Intramolecular Basis Set Superposition Error Research Articles

Quantifying Intramolecular Basis Set Superposition Errors.

We show that medium-sized Gaussian basis sets lead to significant intramolecular basis set superposition errors at Hartree-Fock and density functional levels of theory, with artificial stabilization of compact over extended conformations for a 186 atom deca-peptide. Errors of ∼80 and ∼10 kJ/mol are observed, with polarized double zeta and polarized triple zeta quality basis sets, respectively. Two different procedures for taking the basis set superposition error into account are tested. While both reduce the error, it appears that polarized quadruple zeta basis sets are required to reduce the error below a few kJ/mol. Alternatively, the basis set superposition error can be eliminated using multiresolution methods based on Multiwavelets.

Adiabatic projection: Bridging ab initio, density functional, semiempirical, and embedding approximations.

Modern electronic structure approximations routinely employ reference systems described by approximate Hamiltonians. This work introduces the adiabatic projection formalism for building formally exact corrections to such reference systems. Starting from the real Hamiltonian of a many-electron system, one constructs a reference system Hamiltonian by projecting the kinetic and electron-electron interaction operators onto "interesting" states. The reference system is corrected by density functionals for the difference between the projected and unprojected kinetic and electron-electron energies. These density functionals are constructed from adiabatic connections between the reference and real systems. The Hohenberg-Kohn theorems imply the existence of exact functionals, which can ensure that the reference system's ground-state energy and density match the real system. Adiabatic projection further generalizes Kohn-Sham density functional theory (DFT) and the generalized adiabatic connection [W. Yang, J. Chem. Phys. 109, 10107 (1998)] and recovers these methods for certain choices of projection operators. Other choices of projection operators offer new opportunities, including formally exact and systematically improvable analogues to wavefunction-in-DFT embedding, DFT+U, and semiempirical theories. Numerical results are presented for two representative choices: a projected exchange-correlation correction to small-basis-set coupled cluster theory and a projected kinetic energy density functional correcting basis set errors in DFT. The latter offers performance for dimerization energies approaching the Boys-Bernardi counterpoise correction while also correcting intramolecular basis set superposition errors.

On the Use of Popular Basis Sets: Impact of the Intramolecular Basis Set Superposition Error.

The magnitude of intramolecular basis set superposition error (BSSE) is revealed via computing systematic trends in molecular properties. This type of error is largely neglected in the study of the chemical properties of small molecules and it has historically been analyzed just in the study of large molecules and processes dominated by non-covalent interactions (typically dimerization or molecular complexation and recognition events). In this work we try to provide proof of the broader prevalence of this error, which permeates all types of electronic structure calculations, particularly when employing insufficiently large basis sets.

Anharmonic excited state frequencies of para-difluorobenzene, toluene and catechol using analytic RI-CC2 second derivatives.

Analytic second nuclear derivatives for excited electronic state energies have been implemented for the resolution-of-the-identity accelerated CC2, CIS(D∞) and ADC(2) models. Our efficient implementation with O(N2) memory demands enables the treatment of medium sized molecules with large basis sets and high numerical precision and thereby paves the way for semi-numerical evaluation of the higher-order derivatives required for anharmonic corrections to excited state vibrational frequencies. We compare CC2 harmonic and anharmonic excited state frequencies with experimental values for para-difluorobenzene, toluene and catechol. Basis set problems occur for out-of-plane bending vibrations due to intramolecular basis set superposition error. For non-planar molecules and in plane modes of planar molecules, the agreement between theory and experiment is better than 30 cm-1 on average and we reassign a number of experimental bands on the basis of the ab initio predictions.

D(A)3d(T)3 and d(G)3d(C)3 B-DNA mini-helixes: the DFT/M06-2x and DFT/B97-D3 comparison of geometrical and energetic characteristics.

