Born-Oppenheimer Calculations Research Articles

Electronic and nuclear flux dynamics at a conical intersection

A combined electronic-nuclear wave packet motion is accompanied by temporal changes of probability flux densities. Using a two dimensional model, we study such densities in the vicinity of a conical intersection (CI) between the potential energy surfaces of two electronically excited states. When the dynamics is accompanied by an efficient population transfer, the electronic flux density behaves nearly time-independent although the nuclear flux does not. The second case involves a nuclear motion where the CI is surrounded. There, the nuclear wave packet undergoes a bifurcation, and the electronic density shows characteristics of a rotation. The electronic flux, however, exhibits a constant directional dynamics during the nuclear motion. The geometrical phase which appears in comparing the nuclear dynamics derived from the coupled motion and the Born-Oppenheimer calculation is also seen in the nuclear flux dynamics.

Three-body resonance states just below the antiproton and hydrogen dissociation threshold

We analyze two shallow resonance states below the antiproton hydrogen dissociation threshold with a non-adiabatic three-body calculation. Rearrangement correlation between initial channel and protonium formation channel is explicitly included in the total wavefunction. The lower resonance state is in good agreement with the resonance position and width calculated with the R-matrix theory. The higher resonance state which is newly found is closer to the threshold and much narrower than the former resonance. A polarization effect of the hydrogen atom is found to be indispensable to support the resonance state. The accuracy of the present calculation is evaluated by the extended virial theorem. The resonance states calculated in the present work gives shallower relative energy below the dissociation threshold than the Born-Oppenheimer calculation, suggesting that the electron motion which is ignored in latter calculation would give positive energy because the electron is unbound inside the distance.

Coupled electron-nuclear quantum dynamics through and around a conical intersection.

In solving the time-dependent Schrödinger equation for a coupled electron-nuclear system, we study the motion of wave packets in a model which exhibits a conical intersection (CoIn) of adiabatic potential energy surfaces. Three different situations are studied. In the first case, an efficient non-adiabatic transition takes place while the wave packet passes the region of the CoIn. It is demonstrated that during these times, the nuclear probability density retains its Gaussian shape and the electronic density remains approximately constant. Second, dynamics are regarded where non-adiabatic transitions do not take place, and the nuclear dynamics follows a circle around the location of the CoIn. During this motion, the electronic density is shown to rotate. The comparison with the Born-Oppenheimer nuclear dynamics reveals the geometrical phase being associated with the circular motion. This phase is clearly revealed by an analysis of time-dependent autocorrelation functions and spectra obtained from the numerically exact and the Born-Oppenheimer calculation. The intermediate situation with a small non-adiabatic transition probability is characterized by wave-packet splitting into several fractions.

Unexpected properties of non‐autoionizing doubly excited states of the H2 molecule

We present accurate calculations of the non‐autoionizing and doubly excited states of the H2 molecule using full configuration interaction with Hartree–Fock molecular orbitals and Heitler–London atomic orbitals. We consider the united atom configurations from He(2p2p) up to He(2p8g) and dissociation products from H2(2p + 2p) up to H2(2p + 6ℓ). Born–Oppenheimer calculations are carried out with extended and optimized Slater‐type orbitals for a total of 40 states, 10 for each symmetry, covering the internuclear distances from the united atom to dissociation, which, for some states, is reached beyond 100 a0. Occurrences of repulsive states cleanly interlaced between bound states with many vibrational levels are reported. Some of the potential minima are deep enough to accommodate many vibrational levels (up to 50). Noteworthy large equilibrium minima, like Req = 46.0 a0 in the state dissociating as (2p + 6h) and with 18 vibrational levels. The occurrence of vertical excitations from the singly excited manifolds is analyzed. Several states present double minima generated by avoided crossings, some with a strong ionic character. © 2016 Wiley Periodicals, Inc.

