Hybrid correlation models based on active-space partitioning: correcting second-order Moller-Plesset perturbation theory for bond-breaking reactions

Moller-Plesset second-order (MP2) perturbation theory breaks down at molecular geometries which are far away from equilibrium. We decompose the MP2 energy into contributions from different orbital subspaces and show that the divergent behavior of the MP2 energy comes from the excitations located within a small (or sometimes even the minimal) active space. The divergent behavior of the MP2 energy at large interfragment distances may be corrected by replacing a small number of terms by their more robust counterparts from coupled-cluster (CCSD) theory. We investigated several schemes of such a substitution, and we find that a coupling between the active-space CCSD and the remaining MP2 amplitudes is necessary to obtain the best results. This naturally leads us to an approach which has previously been examined in the context of cost-saving approximations to CCSD for equilibrium properties by Nooijen [J. Chem. Phys. 111, 10815 (1999)]. The hybrid MP2-CCSD approach, which has the same formal scaling as conventional MP2 theory, provides potential curves with a correct shape for bond-breaking reactions of BH, CH(4), and HF. The error of the MP2-CCSD method (measured against full configuration-interaction data) is smaller than that of MP2 at all interfragment separations and is qualitatively similar to that of full CCSD.     

Empty level structure in phenyl and benzyl isocyanates

The energies of the lowest-lying anion states of phenyl (C6H5N=C=O) and benzyl (C6H5CH2N=C=O) isocyanates have been determined experimentally in the gas phase for the first time using electron transmission spectroscopy (ETS), and their localization properties have been evaluated using HF/6-31G, MP2/6-31G*, and B3LYP/6-31G* calculations. The lowest-lying anion state of phenyl isocyanate, mainly of benzene ring character but with some contribution also from the N=C=O pi-system, lies at significantly higher energy than that of other benzenes substituted by pi-functionals, such as benzaldehyde or styrene. The scaling with the use of suitable empirical equations of the virtual orbital energies (VOEs) for orbitals with predominantly pi*(ring) character calculated for the neutral-state molecules leads to vertical attachment energies (VAEs) which closely correspond to those determined experimentally, whereas those calculated for the predominantly pi*(CO) and pi*(NC) orbitals (3rd and 4th LUMO, respectively) are significantly different from the corresponding measured values notwithstanding the fact that the calculations reproduce the shortening of the N=C and C=O double bonds.     

Tensor hypercontraction density fitting. I. Quartic scaling second- and third-order Mo̸ller-Plesset perturbation theory

Many approximations have been developed to help deal with the O(N(4)) growth of the electron repulsion integral (ERI) tensor, where N is the number of one-electron basis functions used to represent the electronic wavefunction. Of these, the density fitting (DF) approximation is currently the most widely used despite the fact that it is often incapable of altering the underlying scaling of computational effort with respect to molecular size. We present a method for exploiting sparsity in three-center overlap integrals through tensor decomposition to obtain a low-rank approximation to density fitting (tensor hypercontraction density fitting or THC-DF). This new approximation reduces the 4th-order ERI tensor to a product of five matrices, simultaneously reducing the storage requirement as well as increasing the flexibility to regroup terms and reduce scaling behavior. As an example, we demonstrate such a scaling reduction for second- and third-order perturbation theory (MP2 and MP3), showing that both can be carried out in O(N(4)) operations. This should be compared to the usual scaling behavior of O(N(5)) and O(N(6)) for MP2 and MP3, respectively. The THC-DF technique can also be applied to other methods in electronic structure theory, such as coupled-cluster and configuration interaction, promising significant gains in computational efficiency and storage reduction.     

