component (␯ϭ1,2,3) of the ith ion. Analogously the forces ⌬ ͑ ͒ϭϪ ͑15͒ acting on the ionic positions will be indicated by capital F R ͳ lim EL ʹ ͉⌬ ͉→ ⌬R letters with the same index notations. R 0 The purpose of the present section is to compute the ⌬ ϩ ͗ ͘ ͑ 1/2⌿͒ forces F acting on each of the M nuclear positions 2ͩ H ͳ lim ⌬ ln J ʹ ͉⌬R͉→0 R ͕R1 ,...,RM͖, being M the total number of nuclei in the system: ⌬ Ϫ ͑ 1/2⌿͒ ͑ ͒ ͳ H lim ⌬ ln J ʹ ͪ , 27 F͑R ͒ϭϪٌ E͕͑c ͖,R ͒, ͑22͒ ͉⌬R͉→0 R a Ra i a with a reasonable statistical accuracy, so that the iteration where the brackets indicate a Monte Carlo like average over ⌬ ⌬ ͑17͒ can be effective for the structural optimization. In this the square modulus of the trial wave function, / R is the ͑ ͒ work we have used a finite difference operator ⌬/⌬R for finite difference derivative as defined in Eq. 23 , and EL a ϭ͗⌿͉ ͉ ͘ ͗⌿͉ ͘ the evaluation of the force acting on a given nuclear position H x / x is the local energy on a configuration x a where all electron positions and spins are given. In analogy with the general expression ͑15͒ of the forces, we can iden- ⌬ E͑R ϩ⌬R ͒ϪE͑R Ϫ⌬R ͒ tify the operators O corresponding to the space-warp ͑ ͒ϭϪ ϭϪ a a a a k F Ra ⌬ E ⌬ Ra 2 R change of the variational wave function: ␯ 2 ⌬ ϩO͑⌬R ͒, ͑23͒ 1/2 O ϭ ln͑J ⌿⌬ ͒. ͑28͒ k ⌬R ⌬R R ⌬ ⌬ where Ra is a three-dimensional vector. Its length R is chosen to be 0.01 a.u., a value that is small enough for neg- The above operators ͑28͒ are used also in the definition of ligible finite difference errors. In order to evaluate the energy the reduced matrix ¯s for those elements depending on the differences in Eq. ͑23͒ we have used the space-warp coordi- variation with respect to a nuclear coordinate. In this way it nate transformation23,24 briefly summarized in the following is possible to optimize both the wave function and the ionic paragraphs. According to this transformation also the elec- positions at the same time, in close analogy with the tronic coordinates r will be translated in order to mimic the Car-Parrinello25 method applied to the minimization prob- right displacement of the charge around the nucleus a lem. Also Tanaka26 tried to perform Car-Parrinello like simulations via QMC, within the less efficient steepest descent ϭ ϩ⌬ ␻ ͑ ͒ ͑ ͒ ¯ri ri Ra a ri , 24 framework. where An important source of systematic errors is the depen- dence of the variational parameters ci on the ionic configu- ͉ Ϫ ͉͒ F͑ r Ra ration R, because for the final equilibrium geometry all the ␻ ͑r͒ϭ . ͑25͒ a ͚M ͉ Ϫ ͉͒ forces f i corresponding to ci have to be zero, in order to ϭ F͑ r Rb b 1 guarantee that the true minimum of the potential energy sur- F(r) is a function which must decay rapidly; here we used face is reached.27 As shown clearly in the preceding section, F(r)ϭ 1/r4 as suggested in Ref. 24. within a QMC approach it is possible to control this condi- The expectation value of the energy depends on ⌬R, tion by increasing systematically the bin length, when the because both the Hamiltonian and the wave function depend thermal bias Tnoise vanishes. In Fig. 3 we report the equilib-

⌬t, by applying the minimization scheme only for those se- ͉ ⌬ ͉Ͼ␴ ͉ ⌬ ͉ lected parameters such that f i / f i cut , until f i / f i Ͻ␴ cut . After this initialization it is then convenient to pro- ceed with the global minimization with all parameters changed at each iteration, in order to explore the variational space in a much more effective way.

D. Different energy scales The SR method performs generally very well, whenever there is only one energy scale in the variational wave function. However if there are several energy scales in the prob-

