Dynamic hyperpolarizabilities in Møller–Plesset perturbation theory

As a formulation for calculating the dynamic polarizabilities and hyperpolarizabilities, two different types of the time‐dependent Møller–Plesset perturbation theory (MPPT) are presented: the MPPT in the quasienergy derivative method (QED–MPPT) and the MPPT in the energylike derivative method (ELD–MPPT). The explicit expressions for the response properties in each of these MPPT up to the quadratic response [μ, α(−ω₁, ω₁), β(−ω_σ, ω₁, ω₂)] at an arbitrary correlated order are given. Calculations of the dynamic polarizabilities and hyperpolarizabilities dependent on one frequency at the second‐order MPPT (MP2), in the QED method (QED–MP2) and in the ELD method (ELD–MP2), are examined for 10‐electron systems: hydrogen fluoride and neon. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 71: 251–271, 1999     

Integral transformation with low-order scaling for large local second-order Møller–Plesset calculations

An algorithm is presented for the four-index transformation of electron repulsion integrals to a localized molecular orbital (MO) basis. Unlike in most programs, the first two indices are transformed in a single step. This and the localization of the orbitals allows the efficient neglect of small contributions at several points in the algorithm, leading to significant time savings. Thresholds are applied to the following quantities: distant orbital pairs, the virtual space before and after the orthogonalizing projection to the occupied space, and small contributions in the transformation. A series of calculations on medium-sized molecules has been used to determine appropriate thresholds that keep the truncation errors small (below 0.01% of the correlation energy in most cases). Benchmarks for local second-order Møller–Plesset perturbation theory (MP2; i.e., MP2 with a localized MO basis in the occupied subspace) are presented for several large molecules with no symmetry, up to 975 contracted basis functions, and 60 atoms. These are among the largest MP2 calculations performed on a single processor. The computational time (with constant basis set) scales with a somewhat lower than cubic power of the molecular size, and the memory demand is moderate even for large molecules, making calculations that require a supercomputer for the traditional MP2 feasible on workstations. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1241–1254, 1998     

Exact size consistency of multireference Møller–Plesset perturbation theory

Single‐reference closed‐shell Møller–Plesset perturbation theory is well known for its size consistency, a quality that is essential for consistent comparisons of calculations on molecules of different size. However, it is far from obvious whether this quality can be retained in the multireference case. In this work it is shown that an exactly size consistently generalization to multireference perturbation theory can be constructed. The central result is that the zeroth‐order Hamiltonian should be constructed using separate projection operators for each excitation level, i.e., it should contain no couplings between different excitation levels. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 72: 549–558, 1999     

Kramers' restricted hartree—fock method for polyatomic molecules using ab initio relativistic effective core potentials with spin—orbit operators

A two-component Kramers' restricted Hartree–Fock method (KRHF) has been developed for the polyatomic molecules with closed shell configurations. The present KRHF program utilizes the relativistic effective core potentials with spin–orbit operators at the Hartree–Fock (HF) level and produces molecular spinors obeying the double group symmetry. The KRHF program enables the variational calculation of spin–orbit interactions at the HF level. KRHF calculations have been performed for the HX, X₂, XY(X, Y = I, Br), and CH₃I molecules. It is demonstrated that the orbital energies from KRHF calculations are useful for the interpretation of spin-orbit splittings in photoelectron spectra. In all molecules studied, bond lengths are only slightly expanded, harmonic vibrational frequencies are reduced, and bond energies are significantly decreased by the spin–orbit interactions.     

Parallel pseudospectral electronic structure: II. Localized Møller–Plesset calculations

We have developed a parallel version of our pseudospectral localized Møller–Plesset electronic structure code. We present timings for molecules up to 1010 basis functions and parallel speedup for molecules in the range of 260–658 basis functions. We demonstrate that the code is scalable; that is, a larger number of nodes can be efficiently utilized as the size of the molecule increases. By taking advantage of the available distributed memory and disk space of a scalable parallel computer, the parallel code can calculate LMP2 energies of molecules too large to be done on workstations. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1030–1038, 1998     

Improved uncoupled Hartree–Fock (IUCHF) perturbation methods and bounds for the second-order energy in coupled Hartree–Fock perturbation theory

Different kinds of improved uncoupled Hartree–Fock methods are proposed for the calculation of second-order perturbation energies. Using these methods inequalities are derived for the error of the uncoupled procedure with geometric approximation.     

Local quantum chemistry: The local space approximation for Møller–Plesset perturbation theory

The local space approximation is an accurate technique for describing a relatively small cluster embedded within an extended system. It has previously been developed for the Hartree-Fock, local density functional, configuration interaction, and coupled cluster electronic structure methods. Here it is extended to Møller-Plesset perturbation theory. © 1995 John Wiley & Sons, Inc.     