We report the comprehensive DFT based comparison of geometrical and energetic parameters of the d(A)3·d(T)3 and d(G)3·d(C)3 nucleic acid mini-helixes performed at B97-D3 and M06-2× levels of theory. We studied the ability of mini-helixes to retain the conformation of B-DNA in the gas phase and under the influence of water bulk, uncompensated charges, and counter-ions. The def2-SV(P) and 6-31G(d,p) basis sets have been used for B97-D3 and M06-2× calculations, correspondently. To estimate basis set superposition error, the recently developed semi-empirical procedure that calls geometrical counterpoise type correction for inter- and intra-molecular basis set superposition error (gcp) has been used in the case of def2-SV(P) basis set. We found that both considered DFT functionals predict very similar results for geometrical ad energetic characteristics. We also found that in contrast to average classical molecular dynamics and data of simple geometrical models, both considered DFT functionals predict the existence of duplex specific geometries. A prediction of interaction energies of d(A)3d(T)3 and d(G)3d(C)3 duplexes accomplished in this study also verifies the applied models and confirms reliability of the new computational gcp technique.

Scalable Electron Correlation Methods. 3. Efficient and Accurate Parallel Local Coupled Cluster with Pair Natural Orbitals (PNO-LCCSD).

A well-parallelized local singles and doubles coupled-cluster (LCCSD) method using pair natural virtual orbitals (PNOs) is presented. The PNOs are constructed using large domains of projected atomic orbitals (PAOs) and orbital specific virtual orbitals (OSVs). We introduce a hierarchy of close, weak, and distant pairs, based on pair energies evaluated by local Møller-Plesset perturbation theory (LMP2). In contrast to most previous implementations of LCCSD methods, the close and weak pairs are not approximated by LMP2 but treated by higher-order methods. This leads to greatly improved accuracy for large systems, in particular when long-range dispersion interactions are important. Close pair amplitudes are optimized using approximate LCCSD equations, in which slowly decaying terms that mutually cancel at long-range are neglected. For weak pairs, the same approximations are used, but in addition, the nonlinear terms are neglected (coupled electron pair approximation). Distant pairs are treated by spin-component scaled (SCS)-LMP2 using multipole approximations. For efficiency reasons, various projection approximations are also introduced. The impact of these approximations on absolute and relative energies depends on the PNO domain sizes. The errors are found to be negligible, provided that sufficiently large PNO domains are used for close and weak pairs. For the selection of these domains the usual natural orbital occupation number criterion is found to be insufficient, and an additional energy criterion is used. For extended one-dimensional systems, the computational effort of the method scales nearly linearly with the number of correlated electrons, but the linear scaling regime is usually not reached in real-life applications for three-dimensional systems. Nevertheless, due to the parallelization that is efficient up to about 100-200 CPU cores (dependent on the molecular size), accurate calculations for three-dimensional molecules with about 100 atoms and augmented triple-ζ basis sets (e.g., cc-pVTZ-F12) can be carried out in 1-3 h of elapsed time (depending on the molecular structure and the number of CPU cores, excluding the time for Hartree-Fock). The convergence of the results with respect to the thresholds and options that control the domain, pair, and projection approximations is extensively tested. Benchmark examples for several difficult and large cases are presented, which demonstrate that the errors of relative energies (e.g., reaction energies, barrier heights) caused by the pair and projection approximations can be reduced to below 1 kJ mol-1. The remaining errors are mainly caused by the finite PAO domains. The larger these are made, the more intramolecular or intermolecular basis set superposition errors are introduced, and the canonical results are approached only very slowly. This problem is eliminated in the explicitly correlated variant of our method (PNO-LCCSD-F12), which will be described in a separate paper, along with a larger set of benchmark calculations.

Using valence bond methods to estimate intramolecular basis set superposition errors

We show that a valence bond type wave function with non-orthogonal orbitals, and enforcing a restriction of which basis functions a given molecular orbital is allowed to be expanded in, can be used to estimate intramolecular basis set superposition errors. The method can be considered as a generalization of the valence bond strictly/extremely localized molecular orbital methods, or as a generalization of the strictly monomer molecular orbital, block-localized wave function, and absolutely localized molecular orbital approaches for intermolecular systems. The method underestimates charge transfer interactions but can be used to estimate the magnitude of basis set superposition errors for relative conformational energies.

Scalable electron correlation methods I.: PNO-LMP2 with linear scaling in the molecular size and near-inverse-linear scaling in the number of processors.