Electron propagator theory study of N-/O-methylglycine conformers

The low-lying conformers of N-/O-methylglycine are studied by ab initio calculations at the B3LYP, MP3, and MP4(SDQ) levels of theory with the aug-cc-pVDZ basis set. The conformers having the intramolecular hydrogen bonds N-H...O=C or O-H...N are more stable than the others. Vertical ionization energies for the valence molecular orbitals of each conformer predicted with the electron propagator theory in the partial third-order quasiparticle approximation are in good agreement with the experimental data available in the literatures. The relative energies of the conformers and comparison between the simulated and the experimental photoelectron spectra demonstrate that there are at least three and two conformers of N- and O-methylglycine, respectively, in the gas-phase experiments. The intramolecular hydrogen bonding O-H...N effects on the molecular electronic structures are discussed for the glycine methyl derivatives, on the basis of the ab initio electronic structure calculations, natural orbital bond, and atoms-in-molecules analyses. The intramolecular hydrogen bonding O-H...N interactions hardly affect the electronic structures of the O-NH2-CH2-C(=O)-O-CH3 and alpha-methylated NH2-CH2-C(CH3)OOH conformers, while the similar intramolecular interactions lead to the significantly lower-energy levels of the highest occupied molecular orbitals for the N-(CH3-NH-CH2-COOH) and beta-methylated (NH2-CH2-CH2-COOH) conformers.     

Rigorous formulation of two-parameter double-hybrid density-functionals

A two-parameter extension of the density-scaled double hybrid approach of Sharkas et al. [J. Chem. Phys. 134, 064113 (2011)] is presented. It is based on the explicit treatment of a fraction of multideterminantal exact exchange. The connection with conventional double hybrids is made when neglecting density scaling in the correlation functional as well as second-order corrections to the density. In this context, the fraction a(c) of second-order M?ller-Plesset (MP2) correlation energy is not necessarily equal to the square of the fraction a(x) of Hartree-Fock exchange. More specifically, it is shown that a(c)≤a(x)(2), a condition that conventional semi-empirical double hybrids actually fulfill. In addition, a new procedure for calculating the orbitals, which has a better justification than the one routinely used, is proposed. Referred to as λ(1) variant, the corresponding double hybrid approximation has been tested on a small set consisting of H(2), N(2), Be(2), Mg(2), and Ar(2). Three conventional double hybrids (B2-PLYP, B2GP-PLYP, and PBE0-DH) have been considered. Potential curves obtained with λ(1)- and regular double hybrids can, in some cases, differ significantly. In particular, for the weakly bound dimers, the λ(1) variants bind systematically more than the regular ones, which is an improvement in many but not all cases. Including density scaling in the correlation functionals may of course change the results significantly. Moreover, optimized effective potentials based on a partially-interacting system could also be used to generate proper orbitals. Work is currently in progress in those directions.     

Resolution of identity approach for the Kohn-Sham correlation energy within the exact-exchange random-phase approximation

Two related methods to calculate the Kohn-Sham correlation energy within the framework of the adiabatic-connection fluctuation-dissipation theorem are presented. The required coupling-strength-dependent density-density response functions are calculated within exact-exchange time-dependent density-functional theory, i.e., within time-dependent density-functional response theory using the full frequency-dependent exchange kernel in addition to the Coulomb kernel. The resulting resolution-of-identity exact-exchange random-phase approximation (RI-EXXRPA) methods in contrast to previous EXXRPA methods employ an auxiliary basis set (RI basis set) to improve the computational efficiency, in particular, to reduce the formal scaling of the computational effort with respect to the system size N from N(6) to N(5). Moreover, the presented RI-EXXRPA methods, in contrast to previous ones, do not treat products of occupied times unoccupied orbitals as if they were linearly independent. Finally, terms neglected in previous EXXRPA methods can be included, which leads to a method designated RI-EXXRPA+, while the method without these extra terms is simply referred to as RI-EXXRPA. Both EXXRPA methods are shown to yield total energies, reaction energies of small molecules, and binding energies of noncovalently bonded dimers of a quality that is similar and in some cases even better than that obtained with quantum chemistry methods such as Mo?ller-Plesset perturbation theory of second order (MP2) or with the coupled cluster singles doubles method. In contrast to MP2 and to conventional density-functional methods, the presented RI-EXXRPA methods are able to treat static correlation.     