High-precision nonadiabatic calculations of dynamic polarizabilities and hyperpolarizabilities for low-lying vibrational-rotational states of hydrogen molecular ions

The static and dynamic electric multipolar polarizabilities and second hyperpolarizabilities of the ${{\mathrm{H}}_{2}}^{+}$, ${{\mathrm{D}}_{2}}^{+}$, and ${\mathrm{HD}}^{+}$ molecular ions in the ground and first excited states are calculated nonrelativistically using explicitly correlated Hylleraas basis sets. The calculations are fully nonadiabatic; the Born-Oppenheimer approximation is not used. Comparisons are made with published theoretical and experimental results, where available. In our approach, no derivatives of energy functions nor derivatives of response functions are needed. In particular, we make contact with earlier calculations in the Born-Oppenheimer calculation where polarizabilities were decomposed into electronic, vibrational, and rotational contributions and where hyperpolarizabilities were determined from derivatives of energy functions. We find that the static hyperpolarizability for the ground state of ${\mathrm{HD}}^{+}$ is seven orders of magnitude larger than the corresponding dipole polarizability. For the dipole polarizability of ${\mathrm{HD}}^{+}$ in the first excited state the high precision of the present method facilitates treatment of a near cancellation between two terms. For applications to laser spectroscopy of trapped ions we find tune-out and magic wavelengths for the ${\mathrm{HD}}^{+}$ ion in a laser field. In addition, we also calculate the first few leading terms for long-range interactions of a hydrogen molecular ion and a ground state H, He, or Li atom.

Analytical energy gradient used in variational Born-Oppenheimer calculations with all-electron explicitly correlated Gaussian functions for molecules containing one π electron

An algorithm for variational calculations of molecules with one π electron performed with all-electron explicitly correlated gaussian (ECG) functions with floating centers is derived and implemented. The algorithm includes the analytic gradient of the Born-Oppenheimer electronic energy determined with respect to the ECG exponential parameters and the coordinates of the gaussian centers. The availability of the gradient greatly accelerates the variational energy minimization. The algorithm is tested in calculations of four electronic excited states, c(3)Π(u), C(1)Π(u), i(3)Π(g), and I(1)Π(g), of the hydrogen molecule at a single internuclear distance specific to each state. With the use of the analytical energy gradient, the present calculations yield new, lowest-to-date, variational energy upper bounds for all four states.

Adjustment of Born-Oppenheimer electronic wave functions to simplify close coupling calculations

Technical problems connected with use of the Born-Oppenheimer clamped-nuclei approximation to generate electronic wave functions, potential energy surfaces (PES), and associated properties are discussed. A computational procedure for adjusting the phases of the wave functions, as well as their order when potential crossings occur, is presented which is based on the calculation of overlaps between sets of molecular orbitals and configuration interaction eigenfunctions obtained at neighboring nuclear conformations. This approach has significant advantages for theoretical treatments describing atomic collisions and photo-dissociation processes by means of ab initio PES, electronic transition moments, and nonadiabatic radial and rotational coupling matrix elements. It ensures that the electronic wave functions are continuous over the entire range of nuclear conformations considered, thereby greatly simplifying the process of obtaining the above quantities from the results of single-point Born-Oppenheimer calculations. The overlap results are also used to define a diabatic transformation of the wave functions obtained for conical intersections that greatly simplifies the computation of off-diagonal matrix elements by eliminating the need for complex phase factors.

Determination of deuteron quadrupole moment from calculations of the electric field gradient inD2and HD

We have carried out an accurate determination of the quadrupole moment of the deuteron nucleus. The evaluation of the constant is achieved by combining high accuracy Born-Oppenheimer calculations of the electric field gradient at the nucleus in the ${\mathrm{H}}_{2}$ molecule with spectroscopic measurements of the quadrupolar splitting in ${\mathrm{D}}_{2}$ and HD. The derived value is $Q=0.285783(30$) fm${}^{2}$.

High-accuracy calculations of the ground, 1 A11′, and the 2 A11′, 2 A31′, and 1 E1′ excited states of H3+

Accurate variational Born-Oppenheimer calculations of the 1 (1)A(1) ('), 2 (1)A(1) ('), 2 (3)A(1) ('), and 1 (1)E(') states of the H(3) (+) ion at the ground-state equilibrium geometry are reported. The wave functions of the states are expanded in terms of explicitly correlated spherical Gaussian functions with shifted centers. In the variational optimization the analytical gradient of the energy with respect to the nonlinear exponential parameters of the Gaussians has been employed. The energies obtained in the calculations are the best variational estimates ever calculated for the four states. One-electron densities for the states, as well as a D(3h)-restricted potential energy surface of the ground state calculated around the equilibrium geometry, are also presented and discussed.