FIG. 3. Plot of the equilibrium distance of the Li2 molecule as a function of lem, some of the variational parameters, e.g. the ones defin- the inverse bin length. The total energy and the binding energy are reported ing the low energy valence orbitals, converge very slowly ͑ ͒ in Tables II and III, respectively. The triangles full dots refer to a simula- with respect to the others, and the number of iterations re- tion performed using 1000 ͑3000͒ iterations with ⌬tϭ0.015HϪ1(⌬t ϭ0.005HϪ1) and averaging over the last 750 ͑2250͒ iterations. For all quired for the equilibration becomes exceedingly large, con- simulations the initial wavefunction is optimized at Li-Li distance 6 a.u. sidering also that the time step ⌬t necessary for a stable convergence depends on the high energy orbitals, whose dy- namics cannot be accelerated beyond a certain threshold. It is easy to understand that SR technique not necessarily be- rium distance of the Li molecule as a function of the inverse comes inefficient for extensive systems with large number of bin length, so that an accurate evaluation of such an impor- atoms. Indeed suppose that we have N atoms very far apart tant quantity is possible even when the number of variational so that we can neglect the interaction between electrons be- parameters is rather large, by extrapolating the value to an longing to different atoms, than it is easy to see that the infinite bin length. However, as it is seen in the picture, stochastic matrix Eq. ͑14͒ factorizes in N smaller matrices though the inclusion of the 3s orbital in the atomic AGP basis and the ⌬t necessary for the convergence is equal to the substantially improves the equilibrium distance and the total single atom case, simply because the variational parameters energy by Ӎ1 mH, this larger basis makes our simulation of each single atom can evolve independently form each less efficient, as the time step ⌬t has to be reduced by a other. This is due to the size consistency of our trial function factor 3. that can be factorized as a product of N single atom trial We have not attempted to extend the geometry optimi- functions in that limit. Anyway for system with a too large zation to the more accurate DMC, since there are technical energy spread a way to overcome this difficulty was pre- 28 difficulties, and it is computationally much more demand- sented recently in Ref. 19. Unfortunately this method is lim- ing. ited to the optimization of the variational parameters in a super-CI basis, to which a Jastrow term is applied, which however cannot be optimized together with the CI coefficients. C. Stochastic versus deterministic minimization In the present work, limited to a rather small atomic In principle, within a stochastic approach, the exact basis, the SR technique is efficient, and general enough to minimum is never reached as the forces f i are known only perform the simultaneous optimization of the Jastrow and the ⌬ statistically with some error bar f i . We have found that the determinantal part of the wave function, a very important method becomes efficient when all the forces are nonzero feature that allows to capture the most nontrivial correlations only within few tenths of standard deviations. Then for small contained in our variational ansatz. Moreover, SR has been enough constant ⌬t, and large enough bin compatible with extended to deal with the structural optimization of a chemi- the computer resources, the stochastic minimization, ob- cal system, which is another appealing characteristic of this tained with a statistical evaluation of ¯s continues in a stable method. The results presented in the following section show manner, and all the variational parameters fluctuate after sev- that in some nontrivial cases the chemical accuracy can be eral iterations around a mean value. After averaging these reached also within this framework. variational parameters, the corresponding mean values repre- However we feel that an improvement along the line sent very good estimates satisfying the minimum energy con- described in Ref. 19 will be useful for realistic electronic dition. This can be verified by performing an independent simulations of complex systems with many atoms, or when a Monte Carlo simulation much longer than the bin used for a very high precision is required at the variational level and single iteration of the stochastic minimization, and then by consequently a wide spread of energy scales has to be in- explicitly checking that all the forces f i are zero within the cluded in the atomic basis. We believe that the difficulty to statistical accuracy. An example is given in Fig. 2 and dis- work with a large basis set will be possibly alleviated by cussed in the preceding section. using pseudopotentials that allows to avoid the high energy On the other hand, whenever few variational parameters components of the core electrons. However more work is ͉ ⌬ ͉Ͼ␴ ␴ are clearly out of minimum, with f i / f i cut , with cut necessary to clarify the efficiency of the SR minimization Ӎ10, we have found a faster convergence with a much larger scheme described here.

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TABLE I. Total energies in variational (EVMC) and diffusion (EDMC) Monte Carlo calculations; the percentages ͓ VMC ͔ ͓ DMC ͔ of correlation energy recovered in VMC Ec (%) and DMC Ec (%) have been evaluated using the ‘‘exact’’ (E0) and Hartree-Fock (EHF) energies from the references reported in the table. Here exact means the ground state energy of the nonrelativistic inﬁnite nuclear mass Hamiltonian. The energies are in hartree.

VMC DMC E0 EHF EVMC Ec (%) EDMC Ec (%) Li Ϫ7.478 06a Ϫ7.432 727a Ϫ7.477 21(11) 98.12͑24͒ Ϫ7.477 91(12) 99.67͑27͒ b Ϫ b Ϫ ͑ ͒ Ϫ ͑ ͒ Li2 Ϫ14.9954 14.871 52 14.990 02(12) 95.7 1 14.994 72(17) 99.45 14 Be Ϫ14.667 36a Ϫ14.573 023a Ϫ14.663 28(19) 95.67͑20͒ Ϫ14.667 05(12) 99.67͑13͒ b Ϫ b Ϫ ͑ ͒ Ϫ ͑ ͒ Be2 Ϫ29.338 54(5) 29.132 42 29.3179(5) 89.99 24 29.333 41(25) 97.51 12 O Ϫ75.0673a Ϫ74.809 398a Ϫ75.0237(5) 83.09͑19͒ Ϫ75.0522(3) 94.14͑11͒ b Ϫ b Ϫ ͑ ͒ Ϫ ͑ ͒ H2O Ϫ76.438(3) 76.068(1) 76.3803(4) 84.40 10 76.4175(4) 94.46 10 b Ϫ b Ϫ ͑ ͒ Ϫ ͑ ͒ O2 Ϫ150.3268 149.6659 150.1992(5) 80.69 7 150.272(2) 91.7 3 C Ϫ37.8450a Ϫ37.688 619a Ϫ37.813 03(17) 79.55͑11͒ Ϫ37.8350(6) 93.6͑4͒ b Ϫ b Ϫ ͑ ͒ Ϫ ͑ ͒ C2 Ϫ75.923(5) 75.406 20 75.8273(4) 81.48 8 75.8826(7) 92.18 14 d Ϫ d Ϫ ͑ ͒ Ϫ ͑ ͒ CH4 Ϫ40.515 40.219 40.4627(3) 82.33 10 40.5041(8) 96.3 3 e Ϫ f Ϫ ͑ ͒ Ϫ ͑ ͒ C6H6 Ϫ232.247(4) 230.82(2) 231.8084(15) 69.25 10 232.156(3) 93.60 21

aExact and HF energies from Ref. 50. dReference 33. bReference 29. eEstimated exact energy from Ref. 43. cReference 51. fReference 52.

