Electron-electron and electron-nucleus correlation effects on exponent values of Gaussian-type functions for quantum protons and deuterons

Electron-electron and electron-nucleus correlation effects on exponent (alpha) values of Gaussian-type functions (GTFs) for quantum protons and deuterons in BH3, CH4, NH3, H2O, and HF molecular systems and their deuterated counterparts were analyzed using the second-order Moller-Plesset (MP2) level of theory of the multicomponent molecular orbital (MCMO-MP2) method. This method can simultaneously determine both nuclear and electronic wave functions. Results showed that the average alpha value (alpha(ave)) of the optimized alpha in single s-type ([1s]) GTF for a proton and a deuteron is similar to that determined using the Hartree-Fock level of the MCMO (MCMO-HF) method. In contrast, due to the electron-nucleus correlation effect, the s- and p-type ([1s1p]) GTFs are delocalized compared with those determined using the MCMO-HF method. For the H-bonded complexes, differences in the interaction energy induced by the H/D isotope effect were clearly evident because the D...Y bond distance for D complex is longer than the H...Y for H complex. Also, the basis set superposition error for the interaction energy in every H complex was similar to that in every D complex. The results here clearly demonstrate that the protonic and deuteronic basis functions based on alpha(ave) values for correlation effects can be applied to the detailed analysis of the quantum effects of protons and the H/D isotope effect in widespread fields that involve H bonds and weak interactions, such as the function of biological molecules, chemical reaction processes, and the design of new materials.     

Simultaneous optimization of Gaussian type function exponents for electron and positron with full-CI wavefunction – application to ground and excited states of positronic compounds with multi-component molecular orbital approach

In order to obtain the best wavefunction of positronic compound with molecular orbital (MO) treatment, the full-configuration interaction (full-CI) fully variational MO (FVMO) method is proposed for multi-component systems, in which all the variational parameters in electronic and positronic wavefunctions are optimized under the full-CI scheme. We have applied the full-CI multi-component FVMO method to the ground and positronic-excited states of [H⁻;e⁺] system. Our treatment gives good improvement in the basis functions for positronic compounds owing to the extension of flexibility in the variational space, though the convergence of electron–positron correlation term is slower than that of conventional electron correlation.     

Gaussian-type function set without prolapse for the Dirac-Fock-Roothaan equation

A Gaussian-type function (GTF) set without a prolapse (variation collapse) is generated for the Dirac-Fock-Roothaan (DFR) equation. The test atom was mercury. The number of primitive GTFs used is between 7 and 62 (abbreviated as 7-62), 6-62, 6-62, 4-36, 4-36, 3-36, and 3-36 for s(+), p(-), p(+), d(-), d(+), f(-), and f(+) symmetries. The respective exponent parameters were determined with even-tempered manner, which requires the minimum and maximum exponents for the respective symmetries. We prepared several sets of these. The total energy (TE) given by the numerical DF (NDF) is -19648.849250 hartree; one of the present sets with largest number of expansion terms gave -19648.849251 hartree. The error (deltaTE) relative to the NDR TE is quite small. We then applied this set to the inert gas atoms Ne (10), Ar (18), Kr (36), Xe (54), Rn (86), and No (102), and also to Es (99) as the representative of the open shell atoms. The absolute values of deltaTE were at most 2.8 x 10(-6) hartree, showing the potential of this set as a universal set.     

Analytical optimization of orbital exponents in Gaussian-type functions for molecular systems based on MCSCF and MP2 levels of fully variational molecular orbital method

We have analyzed the basis function series in molecular systems by optimization of orbital exponents in Gaussian-type functions (GTFs) including the electron correlation effects with multiconfiguration self-consistent field (MCSCF) and M?ller?CPlesset second-order perturbation (MP2) methods. First, we have derived and implemented the gradient formulas of MCSCF and MP2 energies with respect to GTF exponent, as well as GTF center and nuclear geometry, based on the fully variational molecular orbital (FVMO) method. Second, we have applied these electron-correlated FVMO methods to H₂, LiH, and hydrocarbon (CH₄, C₂H₆, C₂H₄, and C₂H₂) molecules. We have clearly demonstrated that the optimized exponent values with electron-correlated methods are different from those with simple Hartree?CFock method, since adequate basis functions for adequate virtual orbitals are indispensable to describe the accurate wave function and geometry for electron-correlated calculations.     

