A local second-order Møller-Plesset method with localized orbitals: a parallelized efficient electron correlation method

Using orthogonal localized occupied orbitals we have developed and implemented a parallelized local second-order M?ller-Plesset (MP2) method based on the idea developed by Head-Gordon and co-workers. A subset of nonorthogonal correlation functions (the orbital domain) was assigned to each of the localized occupied orbitals using a distance criterion and excitations from localized occupied orbitals that were arranged into subsets. The correlation energy was estimated using a partial diagonalization and an iterative efficient method for solving large-scale linear equations. Some illustrative calculations are provided for molecules with up to 1484 Cartesian basis sets. The orbital domain sizes were found to be independent of the molecular size, and the present local MP2 method covered about 98%-99% of the correlation energy of the conventional canonical MP2 method.     

Perturbation expansion theory corrected from basis set superposition error. I. Locally projected excited orbitals and single excitations

The locally projected self-consistent field molecular orbital method for molecular interaction (LP SCF MI) is reformulated for multifragment systems. For the perturbation expansion, two types of the local excited orbitals are defined; one is fully local in the basis set on a fragment, and the other has to be partially delocalized to the basis sets on the other fragments. The perturbation expansion calculations only within single excitations (LP SE MP2) are tested for water dimer, hydrogen fluoride dimer, and colinear symmetric ArM+ Ar (M = Na and K). The calculated binding energies of LP SE MP2 are all close to the corresponding counterpoise corrected SCF binding energy. By adding the single excitations, the deficiency in LP SCF MI is thus removed. The results suggest that the exclusion of the charge-transfer effects in LP SCF MI might indeed be the cause of the underestimation for the binding energy.     

Molecule intrinsic minimal basis sets. I. Exact resolution of ab initio optimized molecular orbitals in terms of deformed atomic minimal-basis orbitals

A method is presented for expressing the occupied self-consistent-field (SCF) orbitals of a molecule exactly in terms of chemically deformed atomic minimal-basis-set orbitals that deviate as little as possible from free-atom SCF minimal-basis orbitals. The molecular orbitals referred to are the exact SCF orbitals, the free-atom orbitals referred to are the exact atomic SCF orbitals, and the formulation of the deformed "quasiatomic minimal-basis-sets" is independent of the calculational atomic orbital basis used. The resulting resolution of molecular orbitals in terms of quasiatomic minimal basis set orbitals is therefore intrinsic to the exact molecular wave functions. The deformations are analyzed in terms of interatomic contributions. The Mulliken population analysis is formulated in terms of the quasiatomic minimal-basis orbitals. In the virtual SCF orbital space the method leads to a quantitative ab initio formulation of the qualitative model of virtual valence orbitals, which are useful for calculating electron correlation and the interpretation of reactions. The method is applicable to Kohn-Sham density functional theory orbitals and is easily generalized to valence MCSCF orbitals.     

Towards a quantum chemical software package utilizing transferable fragments as molecular building blocks

Computer programs have been developed or are under development for the IBM personal computer that enable their users to get information on atomic charges, electrostatic potentials, conformational and other properties of molecular systems containing H, C, N, O, F, Si, P, S, or Cl atoms. The zero-order wavefunction is constructed of strictly localized molecular orbitals with fixed atomic orbital coefficients. The wave function can be refined by optimizing these coefficients, i.e., considering inductive effects via a coupled set of 2 × 2 secular equations within the CNDO /2 approximation. Delocalization and exchange effects are accounted for by expanding the wavefunction on a basis of the aforementioned strictly localized orbitals, instead of conventional atomic orbitals, and solving the corresponding SCF equations. Our method has been applied to the study of large systems. We calculated the electrostatic field of the complex of β-trypsin and basic pancreatic trypsin inhibitor and it has been found that strong field regions more or less coincide with hydration sites. A further potential application of protein electrostatic fields is in NMR spectroscopy. We found a linear correlation between C^αH or backbone NH proton chemical shifts and the protein field at the site of the corresponding proton. At last, we propose a simple method to mimic the bulk around atomic clusters modeling crystalline and amorphous silicon. Based on this method we found a linear correlation between atomic net charges and bond angle distortions in silicon clusters with 35 atoms.     

Molecular spinors from the quasi-relativistic pseudopotential approach

A quasi-relativistic (two-component) pseudopotential SCF method is presented. Kramers symmetry is imposed on the Fock operator. A matrix formulation using a basis of non-relativistic real atomic spin orbitals is given. The molecular spinors obtained have maximum similarity to conventional real molecular orbitals.     

Basis set generation for the SCF calculation

A method for basis set generation for SCF calculations is proposed. Using SCF orbitals and orbital energies obtained in the extended basis set the Fock operator can be expressed as its spectral resolution. The sum of differences between occupied orbital energies and corresponding eigenvalues obtained by the diagonalization of this operator in the new smaller basis set is a criterion of the quality of this new set. The present method consists of the minimization of this sum by changing the parameters that determine the new basis functions. An example of the optimization of the different Gaussian basis sets for the LiH molecule is described.     