We propose to construct electron correlation methods that are scalable in both molecule size and aggregated parallel computational power, in the sense that the total elapsed time of a calculation becomes nearly independent of the molecular size when the number of processors grows linearly with the molecular size. This is shown to be possible by exploiting a combination of local approximations and parallel algorithms. The concept is demonstrated with a linear scaling pair natural orbital local second-order Møller-Plesset perturbation theory (PNO-LMP2) method. In this method, both the wave function manifold and the integrals are transformed incrementally from projected atomic orbitals (PAOs) first to orbital-specific virtuals (OSVs) and finally to pair natural orbitals (PNOs), which allow for minimum domain sizes and fine-grained accuracy control using very few parameters. A parallel algorithm design is discussed, which is efficient for both small and large molecules, and numbers of processors, although true inverse-linear scaling with compute power is not yet reached in all cases. Initial applications to reactions involving large molecules reveal surprisingly large effects of dispersion energy contributions as well as large intramolecular basis set superposition errors in canonical MP2 calculations. In order to account for the dispersion effects, the usual selection of PNOs on the basis of natural occupation numbers turns out to be insufficient, and a new energy-based criterion is proposed. If explicitly correlated (F12) terms are included, fast convergence to the MP2 complete basis set (CBS) limit is achieved. For the studied reactions, the PNO-LMP2-F12 results deviate from the canonical MP2/CBS and MP2-F12 values by <1 kJ mol(-1), using triple-ζ (VTZ-F12) basis sets.

Geometrical Correction for the Inter- and Intramolecular Basis Set Superposition Error in Periodic Density Functional Theory Calculations

We extend the previously developed geometrical correction for the inter- and intramolecular basis set superposition error (gCP) to periodic density functional theory (DFT) calculations. We report gCP results compared to those from the standard Boys-Bernardi counterpoise correction scheme and large basis set calculations. The applicability of the method to molecular crystals as the main target is tested for the benchmark set X23. It consists of 23 noncovalently bound crystals as introduced by Johnson et al. (J. Chem. Phys. 2012, 137, 054103) and refined by Tkatchenko et al. (J. Chem. Phys. 2013, 139, 024705). In order to accurately describe long-range electron correlation effects, we use the standard atom-pairwise dispersion correction scheme DFT-D3. We show that a combination of DFT energies with small atom-centered basis sets, the D3 dispersion correction, and the gCP correction can accurately describe van der Waals and hydrogen-bonded crystals. Mean absolute deviations of the X23 sublimation energies can be reduced by more than 70% and 80% for the standard functionals PBE and B3LYP, respectively, to small residual mean absolute deviations of about 2 kcal/mol (corresponding to 13% of the average sublimation energy). As a further test, we compute the interlayer interaction of graphite for varying distances and obtain a good equilibrium distance and interaction energy of 6.75 Å and -43.0 meV/atom at the PBE-D3-gCP/SVP level. We fit the gCP scheme for a recently developed pob-TZVP solid-state basis set and obtain reasonable results for the X23 benchmark set and the potential energy curve for water adsorption on a nickel (110) surface.

Systematic Study of the Basis Set Superposition Error in Core–Electron Correlation Effects

In a recent paper, Xu et al. [J. Phys. Chem. A 2012, 116, 11668] emphasized the importance of core-electron correlation effects to describe the Si2H6BH3 complex and related systems properly. Unexpected large energy differences between a frozen core and all electron treatment were observed. In the present study, it will be shown that these energy differences are an artifact of an insufficient choice of basis set and can be attributed to an intramolecular basis set superposition error (BSSE). Although the general problem is known, systematic studies on the effect are scarce. Therefore, the BSSE in related systems is investigated. This study shows that the problem of BSSE for core-electron correlation is quite common if inadequate basis sets are applied and that it amounts to 2 kcal mol(-1) on average in binding energies for the given test set (with a maximum of 5.8 kcal mol(-1)).

A geometrical correction for the inter- and intra-molecular basis set superposition error in Hartree-Fock and density functional theory calculations for large systems

