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.     

Calculation of the inner-shell contribution to the correlation energy through DLPNO-CEPA/1 and scaled same-spin second-order Møller–Plesset perturbation theory

The use of two low-cost methods for the prediction of the inner-shells contribution to the correlation energy is analyzed. The Spin-Component-Scaled second-order Møller–Plesset perturbation theory (SCS-MP2) was reparameterized for the prediction of such contributions. The best results are found when only the same spin term is considered (SSS-MP2). The Coupled Electron Pair Approximation (CEPA) using the Domain-based Local Pair Natural Orbital approximation (DLPNO) was also studied for the same purpose. The methods were tested for atomization energies on the W4-11 test set using basis sets up to quadruple zeta quality. The SSS-MP2 proved to be a marked improvement upon MP2 decreasing the root-mean-square-error (RMSE) from 0.443 to 0.302 kcal mol⁻¹. The RMSE of DLPNO-CEPA/1 in the test set is only 0.147 kcal mol⁻¹ and its computational cost is very low considering the intended applications. Furthermore, a linear combination of both methods decreased the RMSE to 0.118 kcal mol⁻¹.     

Møller-plesset calculations taking care of the correlation CUSP

Møller-Plesset calculations to second order have been carried out on the ten-electron systems Ne, HF and H₂O with a new functional, including r₁₂-dependent pair correlation functions, which takes care of the correlation cusp. The calculated second-order pair energies are accurate to within a few millihartree in comparison with the estimated exact values. In particular, second-order energies of 384.2, 380.1 and 362.9 mE_h, have been obtained for Ne, HF and H₂O respectively.     

A parallelized integral-direct second-order Møller–Plesset perturbation theory method with a fragment molecular orbital scheme

We propose a parallelized integral-direct algorithm of the second-order Møller–Plesset perturbation theory (MP2) as a size-consistent correlated method. The algorithm is a modification of the recipe by Mochizuki et al. [(1996) Theor Chim Acta 93:211]. There is no need to communicate the bulky data of integrals across worker processes, keeping the formal fifth-power dependence on the number of basis functions. A multiple integral screening procedure is incorporated to reduce the operation costs effectively. An approximate MP2 density matrix can also be directly calculated through the integral contraction with orbital energies. We implement the MP2 code by accepting Kitauras fragment molecular orbital (FMO) scheme as in the program ABINIT-MP developed by Nakano et al. [(2002) Chem Phys Lett 351:475]. The error in the FMO–MP2 energies is found to be within the order of the chemical accuracy. Timing and parallel acceleration results are shown for test molecules.     

Molecular dynamics, density functional and second-order Møller–Plesset theory study of the structure and conformation of acetylcholine in vacuo and in solution

 Molecular mechanics minimizations based on the CVFF force field and molecular dynamics simulation for a time of 2.5 ns were performed to examine the conformational behaviour and the molecular motion of acetylcholine in vacuo and in aqueous solution. Five low-lying conformations, namely the TT, TG, GG, G*G and GT, were obtained from molecular mechanics computations with the GT structure as the absolute minimum. Molecular dynamics trajectories in vacuo and in water show that only four (GT, GG, G*G and TG) and three (TG, TT and GT) conformations are present in the simulation time, respectively. Density functional B3LYP and second-order M?ller–Plesset (MP2) methods were then used to study all the five lowest-lying conformers of acetylcholine neurotransmitter in vacuo and in water by the polarizable continuum model approach. The B3LYP and MP2 computations show that in the gas phase all minima lie in a narrow range of energy with the G*G conformer as the most stable one. The relative minima GG, GT, TG and TT are located at 1.1 (3.3), 1.8 (4.2), 2.1 (4.5) and 4.3 (7.3) kcal/mol above the absolute one at the B3LYP (MP2) level. The preferred conformation in water is the TG. Solvation reduces the relative energy differences between the five minima in both computations. Received: 4 April 2001 / Accepted: 5 July 2001 / Published online: 30 October 2001     