Accurate description of van der Waals complexes by density functional theory including empirical corrections

An empirical method to account for van der Waals interactions in practical calculations with the density functional theory (termed DFT-D) is tested for a wide variety of molecular complexes. As in previous schemes, the dispersive energy is described by damped interatomic potentials of the form C6R(-6). The use of pure, gradient-corrected density functionals (BLYP and PBE), together with the resolution-of-the-identity (RI) approximation for the Coulomb operator, allows very efficient computations for large systems. Opposed to previous work, extended AO basis sets of polarized TZV or QZV quality are employed, which reduces the basis set superposition error to a negligible extend. By using a global scaling factor for the atomic C6 coefficients, the functional dependence of the results could be strongly reduced. The "double counting" of correlation effects for strongly bound complexes is found to be insignificant if steep damping functions are employed. The method is applied to a total of 29 complexes of atoms and small molecules (Ne, CH4, NH3, H2O, CH3F, N2, F2, formic acid, ethene, and ethine) with each other and with benzene, to benzene, naphthalene, pyrene, and coronene dimers, the naphthalene trimer, coronene. H2O and four H-bonded and stacked DNA base pairs (AT and GC). In almost all cases, very good agreement with reliable theoretical or experimental results for binding energies and intermolecular distances is obtained. For stacked aromatic systems and the important base pairs, the DFT-D-BLYP model seems to be even superior to standard MP2 treatments that systematically overbind. The good results obtained suggest the approach as a practical tool to describe the properties of many important van der Waals systems in chemistry. Furthermore, the DFT-D data may either be used to calibrate much simpler (e.g., force-field) potentials or the optimized structures can be used as input for more accurate ab initio calculations of the interaction energies.     

Role of Hartree-Fock and Kohn-Sham orbitals in the basis set superposition error for systems linked by hydrogen bonds

In this work the effect of the basis set superposition error (BSSE) is explored with the counterpoise method on the occupied and unoccupied Hartree-Fock (HF) and Kohn-Sham (KS) orbitals. Three different systems linked by hydrogen bonds, H(2)O...FH, H(2)O...H(2)O, and H(2)O...CFH(3), were studied by using the basis set families cc-pVXZ and aug-cc-pVXZ (X = D, T, Q). The basis sets were tested with the HF method and two approximations for the exchange-correlation functional of KS: a generalized gradient approximation and a hybrid approach. In addition to these methods, the second-order M?ller-Plesset perturbation theory, MP2, was considered. It was found that the presence of the "ghost" basis set affects the orbitals in two ways: (1) The occupied KS orbitals are more sensitive to the presence of this "ghost" basis set than the occupied HF orbitals. For this reason the BSSE observed in HF is less than that obtained with KS. (2) The unoccupied HF orbitals are more sensitive to the presence of the "ghost" basis set than their corresponding occupied orbitals. Because the MP2 method uses both, occupied and unoccupied HF orbitals, to compute the total energy, the contribution of the BSSE is bigger than that obtained with HF or KS methodologies.     

Improvement of the coupled-cluster singles and doubles method via scaling same- and opposite-spin components of the double excitation correlation energy

There has been much interest in cost-free improvements to second-order M?ller-Plesset perturbation theory (MP2) via scaling the same- and opposite-spin components of the correlation energy (spin-component scaled MP2). By scaling the same- and opposite-spin components of the double excitation correlation energy from the coupled-cluster of single and double excitations (CCSD) method, similar improvements can be achieved. Optimized for a set of 48 reaction energies, scaling factors were determined to be 1.13 and 1.27 for the same- and opposite-spin components, respectively. Preliminary results suggest that the spin-component scaled CCSD (SCS-CCSD) method will outperform all MP2 type methods considered for describing intermolecular interactions. Potential energy curves computed with the SCS-CCSD method for the sandwich benzene dimer and methane dimer reproduce the benchmark CCSD(T) potential curves with errors of only a few hundredths of 1 kcal mol(-1) for the minima. The performance of the SCS-CCSD method suggests that it is a reliable, lower cost alternative to the CCSD(T) method.     

