School of Science and Technology (2015-2019)

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Nazarbayev University School of Science and Technology offers Masters programs in Biology, Computer Science, Physics and Robotics. The programs offer the students the opportunity to conduct their postgraduate studies with world-class faculty at the state-of-the-art campus of Nazarbayev University in Astana. In the School of Science and Technology, students are afforded the opportunity to develop career building skills such as scientific communications, personal study skills and project planning. You are encouraged and allowed independent thinking while integrating knowledge from a variety of disciplines. You will master creativity, being a self-starter, and be facile in acquiring and interpreting data. You will know the scientific method of data gathering and analysis. As a well-trained and knowledgeable consumer of scientific knowledge and technology, you will better prepare to explore your future career aspirations and serve your community, and your country.

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Browsing School of Science and Technology (2015-2019) by Author "Adamowicz, Ludwik"

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    1D states of the beryllium atom: Quantum mechanical nonrelativistic calculations employing explicitly correlated Gaussian functions
    (2011) Sharkey, Keeper L.; Bubin, Sergiy; Adamowicz, Ludwik
    Very accurate finite-nuclear-mass variational nonrelativistic calculations are performed for the lowest five 1D states (1s2 2p2, 1s2 2s1 3d1, 1s2 2s1 4d1, 1s2 2s1 5d1, and 1s2 2s1 6d1) of the beryllium atom (9Be). The wave functions of the states are expanded in terms of all-electron explicitly correlated Gaussian functions. 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 calculations exemplify the level of accuracy that is now possible with Gaussians in describing bound states of a four-electron system where some of the electrons are excited into higher angular states
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    Accuracy limits on the description of the lowest S excitation in the Li atom using explicitly correlated Gaussian basis functions
    (2008) Stanke, Monika; Komasa, Jacek; Kedziera, Dariusz; Bubin, Sergiy; Adamowicz, Ludwik
    We have performed very accurate quantum-mechanical calculations for the two lowest S states of the lithium atom in order to determine the transition energy. In the nonrelativistic part of the calculations performed using the variational method, we explicitly included the nuclear motion. The nonrelativistic wave function was expanded in terms of explicitly correlated Gaussian functions. Next, this wave function was used to calculate the leading 2 relativistic correction is the fine-structure constant and the 3 QED correction. We also estimated the 4 QED correction by calculating its dominating component. The results obtained with Gaussians are compared with the most accurate results obtained recently with the Hylleraas-type basis functions
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    Accurate non-Born-Oppenheimer calculations of the complete pure vibrational spectrum of D2 with including relativistic corrections
    (2011) Bubin, Sergiy; Stanke, Monika; Adamowicz, Ludwik
    In this work we report very accurate variational calculations of the complete pure vibrational spectrum of the D2 molecule performed within the framework where the Born-Oppenheimer (BO) approximation is not assumed. After the elimination of the center-of-mass motion, D2 becomes a threeparticle 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 α2 (where α = 1/c is the fine structure constant) calculated as expectation values of the operators representing these effects
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    Accurate non-Born-Oppenheimer calculations of the complete pure vibrational spectrum of ditritium using all-particle explicitly correlated Gaussian functions
    (2014) Bubin, Sergiy; Stanke, Monika; Adamowicz, Ludwik
    Very accurate variational calculations of the complete pure vibrational spectrum of the ditritium (T2) molecule are performed within the framework where the Born-Oppenheimer approximation is not assumed. After separating out the center-of-mass motion from the total laboratory-frame Hamiltonian, T2 becomes a three-particle problem. States corresponding to the zero total angular momentum, which are pure vibrational states, are spherically symmetric in this framework. The wave functions of these states 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. In the calculations the total energies, the dissociation energies, and expectation values of some operators dependent on interparticle distances are determined.
