On the nature of the Møller-Plesset critical point

It has been suggested [F. H. Stillinger, J. Chem. Phys. 112, 9711 (2000)] that the convergence or divergence of M?ller-Plesset perturbation theory is determined by a critical point at a negative value of the perturbation parameter z at which an electron cluster dissociates from the nuclei. This conjecture is examined using configuration-interaction computations as a function of z and using a quadratic approximant analysis of the high-order perturbation series. Results are presented for the He, Ne, and Ar atoms and the hydrogen fluoride molecule. The original theoretical analysis used the true Hamiltonian without the approximation of a finite basis set. In practice, the singularity structure depends strongly on the choice of basis set. Standard basis sets cannot model dissociation to an electron cluster, but if the basis includes diffuse functions then it can model another critical point corresponding to complete dissociation of all the valence electrons. This point is farther from the origin of the z plane than is the critical point for the electron cluster, but it is still close enough to cause divergence of the perturbation series. For the hydrogen fluoride molecule a critical point is present even without diffuse functions. The basis functions centered on the H atom are far enough from the F atom to model the escape of electrons away from the fluorine end of the molecule. For the Ar atom a critical point for a one-electron ionization, which was not previously predicted, seems to be present at a positive value of the perturbation parameter. Implications of the existence of critical points for quantum-chemical applications are discussed.     

Distributed memory parallel implementation of energies and gradients for second-order Møller-Plesset perturbation theory with the resolution-of-the-identity approximation

We present a parallel implementation of second-order M?ller-Plesset perturbation theory with the resolution-of-the-identity approximation (RI-MP2). The implementation is based on a recent improved sequential implementation of RI-MP2 within the Turbomole program package and employs the message passing interface (MPI) standard for communication between distributed memory nodes. The parallel implementation extends the applicability of canonical MP2 to considerably larger systems. Examples are presented for full geometry optimizations with up to 60 atoms and 3300 basis functions and MP2 energy calculations with more than 200 atoms and 7000 basis functions.     

Fast evaluation of scaled opposite spin second-order Møller-Plesset correlation energies using auxiliary basis expansions and exploiting sparsity

The scaled opposite spin Møller–Plesset method (SOS‐MP2) is an economical way of obtaining correlation energies that are computationally cheaper, and yet, in a statistical sense, of higher quality than standard MP2 theory, by introducing one empirical parameter. But SOS‐MP2 still has a fourth‐order scaling step that makes the method inapplicable to very large molecular systems. We reduce the scaling of SOS‐MP2 by exploiting the sparsity of expansion coefficients and local integral matrices, by performing local auxiliary basis expansions for the occupied‐virtual product distributions. To exploit sparsity of 3‐index local quantities, we use a blocking scheme in which entire zero‐rows and columns, for a given third global index, are deleted by comparison against a numerical threshold. This approach minimizes sparse matrix book‐keeping overhead, and also provides sufficiently large submatrices after blocking, to allow efficient matrix–matrix multiplies. The resulting algorithm is formally cubic scaling, and requires only moderate computational resources (quadratic memory and disk space) and, in favorable cases, is shown to yield effective quadratic scaling behavior in the size regime we can apply it to. Errors associated with local fitting using the attenuated Coulomb metric and numerical thresholds in the blocking procedure are found to be insignificant in terms of the predicted relative energies. A diverse set of test calculations shows that the size of system where significant computational savings can be achieved depends strongly on the dimensionality of the system, and the extent of localizability of the molecular orbitals. © 2007 Wiley Periodicals, Inc. J Comput Chem 2007     

The relative energies of polypeptide conformers predicted by linear scaling second-order Møller-Plesset perturbation theory

We describe an implementation of the cluster-in-molecule(CIM) resolution of the identity(RI) approximation second-order M?ller–Plesset perturbation theory(CIM-RI-MP2), with the purpose of extending RI-MP2 calculations to very large systems. For typical conformers of several large polypeptides, we calculated their conformational energy differences with the CIM-RI-MP2 and the generalized energy-based fragmentation MP2(GEBF-MP2) methods, and compared these results with the density functional theory(DFT) results obtained with several popular functionals. Our calculations show that the conformational energy differences obtained with CIM-RI-MP2 and GEBF-MP2 are very close to each other. In comparison with the GEBF-MP2 and CIM-RI-MP2 relative energies, we found that the DFT functionals(CAM-B3LYP-D3, LC-?PBE-D3, M05-2X, M06-2X and ?B97XD) can give quite accurate conformational energy differences for structurally similar conformers, but provide less-accurate results for structurally very different conformers.     