Correlation regions within a localized molecular orbital approach

A hybrid scheme for the computation of reaction energies in large molecular systems is proposed. The approach is based on localized orbitals and allows for the treatment of different parts of a molecule at different computational levels. The localized orbitals are assigned to regions, and then different local correlation methods, such as local MP2 or local CCSD(T), can be applied to different regions. In contrast to previous hybrid schemes, the molecule does not have to be split into parts and, therefore, it is not necessary to saturate dangling bonds using link atoms. For fixed region sizes, the cost of the high-level calculation becomes independent of the molecular size, and it is demonstrated that O(1) scaling can be achieved. Illustrative applications are presented and the convergence of the results with respect to the size of the regions is investigated for reaction energies, barrier heights, and weakly bound complexes.     

Comparison of localization and delocalization indices obtained with Hartree-Fock and conventional correlated methods: effect of Coulomb correlation

Atomic populations and localization [lambda(A)] and delocalization [delta(A,B)] indices (LIs and DIs) are calculated for a large set of molecules at the Hartree-Fock (HF), MP2, MP4(SDQ), CISD, and QCISD levels with the 6-311++G(2d,2p) basis set. The HF method and the conventional correlation methods [MP2, MP4(SDQ), CISD, and QCISD] yield distinct sets of LIs and DIs. Yet, within the four conventional correlation methods the differences in atomic populations and LIs and DIs are small. Relative to HF, the conventional correlation methods [MP2, MP4(SDQ), CISD, QCISD] yield virtually the same LIs and DIs for molecules with large charge separations while LIs and DIs that differ significantly from the HF values--the LIs are increased and DIs decreased--are obtained for bonds with no or small charge separations. Such is the case in the archetypal homopolar molecules HC(triple bond)CH, H2C=CH2, CH3-CH3, and "protonated cyclopropane" C(3)H(7) (+), in which case the bonding may be atypical. Relative to HF, the typical effect of the conventional correlation methods is to decrease the DI between atoms.     

CH2NH与O(3P)反应的量子化学及电子密度拓扑研究

采用B3LYP、MP2(full)和 QCISD 三种方法在6-311G(d, p)和aug-cc-pVDZ基组水平上对三线态O(3P)原子与CH2NH(s)的反应进行了详细的理论研究. 采用B3LYP和MP2(full)方法对反应势能面上的各驻点进行了几何构型优化, 通过振动频率分析证实了过渡态的真实性, 内禀反应坐标(IRC)跟踪验证了过渡态与反应物和产物的连接关系, 用上述三种方法计算得到了各反应通道的反应势垒. 对反应过程中的一些重要点进行了电子密度拓扑分析研究. 研究结果表明, O(3P)原子进攻CH2NH(s)中的N2原子和C1原子生成CH2NHO(t)和OCH2NH(t), CH2NHO(t)中N2上的H5可迁移到C1上异构化为CH3NO(t).     

Perturbation expansion theory corrected from basis set superposition error II. Charge transfer,pair correlationand dispersion terms

The second-order perturbation theory based on the locally projected molecular orbitals is developed. A few test calculations with cc-pVDZ and aug-cc-pVDZ basis sets are carried out for the dimers, (H₂O)₂ and (HF)₂. The charge transfer terms remove the deficiency of the locally projected self-consistent field method for molecular interaction (LP SCF MO MI), and the potential energy curves calculated with aug-cc-pVDZ are very close to the corresponding curves of the counterpoise-corrected SCF energy. Only after adding the spin-exchanged dispersion type to the dispersion and intra-molecular pair correlation terms, the calculated potential energy curves become close to those of the counterpoise-corrected second-order Møller–Plesset (MP2). Pragmatic approaches for reducing the influence of the basis set superposition error are proposed.     