Accurate Born–Oppenheimer calculations of the low‐lying c3Σ and a3Σ excited states of helium dimer

AbstractAccurate variational Born–Oppenheimer calculations of the two lowest excited Σ triplet states of the helium dimer at their respective equilibrium geometries are reported. The wave functions of the states are expanded in terms of explicitly correlated Gaussian functions with shifted centers. The obtained energies are the best variational estimates ever calculated for these states. One‐electron densities are also presented and discussed. The results are compared with the experimental values and previous calculations. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008

Non–Born–Oppenheimer calculations of the ground state of H3

AbstractWe present quantum–mechanical calculations for the ground state of the H3 system performed without the Born‐Oppenehimer approximation. In the calculations we use explicitly correlated Gaussian basis functions that explicitly depend on all of the interparticle distances. These basis functions allow us to achieve high accuracy while explicitly describing nucleus–nucleus, nucleus–electron, and electron–electron correlation effects. Gaussian basis sets ranging in size from 85 to 950 functions have been optimized using a gradient‐based procedure. The issue of defining and extracting the H3 molecular structure based on the non‐BO wave function is also discussed. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

A new algorithm to handle finite nuclear mass effects in electronic calculations: the ISOTOPE program.

We report the development of a simple algorithm to modify quantum chemistry codes based on the LCAO procedure, to account for the isotope problem in electronic structure calculations. No extra computations are required compared to standard Born-Oppenheimer calculations. An upgrade of the Gamess package called ISOTOPE is presented, and its applicability is demonstrated in some examples.

Reconsidering Car–Parrinello molecular dynamics using direct propagation of molecular orbitals developed upon Gaussian type atomic orbitals

A reconsideration of Car–Parrinello molecular dynamics using only atom centred basis functions is presented by doing direct propagation of molecular orbitals in conjunction with propagation of nuclei. The electronic degree of freedom chosen for the propagation are the coefficients of the linear combination of atomic orbitals development where atomic orbitals are expressed upon Gaussian functions. Considering that the wave function is propagated, we show that only very few iterations are sufficient to calculate nuclear gradients and electronic potential energy with good accuracy, instead of fulfilling the minimisation energy procedure at each step of the dynamics simulation. First tests calculations are presented by considering simple systems (water and monothiooxalic acid molecules and a (H2O)20 water cluster). These tests present the good behaviour of the propagation scheme if one considers the conservation of the total energy and the accuracy of the potential energy surface compared with Born–Oppenheimer calculations.

Nonadiabatic and Born–Oppenheimer calculations of the polarizabilites of LiH and LiD

We present fully nonadiabatic as well as Born–Oppenheimer calculations of the dipole moments and static polarizabilites of LiH and LiD. The nonadiabatic calculations are done in a basis of explicitly correlated gaussian functions. The electronic Born–Oppenheimer values are corrected for vibrational contributions using the numerical Numerov–Cooley method including all orders of anharmonicity. A comparison between the two methods does not suggest any substantial disagreement in absolute values or the isotopic differences of the calculated properties.

Geometric phase effect in isotopomers of X3 systems: Use of a split basis technique for the cone states of HD2

We report a study of the geometric phase (GP) effect in isotopomers of X3 systems using a split basis technique for the nuclear wave functions. The method is used to calculate the vibrational levels in the first electronically excited state of HD2 without consideration and with consideration of the GP effect. It is shown that the GP effect plays an important role and should be treated properly. It is also noted that the GP effect has a more remarkable influence for H3 than for its isotopic variants. Such a mass-type effect shows in the reported generalized Born–Oppenheimer calculations and has apparently not been observed previously. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 80: 454–460, 2000

Adaptive accuracy control for Car-Parrinello simulations

The Car-Parrinello (CP) approach to ab initio molecular dynamics serves as an approximation to time-dependent Born-Oppenheimer (BO) calculations. It replaces the explicit minimization of the energy functional by a fictitious Newtonian dynamics and therefore introduces an artificial mass parameter \(\mu\) which controls the electronic motion. A recent theoretical investigation shows that the CP-error, i.e., the deviation of the CP–solution from the BO-solution decreases like \(\mu^{1/2}\) asymptotically. Since the computational effort increases like \(\mu^{-1/2}\), the choice of \(\mu\) has to find a compromise between efficiency and accuracy. The asymptotical result is used in this paper to construct an easily implemented algorithm which automatically controls \(\mu\): the parameter \(\mu\) is repeatedly adapted during the simulation by choosing \(\mu\) as large as possible while pushing an error measure below a user-given tolerance. The performance and reliability of the algorithm is illustrated by a typical example.