A kinetic energy fitting metric for resolution of the identity second-order Møller-Plesset perturbation theory

A kinetic-energy-based fitting metric for application in the context of resolution of the identity second-order M?ller-Plesset perturbation theory is presented, which is derived from the Poisson equation. Preliminary tests of the applicability include the evaluation of the error in the correlation energy, compared to standard M?ller-Plesset perturbation theory, with respect to the auxiliary basis set employed. We comment on the potential merits of this fitting metric, compared to standard resolution of the identity second-order M?ller-Plesset perturbation theory, and discuss its scaling behavior in the limit of large molecules.     

Application of second-order Møller-Plesset perturbation theory with resolution-of-identity approximation to periodic systems

Efficient periodic boundary condition (PBC) calculations by the second-order M?ller-Plesset perturbation (MP2) method based on crystal orbital formalism are developed by introducing the resolution-of-identity (RI) approximation of four-center two-electron repulsion integrals (ERIs). The formulation and implementation of the PBC RI-MP2 method are presented. In this method, the mixed auxiliary basis functions of the combination of Poisson and Gaussian type functions are used to circumvent the slow convergence of the lattice sum of the long-range ERIs. Test calculations of one-dimensional periodic trans-polyacetylene show that the PBC RI-MP2 method greatly reduces the computational times as well as memory and disk sizes, without the loss of accuracy, compared to the conventional PBC MP2 method.     

Ab initio second-order Møller-Plesset calculation of the vibrational spectra of C4 clusters

Second-order Møller-Plesset calculations on the C₄ potential surface yielded three isomers, a linear triplet and rhombic and tetrahedral singlets. A large discrepancy between observed and calculated frequencies of the Σ_u⁺ vibration of linear ¹²C₄ is outside the expected range of error. It is tentatively suggested that an observed absorption at 1544 cm⁻¹ previously assigned to C₅ might belong instead to C₄.     

Alternative linear-scaling methodology for the second-order Møller-Plesset perturbation calculation based on the divide-and-conquer method

A new scheme for obtaining the approximate correlation energy in the divide-and-conquer (DC) method of Yang [Phys. Rev. Lett. 66, 1438 (1991)] is presented. In this method, the correlation energy of the total system is evaluated by summing up subsystem contributions, which are calculated from subsystem orbitals based on a scheme for partitioning the correlation energy. We applied this method to the second-order Moller-Plesset perturbation theory (MP2), which we call DC-MP2. Numerical assessment revealed that this scheme provides a reliable correlation energy with significantly less computational cost than the conventional MP2 calculation.     

Consistent generalization of the Møller-Plesset partitioning to open-shell and multiconfigurational SCF reference states in many-body perturbation theory

The Møller-Plesset partitioning of the many-electron Hamiltonian is generalized for arbitrary open-shell and multiconfigurational SCF reference states. The method has been initially implemented at the second and third orders for two-configuration generalized valence bond reference wavefunctions in which only two electrons are correlated. Potential curves for the hydrogen molecule, lithium hydride and the helium dication agree excellently with full CI results.     

Auxiliary basis sets for density fitting second-order Møller-Plesset perturbation theory: correlation consistent basis sets for the 5d elements Hf-Pt

Auxiliary basis sets specifically matched to the correlation consistent cc-pVnZ-PP, cc-pwCVnZ-PP, aug-cc-pVnZ-PP, and aug-cc-pwCVnZ-PP orbital basis sets (used in conjunction with pseudopotentials) for the 5d transition metal elements Hf-Pt have been optimized for use in density fitting second-order M?ller-Plesset perturbation theory and other correlated ab initio methods. Calculations of the second-order M?ller-Plesset perturbation theory correlation energy, for a test set of small to medium sized molecules, indicate that the density fitting error when utilizing these sets is negligible at three to four orders of magnitude smaller than the orbital basis set incompleteness error.     