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    Accurate variational calculations of the ground 2Po(1s22s22p) and excited 2S(1s22s2p2) and 2Po(1s22s23p) states of singly ionized carbon atom
    (2011) Bubin, Sergiy; Adamowicz, Ludwik
    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
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    Algorithm for quantum-mechanical finite-nuclear-mass variational calculations of atoms with two p electrons using all-electron explicitly correlated Gaussian basis functions
    (2009) Sharkey, Keeper L.; Pavanello, Michele; Bubin, Sergiy; Adamowicz, Ludwik
    A new algorithm for calculating the Hamiltonian matrix elements with all-electron explicitly correlated Gaussian functions for quantum-mechanical calculations of atoms with two p electrons or a single d electron have been derived and implemented. The Hamiltonian used in the approach was obtained by rigorously separating the center-of-mass motion and it explicitly depends on the finite mass of the nucleus. The approach was employed to perform test calculations on the isotopes of the carbon atom in their ground electronic states and to determine the finite-nuclear-mass corrections for these states
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    An accurate non-Born–Oppenheimer calculation of the first purely vibrational transition in LiH molecule
    (2005) Bubin, Sergiy; Adamowicz, Ludwik; Molski, Marcin
    In this work we study the ground and the first vibrationally excited states of LiH molecule. We performed an extensive nonrelativistic variational calculations of the two states without using the Born–Oppenheimer approximation. The results are analyzed and compared with the data extracted from recent experiments. The 0←1 transition energy obtained in the calculations converged to a value which is less than a wave number above the transition energy estimated from the available experimental data concerning the LiH rovibrational transitions. We discuss the remaining discrepancy and the procedure used to determine the “experimental” transition frequencies
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    An algorithm for calculating atomic D states with explicitly correlated Gaussian functions
    (2011) Sharkey, Keeper L.; Bubin, Sergiy; Adamowicz, Ludwik
    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
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    Analytical energy gradient in variational calculations of the two lowest 3P states of the carbon atom with explicitly correlated Gaussian basis functions
    (2010) Sharkey, Keeper L.; Bubin, Sergiy; Adamowicz, Ludwik
    Variational calculations of ground and excited bound states on atomic and molecular systems performed with basis functions that explicitly depend on the interparticle distances can generate very accurate results provided that the basis function parameters are thoroughly optimized by the minimization of the energy. In this work we have derived the algorithm for the gradient of the energy determined with respect to the nonlinear exponential parameters of explicitly correlated Gaussian functions used in calculating n-electron atomic systems with two p-electrons and n−2 s-electrons. The atomic Hamiltonian we used was obtained by rigorously separating out the kinetic energy of the center of mass motion from the laboratory-frame Hamiltonian and explicitly depends on the finite mass of the nucleus. The advantage of having the gradient available in the variational minimization of the energy is demonstrated in the calculations of the ground and the first excited 3P state of the carbon atom. For the former the lowest energy upper bound ever obtained is reported
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    Assessment of the accuracy the experimental energies of the 1Po 1s22s6p and 1s22s7p states of 9Be based on variational calculations with explicitly correlated Gaussians
    (2012) Bubin, Sergiy; Adamowicz, Ludwik
    Benchmark variational calculations are performed for the six lowest states of the 1Po 1s22snp state series of the 9Be 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 1Po 1s22s6p and 1s22s7p states and suggest their refinement
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    Calculations of low-lying 1P states of the beryllium atom
    (2009) Bubin, Sergiy; Adamowicz, Ludwik
    High-accuracy nonrelativistic variational calculations employing explicitly correlated Gaussian basis functions have been performed to determine the energies and the expectation values of some operators for the lowest four 1P1 states of the beryllium atom. The states correspond to the electron configurations 1s22s1np1, where n=2, 3, 4, and 5. The calculations were performed for both finite and infinite mass of the Be nucleus. The basis set for each state was grown to the level of 5000 Gaussians. With that many functions we achieved a tight energy convergence. The reported values, to the best of our knowledge, are the most accurate ever obtained for the four states
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    Calculations of the ground states of BeH and BeH+ without the Born-Oppenheimer approximation
    (2007) Bubin, Sergiy; Adamowicz, Ludwik
    Non-Born-Oppenheimer variational calculations employing explicitly correlated Gaussian basis functions have been performed for the ground states of the beryllium monohydride molecule BeH and its ion BeH+ , as well as for the beryllium atom Be and its ion Be+ . An approach based on the analytical energy gradient calculated with respect to the Gaussian exponential parameters was employed. The calculated energies were used to determine the ionization potential of BeH and the dissociation energies of BeH and BeH+. Also, the generated wave functions were used to compute various expectation values, such as the average interparticle distances and the nucleus-nucleus correlation functions
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    Charge asymmetry in HD+
    (2005) Bubin, Sergiy; Bednarz, Eugeniusz; Adamowicz, Ludwik
    Expanding the wave functions of the ground and excited states of HD1 ~or pde) in terms of spherically symmetric explicitly correlated Gaussian functions with preexponential multipliers consisting of powers of the internuclear distance, and using the variational method, we performed very accurate nonadiabatic calculations of all bound states of this system corresponding to the zero total angular momentum quantum number ~vibrational states; v50 – 22). The total and the transition energies obtained agree with the best available calculations. For each state we computed the expectation values of the d-p, d-e, and p-e interparticle distances. This is the first time these quantities were computed for HD1 using rigorous nonadiabatic wave functions. While up to the v520 state some asymmetry is showing in the d-e and p-e distances, for v521 and v522 we observe a complete breakdown of the Born–Oppenheimer approximation and localization of the electron almost entirely at the deuteron.