IV. APPLICATION OF THE JAGP TO MOLECULES A. Small diatomic molecules, methane, and water Except from Be and C , all the molecules presented In this work we study total, correlation, and atomization 2 2 here belong to the standard G1 reference set; all their prop- energies, accompanied with the determination of the ground erties are well known and well reproduced by standard quan- state optimal structure for a restricted ensemble of mol- tum chemistry methods, therefore they constitute a good case ecules. For each of them we performed a full all-electron SR for testing new approaches and new wave functions. geometry optimization, starting from the experimental mo- The Li dimer is one of the easiest molecules to be stud- lecular structure. After the energy minimization, we carried ied after H2 , which is exact for any diffusion Monte Carlo out all-electron VMC and DMC simulations at the optimal calculation with a trial wave function that preserves the geometry within the so called ‘‘fixed node approximation.’’ nodeless structure. Li2 is less trivial due to the presence of The basis used here is a double zeta Slater set of atomic core electrons that are only partially involved in the chemical ͑ ͒ orbitals STO-DZ for the AGP part, while for parametrizing bond and to the 2s-2p near degeneracy for the valence elec- the three-body Jastrow geminal we used a double zeta trons. Therefore many authors have done benchmark calcu- ͑ ͒ Gaussian atomic set GTO-DZ . In this way both the anti- lation on this molecule to check the accuracy of the method symmetric and the bosonic part are well described, preserv- or to determine the variance of the internuclear force calcu- ing the right exponential behavior for the former and the lated within a QMC framework. In this work we start from strong localization properties for the latter. Sometimes, in Li2 to move toward a structural analysis of more complex order to improve the quality of the variational wave function compounds, thus showing that our QMC approach is able to we employed a mixed Gaussian and Slater basis set in the handle relevant chemical problems. In the case of Li2 ,a3s Jastrow part, which allows to avoid a too strong dependency 1p STO-DZ AGP basis and a 1s 1p GTO-DZ Jastrow basis in the variational parameters in a simple way. However, both turns out to be enough for the chemical accuracy ͑see Ap- in the AGP and in the Jastrow sector we never used a large pendix C for a detailed description of the trial wave func- basis set, in order to keep the wave function as simple as tion͒. More than 99% of the correlation energy is recovered possible. The accuracy of our wave function can be obvi- by a DMC simulation ͑Table I͒, and the atomization energy ously improved by an extension of the one particle basis set, is exact within few thousand of eV (0.02 kcal molϪ1) ͑Table but, as discussed in the preceding section, this is rather dif- II͒. Similar accuracy have been previously reached within a ficult for a stochastic minimization of the energy. Neverthe- DMC approach,29 only by using a multireference CI like less, for most of the molecules studied with a simple JAGP wave function, that before our work, was the usual way to wave function, a DMC calculation is able to reach the chemi- improve the electronic nodal structure. As stressed before, cal accuracy in the binding energies and the SR optimization the JAGP wave function includes many resonating configu- yields very precise geometries already at the VMC level. rations through the geminal expansion, beyond the 1s 2s HF In the first part of this section some results will be pre- ground state. The bond length has been calculated at the sented for a small set of widely studied molecules and be- variational level through the fully optimized JAGP wave longing to the G1 database. In the second section we will function: the resulting equilibrium geometry turns out to be ϩ ͑ ͒ treat the benzene and its radical cation C6H6 , by taking into highly accurate Table III , with a discrepancy of only account its distortion due to the Jahn-Teller effect, which is 0.001a0 from the exact result. For this molecule it is worth well reproduced by our SR geometry optimization. comparing our work with the one by Assaraf and Caffarel.30

⌬ ⌬ TABLE II. Binding energies in eV obtained by variational ( VMC) and diffusion ( DMC) Monte Carlo calcu- ⌬ lations; 0 is the exact result for the nonrelativistic inﬁnite nuclear mass Hamiltonian. Also the percentages ⌬ ⌬ ͓ VMC(%) and DMC(%)] of the total binding energies are reported.

⌬ ⌬ ⌬ ⌬ ⌬ 0 VMC VMC(%) DMC DMC(%) Ϫ Ϫ ͑ ͒ Ϫ ͑ ͒ Li2 1.069 0.967(3) 90.4 3 1.058(5) 99.0 5 Ϫ ͑ ͒ Ϫ ͑ ͒ O2 Ϫ5.230 4.13(4) 78.9 8 4.56(5) 87.1 9 Ϫ ͑ ͒ Ϫ ͑ ͒ H2O Ϫ10.087 9.704(24) 96.2 1.0 9.940(19) 98.5 9 Ϫ ͑ ͒ Ϫ ͑ ͒ C2 Ϫ6.340 5.476(11) 86.37 17 5.79(2) 91.3 3 Ϫ ͑ ͒ Ϫ ͑ ͒ CH4 Ϫ18.232 17.678(9) 96.96 5 18.21(4) 99.86 22 Ϫ ͑ ͒ Ϫ ͑ ͒ C6H6 Ϫ59.25 52.53(4) 88.67 7 58.41(8) 98.60 13

Their zero-variance zero-bias principle has been proved to be tained by Filippi and Umrigar29 with a multireference wave effective in reducing the fluctuations related to the internu- function by using the same Slater basis for the antisymmetric clear force; however they found that only the inclusion of the part and a different Jastrow factor. From the Table II of the space warp transformation drastically lowers the force statis- atomization energies, it is apparent that DMC considerably tical error, which magnitude becomes equal or even lower improves the binding energy with respect to the VMC val- than the energy statistical error, thus allowing a feasible mo- ues, although for these two molecules it is quite far from the Ӎ ͑ ͒ lecular geometry optimization. Actually, our way of comput- chemical accuracy ( 0.1 eV): for C2 the error is 0.55 2 eV, ing forces ͓see Eq. ͑27͔͒ provides slightly larger variances, ͑ ͒ for O2 it amounts to 0.67 5 eV. Indeed, it is well known that without explicitly invoking the zero-variance zero-bias prin- the electronic structure of the atoms is described better than ciple. the corresponding molecules if the basis set remains the The very good bond length, we obtained, is probably due same, and the nodal error is not compensated by the energy to two main ingredients of our calculations: we have carried difference between the separated atoms and the molecule. In out a stable energy optimization that is often more effective a benchmark DMC calculation with pseudopotentials,32 31 than the variance one, as shown by different authors, and Grossman found an error of 0.27 eV in the atomization en- we have used very accurate trial function as it is apparent ergy for O2 , by using a single determinant wave function; from the good variational energy. probably, pseudopotentials allow the error between the Indeed within our scheme we obtain good results with- pseudoatoms and the pseudomolecule to compensate better, out exploiting the computationally much more demanding thus yielding more accurate energy differences. As a final DMC, thus highlighting the importance of the SR minimiza- remark on the O and C molecules, our bond lengths are in tion described in Sec. III B. 2 2 between the LDA and GGA precision, and still worse than Let us now consider larger molecules. Both C and O 2 2 the best CCSD calculations, but our results may be consid- are poorly described by a single Slater determinant, since the erably improved by a larger atomic basis set, which we have presence of the nondynamic correlation is strong. Instead not attempted so far. with a single geminal JAGP wave function, including implic- Methane and water are very well described by the JAGP itly many Slater determinants,10 it is possible to obtain a wave function. Also for these molecules we recover more quite good description of their molecular properties. For C2 , we used a 2s 1p STO-DZ basis in the geminal, and a 2s 1p than 80% of correlation energy at the VMC level, while DMC yields more than 90%, with the same level of accuracy DZ Gaussian Slater mixed basis in the Jastrow, for O2 we 33–36 employed a 3s 1p STO-DZ in the geminal and the same reached in previous Monte Carlo studies. Here the bind- Jastrow basis as before. In both the cases, the variational ing energy is almost exact, since in this case the nodal energy ͑ energies recover more than 80% of the correlation energy, error arises essentially from only one atom carbon or oxy- ͒ the DMC ones yield more than 90%, as shown in Table I. gen and therefore it is exactly compensated when the atomi- These results are of the same level of accuracy as those ob- zation energy is calculated. Also the bond lengths are highly accurate, with an error lower then 0.005a0 . For Be2 we applied a 3s 1p STO-DZ basis set for the TABLE III. Bond lengths (R) in atomic units; the subscript 0 refers to the AGP part and a 2s 2p DZ Gaussian Slater mixed basis for the exact results. For the water molecule R is the distance between O and H and Jastrow factor. VMC calculations performed with this trial ␪ ͑ ͒ is the angle HOH in deg , for CH4 R is the distance between C and H and ␪ is the HCH angle. function at the experimental equilibrium geometry yield 90% of the total correlation energy, while DMC gives 97.5% of ␪ ␪ R0 R 0 correlation, i.e., a total energy of Ϫ29.333 41(25) H. Al- ͑ ͒ though this value is better than that obtained by Filippi and Li2 5.051 5.0516 2 ͑ ͒ 29 Ϫ ͒ O2 2.282 2.3425 18 Umrigar ( 29.3301(2) H with a smaller basis (3s atomic ͑ ͒ C2 2.348 2.366 2 orbitals not included͒, it is not enough to bind the molecule, ͑ ͒ ͑ ͒ H2O 1.809 1.8071 23 104.52 104.74 17 because the binding energy remains still positive ͓0.0069͑37͒ ͑ ͒ ͑ ͒ CH4 2.041 2.049 1 109.47 109.55 6 CC CC CH CH H͔. Instead, once the molecular geometry has been relaxed, R0 R R0 R ͑ ͒ ͑ ͒ the SR optimization finds a bond distance of 13.5(5)a at the C6H6 2.640 2.662 4 2.028 1.992 2 0 VMC level; therefore the employed basis allows the mol-