Contracted basis sets for electrical property calculations derived from Second-order Møller–Plesset theory atomic natural orbitals

 Using established methods based on correlated atomic natural orbitals (ANOs), sets of contracted polarization functions are derived for use in calculations of atomic and molecular electrical properties (especially electric moments, dipole polarizabilities and related property hypersurfaces). Through test calculations on Ne, Ar, NH₃ and CO₂, these polarization functions are shown to reproduce the accuracy of larger basis sets, to incorporate dynamical electron correlation effects and are economical to use in conjunction with sophisticated electron-correlation treatments. We also show how triple-zeta polarized ANO and double-zeta polarized ANO basis sets are constructed from these contracted polarization functions for use in the calculation of reliable zero-point vibrational averages of electrical properties. Received: 20 December 1999 / Accepted: 15 February 2000 / Published online: 12 May 2000     

When does the non-variational nature of second-order Møller-Plesset energies manifest itself? All-electron correlation energies for open-shell atoms from K to Br

All-electron correlation energies E(c) are not very well known for open-shell atoms with more than 18 electrons. The complete basis-set (CBS) limits of second-order M?ller-Plesset (MP2) perturbation theory energies are obtained for open-shell atoms by computations in large basis sets combined with a knowledge of the MP2/CBS limit for the next larger closed-shell atom with the same valence shell structure. Then higher-order correlation corrections are found by coupled-cluster calculations using basis sets that are not quite as large. The method is validated for the open-shell atoms from Al to Cl for which E(c) is reasonably well established. Then, the method is used to obtain non-relativistic E(c) values, probably accurate to 3%, for the open-shell atoms of the fourth period: K, Sc-Cu, and Ga-Br. These energies are compared with the predictions of 19 density functionals and may be useful for the parameterization of new ones. The results show that MP2 overestimates |E(c)| for atoms heavier than Fe.     

Analysis of self-consistency effects in range-separated density-functional theory with Møller-Plesset perturbation theory

Range-separated density-functional theory combines wave function theory for the long-range part of the two-electron interaction with density-functional theory for the short-range part. When describing the long-range interaction with non-variational methods, such as perturbation or coupled-cluster theories, self-consistency effects are introduced in the density functional part, which for an exact solution requires iterations. They are generally assumed to be small but no detailed study has been performed so far. Here, the authors analyze self-consistency when using M?ller-Plesset-type (MP) perturbation theory for the long range interaction. The lowest-order self-consistency corrections to the wave function and the energy, that enter the perturbation expansions at the second and fourth order, respectively, are both expressed in terms of the one-electron reduced density matrix. The computational implementation of the latter is based on a Neumann series which, interestingly, even though the effect is small, usually diverges. A convergence technique, which perhaps can be applied in other uses of Neumann series in perturbation theory, is proposed. The numerical results thus obtained show that, in weakly bound systems, self-consistency can be neglected since the long-range correlation does not affect the density significantly. Although MP is not adequate for multireference systems, it can still be used as a reliable analysis tool. Though the density change is not negligible anymore in such cases, self-consistency effects are found to be much smaller than long-range correlation effects (less than 10% for the systems considered). For that reason, a sensible approximation might be to update the short-range energy functional term while freezing its functional derivative, namely, the short-range local potential, in the wave function optimization. The accuracy of such an approximation still needs to be assessed.     

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     

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     

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.     

Assessment of the Van Voorhis–Scuseria exchange–correlation functional for predicting excitation energies using time-dependent density functional theory

We assess the performance of the Van Voorhis–Scuseria exchange–correlation functional (VSXC), a kinetic-energy-density-dependent exchange–correlation functional recently developed in our group, for calculating vertical excitation energies using time-dependent density functional theory in a benchmark set of molecules. Overall, VSXC performs very well, with accuracy similar to that of hybrid functionals such as the hybrid Perdew–Burke–Ernzerhof functional and Becke's three parameter hybrid method with the Lee, Yang, and Parr correlation functional, which contain a portion of Hartree–Fock exchange. Received: 29 December 1999 / Accepted: 5 June 2000 / Published online: 11 September 2000     

On the accuracy of numerical Hartree–Fock energies

It is demonstrated that numerical Hartree–Fock (HF) energies reported in the literature in some cases have errors in the milliHartree range. The main cause of these errors is due to the use of too small a value for the practical infinity parameter in the finite difference method for generating the results. By systematically investigating the convergence with respect to the computational parameters, HF energies accurate to at least 1 microHartree are generated for 42 diatomic systems containing first and second row elements, encompassing both cationic, neutral and anionic systems.     