A semi-empirical counterpoise-type correction for basis set superposition error (BSSE) in molecular systems is presented. An atom pair-wise potential corrects for the inter- and intra-molecular BSSE in supermolecular Hartree-Fock (HF) or density functional theory (DFT) calculations. This geometrical counterpoise (gCP) denoted scheme depends only on the molecular geometry, i.e., no input from the electronic wave-function is required and hence is applicable to molecules with ten thousands of atoms. The four necessary parameters have been determined by a fit to standard Boys and Bernadi counterpoise corrections for Hobza's S66×8 set of non-covalently bound complexes (528 data points). The method's target are small basis sets (e.g., minimal, split-valence, 6-31G*), but reliable results are also obtained for larger triple-ζ sets. The intermolecular BSSE is calculated by gCP within a typical error of 10%-30% that proves sufficient in many practical applications. The approach is suggested as a quantitative correction in production work and can also be routinely applied to estimate the magnitude of the BSSE beforehand. The applicability for biomolecules as the primary target is tested for the crambin protein, where gCP removes intramolecular BSSE effectively and yields conformational energies comparable to def2-TZVP basis results. Good mutual agreement is also found with Jensen's ACP(4) scheme, estimating the intramolecular BSSE in the phenylalanine-glycine-phenylalanine tripeptide, for which also a relaxed rotational energy profile is presented. A variety of minimal and double-ζ basis sets combined with gCP and the dispersion corrections DFT-D3 and DFT-NL are successfully benchmarked on the S22 and S66 sets of non-covalent interactions. Outstanding performance with a mean absolute deviation (MAD) of 0.51 kcal/mol (0.38 kcal/mol after D3-refit) is obtained at the gCP-corrected HF-D3/(minimal basis) level for the S66 benchmark. The gCP-corrected B3LYP-D3/6-31G* model chemistry yields MAD=0.68 kcal/mol, which represents a huge improvement over plain B3LYP/6-31G* (MAD=2.3 kcal/mol). Application of gCP-corrected B97-D3 and HF-D3 on a set of large protein-ligand complexes prove the robustness of the method. Analytical gCP gradients make optimizations of large systems feasible with small basis sets, as demonstrated for the inter-ring distances of 9-helicene and most of the complexes in Hobza's S22 test set. The method is implemented in a freely available FORTRAN program obtainable from the author's website.

Model for the fast estimation of basis set superposition error in biomolecular systems.

Basis set superposition error (BSSE) is a significant contributor to errors in quantum-based energy functions, especially for large chemical systems with many molecular contacts such as folded proteins and protein-ligand complexes. While the counterpoise method has become a standard procedure for correcting intermolecular BSSE, most current approaches to correcting intramolecular BSSE are simply fragment-based analogues of the counterpoise method which require many (two times the number of fragments) additional quantum calculations in their application. We propose that magnitudes of both forms of BSSE can be quickly estimated by dividing a system into interacting fragments, estimating each fragment's contribution to the overall BSSE with a simple statistical model, and then propagating these errors throughout the entire system. Such a method requires no additional quantum calculations, but rather only an analysis of the system's interacting fragments. The method is described herein and is applied to a protein-ligand system, a small helical protein, and a set of native and decoy protein folds.

Enthalpy difference between conformations of normal alkanes: effects of basis set and chain length on intramolecular basis set superposition error

The quantum chemistry of conformation equilibrium is a field where great accuracy (better than 100 cal mol−1) is needed because the energy difference between molecular conformers rarely exceeds 1000–3000 cal mol−1. The conformation equilibrium of straight-chain (normal) alkanes is of particular interest and importance for modern chemistry. In this paper, an extra error source for high-quality ab initio (first principles) and DFT calculations of the conformation equilibrium of normal alkanes, namely the intramolecular basis set superposition error (BSSE), is discussed. In contrast to out-of-plane vibrations in benzene molecules, diffuse functions on carbon and hydrogen atoms were found to greatly reduce the relative BSSE of n-alkanes. The corrections due to the intramolecular BSSE were found to be almost identical for the MP2, MP4, and CCSD(T) levels of theory. Their cancelation is expected when CCSD(T)/CBS (CBS, complete basis set) energies are evaluated by addition schemes. For larger normal alkanes (N > 12), the magnitude of the BSSE correction was found to be up to three times larger than the relative stability of the conformer; in this case, the basis set superposition error led to a two orders of magnitude difference in conformer abundance. No error cancelation due to the basis set superposition was found. A comparison with amino acid, peptide, and protein data was provided.

