An accurate three‐dimensional potential energy surface for H3
01 Mar 1978-Journal of Chemical Physics (American Institute of Physics)-Vol. 68, Iss: 5, pp 2457-2465
TL;DR: In this paper, an accurate three-dimensional potential energy surface for H3 has been obtained by the configuration interaction (CI) method, with the energy of the saddle point taken to be zero, are believed to lie within 0.1 kcal/mole of the exact clamped nuclei limit.
Abstract: An accurate three‐dimensional potential energy surface for H3 has been obtained by the configuration interaction (CI) method. The calculated energies, for 156 nuclear configurations, with the energy of the saddle point taken to be zero, are believed to lie within 0.1 kcal/mole of the exact clamped‐nuclei limit. The CI calculations used an extended one‐particle basis set of 4 s‐type, 3 p‐type, and 1 d‐type contracted Gaussian functions, and a nearly complete n‐particle basis set. In order to solve the large secular problem, the direct CI method was adapted to the problem of complete CI for three valence electrons. The properties of the accurate H3 potential surface were used to evaluate ab initio and semiempirical methods for potential surface calculations, with emphasis on their applications to other exchange reactions.
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TL;DR: In this paper, a detailed study of correlation effects in the oxygen atom was conducted, and it was shown that primitive basis sets of primitive Gaussian functions effectively and efficiently describe correlation effects.
Abstract: In the past, basis sets for use in correlated molecular calculations have largely been taken from single configuration calculations. Recently, Almlof, Taylor, and co‐workers have found that basis sets of natural orbitals derived from correlated atomic calculations (ANOs) provide an excellent description of molecular correlation effects. We report here a careful study of correlation effects in the oxygen atom, establishing that compact sets of primitive Gaussian functions effectively and efficiently describe correlation effects i f the exponents of the functions are optimized in atomic correlated calculations, although the primitive (s p) functions for describing correlation effects can be taken from atomic Hartree–Fock calculations i f the appropriate primitive set is used. Test calculations on oxygen‐containing molecules indicate that these primitive basis sets describe molecular correlation effects as well as the ANO sets of Almlof and Taylor. Guided by the calculations on oxygen, basis sets for use in correlated atomic and molecular calculations were developed for all of the first row atoms from boron through neon and for hydrogen. As in the oxygen atom calculations, it was found that the incremental energy lowerings due to the addition of correlating functions fall into distinct groups. This leads to the concept of c o r r e l a t i o n c o n s i s t e n t b a s i s s e t s, i.e., sets which include all functions in a given group as well as all functions in any higher groups. Correlation consistent sets are given for all of the atoms considered. The most accurate sets determined in this way, [5s4p3d2f1g], consistently yield 99% of the correlation energy obtained with the corresponding ANO sets, even though the latter contains 50% more primitive functions and twice as many primitive polarization functions. It is estimated that this set yields 94%–97% of the total (HF+1+2) correlation energy for the atoms neon through boron.
26,705 citations
TL;DR: In this article, a review of the multiconfiguration time-dependent Hartree (MCTDH) method for propagating wavepackets is given, and the formal derivation, numerical implementation, and performance of the method are detailed.
2,053 citations
TL;DR: In this paper, the authors used correlation consistent Gaussian basis sets from double to sextuple zeta quality to estimate the barrier height of the H+H2 exchange reaction.
Abstract: Using systematic sequences of correlation consistent Gaussian basis sets from double to sextuple zeta quality, the classical barrier height of the H+H2 exchange reaction has been calculated by multireference configuration interaction (MRCI) methods. The MRCI calculations for collinear H3 have also been calibrated against large basis set full CI (FCI) results, which demonstrate that the MRCI treatment leads to energies less than 1 μhartree (≤0.001 kcal/mol) above the FCI energies. The dependence of both the H2 and H3 total energies on the basis set is found to be very regular, and this behavior has been used to extrapolate to the complete basis set (CBS) limits. The resulting estimate of the H–H–H CBS limit yields a classical barrier height, relative to exact H+H2, of 9.60±0.02 kcal/mol; the best directly calculated value for the barrier is equal to 9.62 kcal/mol. These results are in excellent agreement with recent quantum Monte Carlo calculations.
1,614 citations
TL;DR: In this article, a more reliable transition state theory that has many of the advantages of conventional TST can also be formulated, and it can be applied to practical problems with an effort that is much closer to that required for conventional transition-state theory than to the effort required for quantal dynamics calculations.
Abstract: In recent years our research group has made a systematic effort to study the validity of transition state theory (TST). We have found that the conventional theory is sometimes remarkably accurate, but in many other cases it leads to large errors. Fortunately we have found that a much more reliable theory that has many of the advantages of conventional TST can also be formulated, and it can be applied to practical problems with an effort that is much closer to that required for conventional transition state theory than to that required for quantal dynamics calculations. The two most important features in the improved approach to transition state theory state theory are the variational determination of the transition state and the incorporation of tunneling contributions by multidimensional semiclassical approximations. 13 refs.
1,186 citations
01 Jan 2010
TL;DR: The authors studies the impact of ambiguity and ambiguity aversion on equilibrium asset prices and portfolio holdings in competitive financial markets, and finds that attitudes toward ambiguity are heterogeneous across the population, just as attitudes toward risk are heterogenous across the populations, but that heterogeneity of attitudes towards ambiguity has different implications than heterogeneity of attitude toward risk, and that investors who have cognitive biases do not affect prices because they are infra-marginal.