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⌬ ⌬ TABLE IV. Binding energies in eV obtained by variational ( VMC) and diffusion ( DMC) Monte Carlo calculations with different trial wave functions for benzene. In order to calculate the binding energies yielded by the ⌬ ⌬ two-body Jastrow we used the atomic energies reported in Ref. 10. The percentages ͓ VMC(%) and DMC(%)] of the total binding energies are also reported.

⌬ ⌬ ⌬ ⌬ VMC VMC(%) DMC DMC(%) ϩ Ϫ ͑ ͒ Kekule´ two-body 30.57(5) 51.60 8 ¯¯ ϩ Ϫ ͑ ͒ Resonating Kekule´ two-body 32.78(5) 55.33 8 ¯¯ Resonating Dewar Kekule´ϩtwo-body Ϫ34.75(5) 58.66͑8͒ Ϫ56.84(11) 95.95͑18͒ Kekule´ϩthree-body Ϫ49.20(4) 83.05͑7͒ Ϫ55.54(10) 93.75͑17͒ Resonating Kekule´ϩthree-body Ϫ51.33(4) 86.65͑7͒ Ϫ57.25(9) 96.64͑15͒ Resonating Dewar Kekule´ϩthree-body Ϫ52.53(4) 88.67͑7͒ Ϫ58.41(8) 98.60͑13͒ Full resonatingϩthree-body Ϫ52.65(4) 88.87͑7͒ Ϫ58.30(8) 98.40͑13͒

ecule to have a van der Waals like minimum, quite far from included in the three-body Jastrow part the same type of the experimental value. In order to have a reasonable de- terms present in the AGP one, namely, both ga,b and ␭a,b are scription of the bond length and the atomization energy, one nonzero for the same pairs of atoms. As expected, the terms needs to include at least a 3s2p basis in the antisymmetric connecting next nearest neighbor carbon sites are much less part, as pointed out in Ref. 37, and indeed an atomization important than the remaining ones because the VMC energy energy compatible with the experimental result ͓0.11͑1͒ eV͔ does not significantly improve ͑see the full resonating has been obtained within the extended geminal model38 by ϩthree-body wave function in Table IV͒. A more clear be- using a much larger basis set ͑9s,7p,4d,2f,1g͒.39 This sug- havior is found by carrying out DMC simulations: the inter- gests that a complete basis set calculation with JAGP may play between the resonance among different structures and describe also this molecule. However, as already mentioned the Gutzwiller-like correlation refines more and more the in Sec. III D, our SR method cannot cope with a very large nodal surface topology, thus lowering the DMC energy by basis in a feasible computational time. Therefore we believe significant amounts. Therefore it is crucial to insert into the that at present the accuracy needed to describe correctly Be2 variational wave function all these ingredients in order to is out of the possibilities of the approach. have an adequate description of the molecule. For instance, in Fig. 4 we report the density surface difference between the B. Benzene and its radical cation nonresonating three-body Jastrow wave function, which breaks the C rotational invariance, and the resonating We studied the 1A ground state of the benzene mol- 6 1g Kekule´ structure, which preserves the correct A symmetry: ecule by using a very simple one particle basis set: for the 1g AGP, a 2s1p DZ set centered on the carbon atoms and a 1s SZ on the hydrogen, instead for the three-body Jastrow, a 1s1p DZ-GTO set centered only on the carbon sites. C6H6 is a peculiar molecule, since its highly symmetric ground state, which belongs to the D6h point group, is a resonance among different many-body states, each of them characterized by three double bonds between carbon atoms. This resonance is responsible for the stability of the structure and therefore for its aromatic properties. We started from a non resonating two-body Jastrow wave function, which dimerizes the ring and breaks the full rotational symmetry, leading to the Kekule´ configuration. As we expected, the inclusion of the resonance between the two possible Kekule´ states lowers the VMC energy by more than 2 eV. The wave function is further improved by adding another type of resonance, that includes also the Dewar contributions connecting third nearest neighbor carbons. As reported in Table IV, the gain with respect to the simplest Kekule´ wave function amounts to 4.2 eV, but the main improvement arises from the further inclusion of the three-body Jastrow factor, which allows to recover the 89% of the total atomization energy at the VMC level. The main effect of the three-body term is to keep the total charge around the carbon sites to approximately six electrons, thus penalizing the double occupation of the p FIG. 4. Surface plot of the charge density projected onto the molecular z ͑ ͒ orbitals. The same important correlation ingredient is present plane. The difference between the nonresonating indicated as HF and resonating Kekule´ three-body Jastrow wave function densities is shown. Notice in the well known Gutzwiller wave function already used for the corresponding change from a dimerized structure to a C rotational 40 6 polyacetylene. Within this scheme we have systematically invariant density profile.