On the vibrational spectra and structural parameters of methyl, silyl, and germyl azide from theoretical predictions and experimental data

The infrared and Raman spectra of methyl, silyl, and germyl azide (XN3 where X=CH3, SiH3 and GeH3) have been predicted from ab initio calculations with full electron correlation by second order perturbation theory (MP2) and hybrid density function theory (DFT) by the B3LYP method with a variety of basis sets. These predicted data are compared to previously reported experimental data and complete vibrational assignments are provided for all three molecules. It is shown that several of the assignments recently proposed [J. Mol. Struct. (Theochem.) 434 (1998) 1] for methyl azide are not correct. Structural parameters for CH3N3 and GeH3N3 have been obtained by combining the previously reported microwave rotational constants with the ab initio MP2/6-311+G(d,p) predicted values. These "adjusted r0" parameters have very small uncertainties of +/-0.003 A for the XH distances and a maximum of +/-0.005 A for the heavy atom distances and +/-0.5 degrees for the angles. The predicted distance for the terminal NN bond which is nearly a triple bond is much better predicted by the B3LYP calculations, whereas the fundamental frequencies are better predicted by the scaled ab initio calculations. The results are discussed and compared to those obtained for some similar molecules.     

Second order Møller-Plesset perturbation theory based upon the fragment molecular orbital method

The fragment molecular orbital (FMO) method was combined with the second order M?ller-Plesset (MP2) perturbation theory. The accuracy of the method using the 6-31G(*) basis set was tested on (H(2)O)(n), n=16,32,64; alpha-helices and beta-strands of alanine n-mers, n=10,20,40; as well as on (H(2)O)(n), n=16,32,64 using the 6-31 + + G(**) basis set. Relative to the regular MP2 results that could be afforded, the FMO2-MP2 error in the correlation energy did not exceed 0.003 a.u., the error in the correlation energy gradient did not exceed 0.000 05 a.u./bohr and the error in the correlation contribution to dipole moment did not exceed 0.03 debye. An approximation reducing computational load based on fragment separation was introduced and tested. The FMO2-MP2 method demonstrated nearly linear scaling and drastically reduced the memory requirements of the regular MP2, making possible calculations with several thousands basis functions using small Pentium clusters. As an example, (H(2)O)(64) with the 6-31 + + G(**) basis set (1920 basis functions) can be run in 1 Gbyte RAM and it took 136 s on a 40-node Pentium4 cluster.     

Linear scaling calculation of maximally localized Wannier functions with atomic basis set

We have developed a linear scaling algorithm for calculating maximally localized Wannier functions (MLWFs) using atomic orbital basis. An O(N) ground state calculation is carried out to get the density matrix (DM). Through a projection of the DM onto atomic orbitals and a subsequent O(N) orthogonalization, we obtain initial orthogonal localized orbitals. These orbitals can be maximally localized in linear scaling by simple Jacobi sweeps. Our O(N) method is validated by applying it to water molecule and wurtzite ZnO. The linear scaling behavior of the new method is demonstrated by computing the MLWFs of boron nitride nanotubes.     

Solvent effects on the S(N)2 reaction: Application of the density functional theory-based effective fragment potential method

The performance of the density functional theory (DFT)-based effective fragment potential (EFP) method is assessed using the S(N)2 reaction: Cl- + nH2O + CH3Br = CH3Cl + Br- + nH2O. The effect of the systematic addition of water molecules on the structures and relative energies of all species involved in the reaction has been studied. The EFP1 method is compared with second-order perturbation theory (MP2) and DFT results for n = 1, 2, and 3, and EFP1 results are also presented for four water molecules. The incremental hydration effects on the barrier height are the same for all methods. However, only full MP2 or MP2 with EFP1 solvent molecules are able to provide an accurate treatment of the transition state (TS) and hence the central barriers. Full DFT and DFT with EFP1 solvent molecules both predict central barriers that are too small. The results illustrate that the EFP1-based DFT method gives reliable results when combined with an accurate quantum mechanical (QM) method, so it may be used as an efficient alternative to fully QM methods in the treatment of larger microsolvated systems.     

