Explicitly correlated Gaussians for high-precision variational calculations of S

In this work we consider an efficient algorithm for variational calculations of quantum few-particle systems in S, P, and D states of the even parity using all-particle explicitly correlated Gaussian (ECG) basis sets. We primarily focus on the description of states where the dominant configuration contains either two particles in p states or a single particle in a d state (all other particles are in s states). The basis functions we consider are products of spherically symmetric ECGs and bipolar harmonics. We introduced a scheme for deriving expressions for matrix elements of the overlap, kinetic and potential energy, as well as their derivatives with respect to the nonlinear parameters of the Gaussians. This allowed us to improve the efficiency of numerical calculations of the matrix elements (which is the most critical part of any code that uses ECGs) by one to two orders of magnitude compared to previous implementations. We provide a complete set of formulas for all basic matrix elements using the formalism of matrix differential calculus and discuss some technical details relevant to their efficient implementation. Lastly, we report a few example calculations of the ^{2}D^{e} states of the Li atom, ^{4}P^{e} and ^{2}D^{e} states of the B atom, and ^{3}P^{e} states of the C atom.

Computer program ATOM-MOL-nonBO for performing calculations of ground and excited states of atoms and molecules without assuming the Born-Oppenheimer approximation using all-particle complex explicitly correlated Gaussian functions.

In this work, we describe a computer program called ATOM-MOL-nonBO for performing bound state calculations of small atoms and molecules without assuming the Born-Oppenheimer approximation. All particles forming the systems, electrons and nuclei, are treated on equal footing. The wave functions of the bound states are expanded in terms of all-particle one-center complex explicitly correlated Gaussian functions multiplied by Cartesian angular factors. As these Gaussian functions are eigenfunctions of the operator representing the square of the total angular momentum of the system, the problem separates and calculations of states corresponding to different values of the total rotational quantum number can be solved independently from each other. Due to thorough variational optimization of the Gaussian exponential parameters, the method allows us to generate very accurate wave functions. The optimization is aided by analytically calculated energy gradient determined with respect to the parameters. Three examples of calculations performed for diatomic and triatomic molecules are shown as an illustration of calculations that can be performed with this program. Finally, we discuss the limitations, applicability range, and bottlenecks of the program.

Lowest

A theoretical ab initio approach for calculating bound states of small atoms is developed and implemented. The approach is based on finite-nuclear-mass [non-Born-Oppenheimer (non-BO)] nonrelativistic variational calculations performed with all-particle explicitly correlated Gaussian functions and includes the leading relativistic and quantum electrodynamics energy corrections determined using the non-BO wave functions. The approach is applied to determine the total and transition energies for the lowest four ^{2}S electronic excitations of the boron atom. The transition energies agree with the available experimental values within 0.2-0.3 cm^{-1}. Previously, such accuracy was achieved for three- and four-electron systems.

Prediction of (1)P Rydberg energy levels of beryllium based on calculations with explicitly correlated Gaussians.

Benchmark variational calculations are performed for the seven lowest 1s(2)2s np ((1)P), n = 2...8, states of the beryllium atom. The calculations explicitly include the effect of finite mass of (9)Be nucleus and account perturbatively for the mass-velocity, Darwin, and spin-spin relativistic corrections. The wave functions of the states are expanded in terms of all-electron explicitly correlated Gaussian functions. Basis sets of up to 12,500 optimized Gaussians are used. The maximum discrepancy between the calculated nonrelativistic and experimental energies of 1s(2)2s np ((1)P) →1s(2)2s(2) ((1)S) transition is about 12 cm(-1). The inclusion of the relativistic corrections reduces the discrepancy to bellow 0.8 cm(-1).

Excited states of positronic lithium and beryllium.

Using a variational method with an explicitly correlated Gaussian basis, we study the e(+)-Li and e(+)-Be complexes in the ground and lowest excited states with higher spin multiplicity. Our calculations provide rigorous theoretical confirmation that a positron can be attached to the excited states: 1s2s2p 4P(o) and 1s²2s2p 3P(o) for e(+)-Li and e(+)-Be, respectively. The result is particularly notable for the e(+)-Be complex, as the excited 3P(o) state lies below the autoionization threshold. We report accurate binding energies, annihilation rates and structural properties of these positron-atom systems. The existence of the ground and metastable excited states with bound positron opens up a new route to the presently lacking experimental verification of stability of a positron binding to any neutral atom.

Prediction of 2D Rydberg energy levels of 6Li and 7Li based on very accurate quantum mechanical calculations performed with explicitly correlated Gaussian functions.

Very accurate variational nonrelativistic finite-nuclear-mass calculations employing all-electron explicitly correlated Gaussian basis functions are carried out for six Rydberg (2)D states (1s(2)nd, n = 6, [ellipsis (horizontal)], 11) of the (7)Li and (6)Li isotopes. The exponential parameters of the Gaussian functions are optimized using the variational method with the aid of the analytical energy gradient determined with respect to these parameters. The experimental results for the lower states (n = 3, ..., 6) and the calculated results for the higher states (n = 7, ..., 11) fitted with quantum-defect-like formulas are used to predict the energies of (2)D 1s(2)nd states for (7)Li and (6)Li with n up to 30.