Møller-Plesset perturbation energies and distances for HeC(20) extrapolated to the complete basis set limit

The potential energy surface for the C₂₀–He interaction is extrapolated for three representative cuts to the complete basis set limit using second‐order Møller–Plesset perturbation calculations with correlation consistent basis sets up to the doubly augmented variety. The results both with and without counterpoise correction show consistency with each other, supporting that extrapolation without such a correction provides a reliable scheme to elude the basis‐set‐superposition error. Converged attributes are obtained for the C₂₀– He interaction, which are used to predict the fullerene dimer ones. Time requirements show that the method can be drastically more economical than the counterpoise procedure and even competitive with Kohn‐Sham density functional theory for the title system. © 2008 Wiley Periodicals, Inc. J Comput Chem, 2009     

Is spin-component scaled second-order Møller-Plesset perturbation theory an appropriate method for the study of noncovalent interactions in molecules?

Testing of the spin-component scaled second-order M?ller-Plesset (SCS-MP2) method for the computation of noncovalent interaction energies is done with a database of 165 biologically relevant complexes. The effects of the spin-scaling procedure (i.e., MP2 vs SCS-MP2), the basis set size, and the corrections for basis set superposition error (BSSE) are systematically examined. When using two-point basis set extrapolations for the correlation energy, augmentation of the atomic orbital basis with computationally costly diffuse functions is found to be obsolete. In general, SCS-MP2 also improves results for noncovalent interactions statistically on MP2, and significant outliers are removed. Moreover, it is shown that effects of BSSE and one-particle basis set incompleteness almost cancel each other in the case of triple-zeta sets (SCS-MP2/TZVPP or SCS-MP2/cc-pVTZ without counterpoise correction), which opens a practical route to efficient computations for large systems. We recommend SCS-MP2 as the preferred quantum chemical wave function based method for the noncovalent interactions in large biologically relevant systems when reasonable coupled-cluster with single and double and perturbative triple excitations (CCSD(T)) calculations cannot be performed anymore. A comparison to MP2 and CCSD(T) interaction energies for n-alkane dimers, however, indicates (and this also holds to a lesser extent for hydrogen-bonded systems) limitations of SCS-MP2 when treating chemically "saturated" interactions. The different behavior of second-order perturbation theory for saturated and for stacked pi-systems is discussed.     

Analytic energy gradient for second-order Møller-Plesset perturbation theory based on the fragment molecular orbital method

The first derivative of the total energy with respect to nuclear coordinates (the energy gradient) in the fragment molecular orbital (FMO) method is applied to second order M?ller-Plesset perturbation theory (MP2), resulting in the analytic derivative of the correlation energy in the external self-consistent electrostatic field. The completely analytic energy gradient equations are formulated at the FMO-MP2 level. Both for molecular clusters (H(2)O)(64) and a system with fragmentation across covalent bonds, a capped alanine decamer, the analytic FMO-MP2 energy gradients with the electrostatic dimer approximation are shown to be complete and accurate by comparing them with the corresponding numeric gradients. The developed gradient is parallelized with the parallel efficiency of about 97% on 32 Pentium4 nodes connected by Gigabit Ethernet.     

Efficient implementation of the interacting quantum atoms energy partition of the second-order Møller–Plesset energy

We describe an efficient implementation of the partition of the second-order Møller–Plesset (MP2) correlation energy within the interacting quantum atoms (IQA) energy decomposition. We simplify the IQA integration bottleneck by considering only the occupied to virtual elements of the second order reduced density matrix, a procedure that reduces substantially the size of the two-electron matrix, which has to be addressed. The algorithmic improvements described herein allow to perform the decomposition of the MP2 correlation energy for medium size molecular systems using moderate computational resources. We expect that the methods developed in this investigation will prove useful to understand electron correlation effects through a real space perspective.     

A hybrid scheme for the resolution-of-the-identity approximation in second-order Møller-Plesset linear-r(12) perturbation theory

In the framework of second-order M?ller-Plesset linear-r(12) (MP2-R12) perturbation theory, a method is developed and implemented that uses an auxiliary basis set for the resolution-of-the-identity (RI) approximation for the three- and four-electron integrals. In contrast to previous work, the two-electron integrals that must be evaluated never involve more than one auxiliary basis function. The new method therefore scales linearly with the number of auxiliary basis functions and is much more efficient than the previous one, which scaled quadratically. A general formulation of MP2-R12 theory is presented for various ansatze, approximations, and orbitals (canonical or localized). The new method is assessed by computations of the valence-shell second-order M?ller-Plesset correlation energy of a few small closed-shell systems. The preliminary calculations indicate that the difference between the new and previous methods is about one order of magnitude smaller than the errors that occur due to basis-set truncations and RI approximations and under the assumptions of generalized and extended Brillouin conditions.     