2 2 TABLE V. Bond lengths (r) for the two lowest B2g and B3g states of the benzene radical cation. The angles ␣ are expressed in degrees, the lengths in

a0 . The carbon sites are numerated from 1 to 6.

2 2 B2g B3g acute obtuse Computational method Ϫ a r(C1 C2) 2.616 2.694 B3LYP/cc-pVTZ 2.649 2.725 BLYP/6-31G*b 2.659͑1͒ 2.733͑4͒ SR-VMCc Ϫ a r(C2 C3) 2.735 2.579 B3LYP/cc-pVTZ 2.766 2.615 BLYP/6-31G*b 2.764͑2͒ 2.628͑4͒ SR-VMCc ␣ a (C6C1C2) 118.4 121.6 B3LYP/cc-pVTZ 118.5 121.5 BLYP/6-31G*b 118.95͑6͒ 121.29͑17͒ SR-VMCc

aReference 45. bReference 44. FIG. 5. Plot of the convergence toward the equilibrium geometry for the c 2 2 This work. B2g acute and the B3g obtuse benzene cation. Notice that both the simulations start from the ground state neutral benzene geometry and relax with a change both in the C-C bond lengths and in the angles. The symbols are the same of Table V. but also the DMC total energy is in perfect agreement with the BLYP/6-31G*, and much better than SVWN/6-31G* that does not contain semi empirical functionals, for which the comparison with our calculation is more appropriate, be- the change in the electronic structure is significant. The best ing fully ab initio. ´ result for the binding energy is obtained with the Kekule The difference of the VMC and DMC energies between Dewar resonating three-body wave function, which recovers the two distorted cations are the same within the error bars; 98.6% of the total atomization energy with an absolute error indeed, the determination of which structure is the real cation ͑ ͒ 41 of 0.84 8 eV. As Pauling first pointed out, benzene is a ground state is a challenging problem, since the experimental genuine RVB system, indeed it is well described by the results give a difference of only few meV in favor of the JAGP wave function. Moreover Pauling gave an estimate for obtuse state and also the most refined quantum chemistry the resonance energy of 1.605 eV from thermochemical ex- methods are not in agreement among themselves.44 A more periments in qualitative agreement with our results. A final affordable problem is the determination of the adiabatic ion- remark about the error in the total atomization energy: the ization potential ͑AIP͒, calculated for the 2B state, follow- ͑ ͒ 42,43 3g latest frozen core CCSD T calculations are able to reach ing the experimental hint. Recently, very precise CCSD͑T͒ a precision of 0.1 eV, but only after the complete basis set calculations have been performed in order to establish a extrapolation and the inclusion of the core valence effects to benchmark theoretical study for the ionization threshold of go beyond the pseudopotential approximation. Without the benzene;45 the results are reported in Table VII. After the latter corrections, the error is quite large even in the CCSD 42 correction of the zero point energy due to the different struc- approach, amounting to 0.65 eV. In our case, such an error ture of the cation with respect to the neutral molecule and arises from the fixed node approximation, whose nodal error taken from a B3LYP/cc-pVTZ calculation reported in Ref. is not compensated by the difference between the atomic and 45, the agreement among our DMC result, the benchmark the molecular energies, as already noticed in the previous calculation and the experimental value is impressive. Notice subsection. ϩ that in this case there should be a perfect cancellation of The radical cation C6H6 of the benzene molecule has 44,45 nodal errors in order to obtain such an accurate value; how- been the subject of intense theoretical studies, aimed to ever, we believe that it is not a fortuitous result, because in focus on the Jahn-Teller distorted ground state structure. In- 2 this case the underlying nodal structure does not change deed the ionized E1g state, which is degenerate, breaks the much by adding or removing a single electron. Therefore we symmetry and experiences a relaxation from the D6h point 2 2 expect that this property holds for all the affinity and ioniza- group to two different states, B2g and B3g , that belong to tion energy calculations with a particularly accurate varia- the lower D2h point group. In practice, the former is the tional wave function as the one we have proposed here. Nev- elongated acute deformation of the benzene hexagon, the lat- ertheless DMC is needed to reach the chemical accuracy, ter is its compressed obtuse distortion. We applied the SR 2 structural optimization, starting from the E1g state, and the 2 2 minimization correctly yielded a deformation toward the TABLE VI. Total energies for the B2g and B3g states of the benzene acute structure for the 2B state and the obtuse for the 2B radical cation after the geometry relaxation. A comparison with a 2g 3g ͑ ͒ one; the first part of the evolution of the distances and the BLYP/6-31G* and SVWN/6-31G* all-electron calculation Ref. 44 is reported. angles during those simulations is shown in Fig. 5. After this equilibration, average over 200 further iterations yields bond VMC DMC BLYP/6-31G* SVWN/6-31G* distances and angles with the same accuracy as the all- 2 Ϫ Ϫ Ϫ Ϫ B2g 231.4834(15) 231.816(3) 231.815495 230.547931 electron BLYP/6-31G* calculations reported in Ref. 44 ͑see 2 Ϫ Ϫ Ϫ Ϫ B3g 231.4826(14) 231.812(3) 231.815538 230.547751 Table V͒. As it appears from Table VI not only the structure

