Nonrelativistic Born-Oppenheimer Research Articles

Three-body potential and third virial coefficients for helium including relativistic and nuclear-motion effects.

The non-additive three-body interaction potential for helium was computed using the coupled-cluster theory and the full configuration interaction method. The obtained potential comprises an improved nonrelativistic Born-Oppenheimer energy and the leading relativistic and nuclear-motion corrections. The mean absolute uncertainty of our calculations due to the incompleteness of the orbital basis set was determined employing complete-basis-set extrapolation techniques and was found to be 1.2%. For three helium atoms forming an equilateral triangle with the side length of 5.6 bohr - a geometry close to the minimum of the total potential energy surface - our three-body potential amounts to -90.6 mK, with an estimated uncertainty of 0.5 mK. An analytic function, developed to accurately fit the computed three-body interaction energies, was chosen to correctly describe the asymptotic behavior of the three-body potential for trimer configurations corresponding to both the three-atomic and the atom-diatom fragmentation channels. For large triangles with sides r12, r23, and r31, the potential takes correctly into account all angular terms decaying as r-l12 r-m23 r-n21 with l + m + n ≤ 14 for the nonrelativistic Born-Oppenheimer energy and l + m + n ≤ 9 for the post-Born-Oppenheimer corrections. We also developed a short-range analytic function describing the local behavior of the total uncertainty of the computed three-body interaction energies. Using both fits we calculated the third pressure and acoustic virial coefficients for helium and their uncertainties for a wide range of temperatures. The results of these calculations were compared with available experimental data and with previous theoretical determinations. The estimated uncertainties of present calculations are 3-5 times smaller than those reported in the best previous works.

Potential-energy curve for the a3Σu+ state of a lithium dimer with Slater-type orbitals

We report state-of-the-art ab initio calculations of the potential energy curve for the $a^3\Sigma_u^+$ state of the lithium dimer conducted to achieve spectroscopic accuracy ($<$1cm$^{-1}$) without any prior adjustment to fit the corresponding experimental data. The nonrelativistic clamped-nuclei component of the interaction energy is calculated with a composite method involving six-electron coupled cluster and full configuration interaction theories combined with basis sets of Slater-type orbitals ranging in quality from double- to sextuple-zeta. To go beyond the nonrelativistic Born-Oppenheimer picture we include both the leading-order relativistic and adiabatic corrections, and find both of these effects to be non-negligible within the present accuracy standards. The potential energy curve developed by us allowed to calculate molecular parameters ($D_e$, $D_0$, $\omega_e$ etc.) for this system, as well as the corresponding vibrational energy levels, with an error of only a few tenths of a wavenumber ($0.2-0.4\,$cm$^{-1}$). We also report an ab initio value for the scattering length of two $^2S$ lithium atoms which determines the stability of the related Bose-Einstein condensate.

What Levels of Coupled Cluster Theory Are Appropriate for Transition Metal Systems? A Study Using Near-Exact Quantum Chemical Values for 3d Transition Metal Binary Compounds.

Transition metal compounds are traditionally considered to be challenging for standard quantum chemistry approximations like coupled cluster (CC) theory, which are usually employed to validate lower level methods like density functional theory (DFT). To explore this issue, we present a database of bond dissociation energies (BDEs) for 74 spin states of 69 diatomic species containing a 3d transition metal atom and a main group element, in the moderately sized def2-SVP basis. The presented BDEs appear to have an (estimated) 3σ error less than 1 kJ/mol relative to the exact solutions to the nonrelativistic Born-Oppenheimer Hamiltonian. These benchmark values were used to assess the performance of a wide range of standard single reference CC models, as the results should be beneficial for understanding the limitations of these models for transition metal systems. We find that interactions between metals and monovalent ligands like hydride and fluoride are well described by CCSDT. Similarly, CCSDTQ appears to be adequate for bonds between metals and nominally divalent ligands like oxide and sulfide. However, interactions with polyvalent ligands like nitride and carbide are more challenging, with even CCSDTQ(P)Λ yielding errors on the scale of a few kJ/mol. We also find that many perturbative and iterative approximations to higher order terms either yield disappointing results or actually worsen the performance relative to the baseline low level CC method, indicating that complexity does not always guarantee accuracy.