Communications: Is quantum chemical treatment of biopolymers accurate? Intramolecular basis set superposition error (BSSE)

The accuracy of quantum chemical treatment of biopolymers by means of density functional theory is brought into question in terms of intramolecular basis set superposition error (BSSE). Secondary structure forms--beta-strands (C5; fully extended conformation), repeated gamma-turns (C7), 3(10)-helices (C10), and alpha-helices (C13)--of homopolypeptides (polyglycine and polyalanine) are used as representative examples. The studied molecules include Ace(Gly)(5)NH(2), Ace(Gly)(10)NH(2), Ace(Ala)(5)NH(2), and Ace(Ala)(10)NH(2). The counterpoise correction procedure was found to produce reliable estimations for the BSSE values (other methods of BSSE correction are discussed). The calculations reported here used the B3LYP, PBE0 (PBE1PBE), and BMK density functionals with different basis sets [from 6-31G(d) to 6-311+G(3df,3pd)] to estimate the influence of basis set size on intramolecular BSSE. Calculation of BSSE was used to determine the deviation of the current results from the complete basis set limit. Intramolecular BSSE was found to be nonadditive with respect to biopolymer size, in contrast to claims in recent literature. The error, which is produced by a basis set superposition, was found to exceed 4 kcal mol(-1) when a medium-sized basis set was used. This indicates that this error has the same order of magnitude as the relative energy differences of secondary structure elements of biopolymers. This result makes all recent reports on the gas-phase stability of homopolypeptides and their analogs questionable.

Communications: Intramolecular basis set superposition error as a measure of basis set incompleteness: Can one reach the basis set limit without extrapolation?

One of only two error sources in the solution of the electronic Schrodinger equation is addressed: The basis set convergence (incompleteness) error (BSIE). The results of ab initio (first principles) correlated methods, for which the Moller-Plesset second order perturbation theory (MP2) was chosen as an example, were extrapolated to the complete basis set (CBS) limit using a Dunning-type basis set series. Basis sets as large as cc-pV5Z and cc-pV6Z were used. A representative molecular set that included nitrogen (N(2)), acetylene (C(2)H(2)), ethylene (C(2)H(4)), carbon dioxide (CO(2)), water (H(2)O), ammonia (NH(3)), hydrogen cyanide (HCN), and ethanol (C(2)H(5)OH) molecules was used for the calculations. The intramolecular basis set superposition error (BSSE) was found to be correlated with BSIE, meaning that intramolecular BSSE can be used as a measure of basis set incompleteness. The BSIE dependence on BSSE could be qualitatively approximated (+/-25%) by a power-law dependence: BSIE = AxBSSE(p), where log(10)(A) = 1.45+/-0.21 and p = 1.27+/-0.09. This leads to the fact that CBS values at the MP2 theory level can be obtained using only one energy value and the corresponding intermolecular BSSE. The same power-law dependence was confirmed for all of the molecular systems studied. The universality of the BSIE versus BSSE dependence presented was checked using Pople-type basis sets. Even the results obtained with 6-311G, 6-311G(**), and 6-311G(2df,2pd) basis sets were found to be nicely described by the same (universal) power law. Benchmark studies of nitrogen and acetylene contraction (compaction) showed that BSIE can be decreased by up to 83% (at the cc-pVTZ level) using the CBS-BSSE strategy described. The presented BSIE versus BSSE dependence can greatly aid in obtaining CBS results for large molecular systems of chemical or biological interest.

An Atomic Counterpoise Method for Estimating Inter- and Intramolecular Basis Set Superposition Errors.

An atomic counterpoise method is proposed to calculate estimates of inter- and intramolecular basis set superposition errors. The method estimates the basis set superposition error as a sum of atomic contributions and can be applied for both independent particle and electron correlation models. It is shown that the atomic counterpoise method provides results very similar to the molecular counterpoise method for intermolecular basis set superposition errors at both the HF and MP2 levels of theory with a sequence of increasingly larger basis sets. The advantage of the atomic counterpoise method is that it can be applied with equal ease to estimate intramolecular basis set superposition errors, for which few other methods exist. The atomic counterpoise method is computationally quite efficient, requiring typically double the amount of computer time as required for calculating the uncorrected energy.

Intramolecular Basis Set Superposition Error Effects on the Planarity of DNA and RNA Nucleobases.