Downloaded 18 Nov 2011 to 147.122.49.74. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 7122 J. Chem. Phys., Vol. 121, No. 15, 15 October 2004 Casula, Attaccalite, and Sorella since the VMC result is slightly off from the experimental TABLE VII. Adiabatic ionization potential of the benzene molecule; our 2 one just by few tenths of eV. The AIP and the geometry estimate is done for the B3g relaxed geometries of the benzene radical ϩ cation, with an inclusion of the zero point motion correction between the determination for the C6H are encouraging to pursue this 2 1 6 B3g state and the A1g neutral molecule ground state, calculated in Ref. 45 approach, with the aim to describe even much more interest- at the B3LYP/6-31G* level. ing and challenging chemical systems. VMCa DMCa CCSD(T)/cc-pVϱZb experimentc

AIP 8.86͑6͒ 9.36͑8͒ 9.29͑4͒ ⌬ Ϫ Ϫ V. CONCLUSION ZPEad Ϫ0.074 0.074 0.074 best estimate 8.79͑6͒ 9.29͑8͒ 9.22͑4͒ 9.2437͑8͒ In this work, we have tested the JAGP wave function on simple molecular systems where accurate results are known. aThis work. b As shown in the preceding section a large amount of the Reference 45. cReference 53. correlation energy is already recovered at the variational level with a computationally very efficient and feasible method, extended in this work to the nuclear geometry optimization. Indeed, much larger systems should be tractable the molecule, by forbidding unphysical H2 dimers with more because, within the JAGP ansatz, it is sufficient to sample a than two electrons. Once the charge is locally conserved, the single determinant whose dimension scales only with the phase of the BCS-AGP wave function cannot have a definite number of electrons. The presence of the Jastrow factor im- value and phase coherence is correctly forbidden by the Ja- plies the evaluation of multidimensional integrals that, so far strow factor. In the present work, the interplay between the can be calculated efficiently only with the Monte Carlo Jastrow and the geminal part has been shown to be very method. Within this framework, it is difficult to reach the effective in all cases studied and particularly in the nontrivial complete basis set limit, both in the Jastrow and the AGP case of the benzene molecule, where we have shown system- terms, although some progress has been made recently.19,46 atically the various approximations. Only when both the Ja- Even if the dimension of the basis is limited by the difficulty strow and the AGP terms are accurately optimized together, to perform energy optimization with a very large number of the AGP nodal structure of the wave function is considerably variational parameters, we have obtained the chemical accu- improved. For the above reasons and the size consistency of racy for most cases studied. From a general point of view the the JAGP we expect that this wave function should be gen- basis set convergence of the JAGP is expected to be faster erally accurate also in complex systems made by many mol- than AGP considering that the electron-electron cusp condi- ecules. The local conservation of the charge around each tion is fulfilled exactly at each level of the expansion. Nev- molecule is taken into account by the Jastrow factor, whereas ertheless, all results presented here can be systematically im- the quality of each molecule is described also by the AGP geminal part exactly as in the H gas example. proved with larger basis set. In particular the Be2 bonding 2 distance should be substantially corrected by a more com- In the near future it is very appealing and promising to plete basis, that we have not attempted so far.39 extend the JAGP study to the DNA nitrogenous bases, whose The usefulness of the JAGP wave function is already geometrical structure is very similar to the benzene ring. In ͑ well known in the study of strongly correlated systems de- particular, we plan to accurately evaluate the energetics re- ͒ fined on a lattice. For instance, in the widely studied Hub- duction potential, ionization energies, electron affinity, etc. bard model, as well as in any model with electronic repul- of DNA bases and base pairs, quantities of great importance sion, it is not possible to obtain a superconducting ground to characterize excess electron and hole transfer which are state at the mean-field Hartree-Fock level. On the contrary, involved in radiation damage as well as in the development as soon as a correlated Jastrow term is applied to the BCS of DNA technologies. wave function ͑equivalent to the AGP wave function in mo- mentum space47͒, the stabilization of a d-wave superconduct- ACKNOWLEDGMENTS ing order parameter is possible, and is expected to be a realistic property of the model.48 More interestingly the presence We grateful acknowledge P. Carloni, F. Becca, and L. of the Jastrow factor can qualitatively change the wave func- Guidoni for fruitful suggestions in writing this manuscript. tion especially at one electron per site filling, by converting a We also thank S. Moroni, S. Zhang, C. Filippi, and D. Cep- BCS superconductor to a Mott insulator with a finite charge erley for useful discussions. This work was partially sup- gap.49 ported by MIUR, COFIN 2003. The same effect is clearly seen for the gedanken experi- ment of a dilute gas of H2 molecules, a clarifying test example already used in the introduction. The AGP wave func- APPENDIX A: STABILIZATION OF THE SR tion is essentially exact for a single molecule ͑at least with TECHNIQUE ͒ the complete basis set , but its obvious size consistent exten- Whenever the number of variational parameters in- sion to the gas would lead to the unphysical result of super- creases, it often happens that the stochastic (Nϩ1)ϫ(N conductivity because the charge around each molecule would ϩ1) matrix be free to fluctuate within the chosen set of geminal orbitals. ͗⌿͉ ͉⌿͘ Only the presence of the Jastrow term added to this wave OkOkЈ s ϭ ͑A1͒ function, allows the local conservation of the charge around k,kЈ ͗⌿͉⌿͘

3 3 3 becomes singular, i.e., the condition number, deﬁned as the 4 3 3 Ϫ Ϫ Ϫ 4 Ϫ

Ϫ Ϫ ␴ϭ␭ ␭ ␭ Ϫ ratio / between its maximum and minimum ei- 10 10 10 N 1 N 10 10 10 b 10 ϫ ϫ ϫ ␭ ϫ s 0 0 ϫ ϫ genvalue 1 , is too large. In that case the inversion of this ϫ 3

8.91 matrix generates clear numerical instabilities which are dif- 2.877 1.132 1.541 Ϫ Ϫ Ϫ Ϫ ﬁcult to control especially within a statistical method. Here ϭ ⌿ ␣ Ok d ln (x)/d k are the operators corresponding to the 3 3

Ϫ Ϫ ␣ ⌿ variational parameters k appearing in the wave function 10 10 b ϭ ϭ ϫ ϫ for k 1,...,N, whereas for k 0 the operator O0 represents 00 00 py