Atomic-pair theorem for universal matrix representatives of molecules and atomic clusters in non-relativistic Born-Oppenheimer approximation

The mathematical underpinnings of a universal atomic-pair based approach to ab initio calculations of the electronic wave functions and energies of molecules and other atomic clusters are reported in non-relativistic Born-Oppenheimer approximation. The general approach utilizes the closure of an orthonormal (Eisenschitz-London) outer product of atomic spectral eigenstates to support totally antisymmetric aggregate eigenstates in the absence of explicit term-by-term atomic-product basis antisymmetry. Meaningful definitions of atomic-pair interaction operators and of their corresponding universal matrix representatives on the atomic product basis are obtained in this way for calculations on physical and chemical aggregates. Symmetric-group-based permutation-symmetry methods for partitioning of aggregate Hamlitonian matrices, performed to date by unitary transformations to remove unphysical “non-Pauli” states after aggregate Hamiltonian matrix construction, are supplemented here with a new two-step procedure. The new approach, which combines explicit antisymmetrization of atomic pair matrices with unitary transformation of aggregate Hamiltonian matrices, is described here in complete mathematical detail. Conditions are established from the new development under which enforcement of pairwise-atomic antisymmetry alone can be sufficient for calculations of electronic energies, providing an efficient approach for practical applications, as well as long-absent theoretical support for related semi-empirical and ad hoc methods widely employed in computational molecular physics and quantum chemistry. An exact expression is obtained for total electronic energies from the solution of the Schrödinger equation in the universal form of sums over pairwise-atomic interaction energies uniquely defined within a molecule in the absence of subjective additional conditions. Wigner rotation matrices are employed to orient pairs of interacting atoms within an aggregate in the implementation of the approach, requiring calculations of the universal pair interaction matrices as functions of radial separations only. Such information can be obtained once and for all from molecular diatomic calculations and retained for subsequent use, avoiding repeated evaluations of the electronic integrals commonly required in conventional variational determinations of accurate potential energy surfaces. Diagonalizations of large Hamiltonian matrices are avoided in favor of optical potential, Krylov-Lanczos, and previously described Stieltjes-Techebycheff methods in determination of aggregate Schrödinger eigenspectra.

Pair Potential with Submillikelvin Uncertainties and Nonadiabatic Treatment of the Halo State of the Helium Dimer.

The pair potential for helium is computed with accuracy improved by an order of magnitude relative to the best previous determination. For the well region, its uncertainties are now below 1millikelvin. The main improvement is due to the use of explicitly correlated wave functions at the nonrelativistic Born-Oppenheimer (BO) level of theory. The diagonal BO and the relativistic corrections are obtained from large full configuration interaction calculations. Nonadiabatic perturbation theory is used to predict the properties of the halo state of the helium dimer. Its binding energy and the average value of the interatomic distance are found to be 138.9(5)neV and 47.13(8)Å. The binding energy agrees with its first experimental determination of 151.9(13.3)neV [Zeller etal., Proc. Natl. Acad. Sci. U.S.A. 113, 14651 (2016)PNASA60027-842410.1073/pnas.1610688113].