Analysis of exponent values in Gaussian‐type functions for development of protonic and deuteronic basis functions

We analyzed the exponent (α) values in Gaussian‐type functions (GTF) for protons and deuterons in BH₃, CH₄, NH₃, H₂O, HF, and their deuterated molecules for the development of nuclear basis functions, which are used for molecular orbital (MO) calculations that directly include nuclear quantum effects. The optimized α (α_opt) value in the single s‐type ([1s]) GTF for protons is changed due to the difference in flexibility of the electronic basis sets. The difference between the energy obtained by using the α_opt value for each molecule and that obtained by using the average α (α_ave) value for these exponents with the 6‐31G(d,p) electronic basis function is only 2 × 10^?5 a.u. The α_ave values of protonic and deuteronic [1s] GTFs by the present calculation are 24.1825 and 35.6214, respectively. We found that the α_ave values enable the evaluation of the total energy and the geometrical changes in hydrogen bonding, such as O…H? O, O…H? N, and O…H? C, while the α_opt value became small by forming a hydrogen bond. The result using only the [1s] GTF for the protonic and deuteronic basis functions is sufficient to explain the differences of energy and geometry induced by the H/D isotope effect, although the total energy of ～5 × 10^?4 a.u. was improved by using the s‐, p‐, and d‐type ([1s1p1d]) GTFs for protons and deuterons. We clearly demonstrate that the protonic and deuteronic basis functions based on the α_ave value enable us to apply the method to other sample molecules (glycine, malonaldehyde, and formic acid dimer). The protonic and deuteronic basis functions we developed treat the quantum effects of protons and deuterons effectively and extend the application range of the MO calculation to include nuclear quantum effects. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Gaussian-type function set without prolapse for the Dirac-Fock-Roothaan equation (II): 80Hg through 103Lr

We present prolapse-free universal Gaussian-type basis sets for 80Hg through 103Lr. The basis set is determined so that the Dirac-Fock-Roothaan total energy should decrease monotonically toward the numerical Dirac-Fock total energy. The difference between the Dirac-Fock-Roothaan total energy and the numerical Dirac-Fock total energy is less than 3 x 10(-6) hartree for 1H through 102No, and less than 5 x 10(-6) hartree for 103Lr. The exponents of the present sets are determined in an even-tempered manner, aiming to give total energy closer to the numerical Dirac-Fock value as the expansion term increases. The recommended set is expanded by (64, 64, 64, 46, 46, 46, 46) terms for (s+, p-, p+, d-, d+, f-, f+) symmetries, respectively. A practical set with (56, 48, 48, 36, 36, 36, 36) terms is also presented.     

Accurate Heitler-London interaction energy for He2

A 75-term Gaussian geminal wave function for the helium atom that has a variational energy within 0.42 μhartree of the exact one is constructed. It predicts an electron density that agrees to better than 0.4% with the predictions of energetically superior Hylleraas wave functions to electron-nucleus distances as large as 6a₀. The first-order Heitler-London interaction energy E⁽¹⁾ between a pair of helium atoms was computed using an antisymmetrized product of this Gaussian geminal wave function for each of the atoms. This interaction energy is an essential component in the exchange-Coulomb model for the He₂ potential. Our E⁽¹⁾ is probably converged to within 0.03 μhartree for interatomic distances between 3.0a₀ and 7.5a₀. The Coulombic part of the interaction energy was checked by computations using even more accurate Hylleraas wave functions for the monomers. Comparison with an E⁽¹⁾ value computed from self-consistent-field atomic wave functions shows that intra-atomic correlation effects range between 4% and 9%.     

Molecular calculations for HeH<Superscript>+</Superscript> with two-center correlated orbitals

A two-center correlated orbital approach was used to calculate the electronic ground state energy for the HeH⁺ molecular ion. The wavefunctions were constructed from the exact solution of the Schrödinger equation for the HeH⁺⁺ problem in prolate-spheroidal coordinates taken together with a Hylleraas type correlation factor. With a simple single term wavefunction, we obtained ground state energy of ?2.95308691 hartree without any variational parameters in the calculation. When a two-configuration-state wavefunction was used and effective charges were allowed to be adjusted, we found an energy of ?2.97384868 hartree, which is to be compared with ?2.97869074 hartree obtained by an 83 term configuration interaction wavefunction or ?2.97364338 hartree by an ab initio calculation (at the MP4(SDQ)/6-311++G(3df, 3dp) level) using the well-known “canned” code.     