Assessment of the accuracy the experimental energies of the 1P(o) 1s(2)2s6p and 1s(2)2s7p states of 9Be based on variational calculations with explicitly correlated Gaussians.

Benchmark variational calculations are performed for the six lowest states of the (1)P(o) 1s(2)2snp state series of the (9)Be atom. The wave functions of the states are expanded in terms of all-particle, explicitly correlated Gaussian basis functions and the effect of the finite nuclear mass is directly included in the calculations. The exponential parameters of the Gaussians are variationally optimized using the analytical energy gradient determined with respect to those parameters. Besides providing reference non-relativistic energies for the considered states, the calculations also allow to assess the accuracy of the experimental energies of the (1)P(o) 1s(2)2s6p and 1s(2)2s7p states and suggest their refinement.

Explicitly correlated gaussian calculations of the 2D Rydberg states of the boron atom.

Accurate non-relativistic variational calculations are performed for the seven lowest members of the (2)D Rydberg series (1s(2)2s2p(2), and 1s(2)2s(2)nd, n = 3, ..., 8) of the boron atom. The wave functions of the states are expanded in terms of all-electron explicitly correlated gaussian basis functions and the effect of the finite nuclear mass is directly included in the calculations allowing for determining the isotopic shifts of the energy levels. The gaussian basis is optimized independently for each state with the aid of the analytic energy gradient with respect to the gaussian parameters. The calculations represent the highest accuracy level currently achievable for the considered states. The computed energies are compared with the available experimental data.

Explicitly correlated Gaussian calculations of the 2P(o) Rydberg spectrum of the lithium atom.

Accurate quantum-mechanical nonrelativistic variational calculations are performed for the nine lowest members of the (2)P(o) Rydberg series (1s(2)np(1), n = 2, ..., 10) of the lithium atom. The effect of the finite nuclear mass is included in the calculations allowing for determining the isotopic shifts of the energy levels. The wave functions of the states are expanded in terms of all-electron explicitly correlated Gaussian functions. The exponential parameters of the Gaussians are variationally optimized with the aid of the analytical energy gradient determined with respect to those parameters. The calculated state energies are compared with the available experimental data.

Accurate variational calculations of the ground 2Po(1s22s22p) and excited 2S(1s22s2p2) and 2Po(1s22s23p) states of singly ionized carbon atom.

In this article we report accurate nonrelativistic variational calculations of the ground and two excited states of C(+) ion. We employ extended and well optimized basis sets of all-electron explicitly correlated Gaussians to represent the wave functions of the states. The optimization of the basis functions is performed with a procedure employing the analytic gradient of the energy with respect to the nonlinear parameters of the Gaussians. The calculations explicitly include the effects due to the finite nuclear mass. The calculated transition energies between the three states are compared to the experimentally derived values. Finally, we present expectation values of some small positive and negative powers of the interparticle distances and contact densities.

Accurate non-Born-Oppenheimer calculations of the complete pure vibrational spectrum of D2 with including relativistic corrections.

In this work we report very accurate variational calculations of the complete pure vibrational spectrum of the D(2) molecule performed within the framework where the Born-Oppenheimer (BO) approximation is not assumed. After the elimination of the center-of-mass motion, D(2) becomes a three-particle problem in this framework. As the considered states correspond to the zero total angular momentum, their wave functions are expanded in terms of all-particle, one-center, spherically symmetric explicitly correlated Gaussian functions multiplied by even non-negative powers of the internuclear distance. The nonrelativistic energies of the states obtained in the non-BO calculations are corrected for the relativistic effects of the order of α(2) (where α = 1/c is the fine structure constant) calculated as expectation values of the operators representing these effects.

Time-dependent density functional study of field emission from nanotubes composed of C, BN, SiC, Si, and GaN.

Field emission from various types of nanotubes is studied by propagating the electronic density in real space and time using time-dependent density functional theory. Capped (5, 5) C, BN, SiC, Si, and GaN nanotubes are considered. The GaN, SiC, and Si nanotubes were found to be significantly better field emitters than C and BN nanotubes, both in terms of current magnitude and sharpness of peaks in the energy spectra. By analyzing the electronic structure of the various systems it is seen that the nanotubes with the highest currents have electron densities that extend significantly from the nanotube in the emission direction.

Refinement of the experimental energy levels of higher 2D Rydberg states of the lithium atom with very accurate quantum mechanical calculations.