Abstract: This paper studies the impact of ambiguity and ambiguity aversion on equilibrium asset prices and portfolio holdings in competitive financial markets. It argues that attitudes toward ambiguity are heterogeneous across the population, just as attitudes toward risk are heterogeneous across the population, but that heterogeneity of attitudes toward ambiguity has different implications than heterogeneity of attitudes toward risk. In particular, when some state probabilities are not known, agents who are sufficiently ambiguity averse find open sets of prices for which they refuse to hold an ambiguous portfolio. This suggests a different cross-section of portfolio choices, a wider range of state price/probability ratios and different rankings of state price/probability ratios than would be predicted if state probabilities were known. Experiments confirm all of these suggestions. Our findings contradict the claim that investors who have cognitive biases do not affect prices because they are infra-marginal: ambiguity averse investors have an indirect effect on prices because they change the per-capita amount of risk that is to be shared among the marginal investors. Our experimental data also suggest a positive correlation between risk aversion and ambiguity aversion that might explain the “value effect” in historical data.
877 citations
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TL;DR: In this article, an accurate method for calculating the nuclear wave functions and vibrational-rotational energies of diatomic molecules with some economy in the number of values of the internuclear potential required is presented.
Abstract: 1. Introduction. The wave equation for the nuclear motion of a diatomic molecule, in the Born-Oppenheimer approximation, is one which is encountered frequently in quantum-theoretical calculations. Numerical methods for its solution have been developed and used [1, 2, 3, 4] over many years for atomic problems where the potential is one obtained by Hartree-Fock self-consistent fields or the Thomas-Fermi-Dirac statistical field methods. Only relatively recently have computational techniques and the application of electronic computers enabled one to obtain accurate theoretical internuclear potentials at enough internuclear distances to calculate the wave functions for the motion of the nuclei and use them to obtain averages, over the nuclear motion, of molecular properties. The present investigation is concerned with obtaining an accurate method for calculating the nuclear wave functions and vibrational-rotational energies of diatomic molecules with some economy in the number of values of the internuclear potential required. An improved formula for the correction of trial eigenvalues, which does not depend so much for its accuracy upon the smallness of the stepsize in the radial coordinate, and an analysis of the convergence of the procedure are given. A computer subroutine was written and numerical results obtained from it are described for a case where exact analytical solutions are known. In what follows, the vibrational quantum number v, v = 0, 1, 2, , will be used as a subscript to index the eigenvalues Ev with the usual convention that Eo < E1 ? E2 ?
1,118 citations
TL;DR: In this article, the ground-state energy of H2 has been extended to include large internuclear distances and accurate potential energy curve for 0.4≤R≤4.0 a.u.
Abstract: Previous calculation of the ground‐state energy of H2 has been extended to include large internuclear distances and accurate potential‐energy curve for 0.4≤R≤10.0 a.u. is presented. For 0.4≤R≤4.0 a.u. expectation values of several operators have also been calculated. The calculation was made using a wavefunction in the form of an expansion in elliptic coordinates. The wavefunction depends on the interelectronic distance but, in contrast to the James—Coolidge expansion, is flexible enough to describe properly the dissociation of the molecule. Extensive calculations have also been made for the repulsive 3Σu+ state (1.0≤R≤10.0) and for the 1Πu state (1.0≤R≤10.0). In the former case a van der Waals minimum has been found at R=7.85 a.u. and 4.3 cm−1 below the dissociation limit. For the 1Πu state the computed binding energy De=20 490.0 cm−1 and the equilibrium internuclear distance Re=1.0330 A are in a satisfactory agreement with the experimental values De=20 488.5 cm−1 and Re=1.0327 A. In this case a van der ...
1,078 citations
TL;DR: In this article, an analytic semi-empirical expression for the ground-state potential energy surface of the H3 system was developed using the valence-bond formulation with the inclusion of overlap and three center terms.
Abstract: An analytic semiempirical expression is developed for the ground‐state potential‐energy surface of the H3 system. Use is made of the valence‐bond formulation with the inclusion of overlap and three‐center terms. The diatomic Coulomb and exchange integrals are estimated from accurate values for the H2 molecule by a modified London—Eyring—Sato procedure, while the three‐center integrals are approximated by simple formulas. A linear, symmetric saddle‐point configuration of minimum energy is found that has properties in approximate agreement, though not identical, with other estimates from theory and experimental rate data. The details of the surface are presented in terms of contour maps for a variety of distances and angles. Because of the inclusion of overlap and three‐center terms, a more realistic energy is expected for the nonlinear configurations than could be obtained from previous potential energy formulas. The utility of the present method for dynamical studies of reaction cross sections is indicated.
610 citations
TL;DR: In this article, Liu and Siegbahn made an accurate least square fit to the potential energy surface start for H+H2 with a root-mean-square error of 0.17 kcal/mol.
Abstract: Liu and Siegbahn’s recent calculations on the potential energy surface start for H+H2 provide us with the most accurately known potential energy surface for any chemical reaction. We have made an accurate least‐squares fit to this surface which satisfies several criteria for use in scattering calculations, including essentially exact agreement with all saddle point properties and being reasonably compact. With eight nonlinear parameters and 15 linear parameters we fit all 267 ab initio points with a root‐mean‐square error of 0.17 kcal/mol and a maximum absolute deviation of 0.55 kcal/mol. The spherical average of the interaction potential is in good agreement with the recent experimental estimate of Gengenbach, Hahn, and Toennies.
526 citations
TL;DR: In this paper, a new method for obtaining the coefficients in a large Cl expansion is proposed, where the expansion coefficients are obtained directly from the list of two-electron integrals by means of an iterative procedure.
468 citations