Variational solution of the three-dimensional Schrödinger equation using plane waves in adaptive coordinates

A series of improvements for the solution of the three-dimensional Schr?dinger equation over a method introduced by Gygi [F. Gygi, Europhys. Lett. 19, 617 (1992); F. Gygi, Phys. Rev. B 48, 11692 (1993)] are presented. As in the original Gygi's method, the solution (orbital) is expressed by means of plane waves in adaptive coordinates u, where u is mapped from Cartesian coordinates, u=f(r). The improvements implemented are threefold. First, maps are introduced that allow the application of the method to atoms and molecules without the assistance of the supercell approximation. Second, the electron-nucleus singularities are exactly removed, so that pseudo-potentials are no longer required. Third, the sampling error during integral evaluation is made negligible, which results in a true variational, second-order energy error procedure. The method is tested on the hydrogen atom (ground and excited states) and the H(2)(+) molecule, resulting in milli-Hartree accuracy with a moderate number of plane waves.     

High precision variational calculations for the Born-Oppenheimer energies of the ground state of the hydrogen molecule

Born-Oppenheimer approximation Hylleraas variational calculations with up to 7034 expansion terms are reported for the 1sigma(g)+ ground state of neutral hydrogen at various internuclear distances. The nonrelativistic energy is calculated to be -1.174 475 714 220(1) hartree at R = 1.4 bohr, which is four orders of magnitude better than the best previous Hylleraas calculation, that of Wolniewicz [J. Chem. Phys. 103, 1792 (1995)]. This result agrees well with the best previous variational energy, -1.174 475 714 216 hartree, of Cencek (personal communication), obtained using explicitly correlated Gaussians (ECGs) [Cencek and Rychlewski, J. Chem. Phys. 98, 1252 (1993); Cencek et al., ibid. 95, 2572 (1995); Rychlewski, Adv. Quantum Chem. 31, 173 (1998)]. The uncertainty in our result is also discussed. The nonrelativistic energy is calculated to be -1.174 475 931 399(1) hartree at the equilibrium R = 1.4011 bohr distance. This result also agrees well with the best previous variational energy, -1.174 475 931 389 hartree, of Cencek and Rychlewski [Rychlewski, Handbook of Molecular Physics and Quantum Chemistry, edited by S. Wilson (Wiley, New York, 2003), Vol. 2, pp. 199-218; Rychlewski, Explicitly Correlated Wave Functions in Chemistry and Physics Theory and Applications, edited by J. Rychlewski (Kluwer Academic, Dordrecht, 2003), pp. 91-147.], obtained using ECGs.     

有机分子的自洽场分子轨道从头计算的通用程序和C,H基组的选择

本文介绍一个适用于中小有机分子的自洽场分子轨道(SCFMO)从头计算通用程序,并对各种C、H基组作了比较研究,提出一个新的最小原子轨道基组。用它试算甲烷分子的电子结构,仅用9个原子轨道(AO)和41个GTO,得到的分子总能量为E=-40.1662hartree,比STO-KG的结果为优。例如STO-6G用9AO-54GTO得到E=-40.1011hartree。甚至比国际通用的Gaussian-70程序所采用的4-31G双ζ基组也有改进,后者用17AO-36GTO得到E=-40.1395hartree。此外,还在S.B.双ζ基组的基础上,在C-H键中间外加浮动键轨道,得到E=-40.1930hartree。     

Density functional theory treatment of electron correlation in the nuclear-electronic orbital approach

This paper presents the nuclear-electronic orbital density functional theory [NEO-DFT(ee)] method for including electron-electron correlation and nuclear quantum effects self-consistently in quantum chemical calculations. The NEO approach is designed to treat a relatively small number of nuclei quantum mechanically, while the remaining nuclei are treated classically. In the NEO-DFT(ee) approach, the correlated electron density is used to obtain the nuclear molecular orbitals, and the resulting nuclear density is used to obtain the correlated electron density during an iterative procedure that continues until convergence of both the nuclear and electronic densities. This approach includes feedback between the correlated electron density and the nuclear wavefunction. The application of this approach to bihalides and acetylene indicates that the nuclear quantum effects do not significantly impact the electron correlation energy, but the quantum nuclear energy is enhanced in the NEO-DFT(ee) B3LYP method. The excellent agreement of the NEO-DFT(ee)-optimized bihalide structures with the vibrationally averaged geometries from grid-based quantum dynamical methods provides validation for the NEO-DFT(ee) approach. Electron-proton correlation could be included by the development of an electron-nucleus correlation functional. Alternatively, explicit electron-proton correlation could be included directly into the NEO self-consistent-field framework with Gaussian-type geminal functions.     