Multi‐state multi‐reference Møller–Plesset second‐order perturbation theory for molecular calculations

This work presents multi‐state multi‐reference Møller–Plesset second‐order perturbation theory as a variant of multi‐reference perturbation theory to treat electron correlation in molecules. An effective Hamiltonian is constructed from the first‐order wave operator to treat several strongly interacting electronic states simultaneously. The wave operator is obtained by solving the generalized Bloch equation within the first‐order interaction space using a multi‐partitioning of the Hamiltonian based on multi‐reference Møller–Plesset second‐order perturbation theory. The corresponding zeroth‐order Hamiltonians are nondiagonal. To reduce the computational effort that arises from the nondiagonal generalized Fock operator, a selection procedure is used that divides the configurations of the first‐order interaction space into two sets based on the strength of the interaction with the reference space. In the weaker interacting set, only the projected diagonal part of the zeroth‐order Hamiltonian is taken into account. The justification of the approach is demonstrated in two examples: the mixing of valence Rydberg states in ethylene, and the avoided crossing of neutral and ionic potential curves in LiF. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Many-body multireference Møller—Plesset and Epstein—Nesbet perturbation theory: Fast evaluation of second-order energy contributions

Multireference perturbation theory is examined in connection with the two partitions in the Møller—Plesset and Epstein—Nesbet schemes. The implementation of an efficient diagrammatic technique is described and two examples of application (diazene and the Cr₂ molecule), involving large variational spaces, are provided. © 1996 John Wiley & Sons, Inc.     

Development of spin‐dependent relativistic open‐shell Hartree–Fock theory with time‐reversal symmetry (I): The unrestricted approach

An open‐shell Hartree–Fock (HF) theory for spin‐dependent, two‐component relativistic calculations, termed the Kramers‐unrestricted HF (KUHF) method, is developed. The present KUHF method, which is formulated as a relativistic counterpart of nonrelativistic UHF, is based on quaternion algebra and partly uses time‐reversal symmetry. The fundamental characteristics of KUHF are discussed in this study. From numerical assessments, it was revealed that KUHF gives a corresponding solution to nonrelativistic UHF; furthermore, KUHF properly describes spin‐orbit interactions. In addition, KUHF can improve the self‐consistent field convergence behavior in spin‐dependent calculations, for example, for f‐block elements.     

Natural bond orbital‐based energy density analysis for correlated methods: Second‐order Møller–Plesset perturbation and coupled‐cluster singles and doubles

Natural bond orbital‐based energy density analysis (NBO‐EDA), which split energies into atomic and bonding contributions, is proposed for correlated methods such as coupled‐cluster singles and doubles (CCSD) and second‐order Møller–Plesset (MP2) perturbation. Applying NBO‐EDA for CCSD and MP2 to ethylene and the Diels–Alder reaction, we are successful in obtaining useful knowledge regarding electron correlation of π‐ and σ‐type orbitals, and clarifying the difference of the reaction barriers and heat of reaction calculated by CCSD and MP2. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

MPI/OpenMP hybrid parallel algorithm for resolution of identity second‐order Møller–Plesset perturbation calculation of analytical energy gradient for massively parallel multicore supercomputers

A massively parallel algorithm of the analytical energy gradient calculations based the resolution of identity Møller–Plesset perturbation (RI‐MP2) method from the restricted Hartree–Fock reference is presented for geometry optimization calculations and one‐electron property calculations of large molecules. This algorithm is designed for massively parallel computation on multicore supercomputers applying the Message Passing Interface (MPI) and Open Multi‐Processing (OpenMP) hybrid parallel programming model. In this algorithm, the two‐dimensional hierarchical MP2 parallelization scheme is applied using a huge number of MPI processes (more than 1000 MPI processes) for acceleration of the computationally demanding O (N ⁵) step such as calculations of occupied–occupied and virtual–virtual blocks of MP2 one‐particle density matrix and MP2 two‐particle density matrices. The new parallel algorithm performance is assessed using test calculations of several large molecules such as buckycatcher C₆₀@C₆₀H₂₈ (144 atoms, 1820 atomic orbitals (AOs) for def2‐SVP basis set, and 3888 AOs for def2‐TZVP), nanographene dimer (C₉₆H₂₄)₂ (240 atoms, 2928 AOs for def2‐SVP, and 6432 AOs for cc‐pVTZ), and trp‐cage protein 1L2Y (304 atoms and 2906 AOs for def2‐SVP) using up to 32,768 nodes and 262,144 central processing unit (CPU) cores of the K computer. The results of geometry optimization calculations of trp‐cage protein 1L2Y at the RI‐MP2/def2‐SVP level using the 3072 nodes and 24,576 cores of the K computer are presented and discussed to assess the efficiency of the proposed algorithm. © 2017 Wiley Periodicals, Inc.     

Auxiliary basis sets for density‐fitting second‐order Møller–Plesset perturbation theory: Weighted core‐valence correlation consistent basis sets for the 4d elements Y–Pd

Auxiliary basis sets (ABS) specifically matched to the cc‐pwCVnZ‐PP and aug‐cc‐pwCVnZ‐PP orbital basis sets (OBS) have been developed and optimized for the 4d elements Y‐Pd at the second‐order Møller‐Plesset perturbation theory level. Calculation of the core‐valence electron correlation energies for small to medium sized transition metal complexes demonstrates that the error due to the use of these new sets in density fitting is three to four orders of magnitude smaller than that due to the OBS incompleteness, and hence is considered negligible. Utilizing the ABSs in the resolution‐of‐the‐identity component of explicitly correlated calculations is also investigated, where it is shown that i‐type functions are important to produce well‐controlled errors in both integrals and correlation energy. Benchmarking at the explicitly correlated coupled cluster with single, double, and perturbative triple excitations level indicates impressive convergence with respect to basis set size for the spectroscopic constants of 4d monofluorides; explicitly correlated double‐ζ calculations produce results close to conventional quadruple‐ζ, and triple‐ζ is within chemical accuracy of the complete basis set limit. © 2013 Wiley Periodicals, Inc.