An exact reformulation of the diagonalization step in electronic structure calculations as a set of second order nonlinear equations

A new formulation of the diagonalization step in self-consistent-field (SCF) electronic structure calculations is presented. It exactly replaces the diagonalization of the effective Hamiltonian with the solution of a set of second order nonlinear equations. The density matrix and/or the new set of occupied orbitals can be directly obtained from the resulting solution. This formulation may offer interesting possibilities for new approaches to efficient SCF calculations. The working equations can be derived either from energy minimization with respect to a Cayley-type parametrization of a unitary matrix, or from a similarity transformation approach.     

Consistent modifications of SINDO1: I. Approximations and parameters

A consistent modification, MSINDO, of the semiempirical MO method SINDO1 is presented. Different basis sets are used for one- and two-center interactions. The treatment of the core matrix elements in the nonorthogonal basis is retained with changes only for hydrogen and 3d orbitals. Orthogonalization corrections are now restricted to nonvanishing core matrix elements in the INDO approximation. The set of atomic parameters is increased, but bond parameters are no longer used. An automatic nonlinear least-squares algorithm with a restricted step constraint is used for the optimization of parameters. Heats of formation are adjusted with inclusion of zero-point energies obtained by a scaling procedure of the force constant matrix. The present version MSINDO provides significant improvements over previous versions. A brief comparison for ground-state properties of the elements H, C, N, O, F, and Na to Cl is given. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 563–571, 1999     

Extended Hartree–Fock self-consistent field with some results for berylliumlike atomic systems

The general methods of deriving the extended Hartree–Fock equations are described. The rules for going over from the energy expression in the ordinary method of calculation to that in an extended one are reformulated and illustrated. The extended Hartree—Fock equations for berylliumlike atomic systems based on the use of nonorthogonal radial orbitals are given and solved. The numerical values of overlap integrals and total energies are given and discussed.     

Integrating steepest-descent reaction pathways for large molecules

Exploring potential energy surfaces of large molecular systems can be quite challenging due to the increased number of nuclear degrees of freedom. Many techniques that are well-suited for small and moderate size systems require diagonalization of the energy second-derivative matrix. Since the cost of this step scales as O(N(atoms)(3)) (where N(atoms) is the number of atomic centers), such methods quickly become infeasible and are eventually rendered cost prohibitive. In this work, the recently developed Euler-based predictor-corrector reaction path integration method [H. P. Hratchian, M. J. Frisch, and H. B. Schlegel, J. Chem. Phys. 133, 224101 (2010)] is enhanced and proposed as a useful alternative to conventional reaction path following schemes in studies on very large systems. Because this integrator does not require Hessian diagonalization, the O(N(atoms)(3)) bottleneck afflicting other approaches is completely avoided. The effectiveness of the integrator in large system studies is demonstrated with an enzyme-catalyzed reaction employing an ONIOM (QM:MM) model chemistry and involving 5368 atomic centers.     

One electron orbitals intrinsic to the reduced hamiltonian

One electron orbitals are determined from the reduced hamiltonian by a simple one-step diagonalization. These reduced hamiltonian orbitals (RHO's) are uniquely determined and virtual orbitals obtained in this procedure are on a par with filled orbitals. These RHO's appear well suited for CI calculations. Minimum basis set calculations are presented for H₂O and compared with similar SCF studies.     

Second quantization for systems with a constant number of particles in the Dirac notation

The relation between the completeness condition for an appropriate one-particle basis set and the occupation number representation (second quantization) is shown for the time-independent case. The explicit expressions for the basic symmetric operators are derived in the Dirac bra–ket notation. The physical meaning of these operators, the algebra as well as the connections with the one-electron density matrix and with the projector on the Fermi sea in the one-electron approximation, follow directly from these expressions. The generalization for a nonorthogonal basis and the algebra for corresponding basic operators are formulated. The connection with the notion of the molecular diagrams of different kinds for the nonorthogonal atomic orbitals is shown. The Mulliken populations and the Chirgwin–Coulson bond orders are equal to the diagonal and offdiagonal elements of the molecular diagram 1, respectively. The matrix elements of the projector on the Fermi sea in the one-electron approximation in the representation of nonorthogonal atomic orbitals are elements of the molecular diagram 2.     

Physical nature of the chemical bond II. Valence atomic orbital and energy partitioning studies of linear nitriles

The valence atomic orbitals (VAO 's) of several linear nitriles are determined using non-empirical SCF –LCAO –MO wave functions expanded in a minimal (CN^?, HCN, FCN, C₂N₂), double-zeta (CN^?, HCN), or double-zeta + polarization (HCN) basis of Slater atomic orbitals (AO 's). The molecular energy of each system (except the double-zeta + polarization HCN system) is partitioned according to the procedure of Ruedenberg to obtain numerical values of nitrile C and N atomic and C?N bond components of the energy. In addition, the nitrile results are compared with minimal AO basis results obtained previously by other authors for homonuclear diatomics, diatomic hydrides and H₂O. The numerical data are used to test the internal self-consistency of the various definitions entering the partitioning method, i.e. whether or not analogous quantities assume similar values in chemically similar situations. The analysis of nitrile SCF –MO wave functions in terms of the set of VAO 's characteristic of the system under consideration is shown to be a promising approach to the problem of extracting useful information from the wave functions. In general, numerical results for the nitrile systems studied are fairly consistent with the concepts on which the partitioning method is based: promotion, quasi-classical interaction, sharing penetration, sharing interference and charge transfer. However, the VAO expansions for several energy components need to be investigated further and possibly revised.     