Accurate Gaussian basis sets for the H2O molecule generated with the molecular improved generator coordinate Hartree–Fock method

The molecular improved generator coordinate Hartree–Fock (MIGCHF) method is used to generate accurate basis sets of primitive Gaussian-type functions for the H₂O molecule. Sequences of increasing size atom centered basis sets are employed to explore the accuracy that can be achieved with this method. Using the O(24s14p8d5f2g1h);H(22s9p5d2f1g) basis set, the HF and second-order electron correlation energies of the H₂O ground state at the experimental geometry are computed as −76.0674680 and −0.3491935 hartree, respectively. The HF energy is in error by 20 μhartree and the second-order correlation energy corresponds to 96.5% of an estimate of the limiting value. The relevance of the present calculations is to show the accuracy that can be achieved in studies of small polyatomic molecules with the MIGCHF method.     

Direct dynamics study on the reaction of acetaldehyde with ozone

The hydrogen abstraction reaction of ozone with acetaldehyde has been studied theoretically over the temperature range 250-2500 K. Two different reactive sites of acetaldehyde molecule, CH(3) and CHO groups have been investigated, and results confirm that the CHO group is a highly reactive site. In this study, the geometries and harmonic vibrational frequencies of all stationary points are calculated at the MPW1K, BHandHLYP, and MPWB1K levels of theory. The minimum energy paths (MEPs) were obtained at the MPW1K/6-31+G(d,p) level of theory. To refine the energies along the MEPs of each channel, single-point energy calculations were performed by a higher-level energy calculation method (denoted as HL). The rate constants were evaluated based on the MEPs from the HL method in the temperature range 250-2500 K by using the conventional transition state theory (TST), the canonical variational transition state theory (CVT), the microcanonical variational transition state theory (muVT), the CVT coupled with small-curvature tunneling (SCT) correction (CVT/SCT), and the muVT coupled with Eckart tunneling correction (muVT/Eckart). The fitted three-parameter Arrhenius expressions of the calculated CVT/SCT and muVT/Eckart rate constants of the H abstraction from CHO group are k CVT/SCT(T) = 4.92 x 10(-27).T 3.77.e(-7867.0/T) and k muVT/Eckart(T) = 2.10 x 10(-27).T(3.90).e(-7706.2/T), respectively. The fitted three-parameter Arrhenius expressions of the calculated CVT/SCT and muVT/Eckart rate constants of the H abstraction from CH3 group are k(CVT/SCT)(T) = 1.27 x 10(-27).T(3.94).e(-14554.1/T) and k muVT/Eckart(T) = 1.62 x 10(-26).T(3.66).e(-15459.8/T), respectively.     

Cellular association of glucosyltransferases in Leuconostoc mesenteroides and effects of detergent on cell association

Most glucosyltransferase (GTF) activity in sucrose-grown cultures of some strains of Leuconostoc mesenteroides is found with the cell pellet after centrifugation. GTFs are known to bind to dextrans, and it was traditionally assumed that cell-associated GTFs were bound to those dextrans that cosedimented with the cells. We used a mutant strain (LC-17), derived from strain NRRL B-1355, which produced dextransucrase in the absence of dextrans, to investigate the extent to which GTFs were bound to cells or dextrans. Much of the GTF activity in glucose-grown cultures of strain LC-17, which do not produce dextran, was located in the cell pellets. Soluble enzyme activity increased when cell suspensions from glucose- or sucrose-grown cultures were incubated with mild nonionic detergents or zwitterionic reagents. Alternansucrase produced by the parent strain B-1355 was almost entirely associated with cells under conditions in which dextrans were or were not produced. Alternansucrase, but not dextransucrase, tended to be enriched in the particulate fraction of B-1355 cells that had been broken in a French press. The distribution of alternansucrase and the effects of detergents on the distribution of GTFs suggest that soluble GTFs sequestered in the cytoplasm, and GTFs bound or adsorbed to the cell membrane are probably the major contributors to the cell-associated GTF activity.     

Correlation energy extrapolation by intrinsic scaling. V. Electronic energy, atomization energy, and enthalpy of formation of water

The method of correlation energy extrapolation by intrinsic scaling, recently introduced to obtain accurate molecular electronic energies, is used to calculate the total nonrelativistic electronic ground state energy of the water molecule. Accurate approximations to the full configuration interaction energies are determined for Dunning's [J. Chem. Phys. 90, 1007 (1989)] correlation-consistent double-, triple- and quadruple-zeta basis sets and then extrapolated to the complete basis set limit. The approach yields the total nonrelativistic energy -76.4390+/-0.0004 hartree, which compares very well with the value of -76.4389 hartree derived from experiment. The energy of atomization is recovered within 0.1 mh. The enthalpy of formation, which is obtained in conjunction with our previous calculation of the dissociation energy of the oxygen molecule, is recovered within 0.05 mh.     