Very accurate variational non-relativistic calculations are performed for four higher Rydberg (2)D states (1s(2)nd(1), n = 8,..., 11) of the lithium atom ((7)Li). The wave functions of the states are expanded in terms of all-electron explicitly correlated Gaussian functions and finite nuclear mass is used. The exponential parameters of the Gaussians are optimized using the variational method with the aid of the analytical energy gradient determined with respect to those parameters. The results of the calculations allow for refining the experimental energy levels determined with respect to the (2)S 1s(2)2s(1) ground state.

An algorithm for calculating atomic D states with explicitly correlated gaussian functions.

An algorithm for the variational calculation of atomic D states employing n-electron explicitly correlated gaussians is developed and implemented. The algorithm includes formulas for the first derivatives of the hamiltonian and overlap matrix elements determined with respect to the gaussian nonlinear exponential parameters. The derivatives are used to form the energy gradient which is employed in the variational energy minimization. The algorithm is tested in the calculations of the two lowest D states of the lithium and beryllium atoms. For the lowest D state of Li the present result is lower than the best previously reported result.

Vibrational transitions of the 7LiH+ ion calculated without the Born-Oppenheimer approximation and with leading relativistic corrections.

We recently presented very accurate calculations of the fundamental vibrational frequency of the (7)LiH(+) and (3)He(4)He(+) ions [Stanke et al. Phys. Rev. A 79, 060501(R) (2009)] performed without the Born-Oppenheimer approximation and included leading relativistic corrections. The accuracy of those calculations was estimated to be of the order of 0.06 cm(-1). In the present work we extend the calculations to the remaining pure vibrational states of (7)LiH(+) and similarly accurate results are generated. They may lead to the experimental search for still unidentified lines corresponding to those transitions.

Isotope shifts of the three lowest 1S states of the B+ ion calculated with a finite-nuclear-mass approach and with relativistic and quantum electrodynamics corrections.

We present very accurate quantum mechanical calculations of the three lowest S-states [1s(2)2s(2)((1)S(0)), 1s(2)2p(2)((1)S(0)), and 1s(2)2s3s((1)S(0))] of the two stable isotopes of the boron ion, (10)B(+) and (11)B(+). At the nonrelativistic level the calculations have been performed with the Hamiltonian that explicitly includes the finite mass of the nucleus as it was obtained by a rigorous separation of the center-of-mass motion from the laboratory frame Hamiltonian. The spatial part of the nonrelativistic wave function for each state was expanded in terms of 10,000 all-electron explicitly correlated Gaussian functions. The nonlinear parameters of the Gaussians were variationally optimized using a procedure involving the analytical energy gradient determined with respect to the nonlinear parameters. The nonrelativistic wave functions of the three states were subsequently used to calculate the leading alpha(2) relativistic corrections (alpha is the fine structure constant; alpha=1/c, where c is the speed of light) and the alpha(3) quantum electrodynamics (QED) correction. We also estimated the alpha(4) QED correction by calculating its dominant component. A comparison of the experimental transition frequencies with the frequencies obtained based on the energies calculated in this work shows an excellent agreement. The discrepancy is smaller than 0.4 cm(-1).

Calculation of transmission probability by solving an eigenvalue problem.

The electron transmission probability in nanodevices is calculated by solving an eigenvalue problem. The eigenvalues are the transmission probabilities and the number of nonzero eigenvalues is equal to the number of open quantum transmission eigenchannels. The number of open eigenchannels is typically a few dozen at most, thus the computational cost amounts to the calculation of a few outer eigenvalues of a complex Hermitian matrix (the transmission matrix). The method is implemented on a real space grid basis providing an alternative to localized atomic orbital based quantum transport calculations. Numerical examples are presented to illustrate the efficiency of the method.

Isotope shift in the electron affinity of lithium.

Very accurate electron affinity (EA) calculations of (6)Li and (7)Li (and (infinity)Li) have been performed using explicitly correlated Gaussian functions and a variational approach that explicitly includes the nuclear motion in the calculations (i.e., the approach that does not assume the Born-Oppenheimer approximation). The leading relativistic and quantum electrodynamics corrections to the electron affinities were also calculated. The results are the most accurate theoretical values obtained for the studied systems to date. Our best estimates of the (7)Li and (6)Li EAs are 4984.9842(30) and 4984.9015(30) cm(-1), respectively, and of the (7)Li/(6)Li EA isotope shift is 0.0827 cm(-1).

Non-Born-Oppenheimer calculations of the BH molecule.

Variational calculations employing explicitly correlated Gaussian basis functions have been performed for the ground state of the boron monohydride molecule (BH) and for the boron atom (B). Up to 2000 Gaussians were used for each system. The calculations did not assume the Born-Oppenheimer (BO) approximation. In the optimization of the wave function, we employed the analytical energy gradient with respect to the Gaussian exponential parameters. In addition to the total nonrelativistic energies, we computed scalar relativistic corrections (mass-velocity and Darwin). With those added to the total energies, we estimated the dissociation energy of BH. The non-BO wave functions were also used to compute some expectation values involving operators dependent on the interparticle distances.