Effects of adiabatic, relativistic, and quantum electrodynamics interactions on the pair potential and thermophysical properties of helium

The adiabatic, relativistic, and quantum electrodynamics (QED) contributions to the pair potential of helium were computed, fitted separately, and applied, together with the nonrelativistic Born-Oppenheimer (BO) potential, in calculations of thermophysical properties of helium and of the properties of the helium dimer. An analysis of the convergence patterns of the calculations with increasing basis set sizes allowed us to estimate the uncertainties of the total interaction energy to be below 50 ppm for interatomic separations R smaller than 4 bohrs and for the distance R = 5.6 bohrs. For other separations, the relative uncertainties are up to an order of magnitude larger (and obviously still larger near R = 4.8 bohrs where the potential crosses zero) and are dominated by the uncertainties of the nonrelativistic BO component. These estimates also include the contributions from the neglected relativistic and QED terms proportional to the fourth and higher powers of the fine-structure constant α. To obtain such high accuracy, it was necessary to employ explicitly correlated Gaussian expansions containing up to 2400 terms for smaller R (all R in the case of a QED component) and optimized orbital bases up to the cardinal number X = 7 for larger R. Near-exact asymptotic constants were used to describe the large-R behavior of all components. The fitted potential, exhibiting the minimum of -10.996 ± 0.004 K at R = 5.608 0 ± 0.000 1 bohr, was used to determine properties of the very weakly bound (4)He(2) dimer and thermophysical properties of gaseous helium. It is shown that the Casimir-Polder retardation effect, increasing the dimer size by about 2 Å relative to the nonrelativistic BO value, is almost completely accounted for by the inclusion of the Breit-interaction and the Araki-Sucher contributions to the potential, of the order α(2) and α(3), respectively. The remaining retardation effect, of the order of α(4) and higher, is practically negligible for the bound state, but is important for the thermophysical properties of helium. Such properties computed from our potential have uncertainties that are generally significantly smaller (sometimes by nearly two orders of magnitude) than those of the most accurate measurements and can be used to establish new metrology standards based on properties of low-density helium.

Multi-Jastrow trial wavefunctions for electronic structure calculations with quantum Monte Carlo

A new type of electronic trial wavefunction suitable for quantum Monte Carlo calculations of molecular systems is presented. In contrast with the standard Jastrow-Slater form built with a unique global Jastrow term, it is proposed to introduce individual Jastrow factors attached to molecular orbitals. Such a form is expected to be more physical since it allows to describe differently the local electronic correlations associated with various molecular environments (1s-core orbitals, 3d-magnetic orbitals, localized two-center sigma-orbitals, delocalized pi-orbitals, atomic lone pairs, etc.). In contrast with the standard form, introducing different Jastrow terms allows us to change the nodal structure of the wavefunction, a point which is important in the context of building better nodes for more accurate fixed-node diffusion Monte Carlo (FN-DMC) calculations. Another important aspect resulting from the use of local Jastrow terms is the possibility of defining and preoptimizing local and transferable correlated units for building complex trial wavefunctions from simple parts. The practical aspects associated with the computation of the intricate derivatives of the multi-Jastrow trial function are presented in detail. Some first illustrative applications for atoms of increasing size (O, S, and Cu) and for the potential energy curve and spectroscopic constants of the FH molecule are presented. In the case of the copper atom, the use of the multi-Jastrow form at the variational Monte Carlo level has allowed us to improve significantly the value of the total ground-state energy (about 75% of the correlation energy with only one determinant and three atomic orbital Jastrow factors). In the case of the FH molecule (fluorine hydride), it has been found that the multi-Jastrow nodes lead to an almost exact FN-DMC value of the dissociation energy [D(0)=-140.7(4) kcal/mol instead of the estimated nonrelativistic Born-Oppenheimer exact value of -141.1], which is not the case with standard nodes, D(0)=-138.3(4) kcal/mol.

Relativistic Correction to the Helium Dimer Interaction Energy

The lowest-order relativistic correction to the helium-helium interaction energy has been calculated for the first time, using two independent methods based on expansions in explicitly correlated Gaussian functions. At the equilibrium interatomic distance of 5.6 bohr, this correction amounts to +15.4 +/- 0.6 mK. As a by-product, a new upper bound of -10.9985 K for the nonrelativistic Born-Oppenheimer interaction energy has been obtained.