Analysis of fourth-order Møller–Plesset limit energies: the importance of three-electron correlation effects

Fourth-order M?ller–Plesset (MP4) correlation energies are computed for 28 atoms and simple molecules employing Dunning's correlation-consistent polarized-valence m-zeta basis sets for m=2, 3, 4, and 5. Extrapolation formulas are used to predict MP4 energies for infinitely large basis sets. It is shown that both total and partial MP4 correlation energies can be extrapolated to limit values and that the sum of extrapolated partial MP4 energies equals the extrapolated total MP4 correlation energy within calculational accuracy. Therefore, partial MP4 correlation energies can be presented in the form of an MP4 spectrum reflecting the relative importance of different correlation effects. Typical trends in calculated correlation effects for a given class of electron systems are independent of the basis set used. As first found by Cremer and He [(1996) J Phys Chem 100:6173], one can use MP4 spectra to distinguish between electron systems with well-separated electron pairs and systems for which electrons cluster in a confined region of atomic or molecular space. MP4 spectra for increasing size of the basis set reveal that smaller basis set calculations underestimate the importance of three-electron correlation effects for both classes by overestimating the importance of pair correlation effects. The minimum size of a basis set required for reliable MP4 calculations is given by a valence triple-zeta polarized basis, which even in the case of anions performs better than a valence double-zeta basis augmented by diffuse functions. Received: 14 June 2000 / Accepted: 16 June 2000 / Published online: 24 October 2000     

Quasirelativistic theory for the magnetic shielding constant. III. Quasirelativistic second-order Møller-Plesset perturbation theory and its application to tellurium compounds

The quasirelativistic (QR) generalized unrestricted Hartree-Fock method for the magnetic shielding constant [R. Fukuda, M. Hada, and H. Nakatsuji, J. Chem. Phys. 118, 1015 (2003); R. Fukuda, M. Hada, and H. Nakatsuji, J. Chem. Phys.118, 1027 (2003)] has been extended to include the electron correlation effect in the level of the second-order M?ller-Plesset perturbation theory (MP2). We have implemented the energy gradient and finite-perturbation methods to calculate the magnetic shielding constant at the QR MP2 level and applied to the magnetic shielding constants and the NMR chemical shifts of 125Te nucleus in various tellurium compounds. The calculated magnetic shielding constants and NMR chemical shifts well reproduced the experimental values. The relations of the chemical shifts with the natures of ligands, and the tellurium oxidation states were investigated. The chemical shifts in different valence states were explained by the paramagnetic shielding and spin-orbit terms. The tellurium 5p electrons are the dominant origin of the chemical shifts in the Te I and Te II compounds and the chemical shifts were explained by the p-hole mechanism. The tellurium d electrons also play an important role in the chemical shifts of the hypervalent compounds.     

Accuracy of the three-body fragment molecular orbital method applied to Møller-Plesset perturbation theory

The three‐body energy expansion in the fragment molecular orbital method (FMO) was applied to the 2nd order Møller–Plesset theory (MP2). The accuracy of both the two and three‐body expansions was determined for water clusters, alanine n‐mers (α‐helices and β‐strands) and one synthetic protein, using the 6‐31G* and 6‐311G* basis sets. At the best level of theory (three‐body, two molecules/residues per fragment), the absolute errors in energy relative to ab initio MP2 were at most 1.2 and 5.0 mhartree, for the 6‐31G* and 6‐311G* basis sets, respectively. The relative accuracy was at worst 99.996% and 99.96%, for 6‐31G* and 6‐311G*, respectively. A three‐body approximation was introduced and the optimum threshold value was determined. The protein calculation (6‐31G*) at the production level (FMO2/2) took 3 h on 36 3.2‐GHz Pentium 4 nodes and had the absolute error in the MP2 correlation energy of only 2 kcal/mol. © 2007 Wiley Periodicals, Inc. J Comput Chem, 2007     

Minimax approximation for the decomposition of energy denominators in Laplace-transformed Møller-Plesset perturbation theories

We implement the minimax approximation for the decomposition of energy denominators in Laplace-transformed Moller-Plesset perturbation theories. The best approximation is defined by minimizing the Chebyshev norm of the quadrature error. The application to the Laplace-transformed second order perturbation theory clearly shows that the present method is much more accurate than other numerical quadratures. It is also shown that the error in the energy decays almost exponentially with respect to the number of quadrature points.