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    Charge asymmetry in pure vibrational states of the HD molecule
    (2009) Bubin, Sergiy; Leonarski, Filip; Stanke, Monika; Adamowicz, Ludwik
    Very accurate variational calculations of all rotationless states also called pure vibrational states of the HD molecule have been performed within the framework that does not assume the Born–Oppenheimer BO approximation. The non-BO wave functions of the states describing the internal motion of the proton, the deuteron, and the two electrons were expanded in terms of one-center explicitly correlated Gaussian functions multiplied by even powers of the internuclear distance. Up to 6000 functions were used for each state. Both linear and nonlinear parameters of the wave functions of all 18 states were optimized with a procedure that employs the analytical gradient of the energy with respect to the nonlinear parameters of the Gaussians. These wave functions were used to calculate expectation values of the interparticle distances and some other related quantities. The results allow elucidation of the charge asymmetry in HD as a function of the vibrational excitation
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    Complete 2 relativistic corrections to the pure vibrational non-Born-Oppenheimer energies of HeH+
    (2008) Stanke, Monika; Ke¸dziera, Dariusz; Bubin, Sergiy; Adamowicz, Ludwik
    We report the implementation of the complete set of the lowest-order relativistic corrections of the order of 2 where is the fine structure constant for calculating vibrational states of diatomic molecular systems within the framework that does not assume the Born-Oppenheimer approximation. To test the accuracy of the approach we have performed calculations for all rotationless vibrational states also called pure vibrational states or S states of the HeH+ ion in the ground electronic state. For the lowest transitions, where very precise experimental results are available, an excellent agreement with the experimental values has been achieved
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    Complete pure vibrational spectrum of HD calculated without the Born-Oppenheimer approximation and including relativistic corrections
    (2011) Bubin, Sergiy; Stanke, Monika; Adamowicz, Ludwik
    All 18 bound pure vibrational levels of the HD molecule have been calculated within the framework that does not assume the Born-Oppenheimer (BO) approximation. The nonrelativistic energies of the states have been corrected for the relativistic effects of the order of α2 (where α is the fine structure constant), calculated using the perturbation theory with the nonrelativistic non-BO wave functions being the zero-order approximation. The calculations were performed by expanding the non-BO wave functions in terms of one-center explicitly correlated Gaussian functions multiplied by even powers of the internuclear distance and by performing extensive optimization of the Gaussian nonlinear parameters. Up to 10 000 basis functions were used for each state
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    Convergence of Experiment and Theory on the Pure Vibrational Spectrum of HeH
    (2006) Stanke, Monika; Kedziera, Dariusz; Molski, Marcin; Bubin, Sergiy; Barysz, Maria; Adamowicz, Ludwik
    Very accurate quantum mechanical calculations of the pure vibrational spectrum of the HeH molecular ion are reported and compared with newly obtained pure vibrational transitions extracted from the available experimental data. The calculations are performed without assuming the Born-Oppenheimer approximation regarding separability of the nuclear and electronic motions and include the first order relativistic mass-velocity and Darwin corrections. For the two lowest transitions, whose experimental energies are established with the highest precision, the calculated and the experimental results show very good agreement
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    Correlated-Gaussian calculations of the ground and low-lying excited states of the boron atom
    (2011) Bubin, Sergiy; Adamowicz, Ludwik
    Benchmark variational calculations of the four lowest 2P and 2S states of the boron atom (including the ground state) have been performed. The wave functions of the states have been expanded in terms of all-particle explicitly correlated Gaussian basis functions and the finite mass of the nucleus has been explicitly accounted for.Variational upper bounds for the nonrelativistic finite- and infinite-nuclear-mass energies of all considered states have been obtained with the relative convergence of the order of 10−7–10−8. Expectation values of the powers of the inter-particle distances and Dirac δ functions depending on those distances have also been computed. These calculations provide reference values that can be used to test other high-level quantum chemistry methods
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    Darwin and mass-velocity relativistic corrections in non-Born-Oppenheimer variational calculations
    (2006) Kedziera, Dariusz; Stanke, Monika; Bubin, Sergiy; Barysz, Maria; Adamowicz, Ludwik
    The Pauli approach to account for the mass-velocity and Darwin relativistic corrections has been applied to the formalism for quantum mechanical molecular calculations that does not assume the Born-Oppenheimer BO approximation regarding separability of the electronic and nuclear motions in molecular systems. The corrections are determined using the first order perturbation theory and are derived for the non-BO wave function of a diatomic system expressed in terms of explicitly correlated Gaussian functions with premultipliers in the form of even powers of the internuclear distance. As a numerical example we used calculations of the transition energies for pure vibrational states of the HD+ ion
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    Darwin and mass-velocity relativistic corrections in the non-Born-Oppenheimer calculations of pure vibrational states of H2
    (2006) Ke¸dziera, Dariusz; Stanke, Monika; Bubin, Sergiy; Barysz, Maria; Adamowicz, Ludwik
    The Darwin and mass-velocity relativistic corrections have been calculated for all pure vibrational states of the H2 using the perturbation theory and very accurate variational wave functions obtained without assuming the Born-Oppenheimer BO approximation. Expansions in terms of explicitly correlated Gaussians with premultipliers in the form of even powers of the internuclear distance were used for the wave functions. With the inclusion of the two relativistic corrections to the non-BO energies the transition energies for the highest states agree more with the experimental results
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