Molecules of utmost importance like DNA and RNA nucleobases are predicted to be nonplanar by a typical ab initio method, such as second order Møller-Plesset perturbation theory (MP2) combined with standard Pople's basis sets. Similarly to the case of other planar aromatic systems, these pitfalls can be explained in terms of intramolecular basis set superposition error (BSSE) effects, induced by local basis set deficiencies. We demonstrate that conventional BSSE correction techniques such as the Counterpoise method can account for this wrong behavior and provide proper correction whenever spurious results occur, mainly in case of thymine, uracil and guanine but also to lower extent for adenine and cytosine. We also show that special care must be taken when assessing the BSSE by means of ghost-orbital calculations for strongly overlapping fragments. Often molecular orbitals in the isolated fragment calculation have a different orientation as in the ghost-orbital calculation. This can lead to bogus derivatives of the CP-correction term, essential to account for geometry and vibrational BSSE effects.

Thermochemical Fragment Energy Method for Biomolecules: Application to a Collagen Model Peptide.

Herein, we first review different methodologies that have been proposed for computing the quantum mechanical (QM) energy and other molecular properties of large systems through a linear combination of subsystem (fragment) energies, which can be computed using conventional QM packages. Particularly, we emphasize the similarities among the different methods that can be considered as variants of the multibody expansion technique. Nevertheless, on the basis of thermochemical arguments, we propose yet another variant of the fragment energy methods, which could be useful for, and readily applicable to, biomolecules using either QM or hybrid quantum mechanical/molecular mechanics methods. The proposed computational scheme is applied to investigate the stability of a triple-helical collagen model peptide. To better address the actual applicability of the fragment QM method and to properly compare with experimental data, we compute average energies by carrying out single-point fragment QM calculations on structures generated by a classical molecular dynamics simulation. The QM calculations are done using a density functional level of theory combined with an implicit solvent model. Other free-energy terms such as attractive dispersion interactions or thermal contributions are included using molecular mechanics. The importance of correcting both the intermolecular and intramolecular basis set superposition error (BSSE) in the QM calculations is also discussed in detail. On the basis of the favorable comparison of our fragment-based energies with experimental data and former theoretical results, we conclude that the fragment QM energy strategy could be an interesting addition to the multimethod toolbox for biomolecular simulations in order to investigate those situations (e.g., interactions with metal clusters) that are beyond the range of applicability of common molecular mechanics methods.

Enthalpy difference between conformations of normal alkanes: Intramolecular basis set superposition error (BSSE) in the case of n-butane and n-hexane

In this paper, an extra error source for high-quality ab initio calculation of conformation equilibrium in normal alkanes-intramolecular basis set superposition error (BSSE)-is discussed. Normal butane (n-butane) and normal hexane (n-hexane) are used as representative examples. Single-point energy difference and BSSE values of trans and gauche conformations for n-butane (and trans-trans-trans and gauche-gauche-gauche conformations for n-hexane) were calculated using popular electron correlation methods: The second-order Moller-Plesset (MP2), the fourth-order Moller-Plesset (MP4), and coupled cluster with single and double substitutions with noniterative triple excitation [CCSD(T)] levels of theory. Extrapolation to the complete basis set is applied. The difference between BSSE-corrected and uncorrected relative energy values ranges from approximately 100 cal/mol (in case of n-butane) to more than 1000 cal/mol (in case of n-hexane). The influence of basis set type (Pople or Dunning) and size [up to 6-311G(3df,3pd) and aug-cc-pVQZ] is discussed.

Assessment of Density Functionals for Intramolecular Dispersion-Rich Interactions.

A range of density functional theory methods, including conventional hybrid and meta-hybrid functionals, a double-hybrid functional, and DFT-D (DFT augmented with an empirical dispersion term) were assessed for their ability to describe the three minima along the ϕGly rotational profile of one particular Tyr-Gly conformer. Previous work had shown that these minima are sensitive to intramolecular dispersion and basis set superposition error, the latter rendering MP2 calculations with small to medium-sized basis sets unsuitable for describing this molecule. Energy profiles for variation of the ϕGly torsion angle were compared to an estimated CCSD(T)/CBS reference profile. The hybrid functionals and the meta-hybrid PWB6K failed to predict all three minima; the meta-hybrid functionals M05-2X and M06-2X and the nonhybrid meta functional M06-L as well as the double-hybrid mPW2-PLYP and the B3LYP-D method did find all three minima but underestimated the relative stability of the two with rotated C-terminus. The best performance was delivered by the most elaborate density functional theory model employed: mPW2-PLYP-D. Only M06-2X and mPW2-PLYP-D predicted the correct order of stability of the three minima.