Accuracy and limitations of second-order many-body perturbation theory for predicting vertical detachment energies of solvated-electron clusters

Vertical electron detachment energies (VDEs) are calculated for a variety of (H(2)O)(n)(-) and (HF)(n)(-) isomers, using different electronic structure methodologies but focusing in particular on a comparison between second-order M?ller-Plesset perturbation theory (MP2) and coupled-cluster theory with noniterative triples, CCSD(T). For the surface-bound electrons that characterize small (H(2)O)(n)(-) clusters (n< or = 7), the correlation energy associated with the unpaired electron grows linearly as a function of the VDE but is unrelated to the number of monomers, n. In every example considered here, including strongly-bound "cavity" isomers of (H(2)O)(24)(-), the correlation energy associated with the unpaired electron is significantly smaller than that associated with typical valence electrons. As a result, the error in the MP2 detachment energy, as a fraction of the CCSD(T) value, approaches a limit of about -7% for (H(2)O)(n)(-) clusters with VDEs larger than about 0.4 eV. CCSD(T) detachment energies are bounded from below by MP2 values and from above by VDEs calculated using second-order many-body perturbation theory with molecular orbitals obtained from density functional theory. For a variety of both strongly- and weakly-bound isomers of (H(2)O)(20)(-) and (H(2)O)(24)(-), including both surface states and cavity states, these bounds afford typical error bars of +/-0.1 eV. We have found only one case where the Hartree-Fock and density functional orbitals differ qualitatively; in this case the aforementioned bounds lie 0.4 eV apart, and second-order perturbation theory may not be reliable.     

Anomeric effects in the symmetrical and asymmetrical structures of triethylamine. Blue-shifts of the C-h stretching vibrations in complexed and protonated triethylamine

Quantum mechanical calculations using density functional theory with the hybrid B3LYP functional and the 6-31++G(d,p) basis set are performed on isolated triethylamine (TEA), its hydrogen-bond complex with phenol, and protonated TEA. The calculations include the optimized geometries and the results of a natural bond orbital (NBO) analysis (occupation of sigma* orbitals, hyperconjugative energies, and atomic charges). The harmonic frequencies of the C-H stretching vibrations of TEA are predicted at the same level of theory. Two stable structures are found for isolated TEA. In the most stable symmetrical structure (TEA-S), the three C-C bond lengths are equal and one of the C-H bond of each of the three CH2 groups is more elongated than the three other ones. In the asymmetrical structure (TEA-AS), one of the C-C bonds and two C-H bonds of two different CH2 groups are more elongated than the other ones. These structures result from the hyperconjugation of the N lone pair to the considered sigma*(C-H) orbitals (TEA-S) or to the sigma*(C-C) and sigma*(C-H) orbitals of the CH2 groups (TEA-AS). The formation of a OH...N hydrogen bond with phenol results in a decrease of the hyperconjugation, a contraction of the C-H bonds, and blue-shifts of 28-33 cm-1 (TEA-S) or 40-48 cm-1 (TEA-AS) of the nus(CH2) vibrations. The nu(CH3) vibrations are found to shift to a lesser extent. Cancellation of the lone pair reorganization in protonated TEA-S and TEA-AS results in large blue-shifts of the nu(CH2) vibrations, between 170 and 190 cm-1. Most importantly, in contrast with the blue-shifting hydrogen bonds involving C-H groups, the blue-shifts occurring at C-H groups not participating in hydrogen bond formation is mainly due to a reduction of the hyperconjugation and the resulting decrease in the occupation of the corresponding sigma*(C-H) orbitals. A linear correlation is established between the C-H distances and the occupation of the corresponding sigma*(C-H) orbitals in the CH2 groups.     

Local explicitly correlated second- and third-order Møller-Plesset perturbation theory with pair natural orbitals

We present an algorithm for computing explicitly correlated second- and third-order M?ller-Plesset energies near the basis set limit for large molecules with a cost that scales formally as N(4) with system size N. This is achieved through a hybrid approach where locality is exploited first through orbital specific virtuals (OSVs) and subsequently through pair natural orbitals (PNOs) and integrals are approximated using density fitting. Our method combines the low orbital transformation costs of the OSVs with the compactness of the PNO representation of the doubles amplitude vector. The N(4) scaling does not rely upon the a priori definition of domains, enforced truncation of pair lists, or even screening and the energies converge smoothly to the canonical values with decreasing occupation number thresholds, used in the selection of the PNO basis. For MP2.5 intermolecular interaction energies, we find that 99% of benchmark basis set limit correlation energy contributions are recovered using an aug-cc-pVTZ basis and that on average only 50 PNOs are required to correlate the significant orbital pairs.