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.     

Determination of the pKa of cyclobutanone: Brønsted correlation of the general base-catalyzed enolization in aqueous solution and the effect of ring strain

The induction of strain in carbocycles, thereby increasing the amount of s-character in the C-H bonds and the acidity of these protons, has been probed with regard to its effect on the rate constants for the enolization of cyclobutanone. The second-order rate constants for the general base-catalyzed enolization of cyclobutanone have been determined for a series of 3-substituted quinuclidine buffers in D(2)O at 25 °C, I = 1.0 M (KCl). The rate constants for enolization were determined by following the extent of deuterium incorporation (up to ～30% of the first α-proton) into the α-position, as a function of time. The observed pseudo-first-order rate constants correlated to the [basic form] of the buffer and yielded the second-order rate constants for the general base-catalyzed enolization of cyclobutanone for four tertiary amine buffers. A Br?nsted β-value of 0.59 was determined from the second-order rate constants determined. Comparison of the results for cyclobutanone to those previously reported for acetone and a 1-phenylacetone derivative, under similar conditions, indicated that the ring strain of the carbocycle appeared to have only a small effect on the general base-catalyzed rate constants for enolization. The similarity of the rate constants for the general base-catalyzed enolization of cyclobutanone to those determined for acetone allowed for an estimation of the limits of the rate constant for protonation of the enolate intermediate of cyclobutanone by the conjugate acid of 3-quinuclidinone (k(BH) = 5 × 10(8) - 2 × 10(9) M(-1) s(-1)). Combining the rate constants for deprotonation of cyclobutanone (k(B)) and protonation of the enolate of cyclobutanone (k(BH)) by 3-quinuclidinone and its conjugate acid, the pK(a) of the α-protons of cyclobutanone has been estimated to be pK(a) = 19.7-20.2.     

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 theoretical study on the importance of steric effects,electronic properties,interaction and solvation energies in the ‘host–guest’ chemistry of protonated azacryptands and halide anions

It is well known that the selectivity of a receptor for an anion depends on the compatibility of the cavity size of the receptor and the size of the anion. In this work the macrobicyclic [H₆L(X)]⁵⁺ (X⁻=F, Cl, Br); are studied theoretically and compared with [H₆L′(F)]⁵⁺ having a smaller cavity size. It was shown that the ideal match between the sizes of the protonated azacryptand and the fluoride ion exists in the [H₆L′(F)]⁵⁺ complex but the [H₆L]⁶⁺ is a better receptor than [H₆L′]⁶⁺ in solution. Thus the results clearly indicate that in some special cases a better receptor is not one whose cavity size has better compatibility with the anion size.     

Scaled opposite-spin second order Møller-Plesset correlation energy: an economical electronic structure method

A simplified approach to treating the electron correlation energy is suggested in which only the alpha-beta component of the second order M?ller-Plesset energy is evaluated, and then scaled by an empirical factor which is suggested to be 1.3. This scaled opposite-spin second order energy (SOS-MP2), where MP2 is M?ller-Plesset theory, yields results for relative energies and derivative properties that are statistically improved over the conventional MP2 method. Furthermore, the SOS-MP2 energy can be evaluated without the fifth order computational steps associated with MP2 theory, even without exploiting any spatial locality. A fourth order algorithm is given for evaluating the opposite spin MP2 energy using auxiliary basis expansions, and a Laplace approach, and timing comparisons are given.