Geometry of (0,2) Landau-Ginzburg orbifolds

Several aspects of (0,2) Landau-Ginzburg orbifolds are investigated. Especially the elliptic genera are computed in general and, for a class of models recently invented by Distler and Kachru, they are compared with the ones from (0,2) sigma models. Our formalism gives an easy way to calculate the generation numbers for lots of Distler-Kachru models even if they are based on singular Calabi-Yau spaces. We also make some general remarks on the Born-Oppenheimer calculation of the ground states elucidating its mathematical meaning in the untwisted sector. For Distler-Kachru models based on non-singular Calabi-Yau spaces we show that there exist “residue” type formulas of the elliptic genera as well.

The non-adiabatic Peierls-Heisenberg model in finite clusters

The exact solution for the Peierls-Heisenberg antiferromagnetic (AF) model in a finite cluster is presented. The spin-spin coupling is retained up to second neighbours. The quantum mechanical character of the phonon variables is maintained in order to properly describe the fluctuations, which have a substantial role in one dimension. The problem of the frustrated AF state is briefly addressed. For the purpose of comparison, Born-Oppenheimer (BO) calculations are also provided. A noticeable result is the fact that the (semiclassical) BO approximation overestimates quantum fluctuations.

Some recent developments in the statistical mechanical theory of ionic solvation

Abstract

Muonic-hydrogen molecular bound states, quasibound states, and resonances in the Born-Oppenheimer approximation.

The Born-Oppenheimer approximation is used as an exploratory tool to study bound states, quasibound states, and scattering resonances in muon (\ensuremath{\mu})--hydrogen (x)--hydrogen (y) molecular ions. Our purpose is to comment on the existence and nature of the narrow states reported in three-body calculations, for L=0 and 1, at approximately 55 eV above threshold and the family of states in the same partial waves reported about 1.9 keV above threshold. We first discuss the motivation for study of excited states beyond the well-known and well-studied bound states. Then we reproduce the energies and other properties of these well-known states to show that, despite the relatively large muon mass, the Born-Oppenheimer approximation gives a good, semiquantitative description containing all the essential physics. Born-Oppenheimer calculations of the s- and p-wave scattering of d-(d\ensuremath{\mu}), d-(t\ensuremath{\mu}), and t-(t\ensuremath{\mu}) are compared with the accurate three-body results, again with general success. The places of disagreement are understood in terms of the differences in location of slightly bound (or unbound) states in the Born-Oppenheimer approximation compared to the accurate three-body calculations.The analytic properties of the function ${\mathit{k}}^{2\mathit{L}+1}$cot\ensuremath{\delta} are used to illustrate the locating of bound states and resonance poles in the complex k or E plane from the scattering data and to deduce the expected widths of resonances of a given L value. The prominent L=3 resonance at 22 eV (\ensuremath{\Gamma}=1.4 eV) in d-(t\ensuremath{\mu}) scattering provides a benchmark for the unsuccessful search within the Born-Oppenheimer approximation for the claimed narrow s-wave d-t-\ensuremath{\mu} resonance at 54.35 eV (\ensuremath{\Gamma}=0.74 eV). Although absolute proof is obviously lacking (because of the approximate calculation), the Born-Oppenheimer results for the s-wave gerade scattering are entirely reasonable; the absence of a centrifugal barrier makes it implausible that there could be resonances at 50--60 eV with narrow widths. We argue that the threshold at 48 eV of the t+(d\ensuremath{\mu}) channel is unlikely to have a major effect. The family of states at 1.9 keV and above are of a different nature. They are, as is already known, molecular states based on the 3d ${\mathrm{\ensuremath{\sigma}}}_{\mathit{g}}$ ``electronic'' potential-energy curve, which is asymptotically the energy of the x(y\ensuremath{\mu}; n=2) system. Born-Oppenheimer calculations of binding energies, and properties depending dominantly on the wave functions in the classically allowed region, agree reasonably well with the accurate three-body computations for the d-t-\ensuremath{\mu} system. Expected disagreement occurs in the probability density ${\mathrm{\ensuremath{\rho}}}_{\mathit{N}}$(0) for the nuclei to be at vanishing internuclear separation (relevant for the fusion rate), in which configuration mixing is more important in the classically forbidden region. Nonetheless, the reported values of ${\mathrm{\ensuremath{\rho}}}_{\mathit{N}}$(0), when compared to the average probability density in the classically allowed region of nuclear motion computed here, appear excessively large. The likelihood that this family of continuum resonances would play a significant role in molecular formation in the muonic cascade is discussed briefly.