Colle-Salvetti-type correction for electron-nucleus correlation in the nuclear orbital plus molecular orbital theory

A Colle-Salvetti (CS)-type electron-nucleus correction in the nuclear orbital plus molecular orbital theory is proposed. The CS-type correction is designed to satisfy the cusp condition for the electron-nucleus interaction. Since the CS-type correction is expressed in terms of the electron and nucleus densities, its evaluation is computationally feasible. Numerical assessment confirms that the CS-type correction performs well for the small G2 set.     

Alternative wavefunction ansatz for including explicit electron-proton correlation in the nuclear-electronic orbital approach

The nuclear-electronic orbital (NEO) approach treats specified nuclei quantum mechanically on the same level as the electrons with molecular orbital techniques. The explicitly correlated Hartree-Fock (NEO-XCHF) approach was developed to incorporate electron-nucleus dynamical correlation directly into the variational optimization of the nuclear-electronic wavefunction. In the original version of this approach, the Hartree-Fock wavefunction is multiplied by (1+G?), where G? is a geminal operator expressed as a sum of Gaussian type geminal functions that depend on the electron-proton distance. Herein, a new wavefunction ansatz is proposed to avoid the computation of five- and six-particle integrals and to simplify the computation of the lower dimensional integrals involving the geminal functions. In the new ansatz, denoted NEO-XCHF2, the Hartree-Fock wavefunction is multiplied by √(1+G?) rather than (1+G?). Although the NEO-XCHF2 ansatz eliminates the integrals that are quadratic in the geminal functions, it introduces terms in the kinetic energy integrals with no known analytical solution. A truncated expansion scheme is devised to approximate these problematic terms. An alternative hybrid approach, in which the kinetic energy terms are calculated with the original NEO-XCHF ansatz and the potential energy terms are calculated with the NEO-XCHF2 ansatz, is also implemented. Applications to a series of model systems with up to four electrons provide validation for the NEO-XCHF2 approach and the treatments of the kinetic energy terms.     

A path from I(h) to C1 symmetry for C20 cage molecule

The symmetry of the C20 cage is studied based on the intrinsical relationship among point groups (Bradley, C. J.; Cracknell, A. P. The Mathematical Theory of Symmetry in Solids; Claredon Press: Oxford, 1972). The structure of the C20 cage with I(h) symmetry is constructed, as are eight other structures with subgroup symmetry. A path from I(h) symmetry to C1 symmetry is obtained for the closed-shell electronic state, and the structure with D2h symmetry is the most stable on this path. Using the D2h structure the correlation energy correction is studied on the condition of restricted excitation space at the CCSD(T) level. We obtain curves on the relation between the orbital numbers and the total energy at the CCSD(T), CCSD, and MP2 level, respectively. The results of these curves obtained from MP2 and CCSD(T) methods have the same tendency, while the results of CCSD gradually diverge with an increase in orbital numbers. When the orbitals used in the calculation reach 460, the total energy is -759.644 hartree at MP2 level and is -759.721 hartree by the CCSD(T) method. From the calculation results, we find that a large basis set can improve the reliability of the MP2 method, and to restrict excitation space is necessary when using the CCSD(T) method.     

Variational transition state theory calculations for the rate constants of the hydrogen scrambling and the dissociation of BH5 using the multiconfiguration molecular mechanics algorithm

The BH5 molecule contains a weak two-electron-three-center bond and it requires extremely high level of theories to calculate the energy and structure correctly. The potential energy of the hydrogen scrambling in BH5 has been generated by the multiconfiguration molecular mechanics algorithm with 15 high-level Shepard interpolation points, which would be practically impossible to obtain otherwise. The high-level interpolation points were obtained from the multicoefficient correlated quantum mechanical methods. The more high-level points are used, the better the shape of the potential energy surface. The rate constants are calculated using the variational transition state theory including multidimensional tunneling approximation. The potential energy curve for the BH5 dissociation has also been calculated, and the variational transition state was located to obtain the dissociation rate constants. Tunneling is very important in the scrambling, and there is large variational effect on the dissociation. The rate constants for the scrambling and the dissociation are 2.1 x 10(9) and 2.3 x 10(12) s(-1) at 300 K, respectively, which suggests that the dissociation is three orders of magnitude faster than the scrambling.