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.     

Core molecular orbital contribution to N2O isomerization as studied using theoretical electron momentum spectroscopy

Core molecular orbital contribution to the electronic structure of N2O isomers has been studied using quantum mechanical density functional theory combined with a plane wave impulse approximation method. Momentum distributions of wave functions for inner shell molecular orbitals of the linear NNO, cyclic and linear NON isomers of N2O are calculated through the (e, 2e) differential cross sections in momentum space. This is possible because this momentum distribution is directly proportional to the modulus squared of the momentum space wave function for the molecular orbital in question. While the momentum distributions of the NNO and cyclic N2O isomers demonstrate strong atomic orbital characteristics in their core space, the outer core molecular orbitals of the linear NON isomer exhibit configuration interactions between them and the valence molecular orbitals. It is suggested that the frozen core approximation breaks down in the prediction of the electronic structure of such an isomer. Core molecular orbital contributions to the electronic structure can alter the order of total energies of the isomers and lead to incorrect conclusions of the stability among the isomers. As a result, full electron calculations should be employed in the study of N2O isomerization.     

大体系多电子相关研究中应用群对称定域轨道的构想

大体系多电子相关研究中应用群对称定域轨道的构想周泰锦，刘爱民（厦门大学化学系，厦门３６１００５）关键词：组态相关，多构型自治叠代，多中心积分，群对称定域轨道，对称约化有关原子簇化合物及化学吸附、过渡态、激发态、催化反应等大体系的量子化学研究，对于探讨...     

Symmetry and equivalence restrictions in electronic structure calculations

A simple method for obtaining MCSCF orbitals and CI natural orbitals adapted to degenerate point groups, with full symmetry and equivalence restrictions, is described. Among several advantages accruing from this method are the ability to perform atomic SCF calculations on states for which the SCF energy expression cannot be written in terms of Coulomb and exchange integrals over real orbitals, and the generation of symmetry-adapted atomic natural orbitals for use in a recently proposed method for basis set contraction.     

On the use of local basis sets for localized molecular orbitals

Two procedures are discussed for the direct variational optimization of localized molecular orbitals which are expanded in local subsets of the molecular basis set. It is shown that a Newton-Raphson approach is more efficient than an iterative diagonalization scheme. The effect of the basis-set truncation on the quality ofab-initio SCF results is investigated for Be, Li₂, HF, H₂O, NH₃, CH₄ and C₂H₆.     

Optimization of auxiliary basis sets for the LEDO expansion and a projection technique for LEDO-DFT

We present a systematic procedure for the optimization of the expansion basis for the limited expansion of diatomic overlap density functional theory (LEDO-DFT) and report on optimized auxiliary orbitals for the Ahlrichs split valence plus polarization basis set (SVP) for the elements H, Li--F, and Na--Cl. A new method to deal with near-linear dependences in the LEDO expansion basis is introduced, which greatly reduces the computational effort of LEDO-DFT calculations. Numerical results for a test set of small molecules demonstrate the accuracy of electronic energies, structural parameters, dipole moments, and harmonic frequencies. For larger molecular systems the numerical errors introduced by the LEDO approximation can lead to an uncontrollable behavior of the self-consistent field (SCF) process. A projection technique suggested by L?wdin is presented in the framework of LEDO-DFT, which guarantees for SCF convergence. Numerical results on some critical test molecules suggest the general applicability of the auxiliary orbitals presented in combination with this projection technique. Timing results indicate that LEDO-DFT is competitive with conventional density fitting methods.     

Development of parallel density functional program using distributed matrix to calculate all-electron canonical wavefunction of large molecules

We developed a new parallel density-functional canonical molecular-orbital program for large molecules based on the resolution of the identity method. In this study, all huge matrices were decomposed and saved to the distributed local memory. The routines of the analytical molecular integrals and numerical integrals of the exchange-correlation terms were parallelized using the single program multiple data method. A conventional linear algebra matrix library, ScaLAPACK, was used for matrix operations, such as diagonalization, multiplication, and inversion. Anderson's mixing method was adopted to accelerate the self-consistent field (SCF) convergence. Using this program, we calculated the canonical wavefunctions of a 306-residue protein, insulin hexamer (26,790 orbitals), and a 133-residue protein, interleukin (11,909 orbitals) by the direct-SCF method. In regard to insulin hexamer, the total parallelization efficiency of the first SCF iteration was estimated to be 82% using 64 Itanium 2 processors connected at 3.2 GB/s (SGI Altix3700), and the calculation successfully converged at the 17-th SCF iteration. By adopting the update method, the computational time of the first and the final SCF loops was 229 min and 156 min, respectively. The whole computational time including the calculation before the SCF loop was 2 days and 17 h. This study put the calculations of the canonical wavefunction of 30,000 orbitals to practical use.