Systematic determination of intersections of potential energy surfaces using a Lagrange multiplier constrained procedure

Two nonrelativistic Born-Oppenheimer potential energy surfaces of distinct space-spin symmetry intersect on a surface of dimension N-1 where N is the number of internal nuclear degrees of freedom. Characterization of this entire surface can be quite costly. An algorithm, employing multiconfiguration self-consistent-field (MCSCF)/configuration interaction (CI) wave functions and analytic gradient techniques, is presented which avoids the determination of the full N-I dimensional surface, while directly locating portions of the crossing surface that are energetically important

Spin-forbidden chemistry within the Breit-Pauli approximation

Abstract Recent advances in the computational techniques available to characterize spin-forbidden processes with the Breit-Pauli approximation are reviewed and the use of these techniques is illustrated. These computational advances include: (i) The use of symbolic matrix element techniques to evaluate matrix elements of the full microscopic spin-orbit Hamiltonian (both the spin-orbit and spin-other-orbit terms) and the dipolar spin-spin Hamiltonian. This approach permits the Breit-Pauli interaction to be characterized in terms of large configurations state functions (CSF) spaces ≥ 106 terms. (ii) The relativistic wavefunctions are determined, directly in the CSF basis, using quasi-degenerate perturbation theory. This approach avoids the computational bottleneck which occurs if the perturbed wavefunction is determined in the eigenstate basis of H0 the non-relativistic Born-Oppenheimer Hamiltonian. (iii) The use of a Lagrange-Newton, analytic gradient-Mcscf/CI wavefunction based, algorithm for determining ...

On the evaluation of the correlation energy and the total non-relativistic born-oppenheimer energy for the water molecule

The results of density-functional correlation energy E c calculations for the first- and second-row atoms and hydrides are presented. For the water molecule various theoretical and empirical estimates of E c and of the total non-relativistic energy are compared.

Hadron-hadron interactions with orthogonal lattice-gauge-field configurations. II. Nucleon-nucleon system.

The application of a model based on lattice-gauge considerations of multiquark interactions to the nucleon-nucleon system is given. The model is a nonrelativistic Born-Oppenheimer approach which includes a confinement mechanism consistent with the local gauge invariance required by QCD. Inclusion of a one-pion-exchange interaction between quarks and a linkage exchange term as suggested in the preceding paper gives a four-parameter model which yields a remarkably good fit to low-energy neutron-proton phase shifts and the static properties of the deuteron. The results overall are similar to those obtained by phenomenological potentials based on conventional meson-exchange ideas such as the Paris potential. The parameters of this model for the nucleon-nucleon system are consistent with the parameters needed to describe the internal excitations of nucleons.

Calculated Potential-Energy Curves for CH+

Ab initio calculations have been performed for states of C${\mathrm{H}}^{+}$ which arise from atomic ${\mathrm{C}}^{+}(^{2}P)$ and $\mathrm{H}(^{2}S)\ensuremath{-}X^{1}\ensuremath{\Sigma}^{+}$, $A^{1}\ensuremath{\Pi}$, $^{3}\ensuremath{\Pi}$, and $^{3}\ensuremath{\Sigma}^{+}$. Potential curves have been constructed over a wide range of internuclear distances. All computed energies are believed to be within 1 eV of their exact nonrelativistic Born-Oppenheimer value, and all energy differences are believed to have an accuracy of better than 0.3 eV. Results, with known experimental values in parentheses, are ${R}_{e}(X^{1}\ensuremath{\Sigma}^{+})=2.136$ (2.137) a.u.; ${D}_{e}^{0}(X^{1}\ensuremath{\Sigma}^{+})=4.11$ (4.21) eV; ${R}_{e}(A^{1}\ensuremath{\Pi})=2.232$ (2.333) a.u.; ${T}_{e}(X\ensuremath{-}A)=24970$ (24146) ${\mathrm{cm}}^{\ensuremath{-}1}$; ${R}_{e}(^{3}\ensuremath{\Pi})=2.130$ a.u. No evidence is found for a quasibound $^{3}\ensuremath{\Sigma}^{+}$ Rydberg state, although such behavior has been predicted for isoelectronic BH.