2 the identity one. 1.173 1.351 Ϫ Ϫ The ﬁrst successful proposal to control this instability was to remove from the inversion problem ͑12͒, required for 3 3

Ϫ Ϫ the minimization, those directions in the variational param- 10 10

b eter space corresponding to exceedingly small eigenvalues ϫ ϫ 0000 000 0000 1.790 7.67 0 1.790 px ␭ 2 i . 1.173 1.351 In this Appendix we describe a better method. As a ﬁrst Ϫ Ϫ step, we show that the reason of the large condition number

3 4 ␴ 4 4

4 is due to the existence of ‘‘redundant’’ variational param- Ϫ Ϫ 4 Ϫ Ϫ Ϫ Ϫ 10 10 10 10 eters that do not make changes to the wave function within a b 10 10 ϫ ϫ ϫ ϫ ϫ pz ⑀ ϫ prescribed tolerance . Indeed in practical calculations, we 2 4.67 4.25 6.838 1.601

7.67 are interested in the minimization of the wave function 3.211 Ϫ Ϫ Ϫ Ϫ within a reasonable accuracy. The tolerance ⑀ may represent

3 therefore the distance between the exact normalized varia- 4 3 Ϫ 3 4 Ϫ Ϫ Ϫ Ϫ

10 tional wave function which minimizes the energy expecta- 10 10 b 10 10 ϫ ϫ s

ϫ tion value and the approximate acceptable one, for which we ϫ ϫ 2 ⑀

4.67 no longer iterate the minimization scheme. For instance, 2.162 6.22 Ϫ Ϫ ϭ1/1000 is by far acceptable for chemical and physical interest. A stable algorithm is then obtained by simply remov- b s 000000 00 6.79 0 1.790 1 1

molecule. The matrix is symmetric to have a spin singlet, therefore we show only the upper part of it. ing the parameters that do not change the wave function by 2 less than ⑀ from the minimization. An efﬁcient scheme to 3 3 3 3 Ϫ Ϫ Ϫ remove the ‘‘redundant parameters’’ is also given. Ϫ 10 10 10

10 ⌿ a ϫ ϫ ϫ Let us consider the N normalized states orthogonal to , s 00000 ϫ 3 but not orthogonal among each other 2.877 1.132 1.541 Ϫ Ϫ Ϫ ͑O Ϫs ͉͒⌿͘ ͉ ͘ϭ k k,0 ͑ ͒ 3 e , A2 i 2 Ϫ ͱ ⌿͉ Ϫ ͒ ͉⌿ ͗ ͑Ok sk,0 10 a ϫ 0 000 0 py ͑ ͒ 2 where sk,0 is deﬁned in Eq. A1 . These normalized vectors

1.351 deﬁne N directions in the N-dimensional variational param- Ϫ eter manifold, which are independent as long as the determi-

3 nant S of the corresponding NϫN overlap matrix Ϫ 10 a

ϫ ϭ͗ ͉ ͘ ͑ ͒ 00 000 0 2.031 ¯s e e A3 px k,kЈ k kЈ 2

1.351 is nonzero. The number S is clearly positive and it assumes Ϫ its maximum value 1 whenever all the directions ei are mu- 3 4 4

Ϫ Ϫ tually orthogonal. On the other hand, let us suppose that Ϫ 10 10 10 a there exists an eigenvalue ¯␭ of ¯s smaller than the square of ϫ ϫ ϫ pz 2 2 the desired tolerance ⑀ , then the corresponding eigenvector 4.25 6.838 1.601 ͉ ͘ϭ͚ ͉ ͘ Ϫ v iai ei is such that Ϫ Ϫ 3 Ϫ 4 ͗ ͉ ͘ϭ ϭ¯␭ ͑ ͒

coefﬁcients of the geminal function expansion in the pairing determinant for the Li v v ͚ a a ¯s , A4

Ϫ i j i, j 10

␭ i, j a 10 ϫ s ϫ 2 where the latter equation holds due to the normalization con- 2.162 6.22 2 Ϫ ͚ ϭ dition iai 1. We arrive therefore to the conclusion that it is possible to deﬁne a vector v with almost vanishing norm a s 1 ¯ ¯¯ ¯¯¯¯¯¯ ¯ ¯¯¯¯¯ ¯¯¯¯¯¯ ¯¯¯¯¯¯¯ ¯¯¯¯¯¯¯¯ ¯¯¯¯¯¯¯¯¯ ¯¯¯¯¯¯¯¯¯¯ ¯¯¯¯¯¯¯¯¯¯¯

1 ͉ ͉ϭͱ␭р⑀ v as a linear combination of ei , with at least some nonzero coefﬁcient. This implies that the N directions ek are a b a b a b linearly dependent within a tolerance ⑀ and one can safely a a a b b b py py pz px pz px s s s s s s 3 3 2 2 2 TABLE VIII. Matrix of the 1 2 2 1 2 2 2 remove at least one parameter from the calculation.

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␭ TABLE IX. Matrix of the coefﬁcients of the pairing function expansion in the three-body Jastrow for the Li2 molecule. As in the previous table only the upper part is reported.

sGa pGxa pGya pGza sGb pGxb pGyb pGzb Ϫ Ϫ ϫ Ϫ4 Ϫ ϫ Ϫ4 Ϫ ϫ Ϫ5 sGa 0.2427 0 0 2.713 10 5.136 10 001.202 10 Ϫ Ϫ ϫ Ϫ3 pGxa ¯ 0.1772 0 0 0 7.997 10 00 Ϫ Ϫ ϫ Ϫ3 pGya ¯¯0.1772 0 0 0 7.997 10 0 ϫ Ϫ2 ϫ Ϫ5 Ϫ ϫ Ϫ3 pGza ¯¯¯1.027 10 1.202 10 008.749 10 Ϫ Ϫ ϫ Ϫ4 sGb ¯¯¯ ¯ 0.2427 0 0 2.713 10 Ϫ pGxb ¯¯¯ ¯ ¯ 0.1772 0 0 Ϫ pGyb ¯¯¯ ¯ ¯ ¯ 0.1772 0 ϫ Ϫ2 pGzb ¯¯¯¯¯¯¯1.027 10

In general whenever there are p vectors vi that are below The indices n and m refer not only to the basis elements but the tolerance ⑀ the optimal choice to stabilize the minimiza- also to the nuclei which the orbitals are centered on. tion procedure is to remove p rows and p columns from the The self consistency problem arises from the last term in ͑ ͒ ͑ ͒ matrix A3 , in such a way that the corresponding determi- Eq. B1 , i.e., when the electron ri belongs to A and r j to B. nant of the (NϪp)ϫ(NϪp) overlap matrix is maximum. If the Jastrow is size consistent, whenever A and B are far From practical purposes it is enough to consider an it- apart from each other this term must vanish or at most gen- erative scheme to find a large minor, but not necessarily the erate a one-body term, that is, in turn size consistent, as we maximum one. This method is based on the inverse of ¯s.At are going to show in the following. In the limit of large each step we remove the ith row and column from ¯s for separation all the ␭m,n off diagonal terms connecting any Ϫ1 which ¯si,i is maximum. We stop to remove rows and col- basis element of A to any basis element of B must vanish. umns after p inversions. In this approach we exploit the fact The second requirement is a sufficiently fast decay of the that, by a consequence of the Laplace theorem on determi- basis set orbitals ␺(r) whenever r→ϱ, except at most for a Ϫ1 nants, ¯sk,k is the ratio between the described minor without constant term Cn which may be present in the single particle the kth row and column and the determinant of the full ¯s orbitals, and is useful to improve the variational energy. matrix. Since within a stochastic method it is certainly not For the sake of generality, suppose that the system A possible to work with a machine precision tolerance, setting contains M A nuclei and NA electrons. The first requirement ⑀ϭ0.001 guarantees a stable algorithm, without affecting the implies that accuracy of the calculation. The advantage of this scheme, 17 compared with the previous one, is that the less relevant ␾ ͒ϭ ␭m,n␺m ͒␺n ͒ ͑ri ,r j ͚ ͑ri ͑r j parameters can be easily identified after few iterations and do m,n෈A not change further in the process of minimization. ϩ ␭m,n␺m ͒␺n ͒ ͑ ͒ ͚ ͑ri ͑r j , B3 m,n෈B APPENDIX B: SIZE CONSISTENCY OF THE THREE- BODY JASTROW FACTOR instead the second allows to write the following expression for the mixed term in Eq. ͑B1͒: In order to prove the size consistency property of the three-body Jastrow factor described in Sec. II C, let us take ␾͑ ͒ϭ ϩ ͑ ͒ into account a system composed by two well separated sub- ͚ ͚ ri ,r j NB ͚ CnPn NA ͚ CmPm , B4 i෈A j෈B n෈A m෈B systems A and B, which are distinguishable and whose di- mensions are much smaller than the distance between them- where the factors Pn are one-body terms defined as selves; in general they may contain more then one atom. In ͑ ͒ this case the Jastrow function J3 10 can be written as J3 ϭeU with TABLE X. Orbital basis set parameters used for the Li2 molecule. Since the 1 1 molecule is homonuclear the parameters of the atom b are the same as the ϭ ␾ ͒ϩ ␾ ͒ U ͚ ͑ri ,r j ͚ ͑ri ,r j atom a. 2 i, j෈A 2 i, j෈B iÞ j iÞ j z1 z2 p ␾ 2.4485 4.2891 0.4278 ϩ ͚ ͚ ␾͑r ,r ͒, ͑B1͒ 1sa i j ␾ 0.5421 1.4143 Ϫ1.5500 i෈A j෈B 2sa ␾ 0.6880 2pxa ¯¯ where we have explicitly considered the sum over different ␾ 0.6880 2pya ¯¯ ␾ ␾ 1.0528 subsystems. As usual, the two particle function (ri ,r j)is 2pza ¯¯ ␾ 0.6386 expanded over a single particle basis ␺, centered on each 3sa ¯¯ ␾ 1.4356 Ϫ0.2044 sGa ¯ nucleus of the system ␾ 0.7969 4.4217 Ϫ1.2689 pGxa ␾ 0.7969 4.4217 Ϫ1.2689 pGya ␾͑ ͒ϭ ␭m,n␺m͑ ͒␺n͑ ͒ ͑ ͒ ␾ 8.98010Ϫ3 Ϫ0.1924 0.3229 ri ,r j ͚ ri r j . B2 pGza m,n

which distinguishes the different components of the two ␭n,m ␺m͑ ͒ ෈ ͚ ͚ ri if n A electrons distance. We found that this two-body Jastrow fac- m෈A i෈A ϭ ͑ ͒ tor is particularly useful for Li , which is much more elon- Pn ͭ . B5 2 ␭n,m ␺m ͒ ෈ gated than the other molecules studied here, for which the ͚ ͚ ͑ri if n B m෈B i෈B usual form in Eq. ͑9͒ has been employed. The optimal parameters obtained for the Jastrow are aϭ0.8796, b Notice that if all the orbitals decay to zero, the size consis- ϭ0.7600. In the expansion of the pairing function for the tency is immediately recovered, since the sum in Eq. ͑B4͒ three body Jastrow term ͓see Eq. ͑10͔͒ we used the following vanishes. Analogously to the derivation we have done to ex- orbitals: tract the one body contribution from the mixed term, the Ϫ 2 ͑ ͒ ␾ ϭ z1r ϩ ͑ ͒ other two terms on the right-hand side of Eq. B1 can be sG e p, C5 rearranged in the following form: Ϫ 2 Ϫ 2 ␾ ϭrជ͑e z1r ϩpe z2r ͒. ͑C6͒ 1 pG ␾͑ ͒ϭ͑ Ϫ ͒ ͚ ri ,r j NA 1 ͚ CnPn The ␭ matrix that connects these orbitals is given in Table 2 i, j෈A n෈A iÞ j IX; this matrix fulﬁlls the same symmetry constraints as in ϩtwo-body terms, ͑B6͒ the case of the paring determinant. In this case the total number of nonzero ␭ is 24 and the symmetry reduces the varia- and the sum in Eq. ͑B1͒ can be rewritten as tional freedom to only eight parameters. The single particle orbitals are reported in Table X, and include other 15 param- ϭ Ϫ ͒ ϩ Ϫ ͒ U ͑N 1 ͚ CnPn ͑N 1 ͚ CnPn eters. n෈A n෈B

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